Woods Hole Mathematics: Perspectives in Mathematics and Physics (Series on Knots and Everything, Vol. 34)

h{Z),D - 4>h{-Z)} = -2-nih (other relations are obviously satisfied). For this, we first verify the identity (23)

4>\z) - 4>\-z) = z. Indeed, e~ipz — eipz

irh f

4> (z)-4> {-z) = - — J^ - _ ^ - ^ _ ^ _ ^ d p = •Kh ( f

r \

2 \JQ

e~ipz

e~ipz

_ irh

J-VL) sinh(Trp) sinh(?r/ip)

2

z

~ sinh(Trp) sinh(7r%)

Using this property, we can transform the commutators, [A + 4>\Z), B - cj>h(-Z)} - [A, B] - [A, h(Z), B] =

= 2mh (-1 - A(_^ ( _ Z ) ) + ^h(zij

= 0.

Analogously, [A + th(Z), D - <j>h{-Z)] = [A, D] - [A, <j>\-Z)} + [\Z),D\ = = 2niH (^(-\-Z))

+ ^\Z)^

= 2nih.

We need to prove also that the morphism preserves the real structure what is obviously equivalent to the realness condition for the function <j>h(z). We have -r-— •wh f

nh f eipz 2 Jn sinh(Trp) sinh(?r/ip) eipi irh t

2 J_n sinh(7rp) sinh(?r/ip)

e~ipi

2 Jn sinh(?rp) sinh(?rfi.p)

For checking Property 5 we must verify that the morphism T^(P) —> T 1/fi (r) commutes with a flip. It means that {A + (j>h{Z))jh = A/h +

17 (f>h(Z/h), (B - 4>xlh{-Z))/h = A/h - 4>h(-Z/h), etc. Therefore, it suffices to prove that 4>h(z)/n = ^/h(z/h).

(24)

Indeed, IT f

*

^

Z/K)

=

e~ipz/h

~2H Ju smh(irP)smh(Trp/h)dp

=

= - 4 / • w ! T- i./ ^i.Qh) =
/Yi-W-Xi)^

Yi+i

Ai+l Bi+i Ci+1 A+i

—Xi

=

\Ei+1J

Di Ei Ai + ^iXi) Bi - 4>h{-Xi)

(26)

\d + *&) 1

We are going to prove that this operator transformation is five-periodic. Assume for a moment that the five-periodicity of Xi is proven. Then, the five-periodicity of Yi is obvious, since l^+i = —Xi. The five-periodicity of, say, Ai follows from the calculation xi+1

= Yi- 4>h{Xi) = -Xi-! - 4>h{xt)

Therefore, \-Xi) = -Xi+1 - X^ Taking into account (23), we have 4> (Xi) = Xi — Xi+i — xt_i

18 Now we can use these identities to transform A^: Ai+5

= A+4 = Bi+3 - H(-Xi+3) = Ei+2 - 4>h(-Xi+3) =

= Ci+1 + 4>h(Xi+1) - h(-Xi+3) = Ai + ^(Xi) + cj>h(Xi+l) - cj>h{-Xi+3) = = Ai + (Xi — Xi-i — Xi+\) + (Xi+i - Xi — Xi+2) + (Xi+4 + Xi+2) = = Ai + Xj + 4 — Xi-i = Ai.

We have shown that the five-periodicity of Ai (and therefore the one of Bi, Ci, Di, Ei, and Yi) follows form the five-periodicity of Xi. Now we need to prove the five-periodicity of Xi. Let us introduce four new algebra elements U=eXi;

V=eYi;

U = eXilh;

V = eYi^h.

They satisfy the following commutation relations UiVi = qViUi, UiV^ViUu

UiVi = qViUi, Ui = (Ui)h,

VijJi = UiVi,

Vi = (Vi)\ (27)

where q=e2"ih,

q=e2wih.

These variables are transformed in an especially simple way, tfi-i = V-1

(28)

Vi-^Uiil+qVi)

(29)

1

Ui-i = V-

(30)

Vi-^Viil+qVi).

(31)

As the first step of the proof, we consider the inverse transformation laws for Xi and Yi\ X^

= -Yi;

y 4 _! =Xi + \Yi).

Equations (28) and (30) are obvious. Then, using the standard formula eA+F(B)

+lA Bi ' F(z)dz^ =eAeJ^

we obtain ( -Kh [ e-^ie-^^-l) , \ TT firh r e-iv(*-«ih) \ —— / • • , . , , — N • . / * sdp = Uiexp — / — — - ; — - d p = V 2 Jn \-lP) smh(Trp) sinh(Trfip) J \ i JQ psmh(np) ) = Ui(l+qVi). = Uiexp

19 The proof of (31) is analogous. Now in order to prove that Xi is five-periodic, it suffices to verify that both Ui and Ui are five-periodic. Indeed, if only the operator Ui is five-periodic, it does not suffice because the logarithm of an operator is ambiguously defined. However, if we have two families of operators U and U, which depend continuously on h, then, assuming the existence of an operator X (depending on h continuously) such that U — ex and U = ex^, this operator appears to be unique. (It can be found as lim( m+n / fi )_ 0 (t/ m C/")/(m + n/h) for any irrational value of H.) The fiveperiodicity of sequence (29) (and (31)) is a direct calculation using (27). Corollary 3.1. 1. Let K be an operator acting in the Hilbert space I<2(R) and having the integral kernel K{x,z)=Fh(z)e-?%k,

(32)

where Fh(z) = exp ( - - / ——————dp \ 4 JQ psmh(7rp)sinh(7rn.p)

(33) J

Then the operator K is unitary up to a multiplicative constant and satisfies the identity K5 = const.

(34)

2. Let h = m/n be a rational number and assume that both m and n are odd. Introduce a linear operator L(u) acting in the space C" and depending on one positive real parameter u through its matrix L{u))=Fh{j,u)q2i\

(35)

where

Fh(j,u) = (1 +uY'n [](1 + q2k-lul'n)-1. fc=0

Then the following identity holds: L(u)L(v + uv)L{v + vu~l + u~l)L{u~lv~l

+ u " 1 ) ! ^ " 1 ) = 1.

(36)

20 3.1. Geodesic length operators The aim of this paragraph is to imbed algebra of geodesic functions (6) into a suitable completion of the constructed algebra T*(5). For any 7, function G 7 (6) can be expressed in terms of the graph coordinates on T^,

G7 = trPZl...Zn = ^ e x p I i Yl j€J

{ a€E(I")

m

Al,<x)za \ ,

(37)

J

where rrtj (7, a) are integer numbers and J is a finite set of indices. In order to find the quantum analogues of these functions, we denote by T a completion of the algebra T'1 containing exZa for any real x. Let for any closed path 7 on 5, the quantum geodesic operator G 7 € T be

G* = i trPZl...Zn *x= J2 exP I \ E K"(7,a)^ + 2nihc^%a)) 1 , £% { ^s(r) J (38) where the quantum ordering * • J implies that we vary classical expression (37) by introducing additional integer coefficients ^ ( 7 , a), which must be determined from the conditions below. Note that the operators {G*} can be considered as belonging to the algebra T . In terms of the generators of T , they are G * = ^ e x p | ^ Yl (rnj(7,a)Z* + 2inc«(1,a))\, j€J { aeE(T) J

(39)

Now let us formulate the defining properties of quantum geodesies. 1. The mapping class group action A(S') (8) preserves the set {G^}, i.e., for any 6 € A(5) and any closed path 7, we have S(G^) = G^. 2. Geodesic algebra. The product of two quantum geodesies is a linear combination of quantum geodesic laminations (QGLs) governed by the skein relation 2 2 . Analogously to the classical case, a QGL is a set of self- and mutually nonintersecting quantum geodesies. 3. Unorientness. Quantum traces of direct and inverse geodesic operators coincide. 4. Exponents of geodesies. A quantum geodesic G^7 with nonnegative winding number n is expressed via G 7 exactly as in the classical case, G^ = 2Tn(G*/2),

(40)

21 where Tn(x) are Chebyshev's polynomials. 5. For any 7 and 7', the operators G* and G*, commute. 6. If closed paths 7 and 7' do not intersect, then the operators GO; and Gy commute. Property 6 implies that all quantum geodesies constituting a QGL mutually commute. We denote the Weyl ordering by a usual normal ordering symbol : •:, i.e., :e

aiea2...

e

» n . s ea1

+

-+an

for

any set

{ffl. . a. ^ _ a J .

Proposition 3.1. / / a (quantum) mapping class group transformations (8) transform a graph simple geodesic 7 into the graph simple geodesic 7, then, for the both corresponding quantum geodesic functions, all 0^(7, a) = 0 in (38). Example 3.1. Quantum geodesies for torus. For the torus, there are three graph simple quantum geodesies, which are exactly (15) in the Weylordered form. Then, the quantum geodesies Gz obtained from Gz by the flip transformation is G% = e - * / 2 - W 2 - z + e x/2-y/2-z + e x/2-y/2 .2cos{nh) + ex/2-Y/2+z +

eX/2+Y/2+Z_

( 4 1 )

The product of two graph simple geodesic functions is G\G\

= e™h/2Ghz + e~iilh/2Ghz.

(42) 19

This algebra is exactly the sog(3) quantum algebra studied in , i.e., it is a finitely generated algebra with the lexicographic basis. Indeed, denoting q = e - i7rfi , [A, B]q = q1l2AB - q~l/2BA, and f = q - q'1, we obtain from (42) [Ghx,GY]q = HGz,

[GY,Gz]q=ZGhx,

[Gz,Gx]q

= ZGY,

(43)

with the only central element (the quantum Markov relation) M=GXGYGZ

- q1/2((Gx)2

+ q-2{GY)2 + (Gz)2).

(44)

3.2. Algebra of quantum geodesies Let 71 and 72 be two graph simple geodesies with one nontrivial intersection. So, for Gj and G^, formula (38) implies, by virtue of Lemma 3.1, the mere Weyl ordering. After some algebra, we obtain (cf. (14)) G\G\ = e -*» f t /2 G |

+ e

i^Gz,

(45)

22

where Gz literally coincides with the Weyl-ordered Gl in the classical case (cf. (14)), whereas Gz contains the quantum correction term, G% = * tr x tr 2 ... (ej,. ® e%)[Xz ®X2Z)...* = : t r ! tr 2 ... (e\j ® e^) [Xz Xz + 2(1 - cos7rft)ejx ® e^2] ... : . Here ejj ® e^ is the standard r-matrix that permutes the spaces "1" and "2," and as a result, the "skein" relation of form (14) appears. Locally, this relation has exactly the form proposed by Turaev

22

, i.e., for intersecting

graph simple geodesies, we have the defining relation

\/ U

2

7

^r

{* 7

w (46)

(The order of crossing Gl and G\ depends on which geodesic function occupies the first place in the product; the rest of the graph remains unchanged for all items in (46)) Note, however, that if the corresponding geodesies 71 and 72 are graph simple, we may turn the geodesic Gz again into the simple geodesic Gz' by performing the quantum flip w.r.t. the edge Z. If we now compare two unambiguously determined expressions: Gz', which must be Weyl ordered, and Gz obtained from the geodesic algebra, we find that Gz = Gz'. This is a hint, which eventually results in the following lemma 3 . Lemma 3.1. There exists quantum ordering * ... * (38), generated by the geodesic algebra, that is consistent with quantum mapping class group transformations (8), i.e., the quantum geodesic algebra must be invariant under the action of the quantum mapping class group. For a single crossing of two simple QGLs, the relation has form (46). Lemma 3.2. If more than one intersection of two QGLs occur, the quantum skein relations must be applied simultaneously at all intersection points. Lemma 3.2 implies the Riedemeister moves for a graph as soon as we set the empty loop to be — e ~mh — e%1Kh. That is, for arbitrary three intersecting geodesies, applying (46) simultaneously at all intersection points,

23 we obtain 3

2

3

/ \

and

2 1

A

3

2 1 =

, ^

2

2

(48)

3

1

!

2

2

2

(48)

Example 3.2. Quantum algebra Mi^. Let us consider the quantum geodesic algebra of torus with two punctures and the basic graph in Fig. 4, which is a particular case of the graph in Fig. 3.

a,

\

x

(

d

/ ^I Fig. 4 There are six graph simple geodesies in Fig. 4: X\ = tr RyRaLxL,i, Y1 = tr LaRb, Y2 = tr RdLc, Zx = tr LyRaLxRc, X2 = trLyLbRxRc, and Zi = tr RyLbRxL,i. (The Weyl normal ordering is assumed.) Then it is straightforward to verify using the skein relations (or directly using formulas (3) and (4)) that the corresponding algebra has the following structure (cf. 2 3 ). Let C{ti,t2,ts) denote the set of cyclic commutators of soq(3) algebra, [ti,ti+1]g = &i+2, i = 1,2,3mod3 (cf. (43)). Then, the algebra

24 Mi > 2 is

C(X1,Z2,Y1);

C(X2,Y1,Z1);

[XX,X2] = \YX,Y2) = 0,

C(XX,Y2,ZX);

C(X2,Z2,Y2); (49)

[ZX,Z2\ = £(YXY2-XXX2).

23

This algebra was studied in in relation with the Kauffman bracket skein quantizations. Algebra (49) possesses two central elements (related to geodesies around the holes), Z1Z2-qY1Y2-q-1X1X2,

(50)

and

XiX2YiY2 - q3/2X2YxZx - q-^2X2Z2Y2 - q^2XxYxZ2 +Y2 + Y2 + q2Z2 + q~2Z2 + q~2X2 + q2X2,

- ql'2XxZxY2 (51)

and admits a lexicographic ordering, as follows from (49).

3.3. Quantizing the Nelson-Regge

algebras

Algebra (17) was quantized by the deformation quantization method in 17 ' 25 . We are now able to implement quantization conditions (18). It is convenient to represent the elements a^ as chords connecting the points of the cyclically ordered set of indices i,j £ {..., m, ( 1 , 2 , . . . , m), 1 . . . }. Then, three variants are possible: if two chords do not intersect, then the corresponding geodesies do not intersect as well and the quantum geodesies commute (Fig. 5a); if two chords have a common vertex, then the corresponding geodesies intersect at one point and the three quantum geodesies dij, ajk, and a ^ (as in Fig. 5b) constitute the quantum subalgebra soq(3); if two chords intersect in the middle point (Fig. 5c), then the corresponding geodesies a^ and a^i, i < k < j < I, have double intersection and satisfy the commutation relation (52)

[a,ij,aki] = £{aikdji — auajk).

j

1

a

I

b

i

l

k

c

j

25

4. Conclusion In this paper, we briefly reviewed the action of the mapping class group on the classical and quantum Teichmller spaces. We considered algebras of classical and quantum geodesies, which are parameterized by the edge lengths of graphs; these lengths coordinatize the Teichmuller space. Elements of the quantum mapping class group satisfy the pentagon relation and preserve the classical and quantum geodesic structures; these elements establish automorphisms between quantum geodesic algebras corresponding to different graphs representing the Riemann surface. In higher dimensions (g > 2 for n = 1), addition restrictions on quantum algebras Mff]W corresponding to moduli spaces must appear (likewise the Schottky problem concerning the period matrix structure.) Using the similar approach, R. Kashaev have found the quantum Liouville central charge using the modular and mapping class group transformations acting in the space M3,i 26. The author expresses his gratitude to the organizers and speakers for the hospitality and the atmosphere of creativity during the Woods Hole meeting. The paper was supported by the Russian Program "Nonlinear Dynamics and Solitons." References 1. L. Chekhov and V. Fock, talk on May, 25 at St. Petersburg Meeting on Selected Topics in Mathematical Physics, LOMI, 26-29 May, 1997. 2. L. Chekhov and V. Fock A quantum Techmiiller space, Theor. Math. Phys. 120 (1245-1259)1999. 3. L. Chekhov and V. Fock Quantum mapping class group, pentagon relation, and geodesies Proc. Steklov Math. Inst. 226, 149-163 (1999). 4. L. Chekhov and V. Fock Observables in 3D gravity and geodesic algebras,Czechoslovak J. Phys. 50, 1201-1208 (2000). 5. E. Verlinde and H. Verlinde, Conformal field theory and geometric quantization, Proc. Superstrings 1989 (Trieste, 1989), (World Scientific, River Edge, NJ, 422-449, 1990). 6. V. V. Fock and A. A. Rosly, Poisson structures on moduli of flat connections on Riemann surfaces and r-matrices, Preprint ITEP 72-92 (1992). 7. V. V. Fock and A. A. Rosly, Flat connections and Poluybles, Theor. Math. Phys. 95, 526-535 (1993). 8. R. C. Penner, The decorated Teichmuller space of Riemann surfaces, Commun. Math. Phys. 113, 299-339 (1988). 9. M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Commun. Math. Phys. 1471-231992.

26

10. L. D. Faddeev, Discrete Heisenberg-Weyl group and modular group, Lett. Math. Phys. 34, 249-254 (1995) . 11. R. M. Kashaev, Quantization of Teichmuller spaces and the quantum dilogarithm, preprint q-alg/9705021. 12. R. M. Kashaev, On the spectrum of Dehn twists in quantum Teichmuller theory, preprint q-alg/0008148. 13. O.Ya. Viro, Lectures on combinatorial presentations of manifolds. Differential geometry and topology (Alghero, 1992), 244-264, (World Sci. Publishing, River Edge, NJ, 1993). 14. K. Strebel, Quadratic Differentials (Springer, Berlin-Heidelberg-New York 1984). 15. V. V. Fock, Combinatorial description of the moduli space of protective structures, hepth/9312193. 16. W. M. Goldman, Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math. 85, 263-302 (1986). 17. J. E. Nelson and T. Regge, Nucl. Phys. B B328, 190 (1989). 18. J. E. Nelson, T. Regge, and F. Zertuche, Homotopy groups and (2 + 1)dimensional quantum de Sitter gravity, Nucl. Phys. B B339, 516-532 (1990). 19. M. Havlfcek, A. V. Klimyk, and S. Posta, Representations of the cyclically symmetric q-deformed algebra soq(3), preprint math.qa/9805048. 20. M. Ugaglia: On a Poisson structure on the space of Stokes matrices, math.ag/9902045. 21. A. Bondal, A symplectic groupoid of triangular bilinear forms and the braid groups, preprint IHES/M/00/02 (Jan. 2000). 22. V. G. Turaev, Skein quantization of Poisson algebras of loops on surfaces, Ann. Scient. Ec. Norm. Sup. ,, S (e)r. 4, 24635-7041991. 23. D. Bullock and J. H. Przytycki, Multiplicative structure of Kauffman bracket skein module quantizations, preprint math.QA/9902117. 24. J. E. Nelson and T. Regge, 2 + 1 quantum gravity,Phys. Lett. B B272, 213216 (1991). 25. J. E. Nelson and T. Regge, Invariants of 2 + 1 gravity, Commun. Math. Phys. 155, 561-568 1993. 26. R. M. Kashaev, Liouville central charge in quantum Teichmuller theory, Proc. Steklov Math. Inst. 226, 62-70 (1999). 27. C Jarlskog in CP Violation, ed. C Jarlskog (World Scientific, Singapore, 1988). 28. L. Maiani, Phys. Lett. B 62, 183 (1976). 29. J.D. Bjorken and I. Dunietz, Phys. Rev. D 36, 2109 (1987).

27

LECTURES ON INDICES AND RELATIVE INDICES ON CONTACT AND CR-MANIFOLDS

CHARLES L. EPSTEIN Department of Mathematics, University of Pennsylvania Philadelphia, PA E-mail: [email protected] The aim of the lectures was to provide sufficient background to discuss recent work, done with Richard Melrose and Gerardo Mendoza on index formulae for Fredholm operators in the Heisenberg calculus. We review the geometry of strictly pseudoconvex CR-manifolds, contact manifolds and Grauert tubes. The calculus of Kohn-Nirenberg pseudodifferential operators is briefly described as well as the basic features of the Heisenberg calculus. A new proof of the Boutet de Monvel index theorem for Toeplitz operators is explained, as well as index theorems for certain classes of elliptic Heisenberg operators, and certain special classes of Fourier Integral Operators. Complete proofs of the results described in these lectures and considerable extensions of these results are found in the monograph by Melrose and myself. 1 These are notes for lectures that I presented at the summer school held at CIRM in Luminy, June 29-July 9, 1999 and Woods Hole, MA October 1999. I have added additional material to make the notes more comprehensible and self contained as well as some additional results in the last two sections.

1. Fredholm operators and Toeplitz operators on the circle Our story begins with a classical theorem which I, somewhat incorrectly, call "The Szego index theorem." The history of this result is unclear, it appears that the first such result was by proved in 1921 by Fritz Noether. 2 Let S1 be the unit circle in the complex plane. A function or distribution on the unit circle can be expanded in a Fourier series. A distribution is the boundary value of a holomorphic function on the unit disk if and only if its negative Fourier coefficients all vanish. Define the sub-space i f 2 ^ 1 ) C L 2 (5 1 ) by

H2{S1) = {f^L2{S1)\f

=

f^aneine}.

71=0

2

l

2

1

Let 7T : L (S ) —> J? (5' ) be the orthogonal projection operator. If / is a bounded measurable function then we define a Toeplitz operator Tf :

28

H2^1)

-> H2^1)

by Tfu = 7r/7r(u).

(1)

The first IT is included as it is sometimes convenient to think of Tf : L2{Sl) -* H2^1). One can easily prove the estimate (2)

\\Tfu\\L2<\\f\\L~\\u\\L2.

This estimate can be read in two different ways. As an operator acting on u, Tf is a bounded operator with ||T/||Op < | | / | | i ~ . Since Tf ~Tg = Ty_9 this implies that P>-r9||op<||/-0||L~

(3) OO

provided f,g <E L ^ ). Thus / H-> Tf is a continuous map from L (S'1) to bounded operators on L2^1) in the operator norm topology. We are interested in a general class of operators called Fredholm operators. 00 1

Definition 1.1. Given Banach spaces X and Y, a bounded linear operator A : X —> Y is called Predholm if it satisfies the following three conditions: (a) The range of A is closed as a subspace of Y, (b) The dimension of the null space of A is finite. (c) The dimension of the co-kernel, Y/AX is finite. The Fredholm index is defined by the formula IndA = dim(ker>4) - dim(cokerA). If the index is non-zero then the operator A is not invertible. On the other hand if Ind A vanishes then there is a compact operator K such that A + K is invertible. For most linear operators it is very difficult to compute the dimension of either the kernel or the co-kernel but their difference, the Fredholm index is often computable. This is a reason that there is so much interest in Fredholm indices. An operator A is Fredholm if and only if there exists a bounded operator B : 7 -> X such that BA - Ix = KUAB -IY

= K2

where K\,Ki are compact operators. In other words, Fredholm operators are operators which are invertible up to a compact error. In fact, K\ and

29

Ki can be always taken to be trace class operators. In that case the index is given by Ind A = Tt[B, A] = T r ^ i - K2).

(4)

The trace of a commutator of finite rank, or more generally trace class operators is zero; the Fredholm index is a measure of the failure of this result for "almost invertible" operators acting on infinite dimensional spaces. If X and Y are finite dimensional then Ind A = dim X- dim Y.

(5)

In certain instances this formula is used metaphorically, in case X and Y are infinite dimensional, to define a renormalized "difference of the dimensions" of X and Y. For example if X and Y are subspaces of a larger space and A is an orthogonal projection then Ind A often gives a reasonable definition for "dimX — dimy." Care must be exercised as there are Fredholm operators from a space to itself with non-zero index. We now recall the basic properties of Fredholm operators. Lemma 1.1. For parts (a) and (b) A and B are Fredholm operators from X toY. (a) If A is Fredholm operator, then there exists e > 0 such that if ||J3|]op < e, then A + B is also Fredholm operator and IndA = lnd{A + B). The index is stable, so we do not need to have perfect knowledge of an operator to compute its index. (b) If A is Fredholm and K : X —* Y is compact, then A+K is Fredholm and IndA = Ind{A + K). Changing a Fredholm operator by a compact operator does not change its index. This means that we are allowed perturbations of arbitrary norm so long as they are compact. (c) If A : X —* Y and B : Y —> Z are Fredholm operators then so is B o A and we have IndBoA = Ind£ + IndA

(6)

(d) If A: X -*Y is a Fredholm operator then A* :Y' -> X' is as well and Ind(A*) = -Ind(A).

(7)

30

(e) Ift —> At is a 1-parameter family of operators for t £ [0,1], continuous, with respect to the operator norm and At is a Fredholm operator for each t, then Ind At is constant. This is an immediate consequence of the fact that the index is a continuous, integer valued function. The proofs of these statements and a good general discussion of Fredholm operators can be found in Hormander. 3 We now analyze Toeplitz operators acting on L2(5X). Proposition 1.1. If f e C°(S1) and f does not vanish, then Tf is a Fredholm operator. To prove this proposition we use the following lemma. Lemma 1.2. If f & C0^1),

then TT[TT, f] is a compact operator.

Proof. If / is an exponential polynomial, say, / =

Y^J=-N

a e%it

j

i

tnen

nlnJj-.L^S^^H^S1) is a finite rank operator. For / G C0^1) let a^f be the Nth Cesaro mean of the Fourier series of / . Recall that for a continuous function lim ||/-<7JV/||L-=0. N—too

Since TT is an orthogonal projection it follows from (3) that ||7r[7r,o-jv/]-7r[7r,/]|| < \\aNf - /|| L ~ -> 0 as N -> oo. Hence n[n, f] is the norm limit of finite rank operators and therefore compact. D Proof. [Proof of Proposition] Since / is non-vanishing the function f~l £ C°(51). As 7r2 = 7T we have the identity: Tf-xTf = irf-1*/* = irf~lfir + T T [ / - \

TTJ/TT = ir

+ Klt

for K\ a compact operator. Hence, Tf-i is the inverse of Tf up to a compact error. Similarly we have T/Tf-i = TT + Ki for another compact operator D K2. The next task is to compute the indTf. Associated to a non-vanishing complex valued function is a topological invariant called the winding number. Orient S1 C C so that counterclockwise is the positive direction and let

31

[51] denote a positive generator for Hi(Sl;Z) and [C*] the counterclockwise unit circle thought of as a generator for H\ (C*; Z). Definition 1.2. Let / : S1 -> C \ {0} be a continuous map then /*[S1] = /e[C*] for an integer k. The number k is the winding number of /, which we denote w#(/). An example of a function with winding number k is fk(eie) — elke. Exercise 1. Show that IndT/fc = — k. From the definition it is clear that the winding number of / only depends on the homotopy class of the map / : 5 1 —> C*. In fact the winding number labels the connected components of the homotopy classes of these maps: w#(/) = w#(g) if and only if / can be homotoped to g. Note that any map / into C* is homotopic to the map /I/I" 1 . Exercise 2. Using the fact that R is the universal cover of S1, prove that w#(/) = k if and only if / is homotopic to fkSuppose that /, g : S1 —> C* then we have a formula for the winding of the product w#(/-ff)=w#(/)+w#(g).

(8)

It is a simple consequence of Lemma 1.2 that Tf oTg — Tfg + K where K is a compact operator. Note the similarity between this formula and (6). The index theorem for Predholm Toeplitz operators is: Theorem 1.1 [Szego index theorem]. If f £ C°(51) and f(eie) ^ 0 for all 9, then Tf is a Fredholm operator, and

Ind7> = - w # ( / ) . Note that the left hand side is an analytic quantity, whereas the right is purely topological. This is the pattern for all index theorems: an analytically defined invariant is shown to equal a topological invariant. One then tries to find a usable formula for the topological quantity. In the case at hand we have Proposition 1.2. // / : S1 —> C* is once differentiate then i

C 0

f'(pie\

(9)

32

Exercise 3. Prove this formula Proof. [Proof of the theorem] Let k = w # (/), then e-ikef(ei9) has winding number zero. As stated above, a map with winding number zero is homotopic to a constant map. Hence there exists a continuous map Ft{eie), (t,eie) G [0,1] x S1, such that Ft : S1 -> C \ {0}, with Fo = e~ikef, F1 = 1. The 1-parameter family of maps Gt = elkeFt{e%e) is then a homotopy between / and /fc. Since Gt is non-vanishing for each t it follows that Tat is a continuous family of Predholm operators. Prom the homotopy invariance of the index we conclude that IndT), = Ind7>. From the exercise above we know that Ind Tfk = —k, which completes the proof of the theorem. • Let us summarize the elements of the above argument. (a) We have a family of operators {Tf : / € C0^1)} parameterized by a "symbol" / . (b) We have an algebraic condition on / which implies Tj is Fredholm; in this case: / ^ 0. (c) Thinking of /, a non-vanishing symbol, as a map from S1 to C* we define a topological invariant; in this case minus the winding number. (d) Using the invariance properties of Fredholm index and the topological invariant, we show they are equal. An important but, simple generalization of Toeplitz operators is to use "symbols," / which take values inn x n complex matrices. Let nn denote the operator 7nS>Idn acting on n-vector valued functions on S1, L2(5'1; C"). We define a matrix-Toeplitz operator by setting Tf = 7T n /7T n .

Using Lemma 1.2 one easily establishes that if / takes values in Gln(C) then Tf-i is an inverse for Tf \range7rn, up to a compact error and therefore in this case Tf : range 7rn —> range 7rn is Fredholm. The cohomology group Hi(G\n; Z) is again one dimensional and so we have an analogous definition for the winding number of / : S1 —> Gln(C). With this understood, the

33

index theorem for a matrix Toeplitz operator is exactly the same as for a scalar Toeplitz operator. If the symbol is differentiable then the index formula reads 2TT

Ind(T/) = ^ | T r [ r 1 / ' ] ^ .

o To get a scalar quantity we need to take a matrix trace. A bit more invariantly, matrix Toeplitz operators act on the sections of oriented vector bundles over the circle. For further discussion of these results see 4 , 5 . Exercise 4 [More challenging exercise]. Prove the matrix-Toeplitz index theorem. 2. Contact and CR-manifolds The archetypal index theorem that most people are familiar with is the Riemann-Roch theorem. One can actually do large parts of the subject of index theory from a slightly different point of view using the Szego index theorem as the starting point. The Riemann-Roch theorem is a theorem about complex curves. In particular it pertains to manifolds with even dimension. The Szego index theorem is for a 1-dimensional manifold. We now consider how to generalize the definition of a Toeplitz operator. The correct context for this analysis turns out to be a contact manifold. Contact manifolds are odd-dimensional, and like the circle, often arise as the boundaries of complex manifolds. Let M be a (2n + l)-dimensional manifold. Let H C TM denote a codimension one sub-bundle, briefly a hyperplanefield.Locally we can choose a one form 6 such that H = ker 9. Definition 2.1. A hyperplane field H C TM defines a contact structure if and only if 8 A [d6]n is nowhere vanishing. A contact structure is the opposite of a foliation. It is a nowhere integrable hyperplane field. The contact field is co-orientable if the line bundle TM/Ti. is orientable. A co-orientation for H is a therefore global, smooth choice of positive half spaces in TM \ 7i. A 1-form annihilating H is positive if it positive on the positive half spaces. If H is co-orientable then it is the kernel of a globally defined 1-form, called a contact form. If 8 is a positive contact form then, for any smooth function p, ep6 is as well. The standard model for a contact manifold is R 2n+1 with the contact structure defined

34

by the 1-form: 1 n Oo = dt +-^2[xjdyj

- yjdXj].

(10)

j=i

There are two basic classes of examples of contact manifolds. One comes from classical mechanics and the other from complex analysis. We begin with the first class. Let X be an (n + l)-dimensional, real manifold. Its cotangent bundle T*X is a symplectic manifold with symplectic form given by n+l

w = 2^dxi

/\d£i,

where (x\,..., xn+\) are local coordinates and ( & , . . . , £n+i) a r e fiber coordinates defined by the local trivialization, {dxi,..., dxn+\}, oiT*X. It is obvious that the 2-form u> is closed. It is a little less apparent, but nonetheless true that w is globally defined and does not depend on the choice of coordinates. This makes the cotangent bundle into a symplectic manifold. There is an action of R + = (0, oo) on T*X by

Mx(x,0-=(x,X0We see that M^(u>) = Xu>. Hence T*X is called a conic, symplectic manifold because the symplectic form is homogeneous of degree one. Let S*X = T*X \ {0}/R + be the co-sphere bundle. If g denotes a Riemannian metric on T*X then we define S*X to be the set of vectors of unit length with respect to this metric. This is called a unit co-sphere bundle. For any choice of g the restriction of the quotient map qg : S*X —» S*X is a diffeomorphism. Note that the quotient by the R + -action is canonical, while the choice of metric is not. Let V = ]C"=!i &<%i be the generator of the M + -action. If we take the interior product of V with w, we get the 1-form: n+l

a = iyw = — 2 J £idxi. t=i

A simple calculation shows that da = to. Restricting a to S*X defines a nowhere vanishing form, let Hg C TS*X denote its null-space. Using the fact that V is transverse to S*X one easily shows that a A [dct]n is nowhere vanishing and hence Hg is a contact structure. Since a is homogeneous of degree 1 under the M + -action and

35

annihilates the infinitesimal generator it defines a conformal class of one forms on the quotient S*X. It is again a simple computation to show that the null-space of this conformal class of one forms, % defines a contact structure on the co-sphere bundle. Indeed the restriction of the quotient map qg : S*X —> S*X is a contact diffeomorphism: qgt, :T-tg—>H. Prom a conic, symplectic manifold we have obtained a contact manifold in a canonical way. The process can be reversed. Suppose that (M,7i) is a co-oriented contact manifold. Let A« be the ray sub-bundle of T*M consisting of positive contact 1-forms. If 9 is a choice of contact form then (p, A) — t > X9P defines a diffeomorphism between M x R + and A-^. The symplectic structure on T*M induces a symplectic structure on A« making it into a conic, symplectic manifold. Using the representation of K-H as M x R + , the induced symplectic form can be written d(\6) = {d\A6 + \de). This symplectic manifold is called the symplectization of the contact manifold, and the previous construction is called the contactization of a conic, symplectic manifold. In symplectic geometry, the first fundamental theorem was proved by Darboux, and it was, in some ways a little bit discouraging. Darboux's theorem states that a symplectic manifold has no local invariants. The technical statement is the following. Theorem 2.1. If (Y, u>) is symplectic, given y £ Y there exists an open neighborhood Uy of y and a diffeomorphism ipy : Uy —» M.2n such that Tpy'i^dxiAdZi)

=w\Uy.

It says that from the point of view of symplectic geometry, a neighborhood of any point in a symplectic manifold looks exactly like a neighborhood of a point in M2n with its standard symplectic structure. There is an analogous statement for contact manifolds. Theorem 2.2. Let M be a contact manifold with contact form 0, for all x £ M, there exists an open neighborhood Ux 3 x and a diffeomorphism i>x:Ux^ R 2n+1 such that

rM = o where 6Q is given by (10).

36

Darboux's theorem says that, even after a contact form is selected, a contact manifold has no local invariants. A reference for this material is Arnold. 6 Now we get to the second class of contact manifolds, those arising in complex geometry. On E 2 n we define the standard complex structure by choosing the sub-bundle T 1 '°C" C TM2n ® C spanned at each point by {dZl,... ,dZn} where dZj = \{dXj - idVj). We also define

(11) The complex structure defines a canonical differential operator on functions: 71

Bf = d/| T O, 1C n = ^

(12)

d~z. fdZj.

j=l

In one complex dimension a hypersurface, that is a real curve, does not have any holomorphic geometry associated with it because any curve in the plane can be mapped, as the boundary value of a holomorphic map, onto a segment of the real line. Once again, in some sense, there are no local invariants. If n > 1 then the complex structure defines a structure on a hypersurface, called a CR-structure. Let p be a smooth function on C" such that dp ^ 0 where p — 0 and M = {/3~1(0)}, be a smooth connected hypersurface in C n . We define the "complex tangent space" of M to be T°'lM = T°'lCn \M n(TM C).

(13)

l

If ~Z is an element of T ^ C " for p G M then Z e T°' M if both Re~Z and Im Z are tangent to M. The differential, dp restricted to M vanishes since p is constant on M. We can write dp = dp + dp. Hence 6 := -dp \M I

is a real, non-vanishing 1-form on M and, ker 6 C TM is a real hyperplane bundle. Clearly T°'lM ® TlfiM C ker6» ® C. When does kerfl defines a contact structure? Writing d6 in complex coordinates we see that:

de =

\{dBp)\M=\Y:^dz^d-Z]\M

The condition we need is that 6A[dO]n~1 ^ 0. There is a neat way to express this condition in terms of complex geometry. The two form id6 defines an hermitian form, called the Levi form: CM : TlfiM x TlfiM -> C, £M(Z,W) = id9(Z,W).

37

Proposition 2.1. The hyperplane bundle, ker# is a contact field if and only if the Levi form is non-degenerate. Exercise 4. Prove this statement In complex analysis it is customary to use a negative defining function. Thus M is considered to be the boundary of the set where X = {z\p(z) < 0}. If the Levi form defined by this choice of p is strictly positive (negative) definite, we say that M is the strictly pseudoconvex (pseudoconcave) bound-* ary of X. Fixing a sign for the defining function determines a co-orientation for the contact-field on M. A simple example is provided by the unit ball. In this case p = \z\2 - 1, the 1-form idp \si is given by n

and n

dd = 2^2 dxj Adyj. A calculation shows that 6 A [d9](n~^ is a positive multiple of the standard volume form on the sphere. The unit sphere can also be thought of as sitting in P". The exterior of the unit sphere is the set p > 0, so — p is a defining function for the exterior of the unit sphere in P". With the respect to this defining function the Levi form of 52™-1 j s negative definite, hence the unit sphere is the strictly pseudoconcave boundary of the exterior of the unit ball. Thus far we have defined the CR-structure induced on a hypersurface in n C . This is a local construction which can therefore be used to define a CRstructure on a real hypersurface in any complex manifold. We extract the main properties of hypersurfaces in complex manifolds to give a definition for an abstract CR-manifold which does not require an embedding into an ambient complex space. Definition 2.2. A CR-manifold M is an odd-dimensional manifold with a sub-bundle r°''McTM®C such that Dimension Non-degeneracy

Fiber dimension of T0<1M = (dimM - l)/2. Th°M D T°
38

Integrability

If Z, W 6 C°° (M, T^M) then, so is [~Z, W].

It is easy to see that these conditions are satisfied if TOllM is defined by (13) for M a hypersurface in a complex manifold. In general, TlfiM ® T°
.Ti,oM

x Ti,oM

^

c

A co-orientation for H fixes the notion of a positive contact form and therefore a sign for CM (if it is definite). A diffeomorphism : M —> N between CR-manifolds is a CR-map if 4>* : T°AM

= T°AN.

A map (j): M —> C" is a CR-embedding if

faT^M = T0'V(iW) where TO>10(-^O is the CR-structure induced on the hypersurface on 4>{M) C C n . More generally we define a CR-embedding of a CR-manifold into CN for any N as an embedding (f>: M —> C w such that

faT^M = r1>(V(M) := T^C" U(M) nr^(M) ® C. In general, for M a real submanifold of CN of co-dimension greater than 1, T1>0CN \M HTM C is not a sub-bundle as the fiber dimension varies from point to point. Unlike the case of symplectic and contact manifolds, CR-manifolds do have local invariants. The Levi form defines the simplest invariant. An hermitian form h has a single invariant, its signature. The signature is a triple of integers (n-,n+, n0) where n = n_ + n+ + no; n+ is the maximum dimension of the subspaces on which h is positive definite and n _ is the maximum dimension of the subspaces on which h is negative definite. For each point p G M the signature of C^ is a CR-invariant. Proposition 2.2. The signature of the Levi form is a CR-invariant. That is: if M and N are CR-manifolds and

39

Exercise 7. Let B\ denote the unit ball in C n and let U be a neighborhood of a point on the boundary. Suppose that <j> :U C\B\ —> C" is an injective, holomorphic map, smooth up to U n B~[. If (j)(U n S2n~l) C 5 2 " - 1 then <j){UC\Bx CBi). If a real hypersurface in C n is transverse to the radial vector field then restricting the 1-form n

a

= Yyxidyj - yidxA 3= 1

to the hypersurface also defines a contact structure. This contact structure is generally different from the contact structure induced on a strictly pseudoconvex hypersurface from the complex structure of C". We finish this section by considering the intersection of the two types of examples. Let X be a real analytic manifold and let X be the complexification of X. Then X embeds into A" as a totally real submanifold. The tubular neighborhood theorem implies that there are diffeomorphisms from a neighborhood of X in X to a neighborhood of the zero section of T*X. (j>:X-> T*X which reduce to the identity on X. One can fix this diffeomorphism so that the induced complex structure on the image of <> / satisfies certain properties. Choose a real analytic Riemannian metric g on X. This defines a function pg(£) = ||£||2 on T*X. The map, can be chosen so that (a) cj>: X —» X is the identity. (b) With the complex structure induced on the image, T0'l<j>{X) = (^•T0'1^, we require that 1: (:r,£) —> (x, —£) is an anti-holomorphic map. (c) Imdpg =a = £ " = 1 iidxi. (d) (dd^r = 0, away from pj x (0). Let X£ = {pg < e 2 }. It follows from the second item above that the contact structure induced on bX€ by the complex structure agrees with the contact structure defined on bXe by the symplectic structure on T*X. The subsets Xe are called Grauert tubes. For more on this construction see 7 or 8 .

40

3. CR-functions and a generalization of Toeplitz operators On a CR-manifold there is a canonical differential operator, analogous to the d-operator denned by Btf := df rTo,iM •

(14) N

If $ = (<j>i,... ,(J>N) is an embedding of M into C then, $ is a CRembedding if and only if dbfa = 0 for alii = 1,..., N. It follows from the Leibniz formula that the kernel of db is an algebra. If M is the boundary of a domain X in C" then the ker db consists of the restrictions of holomorphic functions in X to M. More generally if M is an abstract, compact, strictly pseudoconvex CR-manifold which is the boundary of X, a compact, complex manifold with boundary, then the kerdt again consists of the restrictions to M of holomorphic functions on X. If dim M > 5 then such a compact, complex manifold with boundary always exists. Moreover the induced co-orientation of the contact field agrees with the given coorientation. If dim M = 3 then the existence of such a complex manifold is a very subtle question. Generically such a manifold does not exist, see 9 . A Stein space is a complex space with a very large family of holomorphic functions. Indeed it follows from results of Remmert that any Stein space admits a proper holomorphic embedding into CN, for some N. Grauert showed that a compact, complex manifold with a strictly pseudoconvex boundary is a "proper modification" of a Stein space. Prom work of Heunemann and Ohsawa it follows that a strictly pseudoconvex manifold with boundary X can be augmented along its boundary so that X C X is a relatively compact subset in a proper modification of a Stein space and M is therefore a proper hypersurface in X. If a function / £ ker 9 o n X then / f M is in ker db- Restricting a proper holomorphic embedding of X into C^ to M we obtain a CR-embedding of M into CN. In this case the algebra of CR-functions is large, containing the closure of the algebra generated by the coordinate functions of the embedding. Using work of Boutet de Monvel, Kohn and Harvey and Lawson, one can show that a compact, strictly pseudoconvex CR-manifold has a CR-embedding into CN, for some N if and only if it is the boundary of a strictly pseudoconvex, complex manifold with boundary, see 10, n and 12. We call such a CR-manifold embeddable. We can generalize the construction of a Toeplitz operator defined earlier on the boundary of the unit disk to an embeddable, strictly pseudoconvex CR-manifold. Suppose that M is such a manifold. The ker db is infinite dimensional, containing enough functions to separate points on M. Choose

41 a volume form on M, and let H2(M) denote the L2 closure of ker<9& and S denote the orthogonal projection S : L2{M) -» H2(M). This is called the Szego projector. The Szego projector is not canonically defined as it requires the choice of a volume form. The image of S is canonical, but the null space is not. A complex valued function a on M defines a Toeplitz operator by setting Ta = SaS : H2{M) -» H2(M). This class of operators can easily be enlarged, we can allow A to be a pseudodifferential operator and then define TA by the same formula: TA = SAS. As we shall see this only slightly expands the class of operators that one obtains. Understanding the Szego projector is the key to understanding and further generalizing Toeplitz operators. The Szego projector cannot be a standard pseudodifferential operator. This is a consequence of the symbolic properties of such operators. It turns out that the Szego projector does belong to a slightly exotic calculus of pseudodifferential operators called the Heisenberg calculus. To prepare for our discussion of it we now review the basic facts about standard pseudodifferential operators. 4. Pseudodifferential operators, symbols and radial compactification We begin our study of operators in a very general setting. Given a linear operator A : C£°(R™) —> C°°(Kn), one would like to represent it as integration against a kernel. In a certain sense, this is always possible. Theorem 4.1 [Schwartz kernel theorem]. There exists a unique element KA

fc t-

^JK

X K.

)

such that

Af = JkAf in the weak sense, i.e. < Af, g >=< kA, f(x) ® g(y) > for every

f,g£C?(Rn).

42

Using the variables x, x — y we can represent this as a family of convolution operators: Af(x) = / kA(x,x-

y)f(y)dy

We make the following assumptions on the kernel kA • (a) kA(x, Z) a smooth function on M" with values in tempered distributions on E". (b) kA(x, z) is in C°°(E" x (E" \ {0})) - the only singularities are along the diagonal, i.e. where z = 0. (c) In virtue of our first assumption for each fixed x, kA(x,-) has a Fourier transform. Let a(x, £) be the partial Fourier transform of the kernel, that is > € Coo(En;«S'(En)).

a(a;,0 =< kA(x,z),e-*z

If we assume that the support of kA in z is compact then a(x, £) S C°°(En x R n ). Let (p <= C^°(E) with tp(t) = 1 for |i| < 1 and ip(t) = 0 for \t\ > 2 then we can write kA(x,

z) =
z) + (l-


z).

The first term has compact support in z whereas the second term belong to C°°(En x E n ). Since we are more interested in the singular term we assume, provisionally that kA(x,z) has compact support in z. Indeed, in our applications we work on compact manifolds and then we can always arrange this by using a partition of unity. With these assumptions A has a representation as a Fourier integral:

Af = Ja(x,Z)e**fc)dt. Here, a(x, £) is called the symbol of the linear operator A. For example, let us consider a differential operator

Af(x) =

Yl

aa(x)Daf(x),

\a\
where D = —i(dXl,... ,dXn) and the a are multi-indices. One can check that the symbol of A is

a(a;,0= £ aa(x)e. N
43

The raison d'etre for pseudodifferential operators is to translate analytic properties of linear operators into algebraic properties of the symbol and vice versa. It might appear that with all these assumptions we would have reduced our consideration to a small family of operators. In fact, this is still far too general a class of operators to have a useful theory. To obtain analyzable classes of operators we need to place further restrictions on the symbols. Kohn and Nirenberg introduced the class of symbols that bear their name, in part for the purpose of studying the 3-Neumann problem, see 13-14. These are also called classical or polyhomogeneous, step-1 symbols. A smooth function a(x, £) is a classical symbol if it has an asymptotic expansion: m

a(z,£)~ ] T aj(x,0 for large £,

(15)

where a.j(x, A£) = \ia,j(x, £) and a,- are smooth in K" x (Rn \ {0}) . Here ~ means that for arbitrary multi-indices a, (3 and N <m there exist constants [C^p] such that m

\D<*xDl[a{x,0- Y, aj{x,(,)]\l.

(16)

j=-N

Note that differentiating with respect to £ improves the order of approximation. The number m is called the order of the symbol. Definition 4.1. We denote the set of classical symbols of order m by 5™N(Kn). The first term in (15), am(x,£), is called the principal symbol. The symbol a (a;, £) has an expansion at infinity in the ^-parameter in terms of homogeneous functions. This can be interpreted as a Taylor expansion for the function a(x, £) along the boundary of a compactification of M" x K™ in the ^-directions. Indeed a classical symbol of order zero turns out to be nothing more than a smooth function on this compactified space. We identify Kn with the affine hyperplane xn+i = 1 in R" +1 . Projecting this hyperplane onto the unit sphere in R n + 1 carries lRn to the open upper hemisphere, 5". We can therefore compactify Mn by adding the equator of 5™. This space, denoted by radM", is called the radial compactification of R™. Polar spherical coordinates are defined by:

W= andP=

|

l-

44

The parameters (w,p) define a local coordinates at infinity in rad K". A function on rad R is smooth, near to infinity, if and only if it is a smooth function of the variables (w,p). The boundary of rad K is the set {p = 0}. We now replace Rn x Rn by Rn x r a d l " . Suppose that a is a symbol of order zero. Rewriting the asymptotic expansion we have that o o j=-oo

j=-oo

"

Recalling that a,j is homogeneous of degree j , we can rewrite this as oo

j=o

The conditions in (16) imply that a extends smoothly to R" x rad R and this is the Taylor expansion of the extended function along E" x 6radM . We can therefore identify 5™N(K") with /j-mC°°(]Rn x radM"). So, up to multiplying by pm classical symbols are simply smooth functions on M" x rad R . We have shown that Proposition 4.1. The symbol a(x,£) e ^ ( K n ) if and only if pma{x,Q has a smooth extension to M.n x radM . At the root of this discussion is the dilation structure on K n . The positive reals, R+ acts on K" by M\(£) = A£. A function is homogeneous of degree j precisely when

M*J = XjfThus it is the linear dilation structure of 1R" which defines the smooth functions on the rad R n . Now that we have defined symbols, we turn to the operators they define. A symbol a € S™N defines an operator a(X, D) by

a{X,D)f=-^Ja(x,t)e**fc)dt.

(17)

This is called the left quantization; the function a is called the symbol of the operator a(X, D). An operator with such a representation belongs to the Kohn-Nirenberg class of pseudodifferential operators of order m, which we denoted by *™N(R"). If A G ^™N(Kn) then the order m term in the asymptotic expansion of its symbol is called the principal symbol. It is denoted by am{A).

45

In functional analysis a basic tool for studying an operator is the functional calculus. This is especially powerful for self-adjoint operators. If two self adjoint operators commute then they can be simultaneously diagonalized and again the functional calculus is a very power tool for studying the relationships between the two operators. An underlying motivation for introducing classical pseudodifferential operators is that all such operators are simultaneously approximately diagonalized by the Fourier transform. Moreover we can use this representation to construct an approximate functional calculus for the entire algebra. Recall the definition of Z2-Sobolev spaces, Hs(Rn) for s€R. Definition 4.2. A tempered distribution u belongs to Hs(Rn) if its Fourier transform u is a measurable function such that

\H2H- = fwi)\20- + \Z\2)'<% is finite. Pseudodifferential operators have simple mapping properties on L2-Sobolev spaces. The following theorem summarizes the basic properties of the KohnNirenberg class of pseudodifferential operators. Theorem 4.1. Suppose that A and B are Kohn-Nirenberg pseudodifferential operators. (a) [Symbol filtered algebra] If A e *™(Rn) and Be ^(W1), then AoB e tf£J+m'(Rn). (b) [Composition formula for principal symbols] Let o-m(A),o~mi(B) be the principal symbols of A and B then the principal symbol of the composite is given by the pointwise product:

This is simultaneous diagonalization, to leading order. (c) [Mapping properties] If A € #™N(Rn); then for allseR . tl

\M. ) —> tl

we have

\)!&. ) .

This states that if A £ \&™w(]Rn), m a natural number then its Sobolev space mapping properties are the same as those of a differential operator of order m.

46 Let oo

j = -OO

An operator in ^"^(R™) is called a smoothing operator, its symbol, a(x, £) belongs to C°°(R%,S (&%)). If A is a smoothing operator then A : C-°° -> C°°.

In the context of pseudodifferential operators, the smoothing operators are considered to be negligible. For example a pseudodifferential operator A is considered "invertible" if there is another pseudodifferential operator B such that both AB - Id and BA - Id are smoothing operators. This is, of course not the same thing as being invertible, it is a close as one can come working entirely within the confines of pseudodifferential operators. Definition 4.3. A pseudodifferential operator A of order m is elliptic if am(A) is non-vanishing. It is a consequence of the formula for the symbol of a composition that an elliptic pseudodifferential operator is invertible up to a smoothing error. Proposition 4.2. If A is a pseudodifferential operator of order m then there exists a pseudodifferential operator B of order —m such that AoB — Id and B o A — Id are smoothing operators if and only if A is elliptic. To define pseudodifferential operators on a compact manifold, one needs to check that, for any ip € Diff (Rn) which is the identity outside of a compact set, we have (ip-l)*AiP* G #™N(Mn) provided A € *™N(Kn). This implies that under changes of coordinate, the operator may be changed, but the class of operators, including its order filtration is not changed. A proof of this fact along with all the other results reviewed here can be found in 3 . Definition 4.4. Let X be a manifold. An operator A : C^{X) -> C~°°{X) is called a pseudodifferential operator on X of order m if the following conditions are satisfied. (a) Given ip,ip € C%°(X) such that supp tp n supp tp = 0, then the operator ipAifi is a smoothing operator; tpAil>:C-oo(X)->C°o(X).

47

(b) If supp tp is contained in a coordinate chart U, then with respect to any choice of locaj coordinates on U the operator (pAtp £ \&™N(]Rn). Let T*X denote the cotangent bundle of X. The fibers of X are real vector spaces and therefore have a canonical M+-action. One of the most useful properties of pseudodifferential operators is the fact that if A is a pseudodifferential operator of order m, then its principal symbol am(A) is a function on T*X \ {0}, homogeneous with respect to the canonical I n action on the fibers. If x are local coordinates for X with dual coordinates £ on the fibers of T*X then am(A)(z,A£) = A"V m (,4)(z,0. That the principal symbol is a function is by no means obvious. One needs to examine the transformation law for the symbol of A under a change of local coordinates. In general this dependence is very complicated. It so happens that the leading term in the asymptotic expansion of the symbol transforms as a function on the co-tangent space itself. For the details of this calculation see 3 . Because the principal symbol is well defined as a function we can transfer the notion of ellipticity to a manifold. Definition 4.5. A pseudodifferential operator A of order m defined on X is elliptic if crm(A) is nowhere vanishing on the co-sphere bundle S*X. An elliptic pseudodifferential operator on a compact manifold is invertible up to a compact error. Suppose that the principal symbol crm(A)(x,£) does not vanish for all (z,£) where £ ^ 0. We define a new symbol by setting (At e\' w h e r e V s = \ 1 -r °m{A){x,(,) yi i f s > 2 . If b(X, D) is the quantization of this symbol then it has the following properties: b

*'^=

A o b(X,D)

- I e * K N ( * ) , a n d b(X,D)

oA-Ie

*^{X).

Using the composition formula for pseudodifferential operators, we can actually find an operator B such that AoB-Ie where a-m(B) = b.

tf-~(X),

and B o A - I e * « " ( * ) .

48

A consequence of this almost invertibility is a criterion for a pseudodifferential operator to be Predholm. Proposition 4.3. If A £ \P£J,(X) is elliptic then A is a Fredholm operator as a map A : HS(X) -> Hs-m(X) for any s. It is now easy to show that, on a compact manifold of dimension at east 2, the range of a classical pseudodifferential projection has either finite dimension or finite co-dimension. Let P be an orthogonal projection, acting on functions which is a standard pseudodifferential operator. As it is bounded it must be an operator of order 0. Since P2 — P = 0 it would follow that ao(P)(ao(P)-l) = O. Hence ao(P) is either identically 0 or identically 1. If
49 This decomposes T*X as the Whitney sum of two subbundles T*X =M6»©TJ-. Which in turn defines two bundle maps by projection onto the subbundles, TTi : T*X -> T-1, 7T2 : T*X -> R0. We define a parabolic action of M+ on T*X by letting Mx(O = ATTI(O + A2TT2(£) for £ e TX*X The trajectories look like parabolas with respect to (6,T"L) coordinates. Before we consider the compactification of T*X defined by this action, we examine the action in more detail. To understand this action, we use Theorem 2.2 to introduce Darboux coordinates. These are local coordinates, (t,x,y) on X such that: 9 = dt+ -(x-dy-y-dx),

the Reeb vector field is T = dt- We employ the following notation for covectors n

(&,£') = (£o,Ci,...,6n) ^^odt + Yl^jdxj

+£j+ndyj.

The projections in these coordinates are given by: " 1 1 TTi(&,£') = ]T(& + -yi(,o)dxi + fe+n - -Xi£0)dyi. The coefficients are linear functions on the cotangent space and therefore the symbols of vector fields,

ii + \yjto = e(\(dXi + \yA)),

(18)

Zj+n - 2 ^ 0 = cr(^{dyj - -Xjdt)). We denote these vector fields by: T:=dt,

Xr.= dXl + \dt,

Yr.= dyi -\xjdt.

(19)

50

The contact field is locally spanned by {X,,Y}}. These vector fields satisfy the following commutation relations:

[Xt,T] = [Yk,T] = 0, [Xj,Xk] = [Yj,Yk]=0, [Xj,Yk]=-8jkT. These relations define the Heisenberg Lie algebra. Integrating these vector fields gives the Heisenberg group structure on R 2n+1 . The addition is defined by (t, x, y) © (t1, x', y') = (t + t' + -{y1 • x - x' • y), x + x', y + y'). The parabolic dilation structure "twists" as the base point moves. By this we mean that there do not exist local coordinates (xo, • • • , #2n) such that, in terms of the linear coordinates, {/3j} defined on T*X by the local co-frame {dxo,... ,dx2n} the dilation structure takes the form Mx(x, /?) = (x; A2/J0l Aft,..., A/32n). This is a reflection of the non-integrability of the contact field. Instead we use the vector fields {T,Xj,Yk} to "flatten" the dilation structure. The symbols of these vector fields, {ao,..., <72n}, give local coordinates for the fiber of T*X. They are, in fact, linear coordinates on the fibers. However they do not come from a choice of coordinates on the manifold. With a' = (CTI , . . . , 0"2n), the parabolic action takes a simple form in these coordinates: MX(X,CTO,(T')

=

(x,\2a0,\a).

Using the parabolic dilation structure we define a new compactification on T*X. Let g denote a Riemannian metric on T*X. The smooth hypersurface:

S = {M7ri(Ol2 + MOl2 = l} is transverse to the orbits of M.\. The set underlying the parabolic compactification of T*X is HT*X - T*X U E. That is: we add one point at "infinity" for each trajectory of the dilation structure. The philosophy behind using a dilation structure to define a compactification is that a smooth function in the complement of the zero section which is homogeneous of non-positive, integral degree should have a smooth extension to the compactified space. With this in mind we let P — (ao + k ' l f f ) " *

an

d uo - o-op2,Ui = dip,

i = l,...,2n.

(20)

51

A function near bHT*X is smooth if it is a smooth functions of (p, uo, • • •, U2n). In other words, appropriate subsets of these variables give local coordinates near to bHT*X. Notice that (Top2 and a'p are homogeneous of degree 0 with respect to the parabolic action, and p is homogeneous of degree —1, it is a defining function for the boundary. This new compactification defines a new class of symbols: Definition 5.1. A function a S C°°(T*X) defines a Heisenberg symbol of order m G R if pma has a smooth extension as a function in C°°(HT*X). We denote this set of functions by S™(X). From our discussion of classical symbols it is easily seen that a function a defines a Heisenberg symbol of order m if and only if it has an asymptotic expansion o

(21)

j=-oo

where a,j is //eisen&erg-homogeneous of order j , M*\CLj =

\ia,j.

6. The Heisenberg calculus The introduction of Darboux coordinates identifies a coordinate chart in X with an open set in the Heisenberg group. For the moment we work in the Heisenberg group itself. We can quantize Heisenberg symbols to obtain operators just as before. Instead of formula (17) we use the Weyl quantization formula

aw(X,D)f = ^-L^

IIa(^,Oei<x-y^>f(y)dyd^.

(22)

We use this formula because Weyl quantization has better invariance properties than the left quantization used above. Definition 6.1. An operator A : C£°(X) -> C~°°(X) is a Heisenberg operator of order m on X if For any pair
52

The set of such operators is denoted by ty™(X). The functions {cri,... ,<72n} are homogeneous of order 1 whileCTOis homogeneous of order 2. This shows that, in the Heisenberg calculus, the vector fields tangent to Ti, {X\,... ,Xn,Yi,... ,Yn} define operators of order 1 while the vector field T defines a operator of order 2. If A is a standard pseudodifferential operator and <j> is a diffeomorphism of X then <j>* A<j)~l* is again a standard a pseudodifferential operator. If A is a Heisenberg pseudodifferential operator then in general the conjugated operator is not. A diffeomorphism which preserves the contact structure, i.e. (j)*(9) = ex9 is called a contact transformation. If <j> is a contact transformation then <j)*A~1* is again a Heisenberg operator. The reason for this is quite simple: If X is a diffeomorphism then the map * : T*X -» T*X has a smooth extension to HT*X if and only if <j> is a contact transformation. Observe that the symbols of order zero are defined, in a coordinate invariant way, as the set of smooth functions on a compact manifold with boundary. Exercise 10. Prove that a diffeomorphism <j>* : T*X —> T*X has a smooth extension to HT*X if and only if 0 is a contact transformation. If A G ty™(X) then it has local representations as in (22) where a € S™(U). Such a symbol has an asymptotic expansion in Heisenberg homogeneous terms. It can again be shown that the leading term in the expansion is well defined modulo lower order terms. We call this the Heisenberg principal symbol of A which we denote by a^(A). In the manifold context we interpret the ^-variable as the linear coordinate on the fibers of T*X induced by the local coordinates x. To better understand the link between the class of Heisenberg operators and the Heisenberg group we change variables in (22) as follows: For each x define the (2n + 1) x (2n + 1) matrix

(

i _i 1

i\

2 2\

0 1 0 0 0 1/

where x = (xo,x,x),y = (yo,y,y), and x,x,y,y G R n . Observe that L " 1 = L-x. If © denote the Heisenberg addition operation then x®y~l = Lx(x-y).

53

Now formula (22) reads

aw(X,D)f = =

fL{^,tiy)f{y)dydZ f

]]

<

x

x +v

(24)

2 >*y<*®y- >°>f(y)dyda

where I
Xma^(A)(x,a0,a').

The composition formula for symbols in the Heisenberg calculus is quite different from that in the classical calculus. In order to work out this formula we require a third and final calculus called the isotropic calculus of pseudodifferential operators. This calculus is defined by symbols a which are functions on R 2n . Definition 6.2. A function a 6 C°°(R2n) belongs to S^o if r-ma(u) extends smoothly to the radial compactification of R2n, where r = \u\. Or more prosaically a has an asymptotic expansion o m a(u) ~ r y~^ ajiu) j=-oo

2

where a,- is smooth in K " \ {0} and a,j(\u) = Xja(u), for A > 0. Exercise 11. Show that if a G S™o then for any multi-index a there is a constant Ca so that | ^ a ( « ) | < C Q ( l + M) r o - | Q |

(24)

Exercise 12. Show that for each multi-index a |9 a(M)l lull ,1in " ||a||ro,a = s u P ( i + M ) r o _ | Q | defines a semi-norm on S™o. Show that these semi-norms define a topology on 5™o with respect to which SJ^, is complete.

54

To quantize these symbols we split R2" = R£ xR£ and define an operator acting on <S(Rn) using the Weyl formula:

qw(a)f = J^JJa{Z±2;Z)eH*-vKf(y)dydzm

(25)

If a and b are Schwartz class functions then qw{o) ° qw{b) = Qw{c) where c is another Schwartz class function defined by c(u) = a#b(u) =

i

/ / a(v + u)6(w + u)e2Mv>w)dvdw.

(25)

with w(v,iu) = v • w — v • w, the standard symplectic form on M2n. The restrictions of the symplectic form to R™ x {0} and {0} x Mn are identically zero. These are complementary Lagrangian subspaces. i » a#b is determined by the symplectic structure of The operation (a, b) — R 2n . It defines a non-commutative associative algebra structure on <S(R2n). It is important to note that the symbol of qw{a) ° 1w(b) does not depend on the choice of splitting of R2n as a product of Lagrangian subspaces. Choosing a different such splitting defines a different quantization of S*so which nonetheless has the same composition formula for symbols. Using a standard integration by parts argument one can show that for any m,m' G R the #-product extends to define a map continuous with respect to the symbolic semi- norms, such that qw{a) ° qw{b) = qw{a#b). Exercise 13. Use (24) to extend the #-product to a map # : 5;™ x 5£J,' —> S™^m continuous with to respect the symbolic semi-norms. Exercise 14. If a € 5^, and b € S™o show that a#6 has an asymptotic expansion a#b{x,£) ~ ^ ^ [ i V A j - i V - D e l V z . O & f o , * ? ) r»=x,,=«, as \x\+\£\ -> oo, fe=0

where Dx = -i{dXl,...

,dXn).

Definition 6.3. An isotropic symbol a of order m is elliptic if there is an isotropic symbol b of order — m such that a#6 - 1 G <S(R2n) and 6#a - 1 G 5(R 2n ).

55

An isotropic symbol a of order m is invertible if there is an isotropic symbol b of order — m so that a#b = 6#a = 1. Invertibility is a stronger condition than ellipticity! Exercise 15. If a € S(R2n) show that 1 + a is an elliptic isotropic symbol though it may fail to be invertible. The principal symbol of a Heisenberg pseudodifferential operator is a homogeneous function on the cotangent bundle. This means that it is determined by its restriction to any hypersurface transverse to the orbits of parabolic R+-action. For example the principal symbol is determined by its restriction to p = 1. It turns out to be more convenient to use a different transversal hypersurface: we use instead the bundles of affine hyperplanes E± = {(To = ±1}. We refer to these as the upper and lower hyperplanes, respectively. The subspace GQ = 0 is invariant under the parabolic action and so orbits in this set do not meet either affine plane. However this does not lead to difficulties since these orbits lie in the closure of the orbits which do intersect H±. As the symbols are continuous functions their values on the orbits which lie in (To = 0 are determined. For an operator A e \I>™(X) we let a^(A)(±)(x,a') = a^(A)(x,±l,a'). For each x the affine hyperplanes, Hf are linearly isomorphic to R2n. A function a £ C00^!!^), that is a smooth function on the total space of the affine hyperplane field H^, belongs to Sl^o(H^) if, for each x € X, a(x,-) e S£0(H±). In other words elements of S^H^ are the smooth sections of a vector bundle with fiber Sgo(R2n). If 6 is the contact form then ±d6 induces symplectic structures on H^ . This means that for each x we can define a #-product such that #± : Siso{Hx ) x 5 iso [Hx ) —> SitJ

{Hx ).

Note that we use +dO to define the #-product on H+ and — d6 to define it on H~. This product makes S*S0(H±) into a bundle of algebras. The set of smooth sections of H^ which are Schwartz class functions on each fibre is denoted S(H±). These are ideals in the bundle of isotropic algebras:

5(fr±)#±5rB0(fir±),

SUH^SiH*)

C 5^).

56

With these preliminaries we can state the symbolic composition result for the Heisenberg calculus. Let A Gty™(X)]it is a simple consequence of Heisenberg homogeneity that

a^(A)(±) G Stg{H±). Theorem 6.1. If A £ *™(X) and B € #™'(X) thenAoBe and

Vf+m'(X)

<+m,(AoB)(±)=a«(i)(±)#±C(B)(±). This is a composition formula for the principal symbol; it is a coordinate invariant statement. This theorem appears in 15 and 16. A proof of the composition theorem in the spirit of this discussion is given in l. In 16 and 15 mapping properties for operators in ty^X are proved. A consequence of these results is the following proposition. Proposition 6.1. If A £ ^°H{X) then A defines a bounded operator on L2{X). If A G *™(X) for anm < 0 then A defines a compact operator on L2{X). Definition 6.4. An operator A e <£?™(X) is elliptic if the symbols cr^l(A)(±) are invertible elements of the isotropic algebra. It does not suffice for a^l(A)(±) to be elliptic elements of the isotropic algebra. If A is elliptic then there exists Bo G ^^m(X) such that o-^(A)(±)#±a"_m(B0)(±)

= a"_m(B0)(±)#±a^(A)(±)

= 1.

Using the composition properties of the Heisenberg calculus this is easily shown to imply that there is an element B € }$!~m(X) such that AoB-Id,

BoA-Ide $-°°X.

As noted above we can quantize an isotropic symbol denned on K2" and obtain an operator which acts on 5(R n ). If qw(a) denotes the operator denned by the symbol a then the ^-product satisfies qw (a) o qw (b) = qw (a#b).

(27)

From formula (25) it is evident that this quantization requires a splitting of R2n as R" x W1. This is not an arbitrary splitting, the two factors must be complementary Lagrangian subspaces. Suppose that a £ S™o. Quantization is a useful tool as it allows the application of operator theoretic techniques

57

to determine if a is an invertible element of the isotropic symbol algebra. In Taylor 15 it is show there exists an b G S^J™ such that a#6 = b#a = 1, if and only if "the operator" qw(a) : S(Rn) -* S(Rn) is invertible. Of course this operator depends on the choice of Lagrangian splitting used to define the quantization, however, the invertibility or non-invertibility of qw (a) does not. In general the hyperplane bundles H± do not admit global splittings as products of Lagrangian subbundles. For most applications this is of no real importance: If U C X is a Darboux coordinate neighborhood then we can choose smoothly varying local Lagrangian splittings for H± \u . We can then quantize symbols a^l(A)(±)(x) for x G U, obtaining smooth families of operators qw(^(A)(±)(x)) which act on <S(Rn). These operators are called the model operators for A. As a consequence of the result for S*so the symbols a^l(A)(±)(x) are invertible with respect to the #-product structure if and only if the model operators qw(a^(A)(±)(x)) : S(Rn) — 5(Rn) are invertible. While the "model operators" themselves depend on the choice of Lagragian splitting, their invertibility or non-invertibility does not. For many important operators in the Heisenberg calculus it is possible to explicitly compute the model operators and thereby determine their ellipticity or non-ellipticity. 7. Application of the Heisenberg calculus to several complex variables For the applications of the Heisenberg calculus to several complex variables as well as index theory there is one operator of central importance: the sum of squares: n

£a =

3=1

Y/Xj+Y?+iaT-

If a is real then, with respect to the flat volume form dtdxdy, Ca is a formally self adjoint operator. Its Heisenberg principal symbol is given by

58 This implies that

<72"0Ca)(a)(±) = - f > ? T a . For a moment, let a' = (y, rf) denote variables splitting H± into complementary Lagrangian subspaces. The symbol of Ca becomes:


A-\y\2Ta.

The operator A — \y\2 is of course the quantum harmonic oscillator. Before analyzing this operator we explain the connection between Ca and the CRstructure on the Heisenberg group. On H", there is a natural CR-structure with T°>lW = span^- = i(X, + iYj) = dZj -

iz^}.

With respect to this CR-structure

\{ UJ = Y^ 9jdzj, then <9£(w) = — J2 Zj9j- So the Dt-operator is given by

=

-\(Lx]+Y?+itixi>YiV

=-\dbxi+YJ-™T) --4L-n. This is the flat computation. On any strictly pseudoconvex CR-manifold X we similarly define the Dt-operator on functions by • t = d*b8b.

About any given point x one can choose local coordinates so that, in these coordinates the Dfe-operator on functions is given by •6 = - 4 ^C_ n + l.o.t..

59 Here 1. o. t. are terms of order less than 2 as well as terms of order 2 with coefficients vanishing at x. Thus aJ(D 6 )(±)(x) = -i<7 2 H (£_ n )(±)(z). This explains why the analysis of the sum of squares operator is so important in several complex variables. In addition to 8b on functions there is a complex, analogous to the 8complex which is defined on a CR-manifold, X. For each 0 < q < n we define a vector bundle A°b'qX with fiber A«r(T°'1X)'. Invariantly this is a quotient of the complexified bundle AqT*X ® C. If 6 denotes a contact form with T the corresponding "Reeb" vector field then ( T ^ X ) ' can be identified with the annihilator of T1>0X © CT. If Pi denotes the projection into this annihilator then for / S C°°(X) we have a representation of the 9b-operator: 8bf = P1(df). Similarly for each q we get a projection, Pq into a subbundle of AqT*X ® C which represents A°'qX. If u is a section of A°'q~1X then 8bLJ =

Pq((kj).

It is a consequence of the integrability of the CR-structure that db°Bb = 0. The complex, C°°{X; A°b'°)[r}5>'Coo(X; A^^rf 6 • • • [r]5bC°°(X; A°'n) is called the Kohn-Rossi or 9&-complex. Exercise 16. Show that the integrability of a CR-structure is equivalent to

81 = o. In general if a £ C°°(M,Ab'9), then we get the following expression for dba :

8ba = £ Z j f w

Arf + Y, fPPi+i(<&>")•

Choosing an hermitian metric on A^'1 we can define the formal adjoint of 8b which is denoted 8l : C°°(X;Aob'q+1)

—> C°°(X- A°b'q).

The rib-operator on (0, g)-forms is defined by D6c* = (8;8b + 8b8*b)a.

60 A calculation shows that P i=i

HP

je/3

+ lower order terms, see Beals and Greiner. 16 Using the coordinates introduced above, we obtain that, at the center x of this coordinate system, D a

* = -\ E 0 L > 2 + Y f + ~^n - 2
This implies that at x : o$(CPb«)(±)(x) = -j(T 2 H (£_ {n _ 2g) ® IdAo.q)(±)(x). Once again we see that to use the Heisenberg calculus to analyze D°' 9 , the first step is to analyze the sum of squares operator, C^q-n- We now analyze the quantum harmonic oscillator, as it is the model operator for a"(Co). 8. The quantum harmonic oscillator and •& By using separation of variables, the theory of n-dimensional harmonic oscillator can be reduced to the one dimensional case,

To do this case we introduce the annihilation and creation operators: C = y + d y,

C*=y-dy.

Elementary computations show that C*C + Id = CC* - Id = H, [C, H] = 2C, [C*, H] = -2C*. If H / = A/ then using (29) we deduce that H C " / = (A + 2)C*/, HCf = (A - 2)Cf.

(28) (29)

61 We see that C* increases the energy level, it is called the creation operator. Its formal adjoint, C is called the annihilation operator. It is a standard result that H acting on C£°(R) is an essentially self adjoint operator. Its closure is a self adjoint operator with pure point spectrum, see Taylor. 17 The L2-function /o = e x p ( - y ) , satisfies H/o = /„.

(30)

Is it not difficult to show that

for V e CC°°(R). This implies that the lowest eigenvalue of H is 1. If g is a solution of Hg — g linearly independent of /o then an elementary calculation shows that \g\(y) ~ ey I"1 and therefore the lowest eigenstate is one dimensional, spanned by /o- It is called the ground or vacuum state. We obtain the higher eigenstates by setting

fj(y) = (cyfo. Applying (29) recursively we conclude that H / i = (l + 2 j ) / i . Using the commutation relations and the fact that the ground state is onedimensional one can show that {fj,j — O,--.,oo} defines orthogonal basis for L2(M). Exercise 17. Compute the L 2 -norms of the fj, (fJJj) = (Cj(C*)JfoJO). In the n-dimensional case, we have H n = - A n + \y\2, and/o = e - 5 £ ^ . Analogously to the one dimensional case, we have annihilation and creation operators: Ci = \(vj + dVi),

c; = -l-(yj-dyj)

These operators are related to the harmonic oscillator by

(si)

62

We also have the commutation relations: [Cj,C*k] = 26jk,

[Cj,Hn} = 2Cj,

[C;,H n ] = -2C;.

(32)

The higher eigenfunctions are obtained by applying the creation operators to the ground state. They are indexed by multi-indices fi = (/xj.,..., /zn) with Hi > 0. For such a multi-index we set

u = (c*rfo = (cir---(c*nr»fo. Using the commutation relations one easily shows that H n /o = nf0, H n /^ = (n + 2|/i|)/M where |/i| = ^ M i The spectrum of H n is {n + 2k : k G N U {0}}. We can now analyze the sum of squares operators Theorem 8.1. For a 6 R, the operator Ca is an Heisenberg-elliptic operator provided a<£ {±{n + 2j) | j e N o }. Proof. As noted above 9iy(a2H(£a)(±)) = ^ ( H n T « ) . To prove the theorem it suffices to show that (H n =F a) are invertible operators. Since this operator is self adjoint it is sufficient to show that the L2-null space is empty. In order for (H n p ) u = 0 2

to have an L -solution it is necessary and sufficient that a — ±(n 4- 2j) for some j £ No. This completes the proof of the theorem. D Exercise 18. Show that if a G C \ R then Ca is an elliptic Heisenberg operator. The symbol of D°'9 is diagonal with a" {£>2q-n) along the diagonal. Thus it is an elliptic operator in the Heisenberg calculus if and only if q ^ 0 or n. In the q = 0 case D°'° is not elliptic. This was expected, because kerD°'° = ker db is infinite dimensional for an embeddable strictly pseudoconvex CRmanifold. The model operator for Db' on the upper hyperplane is | ( H n — n) which has a one-dimensional null-space. The model operator on the lower

63

hyperplane is \ (Hn + n) which is invertible. Let ITQ be the operator on L2(E") which is orthogonal projection onto the span of /o. /

, ,s

*>M = /o

<^i

W

/o)

-

2

The Schwartz kernel of TTQ is ||/o||~ (/o(^) ® fo(y))- A computation using the quantization formula shows that

where so = 2"exp[-(|y| 2 + |77|2)].

(33)

If Ob' has a closed range in I? ( which is always true if dim > 5), then using the Heisenberg calculus one can show there are operators Q £ ty~2 and 5 6 f J such that QDb = DbQ =

snb = nbs = o,

I-S,

16

see Beals and Greiner. The operator Q is the partial inverse to D6' and 5 is the orthogonal projection onto the null-space. S is called the Szego projector.

In Darboux coordinates the symbol of the S is given by

^ ) ( ± ) = { 2rieXPHa ' |2) ov A

;

\o

(-).

(+)

'

As expected the (+)-symbol is exactly the symbol of the projection onto the ground state of a harmonic oscillator. It is an element of Heisenberg calculus of order zero. We could at this point generalize the notion of Toeplitz operator to a strictly pseudoconvex manifold. Before doing so, we further analyze the structures underlying the definition of the Szego projector. 9. Fields of harmonic oscillators At the heart of the Szego projector is a field of harmonic oscillators defined on a contact manifold. For the remainder of this section we let X denote a contact manifold with contact field, H. Let 9 denote a choice of contact form and H' the annihilator of the Reeb vector field. In addition J denotes an almost complex structure on fibers of H'. The two form d6 defines a

64

symplectic structure on the fibers of H. Let w denote the dual symplectic form induced on H'. The almost complex structure J is tamed by the symplectic structure if u(JX,X)>0. It is adapted to the symplectic structure if it is tamed and W(JX,JY)=UJ(X,Y)

for all X,Y £H'p,

VpeX.

Exercise 19. Show that if the almost complex structure J is defined by an integrable, strictly pseudoconvex CR-structure with underlying contact field H then it is adapted. The point of making this definition is that there are adapted almost complex structures defined on any orientable contact manifold. If g is a Riemannian metric on T*X then there is a unique endomorphism J of Ti! such that g(X,Y)=w(JX,Y). From the symmetry of g and skew symmetry of w it follows that J is a skew symmetric, real transformation with respect to g. This implies that the spectrum of J is purely imaginary and eigenvalues come in conjugate pairs. Of course to diagonalize J we need to complexify H'• Let A°'x c Ti!<8>C denote the bundle of positive eigenspaces of J and A^' the negative eigenspaces. The real endomorphism J underlying the splitting, H' ® C = Aj'° ® A^'1 is easily seen to be adapted. Let (x, a) denote the coordinates on the cotangent bundle defined by a choice of Darboux coordinates. Definition 9.1. With J an adapted almost complex structure we define a quadratic function on T*X : hj,w(vo,
65

Locally we can choose a symplecto-normal basis for TX : We begin with T, the Reeb vector field fixed by the choice of 9. We then choose Xi,..., Xn, and Yx,... ,Yn such that JXj = Yi,JYj and {Xi,...,

wherej =

= -Xj,

l,...,n

Xn, Y j , . . . , Yn} is an orthonormal basis with respect to gJte(V,W) = d0(JV,W).

Locally the field of harmonic oscillators is the principal symbol of the sum of squares operator, - X^=i[^j + Y2\- In general this operator is is not globally defined even though its principal symbol is. Exercise 20. Suppose that J\ and Ji are adapted almost complex structures. Show that there exists a symplecto-normal basis as above such that if (y,rj) are the dual coordinates defined in T*X then n

hjUu,(y,v) = ^v]

+ rf,

o=

:

(34) +

y2

hj2tu>(y,v) = Yl^ i

h

1?? 2

J-

Here {/J,\, ... ,/xn} are positive numbers. The symbol of the projection onto the ground state of the field of harmonic oscillators is h^

so = e

j

(35)

'o . th

Let Sk be the symbol of the projection onto the k eigenspace of the field of harmonic oscillators. The symbols of generalized Szego projectors vanish on the lower hyperplane. In the sequel we use <JQ(S) to denote (TQ{+){S) with the understanding that
and a%(S) = sk

66

for some choice of adapted almost complex structure J. Definition 9.4. A generalized Szego projector to level N is any projection S £ f ° H such that

, j=0

Remark 9.1. The notion of a generalized Szego projector appears in the work of Boutet de Monvel and Guillemin 18 on the quantization of contact structures. Proposition 9.1. Suppose that S,S' are two generalized Szego projections at level 0. Then the map S : range S' —» range S is a Predholm operator. Remark 9.2. These projections can be quite different and do not, in general have the same principal symbols. Proof. The range of each operator is closed, hence it is itself a Hilbert space. All we have to do is to find an inverse up to compact error. Denote the respective symbols by o-(S) = so, and a(S') = s'o. In virtue of (34) there are Darboux coordinates (x, rf) such that, when we quantize these symbols to obtain operators on L2(Rn), they have Schwartz kernels of the form n = V(x)®V(y),

TT' = V'{x)®V'(y).

Here V and V are pointwise positive functions. The symbol of SS' is a$(SS') = (VX)V{x)®V'(y). Because {V, V) is the L2 inner product of strictly positive functions, it is non-vanishing. We can now define an approximate inverse. With u = (V, V1)2 we define F — S'~ «= vl>0 u "' The principal symbol of the composition is a%(FSS') = o-%(S').

67

As S' [YangeS'— Id we see that the composition FSS' : ranged -> ranged' is the identity up to a compact error. Using a similar construction leads to an approximate right inverse. This proves the proposition. • Definition 9.5. The relative index is defined to be the Fredholm index of the restriction R-Ind(S, S') = Ind(S'S : range S -> range S'). The relative index for generalized Szego projectors satisfies a co-cycle relation: Proposition 9.2. If S, S', S" are generalized Szego projectors at level zero then then R-Ind(S, S") = R-Ind(S,5") + R-Ind(S", S").

(36)

Remark 9.3. Relative indices are a basic tool for relating one kind of index problem to another. The relative index labels the path components of the space of generalized Szego projections. That is two generalized Szego projectors have relative index zero if and only if there is a smooth path through generalized Szego projectors from one to the other. This result is proved in 19. The notion of relative indices appears in the literature on index formulae for boundary value problems. In the context of boundary value problems, the projections are classical pseudodifferential operators often assumed to have the same principal symbol. The relative index was introduced in the context of the Szego projectors defined by integrable CRstructures in 9 and in the generality considered here in 19. We now define generalized Toeplitz operators. Definition 9.6. A generalized Toeplitz operator is an operator of the form

TAf = SASf where S is a generalized Szego projector and A € y™(M). Such operators were considered in Boutet de Monvel-Guillemin 18, but A was assumed to be a classical pseudodifferential operator. As we shall see this is, up to lower order terms, the same class of operators. As before for generalized Szego projectors, we can define generalized Toeplitz operators at level k and to level N.

68

Definition 9.7. A generalized Toeplitz operator at level A; is a operator of the form

TAf = SASf where S is a generalized Szego projector at level k and A E \I>™(M). Definition 9.8. A generalized Toeplitz operator to level N is a. operator of the form TAf = SASf where 5 is a generalized Szego projector to level N and A E ^ ^ ( M ) . In the sequel the unmodified term "generalized Toeplitz operator" refers to a generalized Toeplitz operator at level 0. In our formulation it is very easy to see that Toeplitz operators form an algebra: TATB = SASSBS = S(ASB)S. Since ASB G #*(X) if A,B G $*H{X) the claim follows immediately. Proposition 9.3. Let S be the generalized Szego projection (at level 0) and let A be an element of \Er°, then there exists a smooth function a such that SAS-SMaSeV-^M). Here Ma is the multiplication operator Maf = af. The function a is given by a(x) = J

so#aS(A)(+)#sOLjn.

Hi The symbol So is denned in (35). If A is a classical pseudodifferential operator then a is just the principal symbol of A restricted to the positive contact direction. If a is a smooth function then we denote the operator SMaS by Ta. For a, a smooth function, and B G *f>™(X) a straightforward computation shows that a^(MaB) = a" (BMa) = aa^B). This in turn implies that
69 Applying this to Toeplitz operators we see that TaTb = SMaSMbS = SMabS + S[Ma, S]SMbS. As [Ma,S] 6 * " J it follows that TaTb-TabC*-1.

(37)

A Toeplitz operator, TA = SAS, of any type, is Fredholm if it is a Predholm operator from range S to itself. The index of such an operator is always understood as the index of this restriction. An immediate consequence of this computation is Proposition 9.4. If a £ C°°(X) is non vanishing, then the generalized Toeplitz operator Ta is a Fredholm operator. Proof. Proposition 6.1 implies that operators in ^^(X) are compact. It therefore follows from (37) that Ta-i is an approximate inverse for Ta : L2(X) -> L2(X). a If a G C°°(X) is non-vanishing then it is easy to see that for any generalized Szego projector, S the operator a~1Sa is another generalized Szego projector with the same principal symbol. We can relate the index of Ta to a relative index: Proposition 9.5. If S is a generalized Szego projector at level 0 and a e C°°(X) is non-vanishing then Ind(Ta) = R-Ind(5,

a^Sa).

Exercise 21. Use this proposition and the co-cycle relation to prove that Ind(jTa) does not depend on the choice of S. Proofs of these results can be found in 19 . 10. Vector bundle coefficients and the Atiyah-Singer index theorem So far we have discussed operators acting on functions. We can easily extend this formalism to define pseudodifferential operators acting on sections of vector bundles. This is very important for applications to index theory. We first review the classical case. Let E —» X and F —> X be complex

70

vector bundles with fiber dimensions k and I respectively. Let U be a coordinate neighborhood on X over which E and F are trivialized. In U we have bundle maps identifying

E \v~ U x Cfc,

F \u~ U x C ; .

If P is a linear differential operator carrying sections of E to sections of F then, in terms of these trivializations, P \u is represented by a I x k matrix of differential operators [Pij (x, Dx)\. By analogy with the scalar case we define the symbol of P by formally replacing Dx by the fiber coordinate in the cotangent bundle: o-(P)(x,f) = [Py(x,f)]. If (si,...,Sk) are the coordinates of s G C£°(U;E) with respect to the given trivialization then we can represent the action of P on s using the Fourier transform in the local coordinates by 1

r

k

x

(Ps)i(x) = ^ J e* X ^ ( z , 0 ] 3 ; ( 0 d e .

* = !.•••,*•

Prom this formula it is apparent that for each (a:, £) the matrix [Py(a:,£)] is an element of hom(Ex,Fx). In other words X is the canonical projection. With this as motivation we define pseudodifferential operators of order m carrying sections of E to sections of F as operators A : C°°(X;E) —> C°°(X;F) such that if ip G C%°(U) then, in terms of local trivializations of £ ft/ and F ft/ we have the representation tpAtps = j - ^ n [e*xa(x,£)
71 Using the same local discussion we define Heisenberg operators carrying sections of E to sections of F. The single modification is that, in terms of local trivializations, the section a is represented by a matrix of symbols ay G S™(U). The principal symbol is again defined as the leading term in the asymptotic expansion of a in Heisenberg homogeneous terms. Invariantly it takes values in hom(7r* (E),n*(F)). We denote the Heisenberg operators of order m acting between sections of E and F byty™(X;E, F). If E — F we abbreviate this notation to \&™(.X";.E). Let E —> X be a complex vector bundle over a contact manifold. A generalized Szego projection, at level 0, acting on the sections of E is any projection operator SE € ^HO^! E) s u c n ^ ^ GQ{SE)

= soIdE.

Here SQ is the symbol of a scalar generalized Szego projector. Similarly, we define generalized Szego projections at level k and to level N acting on sections of E as projection operators SE G ^(X;E) with the following properties SE is a projection at level k if cr£ (SE) = Sfc ® Id# N

SE is a projection to level N if ^ ( S B ) = ( ^ Sk) <8> Ids . l

Here Sk denotes the symbol of a scalar, generalized Szego projector at level k. Once again for each family of generalized Szego projectors acting on sections of E there is a corresponding family of Toeplitz operators TA = SEASE where A G $£(M;£). Before considering index formulae for Heisenberg operators we first review the Atiyah-Singer theorem for classical, elliptic pseudodifferential operators. As above we let Y be a compact manifold of dimension n and E, F be vector bundles over Y. Let P e V0(Y;E;F). The principal symbol of P, 0o(P) is a section of Hom(7r*i?) n*F), homogeneous of degree 0, i.e. ao(P)(Y,XO = MP)(x,Z)

for A G R+.

The operator is elliptic if the homomorphism a0(P)(x,0:Ex-^Fx is invertible for all nonzero £. In this case P : L2(Y;E) —> L2{Y;F) is a Fredholm operator. As usual we define its analytic index to be Ind(P) = dimkerP - dimcokerP.

72

The Atiyah-Singer index theorem identifies this index with a topological invariantjconstructed from the triple [E,F,ao(P)}. Let T*Y be one point fiber compactification of T*Y, that is we add one point to each fiber of T*Y. The compactified space T?:T*Y

—>Y

n

is a fiber bundle with fiber S . It is useful to have a second description of this space. Fix a Riemannian metric g on T*Y and define the co-ball bundle B*X = {(x,Z)eT*Y : ||£||<1}. The boundary of the co-ball bundle is the unit co-sphere bundle, S*Y. Two copies of the co-ball bundle glued along their boundaries produces a manifold which is diffeomorphic to T*Y,

f*Y ~ B*Y U B*Y. S'gY

For notational convenience we label the co-ball bundles B±Y. One can think of B\Y as the interior of the unit ball in T*Y and B*_Y as the exterior of the unit ball. If w is the symplectic form on T*Y.then ±un define orientations for B±Y which glue together to define an orientation on T*Y. Using the symbol of the operator we construct a vector bundle over VP -» f * y as follows: VP r s ; y = TT-(J5),

VP \B._Y= n*(F).

To complete the construction we use the symbol of P to glue the two pieces together along their common boundary, the co-sphere bundle, by the equivalence relation:

(x,t,e)~(x,Z,f)iff

= *Q(P)(x,Z)e.

Here we use the fact that <7o(P)(:r,£) is an isomorphism for each £ ^ 0. Remark 10.1. The bundle Vp is homotopy theoretic object, i.e. the interesting geometric invariants of Vp depend only on the homotopy classes of E, F and <JQ{P). For example: If Pt is a continuous, one parameter family of elliptic operators in \I/^ N (Y;£, F) then the bundles Vpt are all equivalent. The important data in Vp is described by the class it defines in KQ(T*Y). The Atiyah-Index theorem identifies Ind(P) with a K-theoretic invariant of VP.

73

The Chern character defines a ring homomorphism from Ch : K0{f*Y) —+ # 2 *(f*Y). The index of P can be expressed in terms of Ch([Vp]) and the Todd class of Y which we denote Td(Y) £ ®H^(Y). For more details on this aspect of the subject see 20 or 21. With these preliminaries we can state: Atiyah-Singer index formula lO.l.If P is an elliptic Kohn-Nirenberg pseudodifferential operator from C°°(Y;E) to C^iY^) then P is a Fredholm operator and Ind(P) = (n* Td(y) A Ch(Vp), [f*Y]).

(38)

This formula looks explicit, but in fact it is quite difficult to use it to compute the index of P except in special cases. If the operator P is of "Dirac type" then a much more explicit formula is available. We observe that Ch(Vp) is a quantity depending on P whereas Td(Y) does not depend on the operator at all. A Dirac operator is defined on a manifold with a spin structure. This is, in a certain sense a refinement of an orientation and there are topological obstructions to its existence. An equally explicit formula is available for a Dirac operator defined by a Spin-C structure. It turns out that an almost complex manifold, Y always has a natural Spin-C structure. This in turn defines a Dirac operator which acts on the complex spinors. For the case of a complex manifold, the complex spinors can be identified with S(Y) = (BqA°'qY and the Spin-C Dirac operator with S c = 3 + 8* : C°°(Y; A°-evenY) —> C°°(Y; A°'oMY). The index of this operator is called the Todd genus of Y\ it is expressed as a cohomological pairing on Y as ind(a c ) = (Td(T1-°y),[y]>. If E —» Y is a complex vector bundle then, by choosing an Hermitian connection on E one defines a twisted, Spin-C Dirac operator 5E, acting on C°°{Y\ E®S(Y)). Again there is a formula for Ind(9j5) a s cohomological pairing on Y : Ind(3B) = (Ch(£) A TdCT^Y), [Y]).

(39)

Here, as above, Ch(jB) is the Chern character of the complex vector bundle E —> Y. While formula 38 looks quite similar to this formula, the latter formula is much simpler because: (1) The integration takes place on Y.

74

(2) Only the Chern character of the bundle E —> Y appears, the symbol of the Dirac operator does not appear explicitly. This special case of the Atiyah-Singer theorem suffices for most of our applications. It is useful to have an explicit formula for the Chern character. Using the Chern-Weil theory it can be expressed in terms of the curvature of a connection defined on E, see 20 and 22 . Let E be an hermitian, complex vector bundle and V B a unitary connection on E. The curvature of the connection is KE = V^; in terms of a local trivialization of E it is represented by a matrix of 2-forms QE- Using the transformation formula for QE> under a change of local framing, as well as the invariance properties of the trace powers it is not difficult to show that

Ch(£)=Tr(exP[^-fiEj). is globally defined. The Bianchi identity implies that it is closed. This sum of forms is a representative of the Chern character of E. Remark 10.2. Let X be a contact manifold with contact field 7i. A choice of almost complex structure on the fibers of Tt defines the bundles Ab'9X. The Whitney sum

q=Q

defines a Spin-C structure on X. The 9&-operator is defined by (14), though it does not define a complex unless the almost complex structure comes from an integrable CR-structure. In any case the operator 3c = Bt + 8*b : C°°(X; Sx) — C°°(X; Sx) is a self adjoint Spin-C Dirac operator. 11. The Boutet de Monvel index formula Let X denote a compact, contact manifold. We would like to prove a result, analogous to (38), for elliptic operators in ^^(X;E). A problem arises at the beginning of the discussion: The condition for a Heisenberg operator A to be elliptic is that its principal symbol cr^A) be invertible in the isotropic algebra. This is not a pointwise condition in T*X but rather that there exist a symbolCT"m (B) so that a^(A)(±)#±alm(B)(±)

= a"_m(B)(±)#±a^(A)(±) = Id.

75

This condition is global in each fiber of the cotangent space. Consider the following example. Let S be a generalized Szego projection (at level 0) and /x be a complex valued function on X. The operator AM = Id +(iS is Predholm if and only if /x does not assume that value — 1. Let

A simple calculation shows that

BllAll = Id +

^-[S,n]S.

As noted above [[i,S] € ^^(X), thus B^ is a left inverse up to a compact error. A similar calculation shows that B^ is also a right inverse, up to a compact error. On the other hand, in appropriate coordinates the principal symbol of A^ is given by forCTo<0.

vo(A»)(x,a)-^

From this formula it is quite clear that if fi(x) is real and less than —1 then CTQ(A^)(X,(7) vanishes on a hypersurface in T*X. Nonetheless it follows from the composition formula for principal symbols that 1 =
(T0

H

(BM)(±)#±<70H(^)(±).

Exercise 22. Provide the details of this computation. Our approach is therefore somewhat indirect. In these notes we restrict our attention to Heisenberg operators of a fairly simple sort. We consider operators of the form Id +A' where A' € * ° {X; E) and the "full" symbol of A' vanishes in the lower half of the cotangent bundle. In other words if we fix a Darboux coordinate system and express A' by the formula (23), then the function a'(x,cro,cr') vanishes identically for negative CJQ. This condition is coordinate invariant. Indeed operators with symbol vanishing in the lower half space are an ideal in the Heisenberg calculus. We denote this ideal by I^.(X;E). The case of a general elliptic element is considerably more complicated; it is treated in 1. To obtain a formula for the index of an operator of the form Id +A' with A' € X^_(X; E) we relate this index to the index of a generalized Toeplitz operator. Boutet de Monvel gave a formula for the index of a Toeplitz operator. Using Boutet's formula and a limiting argument one can obtain a cohomological formula for the index of Id+^4'.

76

The next order of business therefore is Boutet de Monvel's formula for the index of a Toeplitz operator. Let X be a contact manifold and E —> X a complex vector bundle. We fix an adapted almost complex structure, J on the contact field, Ti. and let SE denote a generalized Szego projection at level 0 with principal symbol so®Id£ . As before SQ is the symbol of the projection onto the ground state of the field of harmonic oscillators defined by the choice of almost complex structure. Given a section a € C°°(X;Hom(jB)), let Ta = SECLSE denote the Toeplitz operator it defines. If a is everywhere invertible then the operator Ta is Fredholm from the r a n g e d to itself. The proof used in the scalar case works, mutatis mutandis. Traditionally one would say that a is the symbol of the Toeplitz operator. Using this symbol we construct a vector bundle over X x S1 which we denote by Ea. The base space, X x S1 is identified with X x [0,1]/ ~ where (x, 0) ~ (x, 1). We glue the ends of X x [0,1] x E together using the isomorphism a :

(x,0;e) ~ (x, l;a(x)e). This defines the bundle Ea-+ X x S1. The manifold X = X x Sl has a natural homotopy class of almost complex structures. Since TX = H @ WT, we have TX - H ® RT ® TS1. Let dt denote a coordinate vector field tangent to S1. We extend the almost complex structure defined on H by letting JT = dt. A manifold with an almost complex structure has a natural Spin-C structure. This defines a Dirac operator 9B Q acting A0>evenX (g> Ea In 1 it is shown that the index of the Toeplitz operator Ta equals that of the twisted, Spin-C Dirac operator &Ea • From (39) we deduce: Boutet de Monvel index formula for a Toeplitz operator 11.1. Let E —* X be a complex vector bundle and a G C°°(X;hom(E)) which is everywhere invertible. Let SE denote the Szego projector acting on sections of E. The Toeplitz operator Ta — SE&SE is Fredholm with Ind(rQ) = Ind(S £ J =< Ch{Ea) A Td{TlfiX), [X] > .

(40)

In the special case of X = dD, D C C™, a strictly pseudoconvex domain and E —» X a trivial bundle, the formula simplifies considerably. In this case the Todd class is trivial. For (i £ C°°(X;h.omE), a bundle isomorphism, we have the index formula Ind(SEfiSE) =

cnfTr[fx-ldfi}2n+1. x

77

Here cn is a universal constant, depending only on the dimension. Prom this formula it follows that the index is zero if fiber dimension of E is less than (dimX + l)/2. 12. An index theorem for the Heisenberg calculus Now we turn to the computation of the index of a Heisenberg-elliptic operator of the form A = Ids +A', where A' G T^{X;E). To compute the Ind(A) we fix an adapted almost complex structure J on the contact field. Let hj denote the field of harmonic oscillators fixed by this choice. Let {sk : k G No} denote the symbols of the projectors onto the eigenspaces of hj. These symbols satisfy the relations Sk#hj = /ij#s fc = (n + 2k)sk and sk#Sj = Sj#Sk = Sjksk.

The #-product refers to the #-product defined on a0 = 1 by the symplectic form dual to d8. For each IV > 0 we let S^ G *° (X; E) denote a choice of generalized Szego projectors to level N acting on sections of E. Recall that this means

[S(EN}\2 = S^,

ao«(S)(+) = 5 > ® I d * = *™

These operators belong to X^_(X;E). Without loss of generality we can assume that these projections are orthogonal with respect to a fixed volume form on X and hermitian metric on E. The advantage of working with elements of \I>* (X; E) of this special form is the following result: Theorem 12.1. If A' G 2*(X;£) and ldE+A' is a Fredholm operator then, for sufficiently large N the generalized Toeplitz operator to level N, S1^ {IdE+A')S%r) is Fredholm and Ind(IdB +A') = Ind(5^ ) (Id B +A')S(EN)). The index on the left is of the operator Id B +A' : L2{X; E) —> L2{X; E), whereas the index on the right is of the generalized Toeplitz operator to level N acting on S{PL2(X;E).

78

Proof. For each N we have the identity: Ids +A' = SBN)(ldE +A')SBN) + (Ids S{EN))+ S^A'(UE

-SW) + (Id£ -S™)A'S™

Since A' eI+(X;E)

+ (Id£ -S^)A'(ldE

-S).

it is not difficult to show that

<7 0 H (A')#(Id £ -4 N) ) and ( I d s - 5 ^ ) ^ ^ ' ) tend to zero as symbols of order 0. In fact, for large enough N and t e [0,1] the operators

At = 4 W) (Id £ + J 4')4 N ) + (Ids -5<W))+

t[Sg]A'(ldB

-S{EN)) + (Ids - 4 N V ' 4 V V ) + (W £ -S{EN))A'(IdB

~S(EN))}

are all Fredholm. As the family depends smoothly on t in the operator norm it is evident that Ind(ylo) = Ind(Ai). In other words: Ind(Ids +A') = lnd(S{E\ldE +A')SiE) + (Ids =

S^))

lnd(SiEN)(UE+A')S(EN)).

The last index is of a generalized Toeplitz operator acting on SE 'L2(X; Ep Exercise 23. If /x is a complex valued function and 5 is a level 0 Szego projector, then Ind(Id +nS) = Ind(5(Id +fj,)S) provided that /j. ^ - 1 at any point. This can be proved by directly comparing kernels and co-kernels Exercise 24. If fi € C°°(X; hom(£)) and Ids +/J, is invertible at every point of X then Id^ +/J,SE is Fredholm. In case we have: Ind(Ids +M^E) = Ind(5 E (Id B +n)SE) Theorem 12.1 reduces the problem of computing Ind(Id# +A') to that of computing the index of a Toeplitz operator to level N, for large N and then showing that the resultant formula has a reasonable limit as N —> oo. Theorem 11.1 only gives the formula for the index of a Toeplitz operator at level 0. In fact we can replace SE with a Szego projector at level 0 acting on a larger vector bundle. To accomplish this, and thereby compute the index of the approximating Toeplitz operators, we need to understand the symbols {sk • k > 0} in greater detail. Indeed, it suffices to do this

79

analysis in the model case of IR2n with its standard symplectic and complex structures. The transition to the contact manifold case follows exactly as in the construction of the bundle of isotropic algebras. 13. The structure of the higher eigenprojections If u) is the standard symplectic structure on R 2n , then the #-product on isotropic symbols is given by the oscillatory integral

a#b{w) = - ^ f[a(w + u)b{w + v)e2iu^v)dudv. If a and b belong to Schwartz class then this is an absolutely convergent integral. If a G 5^ o (K 2n ) and b £ 5^o'(lR2n) then a#b £ 5™o+m'(R2n) and it has the following asymptotic expansion: a h

*

~ £ ^ T T ^ -Dn-Dy fc>o K-

Di)ka{x,Z)b{y,r,)\x=yt^v.

(41)

It is well known that if either a or 6 is a polynomial then this sum is finite and gives an exact formula for a#b, see 3 . The differential operator appearing in this formula can be re-expressed in terms of the standard (1,0)- and (0, l)-vector fields on K 2n : Dx-D^-DyD^

1

-[{DX

- i£>£) • (Dy + iDn) - (Dx + iDtf • (Dy - iDn)\

= dz • dn, - dz • dw. Prom this formula it is apparent that if a and b are both holomorphic (or both anti-holomor-phic) polynomials, then only the k = 0 term in (41) is non-zero. This shows that the holomorphic and antiholomorphic polynomials are subalgebras in the #-product structure. In fact the #-product and ordinary pointwise product agree on these subalgebras. Let Zj = Xj — i£j and 2j = Xj + i£j. Let 2]t denote the set of multi-indices of length k and Wk and Wk denote linear spans of {za : a S Ik) and {za : a\ S Ik} respectively. Note that as vector spaces Wk ~ Wk#s0 and Wk ~ SoifWk-

(42)

Given the symplectic form LJ we define a linear functional on »S(K2n) by setting Tr(a) / aw". R2n

(43)

80

Exercise 25. If a € <S(R2n) show that qw{a) is a trace class operator and Tr(a) = Tr(qw(a)). Exercise 26. Show that if a, b £ 5(K 2n ) then prove directly that Tr(a#b) = Tr(b#a). This shows that Tr behaves like a trace on this algebra. Using the trace functional we define inner products on the vector spaces W k:
J=I

The projection on the ground state of the harmonic oscillator has symbol So = 2ne~h. It satisfies the symbolic identities: /i#so = so#h = ns0,

so#so = so,so'= so.

The symbol of the projection onto the eigenspace with eigenvalue n -\- 2k is Sfe, and it satisfies ft#Sfc = (n + 2A;)sfc,

sfc#sfc = sfc = s£.

These identities are easily proved using the quantization of the symbols. Let Cj=qw{-jZj),

j =

l,...,n

denote the annihilation operators defined in (31), and vo the ground state for H n = qw(h). As noted above the eigenspace of Hn, with eigenvalue n + 2k, is spanned by: {C*av0 : \a\ = *;}. Moreover, {C*avQ,C*0vo)

= ca5aP.

Exercise 27. Prove these formulae using integration by parts and the commutator relations.

81 Using these facts we obtain the following formula for Sfc. Sk=

1 ^

c

•

As the eigenspaces of H n are orthogonal we have Sk#si = 0 if k ^ 0. If we let s W=£ S f c t hen S W# S W= S W.

Thus s(N) is the symbol of the projection to level N. To complete our discussion we define the vector valued symbols Qk = {—7= -a elk)If / = p#so £ W^c#so then % # / is so times the vector of coefficients of p with respect to the orthonormal basis {2Q[cQ]~2 : a £ Ik)- That is qk defines an isomorphism from Wfc#s0 C <S(K2n) to CI / Q L These vector valued symbols satisfy the identities: Qi#Qk =6lkIdWl®s0, q*k#Qk = Sfc.

(44)

If a G <S(K 2n ) t h e n

fc,/=o Qei fc /3eii

CQC

^

= E E aa(i-FL={za#so#zf>)

(44)

where

aQ/3 = T r [ - ^ = ( s o # 2 a # a # z 0 # s o ) l .

(46)

Let

WW = 0W fc #5 O andgW = 0g f c , fc=0 fc=0

from the remarks above it is clear that we can interpret the matrix a^N^ = aap as an element of hom(W^JV'). Using the relations in (44) we see that gW#sW#a#SW#gW*

= a<"> ® s0.

(47)

82

Symbols a G <S(IR2n;hom(V)) for a complex vector space, V are treated analogously. The matrices { a ^ } are then homomorphisms of V <8> W ^ . These considerations are easily transferred to the context of a contact manifold X with a choice of adapted almost complex structure. Indeed one can simply regard this discussion as taking place in a single fiber of the cotangent space. The data then varies smoothly as the base point varies. A few additional remarks should clarify this construction. The vector spaces Wk are canonically defined by the choice of almost complex structure and therefore define global vector bundles over X. We denote these vector bundles by Wfc. Complex multiples of the globally denned symbol so then define a trivial line bundle; the bundles Wk <8> so are also globally defined and isomorphic to Wfc. The symbols {%} are globally denned and take values in hom(C, Wfcso)- When quantized they define operators from C°°(X) to C°°(X; Wk ® so)- Formula (47) implies that s<w>#(l + a')#s (Ar) can be interpreted as the symbol of a Toeplitz operator at level zero acting on sections of the vector bundle with fiber W ^ = (B%=0Wk- In order to treat an operator acting on sections of a vector bundle E —> X we replace {Wfc ® so} by {E Wfc ® so} and {qk} by {Id£ ®qk}For a complex vector bundle F let SE®F denote a level 0 Szego projector acting on C°°(X; E®F). Quantize the symbols {Id^ ®qk} in the Heisenberg calculus to obtain operators Qfc : C°°(X; E) -» C°°(X; E ® Wfe). From the relations in (44) it follows that a

o(QkQT)

— dki(ro(SE®Wk®sa),

(48)

As these are principal symbol calculations they remain valid if we replace Qfcby Qi = SE9W1Q'ISI,E.

(49)

From these symbolic calculations it follows easily that QiN) = QlUQi is a Fredholm map from range 5^ —> range SE&W(.N) . The operator Q^S^ildE+A^S^Q^* is a level 0 Toeplitz operator acting on C°°(X; E ® W ^ ) . The computations above imply that )S{BN)QW*) = a%(SE0WiN)aWsmwW)

(50)

83

where a^ e C°°(X;Hom(.E W(JV)) is defined (fiberwise) in (45) and (46). Prom the formulae (7) and (6), Theorem 12.1 and (50) it follows that, for large N, we have the identities Ind(IdB +A') = Ind(5^V)(IdE +A')S{EN)) = lnd(Q^S(EN\ldE l

1

= Ind(S j50y^(N) a -

+A')S^N)Q^*)

(51)

SElgW(N)).

The operator on the last line is a level 0 Toeplitz operator, its index can be computed using (40). To carry through this computation and obtain a cohomological formula for the index requires a diuscussion of the differential geometry of the bundle of isotropic algebras and especially of the subbundles {Wfc#so}- As a careful discussion would require many additional pages we limit ourselves to a few general remarks. A complete discussion can be found in *. One can find a connection V on the bundle of isotropic algebras with many useful properties. The connection is a derivation with respect to the #-product structure: V(a#6) = (Va)#6 + a#(V6). The symbol of the harmonic oscillator and the entire functional calculus it generates are flat sections with respect to this connection. In particular Vsfc = 0 for A; = 0 , 1 , . . . . Finally the connection carries sections of the subbundles {W/c#so} to sections of these subbundles. Taken together these properties allow a very explicit computation of the contribution of the symbol aft (A1) to the index formula. For each N the Chern character appearing in (40) takes the form

Ch(EaiN)) = Tr (exp [ ^ o ( w ) ] ) •

(52)

The curvature forms on the bundles Ea(N) are defined by restricting the curvature form from an infinite dimensional bundle over X x S1 defined by aft (Id + A'). The trace operation defined in (43) transplants to define a fibrewise trace on sections of S(H+). Because a' = ao(A') is rapidly decreasing on the fibers of H+ it is not difficult to show that the expression in (52) has a limit as N tends to infinity. In the limit, the matrix trace in (52) is replaced by this algebra trace. For example the principal term is of the form Cn TV[(Id B +a

/

)- 1 #Va'] 2n+1 .

84

Here (Id^+a')" 1 is the inverse with respect to the ^-product. This is a straightforward computation as Va' is a (homB)-valued, Schwartz class 1-form on H+. If we let Ch(a') denote the limiting Chern character then Ind(Id +A') = (Ch(a') A TdCT1-0*), [X]). Two sorts of difficulties arise in the analysis of general elliptic elements in the Heisenberg calculus: The first problem one encounters is with symbols that do not vanish to high order along the set {cr0 = 0}. In this case the algebra trace used above to define Ch(a') no longer converges absolutely. Instead one uses a ^-function regularization to extend the trace to these symbols. Unfortunately this leads to additional terms in the index formula which are essentially Wodzicki residue traces. The final difficulty is with symbols that do not reduce to the identity on {a0 = 0}. These are reduced to the previous case using a K-theoretic argument. The index formula for these last two classes of operators is a less explicit than for the case treated above. The interested reader is referred to the monograph 1. Philosophically, the proof in the general case shows that, insofar as index computations are concerned, the Heisenberg algebra (stably)retracts onto the subalgebra of Toeplitz operators. Using this principle, the indices of many operators can be computed quite explicitly. We close this section with such an example. Let 8 denote a contact form and T the associated Reeb vector field. Let J denote a choice of adapted almost complex structure on Ti and L a self adjoint differential operator with principal symbol hj. If {X\,..., Xn, Y\,..., Yn] is a symplecto-normal basis with respect to J and d6 then locally

L = -£[X? + Y?] + lo.t.. Let S denote a Szego projector compatible with J. For a positive integer N let a be a smooth L/(JV)-vamed function denned on X. Define a differential operator acting on C-^-valued functions by Ca = (L + inT) ® IdCiv +ia ® T. It is not difficult to show that Ca is a Fredholm operator. Using symbolic computations in the Heisenberg algebra one can show that Ind(£ a ) = Ind(5 ® UCN MaS® Id C "), where the operator on the right is a Toeplitz operator acting on range(5' ® IdcAf). As a corollary of this formula we obtain:

85

Corollary 13.1. If (X,Ti) is a compact contact manifold then there exist second order, sub-elliptic differential operators with non-zero index. Remark 13.1. This result should be contrasted with the classical result that an elliptic differential operator on an odd dimensional manifold always has index zero. 14. Grauert tubes and the Atiyah-Singer index theorem Boutet de Monvel's proof of the index theorem for Toeplitz operators uses essentially the full Atiyah-Singer index theorem. However in 23 he outlines an method for reversing the direction of the argument, explaining how one could prove the Atiyah-Singer theorem from the index theorem for Toeplitz operators. In x we give a new proof of the Toeplitz index formula which reduces the computation of the Toeplitz index to the computation of the index of a twisted, Spin-C Dirac operator. For this special case, the AtiyahSinger theorem can be proved by relatively elementary means, see 24. In this section we give an overview of this proof of the Atiyah-Singer theorem. Recall that in section 1 we defined the notion of a Grauert tube. Let Y denote a real analytic manifold and 7r: T*Y —> Y the cotangent bundle. For the purposes of this discussion we think of the Grauert tubes as a nested family of neighborhoods, {X£ : 0 < e < e0} of the zero section in T*Y with an adapted complex structure as described in section 1. In the adapted complex structure the zero section of T*Y and the fibers of n are totally real submanifolds. The Grauert tubes are Stein manifolds. Let Qn(Xe) denote the holomorphic (n,0)-forms on the Grauert tube. If w e nn(Xe)nC°°(X'c;An-0) then its restriction to a fiber of n is a complex n-form which can be integrated over the fiber, thus defining a map:

Pe(u)(y) =

J w. •K-i(y)C\Xe

This map is a Fourier integral operator with complex phase. In 25 it is shown that there exists an 0 < c\ < CQ such that Pe : fin(A-£)nC°°(A7;An'0) —• C°°{Y) is an isomorphism if e < ei. By considering holomorphic (n, 0)-forms with polynomial growth at bX£ one can extend Pe as isomorphism onto C~°°(Y).

86

Let Q,n(bXe) denote the boundary values of holomorphic (n, 0)-forms. Forms smooth up to the boundary have smooth boundary values whereas forms with polynomial growth have distributional boundary values. For 77 G D.n(bXe), let u)v denote its extension to Xc as a holomorphic (n, 0)form. Using, for example, the Henkin-Ramirez kernel one can obtain an operator: Gt : fl(bXt) —» C-°°(Y) which satisfies

GM = Pe(wv). From the result above for Pe it follows that, for e < t\, this map is an isomorphism and its restriction to smooth forms is an isomorphism onto C°°(Y). A simpler proof of this result has recently be found by Raul Tataru. 26 These operators are again FIOs with complex phase and satisfy very precise mapping properties with respect to L2-Sobolev space. If E —» Y is a complex vector bundle then Ee = IT* (E) |>e is also a smooth complex vector bundle. Since Xe is a Stein manifold it follows from the Oka-Grauert Principle, see 27, that there is a unique holomorphic bundle £€ —> Xe equivalent to Et. Since we can obtain the holomorphic structure by restricting £ci to X€ for an e < e' < eo it is clear that the holomorphic structure is smooth up to bXe. Let fln(bXe; £e) denote the boundary values of holomorphic (n,0)-forms with values in £e. The method used to prove that Ge is an isomorphism for e < e\ applies mutatis mutandis to show that the analogous map Gt,E-Mn(bXe;Se)-^

C-°°(Y;E)

is also an isomorphism for e < ei. As before the restriction of Gt<E to smooth (n, 0)-forms gives an isomorphism onto C°°(Y\ E). For such a bundle we let SCIE denote the classical Szego projector onto boundary values of holomorphic sections. Let F —> Y be a second complex vector bundle and Tt the holomorphic representatives of the restrictions of n*(F) and SC:F the classical Szego projector onto boundary values of holomorphic sections of J-c. Finally let A be a classical elliptic pseudodifferential operator: A:COO(Y;E)-*COO{Y;F). For purposes of index computations there is no loss in generality in assuming that A is of order zero, let a0 eC°°(T*Y\{0};hom(ir*(E),iT*(Y))) denote

87 its principal symbol. We define a Toeplitz operator: T£,ao = S€,FMaoSe,E

: nn(bXe;£e)

—» fi"(6*£; JF£).

This is slightly different from what we considered before as we now allow the operator to act between sections of different vector bundles. On the other hand, the Szego projectors are now defined by integrable, almost complex structures. Using results in 2 8 one can show that for e < ei A€ = GetFTeiaoGl

E

is a pseudodifferential operator with the same principal symbol as A. As GCIE and Ge,F are isomorphisms we obtain:

Theorem 14.1. If E —> Y and F —> Y are complex vector bundles and A s ^ ° N ( y ; E, F) is an elliptic operator with principal symbol ao then for sufficiently small e we have lnd(A) = Ind(T tlO0 ). As before Ind(T 6O0 ) can also be identified with a relative index Ind(T£,O0) = R-Ind(S £ ,.E,ao ll5 e,Fao)' The theorem therefore has the satisfying philosophical consequence of identifying the indices of all elliptic operators with relative indices of generalized Szego projectors. A cohomological formula for Ind(T £]ao ) leads immediately to a formula for Ind(A). For the case E = F such a formula is given in (40). The general case has been considered by Leichtnam, Nest and Tsygan. 2 9 .

15. The contact degree and index of FIOs In this section (X, 7i) denotes a compact, contact manifold. Recall that a diffeomorphism T/>, of X is a contact map if "4>*HX = Hijt^x). We call an orientation preserving contact diffeomorphism which preserves denote the co-orientation of H a contact transformation. Let M(X,H) isotopy classes of contact transformations; this is the contact mapping class group. As a final application of the ideas presented above we define a homomorphism c-deg : M (X, H) —> Z and give a formula for c-deg which is analogous to (40). If X = S*Y for a compact manifold Y then, using a construction similar to that used in the previous section, we can relate

88

this integer to the index of a certain class of Predholm Fourier integral operators. Most of these results can be found in 19. Choose an adapted almost complex structure J, for the fibers of H. Let S denote a generalized Szego projector with principal symbol defined by the field of vacuum states for the corresponding field of harmonic oscillators. If ip is a contact transformation then is also a generalized Szego projector, though its principal symbol is in general different from that of S. We define the contact degree as follows c-deg(ip) = R-Ind(5,5v,). Using the stability properties of the Predholm index described in Lemma 1.1, it follows immediately that c-deg(ip) depends only on the equivalence class of ip in M(X, Ti). Using the co-cycle relation (36) one can prove that c-deg(f/>) does not depend on the choice of S and that if ip and 4> are two contact transformations then c-deg('0 o (j>) = c-deg(ip) + c-deg(). In other words c-deg : M(X,H) —> Z is a homomorphism. Exercise 28. Prove this statement. The next order of business is to find a formula for c-deg(V'). For an arbitrary pair S, S" of generalized Szego projectors, the relative index is a very delicate invariant and the value of R-Ind(5, S") is difficult to compute. However in the special case that S' = S^ the relative index is much more robust and is essentially a topological invariant. We use a construction similar to that used to relate the index of a Toeplitz operator to the index of a Dirac operator on a mapping torus. Define the mapping torus Z^Xx[0,l]/(i,0)~(^),l). Since ip is a contact transformation the bundle H lifts to define a codimension 2 subbundle Hip C TZ^. This bundle has a conformal symplectic structure and therefore we can define an almost complex structure J on H^. If t denotes the parameter and T a vector field tangent to the fibers of Z^, —> Sl transverse to H^ © M.dt then we can extend the almost complex structure defined on H.^ to an almost complex structure denned on TZ^ by letting JT = dt.

89 The almost complex structure on TZ$ defines a canonical Spin-C structure on Z^. Let 9,/, denote the Spin-C Dirac operator defined by a choice of hermitian structure. We think of 9,/, as acting from even to odd spinors. In 19 the following result is proved Theorem 15.1. If ip is a contact transformation of(X,7i) then c-degip =Ind(cV).

(53)

To prove this result we introduce a 'resolution' of the range of S which is an acyclic differential complex very similar to the db-complex. Recall however that the almost complex structure J, on % is not required to be integrable. This complex defines a self adjoint, Fredholm Heisenberg pseudodifferential operator Do- The same construction applied to S^ leads to the operator D\ = (ip-1)*Doip* which is therefore isospectral to Do. These operators can be put into a continuous family Dt. Since the ends are isospectral, the spectral flow of the family is well defined; denote it by sfQDt]). The first step is to show that R-Ind(5,5^) = sf([A])The next problem is to deform the family [Dt] to a family of Spin-C Dirac operators [9t] through Fredholm operators. Technically this is the most challenging step. This is because the operators in the family [Dt] while Fredholm are not even Heisenberg elliptic, whereas Dirac operators are classical elliptic differential operators. To control this very singular perturbation problem we introduction an extension of the Heisenberg calculus which includes both the classical calculus and the Heisenberg calculus as subalgebras. Once this is accomplished, it follows from the stability of the Fredholm index that sf([Z?t]) = sf([9t]). It is then a fairly standard result that sf([9t]) = Ind(8tf). For the case that the dimX = 3 one can use the Atiyah-Singer index theorem and Hirzebruch signature formula to obtain a simple explicit formula for c-deg(V'). In this case

90 Here 11 denotes a trivial line bundle. Using formula (18) in that

19

we deduce

Ind(o>) =4 P l (T^)[^]

\ (54) sig Z 12 ( ^)' where p\ is the first Pontryagin class and sig(i^) is the signature of the 4-manifold Z^. Let P : Z^ —-> S1 be the canonical projection and set =

Zo =

P-^O.TT])

and Zi = P " 1 ^ * - ] ) .

The manifolds with boundary ZQ and Z\ are diffeomorphic and, as oriented manifolds,

z,p ~ z 0 [_J —ZW boundary

that is we reverse the orientation of Z\. Let sig(Zo) = sig(Zi) denote the index of the non-degenerate pairing defined by the cup product on the image of H2(Zi,bZi) in H2(Zi). The Novikov addition formula states that sig(^) = sig(Z o )-sig(Zi) = O, 30 31

see , . This proves the following result Theorem 15.2. If X is a 3-dimensional contact manifold then the contact degree is the zero homomorphism. Remark 15.1. I would like to thank Rafe Mazzeo for pointing out the connection, for this case, between the contact degree and the signature, and Dennis Sullivan for telling me about the Novikov addition formula. Remark 15.2. In 9 the relative index is defined for a pair of embeddable CR-structures denned on a 3-dimensional contact manifold. There it is shown that the index descends to define an invariant on the "Teichmuller" space of CR-structures. This is the space of equivalence classes of complex structures on the contact field where two structures are equivalent if one is the push forward of the other by a contact map isotopic to the identity. The contact mapping class group acts on this Teichmuller space. Theorem 15.2 implies that the relative index actually descends to define an invariant on the moduli space itself. As a second application of Theorem 15.1 we show that the contact degree is related to the index of a certain class of Fourier integral operators. For

91 this application we need to restrict X = S*Y, for Y a compact manifold. If V : S*Y —> S*Y is a contact transformation then it defines a conic Lagrangian submanifold of T*Y\ {0} x T*Y\ {0}. To see this identify S*Y with a unit cosphere bundle and extend the map ip : S*Y —» S*Y to be homogeneous of degree 1. Denote this extension by \I>. It is a canonical transformation of the cotangent bundle, its graph, A,/, is therefore a conic, Lagrangian submanifold. Such a submanifold defines a class of Fourier integral operators, see 32 or 33 . Roughly speaking an operator belongs to this class if the wave front set of its Schwartz kernel is contained in A,/,. For example, pseudodifferential operators are among the Fourier integral operators defined by the identity map. Let {Xe} denote the Grauert tubes introduced in the previous section. As noted in section 1, S*Y is contact equivalent to bXc for any e < eoTherefore a contact transformation of S*Y can be thought of as defining a contact transformation of bXe. Let ip : bXe —> bXe be such a map. We define an operator on C°°(Y) by setting F^u = Gcip*G*{u). This is a Fredholm Fourier integral operator associated to the Lagrangian submanifold A,/,. In 19 the index of this operator is computed. Theorem 15.3. If ip : S*Y —> S*Y is a contact transformation then Ind(i^) = c-deg(ip). As a corollary of this result and Theorem 15.2 we have Corollary 15.1. If dim Y = 2 then Ind(F,p) = 0 for every contact transformation. The detailed proofs of these results are in

19

and 1.

Remark 15.3. Thus far no example of a contact manifold X and contact transformation ip '• X —> X such that c-deg(V') ^ 0 has been found. Acknowledgments I would like to thank the organizers of the summer school at CIRM in Marseilles, B. Coupet, J. Merker and A. Shukov as well as the the organizers of the meeting in Woods Hole, Nils Tongring and Dennis Sullivan for giving me the opportunity to present these lectures. I would also like to thank Andy Solow for making the Math meetings in Woods Hole a possibility. I am most grateful to Hyunsuk Kang

92 for providing a careful transcript of the original lectures. Finally I would like to thank my collaborators, Richard Melrose and Gerardo Mendoza for allowing me to write this expository account of our joint work. The research described in these notes was partially supported by the National Science Foundation. References 1. Charles L. Epstein and Richard Melrose. The Heisenberg algebra, index theory and homology. preprint, 2003. 2. Fritz Noether. Uber eine Klasse singularer Integralgleichungen. Math. Ann., 82:42-63, 1921. 3. L. Hormander. The Analysis of Linear Partial Differential Operators, volume 3. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985. 4. Israel Gohberg, Seymour Goldberg, and Marinus A. Kaashoek. Classes of Linear Operators, vol. I and II. Birkhauser, Basel-Boston-Berlin, 1990. 5. P.D. Lax. On factorization of matrix valued functions. Comm. Pure Appl. Math., 29:683-688, 1976. 6. V.I. Arnold. Mathematical Methods in Classical Mechanics, volume 60 of GTM. Springer Verlag, Berlin and New York, 1978. 7. Laszlo Lempert and Robert Szoke. Global solutions of the homogeneous complex Monge-Ampre equation and complex structures on the tangent bundle of riemannian manifolds. Math. Ann., 290:689-712, 1991. 8. V.W. Guillemin and M. Stenzel. Grauert tubes and the homogeneous MongeAmpere equation. J. Differential Geom., 1991. 9. Charles L. Epstein. A relative index on the space of embeddable CRstructures, I, II. Annals of Math., 147:1-59, 61-91, 1998. 10. L. Boutet de Monvel. Integration des equations Cauchy-Riemann induites formelles. Seminar Goulaouic-Lions-Schwartz, pages IX.1-IX.13, 1974-75. 11. J.J. Kohn. The range of the tangential Cauchy-Riemann operator. Duke J., 53:525-545, 1986. 12. F. Reese Harvey and H. Blaine Lawson. On the boundaries of complex analytic varieties. Ann. of Math. (2), 106:223-290, 1977. 13. J.J. Kohn and L. Nirenberg. On the algebra of pseudo-differential operators. Comm. Pure Appl. Math., 18:269-305, 1965. 14. J.J. Kohn and L. Nirenberg. Non-coercive boundary value problems. Comm. Pure Appl. Math., 18:443-492, 1965. 15. M.E. Taylor. Noncommutative microlocal analysis, part I, volume 313 of Mem. Amer. Math. Soc. AMS, 1984. 16. R. Beals and P. Greiner. Calculus on Heisenberg Manifolds, volume 119 of Annals of Mathematics Studies. Princeton University Press, 1988. 17. M.E. Taylor. Noncommutative harmonic analysis, volume 22 of Mathematical Surveys and Monographs. AMS, Providence, R.I., 1986. 18. L. Boutet de Monvel and V. Guillemin. The spectral theory of Toeplitz operators, volume 99 of Ann. of Math. Studies. Princeton University Press, 1981.

93 19. Charles L. Epstein and Richard Melrose. Contact degree and the index of Fourier integral operators. Math. Res. Letters, 5:363-381, 1998. 20. Friedrich Hirzebruch. Topological methods in algebraic geometry, volume 131 of Grundlehren der mathematishen Wissensckaften. Springer Verlag, 1978. 21. H. Blaine Lawson Jr. and Marie-Louise Michelson. Spin Geometry, volume 38 of Princeton Mathematical Series. Princeton University Press, 1989. 22. S. S. Chern. Complex Manifolds Without Potential Theory. Van Nostrand Reinhold Co., New York, 1967. 23. L. Boutet de Monvel. On the index of Toeplitz operators of several complex variables. Invent. Math., 50:249-272, 1979. 24. N. Berline, E. Getzler, and M. Vergne. Heat Kernels and Dirac Operators, volume 298 of Grundlehren der mathematischen Wissenschaften. SpringerVerlag, Berlin Heidelberg New York, 1992. 25. Charles L. Epstein and Richard Melrose. Shrinking tubes and the d-Neumann problem, preprint, 1990. 26. Raul Tataru. Adiabatic limit and SzegS projections. MIT PhD Thesis, 2003. 27. J. Leiterer. Holomorphic vector bundles and the Oka-Grauert principle. In S.G. Gindikin and G.M. Khenkin, editors, Several Complex Variables, IV, volume 10 of Encyclopedia of Mathematical Sciences, chapter 2. Springer Verlag, 1990. 28. V.W. Guillemin. Toeplitz operators in n dimensions. Int. Eq. Op. Theory, 7:145-205, 1984. 29. Eric Leichtnam, Ryszard Nest, and Boris Tsygan. Local formula for the index of a fourier integral operator, to appear JDG, pages 1-25, 2002. 30. M.F. Atiyah, V.K. Patodi, and I.M. Singer. Spectral asymmetry and Riemannian geometry, I. Math. Proc. Camb. Phil. Soc, 77:43-69, 1975. 31. M.F. Atiyah and I.M. Singer. The index of elliptic operators, III. Ann. of Math., 87:546-604, 1968. 32. L. Hormander. Fourier integral operators, I. Ada Math., 127:79-183, 1971. 33. J. Briining and V.W. Guillemin (Editors). Fourier integral operators. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1994.

94

BIOLOGIC II

LOUIS H. KAUFFMAN Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 South Morgan St., Chicago IL 60607-7045, U.S.A. E-mail: [email protected] In this paper we explore the boundary between biology, topology, algebra and the study of formal systems (logic).

1. Introduction This paper concentrates on relationships of formal systems with biology. In particular, this is a study of different forms and formalisms for replication. It is a sequel to Kauffman10 and contains much of the material in that paper plus new material about projectors in the Temperley Lieb algebra. In living systems there is an essential circularity that is the living structure. Living systems produce themselves from themselves and the materials and energy of the environment. There is a strong contrast in how we avoid circularity in mathematics and how nature revels in biological circularity. One meeting point of biology and mathematics is knot theory and topology. This is no accident, since topology is indeed a controlled study of cycles and circularities in primarily geometrical systems. In this paper we will discuss DNA replication, logic and biology, the relationship of symbol and object, the emergence of form. It is in the replication of DNA that the polarity (yes/no, on/off, true/false) of logic and the continuity of topology meet. Here polarities are literally fleshed out into the forms of life. We shall pay attention to the different contexts for the logical, from the mathematical to the biological to the quantum logical. In each case there is a shift in the role of certain key concepts. In particular, we follow the notion of copying through these contexts and with it gain new insight into the role of replication in biology, in formal systems and in the quantum

95 level (where it does not exist!). In the end we arrive at a summary formalism, a chapter in boundary mathematics (mathematics using directly the concept and notation of containers and delimiters of forms - compare Bricken & Gullichsen3 and Spencer-Brown11) where there are not only containers <>, but also extainers >< - entities open to interaction and distinguishing the space that they are not. In this formalism we find a key for the articulation of diverse relationships. The boundary algebra of containers and extainers is to biologic what boolean algebra is to classical logic. Let C = < > and E =>< then EE = > < > < = > C < and CC = < > < > = < E > Thus an extainer produces a container when it interacts with itself, and a container produces an extainer when it interacts with itself. The formalism of containers and extainers is a chapter in the foundations of a symbolic language for shape and interaction. With it, we can express the form of DNA replication succinctly as follows: Let the DNA itself be represented as a container DNA = < > .

(1)

We regard the two brackets of the container as representatives for the two matched DNA strands. We let the extainer E =>< represent the cellular environment with its supply of available base pairs (here symbolized by the individual left and right brackets). Then when the DNA strands separate, they encounter the matching bases from the environment and become two DNA's. D N A = < > — > < £ • > — > < > < > = DNA DNA.

(2)

Life itself is about systems that search and learn and become. Perhaps a little symbol like E —X with the property that EE =><>< produces containers <> and retains its own integrity in conjunction with the autonomy of <> (the DNA) could be a step toward bringing formalism to life. These concepts of concatenation of extainers and containers lead, in Section 6, to a new approach to the structure of and generalizations of the Temperley Lieb algebra. In this Section we discuss how projectors in the Temperley Lieb algebra can be regarded as topological/algebraic models of self-replication, and we take this point of view to characterize multiplicative elements P of the Temperley Lieb algebra such that PP = P. What emerges here is a topological view of self-replication that is different in principle from the blueprint-driven self-replications of logic and from the environmentally driven self-replication described above as an abstraction

96

of DNA action. This topological replication is a direct descendant of the fact that you can get two sticks from one stick by breaking it in the middle. Here we obtain more complex forms by allowing topological deformation of the stick before it is broken, but to see how this works the reader should go to Section 6.1. 2. Replication of DNA We start this essay with the question: During the replication of DNA, how do the daughter DNA duplexes avoid entanglement? In the words of John Hearst6, we are in search of the mechanism for the "immaculate segregation." This question is inevitably involved with the topology of the DNA, for the strands of the DNA are interwound with one full turn for every ten base pairs. With the strands so interlinked it would seem impossible for the daughter strands to separate from their parents. A key to this problem certainly lies in the existence of the topoisomerase enzymes that can change the linking number between the DNA strands and also can change the linking number between two DNA duplexes. It is however, a difficult matter at best to find in a tangled skein of rope the just right crossing changes that will unknot or unlink it. The topoisomerase enzymes do just this, changing crossings by grabbing a strand, breaking it and then rejoining it after the other strand has slipped through the break. Random strand switching is an unlikely mechanism, and one is led to posit some intrinsic geometry that can promote the process. In Kauffman6 there is made a specific suggestion about this intrinsic geometry. It is suggested that in vivo the DNA polymerase enzyme that promotes replication (by creating loops of single stranded DNA by opening the double stranded DNA) has sufficient rigidity not to allow the new loops to swivel and become entangled. In other words, it is posited that the replication loops remain simple in their topology so that the topoisomerase can act to promote the formation of the replication loops, and these loops once formed do not hinder the separation of the newly born duplexes. The model has been to some degree confirmed Zechiedrich, et al.12 The situation would now appear to be that in the first stages of the formation of the replication loops Topo I acts favorably to allow their formation and amalgamation. Then Topo II has a smaller job of finishing the separation of the newly formed duplexes. In 1 we illustrate the schema of this process. In this Figure we indicate the action of the Topo I by showing a strand being switched in between two replication loops. The action of Topo II is only

97

stated but not shown. In that action, newly created but entangled DNA strands would be disentangled. Our hypothesis is that this second action is essentially minimized by the rigidity of the ends of the replication loops in vivo.

I replication loops

)'/

\ r

i 7

\

top

°'* r i

ytopo n \ > o o o o c

^'"'•Wi'-W vi V \ x \ *>

\r

'

DNA

to on )l Vs. P *"Nr-v^>v-Nr-x-^r It \\ — TOOOwCX DNA >^^y Figure 1. DNA Replication

In the course of this research, we started thinking about the diagrammatic logic of DNA replication and more generally about the relationship between DNA replication, logic and basic issues in the foundations of mathematics and modelling. The purpose of this paper is to explain some of these issues, raise questions and place these questions in the most general context that we can muster at this time. The purpose of this paper is there-

98 fore foundational. It will not in its present form affect issues in practical biology, but we hope that it will enable us and the reader to ask fruitful questions and perhaps bring the art of modelling in mathematics and biology forward. To this end we have called the subject matter of this paper "biologic" with the intent that this might suggest a quest for the logic of biological systems or a quest for a "biological logic" or even the question of the relationship between what we call "logic" and our own biology. We have been trained to think of physics as the foundation of biology, but it is possible to realize that indeed biology can also be regarded as a foundation for thought, language, mathematics and even physics. In order to bring this statement over to physics one has to learn to admit that physical measurements are performed by biological organisms either directly or indirectly and that it is through our biological structure that we come to know the world. This foundational view will be elaborated as we proceed in this paper.

3. Logic, Copies and DNA Replication In logic it is implicit at the syntactical level that copies of signs are freely available. In abstract logic there is no issue about materials available for the production of copies of a sign, nor is there necessarily a formalization of how a sign is to be copied. In the practical realm there are limitations to resources. A mathematician may need to replenish his supply of paper. A computer has a limitation on its memory store. In biology, there are no signs, but there are entities that we take as signs in our description of the workings of the biological information process. In this category the bases that line the backbone of the DNA are signs whose significance lies in their relative placement in the DNA. The DNA itself could be viewed as a text that one would like to copy. If this were a simple formal system it would be taken for granted that copies of any given text can be made. Therefore it is worthwhile making a comparison of the methods of copying or reproduction that occur in logic and in biology. In logic there is a level beyond the simple copying of symbols that contains a non-trivial description of self-replication. The schema is as follows: There is a universal building machine B that can accept a text or description x (the program) and build what the text describes. We let lowercase

99

x denote the description and uppercase X denote that which is described. Thus B with x will build X. In fact, for bookkeeping purposes we also produce an extra copy of the text x. This is appended to the production X as X, x. Thus B, when supplied with a description x, produces that which x describes, with a copy of its description attached. Schematically we have the process shown below. B, x —> B, x; X, x

(3)

Self-replication is an immediate consequence of this concept of a universal building machine. Let b denote the text or program for the universal building machine. Apply B to its own description. B,b—>B,b;B,b

(4)

The universal building machine reproduces itself. Each copy is a universal building machine with its own description appended. Each copy will proceed to reproduce itself in an unending tree of duplications. In practice this duplication will continue until all available resources are used up, or until someone removes the programs or energy sources from the proliferating machines. It is not necessary to go all the way to a universal building machine to establish replication in a formal system or a cellular automaton (see the epilogue to this paper for examples). On the other hand, all these logical devices for replication are based on the hardware/software or Object/Symbol distinction. It is worth looking at the abstract form of DNA replication. DNA consists in two strands of base-pairs wound helically around a phosphate backbone. It is customary to call one of these strands the "Watson" strand and the other the "Crick" strand. Abstractly we can write DNA =< W\C >

(5)

to symbolize the binding of the two strands into the single DNA duplex. Replication occurs via the separation of the two strands via polymerase enzyme. This separation occurs locally and propagates. Local sectors of separation can amalgamate into larger pieces of separation as well. Once the strands are separated, the environment of the cell can provide each with complementary bases to form the base pairs of new duplex DNA's. Each strand, separated in vivo, finds its complement being built naturally in the environment. This picture ignores the well-known topological difficulties

100

present to the actual separation of the daughter strands. The base pairs are AT (Adenine and Thymine) and GC (Guanine and Cytosine). Thus if < W | = < .. .TTAGAATAGGTACGCG... |

(6)

\C > = \...AATCTTATCCATGCGC...

(7)

Then >.

Symbolically we can oversimplify the whole process as < W\ + E —>< W\C >= DNA E+\C>—><W\C>=DNA < W\C>—-*< W\ + E + \C>=< W\C>< W\C>

(8) (9) (10)

Either half of the DNA can, with the help of the environment, become a full DNA. We can let E —> \C >< W\ be a symbol for the process by which the environment supplies the complementary base pairs AG, TC to the Watson and Crick strands. In this oversimplification we have cartooned the environment as though it contained an already-waiting strand \C > to pair with < W\ and an already-waiting strand < W\ to pair with \C > . In fact it is the opened strands themselves that command the appearance of their mates. They conjure up their mates from the chemical soup of the environment. The environment E is an identity element in this algebra of cellular interaction. That is, E is always in the background and can be allowed to appear spontaneously in the cleft between Watson and Crick: < W\C >—>< W\\C >^< W\E\C> ((10) —4<W\\C><W\\C>—+<W\C><W\C> ' This is the formalism of DNA replication. Compare this method of replication with the movements of the universal building machine supplied with its own blueprint. Here Watson and Crick ( < W\ and \C > ) are each both the machine and the blueprint for the DNA. They are complementary blueprints, each containing the information to reconstitute the whole molecule. They are each machines in the context of the cellular environment, enabling the production of the DNA. This coincidence of machine and blueprint, hardware and software is an important difference between classical logical systems and the logical forms that arise in biology.

101 4. Lambda Algebra - Replication Revisited One can look at formal systems involving self-replication that do not make a distinction between Symbol and Object. In the case of formal systems this means that one is working entirely on the symbolic side, quite a different matter from the biology where there is no intrinsic symbolism, only our external descriptions of processes in such terms. An example at the symbolic level is provided by the lambda calculus of Church and Curry2 where functions are allowed to take themselves as arguments. This is accomplished by the following axiom. Axiom for a Lambda Algebra: Let A be an algebraic system with one binary operation denoted ab for elements a and b of A. Let F(x) be an algebraic expression over A with one variable x. Then there exists an element a of A such that F(x) = ax for all x in A. An algebra (not associative) that satisfies this axiom is a representation of the lambda calculus of Church and Curry. Let b be an element of A and define F(x) — b(xx). Then by the axiom we have a in A such that ax = b(xx) for any x in A. In particular (and this is where the "function" becomes its own argument) aa = b(aa).

(12)

Thus we have shown that for any b in A, there exists an element x in A such that x = bx. Every element of A has a "fixed point." This conclusion has two effects. It provides a fixed point for the function G(x) = bx and it creates the beginning of a recursion in the form aa = b{aa) = b(b(aa)) = b{b{b(aa))) = ...

(13)

The way we arrived at the fixed point aa was formally the same as the mechanism of the universal building machine. Consider that machine: B,x—>X,x

(14)

We have left out the repetition of the machine itself. You could look at this as a machine that uses itself up in the process of building X. Applying B to its own description b we have the self-replication B,b—>B,b.

(15)

102

The repetition of x in the form X, x on the right hand side of this definition of the builder property is comparable with ax = b(xx)

(16)

with its crucial repetition as well. In the fixed point theorem, the arrow is replaced by an equals sign! Repetition is the core of self-replication in classical logic. This use of repetition assumes the possibility of a copy at the syntactic level, in order to produce a copy at the symbolic level. There is, in this pivot on syntax, a deep relationship with other fundamental issues in logic. In particular this same form of repetition is in back of the Cantor diagonal argument showing that the set of subsets of a set has greater cardinality than the original set, and it is in back of the Godel Theorem on the incompleteness of sufficiently rich formal systems. The pattern is also in back of the production of paradoxes such as the Russell paradox of the set of all sets that are not members of themselves. There is not space here to go into all these relationships, but the Russell paradox will give a hint of the structure. Let "ab" be interpreted as "b is a member of a". Then RX = -<(XX) can be taken as the definition of a set R such that X is a member of R exactly when it is not the case that X is a member of X. Note the repetition of X in the definition RX = ->(XX). Substituting R for X we obtain RR = ->(RR), which says that R is a member of R exactly when it is not the case that R is a member of R. This is the Russell paradox. From the point of view of the lambda calculus, we have found a fixed point for negation. Where is the repetition in the DNA self-replication? The repetition and the replication are no longer separated. The repetition occurs not syntactically, but directly at the point of replication. Note the device of pairing or mirror imaging. A calls up the appearance of T and G calls up the appearance of C. < W\ calls up the appearance of \C > and \C > calls up the appearance of < W\. Each object O calls up the appearance of its dual or paired object O*. O calls up O* and O* calls up O. The object that replicates is implicitly a repetition in the form of a pairing of object and dual object. OO* replicates via O —» OO*

(17)

O* —+ OO*

(18)

103

whence OO* —> O O* —• OO* OO*.

(19)

The repetition is inherent in the replicand in the sense that the dual of a form is a repetition of that form.

5. Quantum Mechanics We now consider the quantum level. Here copying is not possible. We shall detail this in a subsection. For a quantum process to copy a state, one needs a unitary transformation to perform the job. One can show, as we explain in the last subsection of this section, that this cannot be done. There are indirect ways that seem to make a copy, involving a classical communication channel coupled with quantum operators (so called quantum teleportation13). The production of such a quantum state constitutes a reproduction of the original state, but in these cases the original state is lost, so teleportation looks more like transportation than copying. With this in mind it is fascinating to contemplate that DNA and other molecular configurations are actually modelled in principle as certain complex quantum states. At this stage we meet the boundary between classical and quantum mechanics where conventional wisdom finds it is most useful to regard the main level of molecular biology as classical. We shall quickly indicate the basic principles of quantum mechanics. The quantum information context encapsulates a concise model of quantum theory: The initial state of a quantum process is a vector \v > in a complex vector space H. Observation returns basis elements (3 of H with probability | \2/

(20)

where < v \w >= v*w with v* the conjugate transpose ofv. A physical process occurs in steps \v >—> U\v >= \Uv > where U is a unitary linear transformation. Note that since < Uv \Uw >=< v\w > when U is unitary, it follows that probability is preserved in the course of a quantum process. One of the details for any specific quantum problem is the nature of the unitary evolution. This is specified by knowing appropriate information about the classical physics that supports the phenomena. This information

104 is used to choose an appropriate Hamiltonian through which the unitary operator is constructed via a correspondence principle that replaces classical variables with appropriate quantum operators. (In the path integral approach one needs a Lagrangian to construct the action on which the path integral is based.) One needs to know certain aspects of classical physics to solve any given quantum problem. The classical world is known through our biology. In this sense biology is the foundation for physics. A key concept in the quantum information viewpoint is the notion of the superposition of states. If a quantum system has two distinct states \v > and \w >, then it has infinitely many states of the form a\v > +b\w > where a and b are complex numbers taken up to a common multiple. States are "really" in the projective space associated with H. There is only one superposition of a single state \v > with itself. Dirac5 introduced the "bra-(c)-ket" notation < A\B >= A*B for the inner product of complex vectors A,B £ H. He also separated the parts of the bracket into the bra < A | and the ket \B > . Thus < A\B >=< A\ \B> .

(21)

In this interpretation, the ket \B > is identified with the vector B G H, while the bra < A | is regarded as the element dual to A in the dual space H*. The dual element to A corresponds to the conjugate transpose A* of the vector A, and the inner product is expressed in conventional language by the matrix product A*B (which is a scalar since B is a column vector). Having separated the bra and the ket, Dirac can write the "ket-bra" \A > < B | = AB*. In conventional notation, the ket-bra is a matrix, not a scalar, and we have the following formula for the square of P = \A >< B | :

P2 = \A >< B \\A >< B | = A(B*A)B* = {B*A)AB* =< B \A > P. (22) Written entirely in Dirac notation we have

P2 = \A >< B\\A >< B\ = \A >< B\A> \A >P.

[(22)

'

The standard example is a ket-bra P = \A >< A\ where < A \A >= 1 so that P2 = P. Then P is a projection matrix, projecting to the subspace of H that is spanned by the vector \A >. In fact, for any vector \B > we have

P\B >= \A>= \A >< A \B >=< A \B > \A > .

(24)

105 If {|Ci >, |C2 >, •' • \Cn >} is an orthonormal basis for H, and P, = |Cj >< Ci|, then for any vector \A > we have |A >=< Cx \A > | d > + • • • + < Cn \A > \Cn > .

(25)

Hence < B |A >=< d\A >< B \d >+•••+< Cn\A >< B \Cn > = + ---+< Cn \A > =< B | [|d >< Ci | + • • • + |C n >< Cn |] |4 > =.

. .

[M)

We have written this sequence of equalities from < j B | A > t o < B | l | j 4 > to emphasize the role of the identity E£=1flfe = E ] J = 1 | C f c x C 7 f c | = l

(27)

so that one can write

< B \A >=< B 11 \A >=< B IEJUIC* X Ck \\A > = Eg=1.

l ; (27)

In the quantum context one may wish to consider the probability of starting in state \A > and ending in state \B > . The square of the probability for this event is equal to | < B \A > | 2 . This can be refined if we have more knowledge. If it is known that one can go from A to C* (i = 1, • • • , n) and from Ci to B and that the intermediate states \d > are a complete set of orthonormal alternatives then we can assume that < C, \d >= 1 for each i and that Ej|Cj >< d | = 1. This identity now corresponds to the fact that 1 is the sum of the probabilities of an arbitrary state being projected into one of these intermediate states. If there are intermediate states between the intermediate states this formulation can be continued until one is summing over all possible paths from A to B. This becomes the path integral expression for the amplitude .

5.1. Quantum Formalism and DNA

Replication

We wish to draw attention to the remarkable fact that this formulation of the expansion of intermediate quantum states has exactly the same pattern as our formal summary of DNA replication. Compare them. The form of

106 DNA replication is shown below. Here the environment of possible base pairs is represented by the ket-bra E = \C >< W |.

<W\O-^<W\\C>—*<W\E\C>

—•<w\\c><w\\c>—+ <w\c><w\c>

. .

{

'

v

;

Here is the form of intermediate state expansion.

—>-^ ^^Zk. We compare E = \C><W\

(31)

1 = Efc \Ck X Ck |.

(32)

and

That the unit 1 can be written as a sum over the intermediate states is an expression of how the environment (in the sense of the space of possibilities) impinges on the quantum amplitude, just as the expression of the environment as a soup of bases ready to be paired (a classical space of possibilities) serves as a description of the biological environment. The symbol E = \C >< W | indicated the availability of the bases from the environment to form the complementary pairs. The projection operators \Ci X Ci | are the possibilities for interlock of initial and final state through an intermediate possibility. In the quantum mechanics the special pairing is not of bases but of a state and a possible intermediate from a basis of states. It is through this common theme of pairing that the conceptual notation of the bras and kets lets us see a correspondence between such separate domains.

5.2. Quantum Copies are not Possible Finally, we note that in quantum mechanics it is not possible to copy a quantum state! This is called the no-cloning theorem of elementary quantum mechanics13. Here is the proof: Proof of the No Cloning Theorem: In order to have a quantum process make a copy of a quantum state we need a unitary mapping U : H®H —> H <8> H where H is a complex vector space such that there is a fixed state \X > S H with the property that U(\X>\A>)

= \A>\A>

(33)

107 for any state \A>e H. (\A> \B > denotes t h e tensor product \A> ®\B > .) Let

T{\A >) = U(\X > \A >) = \A > \A > .

(34)

Note that T is a linear function of \A > . Thus we have IT\O>= | 0 > | 0 > = |00>,

(35)

T | l > = |1 > | 1 > = |11 >,

(36)

T(Q|0 > +/3\l >) = (a|0 > +/J|l >)(a|0 > +/3|1 >).

(37)

But T(a|0> +/3|l >) = a|00 >+/?|11 > .

(38)

a|00 > +p\ll > = (a|0 > +/3|1 >)(a|0 > +/3|1 >) = a 2 |00 > +/? 2 |H > +a/?|01 > +/3a|10 >

,„„, ^ ;

Hence

From this it follows that a/? = 0. Since a and /? are arbitrary complex • numbers, this is a contradiction. The proof of the no-cloning theorem depends crucially on the linear superposition of quantum states and the linearity of quantum process. By the time we reach the molecular level and attain the possibility of copying DNA molecules we are copying in a quite different sense than the ideal quantum copy that does not exist. The DNA and its copy are each quantum states, but they are different quantum states! That we see the two DNA molecules as identical is a function of how we filter our observations of complex and entangled quantum states. Nevertheless, the identity of two DNA copies is certainly at a deeper level than the identity of the two letters "i" in the word identity. The latter is conventional and symbolic. The former is a matter of physics and biochemistry.

6. Mathematical Structure and Topology We now comment on the conceptual underpinning for the notations and logical constructions that we use in this paper. This line of thought will lead to topology and to the formalism for replication discussed in the last section.

108 Mathematics is built through distinctions, definitions, acts of language that bring forth logical worlds, arenas in which actions and patterns can take place. As far as we can determine at the present time, mathematics while capable of describing the quantum world, is in its very nature quite classical. Or perhaps we make it so. As far as mathematics is concerned, there is no ambiguity in the 1 + 1 hidden in 2. The mathematical box shows exactly what is potential to it when it is opened. There is nothing in the box except what is a consequence of its construction. With this in mind, let us look at some mathematical beginnings. Take the beginning of set theory. We start with the empty set <j> = { } and we build new sets by the operation of set formation that takes any collection and puts brackets around it: abed —> {a,b,c,d}

(40)

making a single entity {a, b, c, d} from the multiplicity of the "parts" that are so collected. The empty set herself is the result of "collecting nothing". The empty set is identical to the act of collecting. At this point of emergence the empty set is an action not a thing. Each subsequent set can be seen as an action of collection, a bringing forth of unity from multiplicity. One declares two sets to be the same if they have the same members. With this prestidigitation of language, the empty set becomes unique and a hierarchy of distinct sets arises as if from nothing. — { }— {{}}-^{{}>{{}}}—>••• (41) All representatives of the different mathematical cardinalities arise out of the void in the presence of these conventions for collection and identification. We would like to get underneath the formal surface. We would like to see what makes this formal hierarchy tick. Will there be an analogy to biology below this play of symbols? On the one hand it is clear to us that there is actually no way to go below a given mathematical construction. Anything that we call more fundamental will be another mathematical construct. Nevertheless, the exercise is useful, for it asks us to look closely at how this given formality is made. It asks us to take seriously the parts that are usually taken for granted. We take for granted that the particular form of container used to represent the empty set is irrelevant to the empty set itself. But how can this

109 be? In order to have a concept of emptiness, one needs to hold the contrast of that which is empty with "everything else". One may object that these images are not part of the formal content of set theory. But they are part of the formalism of set theory. Consider the representation of the empty set: { }. That representation consists in a bracketing that we take to indicate an empty space within the brackets, and an injunction to ignore the complex typographical domains outside the brackets. Focus on the brackets themselves. They come in two varieties: the left bracket, {, and the right bracket, }. The left bracket indicates a distinction of left and right with the emphasis on the right. The right bracket indicates a distinction between left and right with an emphasis on the left. A left and right bracket taken together become a container when each is in the domain indicated by the other. Thus in the bracket symbol { }

(42)

for the empty set, the left bracket, being to the left of the right bracket, is in the left domain that is marked by the right bracket, and the right bracket, being to the right of the left bracket is in the right domain that is marked by the left bracket. The doubly marked domain between them is their content space, the arena of the empty set. The delimiters of the container are each themselves iconic for the process of making a distinction. In the notation of curly brackets, { , this is particularly evident. The geometrical form of the curly bracket is a cusp singularity, the simplest form of bifurcation. The relationship of the left and right brackets is that of a form and its mirror image. If there is a given distinction such as left versus right, then the mirror image of that distinction is the one with the opposite emphasis. This is precisely the relationship between the left and right brackets. A form and its mirror image conjoin to make a container. The delimiters of the empty set could be written in the opposite order: }{. This is an extainer. The extainer indicates regions external to itself. In this case of symbols on a line, the extainer }{ indicates the entire line to the left and to the right of itself. The extainer is as natural as the container, but does not appear formally in set theory. To our knowledge, its first appearance is in the Dirac notation of "bras" and "kets" where Dirac takes an

110 inner product written in the form < B\A > and breaks it up into < B | and \A > and then makes projection operators by recombining in the opposite order as \A >< B |. See the earlier discussion of quantum mechanics in this paper. Each left or right bracket in itself makes a distinction. The two brackets are distinct from one another by mirror imaging, which we take to be a notational reflection of a fundamental process (of distinction) whereby two forms are identical (indistinguishable) except by comparison in the space of an observer. The observer is the distinction between the mirror images. Mirrored pairs of individual brackets interact to form either a container C = {}

(43)

E =}{.

(44)

or an extainer

These new forms combine to make: CC = {}{} = {£}

(45)

EE=}{}{=}C{.

(46)

and

Two containers interact to form an extainer within container brackets. Two extainers interact to form a container between extainer brackets. The pattern of extainer interactions can be regarded as a formal generalization of the bra and ket patterns of the Dirac notation that we have used in this paper both for DNA replication and for a discussion of quantum mechanics. In the quantum mechanics application {} corresponds to the inner product < A\B >, a commuting scalar, while }{ corresponds to \A >< B |, a matrix that does not necessarily commute with vectors or other matrices. With this application in mind, it is natural to decide to make the container an analog of a scalar quantity and let it commute with individual brackets. We then have the equation EE =}{}{=}C{= C}{= CE.

(47)

By definition there will be no corresponding equation for CC. We adopt the axiom that containers commute with other elements in this combinatorial algebra. Containers and extainers are distinguished by this property. Containers appear as autonomous entities and can be moved about. Extainers are open to interaction from the outside and are sensitive to their

111 surroundings. At this point, we have described the basis for the formalism used in the earlier parts of this paper. If we interpret E as the "environment" then the equation }{= E = 1 expresses the availability of complementary forms so that

0 — {E} — {}{}

(48)

becomes the form of DNA reproduction. We can also regard EE = {}E as symbolic of the emergence of DNA from the chemical substrate. Just as the formalism for reproduction ignores the topology, this formalism for emergence ignores the formation of the DNA backbone along which are strung the complementary base pairs. In the biological domain we are aware of levels of ignored structure. In mathematics it is customary to stop the examination of certain issues in order to create domains with requisite degrees of clarity. We are all aware that the operation of collection is proscribed beyond a certain point. For example, in set theory the Russell class R of all sets that are not members of themselves is not itself a set. It then follows that {R}, the collection whose member is the Russell class, is not a class (since a member of a class is a set). This means that the construct {R} is outside of the discourse of standard set theory. This is the limitation of expression at the "high end" of the formalism. That the set theory has no language for discussing the structure of its own notation is the limitation of the language at the "low end". Mathematical users, in speaking and analyzing the mathematical structure, and as its designers, can speak beyond both the high and low ends. In biology we perceive the pattern of a formal system, a system that is embedded in a structure whose complexity demands the elucidation of just those aspects of symbols and signs that are commonly ignored in the mathematical context. Rightly these issues should be followed to their limits. The curious thing is what peeks through when we just allow a bit of it, then return to normal mathematical discourse. With this in mind, lets look more closely at the algebra of containers and extainers. Taking two basic forms of bracketing, an intricate algebra appears from their elementary interactions:

112 E = ><

(49) H=}< are the extainers, with corresponding containers: <>,

[], [>, <]•

(50)

These form a closed algebraic system with the following multiplications: EE=><><=<>E F F = ] [ ] [ = []F GG = >[>[=[>G HH=]<] <=<}H

(49) 0)

EF = ><}{= <}G EG = X>[=OG EH = ><]<= <)E

(52)

^

and

FE = ] [ > < = \>H

FG=}[>[=[>F FH=}[]<=[]H

(53)

GE = > [><= [> E G F = >[][=[] [ ] < = [}E

(54)

HE=] <><=<> H HF =]<][= <]F HG=) <>[=<> F

(55) (56)

Other identities follow from these. For example, EFE = ><}[><=<}[> E.

(57)

This algebra of extainers and containers is a precursor to the Temperley Lieb algebra, an algebraic structure that first appeared (in quite a different way) in the study of the Potts model in statistical mechanics 1 . We shall

113

forgo here details about the Temperley Lieb algebra itself, and refer the reader to Kauffman15 where this point of view is used to create unitary representations of that algebra for the context of quantum computation. Here we see the elemental nature of this algebra, and how it comes about quite naturally once one adopts a formalism that keeps track of the structure of boundaries that underlie the mathematics of set theory. The Temperley Lieb algebra TLn is an algebra over a commutative ring k with generators {1, U\, U2, •••, Un-i} and relations

v? = suu

(58)

UiUi±1Ui = Uu UiUj =

UjUi,\i-j\>l,

where 6 is a chosen element of the ring k. These equations give the multiplicative structure of the algebra. The algebra is a free module over the ring k with basis the equivalence classes of these products modulo the given relations. To match this pattern with our combinatorial algebra let n = 2 and let Ui= E = X , U2 = F =][ and assume that 1 =<] = [> while 6 =<>= []• The above equations for our combinatorial algebra then match the multiplicative equations of the Temperley Lieb algebra. The next stage for representing the Temperley Lieb algebra is a diagrammatic representation that uses two different forms of extainer. The two forms are obtained not by changing the shape of the given extainer, but rather by shifting it relative to a baseline. Thus we define diagrammatically U = Ui and V = t/2 as shown below: U=

>< ><

V=

_ (59)

UU =

=<>

><>K

uvu =

><:

> <

=

=<>U

—u

In this last equation UVU = U we have used the topological deformation of the connecting line from top to top to obtain the identity. In its typographical form the identity requires one to connect corresponding endpoints of

114 the brackets. In Figure 2 we indicate a smooth picture of the connection situation and the corresponding topological deformation of the lines. We have deliberately shown the derivation in a typographical mode to emphasize its essential difference from the matching pattern that produced (60)

EFE = ><}[><=<}[> E.

By taking the containers and extainers shifted this way, we enter a new and basically topological realm. This elemental relationship with topology is part of a deeper connection where the Temperley Lieb algebra is used to construct representations of the Artin Braid Group. This in turn leads to the construction of the well-known Jones polynomial invariant of knots and links via the bracket state model 9 . It is not the purpose of this paper to go into the details of those connections, but rather to point to that place in the mathematics where basic structures apply to biology, topology, and logical foundations.

U

V

^C uvu II

—DC— Figure 2. A Topological Identity

=u

115

It is worthwhile to point out that the formula for expanding the bracket polynomial can be indicated symbolically in the same fashion that we used to create the Temperley Lieb algebra via containers and extainers. We will denote a crossing in the link diagram by the letter chi, X- The letter itself denotes a crossing where the curved line in the letter chi is crossing over the straight segment in the letter. The barred letter denotes the switch of this crossing where the curved line in the letter chi is undercrossing the straight segment in the letter. In the bracket state model a crossing in a diagram for the knot or link is expanded into two possible states by either smoothing (reconnecting) the crossing horizontally, X, or vertically X . The vertical smoothing can be regarded as the extainer and the horizontal smoothing as an identity operator. In a larger sense, we can regard both smoothings as extainers with different relationships to their environments. In this sense the crossing is regarded as the superposition of horizontal and vertical extainers. The crossings expand according to the formulas

X = Ax + A~1>< X = A~1x + A><.

(61) (62)

The verification that the bracket is invariant under the second Reidemeister move is then seen by verifying that XX = -•

(63)

For this one needs that the container < > has value -A2 - A'2 (the loop value in the model). The significant mathematical move in producing this model is the notion of the crossing as a superposition of its smoothings. It is useful to use the iconic symbol > < for the extainer, and to choose another iconic symbol X for the identity operator in the algebra. With these choices we have x x

=

x

(64)

Thus XX

= (Ax + A-1 ><)(A" x x + A ><) = AA~ x xx + A 2 x x +A~2 x x + AA~l ><>< = ^ + A2><+A-2><+6>< = ~ + {A2 + A-2 + 6)><

(64) {b5)

116 Note the use of the extainer identity > < > < = > 8 <— 6 X . A t this stage the combinatorial algebra of containers and extainers emerges as the background to the topological characteristics of the Jones polynomial.

6.1. Projectors and Meanders In the Temperley Lieb algebra the generators satisfy the formula U2 = SU where 6 is the value of a loop. In fact, there are also elements P that satisfy the equation P2 = P. See Figure 2.1. Note that the identity PP = P is topological. Once PP has been constructed, there is a deformable string that contracts and yields P once again. One can view the equation PP = P as a form of self-reproduction by taking it in the order P —> PP. That is, one starts with P in contracted form, allows it to undergo the production of the little wiggle in the middle, and then cuts the resulting form apart to form two copies of P. See Figure 4.

\

P

PP = P Figure 3. P 2 = P

What is the secret by which we have obtained this self-reproduction in a topological/algebraic context? The reader should look closely once more at P and discern that P can be written as a product P = AB where A

117

vy ^ ^ \y Expand

Cut/Divide

I

I

\

P, P Figure 4. p reproduces itself.

has three top strands and one bottom strand, while B has one top strand and three bottom strands. We will say that A is of type (3,1), while B is of type (1,3). See Figure 5. Now we can also compose and form BA as in Figure 5, and we see at once that BA is topologically equivalent to a single (1,1) strand. We will write BA = I to denote this single strand. We see that it is the equivalence BA = I that makes for the identity PP = P, for we have PP = ABAB = A(BA)B = A{I)B = AB = P

(66)

where the identity A{I)B = AB is simply stating the contractibility of a single strand added in the middle of a product. By the same token, the "genetic" information in this self-production is contained in the factoring of the identity line / into the parts B and A. We now see how to generalize the construction to make infinitely many examples of projectors P such that PP = P. To make such an example we choose a deformation M of the identity line / , and then cut it to obtain a factorization / = BA. We then define P by the equation P = AB. See Figure 6 for an illustration of this process. There are infinitely many deformations of the identity, each giving rise to a factorization via cutting the deformation in half, and each giving rise to distinct elements P in the multiplicative Temperley Lieb algebra that have the property PP = P. It is quite fascinating to note that this solution to the nature of elements P with PP = P depends crucially on a combinatorial and topological view of the algebra. In Figure 2.5 we have illustrated how to take such an element and, by pairing maxima and minima, write it as a product in the standard

118

P

=

AB

•i:-r

BA = l

Figure 5. P = AB, BA = /.

generators Uk of the Temperley Lieb algebra. Here Uk consists of a paired maximum and minimum with the minimum connecting upper strands k and k + 1 (ordered from left to right), and the maximum connecting lower strands k and k + 1. The remaining strands proceed from top to bottom. While it is easy to obtain such a factorization, it is a difficult problem to characterize idempotent elements in terms of these generators. To go to the complete solution from the hints given here will be the subject of another paper, but we should point out that it is possible to generalize the element I to Ik, a collection of k parallel lines. Cutting a deformation of Ik to obtain an factorization Ik = BA and an element P = AB gives the general solution to the problem of finding all multiplicative elements in the Temperley Lieb algebra with PP = P. A modification of this approach yields a characterization of all elements Q with QQ = 6rQ for some positive integer r. The basic structure behind this classification is the meander, a simple closed curve in the plane that has been bisected by a straight line. Here we have illustrated the concept with an open meander

119

M

Cut[M] = BA

]

I P-AB

r\fey(\l(f^\ Figure 6. Constructing a new P with PP = P.

consisting in a cutting of a deformation of the straight line I. It is remarkable that the classification of meanders is clearly formally related to the classification of folded molecules and, from this point of view, also related to the structure of self-reproduction. This section has been an abstract foray into the possibilities of topological genetics. 6.2. Protein Folding and Combinatorial

Algebra

The approach in this section derives from ideas in Kauffman and Magarshak8. Here is another use for the formalism of bras and kets. Consider a molecule that is obtained by "folding" a long chain molecule. There is a set of sites on the long chain that are paired to one another to form the folded molecule. The difficult problem in protein folding is the determination of the exact form of the folding given a multiplicity of possible paired sites. Here we assume that the pairings are given beforehand, and consider the abstract structure of the folding and its possible embeddings in three dimensional space. Let the paired sites on the long chain be designated by labelled bras and kets with the bra appearing before the ket in the chain

120

W&J \

-

I IMKJII P.AB

l/WV

V V I I U5 U4 U6

\jQv\lU3U5U7

.A A A A

L'iiXX\

U2 U4 U6 U8

U5U7

P = U5 U4 U6 U3 U5 U7 U2 U4 U6 U8 U5 U7 U1 U6 Figure 7. Writing P as a product of standard generators.

order. Thus < A\ and |A > would denote such a pair and the sequence C -< a\ < b\ < c\\c > \b >< d\\d > \a > < e||e >

(67)

could denote the paired sites on the long chain. See Figure 8 for a depiction of this chain and its folding. In this formalism we do not assume any identities about moving containers or extainers, since the exact order of the sites along the chain is of great importance. We say that two chains are isomorphic if they differ only in their choice of letters. Thus < a\ < b\\b > \a > and < r\ < s\\s > \r > are isomorphic chains. Note that each bra ket pair in a chain is decorated with a distinct letter. Written in bras and kets a chain has an underlying parenthesis structure that is obtained by removing all vertical bars and all letters. Call this P(C) for a given chain C. Thus we have P(C) = P(< a\ < b\ < c\\c > \b >< d\\d > \a >< e\\e >) = « < » < > > < > . (68) Note that in this case we have P(Chain) is a legal parenthesis structure in the usual sense of containment and paired brackets. Legality of parentheses is denned inductively:

121 (1) <> is legal. (2) If X and Y are legal, then XY is legal. (3) If X is legal, then < X > is legal. These rules define legality of finite parenthetic expressions. In any legal parenthesis structure, one can deduce directly from that structure which brackets are paired with one another. Simple algorithms suffice for this, but we omit the details. In any case a legal parenthesis structure has an intrinsic pairing associated with it, and hence there is an inverse to the mapping P. We define Q(X) for X a legal parenthesis structure, to be the result of replacing each pair • • • < • . . > . . . in X by • • • < A\ • • • \A > • • • where A denotes a specific letter chosen for that pair, with different pairs receiving different letters. Thus < 2 ( < < > > ) = < a | < 6 | | 6 > | a > . Note that in the case above, we have that Q(P(C)) is isomorphic to C. A chain C is said to be a secondary folding structure if P(C) is legal and Q(P(C)) is isomorphic to C. The reader may enjoy the exercise of seeing that secondary foldings (when folded) form tree-like structures without any loops or knots. This notion of secondary folding structure corresponds to the usage in molecular biology, and it is a nice application of the bra ket formalism. This also shows the very rich combinatorial background in the bras and kets that occurs before the imposition of any combinatorial algebra.

a b c

Figure 8.

c

b

d

d

a

e

Secondary Structure < a\ < b\\c > \b X

e

d\\d > \a X

e\\e >

122 Here is the simplest non-secondary folding: L=\b>

(69)

.

Note that P(L) = « » is legal, but that Q{P(L)) = Q(<<>>) = < a\ < b\\b > \a > is not isomorphic to L. L is sometimes called a "pseudo knot" in the literature of protein folding. Figure 9 should make clear this nomenclature. The molecule is folded back on itself in a way that looks a bit knotted.

A

B

A

B

A B Figure 9. Tertiary Structure - < a\ < b\\a > \b >

With these conventions it is convenient to abbreviate a chain by just giving its letter sequence and removing the (reconstructible) bras and kets. Thus C above may be abbreviated by abccbddaee. One may wonder whether at least theoretically there are foldings that would necessarily be knotted when embedded in three dimensional space. With open ends, this means that the structure folds into a graph such that there is a knotted arc in the graph for some traverse from one end to the other. Such a traverse can go along the chain or skip across the bonds joining the paired sites. The answer to this question is yes, there are folding patterns that can force knottedness. Here is an example of such an intrinsically knotted folding. ABCDEFAGHIJKBGLMNOCHLPQRDIMPSTEJNQSUFKORTU. (70) It is easy to see that this string is not a secondary structure. To see that it is intrinsically knotted, we appeal to the Conway-Gordon Theorem4 that tells us that the complete graph on seven vertices is intrinsically knotted. In

123

closed circular form (tie the ends of the folded string together), the folding that corresponds to the above string retracts to the complete graph on seven vertices. Consequently, that folding, however it is embedded, must contain a knot by the Conway-Gordon Theorem. We leave it as an exercise for the reader to draw an embedding corresponding to a folding of this string and to locate the knot! The question of intrinsically knotted foldings that occur in nature remains to be investigated. 7. Cellular Automata Some examples from cellular automata clarify many of the issues about replication and the relationship of logic and biology. Here is an example due to Maturana, Uribe and Varela14. See also Varela16 for a global treatment of related issues. The ambient space is two dimensional and in it there are "molecules" consisting in "dots" (See Figure 10). There is a minimum distance between the dots (one can place them on a discrete lattice in the plane). And "bonds" can form with a probability of creation and a probability of decay between molecules with minimal spacing. There are two types of molecules: "substrate" and "catalysts". The catalysts are not susceptible to bonding, but their presence (within say three minimal step lengths) enhances the probability of bonding and decreases the probability of decay. Molecules that are not bonded move about the lattice (one lattice link at a time) with a probability of motion. In the beginning there is a randomly placed soup of molecules with a high percentage of substrate and a smaller percentage of catalysts. What will happen in the course of time? In the course of time the catalysts (basically separate from one another due to lack of bonding) become surrounded by circular forms of bonded or partially bonded substrate. A distinction (in the eyes of the observer) between inside (near the catalyst) and outside (far from a given catalyst) has spontaneously arisen through the "chemical rules". Each catalyst has become surrounded by a proto-cell. No higher organism has formed here, but there is a hint of the possibility of higher levels of organization arising from a simple set of rules of interaction. The system is not programmed to make the proto-cells. They arise spontaneously in the evolution of the structure over time. One might imagine that in this way, organisms could be induced to arise as the evolutionary behavior of formal systems. There are difficulties, not the least of which is that there are nearly always structures in such systems whose probability of spontaneous emergence is vanishingly small.

124

•)

—

«>

Figure 10. Proto-Cells of Maturana, Uribe and Varela

A good example is given by another automaton - John H. Conway's "Game of Life". In "Life" the cells appear and disappear as marked squares in a rectangular planar grid. A newly marked cell is said to be "born". An unmarked cell is "dead". A cell dies when it goes from the marked to the unmarked state. A marked cell survives if it does not become unmarked in a given time step. According to the rules of Life, an unmarked cell is born if and only if it has three neighbors. A marked cell survives if it has either two or three neighbors. All cells in the lattice are updated in a single time step. The Life automaton is one of many automata of this type and indeed it is a fascinating exercise to vary the rules and watch a panoply of different behaviors. For this discussion we concentrate on some particular features. There is a configuration in Life called a "glider". See Figure 11. This illustrates a "glider gun" (discussed below) that produces a series of gliders going diagonally from left to right down the Life lattice. The glider consists in five cells in one of two basic configurations. Each of these configurations produces the other (with a change in orientation). After four steps the glider reproduces itself in form, but shifted in space. Gliders appear as moving entities in the temporality of the Life board. The glider is a complex entity that arises naturally from a small random selection of marked cells on the Life board. Thus the glider is a "naturally occurring entity" just like the proto-cell in the Maturana-Uribe-Varela automaton. But Life contains potentially much more complex phenomena. For example, there is the "glider gun" (See Figure 11) which perpetually creates new

125 gliders. The "gun" was invented by a group of researchers at MIT in the 1970's (The Gosper Group). It is highly unlikely that a gun would appear spontaneously in the Life board. Of course there is a tiny probability of this, but we would guess that the chances of the appearance of the glider gun by random selection or evolution from a random state is similar to the probability of all the air in the room collecting in one corner. Nervertheless, the gun is a natural design based on forms and patterns that do appear spontaneously on small Life boards. The glider gun emerged through the coupling of the power of human cognition and the automatic behavior of a mechanized formal system. Cognition is in fact an attribute of our biological system at an appropriately high level of organization. But cognition itself looks as improbable as the glider gun! Do patterns as complex as cognition or the glider gun arise spontaneously in an appropriate biological context?

• i>Mi«ki|*MMprlr<<|riljl«>>rMM»»H«iMi I^^^^^E^^R • • • * • » • • • * a^^^^^^ * • * ^ ^ ^ ^ ^ ^ « * i^^R** * A * < * i * • « • »•«•«•••> "iiti»i|iti|iii|P
. *

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.;,.. H ,.. j ,.. i ,. t ,.:,,,|,. t .. i ,., i .. tv l. i . i ,.. i ,. t . t , t .::...:,..:,..;H,,, i .,. H .^,^ t

Figure 11. Glider Gun and Gliders

There is a middle ground. If one examines cellular automata of a given type and varies the rule set randomly rather than varying the initial conditions for a given automaton, then a very wide variety of phenomena will present themselves. In the case of molecular biology at the level of the DNA there is exactly this possibility of varying the rules in the sense of varying the sequences in the genetic code. So it is possible at this level to produce a wide range of remarkable complex systems. 7.1. Other Forms of Replication Other forms of self-replication are quite revealing. For example, one might point out that a stick can be made to reproduce by breaking it into two

126

pieces. This may seem satisfactory on the first break, but the breaking cannot be continued indefinitely. In mathematics on the other hand, we can divide an interval into two intervals and continue this process ad infinitum. For a self-replication to have meaning in the physical or biological realm there must be a genuine repetition of structure from original to copy. At the very least the interval should grow to twice its size before it divides (or the parts should have the capacity to grow independently). A clever automaton, due to Chris Langton, takes the initial form of a square in the plane. The rectangle extrudes a edge that grows to one edge length and a little more, turns by ninety degrees, grows one edge length, turns by ninety degrees grows one edge length, turns by ninety degrees and when it grows enough to collide with the original extruded edge, cuts itself off to form a new adjacent square, thereby reproducing itself. This scenario is then repeated as often as possible producing a growing cellular lattice. See Figure 12.

Figure 12. Langton's Automaton

The replications that happen in automata such as Conway's Life are all really instances of periodicity of a function under iteration. The gilder is an

127

example where the Life game function L applied to an initial condition G yields L5(G) = t(G) where t is a rigid motion of the plane. Other intriguing examples of this phenomenon occur. For example the initial condition D for Life shown in Figure 8 has the property that L48(D) = s(D) + B where s is a rigid motion of the plane and s(D) and the residue B are disjoint sets of marked squares in the lattice of the game. D itself is a small configuration of eight marked squares fitting into a rectangle of size 4 by 6. Thus D has a probability of 1/735471 of being chosen at random as eight points from 24 points.

........jjjj............. • • ^^Mi • • . . . . . ^^^^TT¥^^^^ • • • • • • • .

»

M||IL

.

^m^ ^

• i^^^^h • ^^^^H • . . . . i^^^^% • • • i^^^^j^^^^a . . . . • • • • . . . . . . . .

^^^^^^^^^^t . . • •

Figure 13. Condition D with geometric period 48

Should we regard self-replication as simply an instance of periodicity under iteration? Perhaps, but the details are more interesting in a direct view. The glider gun in Life is a structure GUN such that L30(GUN) = GUN + GLIDER. Further iterations move the disjoint glider away from the gun so that it can continue to operate as an initial condition for L in the same way. A closer look shows that the glider is fundamentally composed of two parts P and Q such that LW(Q) is a version of P and some residue and such that Ll5(P) = P* + B where B is a rectangular block, and P* is a mirror image of P, while L15{Q) = Q* + B' where B' is a small nonrectangular residue. See Figure 9 for an illustration showing the parts P and Q (left and right) flanked by small blocks that form the ends of the gun. One also finds that L15(B + Q*) = GLIDER +Q + Residue. This is the internal mechanism by which the glider gun produces the glider. The extra

128

blocks at either end of the glider gun act to absorb the residues that are produced by the iterations. Thus the end blocks are catalysts that promote the action of the gun. Schematically the glider production goes as follows: P + Q—>P* + £ + Q*

(71)

B + Q* —» GLIDER + Q

(72)

whence P+Q —> P* + B + Q* —> P + GLIDER + Q = P + Q + GLIDER. (73)

The last equality symbolizes the fact that the glider is an autonomous entity no longer involved in the structure of P and Q. It is interesting that Q is a spatially and time shifted version of P. Thus P and Q are really "copies" of each other in an analogy to the structural relationship of the Watson and Crick strands of the DNA. The remaining part of the analogy is the way the catalytic rectangles at the ends of the glider gun act to keep the residue productions from interfering with the production process. This is analogous to the enzyme action of the topoisomerase in the DNA. .4-.j.4..!.~j...j.44-^

...j...j...j...:...j...j...:. D j...j..^

:::!::rt!::iffi

Figure 14. Compose the Glider Gun

The point about this symbolic or symbiological analysis is that it enables us to take an analytical look at the structure of different replication scenarios for comparison and for insight. 8. Epilogue - Logic and Biology

We began with the general question: What is the relationship of logic and biology. Certain fundamentals, common to both are handled quite

129

differently. These are certain fundamental distinctions: The distinction of symbol and object (the name and the thing that is named). The distinction of a form and a copy of that form. In logic the symbol and its referent are normally taken to be distinct. This leads to a host of related distinctions such as the distinction between a description or blueprint and the object described by that blueprint. A related distinction is the dichotomy between software and hardware. The software is analogous to a description. Hardware can be constructed with the aid of a blueprint or description. But software intermediates between these domains as it is an instruction. An instruction is not a description of a thing, but a blueprint for a process. Software needs hardware in order to become an actual process. Hardware needs software as a directive force. Although mutually dependent, hardware and software are quite distinct. In logic and computer science the boundary between hardware and software is first met at the machine level with the built-in capabilities of the hardware determining the type of software that can be written for it. Even at the level of an individual gate, there is the contrast of the structure of that gate as a design and the implementation of that design that is used in the construction of the gate. The structure of the gate is mathematical. Yet there is the physical implementation of these designs, a realm where the decomposition into parts is not easily mutable. Natural substances are used, wood, metal, particular compounds, atomic elements and so on. These are subject to chemical or even nuclear analysis and production, but eventually one reaches a place where Nature takes over the task of design. In biology it is the reverse. No human hand has created these designs. The organism stands for itself, and even at the molecular level the codons of the DNA are not symbols. They do not stand for something other than themselves. They cooperate in a process of production, but no one wrote their sequence as software. There is no software. There is no distinction between hardware and software in biology. In logic a form arises via the syntax and alphabet of a given formal system. That formal system arises via the choices of the mathematicians who create it. They create it through appropriate abstractions. Human understanding fuels the operation of a formal system. Understanding imaged into programming fuels the machine operation of a mechanical image of that formal system. The fact that both humans and machines can operate a given formal system has lead to much confusion, for they operate it quite differently. Humans are always on the edge of breaking the rules either through error

130 or inspiration. Machines are designed by humans to follow the rules, and are repaired when they do not do so. Humans are encouraged to operate through understanding, and to create new formal systems (in the best of all possible worlds).

Here is the ancient polarity of syntax (for the machine) and semantics (for the person). The person must mix syntax and semantics to come to understanding. So far, we have only demanded an adherence to syntax from the machines. The movement back and forth between syntax and semantics underlies all attempts to create logical or mathematical form. This is the cognition behind a given formal system. There are those who would like to create cognition on the basis of syntax alone. But the cognition that we all know is a byproduct or an accompaniment to biology. Biological cognition comes from a domain where there is at base no distinction between syntax and semantics. To say that there is no distinction between syntax and semantics in biology is not to say that it is pure syntax. Syntax is born of the possibility of such a distinction. In biology an energetic chemical and quantum substrate gives rise to a "syntax" of combinational forms (DNA, RNA, the proteins, the cell itself, the organization of cells into the organism). These combinational forms give rise to cognition in human organisms. Cognition gives rise to the distinction of syntax and semantics. Cognition gives rise to the possibility of design, measurement, communication, language, physics and technology. In this paper we have covered a wide ground of ideas related to the foundations of mathematics and its relationship with biology and with physics. There is much more to explore in these domains. The result of our exploration has been the articulation of a mathematical region that lies in the crack between set theory and its notational foundations. We have articulated the concepts of container <> and extainer >< and shown how the formal algebras generated by these forms encompass significant parts of the logic of DNA replication, the Dirac formalism for quantum mechanics, formalism for protein folding and the Temper ley Lieb algebra at the foundations of topological invariants of knots and links. It is the mathematician's duty to point out formal domains that apply to a multiplicity of contexts. In this case we suggest that it is just possible that there are deeper connections among these apparently diverse contexts that are only hinted at in the steps taken so far. The common formalism can act as compass and guide for further exploration.

131 Acknowledgments

The author thanks Sofia Lambropoulou for many useful conversations in the course of preparing this paper. The author also thanks Sam Lomonaco, John Hearst, Yuri Magarshak, James Flagg and William Bricken for conversations related to the content of the present paper. Most of this effort was sponsored by the Defense Advanced Research Projects Agency (DARPA) and Air Force Research Laboratory, Air Force Materiel Command, USAF, under agreement F30602-01-2-05022. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright annotations thereon. The views and conclusions contained herein are those of the author and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Defense Advanced Research Projects Agency, the Air Force Research Laboratory, or the U.S. Government. (Copyright 2003.)

References 1. R. J. Baxter, "Exactly Solved Models in Statistical Mechanics", Academic Press, 1982. 2. H.P. Barendregt, "The Lambda Calculus Its Syntax and Semantics",North Holland, 1981 and 1985. 3. W. Bricken and E. Gullichsen, An Introduction to Boundary Logic with the Losp Deductive Engine, Future Computing Systems 2(4),(iQ89) 1-77. 4. J. H. Conway and C. McA Gordon, Knots and links in spatial graphs, J. Graph Theory, Vol. 7 (1983), 445-453. 5. P. A. M. Dirac, "Principles of Quantum Mechanics", Oxford University Press 1958. 6. J. E. Hearst, L.H. Kauffman, and W. M. McClain, A simple mechanism for the avoidance of entanglement during chromosome replication, Trends in Genetics, June 1998, Vol. 14, No. 6, 244-247. 7. L. H. Kauffman, Knot Logic, In "Knots and Applications" ed. by L. Kauffman, World Scientific Pub. Co., (1994), 1-110. 8. L. Kauffman and Y. Magarshak, Graph invariants and the topology of RNA folding, Journal of Knot Theory and its Ramifications, Vol3, No.3, 233-246. 9. L. H. Kauffman, State Models and the Jones Polynomial, Topology 26 (1987), 395-407. 10. L. H. Kauffman, Biologic, in AMS Contemporary Mathematics Series Vol. 304, (2002), pp 313-340. 11. G. Spencer-Brown, "Laws of Form", Julian Press, New York (1969). 12. E. L. Zechiedrich, A. B. Khodursky, S. Bachellier, R. Schneider, D. Chen, D. M. J. Lilley and N. R. Cozzarelli, Roles of topoisomerases in maintaining

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13.

14. 15. 16.

steady-state DNA supercoiling in Escherichia coli, J. Biol. Chem. 275:81038113 (2000). S. Lomonaco Jr, A Rosetta Stone for Quantum Mechanics with an Introduction to Quantum Computation, in "Quantum Computation: A Grand Mathematical Challenge for the Twenty-First Century and the Millennium," AMS, Providence, RI (2000) (ISBN 0-8218-2084-2). H. R. Maturana, R. Uribe and F. G. Varela, Autopoesis: The organization of living systems, its characterization and a model, Biosystems, Vol. 5, (1974), 7-13. L. H. Kauffman, Quantum Computing and the Jones Polynomial (to appear in "Proceedings of AMS special session on Quantum Computing", edited by S. Lomonaco Jr). F. J. Varela, "Foundations of Biological Autonomy", North Holland Press (1979).

133

OPERADS, MODULI OF SURFACES AND QUANTUM ALGEBRAS

RALPH M. KAUFMANN* Oklahoma State University SHUwater OK, USA E-mail: kaufmann@math. okstate. edu We review the relationship of moduli spaces of surfaces to operads and deformations of algebras. We start by recalling basic facts about operads, examples of them, and their relations to algebras. In particular, we regard the Arc operad as well as several suboperads which can be thought of as cacti operads. These play an essential role in the string topology of Chas and Sullivan. It is recalled that spineless cacti and cacti homotopy equivalent to the little discs and framed little discs operads featured prominently in deformation theory. Furthermore, we analyze operations on operads and use the results to relate the spineless cacti operad to the renormalization Hopf algebra of Connes and Kreimer. Finally, we give a cell decomposition for spineless cacti and show that the cellular chains operate on the Hochschild complex of an associative algebra. This gives a solution to Deligne's conjecture about the Hochschild complex and furthermore directly relates the Gerstenhaber structures on the loop space and the Hochschild complex.

1. Introduction In recent years, there has been a dynamic and vibrant interplay between physics and mathematics which is now spreading to biology. As a result of this exchange the ideas around topological and conformal field theory as well as string theory have found a mathematical manifestation in constructions which use the geometry of moduli spaces to describe deformations of algebras. In this setting, there are two main questions: which geometries govern specific deformations; and given specific geometries, what kind of algebraic objects do they represent? The deformations are usually given in terms of an expansion in one or more deformation parameters and the coefficients of these expansions can be viewed as multilinear functions whose relations are of particular interest. For their study operads provide the •Partially supported by NSF grant DMS-#0070681

134 right tool being at once an organizing principle, a theoretical framework and tertium comparationis. For persons not already inclined towards operads one could say operads are very much like Steenrod operations in the following sense. They both have three aspects: 1) There is an algebraic description. Pertaining to the Steenrod operations this is the fact that they form an algebra. For operads the counterpart is the definition of an operad in terms of its constituents and operations. 2) They can be viewed as universal operations, i.e. regarding the Steenrod algebra as operations on cohomology and for operads considering the classes of algebras over this operad. 3) There are realization in concrete models. In passing from geometry to algebra one usually uses geometric moduli to produce and describe the structure of deformations. For the physically inclined one could state that one starts out with an interpretation of Feynman rules. These can usually be translated into a graph theoretical picture together with a geometrical description such as a stratification of a moduli space indexed by these graphs. This would be a safe place to start for a critical mathematician. Appealing once more to physics; the idea of the path integral, like a usual integral, also encompasses an additivity property in the domain. This translates to glueing or pasting operations which have been known in mathematics as operads. Combining these two ingredients with the idea of strings one is mathematically led to consider operads of surfaces as we discuss below. The basic example is that of topological field theory where graphs actually represent surfaces with labelled boundaries and the glueing operation is the glueing along the boundaries - this is discussed below in detail. The relation to string theory is the apparent interpretation that the boundaries are strings and as they move they sweep out the surface. To make this interpretation rigorous, one has to introduce orientations and look at the cobordism category which allows to speak of inputs and outputs. With these inputs and outputs a surface looks like a thickened Feynman graph. These surfaces are naturally the geometric part of the picture; the algebra comes in form of a functor to the category of vector spaces as in Atiyah's1 version of topological field theory. Here the geometric fact that any surface can be decomposed into pairs of pants, cylinders and discs is translated into the fact that on the vector space there should be an associative commutative multiplication, a nondegenerate bilinear form and an identity for the multiplication. Starting with this topological picture one can put more structure on the

135

surfaces which will in turn give more operations on the algebraic side. This leads to deformations or augmentation of the basic structure of an associative commutative algebra. One can think for instance of surfaces together with conformal structures. In the algebro-geometric setting one is led in this case either to consider the open moduli spaces M 9 n of n-punctured surfaces of genus zero or their Deligne-Mumford compactification M 9 , n . In the latter case, one arrives at Gromov-Witten invariants37 or quantum cohomology which is, as the name states, a deformation of the commutative associative algebra. If one looks at the algebras that are governed by MQIU then one arrives at so-called gravity algebras or Lie^ in the genus 0 case which are deformations of Lie algebras. The fact that Lie algebras instead of commutative algebras appear has a very nice interpretation in terms of operads, since the operad H*(Mo,n) is Koszul dual to .ff*(Mo,n)18'20 just as the operads for commutative and Lie algebras are Koszul duals of each other. An augmentation can be made by adding for instance G-bundles to the picture for a finite group G. In this case one arrives at G-equivariant theories27 which are well suited to study so-called "stringy aspects" of global quotients such as symmetric products28. Adding conformal structure one arrives at the G-equivariant setting of GW-invariants26. For general orbifolds the corresponding invariant structures are contained in13. Staying on the differential-topological or analytic side one can keep the boundaries and add punctures for instance. The natural spaces to study are then Teichmuller spaces and their moduli space quotients. One especially promising aspect of this approach is the natural appearance of graphs in this theory. They arise in the theory of Strebel differentials as well as in Penner's work41'42. More recently another beautiful view of the topology of movements of strings in terms of operations on loop spaces has been put forth and developed8'9'45. This or some aspects of the theory also have an interpretation in terms of surfaces decorated by weighted arcs which again form an operad, the arc operad34. In the genus zero case this algebra of string topology is captured by the cacti operad of Voronov49 which is a suboperad of the arc operad34'29. The algebraic structure that appears is that of a Gerstenhaber Batalin-Vilkovisky algebra up to homotopy (basically this is an odd Poisson algebra whose bracket is the odd commutator of a differential that is not a derivation, but a degree two differential operator; see §4.12 for the definitions). This structure is closely related to the gravity algebra structure16 as is the space defining the arc operad to the open moduli space42.

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For the second type of question - what kind of geometry governs specific deformations - the following two problems are key examples: deformation quantization and Deligne's conjecture about the Hochschild cochain complex. It has surprisingly turned out that they are closely related32'48. The solving of the deformation quantization32 was done through geometry in terms of integrals over configuration spaces. Deligne's conjecture, on the other hand, is directly a question about the geometry governing deformations. It was realized by Gerstenhaber16 that deformations of the multiplication of the product in an associative algebra into an associative star product are governed by the Hochschild complex of this algebra. He proved that there is a surprising structure, namely that of an odd Poisson algebra, on the homology level. This structure is actually derived from the cochain level. Since the operad governing Gerstenhaber algebras is the homology of a geometric object, that of little discs, it was conceivable that this correspondence could be lifted to the chain level. This is the content of Deligne's conjecture which we prove below. This conjecture has been proven in various wayS32,48,39,50,36 ^for a £UJJ lev-lew see40) ^y ^ggj. cally choosing adequate chain models. The virtue of our approach lies in its directness and that it also establishes the surprising and intriguing fact that the Gerstenhaber structure on the loop spaces and the Gerstenhaber structure on the Hochschild cochains have a common natural interpretation in terms of surfaces. The formal similarities of the Gerstenhaber structure on the arc operad or rather of the suboperad of spineless cacti29 with the Gerstenhaber structure on the Hochschild cochains were first observed by Gerstenhaber himself and have given rise to our new proof of Deligne's conjecture in terms of spineless cacti30 which is reviewed below. In fact, spineless cacti are homotopy equivalent to the little discs operad and here is a cell decomposition of the spineless cacti operad indexed by trees which directly give the operation in the Hochschild cochain complex. Furthermore, the spineless cacti can be thought of as surfaces with weighted arcs that satisfy certain natural restrictions and as such can be seen as the substructure of moduli space giving rise to the bracket operations. This fits also well with a path integral description of deformation quantization by5. On the other hand, cacti and spineless cacti49'29 are the operads which correspond to the "string topology"8 and provide the BV and Gerstenhaber structure, respectively. A realization of the Gerstenhaber structure in this setting is given by the Goldman bracket8. In another direction, we show that the top-dimensional cells of the cell decomposition which provides the solution to Deligne's conjecture is related to the pre-Lie operad and to the

137 renormalization Hopf algebra of Connes and Kreimer11. It is thus tempting to view the Arc operad34 as an underlying "string mechanism" for all of these structures. The paper is organized as follows: After fixing some notation in section 2, we introduce and discuss several types of trees in section 3. Trees are the language in which operads are most easily spoken about. In the fourth section, we review the notion of an operad and give many examples of them. These examples include the ones of functions and trees, which are archetype of operads. We also introduce all the operads necessary for our discussion of Deligne's conjecture. Section 5 is devoted to a review of the construction of the Arc operad34 which can be seen as a combinatorial version of the operad of moduli spaces of genus g curves with n punctures. In a sense this operad describes hyperbolic field theory. Besides a direct relation discussed in the same volume by Penner, abstractly there is a relation due to the BV nature of the operad which we also review as well as giving a string interpretation for the arc operad. The section 6 deals with the cacti operads49'29. We show how they are naturally sub-operads of the arc operad thus relating the arc operad to the string topology8. In section 7, we give a natural cell decomposition for spineless cacti which is the basis of our solution of Deligne's conjecture. Paragraph 8 is an intermezzo about universal operations for operads, if one wishes meta-operads. This explains the natural appearance of insertion operads in terms of foliage operators and leads to the definition of preLie, Lie and Hopf algebras for operads. In section 9 we give our proof of Deligne's conjecture based on our cell model of the little discs operad which is given by the cellular chain operad of the natural cell decomposition for the spineless cacti operad or better the chain operad of normalized spineless cacti. We do this by providing two natural ways of letting these chains operate on the Hochschild cochains of an associative algebra. The first operation is of the same type as the operation of the chains of the arc operad on themselves self and is related to string topology. The second realization uses the fact that there is a geometric interpretation of the Hochschild cochains as chains of the spineless cacti operad. This view is implemented by the foliage operator introduced in the previous section. Section 10 contains the application of our results of section 8 and 7 to the operad of pre-Lie algebras. Putting together the previous results we obtain the renormalization Hopf-algebra of Connes and Kreimer as the symmetric group coinvariants of the Hopf algebra of an operad of cells.

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The cells are the symmetric top-dimensional cells of our cell decomposition of spineless cacti suitably shifted. Furthermore section 10 contains a proof of a generalization of Deligne's conjecture to operad algebras and mentions the generalization of it to cyclic cohomology of certain types of algebras. In the last section §11 we speculate on the Ax generalization of our results as well as on possible relations of the arc formalism to other "quantum phenomena" and additional subjects. Acknowledgments I would like to thank Muriel Livernet and Bob Penner for their wonderful collaboration34. Furthermore, I would like to thank Nils Tongring for organizing the extremely stimulating Woods Hole meetings. The visits at the Max-Planck Institute for Mathematics and the IHES, were instrumental in the carrying out of the presented research; especially the discussions with Maxim Kontsevich. Also the discussions about cacti with Sasha Voronov were the starting point for our investigations into the several types of cacti. My thanks also goes to Wolfgang Luck and Peter Teichner for their hospitality and interest. In fact the first steps for our solution to Deligne's conjecture were taken during a visit of the SFB in Miinster and the final step came to me during a visit to San Diego in response to questions of Peter Teichner. Finally, I wish to thank Jim Stasheff for very enlightening discussions and encouragement as well as Murray Gerstenhaber for pointing out the similarities between his constructions and those of the arc operad34 which was initial spark for the work on Deligne's conjecture. The author also acknowledges financial support from the NSF under grant #DMS-0070681. 2. Notation We denote by §„ the permutation group on n letters and by Cn the cyclic group of order n. Also, let k be a field. We will tacitly assume that everything is in the super setting, that is Z/2Z graded. For all formulas, unless otherwise indicated, the standard rules of sign38 apply. 3. Trees Trees are a very useful organizational tool when dealing with operads. The different types of operads we study give rise to different types of trees. In the following, we introduce the types of trees we will need for our discussion.

139

3.1. General

Definitions

Definition 3.1.1. A graph is a one dimensional simplicial-complex. We call the O-dimensional simplices vertices and denote them by V(T). The 1-dimensional simplices are called edges and denoted by E(Y). A tree is an isomorphism class of a connected simply-connected graph. A rooted tree is a tree with a marked vertex. We call a rooted tree planted if the root vertex lies on a unique edge. In this case we call this unique edge the "root edge". We usually depict the root by a small square, denote the root vertex by root(r) £ V(T) and the root edge by eToot(j) £ E(T). Notice that an edge e of a graph or a tree is gives rise to a set of vertices ®ie) = {wi!y2}- In a tree the set de = {v\,v2} uniquely determines the edge e. An ordered edge is an edge together with an orientation of that edge. On a tree to give an orientation to the edge e given by the boundary vertices {v\,v2} is equivalent to giving an order (v\,v2). If we are dealing with trees, we will denote the edge corresponding to {vi,v2} just by {v\,v2} and likewise the ordered edge corresponding to (vi,v2) just by (vi,v2). An edge that has wasa vertex is called an adjacent edge to v. An edge path on a graph T is an alternating sequence of vertices and with vt € V(J),et € E(T), s.t. dfa) = {vi,vi+i}. edges vi,ei,v2,e2,v3,... Definition 3.1.2. Given a tree r and an edge e £ E{T) one obtains a new tree by contracting the edge e. We denote this tree by con(r, e). More formally let e = {ui,i>2}, and consider the equivalence relation ~ on the set of vertices which is given by Viu S V(j) : w ~ w and i>i ~ v%. Then con(T,e) is the tree whose vertices are V ( T ) / ~ and whose edges are E(r)\{e}/ ~ where ~ ' denotes the induced equivalence relation {wi, w2} ~ ' {w[,w2} if w\ ~ ' w[ and W2 ~ w'2 or w2 ~ w[ and w\ ~ w'2• 3.2. Structures on Rooted Trees A rooted tree has a natural orientation, toward the root. In fact, for each vertex there is a unique shortest edge path to the root and thus for a rooted tree r with root vertex root G V(T) we can define the function N : V(T) \ {root} -> V(T) by the rule that N(v) = the next vertex on unique path to the root starting at v This gives each edge {^1,^2} with v2 = N(vi) the orientation

(vi,N(vi)).

140

We call the set {(w, v)\w e 7V~x(u)} the set of incoming edges of v and denote it by G (v) and call the edge (v, N(v)) the outgoing edge of v. Definition 3.2.1. We define the valence of v to be \v\ := {N'1^)]. The set of leaves Vieaf of a tree is defined to be the set of vertices which have valence zero, i.e. a vertex is a leaf if the number of incoming edges is zero. We also call the outgoing edges of the leaves the leaf edges and denote the collection of all leaf edges by Eieaf. CAVEAT: Our |t>| is the number of incoming edges, which is the number of adjacent edges minus one for all edges except the root edge where |v| is indeed the number of adjacent edges.

Remark 3.2.2. For a rooted tree there is also a bijection which we denote by out: V(T) \ {root} -> E(T). It associates to each vertex except the root its unique outgoing edge v i-+ (v, N(v)). Definition 3.2.3. An edge e' is said to be above e if e lies on the edge path to the root starting at the vertex of e' which is farther from the root. The branch corresponding to an edge e is subtree of made out of the all edges which lie above e (this includes e) and their vertices. We denote the resulting tree by br(e). 3.3. Planar Trees Definition 3.3.1. A planar tree is a pair (r,p) of a tree r together with a so-called pinning p which is a cyclic ordering of each of the sets given by the adjacent edges to a fixed vertex. 3.4. Structures on Planar Trees A planar tree can be embedded in the plane in such a way that the induced cyclic order from the natural orientation of the plane and the cyclic order of the pinning coincide. The set of all pinnings of a fixed tree is finite and is a principal homogeneous set for the group §(T) := x v 6 V ( r ) § | v | where each factor acts S|u| by permutations on the set of cyclic orders of the edges adjacent to v. This action is given by symmetric group action permuting the \v\ + 1 edges of v modded out by the subgroup of cyclic permutations which act trivially on the cyclic orders §\v\ ~ §\v\+i/C\v\+i.

141 3.5. Planted Planar Trees Given a rooted planar tree there is a linear order at each vertex except for the root. This order is given by the cyclic order and designating the outgoing edge as the smallest element. The root vertex has only a cyclic order, though. On a planar planted tree there is a linear order at all of the vertices, since the root now has only one incoming edge and no outgoing one. Furthermore, on such a tree there is a path which passes through all the edges exactly twice -once in each direction- by starting at the root going along the root edge and at each vertex continuing on the next edge in the cyclic order and finally terminating in the root vertex. We call this path the outside path. By omitting recurring elements this yields a linear order -<(r'p) starting with the root edge on the set V(r) II E{T). This order induces an order on the set of vertices V(r), on the set of all edges E(T), as well as a linear order for all the vertices incident to the vertex v <J whose smallest element is the outgoing edge. We omit the superscript for -<(r'p) if it is clear from the context. 3.6. Labelled Trees Definition 3.6.1. For a finite set 5 an 5 labelling for a tree is an injective map L : S —> V(r). An S labelling of a tree yields a decomposition into disjoint subsets of V(r) = Vi II Vu with Vi = L(S). For a planted rooted tree, we demand that the root is not labelled: root G Vu. An n-labelled tree is a tree labelled by n := { 1 , . . . ,n}. For such a tree we call Vi := L(i). A fully labelled tree r is a tree such that Vi = V(r). A leaf labelled tree r is a labelled tree in which exactly the leaves are labelled Vt = Vieaf. 3.7. Black and White Trees Definition 3.7.1. A black and white graph (b/w graph) r is a graph together with a function color : V(T) —> {0,1}. We call the set VW(T) := color"1 (0) the set of white vertices and call the set Vb(r) := color"1 (1) the set of black vertices. By a bipartite b/w tree we understand a b/w tree whose edges only connect vertices of different colors.

142 An S labelled b/w tree is a b/w tree in which exactly the white vertices are labelled, i.e. V\ = Vw and Vu = VbFor a rooted tree we call the set of black leaves the tails. A rooted b/w tree is said to be without tails, if all the leaves are white. A rooted b/w tree is said to be stable if there are no black vertices of valence 1. A rooted b/w tree is said to be fully labelled, if all vertices except for the root and the tails are white and labelled. Definition 3.7.2. For a black and white bi-partite tree, we define the set of white edges EW(T) to be the edges (vi,N(vi)) with N(vi) e Vw and call the elements white edges and likewise we define Eb with elements called black edges, so that there is a partition E{T) = EW(T) U.Eb(r). Notation 3.7.3. For a planar planted b/w tree, we understand the adjective bipartite to signify the following attributes: both of the vertices of the root edge are black, i.e. root and the vertex N~1(root) is black and the tree after iteratively contracting tails is bipartite otherwise. By iterative contraction of tails, we mean that the operation of contracting the tail edges is repeated until there are no tail edges left. The root edge is considered to be a black edge. Also in the presence of tail, all non-white tail edges are considered to be black. Definition 3.7.4. For a planar planted r b/w bi-partite tree we understand by the branch of e = (b,N(b), b € Vj,(r) to be the planar planted bi-partite rooted tree which given by the branch of e where the color of N(b) is changed to black and this black vertex is the root. In the case that the tree to which e is labelled the branch of e is the tree which is labelled by the set of labels of its white edges - we stress that this does not include the root N(b). If the tree r is labelled then the tree br(e) for any e € E(T) is a labelled tree with the labelling induced by that of r. Notice that by definition the root of br(e) will be unlabelled. 3.8. Notation I N.B. A tree can have several of the attributes mentioned above; for instance, we will look at bipartite planar planted rooted trees. To fix the set of trees, we will consider the following notation. We denote by T the set of all trees and use sub and superscripts to indicate the restrictions. The superscript r,pp, nt will mean rooted and planar planted, without tails while

143 the subscripts b/w, bp, st will mean black and white, bi-partite, and stable, where bi-partite and stable insinuate that the tree is also b/w. E.g. Tr The set of all rooted trees T£? The set of planar planted b/w trees -jvv rpjjg ge |. of p j a n a r planted bipartite trees Furthermore we use the superscripts / / and II for fully labelled and leaf labelled trees. E.g. Tli The set of all rooted leaf labelled trees We furthermore use the notation that T(n) denotes the n-labelled trees and adding the sub and superscripts denotes the n-labelled trees of that particular type conforming with the restrictions above for the labelling. Likewise T(S) for a set S are the S labelled trees conforming with the restrictions above for the labelling. E.g. T£? (n) The set of planar planted b/w trees with n white vertices which are labelled by the set { 1 , . . . , n}. 3.9. Notation II Often we wish to look at the free Abelian groups or free vector spaces generated by the sets of trees. We could introduce the notation Free(T, Z) and Free(T, k) with suitable super and subscripts, for the free Abelian groups or vector spaces generated by the appropriate trees. In the case that there is no risk of confusion, we will just denote these freely generated objects again by T with suitable sub and superscripts to avoid cluttered notation. If we define a map on the level of trees it induces a map on the level of free Abelian groups and also on the k vector spaces. Likewise by tensoring with k a map on the level of free Abelian groups induces a map on the level of vector spaces. Again we will mostly denote these maps in the same way. 3.10. Notation III If we will be dealing with operads of trees, we will consider the collection of the T(n) with the appropriate sub and superscripts. Again to avoid cluttered notation when dealing with operads, we also denote the whole collection of the T(n) just by T with the appropriate sub and superscripts. 3.11. The Map cppin : Tr -+ 7^ p There is a map from planted trees to rooted trees given by contracting the root edge. This map actually is a bijection between planted and rooted

144

trees. The inverse map is given by adding one additional vertex which is designated to be the new root and introducing an edge from the new root to the old root. We call this map plant. Also, there is a map pin from the free Abelian group of planted trees to that of planted planar trees given by.

pin(r) =

]T

(r,p)

p S Pinnings(i-)

Finally there is a map from planted planar trees to planted-planar bipartite trees. We call this map bp. It is given as follows. First color all vertices white except for the root vertex which is colored black, then insert a black vertex into every edge. In total we obtain a map cppin := bpo pin o plant: Tr —> 77? that plants, pins and colors and expands the tree in a bipartite way. Using the map cppin, we will view T r as a subgroup of TF?'. The image of Tr coincides with the set of invariants of the actions §(r). We will call such an invariant combination a symmetric tree. Remark 3.11.1. The inclusion above extends to an inclusion of the free Abelian group of fully labelled rooted trees to labelled bi-partite planted planar trees: cppin : TT^l{n) —* 77? (n). 3.12. The Map st^ : Taptp -> T™ We define a map from the free groups of stable b/w planted planar trees to the free group of bi-partite b/w planted planar trees in the following way: First, we set to zero any tree which has black vertices whose valence is greater than two. Then, we contract all edges which join two black vertices. And lastly, we insert a black vertex into each edge joining two white vertices. We call this map st^. Notice that stoo preserves the condition of having no tails and induces a map on the level of labelled trees. This nomenclature is chosen since this map in a certain precise sense forgets the trivial A^ structure of an associative algebra in which all higher multiplications are zero. 4. Operads In this section we briefly review the notion of an operad and give the main examples of operads for the algebras we will be considering.

145 First let us fix a symmetric monoidal category (C,
4.1. Operads Definition 4.1.1. An operad in C is a collection of objects O := {O(n) : O(n) £ C, n > 1} together with an §„ action on O(n) and maps °i : O(m)®O(n)

-> O{m + n-l),i

£ {l,...m}

(4.1)

which are associative and § n -equivariant and an element id £ 0(1) that satisfies for all opn £ O(n),i £ { 1 , . . . , n} Oi(opn,id) = ox(id,opn)

- opn

i) Associativity: for opk £ O(k),op[ € 0(1) and op'^ £ O(m)

{

(opk OJ op'^) o i+m _x op[)

if 1 < j < i

opk °i (op[ Oj-i+i op'm)

if i < j < i + I

(opk Oj_i+i op\) OJ op'^

iii + l<j

ii) Equivariance: opm £ O(m) and opn G O(n) am(opm)

°i ar'n(opn) = °m °i o'n(opm o e r m ( i ) opn)

where am Oi <j'n £ S m + n _ i is the block or iterated permutation (l)2,...,i-l,(l'l...,m/))i + l...,n)i-»

2 , . . . , i - 1, a'm(l',

an(l,

. . . , m ' ) , i + l...,n)

induces on ( 1 " , . . . , (m + n — 1)") where

{

1< j
j

(j-i j —n

+ 1)'

i <j
+ n— 1

(4.2)

146 That is the permutation which permutes the j " with i < j < i + n — 1 according to an and then permutes the all of the j " according to am treating the previous j " as a block in the position i. As an example of a block permutation let's regard (123) o2 (12) this is the permutation /1324\

/1234\ _ /1234\

V324V ° V1324J ~ U 2 4 1 /

We call the operads in Set, Top, Chain, Vectk combinatorial, topological operads, chain operads and linear operads, respectively. We also call O(n) the n-th component of O. Remark 4.1.2. Both the three different cases for associativity and the block permutation can be naturally understood in the examples of functions and trees. Definition 4.1.3. A morphism of two operads O, O' in the same monoidal category is a collection of morphisms from O(n) —> O'(n) which respect all the structures, i.e. respect the glueings and are §„ equivariant. Such a morphism is also called an operadic morphism. A suboperad is an injective morphism of operads. In this case, we call O a suboperad of O'. 4.2. Induced Operads As we mentioned before, the categories we are interested in are Set,Top,Chain and Vectk- Now there is the singular homology functor H* : Top —> Vectk which given a topological operad yields a linear operad. Using the functor of singular chains C» : Top —> Chain gives a map from topological operads to chain complexes and finally taking homology H* : Chain —> Vectk of a chain complex yields a functor from chain operads to linear operads. For the intermediate level, the chain level, there might be other operad structures depending on the choice of model for the chains which is compatible with the operadic compositions. Given a CW complex one can try to define an operad structure on the level of CC* of cellular chains, and given a triangulated space one can consider the simplicial chains C^. In these cases one has to additionally

147

check that the compositions on the chains are indeed (a sum) of chains to obtain an acceptable model. There is also a functor of T : Set t-> Chain which associates the free Abelian group to a set, however, usually there are some signs that appear in geometric situations coming from different orientations. In a given case, there might also be a candidate of a possible non-trivial differential. 4.3.

The Fundamental Examples

There are two fundamental examples which help to explain the notion of an operad. If one wishes, an operad is the abstraction of the algebraic properties of these examples. They are the operad of functions and the operad of leaf labelled trees. 4.4. The Operad of Functions Fix a set X and regard Funct(n) := Map(Xn,X). action induced by permutation of the variables: a(f)(xi,...,xn)

The §„ action is the

:= / ( ^ ( i ) , . . . , ^ ^ ) )

and the maps Oj are defined to be the substitution maps ( / ° i S ) (^li- • -Zm+n-l)

•= f{z\,

. , Z j + n _ i ) , Zi+n, . . . , Z m + n _ l )

Zi-i,g(Zi,..

This map is the following substitution: say / is a function of the variables Xi and g is a function of the variables yj then setting Xi = g(yi,..., yn) yields f(xi,..., xm)oig(y1 , . . . , y n ) = f(xi,... .Xi-i, g(yi,..., y n ) , xi+i , . . . , x m ) =

with

{

Xj

(f°ig)(zi,...,zm+n-i)

1< j
(4.3)

Vj-i+i i<j
148 °x{f) °i cry{g) = az(f oCTi(i) g)

The permutation which can be constructed from this condition is in fact unique and is exactly the block permutation ax Oj ay for the general case. The associativity translates to the fact that if one does two substitutions the order in which the substitutions are performed does not matter. Notice that this gives the three cases in the general definition. The first and the third case correspond to the substitutions in which two of the variable x, say Xi and Xj of the function / are substituted by functions g\ and g j . The second case corresponds to a nested substitution for three functions f,g,h. The two sides of the equation being either first substituting g into / and then substituting h into the outcome of this substitution or first substituting h into g and then substituting the outcome of this substitution into / . 4.5. Rooted Leaf Labelled Trees Another useful primordial example is that of rooted trees with labelled leaves Tu. The n-th component of this operad are rooted trees with n labelled leaves. The §„ action on the n-th component is given by permuting the labels. The operation of grafting defines the compositions. The composition r ot T' is defined to be the rooted tree obtained by grafting the tree T' to the vertex v» of r. To graft two rooted leaf labelled trees r and r' at the vertex Vi of r identify the root of r' with the vertex Vi. The root of the tree is the image of the root of T. We label the leaves of the composition analogously to the example 4.4 in equation (4.3). The § n equivariance then follows naturally. An example of grafting is as depicted in figure 1. The condition of associativity which is met can be rephrased as stating, that the result of several graftings does not depend on the order in which they are performed. There are two basic situations for associativity which are depicted in figure 2. They correspond to the first and third case, and the second case. Remark 4.5.1. The grafting procedure still makes sense for planar trees with labelled leaves, by keeping the linear order at the grafted vertex.

149

Y

•-

%

Figure 1. Example of grafting two trees

t

-w- w w- # I

II

III

IV

Figure 2. The first type of associativity for gluing is depicted in I. II shows the result of either order of gluing. The second type of associativity is depicted in III. The result of either order of gluing for this case is depicted in IV.

Definition 4.5.2. In the case of planar planted trees, we fix by convention that for planted planar trees r oi r' denotes the planted planar tree in which after the grafting the image of the root edge of r' is contracted. An example of his procedure is as depicted in figure 3. Definition 4.5.3. In the case of planar planted bi-partite trees, the grafting operation Oj for Vi a leaf is denned to be the grafting operation for planar planted trees followed by the contraction of the image of the outgoing edge Of Vi.

An example of his procedure is as depicted in figure 4. Notice that at the moment, since we only defined how to glue for leaves, this is only a partial operad structure (cf. §4.10.5 ); the full operad structure is explained in §4.7.

150

\'

-••'.••/

s

i

\,'

x

i/

\.

i"-

>^

Figure 3. Grafting two planted planar trees

<3>

°J$>

I

°3

T

J2>

<=on(con(

<0>

,e),e')

=

<0>

(i

•^''^



•

•

1

II

Figure 4. Grafting two bi-partite planted planar trees

4.6. Bordered Surfaces and Corollas We can also consider bordered topological surfaces with or without punctures and genus. Here the space E(n) is the space of homeomorphism classes of bordered surfaces (£,d£) whose boundary is homeomorphic to the disjoint union of n + 1 circles which are labelled by 0,..., n: <9£ ~homeo Ui=0,...,nSJ.

The operad structure is denned via glueing of surfaces with boundary along their boundaries. I.e. £ Oj £' is the surface obtained by glueing the boundary 0 of £' to the boundary i of £ as depicted in Figure 5. The § n action is given by permuting the labels, and the labelling after the grafting is again denned analogously to equation (4.3). This guarantees the §„ equivariance and the associativity. Another way to graphically encode the same operad is to represent each surface by a tree which has a root vertex and n labelled leaves. Such a tree

151 2

D

°

!

'

°

Figure 5. Example of grafting two surfaces

is called a (labelled) corolla. The operad structure comes from glueing corollas T, T' and then contracting the edge, (root(T'),Vi). The S n action is by permutation the labels and the labelling of the composition is again denned according to (4.3).

1

1 m Np*^

1 ... n

\f V m+n-1 V^=ccm( V3>jn,e) = N ^ 1

Figure 6. Example of grafting two corollas

In order to make the two pictures meet, we plant the corolla. This planted corolla then has one internal vertex representing the surface, a root which represents the boundary 0 and leaves representing the other boundaries. In this picture, we can view the surface as a thickening of the (planted) corolla. 4.7. Tree Insertion

Operads

Lately, another type of operad of trees has appeared36'7 where the glueings are not restricted to the leaves. This means that there is a grafting procedure into inner vertices as well. The natural origin of these operads is discussed in detail section 8. We will need the following variant of an insertion operad structure30. It is defined on the collection of free Abelian groups T^{n) that is n-labelled rooted planar planted bipartite trees. Recall that this means that there are n white vertices which are labelled from 1 to n. The description in words is as follows: there are three steps. First

152

cut off the branches corresponding to the incoming edges of u,. Notice that these branches have a linear order according to the linear order of the edges adjacent to Vi. Second graft r ' as a planar planted tree onto the remainder of T at the vertex Vi which is now a leaf. This grafting is in the sense of bi-partite planar planted trees. Lastly sum over the possibilities to graft the cut off branches onto the white vertices of the resulting planar tree which before the grafting belonged to r'. Since we are dealing with planted planar trees, the grafting entails a choice of the linear order at the vertices at which we graft after the grafting. We only sum over those choices in which the order of the branches given by the linear order at Vi is respected by the grafting procedure. I.e. the branches after grafting appear in the same order on the grafted tree as they did in r. An example of such an insertion is depicted in figure 7 To make this definition precise:

Definition 4.7.1.

Definition 4.7.1. Given r e ^^(m) and r ' e T^(n), we define the tree

ToiT':=

Yl (gr:N-^vi)^Vw(T'),P(gr))

(Tgr,p(gr))eTbppp(m + n-l)

where (1) gr is a bijection. (2) Tgr is the tree whose vertices are V(T O, T') := (Vr \ {vi}) II (Vf) \ {root(r'), N^1 (root(T')}) and the following edges: the root edge of T' and the outgoing edge of Vi are deleted, all edges not incident to v; or N~1(root(T>)) are kept, the edges (w, N~l(root(T'))) are replaced by edges (w,N(vi)). The order of the edges at N(vi) is given by first enumerating the edges incident to N(vi) which came before the edge (vi,N(vi)), then the edges (w,N(vi)) according to the order of incidence of (w,N~1(root(r1))) at N~1(root(Tl) and lastly the rest of the edges incident to N(vi) which came after the edge ((UJ), N(vi)). This is precisely the grafting of r' onto the tree r with cutoff branches at Vi. Finally the incoming edges of Vi are connected N'1^). according to gr, i.e. give rise to edges (v,gr(v)) for v € (3) p(gr) is a pinning of rgr and (4) the / on J2' indicates that the sum is over all compatible pairs of a bijection gr and a pinning p(gr) that preserve the linear order. Here compatibility is the following: Let -
153 the linear order on the edges r' as well as for the edges of r that are kept is respected and e\ = {vi,Vi) -< e^ = {vk,Vi) in T implies that {vi,gr(vi)) -
into the concatenations, which are dictated by the orientations of the cells. One such consistent choice of sign is provided as follows: we order the white edges of r and t' according to the linear order of the respective trees. Now the white edges of r ot T' are correspond exactly to the union of white edges of r and r'. We now define the sign to be the sign of the shuffle which shuffles the edges of r' into their position, i.e. the shuffle from EW(T) II EW{T) to EW(T OJ r') where we regard Ew as ordered. There are also other natural choices as discussed in §7.7.

Remark 4.8.1 With the above compositions is a suboperad of

Remark 4.8.1. With the above compositions T^'nt

is a suboperad of T^f.

Figure 7. Example of the insertion of a bi-partite planted planar tree

4.9 Other Tree Insertion Operads and Compatibilities There area two tree insertion operads structures already present in the literatureFigure 7. Example of the insertion of a bi-partite planted planar treeplanted stable b/w trees without tails the case ofa 4.9. Other Tree Insertion Operads and Compatibilities

There are two tree insertion operads structures already present in the literature on rooted trees Tr>fl 7 and, historically the first, on planar planted stable b/w trees without tails T™wst'nt™. In the gluing for Tr one omits mention to the order and in the case of ^bfw " o n e a ^ s o a U° w s glueing to the images of the black vertices of r'.

154

Also in the first case the basic grafting of trees is used (no contractions) while in the second case the grafting for planted trees is used, i.e. the image of the root edge is contracted, but not the outgoing edge of Uj. The signs for the first gluing are all plus7 and in the second gluing are dictated by a chain interpretation36 see also below. Remark 4.9.1. The operad structure denned above restricted to T£?'nt is compatible with those of Tr'fl and T^st'nt under the maps cppin and

st .

In 7.7, signs for the operad are denned by giving a cell interpretation. The result is that there is an orientation for top-dimensional cells corresponding to a choice of signs which makes cppin into an operadic map and an orientation of cells which fixes the signs in such a way that stoo is an operadic map, see §10.3 and §9.9. There are topological versions of this type of insertion glueing49'34'29 as explained in §5 and §6. 4.10. Variations of Operads There are some variations of the structure of operads which we will need. Remark 4.10.1. One can also consider operads as collections O(n) starting at n = 0. Definition 4.10.2. If the requirement of having an identity id omitted the resulting structure is called a non-unital operad. Omitting the S n action and the §„ equivariance yields the structure of a non-S operad. It is clear that when considering operads one can consider indexing by arbitrary sets instead. We will also need the following weakening of the structure of an operad: Definition 4.10.3. need not hold.

29

A quasi-operad is an operad where associativity

Remark 4.10.4. If a quasi-operad in the topological category is homotopy associative then its homology has the structure of an operad. In certain cases, like the ones we will consider, the structure of an operad already exists on the level of a particular chain model. Definition 4.10.5. Lastly in a partial operad the concatenations need not all be defined on the whole components O(n), but when they are defined the axioms hold.

155 Definition 4.10.6. A cyclic operad is an operad together with an action of § n + i on each O(n) where § n+ i acts on the set {0,1,... , n} extending the action of S n on {l,...,n} and additionally satisfying the following compatibility. Let * denote the action of the cycle (012... n) £ S n+ i then for a € O(n) (aon&)* = (-l)laHblk*Oia* (4.4) where we denoted the possible Z/2Z degree of an element a by \a\. The additional axiom means that the root is not distinguished any more. Examples are the S(n) and the operad of rooted trees with grafting on leaves and also the operads M9:n, M9tn and Arc (see below). In the operad S(n) the cyclic nature is inherent since there is no topological distinction between the boundary components. An example illustrating the equation (4.4) using leaf labelled trees together with a labelling of the root by 0 is given in figure 8

AIA •

1 • 0

.

•

n

«

n n+1

•• * .MAY* • , \ R y T '

m

n-l

o

m

o -™- 1

i 0

* 1

0

n+m 1

o

i 1

"

i 1

• °

* n+m-1

Figure 8. An illustration of the equation (4.4)

4.11. Algebras Over Operads If we take the example 4.4 to be linear functions of a fixed vector space V onto itself, we obtain the linear operad Tiomy with Homyin) := Homk-Hn{V®n,V) Definition 4.11.1. An algebra V over a linear operad O is a vector space V over k together with an operadic map O —» Homy • The idea of the definition is that each element of O(n) defines a n-ary multilinear operation on V. The C/r-operation one has in mind is an algebra multiplication V®2 -> V.

156 Definition 4.11.2. An algebra over a cyclic operad O is vector space V together with a symmetric non-degenerate bilinear form (, ) and a map of operads : O —> Homy s.t. (vo, (opn)(vi,..., for opn €

vn)} = (vn,
vn-i))

0(n).

Remark 4.11.3. The concept of an algebra over a cyclic operad illustrates the idea of a cyclic operad. The prime example being Homy- Now if V has a non-degenerate bi-linear form then Homv(n) = V®n®V* = V® n+1 . This isomorphism makes the action of § n +i perspicuous. Sometimes this is stated as that S n+ i also permutes the symbol of the function. As an example Frobenius algebras -that is associative, commutative algebras with a non-degenerate bi-linear pairing, which is invariant with respect to the pairing (a, 6c) = (ab, c)- are algebras over the cyclic operad £(n) where the S n+ i action permutes all labels 0 , 1 , . . . , n. 4.12. Operads Classifying Algebras More examples of operads come from operads classifying algebras; i.e. operads s.t. each algebra over them is of a certain type -like associative commutative etc- and vice-versa each algebra of the given type is indeed an algebra over this operad. 4.13. The Operad for Commutative

Algebras

The operad COM of associative, commutative algebras over a field k is given by Com{n) := k as the trivial S n module With glueing maps given by the identification k®kk~k. It is easily seen that this operad coincides with the operad S(n) or that of corollas if one takes their A;-linear span. For an algebra over this operad the operation for 1 £ k = Com(2) defines a map V®2 —> V : v\ ® V2 >-> t>i • i>2 which has to be commutative since it is §2 invariant. Furthermore, it is associative from glueing two two-corollas in the two possible different ways to obtain the three-corolla. The element 1 € k = Com(n) represented by the n-corolla is necessarily the map V®n »-> V : vi • • • vn i-> v\ vn which is uniquely defined since the algebra is associative and the multiplication is independent of the order due to the §„ invariance.

157 4.14. The Operad for Associative Algebras The operad ASSOC of associative algebras over a field k is given by Assoc(n) := The regular representation of § n with glueing maps given by composition of permutations as in the axiom of § n equivariance of 4.1.1. It is easily seen that this operad coincides with planar corollas. An algebra over this operad has two multiplications • and - op coming from ASSOC (2). The multiplication • is associative and the planar ncorollas corresponding to the basis element eCT of the regular representation of S n represent the maps v\ (g> • • • ® vn >-> va{\) • • • • • i>CT(n) This operad is the same as the natural one for planar planted corollas. 4.15. The Operad for Gerstenhaber Algebras Definition 4.15.1. A Gerstenhaber algebra A is a graded vector space together with two operations, a graded commutative and associative product • of degree 0, a bracket { • } of degree 1 (which is a Lie graded bracket on HA, the suspension of A sometimes also called an odd Lie-bracket) such that the bracket is odd Poisson for the multiplication •. More precisely: if we denote the degree of x £ A by \x\ and by |sa:| = |x| + 1 (the degree of x in EA), then the following equations hold: {x-v) = x-{y-z)

= (x-y)-z

{x*y} = {x.{y.

(-l)WMyx -(-l)MM{yx}

z}} = {{x . y} . z) + (-l)l-ll^{j/ . {x . z}}

{x*yz}

= {x*y}-z

+ (-l)^My

• {x • z).

The penultimate equation is sometimes called odd Jacobi and the last equation is the odd Poisson property. The idea is that • is of degree 1. We will actually see later on that • can be interpreted geometrically as a one dimensional simplex. Definition 4.15.2. The little discs operad £>2 is the operad given by D2{n) := {configurations of n discs labelled from 1 to n which are embedded into the unit disc of M2} together with the glueing described below and the § n action by permuting the labels.

158 Let us call the unit disc the outer disc. The glueing o; is denned by scaling the second configuration by a homothety so that the diameter of the outer disc coincides with the diameter of the i-th disc of the first configuration and then glueing in the scaled second configuration into the i-th disc of the first configuration. This is done by identifying the scaled outer disc of the second configuration and the i-ih disc of the first configuration and erasing the identified boundary.

f&^\

f^\

f^o\ t erase

/
^N

insert

scale to

fit

/ ^

^ ^

insertion w/ identified border Figure 9. The composition maps for little discs

The following proposition is well known3'4. Proposition 4.15.3. (Cohen)Any Gerstenhaber algebra is an algebra over the homology of the little discs operad and vice versa. 4.16. The Operad for Batalin-Vilkovisky

(BV) Algebras

Definition 4.16.1. A Batalin-Vilkovisky (BV) algebra is an associative super-commutative algebra A together with an operator A of degree 1 that satisfies

159 A2 = 0 A(abc) = A(ab)c + (-l)^aA(bc) + (-l) |sa||6l 6A(ac) - A(a)bc -(-l)l a laA(6)c - (-l)lal+l6la6A(c) Proposition 4.16.2. For any BV-algebra (A, A) define {a • b} := (-l)l a l A(a6) - (-l)l°l(A(o))6 - aA(6)

(4.5)

Then (A, { • }) is a Gerstenhaber algebra. We call a triple (A, { • }, A) a GBV-algebra if (A, A) is a BV algebra and "{ • } : A ® A —> A satisfies the equation (4.5)". By the above proposition (^4, { • }) is a Gerstenhaber algebra. Definition 4.16.3. The framed little discs operad fD^ is the operad given by fD2(n) := {configurations of n discs labelled from 1 to n which are embedded into the unit disc of R2 together with an orientation (i.e. an angle 6 6 [0, 2TT]) of each of the n discs} together with the permutation action of §„ on the labels and the glueing ot which is given by first rotating the second configuration by the angle #,, then scaling the configuration and finally inserting it. Proposition 4.16.4. (Getzler) Any BV algebra is an algebra over the homology of the framed little discs operad and vice versa. 4.17. The Pre-Lie Operad Definition 4.17.1. (Gerstenhaber) A pre-Lie algebra is a graded vector space V together with a bilinear operation * that satisfies (x * y) * z — x * (y * z) = (—l)'y"x'((a; * z) * y - x * (z * y))

Proposition 4.17.2. (Gerstenhaber) Define {a»b}:=a*b-

(_l)(H+1Hlfel+1)6 * a

(4.6)

Then (V, { • }) is an odd Lie algebra. In the case of no signs the operad which defines pre-Lie algebras is actually isomorphic to an insertion operad of trees. It is the operad structure on Tr^1 discussed in 4.7. Proposition 4.17.3. (Chapoton-Livernet)7 Any pre-Lie algebra is an algebra over the operad of labelled rooted trees Tr^1 and vice versa.

160 To keep the signs is more tricky, but by using a cell interpretation in terms of the symmetric top-dimensional cells of spineless cacti (CCtop(Cact1))s we are able to identify the operad for graded pre-Lie algebras as an operad of trees see Theorem 10.2.3. Proposition 4.17.4. Any graded pre-Lie algebra is an algebra over the operad of rooted trees Tr'fl with a grading the choice of signs Nat (see 7.7) and vice versa. More precisely any graded pre-Lie algebra is an algebra over the operad (CClop(Cact1)f Remark 4.17.5. We summed up the results of the previous sections in the table 4.17 Operad

Algebras

COM ASSOC Tr>fl {CCl^iCact^f i/* (little discs) Hi, (framed little discs)

commutative algebras associative algebras pre-Lie algebras graded pre-Lie algebras Gerstenhaber algebras BV algebras

Remark 4.17.6. It is interesting to point out, that the pre-Lie structure essentially lives on the chain level, while the Gerstenhaber structure only lives on the homology level. This is the case, since the relation for the associator does hold on the chain level, while the derivation properties involve homotopies.

4.18. Operads of Surfaces with Extra Structure Starting with the operad of topological surfaces with boundary E(n) one can endow the surfaces with extra structures and then take care to glue these extra structures. Essentially this means that one studies moduli spaces of surfaces.

4.19. Operads of Moduli Spaces of Curves Going over to an algebraic point of view, one replaces boundaries by punctures and considers the following moduli spaces.

161 The Deligne-Mumford compactifications M9,n of the moduli spaces of curves of genus g with n marked points form an operad. The glueing is essentially glueing of two curves at two chosen marked points. If one allows self- glueing one obtains a so-called modular operad22. The theory of Gromov-Witten invariants yields the statement that the cohomology of a smooth variety is an algebra over the modular operad H*(MgiTl)The spaces Mo,« form a suboperad (not modular) and the cohomology of a smooth variety as an algebra over H*(M0,n) is usually called quantum cohomology. One can also consider the open moduli spaces M9:n and these also form a modular operad22. The algebras over the suboperad JFf»(M0,n) are called gravity algebras. The relationship between Hf (Mo,n) and i7*(Mo,n) is that they are Koszul dual to each other as quadratic operads18. 4.20. Arc Operads Staying in the topological realm of surfaces with borders, we will be interested in the extra structure of adding arcs to the surface, which can be viewed as a version of hyperbolic field theory34 and the next section. 5. The Arc Operad 5.1. The space There is an operad based on bordered surfaces with arcs projectively weighted by non-negative real numbers34, which is an extension of the bordered surface operad. This operad is called the Arc operad and we will briefly recall its definition34 here. We fix a surface of genus g with r punctures and n + 1 boundary components that are labelled from 0 t o n and call it F = F^n+1. We also fix a window which is a closed proper subset Wi C diF, for each boundary component diF. Definition 5.1.1. An essential arc in F is an embedded path a in F whose endpoints lie among the windows, where we demand that a is not isotopic rel endpoints to a path lying in dF. Two arcs are said to be parallel if there is an isotopy between them which fixes each di — Wi pointwise (and fixes each Wt setwise) for i = 1,2,..., r. An arc family in F is the isotopy class of an unordered collection of disjointly embedded essential arcs in F, no two of which are parallel. Thus, there is a well-defined action of the pure mapping class group on arc families. Where the pure mapping class

162

group PMC = PMC(F) is the group of isotopy classes of all orientationpreserving homeomorphisms of F which fix each di - Wi pointwise (and fix each Wi setwise), for each i = 1,2,..., r. The arc families have a natural partial order which is given by inclusion. This allows to build a simplicial cell complex whose k skeleton is composed of simplices indexed by arc families with k+1 arcs. These are attached to the k-1 skeleton by the face maps given by deleting one arc from the collection. Definition 5.1.2. We define Arc'(F) to be the complex obtained in the above manner and Arc(F) to be the topological space obtained as quotient of Arc'(F) by the action of PMC. We also define DArc(F) := Arc'{F) x K>o. We would like to remark that the points of Arc(F) can and should be thought of as a mapping class group orbit of a projectively weighted (by positive real numbers) arc family on F and points of DArc(F) as mapping class group orbits of weighted (by positive real numbers) arc families. Any point of Arc(F) lies in the interior of some cell of minimal dimension or on a vertex of the complex. If the vertex lies inside a cell, then assigning barycentric coordinates. We can now look at the point as being given by positive real co-ordinates Wj assigned to the arcs which make up the family; we call the Wj weights. If we drop the condition that the sum of the coordinates is one, then we obtain DArc(F) by picking the coordinate on E>o to be given by the sum and the coordinates on Arc(F) to be given by the normalization. Finally, viewing Arc(F) as the quotient of DArc(F) by the action of R>o scaling all coordinates at the same time, we obtain projective weights. If it lies on a vertex the point corresponds to a single arc which we can think of having any non-zero weight. We will represent such a point by choosing a representative family and representative weights. The underlying topological space for the n-th component our operad is an open subset of \Jg s Arc(F°n+1). Definition 5.1.3. An arc family is said to be exhaustive if for each boundary component <%, for i = 1,2,..., r, there is at least one component arc in a with its endpoints in the window Wi. Likewise, a PMC-orbit of arc families is said to be exhaustive if some (that is, any) representing arc family is so. Define the topological spaces Arc3g(n) = {[a] € Arc(Fgn+1) : a is exhaustive}

(5.1)

163

5.2. The Operad Structure alias the Glueing Maps. The definition of the glueing maps is best and most naturally done in the setting of partial measured foliations34. The basic idea however is the following: First we glue two surfaces in the standard operadic fashion 4.5, i.e. boundary 0 of the second surface to the boundary of the first surface. Secondly, we have to give a procedure, how to glue together the bands. To this end we would like to think of the weighted arcs as bands of width given by the weights; this can be done by thickening the arcs into a partially measured foliation - which we view as a collection of bands. Now we arrange the bands in the window, such that in a neighborhood of the window the bands looks is depicted in figure 10 IV. In this way, the bands or the partially measured foliation can equivalently be thought of as a partition on an interval as depicted in figure 11 I. Now if the sum of the weights of the arcs hitting the two boundary components of the two surfaces that are to be glued happen to coincide, we can "splice" the bands according to the largest common refinement of the two partitions; this is depicted in figure 12 I. 5.3. Several Models for Arcs To elucidate the role of the windows, we would like to briefly recall34 several geometric models for the common underlying combinatorics of arc families.

1 1 JR I u v w uvw uvw uvw I II III IV Figure 10. I. Arcs running to a point on the boundary; II. Arcs running to a point at infinity; III. Arcs in a window; IV. Bands in a window

For the first such model, let us choose a distinguished point di £ <%, for i = 1,2,..., r, and consider the space of all complete finite-area metrics on F of constant Gauss curvature —1 (so-called "hyperbolic metrics") so that each d* = di — {di} is totally geodesic (so-called "quasi hyperbolic

164 metrics") on F. To explain this, consider a hyperbolic metric with geodesic boundary on a once-punctured annulus A and the simple geodesic arc a in it asymptotic in both directions to the puncture; the induced metric on a component of A — a gives a model for the quasi hyperbolic structure on Fx = F — {di}\ near df. The first geometric model for an arc family a in F is as a set of disjointly embedded geodesies in Fx, each component of which is asymptotic in both directions to some distinguished point di\ see part II of figure 10. In the homotopy class of each <%, there is a unique geodesic d* C Fx. Excising from F — U{94x}i any component which contains a point of d x , we obtain a hyperbolic structure on the surface F* C Fx with geodesic boundary (where in the special case of an annulus, F* collapses to a circle). Taking aD F*, we find a collection of geodesic arcs connecting boundary components (where in the special case of the annulus, we find two points in the circle). This is our second geometric model for arc families. We may furthermore choose a distinguished point pi £ d* and a regular neighborhood Ui of Pi in d*, for i = l,2,...,r. Provided pt ^ a, we may take Ui sufficiently small that Ui n a = 0, so the arc Vi = d* — Ui forms a natural "window" containing add*. There is then an ambient isotopy of F* which shrinks each window Vj down to a small arc Wi C d*, under which a is transported to a family of (non-geodesic) arcs with endpoints in the windows Wi. In case pi does lie in a, then let us simply move pi a small amount in the direction of the natural orientation (as a boundary component of F*) along d* and perform the same construction; see part III of figure 10. This leads to our final geometric model of arc families, namely, the model we used to define our spaces: arcs in a bounded surface with endpoints in windows. This third model is in the spirit of train tracks and measured foliations44 as we shall see and is most convenient for describing the operadic structure. In this picture there is also then a unique orientation-preserving mapping34 c[Q) : di(a) -» S 1

(5.2)

which maps the (class of the) first point of di(a) in the orientation of the window Wi to 0 £ S1, contracts the parts of the boundary outside the window and not hit by the bands, and maps the boundary points which are part of the bands to S1 according to the normalized partial measure on the bands, which in our rudimentary discussion is just given by the weight

165 and the normalizations means that the total weight is one. If we do not normalize, the map will take us to a circle of radius given by the total weight. 5.4. Pictorial Representations of Arc Families As explained before, there are several ways in which to imagine weighted arc families near the boundary. They are illustrated in figure 10. It is also convenient to view arcs near a boundary component as coalesced into a single wide band by collapsing to a point each interval in the window complementary to the bands; this interval model is illustrated in figure 11, part I. It is also sometimes convenient to further take the image under the maps 5.2 to produce the circle model as is depicted in figure 11, part II.

.

. u v

1

w

w^-—>/

/ V —<J v

s'

N

u \

II Figure 11. I. Bands ending on an interval; II. Bands ending on a circle

5.5. Glueing Weighted Arc Families. Given two weighted arc families (a') in F£ m + 1 and (/3') in Ffc n + 1 so that Hi(di(a')) — lio(do((3')), for some 1 < i < m, we shall next make choices to define a weighted arc family in F*+£ m+n as follows. First of all, let do,d\,... ,dm denote the boundary components of Fg,m+i > l e t 9Q, d[,..., d'n denote the boundary components of F^ n+1, and fix some index 1 < i < m. Each boundary component inherits an orientation in the standard manner from the orientations of the surfaces, and we may choose any orientation-preserving homeomorphisms £ : di —* S1 and r\ : d'o —> S1 each of which maps the initial point of the respective window to the base-point 0 € S1. Glueing together di and d'Q by identifying x € Sl with y € Sl if £(x) = r](y) produces a space X homeomorphic to

166 Fg+htm+n* where the two curves di and d'o are thus identified to a single separating curve in X. There is no natural choice of homeomorphism of X with F*+l m+n, but there are canonical inclusions j : F* m+1 —> X and

k:Fln+1^'x.

We enumerate the boundary components of X in the order do,&i,... ,di-i,d[,d2,.

..d'n,di+i,di+2,--

-dm

and re-index letting dj, for j = 0 , 1 , . . . , m + n — 1, denote the boundary components of X in this order. Likewise, first enumerate the punctures of Fgm+l in order and then those of Ffc n + 1 to determine an enumeration of those of X, if any. Let us choose an orientation-preserving homeomorphism H : X —» F*+lim+n which preserves the labelling of the boundary components as well as those of the punctures, if any. In order to define the required weighted arc family, consider the partial measured foliations Q in Fgm+1 and H in Ffc n + 1 corresponding respectively to (a1) and (/?'). By our assumption that Hi(di(a')) = fJ-o{do(P')), we may produce a corresponding partial measured foliation T in X by identifying the points x £ <%(«') and y £ do{f3') if c\a){x) = cf\y). The resulting partial measured foliation T may have simple closed curve leaves which we must simply discard to produce yet another partial measured foliation T' in X. The leaves of T' thus run between boundary components of X and therefore, as in the previous section, decompose into a collection of bands Bi of some widths Wi, for i = 1,2,.../. Choose a leaf of J-1 in each such band Bi and associate to it the weight Wj given by the width of Bi to determine a weighted arc family (5') in X which is evidently exhaustive. Let (7') = H{5') denote the image in F*^ m+n under H of this weighted arc family. It is a fact that this is well defined also on PMC orbits 34 . Definition 5.5.1. Given [a] € Arcsg(m) and [/?] G Arc^n) and an index 1 < i < m, let us choose respective deprojectivizations (a') and (/3') and write the weights w(a')

=

(uo,u1,...,um), (vo,vi,...,vn).

w{/3') =

Define Po=

^2

Vi,

{6e/3:06ndo^0}

Pi=

^2

Ui,

167 where in each sum the weights are taken with multiplicity, e.g., if a has both endpoints at do, then there are two corresponding terms in poSince both arc families are exhaustive, /?, ^ 0 7^ po, and we may re-scale pow{a') = PiW(0') =

(pouo,poui,...,poum), (piVO,PiVl,...,PiVn),

so that the 0 th entry of piiu(P') agrees with the ith entry of pow(a'). Thus, we may apply the composition of 5.5 to the re-scaled arc families to produce a corresponding weighted arc family (7') in F^fim+n, whose projective class is denoted [7] € Arc^^m + n — 1). We let

M °i \P\ = hi in order to define the composition Oj : Arcsg(m) x Arc^n) —> Arc'^im 5.6. A Pictorial Representation

+ n - 1), for any i = 1,2,..., m. of the Glueing

A graphical representation of the glueing can be found in figure 12, where we present the glueing in three of the different models.

LU

111 W W tVf U S

1

'

SI

11

"I

Figure 12. The glueing: I. in the interval picture, II. in the windows with bands picture and III. in the arcs running to a marked point version.

Definition 5.6.1. For each n > 0, let Arccp(n) = Arc%(n) (where the "cp" stands for compact planar), and furthermore, define Arc(n) of the union over all Arc3g{n), which we give the direct limit topology as g, s —> 00. Theorem 5.6.2. The compositions Oj of Definition 5.5.1 imbue the collection of spaces Arc(n) with the structure of a topological operad under the natural Sn-action on labels on the boundary components. The operad has

168 a unit 1 £ Arc(l) given by the class of an arc in the cylinder meeting both boundary components, and the operad is cyclic for the natural Sn+i-action permuting the labels of the boundary components. Furthermore the spaces Arccp form a cyclic suboperad. 5.7. The Deprojectivized Spaces T>Arc For the following it is convenient to introduce deprojectivized arc families. This amounts to adding a factor R>0 for the overall scale. Let VArc{n) = Arc(n) x K > 0 be the space of weighted arc families; it is clear that VArc(n) is homotopy equivalent to Arc(n). As the definition 5.5.1 of gluing was obtained by lifting to weighted arc families and then projecting back, we can promote the compositions to the level of the spaces T>Arc(n). This endows the spaces T>Arc(n) with a structure of a cyclic operad as well. Moreover, by construction the two operadic structures are compatible. This type of composition can be compared to the composition of loops, where such a rescaling is also inherent. In our case, however, the scaling is performed on both sides which renders the operad cyclic. In this context, the total weight at a given boundary component given by the sum of the individual weights u>t of incident arcs makes sense, and thus the map 5.2 can be naturally viewed as map to a circle of radius ^2t u>t. 5.8. Notation We denote the operad on the collection of spaces Arc(n) by Arc and the operad on the collection of spaces VArc(n) by VArc. By an "Arc algebra", we mean an algebra over the homology operad of Arc. Likewise, Arccp and VArccp are comprised of the spaces Arccp(n) and T>ArcCp{ri) respectively, and an "Arccp algebra" is an algebra over the homology of Arccp 5.9. Suboperads and PROPS There are several natural suboperads for the arc operad, given by imposing certain conditions on the arcs. For example, one may specify a symmetric (n + l)-by-(n + 1) matrix A^ as well as an (n + l)-vector R^ of zeroes and ones over Z/2Z and consider the subspace of Arc(Fg n + 1 ) where arcs are allowed to run between boundary components i and j if and only if A\™ ^ 0 and are required to

169 meet the boundary component k if and only if R£ ¥" 0- F° r instance, the case of interest for cacti corresponds to A^ the matrix whose entries are all one, and R^ the vector whose entries are also all one. For a class of examples of PROPS (the generalization of operads with arbitrary inputs and outputs40), consider a partition of {0,1,... n} = 1^ U O<"\ into "inputs" and "outputs", where A\f = 1 if and only if {i, j}nl^ and {i, j} (~1 O^ are each singletons, and R^ is the vector whose entries are all one. This are the PROPS or di-operads which are of interest in string topology. In words these are the arc families in which the boundaries can be and are partitioned into incoming and outgoing in such a way that there are only arcs running between incoming and outgoing boundaries. Definition 5.9.1. The trees suboperad is defined for arc families in surfaces with g = s = 0 in the notation explained above by the allowed incidence matrix A^n\ whose non-zero entries are aoi = 1 — a^, for i = l , . . . , n , and required incidence relations R^n\ whose entries are all equal to one. This is a suboperad of Arccp, and it has a representation in terms of labelled trees29. Dropping the requirement that g = s = 0, we obtain a suboperad of Arc called the rooted graphs or Chinese trees suboperad. 5.10. Linearity

Condition

We say that an element of the (Chinese) trees suboperad satisfies the Linearity Condition if the linear orders match, i.e., the bands hitting each boundary component in their linear order are a subchain of all the bands in their linear order derived from the 0-th boundary. It is straightforward to check that this condition is stable under composition. We call the suboperad of elements satisfying the Linearity Condition of the (Chinese) trees operad the (Chinese) linear trees operad. The following proposition34 clarifies the role of this condition. Proposition 5.10.1. The suboperad generated by (Chinese) linear trees and Arccp(l) inside Arc coincides with (cyclic Chinese) trees, where cyclic Chinese trees are those arc families in Chinese trees in which the cyclic orders match.

170

5.11. String Interpretation One way that one can view the Arc operad is as tracing n closed strings as they move, split and recombine to form one loop. If one traces fixed base points of the strings and keeps track of the length of the strings that the closed strings may break into, one arrives exactly at the arc picture. Here one arc of weight w or better band of width w depicts the movement of a piece of string of length w. Starting with n strings that may move, break and recombine to become one string, one obtains the trees suboperad. If the strings have in this way swept over a genus zero surface (with no punctures) this information is enough to recover the surface, since one only needs to know the number of boundary components. If there is nontrivial topology, however, this information may not be enough, but one can consider all graphs of the traces of the base point on a given fixed topological surface. This is what the arc operad does, cf. Figure 13. If one wishes to see n strings move and recombine into m strings one arrives at the notion of the props discussed above. This corresponds to having a kind of Morse function for the surface which gives the state of the strings at a given time. In this view, the singular level sets are the important ingredients and a careful analysis of this picture leads to the Cacti and the considerations relevant to string topology8. On the other hand, from the point of view of closed string field theory51 there is actually no big distinction of incoming and outgoing circles in the sense of our prop definitions, as strings may also annihilate. Thus even in genus zero, we do not need to restrict to the trees in the arc operad, but see all of Arccp and of course in general all of Arc. The depiction of a pair of pants according to closed string field theory is not a figure eight as before, but a theta shape, as indicated in figure 13 and figure 28. This type of shape has recently also played a role in the geometry of the so-called stringor bundles46. 5.12. Relation to Moduli Spaces In this section, we would like to very briefly digress on the relation of the spaces Arc(F) and the Arc operad to moduli spaces, due to Penner 42 - 43 . Assume that r ^ 0 (and allow both cases s = 0 or s > 0). Enumerate the (smooth) boundary components of F as di, where i = l , . . . , r and set d = U{9j}i. Let Hyp(F) be the space of all hyperbolic metrics with

171

i

ii

Figure 13. Movements of strings combining I and breaking, annihilating and combining II

geodesic boundary on F. Define the moduli space to be M = M(F) = [HyP(F)(([di)}/~ 1

where ~ is the equivalence relation generated by push-forward of metric under orientation- preserving diffeomorphism

/.(r,(6)i) = (/.(r),/(&)D where F H / , (F) is the usual push-forward of metric on Hyp(F). Now let M(.F)/M>o denote the quotient of the moduli space of the bordered surface F by the action of M>o by homothety on the tuple of hyperbolic lengths of the geodesic boundary components. Denote by Arc#(g, r)s the space of quasi-filling arc families, which means that complementary regions are either a polygon or a once punctured polygon. Notice that for g = s = r = 0 Arc# (0,0)° = Arccp and in general Arc#(g,r)s C Arcsgr. Theorem 5.12.1. 42 For any bordered surface F ^ F§2, Arc#(F) is proper homotopy equivalent to M(F)/R>o. Furthermore, each of M(F)/R>0 and Arc#(F) admit natural (5 1 ) r actions (moving distinguished points in the boundaries for the former and "twisting" arc families around the boundary components for the latter); the proper homotopy equivalence in the Theorem above is in fact a map of (51)r-spaces. There is also an interpretation in terms of ribbon graphs and Strebel differentials, see §6.15. It is astonishing that in the case of genus zero with no punctures, there is another relationship on the algebraic level. It was shown by Getzler17

172 that the homology operad of the spaces M0,n yields the notion of a gravity or LQO algebra. The corresponding cohomology classes are also present in our Arc picture through the BV structure discussed below, since this structure allows to define the higher brackets17 as well. Actually there are good candidates for arc families leading to these brackets which we will discuss elsewhere35. In a sense, what the Arc operad does is to replace the conformal field theory given by the operads H*(M9in) by its hyperbolic counterpart. The benefit, as always when dealing with hyperbolic aspects of moduli theory, the formulas become discrete and all spaces tend to have a PL-structure which makes everything very manageable. The BV structure below legitimizes this point of view, since it is what one would expect from the non-compactified moduli. These are in turn related to the operad constituted by the Deligne-Mumford compactified spaces by Koszul duality for quadratic operads. It is also interesting that the cell decomposition of trees we present in §7.6.5 is also related to Strebel differentials via the indexing set of trees36. 5.13. Arc Families and their Induced Operations. The points in Arccp(l) are parameterized by the circle, which is identified with [0,1], where 0 is identified to 1. To describe a parameterized family of weighted arcs, we shall specify weights that depend upon the parameter s € [0,1]. Thus, by taking s € [0,1] figure 14 describes a cycle 5 € Ci(l) that spans

I

Hi{Arccp(l)).

II

Figure 14. I. The identity and II. the arc family 6 yielding the BV operator As stated above, there is an operation associated to the family S. For instance, if F\ is any arc family F\ : ki —> Arccp, 5Fi is the family parameterized by / x k\ —> Arccp with the map given by the picture by inserting

173 F\ into the position 1. By definition, A = -S e Ci(l). In C*(2) we have the basic families depicted in figure 15 which in turn yield operations on C*.

The dot product

The s l a r

5 (a>t>)

Figure 15. The binary operations

To fix the signs, we fix the parameterizations we will use for the glued families as follows: say the families F\, F2 are parameterized by F\ : ki —> Arccp and F2 : fc2 —» Arccp and / = [0,1]. Then i*\ • F2 is the family parameterized by k\ x k2 —> ^4rccp as defined by figure 15 (i.e., the arc family F\ inserted in boundary a and the arc family F2 inserted in boundary b). Interchanging labels 1 and 2 and using * as a chain homotopy as in figure 16 yields the commutativity of • up to chain homotopy d(Fi * F2) = {-1)\F^F^F2 •Fl~F1-F2

(5.3)

Notice that the product • is associative up to chain homotopy. Likewise F\ *F2 is defined to be the operation given by the second family of figure 15 with s e / = [0,1] parameterized over k\ x / xk2 —> Arccp. By interchanging the labels, we can produce a cycle {Fx,F2} as shown in figure 16 where now the whole family is parameterized by k\ x / x k2 —> Arccp. {FUF2}

:= Fi *F2 - (-i)(l*l+i)
Remark 5.13.1. We have defined the following elements in C*: 5 and A = -6 in Ci(l);

174 a*b

(lal+i)(lbl+l) -f-1) b*a Figure 16. The definition of the Gerstenhaber bracket

• in Co (2), which is commutative and associative up to a boundary. * and {-, - } in Ci(2) with d(*) = T • — and {-, —} = * — T*. Note that 6, • and {—, —} are cycles, whereas * is not. 5.14. The BV Operator The operation corresponding to the arc family 8 is easily seen to square to zero in homology. It is therefore a differential and a natural candidate for a derivation or a higher order differential operator. It is easily checked that it is not a derivation, but it is a B V operator. Proposition 5.14.1. The operator A satisfies the relation of a BV operator up to chain homotopy. A2 ~ 0 A{abc) ~ A(ab)c+{-l)MaA{bc) + (-l)|sQllfcl&A(ac) - A(a)6c (5.4) -(-l) | Q laA(6)c- (-l)|al+lbla6A(c)

175 Thus, any Arc algebra and any Arccp algebra is a BV algebra.

Lemma 5.14.2. S(a, b, c) ~ (-l) (|a|+1)|fc| M(a, c) + 6(a, b)c - 5{a)bc

(5.5)

Proof. The proof is contained in figure 17. Let a : ka —> Arccp, b : kb —> Arccp and c : kc —> Arccp, be arc families then the two parameter family filling the square is parameterized over / x / x ka x kb x kc. This family gives us the desired chain homotopy. • Given arc families a : ka —» Arccp, b : kb —> Arccp and c : kc —> ^4rccp, we define the two parameter family defined by the figure 18 where the families in the rectangles are the depicted two parameter families parameterized over Ixlxkaxkb>
176

/sj

1-s

\

s

[ ®

1

(lal+l)lbl

-(-1) b5(a)c =

5(a,b)c

^

-6(a)bc

1

- ^

(lal+l)lbl

-(-1) b5a,c) ^

8 fa.b.c)

Figure 17. The basic chain homotopy responsible for BV

The glueing * o2 * in arc families is simpler and yields the glueing depicted in figure 22 to which we apply a normalizing homotopy — by changing the weights on the bands emanating from boundary 1 from the pair (2s, 2(1 - s)) to (s, 1 — s) using pointwise the homotopy ( ^ 2 s , ^ ( 1 - s)) for te [0,1]: Combining figures 20 and 22 while keeping in mind the parameteriza-

177

<

t

I

Figure 18. The homotopy BV equation

Figure 19. The first iterated glueing of *

tions we can read off the pre-Lie relation: Fi * (F2 * F3) - (Fi * F2) * F3 ~

(_l)(l^l+i)(l^l+i) (j p 2 * (Fl * p3) _ (i r 2 „ F l ) * F3)

(5.6)

178

f

V 2s=t+l

l

/

/

._,

s-1

vn

VI 2s>t+l

Figure 20. The glued family after normalization

t

I

n

s=0

2s<

I

12;

I—t

t!

2s

1

;yi^)

m 2s=l

M

t

2s

1

1-t

2<<-s)

iv

v

vi

vn

l<2s
2s=t+l

2sw+l

s=l

t

2s

!l

;

1—t

2u-s)

t

2s

1

;

I-I

20-*} 12&

I

1 ! M

;

; 2a|s)

I

1

;

2

M

;

Figure 21. The different cases of glueing the bands

which shows that the associator is symmetric in the first two variables and thus following Gerstenhaber [G] we obtain: Corollary 5.15.1. { F i , ^ } satisfies the odd Jacobi identity.

179

Figure 22. The other iteration of *

6. Species of Cacti and their Relations to other Operads 6.1. Configurations of Loops and their Graphs There are several species of cacti29 to which we refer the reader for details. We briefly recall the main definitions here. In words, the cacti operad which was introduced by Voronov49 has as its n-th component defined in connected, planar tree-like configurations of parameterized loops (of possibly different circumference), together with a marked point on the configuration. The spineless cacti29 are the suboperad where the zero of the parametrization corresponds to the lowest intersection point. There are also normalized versions of these configurations29 where the circumference of each loop is fixed to be one. To give a more precise definition we need the following definitions. We denote the standard circle of radius r by S^ := {(x,y) C K2 : x2 + y2 = r2}. Definition 6.1.1. A configuration of n parameterized loops is a collection (Ji,... ,ln) of n orientation preserving continuous injections —called loops— U : S^ —> K2 considered up to isotopy. Where the isotopy is required to fix the incidence conditions, that is if h\ : S^ x I are the isotopies and hl0(p) = h(p) = lj(q) = h{{q) then for all t: h\(p) = h{(q) and viceversa if hi(p) = k{p) ^ l^q) = hl{q) then for all t: h\(p) ^ h{{q). A pointed configuration is a configuration together with a marked component k and marked point on this component * € £*.. Definition 6.1.2. For a configuration of n parameterized loops, with only finitely many intersection points, we can define a bipartite b/w graph as follows: There is one white vertex for each loop and one black vertex for each intersection point. We join a white vertex and a black vertex by an edge, if the intersection point corresponding to the black vertex lies on the loop corresponding to the white vertex. We call this black and white graph

180

the graph of the configuration. We also endow each vertex with the cyclic order coming from the orientation in the plane. For a pointed configuration, we include one more black vertex for the marked point and a second black vertex, the root and draw edges from the vertex for the marked point to each white vertex of components the marked point lies on. We also endow the vertex for the marked point by the linear order given by the cyclic order induced from the plane and the choice of the smallest element being the root edge followed by the edge for the marked component. 6.2. Cacti and Spineless Cacti Definition 6.2.1. The cacti operad which was introduced by Voronov49 has as its n-th component pointed configurations of n parameterized loops whose image is connected and whose graph is a tree. The space Cact(n) is endowed with the action of §„ by permuting the labels. Definition 6.2.2. Notice that the tree (graph) of a cactus is actually a bi-partite planar planted tree without tails which thus has a linear order on all of the vertices. We choose to plant the tree to reflect the linear order at the root. We will allow ourselves to talk about the image of a cactus in M2 by picking a representative (li,-..,ln) and considering Ui'('^'ri) keeping in mind that this is only defined up to isotopy. The loops, and also the inside of these loops, are sometimes called lobes — again the above remark applies. whose loops have radii r< Definition 6.2.3. Given a cactus (h,...,ln) there is a surjective orientation preserving map from S ^ y . r . >-» Ui K^U^ whose only multiple points are the intersection points of the loop. This map is denned as follows. Start at the marked point (the global zero) and go around the marked loop counterclockwise; if a double point is hit continue on the next loop to the right (i.e. the next in the cyclic order) and continue in this manner until one returns to the marked point. We will call this map the "outside circle" and sometimes refer to the marked point as the "global zero", since it is the image of 0 £ S^. 6.3. Glueing for Cacti We define the following operations

181

Oj : Cacti(n) x Cacti(m) —> Cacti(n + m— 1)

(6-1)

by the following procedure: given two cacti without spines we reparameterize the outside circle of the second cactus to have length r, which is the length of the i-th circle of the first cactus. Then glue in the second cactus by identifying the outside circle of the second cactus with the i-th circle of the first cactus. Proposition 6.3.1. The glueings above together with the § n action on Cacti(n) by permuting the labels imbue the collection Cacti of the Cacti(n) with the structure of an operad. Endowing the spaces Cacti(n) with the topology as subspaces of VArc, see below, turns the operad Cacti into an operad of topological spaces. Definition 6.3.2. The spineless variety of cacti is obtained by postulating that the local zeros defined by the parameterizations of the loops coincide with the first intersection point of the perimeter with a loop (sometimes called a "lobe") of the cactus. Here the first intersection point is the point given by the black vertex which lies on the outgoing edge of the white vertex representing the parameterized loop under consideration. This suboperad inherits the permutation action of S n on the labels. Proposition 6.3.3. The symmetric group actions permuting the label together with the restriction of the glueing for Cacti OJ : Cact(n) x Cact(m) —> Cact(n + m - 1)

(6.2)

makes Cact into a topological operad which is a suboperad of Cacti.

6.4. The Chord Diagram and Planar Planted Tree of a Cactus There is another representation of a cactus. If one regards the outside loop, then this can be viewed as a collection of points on an S1 with an identification of these points, plus a marked point corresponding to the global zero. We can represent this identification scheme by drawing one chord for each two points being identified as the beginning and end of a circle. This chord diagram comes equipped with a decoration of its arcs by their length or alternatively can be thought of as embedded in 1R2. To obtain a cactus from such a diagram, one simply has to collapse the chords. This type of chord diagram of the outside circle is explicit in the embedding

182

of cacti into the arc operad where the perimeter is indeed the outside circle, as explained below. The local zeros will then be extra points on the outside circle, which coincide with the beginnings of the chords in the spineless case. There is a special case for the chord diagram which is given if there is a closed cycle of chords. This happens only if three or more lobes intersect at the global zero. Here one can delete the first chord, if so desired. It does play a role however in the completed chord diagram29 which parameterizes the fiber of the map forgetting the n-th lobe29. This kind of representation is reminiscent of Kontsevich's formalism of chord diagrams2 as well as the shuffle algebras and diagrams of Goncharov23. We wish to point out that although the multiplication is similar to Kontsevich's and also could be interpreted as cutting the circle at the global zero resp. the local zero, it is not quite the same. However, the exact relationship and the co-product deserve further study. Lastly, we can recover a planar tree as the dual tree of the chord diagram. This is the dual tree on the surface with boundary the outside circle, i.e. one vertex for each chamber inside the circle and an edge for chambers separated by a chord. This planar tree is the tree obtained from the tree describing the cactus by contracting all black edges. The special case corresponds again to the case where three or more lobes intersect at the global zero. If one chooses to keep the whole cycle of chords the dual tree the rooted tree which is obtained by contracting the root edge of the planted tree. If one also removes the first chord in the cycle, then the tree is the rooted tree which is obtained by contracting the root edge of the planted tree and the next edge which appears in the outside path. A representation of a cactus without spines in all possible ways including its image in the Arc operad can be found in figure 23. 6.5. Normalized Cacti and Normalized Spineless Cacti Definition 6.5.1. The spaces Cactil(n) are the subspaces of Cacti(n) with the restriction that all the radii of the lobes are fixed to one. The elements are called normalized spineless cacti. 6.6. Gluing for Normalized Cacti We define the following operations o4 : Cacti1 (n) x Cacti1 (m) -> Cacti1 (n + m - 1)

(6.3)

183

R4

R3

5

I

( 5 \/R2-n Wiy

R4

2

, t

5\

i

\

Rl-s-V"^/ s II

R3 (5's *! 4

IV

s

YR,' /

T2

RJ

I

Rl-s-v

R3

'

III

(^<®\V\Q X^J_R4AX V

Figure 23. I. A cactus without spines; II Its planted planar bi-partite tree; III Its dual tree; IV Its chord diagram; V Its image in Arc

by the following procedure: given two normalized cacti we reparameterize the i-th component circle of the first cactus to have length m and glue in the second cactus by identifying the outside circle of the second cactus with the i-th circle of the first cactus. Here we match the global zero of the second cactus to the local zero of the i-th lobe. These glueings do not endow the normalized spineless cacti with the structure of an operad, but with the slightly weaker structure of a quasioperad defined in Definition 4.10.3. Theorem 6.6.1. 29 The glueings above together with the S n action on Cacti1 {n) by permuting the labels imbue the collection of Cacti1 (n) with the structure of a quasi-operad.

184 Furthermore the quasi-operad of normalized cacti is homotopy associative. It is homotopic as a quasi-operad to the operad of cacti and is quasiisomorphic to the operad of cacti. I. e. the homology operads of normalized cacti and cacti are isomorphic. Definition 6.6.2. The spaces Cact1{n) are the subspaces of Cact(n) with the restriction that all the radii of the lobes are fixed to one. The elements are called normalized spineless cacti. Theorem 6.6.3. 2 9 The symmetric group actions permuting the label together with the restriction of the glueing for Cacti1 OJ : Cactl{n) x Cactl(m) -> Cactl{n + m - 1) 1

(6.4) 1

makes Cact into a quasi-operad which is a quasi-suboperad of Cacti . Furthermore the quasi-operad of normalized spineless cacti is homotopy associative. It is homotopic as a quasi-operad to the operad of spineless cacti and is quasi-isomorphic to the operad of spineless cacti. I. e. the homology operads of normalized cacti and cacti are isomorphic. 6.7. Scaling of a Cactus and Protective Cacti Cacti and spineless cacti both come with a universal scaling operation of ]R>o which simultaneously scales all radii by the same factor A £ K>o. This action is a free action and the glueing descends to the quotient by this action. We sometimes call these operads projective cacti or spineless projective cacti. 6.8. Left, Right and Symmetric Cacti Operads For the glueing above one has three basic possibilities to scale in order to make the size of the outer loop of the cactus that is to be inserted match the size of the lobe into which the insertion should be made. (1) Scale down the cactus which is to be inserted. This is the original version - we call it the right scaling version. (2) Scale up the cactus into which will be inserted. We call it the left scaling version. (3) Scale both cacti. The one which is to be inserted by the size of the lobe into which it will be inserted and the cactus into which the insertion is going to be taking place by the size of the outer loop of the cactus which will be inserted. We call this it the symmetric scaling version.

185

All of these versions are of course homotopy equivalent and in the quotient operad of Cacti by overall scalings, the projective cacti Cacti/W>o they all descend to the same glueing. The advantages of the different versions are that version (1) is the original one and inspired by the rescaling of loops, i.e. the size of the outer loop of the first cactus is constant. Version (2) has the advantage that cacti whose lobes have integer sizes are a suboperad. We will use this later on. And version (3), the symmetric version, is the one we also used in VArc and as shown below, in this version there is an embedding of the cacti operad into the cyclic operad VArc. 6.9. Cacti as a Suboperad of T>Arc In the following sections we will show how to realize our species of cacti naturally as suboperads of VArc. This has the advantage of making their topology transparent. 6.10. Framing of a Cactus We will give a map of cacti into Arc called a framing. First notice that a cactus can be decomposed by the initial point and the intersection points and the local zeros into a sequence of arcs following the natural orientation given by the data. These arcs are labelled by their lengths as parts of UiS^). To frame a given cactus, draw a pointed circle around it and run an arc from each arc of the cactus to the outside circle respecting the linear order given by the outside loop, i.e. starting with the initial arc of the cactus as the first arc emanating from the outside circle in its orientation. Label each such arc by the parameter associated to the arc of the cactus. We can think of attaching wide bands to the arc of the cactus. The widths of the bands are just the lengths of the arcs to which they are attached. Using these bands we identify the outside circle with the circumference of the cactus. Notice that this "outside" circle appears in the glueing formalism for cacti. The marked points on the inside boundaries correspond to the local zeros of the inside circles viz. lobes of the cactus. Two examples of this procedure are provided in figure 24. Remark 6.10.1. If one frames a spineless cactus, then the image is in the linear trees.

186

Figure 24. Framings of cacti

6.11. The Loop of an Arc Family

Given a surface with arcs we can forget some of the structure and in this way either produce a collection of loops or one loop which is given by using the arcs as an equivalence relation. 6.12. The Boundary Circles

Given an exhaustive weighted arc family (a) in the surface F, we can consider the measure-preserving maps cf > : di{a) - S^

(6-5)

where 5^ is a circle of radius r and m, = n%(di{a)) is the total weight of the arc family at the i-th boundary. Combining these maps, we obtain

c:d{a)-^Y[_Slmi.

(6.6)

i

Choosing a measure on OF as in §1 to identify d(a) with OF, we finally obtain a map

circ-.dF^Hs1^ i

(6.7)

187 Notice that the image of the initial points of the bands give well-defined base-points 0 G S^. for each i. 6.13. The Equivalence Relations Induced by Arcs On the set d(a) there is a natural reflexive and symmetric relation given by p ~/ 0 ; q if p and q are on the same leaf of the partial measured foliation. Definition 6.13.1. Let ~ be the equivalence relation on \Ji S^. generated by ~/ 0 (- In other words p ~ q if there are leaves lj, for j = 1,... m, so that p £ c(d(h)), q G c(5(U)andc(^))nc(a(Zj+1))^0. Remark 6.13.2. It is clear that neither the image of circ —which will denote by circ((a))— as a collection of parameterized circles nor the relation ~ depends upon the choice of measure on dF. Definition 6.13.3. Given a deprojectivized arc family (a) G VArc, we define Loop((a)) = circ((a))/ ~ and denote the projection map it : circ((a)) —> Loop((a)).

Furthermore, we define two maps taking values in the monoidal category of pointed spaces: n

int((a)) = LJ(7r(c|a)(ai(a)),7r(*i))

(6.8)

ext((a)) = (7r(4Q)(ao(a)),7r(*o)) (6.9) and call them the internal and external loops of (a) in Loop((a)). We denote the space with induced topology given by the collection of images Loop((a)) of all (a) G VArc(n) by Loop{n). Notice that there are n + 1 marked points on Loop((a)) for (a) G VArc(n). Examples of loops of an arc family are depicted in figures 25-27. In figure 27 I the image of the boundary 1 runs along the outside circle and then around the inside circle. The same holds for the boundary 3 in figure 27 II. In both 27 I and II, the outside circle and its base-point are in bold. Remark 6.13.4. There are two types of intersection points for pairs of loops. The first are those coming from the interiors of the bands; these points are double points and occur along entire intervals. The second type of multiple point arises from the boundaries of the bands via the transitive closure; they can have any multiplicity, but are isolated.

188

Figure 25. An arc family whose loop is a cactus

Figure 26. An arc family whose loop is a cactus without spines

6.14. From Loops to Area 6.14. From Loops to Arcs are family satisfies g = s = 0, then its Loop If the underlying surface of an arc family satisfies g — s = 0, then its Loop together with the parameterizations uniquely determines the arc family. In other words, the map frame is a section of Loop. Definition 6.14.1. A configuration of circles is the image of a surjection P '• Hi 4 ; —> i of metric spaces such that each point of L lies in the image of at least two components and the intersections of the images of more than two components are isolated. Let Config(n) be the space of all such configurations of n + 1 circles with the natural topology. We call a configuration of circles planar, if L can be embedded in the plane with the

189

\|

a

a

/ I

all • are to be identified II Figure 27. Loops of arc families not yielding cacti: I. genus 0 case; II. genus 2 with one puncture.

natural orientation for all images S} : i ^ 0 coinciding with the induced orientation and the opposite orientation for SQ . We call the space of planar configurations of n + 1 circles Configp(n). Proposition 6.14.2. The map Loop : VArc(n) —> Configp(n) is surjective. Definition-Proposition 6.14.3. The deprojectivized arc families such that n \c(d0)(a)) = Loop((a)) constitute a suboperad of Arc. We call this suboperad Coop. Proposition 6.14.4. // (a) € Coop then Loop((a)) is a cactus. Furthermore, the operad Coop is identical to the operad of Chinese trees. Using the symmetric scaling version of cacti (see 6.8), we obtain: Theorem 6.14.5. The framing of a cactus is a section of Loop and is thus an embedding. This embedding identifies (normalized and/or spine-

190 less) cacti as (normalized and/or linear) trees. Loop

» VArc

(spineless) cacti

"I

(spineless) cacti/M>o

frame

rame )

1-

j^rc

wLoop

where nLoop is defined by choosing any lift. By inspection of the diagrams for the Gerstenhaber relations we see that up to homotopy all are denned in the image of Cact and the BV structure can be defined in the image of Cacti. Thus we have: Corollary 6.14.6. 29 . The chains of Cact carry the structure of a Gerstenhaber algebra up to homotopy. The chains of Cacti carry the structure of a GBV algebra up to homotopy. Remark 6.14.7. Actually, up to homotopy all the structures can be obtained using normalized cacti, see below §7. 6.15. Configurations, Loops and Ribbon Graphs Although we did not use the notation of ribbon graphs, it is easy to see that our configurations are essentially ribbon graphs with marked points on some cycles. For this discussion it is easier to restrict to Arc# and s = 0 (no punctures). In this case the graph we obtain from loop is the dual graph on the surface, which is a ribbon graph. Also on each cycle there is a marked point and a parametrization. In this description it is also easy to see that for s = 0 Arc# is homotopy equivalent to the decorated moduli space by using Strebel differentials. 6.16. Comments on an Action on Loop Spaces Given a manifold M we can consider its loop space LM. Using the configuration we have maps Arc{n)xLMn L°^id

Config{n)xLMn J- LConfia^M

A LM (6.10)

191

where LCon^%9^n^M are continuous maps of the images L of the configurations into M, i.e., such a map takes a configuration p : ^ S ^ . —> L and produces a continuous / : L —> M; the maps i,e are given by *(/) = (P : II, S ^ - £, / ( P ( S J J . • • •, /(P(5^))) and e(/) = / ( p ( S j J ) . One would like apply a Pontrjagin-Thom construction10'49 so that the maps i and e would in turn induce maps on the level of homology "Ht{Arc(n)) ® H*(LMn) ~ ff,(D.4rc(n)) ® H*(LMn) H*{Config{n)) ® Ht{LMn) -^ Ht{LConfi^n)M)

Lo

^'

% H,(LM)n

(6.11)

where r is the "Umkehr" map. If we restrict ourselves to this subspace for which the validity of the argument above has been established8'49'10'6 we obtain Proposition 6.16.1. The homology of the loop space of a compact manifold is an algebra over the suboperad of quasi-filling Chinese trees. 6.17. Remarks (1) This also holds for the appropriate PROP or di-operad setting, in which the arcs only run from distinguished inputs to outputs. (2) It is clear that one desideratum is the extension of this result to all of Arc#. (3) The first example of an operation of composing loops which are not cacti would be given by the Loop of the pair of pants with three arcs as depicted in Figure 28. This kind of composition first appeared in the considerations of closed string field theory. (4) If the image of Loop is not connected, then the information is partially lost. This can be refined however by using a prop version of our operad. (5) Factoring the operation of Arc through Loop has the effect that the internal topological structure is forgotten; thus, the torus with two boundary components has the same effect as the cylinder, for instance. This amounts to a certain stabilization. 7. Little Discs, Spineless Cacti and the Cellular Chains of Normalized Spineless Cacti One motivation for studying spineless cacti is that they give a well adapted chain model for the little discs operad. In fact, we will show below how the

192

i

°

Figure 28. I. The Loop of a symmetric pair of pants; II. A closed string field theory picture of a pair of pants

chains of spineless cacti operate naturally on the Hochschild complex of an associative algebra using the chain decomposition of this section. 7.1. Cacti as Semi-Direct Products of Normalized Cacti In this paragraph we will make the relationship between cacti and normalized cacti explicit using the notion of semi-direct product of quasioperads29. Since we will be dealing with one explicit example of this structure only, where all the relevant operations are explicitly defined29, we will not review the general constructions here. We include this section to show that the normalized cacti are indeed associative up to homotopy. We will also use this description of normalized cacti to obtain the operadic structure on its cell decomposition in §7 7.2. The Scaling Operad We define the scaling operad 7£>o to be given by the spaces TZ>o(n) := M™0 with the permutation action by §„ and the following products in,

• • -,rn) Oj (r'1,...,r'm)

= (n,..

. r < _ i , ^ r i > • • • > •^r'm,ri+1,.

..rn)

where R = ]T}fcLi r'k. It is straightforward to check that this indeed defines an operad. 7.3. The Perturbed Compositions We define the perturbed compositions of>0 : Cacti1 (n) x n>Q(m) x Cacti1 (m) -> Cactix(n + m - 1)

(7.1)

via the following procedure: Given (c,?,^) we first scale c' according to f1, i.e. scale the j - t h lobe of c' by the j-th entry r^ of f for all lobes. Then

193 we scale the i-th lobe of the cactus c by R = £) • r, and glue in the scaled cactus. Finally we scale all the lobes of the composed cactus back to one. We also use the analogous perturbed compositions for Cact1. 7.4. The Perturbed Multiplications in Terms of an Action We can also describe, slightly more technically, the above compositions in the following form. Fix an element r := ( n , . . . ,r n ) G M>0 a n d set R = J2i ri aQ d a normalized cactus c with n lobes. Denote by f(c) the cactus where each lobe has been scaled according to f, i.e. the j'-th lobe by the j - t h entry of r. Now consider the chord diagram of the cactus f(c). It defines an action on S1 via p . b —> bR

—> bn —> 5

(7.2)

Where cont? acts on S}^ in the following way. Identify the pointed S^ with the pointed outside circle of the chord diagram of f(c). Now contract the arcs belonging to the i-th lobe homogeneously with a scaling factor ^-. Using this map on the i-lobe of a normalized (spineless) cactus which we think of as an JS*1 with base point given by the local zero together with marked points, where the marked points are the intersection points, we obtain maps Pi : Cactl(n) x U>0(m) x Cactl{m) -> Cactl(ri) Pi : Cacti1(n) x U>0{m) x Cacti1(m) -> Cacti1 {n)

(7.3)

What this action effectively does is move the lobes and if applicable the root of the cactus c which are attached to the i-th lobe according to the cactus r(c') in a manner that depends continuously on r*and c'. With this action we can write the perturbed multiplication as of>0 : Cacti1 (n) x 1Z>0(m) x Cacti1 (m) idxidxA Cactii(nj

x

fi>0{m) x Cacti1 {m) x Cacti1(m)

p

l^Xd Cacti1 (n) x Cacti1 (m) - ^ Cacti1 (n + m - 1) (7.4)

Theorem 7.4.1. The operad of spineless cacti is isomorphic to the operad given by the semi-direct product of their normalized version with the scaling operad. The latter is homotopic through quasi-operad maps to the direct product as quasi-operads. The same statements hold true for cacti.

194 Cact = TZ>0 K Cact1 ~ Cact1 x TZ>0 Cacti ^ Tl>0 K Cacti1 ~ Cacti1 x 7 l > 0

(7.5)

as operads where the operadic compositions are given by

(f, c) Oi (?, c') = (f Oi P, C of d)

(7.6)

Prom this description we obtain several useful corollaries. Corollary 7.4.2. The quasi-operads of normalized cacti and normalized spineless cacti are homotopy associative and thus their homology quasioperads are actually operads. Corollary 7.4.3. The quasi-operads of normalized cacti and normalized spineless cacti are weakly homotopy equivalent to cacti respectively spineless cacti. Furthermore the quasi-operads of normalized cacti and normalized spineless cacti are homotopy equivalent to cacti respectively spineless cacti as quasi-operads. And lastly: Corollary 7.4.4. Normalized cacti and normalized spineless cacti are quasi-isomorphic to cacti respectively spineless cacti. I.e. there homology operads are isomorphic.

7.5. Cact(i) and the (Framed) Little Discs Operad We would like to collect the following facts. Theorem 7.5.1. 29 The operad Cact is (weakly) homotopy equivalent to the little discs operad. We proved this fact 29 by using the recognition principle of Fiedorowicz15. The yloo structure is given by the so-called spineless corolla cacti, whose defining property is that all base points coincide. They correspond to arc families of genus zero with no punctures and exactly one arc from boundary i to boundary 0. The braid structure is shown to hold by using the diagram for the associator. Finally the contractibility of the universal cover follows from the fact that by contracting the n + 1-st lobe of a cactus with n+1 lobes Cact(n-\-1) is homotopy equivalent to the universal

195 fibration over spineless cacti with n lobes whose fiber over a cactus is the image of that cactus. Theorem 7.5.2. 49 ' 29 The operad Cacti is (weakly) homotopy equivalent to the framed little discs operad. We also introduced the notion of semi-direct and bi-crossed products for quasi-operads29 which are the suitable generalization of the same notions for groups. With this notion the relationship between cacti and spineless cacti can be formulated precisely. Theorem 7.5.3. 29 The operad of cacti is the bi-crossed product of the operad of spineless cacti with the operad S1 based on S1 and furthermore this bi-crossed product is homotopic to the semi-direct product of the operad of cacti without spines with the circle group S1 which is homotopy equivalent as quasi-operads to the semi-direct product. Cacti S* Cact tx S1 ~ Cact x S1

(7.7)

This fact that should be compared the fact Theorem 7.5.4. 47 The framed little discs operad is the semi-direct product of the little discs with S1. For any monoid, there is a notion of an associated operad47'29. In this case the semi-direct product of quasi-operads29 actually yields and operad47'29. Remark 7.5.5. This theorem47 together with Theorem 7.5.3 and Theorem 7.5.1 imply Voronov's Theorem. 7.6. A Cell Decomposition for Spineless Cacti Recall that the spaces Cactl{n) are the subspaces of Cact(n) with the restriction that all the radii of the lobes are fixed to one. This space inherits the obvious action by §„ of permuting the labels. Definition 7.6.1. The topological type of a spineless normalized cactus in Cact1 (n) is denned to be the tree r e T^'nt (n) which is its b/w graph together with the labelling induced from the labels of the cactus and the linear order induced on the edges, by the embedding into the plane and the position of the root.

196

Definition 7.6.2. We define Tbppp'nt(n)k to be the elements of T^'nt{n) with \EW\ = k.

Let A" denote the standard n-simplex. Definition 7.6.3. For r € 7™'nt we define A(r) := xveVm{T)AW

(7.8)

Notice that dim(|A(r)|) = \EW{T)\. Theorem 7.6.4. The space Cactl(n) is a CW complex whose k-cells are indexed by r S T^''nt(n)k with the cell C(T) ~ |A(r)|. Moreover the map T H-> C(T) is a map of differential operads and it identifies Tpp'nt(n)k with CCkiCact1^)), where CCk are the dimension k cellular chains. Proof. Given an element in Cact1, we can view it as given by its topological type and a labelling of the arcs of its underlying arc family with the condition that the sum of all labels at each boundary is one. The number of incident arcs at each boundary i is \vi\ and the condition of the weights summing to one translates to the weights being in |Al"'l|. Vice versa given an element on the right hand side, the summand determines the topological type and it is obvious that any tree in Tpp'nt can be realized. Then the barycentric coordinates in the standard orientation define weights to the arcs in their fixed orientation of incidence. We orient the cells |A(r)| in the natural orientation induced from the linear order on the white edges. This makes the glueing well defined which can, for instance, be seen from the definition of the arc complex. • Keeping track of the homotopies which are explicitly given in §7.1, it is evident that CC^Cact1^)) is indeed a chain operad. From our previous analysis about the structure of Cact as a semi-direct product see §7.1 we thus obtain: Theorem 7.6.5. The glueings induced from the glueings of spineless normalized cacti make the collection CO^Coct1 (n)) into a chain operad. And since Cact,Cact1 an&D% are allhomotopy equivalent CC^cact1), is a model for the chains of the little discs operad. Remark 7.6.6. If one would like a topological operad in the background, one can choose any chain model Chain(TZ>o) for the scaling operad, then use the mixed chains for Cact i.e. CC+iCact1) C7mm(7£>o). It follows

197 that the inclusion of the cellular chains of Cact1 into the mixed chains is an inclusion of operads up to homotopy. Finally, given an operation of the CC^iCact1) we can let the mixed chains of Cact act by letting the mixed chains of bidegree (n, 0) act as the component of Cact1 and sending all the others to zero.

7.7. Orientations of Chains To fix the generators and thereby the signs for the chain operad we have several choices, each of which is natural and has appeared in the literature. To fix a generator g(r) of CC*{Cactl) corresponding to the cell indexed by T 6 ^~bp''nt(n) w e need to specify an orientation for it, i.e. a parameterizations or equivalently an order of the white edges of the tree the arc family it represents, i.e. a parameterizations or . The first orientation which we call Nat is the orientation given by the natural orientation of the arc family or equivalently the natural orientation for a planar planted tree. I.e. fixing the order of the white edges to be the one given by the embedding in the plane. We will also consider the orientation Op which is the enumeration of the white edges which is obtained by starting with the incoming edges of the white vertex labelled one, in the natural orientation of that vertex, then continuing with the incoming white edges into the vertex two, etc. until the last label is reached. Lastly, for top-dimensional cells, we will consider the orientation of the edges induced by the labels, which we call Lab. It is obtained from Nat as follows: for r e T^f^^1 let a € S n be the permutation which permutes to their natural order induced by the order -<(T\ the vertices v\,...,vn Then let the enumeration of Ew be a(Nat), where the action of a on Ew is given by the correspondence out and the correspondence between black and white edges via (v,N(v)) *—» (N(v),N2(v)) for top dimensional cells. To compare with the literature it is also useful to introduce the orientations Nat, Lab, and Op which are the reversed orientation of Nat, Lab and Op, i.e. reading them from right to left.

7.8. The Differential on T£?'nt There is a natural differential on T^'nt which it inherits from its interpretation as CC*(Cact) see below. Recall that for a planted planar tree there is a linear order on all edges and therefore a linear order on all subsets of edges.

198

Definition 7.8.1. Let r S T™'nt. We set Eangle = E{T) \ (Eleaf(r) U {sroot}) and we denote by nums : Eangie —> {1, • • •, N} the bijection which is induced by the linear order -<(T'p). Definition 7.8.2. Let r € Tbppp'nt, e e E ^ i e , e = {w,&}, with iu € Vw and 6 € Vi,. Let e - = {u;, 6-} be edge preceding e in the cyclic order -
0( r ) := J2 (-l) n u m E ( e K 1 5 e (r)

i

(7.9)

• Figure 29. The collapsing of an angle.

Denote by T^'nt{n)k

the elements of Tbppp>nt{n) with k white edges.

Proposition 7.8.4. The map d : Tbppp'nt{n)k -» Tb^>'nt(n)k-1 tial for ?f,pP'n and turns T^f'™ into a differential operad.

is a differen-

Proof. The fact that d reduces the number of white edges by one is clear. The fact that d2 = 0 follows from a straightforward calculation. Collapsing two angels in one order contributes negatively with respect to the other order. The compatibility of the multiplications Oj is also straightforward. All these properties will also follow from the chain interpretation of the trees in §7. •

199 Theorem 7.8.5. For the choice of orientation Nat and the induced operad structure Oj the map T — t > g(r) where g(r) is the generator corresponding to C(T) fixed in 7.7 is a map of differential operads and it identifies T^'nt(n)k with CCk{Cactl{rij), where CCk are the dimension k cellular chains. The analogous statement holds true when passing to operads indexed by sets for both Cact and Tbppp'nt. 7.9. The Operadic Action of Tg*'nt A natural way to let Tpp'nt act on a complex (0,5) is given by building a mixed complex by identifying the white edges of a tree with elements from Cact1 and the vertices with elements from O. First we notice that if we are dealing with planted planar trees, we have the total linear order -< on the set of vertices and edges. For an action of the operad T£?'nt on a graded space O = ^ O(n), we will consider maps p : 7fpp'nt(k) ® O(m) ® • • • ® O(nk) -» O(m)

(7.10)

T®/i---®/fcH->r(/i<8>---/fc)

(7.11) n

Actually, T ( / I ® • • • ® fk) will be zero unless \vi\ = n, and m = 2 * ~ k. In the graded case, we have to fix the order of the tensor product on the l.h.s. of the expression (7.11). We do this by using -
II EW(T)}\

a n d let num

: \{V(T)

U E(T)}\

->{1,...,N}

be the bijection which is induced by -
(O(nj)

iinum-\i)=vj

\Li

if num~1(i) is a white edge

( ? i 2 )

We then define the order on tensor product on the l.h.s. of the expression (7.11) to be given by W := Wx ® • • • ® WN Another way would be to include the sign which is necessary to permute the l.h.s. of 7.11 into W into the operation p. 7.10. The Action of the Symmetric Group The action of the symmetric group is induced by permuting the labels and permuting the elements of O respectively. This induces a sign by permutation on W.

200

Remark 7.10.1. This treatment of the signs is essential if one is dealing with operads and wishes to obtain equivariance with respect to the symmetric group actions. In general the symmetric group action on the endomorphism operads will not produce the right signs needed in the description of the iterations of the universal concatenation o of §8. In particular this is the case for Gerstenhaber's product on the Hochschild cochains. The above modification however leads to an agreement of sings for the action of the symmetric group for the subcomplex of the Hochschild complex generated by products and the brace operations, see below §7.9 and §9. Another approach is given by viewing the operations not as endomorphisms of the Hochschild cochains but rather maps of the Hochschild cochains twisted by tensoring with copies of the line Li36 and §9. If one is not concerned with the action of the symmetric group, then one can forgo this step. 7.11. The Action of Chain(Arc) Topology

on Itself and String

A good example of the type of action described above is the action of the chains of the Arc operad on themselves34. For the homotopy Gerstenhaber structure we need an action of CC*{Cactl) on any choice of chain model for Arc or any of the suboperads which are stable under the linear trees suboperad. The action p is just given by the glueing in Arc. We get agreement with the signs of the operation on Arc which agree with those of string topology8, if we denote the action of ri as *op and T| as •, see §5.13 for the operations and figure 31 for the definitions of the trees. For the homotopy GBV structure we should consider the chains CC^Cacti1) and again any choice of chain model for Arc or any of the suboperads which are stable under the action of the trees suboperad. 8. Structures on Operads and Meta-Operads Before going into the statement and proof of Deligne's conjecture, we would like to digress once more on operads. This helps to explain some choices of signs and explains the naturality of the construction of insertion operads which gives a special role to spineless cacti as their topological incarnation as well as to Arc as a natural generalization. This analysis also enables us to relate spineless cacti to the renormalization Hopf algebra of Connes and Kreimer11. In particular for a given linear operad or operad which affords a direct sum, we denned a Hopf algebra30. The symmetric group coinvariants of the Hopf algebra of the suboperad

201

of symmetric top-dimensional cells of the normalized spineless cacti are exactly the Hopf algebra of Connes and Kreimer. Given any operad there are certain universal operations, i.e. maps of the operad to itself. We will first ignore possible signs and comment on them later on. 8.1. The Universal

Concatenations

Given any operad, we have the structure maps OJ

: O(m) ® O(n) -> O(m + n - 1)

and the concatenations of these, which can be described by theirflowcharts. These are given by r € T£?^1. More precisely given k elements opk £ O(nk), we can concatenate them with the Oj to produce a tree flow chart where the inputs are the leaves and tails and the inner vertices are labelled by the operations opk, where the a vertex Vk labelled by opk necessarily has valence nk • The number of leaves and the degrees n — i of the opi satisfy the condition wt(r) = y j |v| = y j n i

=

#leaves + #inner vertices = #leaves + k (8.1)

V

Notice, we might have white leaves, which allows one to consider operads also with a 0 component such as CH* see below. So let T € Tb/w W an<^ ^ ni '• i 6 l,..-k := \vi\ then there is an operation o(T)(O(ni)®-.-®O(n fc ))-»O(m)

(8.2)

by labelling the vertex f, by opni £ O(rii). Notice that, we used the linear order on a planted planar tree in order to associate the functions to the non-leaf vertices. In general lifting the restriction on the n,, we define the operations o(r) to be zero of \vi\ ^ Hi. The above considerations give rise to a partial non-E operad operation of T^jM which can be made into an operation of the operad T^w by using §„ equivariance. The partial concatenations Oj insert a tree with A;-tails into the vertex n, if |tij| = k, by connecting the incoming edges of Vi to the tail vertices in the linear order at Vi and contracting the tail edges. Definition 8.1.1. We will fix that for O in Set the direct sum which we again denote by O is given by the free Abelian group generated by O which

202

we consider to be graded by the arity of the operations op 6 O minus one. If the operad O is in Chain we can take the direct sum of the components as Z-modules. In the case of an operad O in the category Vectk we consider its direct sum to be the direct sum over k of its components. In all these cases, we call O the direct sum and say O affords a direct sum and write O = © n e N Op(n). In all these cases we can consider O to be graded by N with the degree of Op(n) being n — 1. Remark 8.1.2. The above definition allows one to can make sense, formal linear combinations of operad elements with coefficients ±1. We could extend the use of the expression to afford a direct sum to mean that, the category which the operad is defined allows one to construct direct sums which are Z modules. If we consider an operad which affords a direct sum and let O be its direct sum then we obtain an operadic map.

Tfff-> Hom(O,O) In this sense one can say that Tffij partial-operad.

is the universal concatenation

8.2. The pre-Lie Structure of an Operad In an operad which affords direct sums, one can define the analog of the o product and the iterated brace operation (cf. 1 9 ' 2 5 ), see above. Definition 8.2.1. Given any operad O in Set, Chain or Vectk, we define the following map O(m) 0{n) -> O(m + n - 1)

(8.3)

m

opm ® OPn , - 2 ( - l ) ( < - 1 ) ( n + 1 ) p p m °i opn

(8.4)

i=l

This extends to a map o:O®O-*O

(8.5)

which we call the o product. We call the map which is obtained from in the same fashion as o, but with the omission of the signs (—l)(l~1)(™+1) the ungraded o product.

203

Following Gerstenhaber's calculation16'40 (essentially using associativity), we immediately have the following proposition Proposition 8.2.2. The product o defines on O := 0 i e N O(n) the structure of a graded pre-Lie algebra. Omitting the sign (—l)(J-1)(n+1) in the sum yields the structure of a non-graded pre-Lie algebra. We do not rewrite the proof here, but in graphical notation the proof follows from figure 30 below. Without signs this notation is related to the one that can be found for rooted trees7, for the case with signs see §7.7 and §9.8. 8.3. The Insertion Operad The interesting property of the operation o is that it effectively removes the dependence on the number of inputs of the factors. Given an operad in Chain we can also define other operations similar to o which are in natural correspondence with T pp . In fact these operations all appear in the iterations of o. They are given by inserting the operations into each other according to the scheme of the tree and then distributing tails so that the equation (8.1) is satisfied. Examples of this are given in figure 30. Here the first tree yields the operation /i o / 2 , i.e the insertion (at every place) of fi into f\. Iterating this insertion we obtain expression II which shows that inserting / 3 into f\ o f2 gives rise to three topological types: inserting /3 in front of f2, into f2 and behind f2. In the opposite iteration one just inserts f2 o fj, into f\ which gives a linear insertion of f2 into /i and f% into f2 • From the figure (up to signs) one can read off the symmetry in the entries 2 and 3 of the associator.

I

II

III

Figure 30. I. h o / 2 II. (fx o f2) o f3 and III. /j o (/ 2 o / 3 )

We will not care about signs at the moment, they follow from §7.7 of from 9.8.

204

8.4. Notation There are some standard trees, which are essential in our study, these are the n-tail tree ln, the white n-leaf tree r n , and the black n-leaf tree T£, as shown in figure 31

| V

JV

1

•

ii

I

II

III

V

Figure 31. I The n-tail tree /„, II. The white n-leaf tree rn, III. The black n-leaf tree

Essentially, if we would not like to a priori specify the number of leaves, i.e. inputs and degrees of the operations, we have to consider trees with all possible decorations by leaves. For this we need foliation operators in the botanical sense. To avoid confusion with the mathematical term "foliation", we choose to abuse the English language and call these operations "foliage" operators. Recall that there is an operation of T£?J on homogeneous elements of O of the right degree. We extend this operation to all of O by extending linearly and setting to zero expressions which do not satisfy degree condition that opk G O{\vk\) Definition 8.4.1. Let ln be the tree in T^ with one white vertex labelled by v and n tails as depicted in figure 31. The foliage operator F : T™wst'nt -> T™wst is denned by the following equation

F(r) := J2lnovT Notice that the right hand side is infinite, but since 1^T^st is graded by say the number of leaves, and F(T) is finite for a fixed number of black leaves the definition does not pose any problems. Furthermore, one could let F take values in TbA^IMl w n e r e * keeps track of the number of tails which would make the grading explicit.

205

Also notice that F : T™nt -> 1™ and F : 1™*>fl -

T™f.

1

Recall that there is an operation of T^f on homogeneous elements of O of the right degree. We extend this operation to all of O by extending linearly and setting to zero expressions which do not satisfy degree count, i.e. satisfy the equation (8.1). Given r in T^?^1 '* (n), we can then define the operation ° T ( / I ® • • • ® /«) := F{T)(/I

® • • • ® /„)

Notice that, although F(r) is an infinite linear combination, for given /i i • • • > In the expression on the right hand side is finite. The following is almost automatic. Theorem 8.4.2. Any chain operad which affords a direct sum is an algebra over the operad Tpp = T^f^1 with the insertion product. In fact, following Remark 8.4.3, we are forced to look at the insertion product. Remark 8.4.3. Thinking about F as a formal power-series, e.g. TpFjl[[t\], we can define a product * by the formula F(n) o F(T2) := F(n *

T2)

in (8.6)

Now, if the * is thought of as operadic, i.e. T\ * T% = 7 ( * , T I , T 2 ) , then by linearity and associativity, we know how to define 7(7-, t\,..., tn) for any T G Tr c Tpp, fixing the operations of the pre-Lie operad.

8.5. The Hopf Algebra of an Operad We have seen in 8.2 that any operad that affords a direct sum gives rise to a pre-Lie algebra. Now the defining property for a pre-Lie algebra is that the commutator of its product gives a Lie or in the graded case an odd Lie algebra. Definition 8.5.1. Given an operad O, which affords a direct sum, we define its pre-Lie algebra PL(O) to be the pre-Lie algebra (0, o), its Lie algebra L(O) to be the Lie algebra (O, [ ])using the Lie bracket [a, b] := aob = boa,its Gerstenhaber algebra G(O) to be the Gerstenhaber algebra (O,{ }) where { } is defined as usual via {a, b} := a*6-(-l)d a l+ 1 )(l f) l+ 1 )6* o. Lastly the Hopf algebra of an operad H(O) is defined to be U*(L(O)), i.e. the dual of universal enveloping algebra of its Lie algebra.

206

9. Spineless Cacti as a Natural Solution to Deligne's Conjecture 9.1. The Hochschild Complex, its Gerstenhaber Structure and Deligne's Conjecture Let A be an associative algebra over a field k. We define CH*(A, A) := 0,> o Hom(A®«,ii) There are two natural operations o; : CHm(A, A) ® CHn{A, A) -> CHm+n~l(A, n

m

A)

m+n

U : CH (A, A) ®CH {A,A) -» CH

(9.1) (9.2)

(A,A)

where the first morphism is as in 4.4 , i.e. for / £ CHP(A, A) and g £

cm {A, A). foig(xi,...,

Xp+q-i)

= f ( x i , . . . , Xi-i,g(xi,...,

Xi+q-i),

x

i + q

, . . . , a;p+9-i)

and the second is given by the multiplication f ( a i . . . , a m ) U g(bi, . . . , b n ) = f ( a i . . . , am)g(bi , . . . , b n ) 9 . 2 . The Differential on CH* The Hochschild complex also has a differential which is also derived from the algebra structure. A) then Given feCHn(A, d(f)(ai, • • .,an+i)

:= a i / ( a 2 , . . . , an+i) - f(aia2,. •. , a n + i ) +

• • • + ( - l ) n + 1 / ( a i , • • • - anan+1) + (-l)n+2f(au

..., an)an+l

Definition 9.2.1. The Hochschild complex is the complex (CH*, d), its cohomology is called the Hochschild cohomology and denoted by HH*(A, A). 9.3.

The Gerstenhaber Structure

Gerstenhaber16 introduced the o operations: for / £ CHP(A, A) and g e CH«(A,A)

fog:=J2(-l)(i-1}ig+1)f°i9 t=i

and defined the bracket {/•g}:=f°9-(-i)ip+1)i9+1)gof

207

and showed that this is indeed induces what is now called a Gerstenhaber bracket, i.e. odd Poisson for U, on HH*(A, A). 9.4. Deligne's Conjecture Since HH*(A, A) has the structure of a Gerstenhaber algebra one knows from general theory that thereby HH* (A, A) is an algebra over the homology operad of the little discs operad. The question of Deligne was: Can one lift the action of the homology of the little discs operad to the chain respectively cochain level? Or in other words: is there a chain model for the little discs operad that operated on the Hochschild cochains which reduces to the usual action on the homology/cohomology level? This question has an affirmative answer in many ways by picking a suitable chain model for the little discs operad 32 ' 48 ' 39 ' 50 . A review of these constructions is also available40. We will provide a new and in a sense natural and minimal positive answer to this question, by giving an operation of CC^iCact1) on the Hochschild cochains. There is a certain minimal set of operations necessary for the proof of such a statement which is given by iterations of the operations U and Oj. These are, as we argue below in bijective correspondence with trees in T£?'nt, our model for the chains of the little discs operad CG^Cact 1 ), has chains which are exactly indexed by these trees. Furthermore, the top dimensional cells which control the bracket are the universal concatenation operad. And lastly we will show that the differential of deleting arcs can be seen as a topological version of the Hochschild differential. This makes our new solution natural and minimal. 9.5. The Operation of CC*(Cactx) on HomeH For O = TiomcH, we define the map p of ed. 7.11, to be given by the operadic extension of the maps which send the tree r n to the nonintersecting brace operations i.e. for homogeneous / , gi of degrees | / | and

M.-N=l/l + EilSi|-«

208 f{9i,---,9n}(xi,...,xN)

:=

±

^ l < i i < • • • < » „ < |/| :

ij + \9j\<

f(xi,...,

xh-i,gi(xh,...,

ij+i

i i l + | 9 l | ) , . . . , a;in_i, gn(xin,...,

xin+\gn\),

• • •, %N) (9.3)

where the sign of the shuffle of the gj and xt which is determined by considering the shifted degrees, i.e. the Xi to have degree 1 and the gj to have degree \gj\ + 1. Notice that f{g} = f o g. Brace operations have first been considered by Getzler19. In order to make signs match with those of Gerstenhaber16, we will have to consider the opposite orientation for W i.e. W := W^ • • • ® W\. To implement this change of sign we define signw (r) to be the sign obtained by passing from W to W. This basically means that in the orientation of W one would regard the operation o^U 033 on the Hochschild complex, where fo°Pg = (-l)P^+P+1g o / and / Uop g := {-l)pqg U / . The action of the tree r n is given by: / ® Lei ® gi ® • • • ® Len ® gn K-> (_i)«9» w W/{ f f l ) . ..,£„}

(9.4)

The action of r^ is given by 5i ® • • • ® gn •-» (-i) si9 " W(T) ^i u • • • u
209 The generators of this suboperad are in 1-1 correspondence with elements of T^'nt. Such a tree represents a flow chart. The functions to be acted upon are to be inserted into the white vertices. A black vertex signifies the multiplication of the incoming entities, while a white vertex represents the brace operation of the elements attached to that vertex outside the brace and incoming elements inside the brace. Notice that in the flow chart of an expression of the type /{(gi), (#2 • 93 • 94{hi, ^2})} the symbols "{" and "," correspond to the white edges. Proposition 9.6.1. The association of a flow chart is a non-T, operadic isomorphism between Brace and T^T'nt of operads with a differential. Definition 9.6.2. We define an action of the symmetric group on Brace, by considering the symbols "{" and "," to be each of degree one. The following propositions follow from straightforward computation30. Proposition 9.6.3. With the above action of the symmetric group on Brace the isomorphism of 9.6.1 is an isomorphism of operads. Proposition 9.6.4. The above procedure gives an operation ofCC^Cact1) on CH*(A,A) and an operadic isomorphisms of Brace and CC^Cact1). 9.7. The Differential If we denote the differential on CC*(Cactl) as d and the differential of CH as S then the action of CC^Cact1) on HomcH commutes with the differential. On the space W there is a natural differential d\y '•= 5 + d. The calculations for the chains of the arc operad34 and the straightforward generalization to action of r n yield the following proposition. Proposition 9.7.1. po(dw)

= Sop

(9.5)

9.8. Another Approach to Signs and Actions Another way to fix the signs for the symmetric group actions on the Hochschild complex36 is achieved by tensoring with one dimensional spaces L\ and L2 of degrees —1 and —2 and their duals L\ and L\. S Also it is useful to deal with operads indexed by arbitrary sets. For a graded vector space A and an indexing set I one36 defines C = C*{A; A) = QrHomiA91, A) ® (L*2 ® Li)®7

210 where the sum is taken over all non-empty complectly ordered finite sets, and Horn is the internal Horn in the tensor category Vectz of Z-graded vector spaces (with Koszul rule of signs). The Gerstenhaber bracket is a map then a map from C ® C —> C ® (L2 ® ^i) 3 6 - Since we will not be in the .Aoo setting, we can omit the reference to the lines Li. 9.9. A Second Approach to the Operation of CC+{Cacbx) Another way to make CC*{Cactl) or Tj™'nt act is by using the foliage operator. This approach36 stresses the fact that a function / G CHq(A, A) is naturally depicted by r n . Notice for instance the compatibility of the differentials.

9.10. Natural Operations on CH* and their Tree Depiction Given elements of the Hochschild cochain complex there are two types of natural operations which are defined for them. Suppose /j is a homogeneous element, then it is given by a function / : A®n —» A. So treating the cochains as function, we have the operation of insertion, as in 4.4. The second type of operation comes from the fact that A is an associative algebra; therefore, for each collection / 1 , . . . , /„ € CH* (A, A) we have the n! ways of multiplying them together. We will encode the concatenation of these operations into a black and white bipartite tree as follows: Suppose that we would like to build a cochain by using insertion and multiplication on the homogeneous cochains /l) • • •) In- First we represent each function /j as a white vertex with | / j | inputs and one output with the cyclic order according to the inputs 1 , . . . , | / j | of the function. For each insertion of a function into a function we put a black vertex of valence having as input edge the output of the function to be inserted and as an output edge the input of the function into which the insertion is being made. For a multiplication of k > 2 functions we put a black vertex whose inputs are the functions which are to be multiplied in the order of their multiplication. Finally we add tails to the tree by putting a black vertex at each input edge which has not yet been given a black vertex, and we decorate the tails by variables a\,... as according to their order in the total order of the vertices of the rooted planted planar tree. It is clear that this determines a black and white bipartite tree. A rooted planted planar bipartite black and white tree whose tails are all black and decorated by variables <2i,... a3 and whose white vertices are labelled by homogeneous elements /„ 6 CH^(A, A) determines an element

211 in CHS(A,A) by using the tree as a "flow chart", i.e. inserting for each black vertex of valence one and multiplying for each black vertex of higher valence. Notice that, since the algebra is associative, given an ordered set of elements there is a unique multiplication. Remark 9.10.1. The possible ways to compose k homogeneous elements of CH* (A, A) using insertion and cup product are bijectively enumerated by black and white bipartite planar rooted planted trees with tails and k white vertices labelled by k functions whose degree is equal to the valence of the vertex. We will fix A and use the short hand notation CH := CH*(A, A). For an element / £ CH, we write f^ for its homogeneous component of degree d. If we would like to consider non-homogeneous elements, then given a tree we can only use the homogeneous components of the elements of CH with the right degree. This leads to: Definition 9.10.2. For r £ Tbppp(n) and / i , . . . , / n £ CH we let r (/i> • • • i fn) be the operation obtained in the above fashion by decorat• ing the vertex Vi with label i with the homogeneous component of f\ Notice that the result is zero if any of the homogeneous components / / vanish. Remark 9.10.3. Up to the signs which are discussed below this gives an operation of CC* (Cact) on the Hochschild complex. 9.11. The Operation of T^'nt Definition 9.11.1. For a tree r £ T^p'nt(n) with n white vertices we define a map OP(T) £ Hom(C#® n , CH) = HomCH{n) by OP(T)(/I,

• • •, fn) •= ±oP(ins{F(T), ( / i , . . . , /„))

here ins inserts the function ft into the label i and the signs are discussed in §7.7 and §9.8. Proposition 9.11.2. The Hochschild cochains are an algebra over Tpp'nt. 9.12. The Differential Again the differentials are compatible. This can be checked by a straightforward calculation, see §10.6 below. It is also implicit in the work of Kontsevich and Soibelman36.

212

Notice that in the tree formalism a black edge is inserted upon differentiating, while in Arc an arc is erased which in Cact corresponds to contracting an arc. Both these operations reduce the number of parameters by one. Of course omission of a parameter and insertion of a degree one space amounts to the same signs. Remark 9.12.1. The considerations of this section naturally lead to brace operations in far more general setting. This is explained in detail in §10.5 9.13. A Solution Of Deligne's Conjecture from Spineless Cacti In this section, we sum up the rather technical results of the previous ones. 9.14. The Action Using the cell decomposition 7.6.4 and the interpretation of CC*(Cactl) as 7^pp'nt; CC*(Cactl) acts naturally on the Hochschild complex, with the signs being fixed by one of the schemes above. 9.15. Deligne's

Conjecture

Notice that we have proven that Cact is homotopy equivalent to the little discs operad29 as well as to Cact1 (see Corollary 7.5.1) and hence the cellular chains of Cact1 give us a model of .0(2). Prom our previous analysis: Theorem 9.15.1. Deligne's conjecture is true for the chain model of the little discs operad provided by CCf(Cactl) and moreover C H* {A, A) is even a dg-algebra over CC% (Cact1). Remark 9.15.2. This operad of spineless cacti and its cellular chains thus give a simple minimal topological description of the Gerstenhaber structure of the Hochschild complex. Remark 9.15.3. As mentioned in §7.6.6 choosing a chain model Chain(Cact) of Cact by fixing a chain model for the scaling operad, we can let Chain(Cact) by sending all cells of Chain(Cact) which are not the product of a cell of CC*(Cactl) and a zero dimensional cell of M™0 to zero and letting the cells of CC^Cact1) times a zero dimensional cell of M"o i.e. cells of the type A(T) X pt act via r.

213 Remark 9.15.4. As noticed before, the fact that the chain operads oiCact and Cact1 possess the structure of a Gerstenhaber algebra up to homotopy, means that this structure exists also on HomCH*(A,A) UP t o homotopy and on TtorriA on the nose - the latter being Gerstenhaber's original theorem16. It is interesting to note that our homotopies can be seen as a natural geometric depiction of the homotopies Gerstenhaber used. 10. The Relation of Cact to Connes-Kreimer's Hopf Algebra and Generalizations 10.1. Connes-Kreimer's of an Operad

Hopf Algebra as the Hopf Algebra

Connes and Kreimer11 defined a Hopf algebra based in order to explain the procedure of renormalization in terms of the antipode of this Hopf algebra. This Hopf algebra was described directly, but also as the dual to the universal enveloping algebra of certain Lie algebra which is the Lie algebra associated to the free pre-Lie algebra in one generato7r. Definition 10.1.1. By the §„ coinvariants of an operad which affords a direct sum, we mean © ng fsj(^( n ))sn- Here ® is the shorthand notation explained in §8.1.1. In our notation we can rephrase the results11'7 about this Hopf algebra as Proposition 10.1.2. The renormalization Hopf algebra of Connes and Kreimer HCK is the Hopf algebra of S n coinvariants of H(Tr'fl) which agrees with the S n coinvariants of H(Vl). For the reader unfamiliar with this particular Hopf algebra this can also be a definition. 10.2. The Top Dimensional Cells of Spineless Cacti and the Pre-Lie Operad We denote the top-dimensional cells of CCn(Cact(n)) by CC*op(n). These cells again form an operad and they are indexed by trees with black vertices of valence one (recall that means one input). Furthermore, the symmetric combinations of these cells which are the image of Tr^1 under the embedding cppin form an sub-operad. From our previous description, one obtains30

214

Lemma 10.2.1. In the orientation Lab for the top-dimensional cells for TGTr'fl(n),T'Tr'fl(m) cppin(T) Oj cppin(r') = ±cppin(r of T') Definition 10.2.2. Let QPl be the quadratic operad in the category Vectz obtained as the quotient of free operad T generated by the regular representation of §2 by the quadratic relations defining a graded pre-Lie algebra, i.e. the quotient of F by the ideal R generated by the graded S3 submodule generated by the relation r = {x\ *X2)*xz — x\ *{x2*xz) — {—l)^X2^Xl^{{x\* £3) * X2 — x\ * {xz * X2)). Where T and R are considered to be graded by given the degree n — 1 to !F{n). Theorem 10.2.3. The operad CCl?p(ri)^ k is isomorphic to the operad QPl for graded pre-Lie algebras. Furthermore the shifted operad (CC%>P ® L®Ew)s(n) ® k is isomorphic to the operad VI defining pre-Lie algebras. Here we have used the notation of tensor products indexed by arbitrary sets 14 and used the "shifted line" L which a the free object generated by an element Proof. In view of Lemma 10.2.1 and the definition of the map cppin the second statement follows from the operadic isomorphism of Tr^1 and the pre-Lie operad VI7• This also proves the first statement up to signs. The matching of the signs is guaranteed by the shift. The fact that the relation r holds and generates the respective ideal is explicitly verified in the presentation of Gerstenhaber structure on the chains of the arc operad 34 • Remark 10.2.4. These statements also hold over Z. Thus, from now on we will omit the explicit tensoring with k. As an immediate consequence, we obtain: Corollary 10.2.5. The direct sum of an operad which affords a direct is an algebra over the symmetric top dimensional chains of the little disc operad of the chain model provided CC^Cact1) as well as over the shifted chains (CC%>P)S ® L®Ew. of degree -1. Recall that Ew is the set of white edges. Corollary 10.2.6. The pre-Lie algebra of Sn coinvariants {{CCi£p ® L®Ew)s{ri))§n is isomorphic to the free pre-Lie algebra in one generator.

215

Likewise the graded pre-Lie algebra of§n coinvariants (CC£ op ) s (n))s n is isomorphic to the graded free pre-Lie algebra in one generator Proof. The first statement follows from the references7'11 and thus so does the second up to signs. These are guaranteed to agree by the shifting D procedure and Theorem 10.2.3. 10.3. A Cell Interpretation of HCK As shown in Theorem 10.2.3 there is cell and thus a topological interpretation of the pre-Lie operad and the graded inside Cact1 and thus inside the Arc operad. In this interpretation HCK is also the Hopf algebra of the coinvariants of the shifted chain operad CCt°p(Cact)s L®Ew. Corollary 10.3.1. HCK is equal the Hopf algebra of S n coinvariants of the sub-operad of top-dimensional symmetric combinations of shifted cells CCtop{Cactl)s ® L®Ew of the shifted cellular chain operad of normalized spineless cacti CC^Cact1) L®E™. It is interesting to note that also the G and BV structures34 are inside the symmetric (graded symmetric) combinations. 10.4. Comments on Operads and HCK We have shown that any operad is an algebra over the operad Tr'fl in a natural way and thus the Hopf algebra HCK naturally appears in any context involving operads, such as Deligne's conjecture. We have furthermore shown that there is a topological incarnation of the insertion product, which is based on surfaces. In this setting, we have constructed a chain representation of the algebra HCK- This links the algebra HCK and its underlying bracket for instance to string topology. Remark 10.4.1. We expect to obtain other interesting examples of such Hopf algebras by considering other tree operads. 10.5. Operad Algebras and a Generalized Deligne Conjecture Definition 10.5.1. We define an operad algebra to be an operad O which affords a direct sum together with an element U G O(2) which is associative, i.e. define a U b to be (-l) | a | (U oi a) O|a+1| b then (a U 6) U c = a U (6 U c), recall that \a\ = n - 1 if a e O(n).

216

Definition 10.5.2. We formulate the generalized Deligne conjecture as the statement that the direct sum of any operad algebra which affords a direct sum is an algebra over the chains of the little discs operad in the sense that there is a map of differential operads of CC*(Cactl) action as specified in §7.9. Definition 10.5.3. For / € O(m),gi £ O(rii), we define the generalized brace operations f{gi,---,9n}--=

53

±(---((/°ti3i)oi2ff2)oi3---)°in3n

1 < i\ < • • • < in < m :

ij + \9j + 1| < ij+1

(10.1)

where the sign is defined to be the same one as in equation (9.3) Lemma 10.5.4. There is an operadic action ofT^J''nt of any operad algebra. Proof. We can view the bipartite tree as a flow chart. For the white vertices, we use the brace operations above and for black vertex with n incoming edges, we use the operation of applying U n — 1 times. Notice that the order in which we perform these operations does not matter, since we took U to be associative. • Definition-Proposition 10.5.5. Generalizing Gerstenhaber's16 definition to an operad algebra, we define a differential on the direct sum by df = /oU-(-l)l/lUo/.

Proof. The fact that this is a differential follows from the calculations of • Gerstenhaber16. 10.6. Differential on Trees with Tails Definition 10.6.1. For a tree r with tails in 7^ p and vertex v G Vb\{vrOot} we define r+ to be the b/w tree obtained by adding a black vertex b+ and an edge e + := {b+, v}, if \v\ ^ 0 and if \v\ — 0, the tree obtained by adding two vertices b+ and bst and two edges e+ = {b+,v} and est = {bst,v} to r.

217 We call a linear order -<' on T | compatible with the order -< on r if a) e+ -<' est if applicable and b) the order induced on r by -<' by contracting e + and est (if applicable) coincides with -<. We define Ew-int to be the white internal edges, i.e. white edges which are not leaves and set E^angie : E(r) \ Ew-int. For a linear order -
:= (—i)l{eleesb_anff,e,c-:'t;}|

and set

5,(r):=

5^

«5n(V)(T+,-!')

compatible-^'

we recall, that tail edges are considered to be black. Finally we define

(10.2) v£Vb\{vroot}

Remark 10.6.2. There is again a tree depiction for the operations of insertion an cup product. This is analogous to tree picture explained in 9.10 where we now replace functions by elements of the operad. The tree differential then describes the insertion of the new edges at all angles corresponding to the black vertices which amounts to inserting a U product. Using this interpretation and the tree notation for the known calculations 16>25 it is straightforward to check that the tree differential (10.2) denned above agrees with the differential induced by the differential on the operad, which we denned in 10.5.5. Our differential also agrees with differential induced by the differential of Kontsevich and Soibelman36 via st^. Theorem 10.6.3. The generalized Deligne conjecture holds. Proof. By the preceding Lemma 10.5.4, we have an operadic action of T£V'nt and thus an action of the chains CC^Cact1) which is a chain model for the little discs operad. The compatibility of the differentials follows directly from their definitions by a straightforward calculation as remarked D above. 10.7. A Cyclic Version of Deligne's

Conjecture

We have shown that spineless cacti naturally act on the Hochschild cohomology of an associative algebra thereby providing a solution to Deligne's conjecture. Recently31 we have generalize this fact to an action of a cell

218 model of cacti on the Hochschild cohomology of an associative algebra which is isomorphic as a bi-algebra to its dual. This means that there is a BV structure on the Hochschild cochains up to homotopy and a BV structure inducing the Gerstenhaber structure on the Hochschild cohomology of such an algebra. Our treatment is again of a more general nature. It is easy to abstract from it to a general setup of cyclic operad algebras. 11. Outlook and Speculations 11.1. Operation on the Hochschild Complex of an A ^ algebra There is a natural action of 7 ^ s t ' n * on CH*(A, A)36 was constructed which allows to solve Deligne's conjecture. Here the black vertices stand for the higher multiplications /xn of the AQO structure. The solution was then established by constructing a quasi-isomorphism of the free operad of M onto the operad of the Fulton-MacPherson compactification of the configurations of R2. This beautiful construction is however indirect, as one has to invert the quasi-isomorphism and furthermore the map to the FultonMacPherson compactification is rather involved and has considerably many choices. We would hope to find a direct interpretation of ^ ^ s t ' " * 11.2. A Putative Cell Decomposition In our situation an A,*, version of Deligne's could be established, if we had a cellular decomposition of a suitable version of Cact (e.g. Cact- where this is the space of cacti whose lobes have radius not greater than 2ht^) such that the cells are indexed by T^st'nt and are given by

cell(r) := J J CH x J ] KM v£Vw

(11.1)

VEVB

where C\v\ is the |i>|-dimensional cyclohedron and K\v\ is the \v\— dimensional Stasheff polytope or associahedron40. be the CW complex glued from the cells cell(r) using Let Cell(l£^st'nt) the natural differential S for the cyclohedra and associahedra and let d be the tree differential36. It is then straightforward to show that: 11.2.1. The Proposition CC.{Cell{TVfwat>nt))

differential

d

and

5

agree

on

219 This leads us to the conjecture Conjecture 11.2.2. There is a suitable suboperad ofCact which is quasiisomorphic to Cact and whose cell decomposition is given by eq. (11.1). 11.3. Truncation of Simplices and Stasheff Polytopes A good candidate for the space for which the cell decomposition should be possible is the space Cact- of restricted cacti. It is straightforward to see that this space is indeed a suboperad. To find the cell decomposition, we should have a special realization of cyclohedra and associahedra. In particular, we need a realization of an n associahedron in the hyperplane ^ j £ j = 2"" 1 C R"" 1 and the n cyclohedron in R 2 n + 1 . The corners for the Associahedron Kn should be given by n tuples indexed by a binary tree and should be 2n~ht^ which are the weights obtained by using the homotopy associative operation • on VArc using the bracketing given by a binary tree and reading off the total weights on the boundaries di(F). For n = 2 this is the point (1,1) for n = 3 one can take the interval on the line through (1,1,2) and (2,1,1). For n = 4 the corners of the pentagon should be (1,1,2,4), (2,1,1,4), (2,2, 2,2), (4,1,1, 2) and (4, 2,1,1). The cyclohedra should come from blowing up the n dimensional simplex A" which corresponds to the r n , so that the tree boundary for the cyclohedra coincides with the geometric boundary of the operad. Conjecture 11.3.1. We conjecture that there is a truncation of simplices with the above corners that yields the above realizations of associahedra and cyclohedra. This has been checked for low dimensions, but we currently lack a general scheme. One step in this direction would be the construction of an explicit map of the compactification of the configuration space of n points on S1 with one point fixed at the origin to the n-th cyclohedron. 11.4. Relations to the Fulton-MacPherson Compactification Finally, we expect a relation of spineless Cacti or the Arc operad to the Fulton-MacPherson compactification of the configuration space of points in R2. The idea is to use a variant of polar coordinates and then to keep track

220

of the collision speeds in the width of the bands. A thorough analysis of this fact should result in a positive answer to a conjecture made by Kontsevich and Soibelman36. This will be elaborated on elsewhere. 11.5. Actions of Arc Denote by Arc# the suboperad of Arc of surfaces without punctures whose image under Loop is a graph whose genus and number of cycles coincides with the genus of the surface and the number of its boundary components. We also expect that Arc# operates on CH*(A, A), for an algebra A together with a non-degenerate invariant as a cyclic operad. This has two steps, first the non-trees and second higher genus. The operation of Cacti has recently been established31. Conjecture 11.5.1. We conjecture that a suitable chain model of Arc# acts on CH*. For the action of loop spaces there are several conjectures. Conjecture 11.5.2. We conjecture that of Arc# acts on LM, the loop space of a compact manifold M. If we consider all of Arc then we can obtain the surfaces whose loop has the wrong genus or number of cycles as images of the stabilization with respect to the genus operator34. The map Loop is not sensitive to this stabilization and thus if there is an action of Arc which factors through the map Loop, then one is essentially dealing with the stabilized moduli space. We consider the suboperad Arcc of Arc given by surfaces without punctures whose Coop is connected. The above considerations and the fact that the sequences39 have an interpretation in terms of loops of arc families on higher genus surfaces leads us to conjecture: Conjecture 11.5.3. We conjecture that if the Arcc operad acts on M by a map that factors through Loop then M has the homotopy type of an infinite loop space. In particular the stabilization of Arcc has the homotopy type of an infinite loop space. In this conjecture, we can probably replace Arcc by the sub-operad of Chinese trees34. Lastly,

221 Conjecture 11.5.4. We conjecture that the Arc operad will act on LM by a combinatorial Gromov-Witten type setup. This we understand as follows. If we wish to concatenate loops in the free loop space LM that do not intersect, we have to be able to move the loops into the a position in which they do. Regarding the image of a surfaces with arcs in M whose boundaries 1 , . . . , n are exactly the loops to be multiplied will give us a way to move and multiply them into the loop 0. The versions of the discussed Props yield straightforward generalizations. With a suitable version of mapping spaces and (virtual) fundamental classes one could hope to construct combinatorial invariants, pulling back families of loops and integrating over the moduli space. 11.6. Rankin-Cohen Brackets Recently12 it was discovered that the Rankin-Cohen brackets can be realized inside the foliation Hopf Algebra introduced by Connes and Moscovici. Since these brackets have a conjectured form in terms of naturally grown trees and due to the relationship of Arc to moduli spaces we formulate: Conjecture 11.6.1. We conjecture that Rankin Cohen brackets are also realizable on Cact. 11.7. Open Ends and Questions It still remains to find out the exact relationship of cacti to chord diagrams, the dihedral algebra23 and polylogarithms and higher zeta values. For the latter the clarification of the relationship to configuration spaces and cacti should be key as well as the relationship to moduli spaces. We wish to conclude by remarking that the new feature of the Chinese tree operad is that the chord diagrams29 are no longer planar. Cutting at zero, we obtain not only rainbow diagrams as for genus zero, but (for) high enough genus any trivalent diagram of the types depicted in figure 32 which are known from high energy physics and knot theory2 and are making their appearance in biology in the form of folding problems for RNA. References 1. M. Atiyah, Topological quantum field theories, Inst. Hautes Etudes Sci. Publ. Math. 68 (1988), 175-186. 2. D. Bar-Natan. On the Vassiliev knot invariants. Topology 34 (1995), 423-472.

222

I

II Figure 32. I. A genus one arc family, its cactus, cord diagram and rainbow diagram, II A genus one diagram giving a braid move

3. F. R. Cohen. The homology of C n +i -spaces, n > 0. In The homology of iterated loop spaces., volume 533 of Lecture Notes in Mathematics. Springer, 1976. 4. F. R. Cohen. Artin's braid groups, classical homotopy theory, and sundry other curiosities. In Braids (Santa Cruz, CA, 1986), volume 78 of Contemp. Math., pages 167-206. Amer. Math. Soc, Providence, RI, 1988. 5. A. Cattaneo and G. Felder. A path integral approach to the Kontsevich quantization formula. Comm. Math. Phys. 212 (2000), no. 3, 591-611. 6. R. L. Cohen and V. Godin. A polarized view of string topology. Preprintmath. AT/0303003 7. F. Chapoton, M. Livernet. Pre-Lie algebras and the rooted trees operad. Internat. Math. Res. Notices 2001, no. 8, 395-408. 8. M. Chas and D. Sullivan. String Topology. Preprint math.GT/9911159 9. M. Chas and D. Sullivan. Closed string operators in topology leading to Lie bialgebras and higher string algebra. Preprint math.GT/0212358 . 10. R. L. Cohen and J.D.S. Jones A homotopy theoretic realization of string topology Math. Ann. 324 (2002), no. 4, 773-798. 11. A. Connes and D. Kreimer Hopf Algebras, Renormalization and Noncommutative Geometry. Commun. Math. Phys. 199 (1998) 203-242. 12. A. Connes and H. Moscovici. Rankin-Cohen Brackets and the Hopf Algebra of Transverse Geometry. Preprint math.QA/0304316 . 13. W. Chen and Y. Ruan Orbifold Gromov-Witten Theory Preprint math.AG/0103156. 14. P. Deligne. Categories tannakiennes. The Grothendieck Festschrift, Vol. II, 111-195, 15. Z. Fiedorowicz. The symmetric Bar Construction. Preprint (available at http://www.math.ohio-state.edu/~fiedorow/) and Z. Fiedorowicz. Constructions of En operads. Preprint math.AT/9808089,

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1999. 16. M. Gerstenhaber. The cohomology structure of an associative ring, Ann. of Math. 78 (1963), 267-288. 17. E. Getzler. Two-dimensional topological gravity and equivariant cohomology. Comm. Math. Phys. 163 (1994), 473-489. 18. E. Getzler. Operads and moduli spaces of genus 0 Riemann surfaces. The moduli space of curves (Texel Island, 1994), 199-230, Progr. Math., 129, Birkhauser Boston, Boston, MA, 1995. 19. E. Getzler. Cartan homotopy formulas and the Gauss-Manin connection in cyclic homology. Quantum deformations of algebras and their representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992), 65-78, Israel Math. Conf. Proc, 7, Bar-Ilan Univ., Ramat Gan, 1993. 20. V. Ginzburg and M. Kapranov. Koszul duality for operads. Duke Math. J. 76 (1994), 203-272. 21. E. Getzler. Operads and moduli spaces of genus 0 Riemann surfaces. The moduli space of curves (Texel Island, 1994), 199-230, Progr. Math., 129, Birkhuser Boston, Boston, MA, 1995. 22. E. Getzler and M.M. Kapranov. Modular operads. Compositio Math. 110 (1998), no. 1, 65-126. 23. A.B. Goncharov Multiple zeta-values, Galois groups, and geometry of modular varieties. Talk at the European Congress of Math., 2000. Preprint math.AG/0005069 24. Yu. I. Manin and A. B. Goncharov. Multiple zeta-motives and moduli spaces M 0 , n Preprint math.AG/0204102. 25. M. Gerstenhaber and A. A. Voronov. Higher-order operations on the Hoehschild complex. Funktsional. Anal, i Prilozhen. 29 (1995), no. 1, 1-6, 96; translation in Funct. Anal. Appl. 29 (1995), no. 1, 1-5. 26. T. Jarvis, R. Kaufmann and T. Kimura, Pointed Admissible G-Covers and G-equivariant Cohomological Field Theories, Preprint math.AG/0302316. 27. R. Kaufmann. Orbifolding Frobenius algebras. math.AG/0107163, Internat. J. of Math. 14 (2003), 573-619 28. R. M. Kaufmann. Second quantized Frobenius algebras. To apperar in Comm. Math. Phys. 29. R. M. Kaufmann. On several varieties of cacti and their relations. Preprint MPI-2002-113, math.QA/0209131. 30. R. M. Kaufmann. On Spineless Cacti, Deligne's Conjecture and ConnesKreimer's Hopf Algebra.. Preprint. math.QA/0308005. 31. Kaufmann, Ralph M. A proof of a cyclic version of Deligne's conjecture via Cacti. Preprint, math.QA/0403340. 32. M. Kontsevich. Operads and Motives in Deformation Quantization. Lett.Math.Phys. 48 (1999) 35-72. 33. Kadeishvili, T. The structure of the A(oo)-algebra, and the Hoehschild and Harrison cohomologies. Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 91 (1988). 34. R. M. Kaufmann, M. Livernet and R. B. Penner. Arc Operads and Arc Algebras. Geometry and Topology 7 (2003), 511-568.

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35. R. M. Kaufmann, M. Livernet and R. B. Penner. Homological arc operads. In preparation. 36. M. Kontsevich and Y. Soibelman. Deformations of algebras over operads and Deligne's conjecture. Conference Moshe Flato 1999, Vol. I (Dijon), 255-307, Math. Phys. Stud., 21, Kluwer Acad. PubL, Dordrecht, 2000. 37. Yu. Manin. Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces. AMS 1999. 38. Yu. Manin. Gauge Field Theory and Complex Geometry. Springer, BerlinHeidelberg-New York, 1988. 39. J. E. McClure and J. H. Smith, Jeffrey H. A solution of Deligne's Hochschild cohomology conjecture. Recent progress in homotopy theory (Baltimore, MD, 2000), 153-193, Contemp. Math., 293, Amer. Math. Soc, Providence, RI, 2002. 40. M. Markl, S. Shnider and J. Stasheff. Operads in algebra, topology and physics. Mathematical Surveys and Monographs, 96. American Mathematical Society, Providence, RI, 2002. x+349 pp. 41. R. C. Penner. The simplicial compactification of Riemann's moduli space. Proceedings of the 37th Taniguchi Symposium, World Scientific (1996), 237252. 42. R. C. Penner. Decorated Teichmuller theory of bordered surfaces. Preprint math.GT/0210326. 43. R. C. Penner. Cell decomposition and compactification of Riemann's moduli space in decorated Teichmuller theory.. This volume. 44. R. C. Penner with J. L. Harer. Combinatorics of Train Tracks, Annals of Mathematical Studies 125, Princeton Univ. Press (1992); second printing (2001). 45. D. Sullivan. Open and Closed String field theory interpreted in classical Algebraic Topology. Preprint math.QA/0302332 . 46. S. Stolz and P. Teichner. What is an elliptic object?. Preprint. 47. P. Salvatore and N. Wahl. Framed discs operads and the equivariant recognition principle. Preprint math.AT/0106242 48. Tamarkin, D., Another proof of M. Kontsevich formality theorem, math/9803025, and Formality of Chain Operad of Small Squares, math/9809164. 49. A. A. Voronov Notes on universal algebra. Preprint math.QA/0111009 50. A. A. Voronov Homotopy Gerstenhaber algebras. Conference Moshe Flato 1999, Vol. II (Dijon), 307-331, Math. Phys. Stud., 22, Kluwer Acad. PubL, Dordrecht, 2000 51. B. Zwiebach. Closed string field theory: quantum action and the BatalinVilkovisky master equation. Nuclear Phys. B 390 (1993), no. 1, 33-152.

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FRAGMENTS OF NONLINEAR GROTHENDIECK-TEICHMULLER THEORY

PIERRE LOCHAK Institut de Mathematiques de Jussieu 175 rue du Chevaleret, F-75013 Paris Email:[email protected] After giving a short survey of the theory as it exists to-date, we discuss the specifity of what we call the nonlinear version of Grothendieck-Teichmiiller theory, in contrast with the pronilpotent motivic approach.

1. Introduction This note contains no new result; its main goal is to detail the meaning of the adjective 'nonlinear' occurring in the title, which we do in section 3. Moreover, as this same title indicates, the first section (supplemented by the Appendix) can hopefully be seen as providing a very short and incomplete introduction to the existing part of the theory. As for the short second section it has been included for the sake of pointing out a few basic phenomena and can perhaps be useful to the newcomer as it lists the main entry points into the recent literature. In connection with this we note once and for all that we have considered that the existing extremely efficient electronic data bases make it both hopeless and useless to aim at any kind of exhaustivity. Relevant references can nowadays easily be retrieved by using just the name of an author or an appropriate keyword. Grothendieck-Teichmiiller theory was conceived or dreamt of by A.Grothendieck in his Esquisse d'un programme (now available in [21]), following his Longue marche a travers la theorie de Galois. A few seminal papers, especially [9], [27] and [10], started giving flesh to the vision. The theory is still very much in flux and there are several possible frameworks or versions which are connected in particular by various kinds of 'linearization' processes. This is in part what we will discuss in section 3, from a biased and prospective viewpoint. Clearly, linearizing a situation is at present a natural, indeed almost irresistible mathematical inclination, and the linear

226

toolbox is much more varied, sophisticated and powerful than the nonlinear one. So it is not only unavoidable but also necessary and fruitful. But it does leave aside part of the richness of the situation, as we will try to suggest by recalling and assembling concrete and fairly elementary facts. The adjective 'linear' should clearly here be taken in a rather extended acception, with partial synonyms such as 'abelian', '(pro)nilpotent', '(pro)solvable' or even 'motivic'. In turn 'nonlinear' can translate in several ways, such as 'profinite' or 'anabelian'. As a partial illustration of this dichotomy we quote the following sentence from the introduction of [10]: "Cette relation etroite avec la cohomologie indique que l'etude du TTI rendu nilpotent est loin du reve 'anabelien' de A.Grothendieck. Elle permet par contre de s'appuyer sur sa philosophic des motifs". We will see that on the nonlinear side, there may be a - as yet modest and very much incomplete - new landscape which is slowly emerging, perhaps in partial conformity with the vision outlined in the Esquisse. 2. A Short Reminder and A Reading Guide We will be concerned here with what we call the nonlinear version of Grothendieck-Teichmuller theory, obviously as it is available to-date (2003), which means at a very incomplete stage of development. Apart from the Esquisse and the seminal papers already mentioned in the introduction we refer the reader to [49], [36], the introduction to [24] and references therein for very imperfect and partial accounts. Clearly there does not and cannot exist at present a satisfactory survey of a largely unchartered landscape. Getting started in a nutshell, skipping the necessary motivating questions: We first consider the collection of the fine moduli spaces of curves Mgtn for varying finite hyperbolic type (g, n), together with their stable completions MSjn. These objects were constructed in [13] as algebraic stacks over Z but we confine attention to the generic fiber and view them as separated regular Deligne-Mumford Q-stacks. The reader who finds this introit a little abrupt can go for instance to [30], [46], [13] and the many other texts which introduce these objects in a more or less analytic or algebraic fashion. Here is a short da capo, in order to fix more notation and make the path from topology into arithmetic geometry perhaps more transparent. One can construct the space Mgtn (g > 0, n > 0, 2g - 2 + n > 0) as a complex orbifold of dimension 3g - 3+n classifying Riemann surfaces of genus g with n marked points, and

227

then compactify it by adding in Riemann surfaces with nodes. The orbifold fundamental group of Mg,n coincides with the mapping class group of the topologists: Tv°rb(Mgtn) = F*°£. On the algebraic side, as mentioned above, one constructs Mg,n as a Q-stack which can be completed by gluing a divisor at infinity parameterizing the stable curves. The theory of the fundamental group for stacks parallels, but with interesting differences, Grothendieck's classical (SGA 1) theory for schemes (see [43] and [38]); for the specific case of the moduli stacks of curves, which are classifying spaces, we also refer to the contribution of T.Oda in [21], which uses simplicial techniques. In particular, if X is a geometrically connected separated Deligne-Mumford stack over Q, one can consider its geometric (stack) fundamental group 7rfom(X) = TTI(X® Q). After fixing an embedding Q c C, an extended version of Lefschetz principle implies that ^eom (X) is canonically isomorphic to the profinite completion of ff°rb(Xan), where Xan is the complex orbifold obtained by analytification of X ® C. Here and above, TT°rb is the fundamental group defined and studied by W.Thurston. In particular Ttfeom(Mgin) = T9in, where Tg,n denotes the profinite Teichnriiller modular group. Note that in the absence of superscript groups will be profinite by default. The M3in and their stable completions fit together into a category or modular tower which we denote by M. This means that one can define smooth morphisms of geometric origin between the M g n 's. There are actually several (existing or not yet existing) versions, the simplest and most classical being derived from the stable stratification. Recall that this is the stratification of Mgn. One can also add in the point erasing morphisms, that is the fibration M ffin+1 —> Mg
228

M9>n —+ Mg[n] which is orbifold unramified of group Sn (the permutation group on n objects) and gives rise to yet another set of morphisms. The long and the short is that the above defines, modulo a lot of 'technical details' a version of M which is actually fairly simple minded for at least two reasons: a) the morphisms live at infinity, b) they are defined over Q (not only Q), essentially because they feature algebraic counterparts of familiar topological gestures: pasting, pinching, erasing etc. It seems fair to say however that this is the only version of the modular tower containing all the M 9in 's which has really been studied and understood in this context, that is in relation with the Grothendieck-Teichmiiller group (see below). It gives rise to what we call the (Teichmiiller) lego at infinity, for which we refer to [24] and [44] from the viewpoint of Grothendieck-Teichmiiller theory and to [16], [5] and references therein from other perspectives. In the case of genus 0 as initiated in [9], it is sometimes referred to as the 'geometry of associativity' because it was Drinfeld's groundbreaking idea to, so to speak not take associativity for granted (keywords: quasi-Hopf algebras, braided categories, McLane coherence relations, universal scattering matrix, Yang-Baxter equations, gravity operad etc. etc.). Last but not least, we note that this is not the version of M and the hypothetical attending lego which Grothendieck seems to have in mind in the Esquisse, where he refers explicitly to curves with automorphisms, to which we will briefly return below. Not much is known to-date beyond this version of the modular tower which again a) lives at infinity, b) is denned over Q. In fact a) and b) are far from independent and it may well be that it is essentially the largest possible tower which is entirely denned over Q. At any rate and for the time being, having built a more or less expensive version of M over Q, one applies the geometric fundamental group functor 7rfeom, from the category (Q — Stacks) of Q-stacks with Q-morphisms to the category (Grps) of finitely generated profinite groups with continuous homomorphisms. In other words if X is a Q-stack (say, nice: separated, D-M, quasicompact, geometrically connected), one sets as above ^seom ^JQ _ ^j (X ® Q), which is a finitely generated profinite group. One can then indeed regard ^eom as a covariant functor from (Q - Stacks) to (Grps). Letting this functor act on our modular tower, we get the Teichmiiller tower F = nfeom(M) = TTI (M (g> Q), which is thus no more and no less than the collection of the (profinite) Teichmiiller modular groups r S ) n with varying type (g, n) and morphisms coming functorially from the morphisms in M. The next step consists in considering the group Out(nfeom), that is

229 the outer automorphism group of the functor irfeom, meaning the group of automorphisms modulo inner automorphisms on the right-hand side, i. e. in (Grps). Let GQ denote as usual the absolute Galois group of Q. By a fairly easy extension to stacks of Grothendieck's short exact sequence (SGAl, §IX.6), we get a morphism GQ —> Out(Trfeom). This map is injective, i.e. the outer Galois action is faithful as soon as we consider a big enough version of (Q — Stacks). We were a little fuzzy above as to which geometric objects we wish to include in ( — Stacks) partly because Belyi's theorem (recalled in the Appendix) immediately implies a much more drastic assertion: as soon as ( — Stacks) contains the single object P Q \ {0,1, oo}, the action is faithful, that is the above map from GQ to Out(irfeom) is an injection. To see this, just apply theorem Al to the set of all elliptic curves defined over numberfields, that is with j-invariants in Q. From now on we will sometimes write P* = P 1 \ {0,1, oo} for brevity. Note that the action of GQ is faithful when restricted to other kinds of coverings of P*, for instance on trees (see the article of L.Schneps in [11] as well as later papers by L.Zapponi). The fact that the 'enormous' and complicated profinite group GQ acts naturally and faithfully on various types of seemingly simple looking topological objects has been realized only in the last two decades and has by now come up in a variety of situations. Before going back to our central geometric object, namely the modular tower, we mention an important recent and as yet unpublished result of F.Pop. We have just seen that the map GQ —> Out(Trfeom) is injective. In fact, not only is the above map an injection, but under rather mild conditions it is an isomorphism! This means the following: consider C C (Q - Stacks) a full subcategory and the restriction of K^0"1 to C. We still get a natural map GQ —> Out(irfe°m(C)) which is still an injection under very mild conditions (e.g. if C contains P*). Moreover, and somewhat informally, as C gets smaller, the target gets larger. Pop's result says that actually, even for a rather 'small' sample of geometric objects C, the above map is an isomorphism. Here is a sample statement: Define C to be the category whose objects are quasi-projective varieties defined over Q which are the complements in the projective plane of (not necessarily irreducible) curves; the morphisms are the dominant Q-morphisms. With this definition of the geometric category C we have (F.Pop, 2000): The natural map: GQ —> 0u£(7ifeor" (C)) is an isomorphism.

This is a striking and beautiful result, the first in this field which connects a purely arithmetic object (on the left-hand side) to a purely geometric object (on the right-hand side). Only we note for further reference that

230

nothing is known about the right-hand side, apart from being proved to be isomorphic to the left-hand side. Now let us go back to the modular tower M and consider the group Out(7rfeom (M)) = Out(T_) where the outer automorphisms are equivariant with respect to the morphisms in the Teichmuller tower F. One has GQ •—> Out(T_) since P* ~ Mo,4 appears as an object of M. In the colorful words of the Esquisse: 'L'action est deja fidele au premier etage'. Moreover the action of GQ enjoys a well-known property: it preserves inertia groups and the inertia groups in F g i n associated to the components of the divisor at infinity of the stable completion of M9tn are nothing but Dehn twists. Concrete conclusion: the action of GQ on the F 3)n 's maps Dehn twists to conjugates of powers of themselves; in other words they preserve the conjugacy classes of the procyclic groups they generate. Here we are talking about the inertia 'at infinity' associated to the classical situation where a scheme X can be written as X = X\D with X proper and D a divisor with (strict) normal crossings. This is discussed in general in [19]; the adaptation to stacks is not completely banal but this is not by itself a typical stacky phenomenon. In fact other types of inertia appear for stacks (see below for references) which have just started being explored in this context. For the moment and in view of the above, we define IT = Out*(T) c Out(T) to be the subgroup of inertia preserving outer automorphisms in that sense. This is by definition the Grothendieck- Teichmuller group, at least in its present, all genera, profinite version, and for the version of the modular tower M outlined above. As a first concrete approach, and in order to find 'coordinates' for IF, one notes that T = Out*(T) C Out*(ir{eom(M0,4)). Then Mo,4 ~ P* (here over Q) and 7r?eom(P*) = 7n(P* ® C) = n[(C \ {0,1}) = F2, the profinite completion of the free group on two generators, since obviously 7i^op(C \ {0,1}) ~ -F2. We remark that the importance of P* was already mentioned in SGA 1 ('le probleme des trois points') where Grothendieck notices that there was (and apparently still is, after four decades) no other way to compute its geometric fundamental group (say over Q) than to use Lefschetz principle as above (TTI(C \ {0,1}) is the profinite completion of 7TjOp(C \ {0,1})), thus reducing oneself to elementary topology. In other words there is still no algebraic way of computing -say- TTI( \ {0,1})..At any rate proceeding as above we get JT C Out*(F2) and it is easy to see that the latter group can be parametrized by pairs F = (A, /) C Z* x F!2 where Fl^ denotes the derived subgroup of F
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generate F2, the action of F = (A,/) is given explicitly by: F(x) = xx, We also note that the outer automorphism groups we F(y) = f~lyxf. are considering, in particular Out*(F2) and the Grothendieck-Teichmiiller group F are naturally endowed with the profinite topology because they are automorphism groups of topologically finitely generated groups; one uses here the fact that in such a group, characteristic sugbroups form a cofinal sequence. Now the really amazing and central point, foreshadowed in the Esquisse, is that IT defined as above is in a certain sense explicitly computable: it is given as a subgroup of Out*(F2) by a small number of relations (say four) which translate into equations on the pair (A, / ) . In fact IT has been computed in [24] and [44], adding one, perhaps not independent relation to the genus 0 version introduced in [9]. Note that the term 'relation' which is commonly used here should not be misleading; IT is given as a subgroup, not a quotient of Out*(F2). We refer the reader to the articles quoted above for (much) more on this, but it may be good to mention Grothendieck's 'principe des deux premiers etages' ('two levels principle') at this point. It says that the generators of T can be found at the first level of M (moduli spaces of dimension 1) and the relations at the first and second level (moduli spaces of dimensions 1 and 2). Indeed we just found the generators by looking at P*, that is Mo ,4 and the first level consists of Mo,4 and M\^ (moduli of elliptic curves) which moreover are tightly related. We also notice that this principle (discussed from a geometric viewpoint in [32]) reminds one of the way one computes the fundamental group of a cellular complex in topology. Essentially by definition there is a natural inclusion GQ C IT; see the contribution of Y.Ihara in [11] for a proof from the viewpoint of algebraic geometry. Whether or not this inclusion is strict is a main driving question of this young field. Note that the situation is in some sense opposite to that of F.Pop's result quoted above. Throwing in more, or say different objects he proves the remarkable isomorphism: GQ ~ Out(nfeom(C)), giving in principle a geometric characterization of the arithmetic Galois group GQ. Yet as mentioned above we know nothing concrete about the right-handside, so it does not immediately help study the left-hand-side. On the other hand, using the Teichmiiller tower as above, we get that GQ C IT = Out*(T_). Here in some sense we are able to 'compute' the right-hand side, but we do not know whether or not the inclusion is an isomorphism. Although it took quite some time to complete the picture sketched above, it should still be considered as a rather primitive stage of the the-

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ory. True we have used moduli spaces of curves of all finite types, especially all genera, and we have used the full profinite completions, two important positive features. But we have used essentially only the structure of the modular tower at infinity. So we get what can be called a (GrothendieckTeichmuller) lego at infinity or parabolic lego to take up the terminology of the Thurston-Bers classification of diffeomorphisms. Prom this point of view, one can ask for a different and probably much more subtle sort of lego, connected in particular with the automorphisms of curves (so that it could be termed elliptic lego) which is actually the only one mentioned in the Esquisse and would encode a lot more arithmetic than the one at infinity. In particular, one has to enrich the modular tower in a drastic way, probably throwing in objects and morphisms which are defined over Q, not only over Q. Each such morphism is actually defined over a finite extension K of Q and leads to an equivariant action of GK , that is an open subgroup of GQ which however effectively depends on the particular morphism one is looking at. This is elementary Galois theory, but the counterpart is lacking at present on the Grothendieck-Teichmuller side. Since the exploration of these tracks is hardly beginning we prefer to stop here and refer the reader to recent texts which contain first results and try to isolate relevant features and objects (see in particular [33], [38], [39], [50], [51]). Remark: The Grothendieck-Teichmuller action is studied in [24] and [44] in the context of the modular tower M and the attending Teichmiiller tower F = 7i^eom (M) in the version described at the beginning of this section, that is 'at infinity' and 'over Q'. This is the only case to-date where a reasonably complete picture has been obtained outside of genus 0. These articles may not however look so user friendly and it might be helpful to the potential reader to get a few general clues as to the techniques. A main point is that 'complexes of curves' as introduced by W.Harvey in the sixties and actively studied by topologists since then describe the structure at infinity of moduli spaces of curves (see [23] for an introduction). In genus 0, they translate into the much simpler McLane's coherence relations for braided categories (see the contribution of L.Schneps in [46] for a careful description of this translation; it is somehow implicit in [9]). A version of these complexes, introduced by A.Hatcher and W.Thurston in a classical paper, is the main tool of [24]. In this way, one captures the structure at infinity of the moduli spaces based on the so-called maximally degenerate points but not the more refined attending tangential base points. Algebraic translation (restricted to the genus 0 situation) in terms of braided categories: one cannot modify the commutativity operator (the

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universal /^-matrix) but concentrates on the 'geometry of associativity', as pioneered in [9]. Galois translation: cne must leave aside the cyclotomic action. Another difficulty which is left aside in [24] is connected with an important map at the first level of the modular tower (already mentioned in [9] for the same reasons): it maps an elliptic curve viewed as a twofold cover of the projective line to the projective line with the four ramification (i.e. Weierstrass) points marked. This implies a change in the uniformizing parameter at the origin of the elliptic curve. In turn this change of uniformizing parameter results in the occurrence of the Kummer character p2 at 2 in the Galois action, as can be inferred from the asymptotics of the Legendre modular function. In particular one needs to devise a Grothendieck-Teichmuller analog of pi in order to pass from the Galois to the Grothendieck-Teichmuller action. Putting these two restrictions together one finds that [24] deals with a subgroup A c F which contains the subgroup of the elements of GQ with x = 1 (x = cyclotomic character), P2 = 0In [44], the strategy of [24] is refined in two ways, in order to get the action of the full Galois group and the corresponding version F of the Grothendieck-Teichmiiller group: First one needs to take the cyclotomic action into account, which necessitates passing from maximally degenerate points to the more refined tangential base points. Topological translation: one has to rigidify, adding 'seams' to the 'pants', which leads to objects which in [44] are called 'quilts' and appear elsewhere under various names in topological contexts (cf. [16]). Second [44] contains a definition and study of the analog of p^. Putting these improvements together, one recovers the full IT action on the Teichmuller tower F. 3. Glimpses of the motivic theory Much of what can be called the linearized part of the theory has now been cast in the framework of mixed Tate motives. This is also where the genus 0 prounipotent Grothendieck-Teichmuller group, which is actually the version which was originally introduced by V.Drinfeld, connects with multiple zeta values. The few words below are meant to motivate the reader for further exploration and to point out a few general features in view of the discussion in the next section. An important starting point was an insight of P.Deligne and Z.Wojtkowiak in the early eighties about the monodromy of polylogarithms; this is how mixed Tate motives were brought to bear on the subject in [10]

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(cf. the quotation in the introduction). At that time the category of MT motives was still conjectural and those were regarded in [10] as "systems of realizations", essentially the Hodge and ^-adic realizations. When k is a numberfield the abelian category MT(k) of MT motives over k was constructed in the early nineties by M.Levine ([34]) and in the late nineties it was pointed out by A.B.Goncharov that one could easily from there define the full subcategory MT{Ok{S)) of MT motives which are unramified (in a suitable sense) outside the set S of primes in the ring Ok of integers of k. These constructions have been used especially for k = Q and 5 = 0 (Ofc(S) = Z). A recent detailed account can be found in [12]. Among the objects of MT(Z), one finds in particular the moduli spaces Moin (n > 3) of genus 0 curves, and among their relative periods all the multiple zeta values ([20]) occur, and may even give all the periods of MT{2J). In fact the Mo,n's conjecturally generate the abelian category MT(Z). Another and indeed more classical way of retrieving the multiple zeta values is via iterated integrals, or what amounts to the same 7r"m(P£), the prounipotent completion of the fundamental group of P£. That the latter group can be viewed as an object of MT(Q) is explained in detail in [12], amplifying [10] and previous work by A.B.Goncharov and others. Once the category MT(B) of MT motives over a base scheme B has been constructed, one gets (unconditional, i.e. free from conjectures!) access to a paradise where things indeed go smoothly and are often reduced to linear algebra, modulo categorical work. Note that as first explained in detail by M.Levine, the construction of MT(B) as an abelian category with all the desired properties rests in an essential way on the truth of the Beilison-Soule vanishing conjecture for B, which gives control on the extension classes Ext1(Q(n), m. Taking the graduation Grw associated to this weight filtration, by the very definition of Tate

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objects, the graduate pieces are sums of copies of pure objects Q(n) of the appropriate weight. For compatibility reasons with Hodge theory, or say an appropriate interpretation of Cauchy formula, one assigns Q(l) the weight - 2 , so that Q(n) has weight -2n and for any M £ MT(k) and n £ Z, Gr™2n{M) is a sum of copies of Q(n), which one can write explicitly as Gr%n(M) = Q(n) ® Hom(Q(n), Gr™2n{M)) = Q(n) w n (M), where the latter equality defines wn. Denoting by w* the direct sum of the u>n one gets a functor M —> - w»(M) from MT{k) to the category of graded vector spaces, which respects the tensor product. Finally there is a natural action of the multiplicative group G m on the pure objects, with Q(l) giving the standard (identity) representation, hence Q(n) the n-th power (A H-> A"). Note that the category of finite dimensional graded vector spaces is equivalent to that of finite dimensional representations of GTO (here one can consider Q-vector spaces and regard G m as a group scheme over Q). By forgetting the graduation we pass from a;* to w, which is a <S)-functor from MT(k) to the category of finite dimensional vector spaces, exhibiting MT(k) as a Tannakian category with fiber functor w. So by the Tannakian dictionary, MT{k) is equivalent to the category of the representations of a proalgebraic fc-group Gu = Aut®(u>). There is a natural surjection Gu —> G m where G m acts on Q(l) = and in fact on all pure objects (sums of copies of Q(n)). The kernel is a group Uu which is prounipotent. Indeed by definition it respects the filtration W, or rather its image via ui and it acts trivially on the associated graduation Grw, since it acts trivially on pure objects. Temporary conclusion: MT(k) is equivalent to the category of the finite dimensional representations of a linear proalgebraic group G^, which is an extension of GTO by a prounipotent group Uw. This is an extremely general result, which rests solely on the vanishing of the appropriate Sxi-groups, leading so to speak to triangular matrices. Further, the functor which to a prounipotent algebraic group associates its Lie algebra is an equivalence of categories. This is explained in many places; apart from the papers we quoted already, a nice place is the Appendix of the classical paper by D.Quillen on Rational Homotopy Theory. Gain: the Lie algebra is easier to deal with for computations and it is explicitly an affine space, i.e. the spectrum of a polynomial ring. Things are even better because of the vanishing of the higher Extgroups, which is part of the Beilinson-Soule conjecture. The vanishing of Ext2 of pure objects is actually enough to conclude that Uu is free. Moreover, generators are provided by suitably lifting bases of the vector spaces £'a;t1(Q(0),Q(n)) for varying n > 1. We refer to e.g. [25] or [12]

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for detail and will briefly return to this fact in §3.1 below. As a result, the category MT(k) (as well as MT(Ok{S))) is equivalent to that of the linear representations of an algebraic group G = Gw which is an extension of G m by a free prounipotent group U = U^. We insist that this important structure result is a consequence of very general principles which it may be useful to repeat in a few words: First there is a natural filtration which is derived from the existence of the weight filtration on the Hodge realisation and the fact that the latter is fully faithful. Next the naturality of the filtration implies that it is respected by the action of the motivic galois group. Moreover the very definition of a MT category shows that the pure objects, hence the graded pieces of the graduation associated to the filtration, are copies of Q(n), so that the Galois group of the pure motives is isomorphic to Gm. Finally, in order to construct the category MT(B), one uses the truth of the Beilinson-Soule conjecture for the given base scheme B, which immediately implies the freeness of the unipotent radical U of the Galois group. It is not our purpose here to detail the conjectures and results which are more or less tightly connected with the motivic viewpoint. Fortunately this has been done recently in several complementary ways. We refer to [1] for a careful analysis of the conjectures and their intricate connections from a motivic viewpoint (see also [25]). A detailed and prospective discussion of the Galois picture can be found in [28] and [14] discusses various aspects of the MZV's in this light. In particular the Galois and Hodge-de Rham aspects of the prounipotent version of the Grothendieck-Teichmiiller group appear quite clearly in [28] and [14], along with many references. The group GT introduced by V.Drinfeld in [9] is again an extension of Gm by a pronipotent group GT1 and in fact, by concatenating several conjectures together, one is led to predict that it should be isomorphic with the Galois group of MT(Z), the two groups being viewed as proalgebraic groups over Q. This would in particular unravel the structure of GT since as explained above the Galois group of MT(Z) is an extension of G m by a free prounipotent group whose generators are determined by the (known) structure of K(Z) <8> Q, the nontorsion part of the if-theory of Z. Such an identification would have enormous consequences of all kinds, both on the Galois and on the Hodge side, including for instance on the transcendence properties of the MZV's. Apart from the papers already quoted we refer to several recent papers by A.B.Goncharov for much more on this and related themes.

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4. Nonlinear? Indeed how can this adjective be understood in this context? We will discuss a series of topics coming from various regions of mathematics which are all relevant in the context of Grothendieck-Teichmuller theory and will hopefully provide a somewhat coherent and perhaps tantalizing picture. Let us also make it clear again that the 'linear' (in one way or another) techniques are at present more sophisticated and in some sense more powerful than the nonlinear ones, a statement which after all may also apply to such fields as dynamical systems or mathematical physics in general. 4.1. Pro finite versus

pronilpotent:

A first and basic contrast lies between the types of profinite groups mentioned in the title. In this subsection we briefly take up this profinite setting which was implicitly used in Section 1 and will return to the proalgebraic setting, already mentioned in Section 2, in the next item. These two subsections thus deal with various topologies on groups. One of our main objects of study is of course the study of arithmetic actions on geometric fundamental groups, that is of maps of type Gk —> Out(Trfeom(X)) with X a scheme (stack) over the field k (replace Out by Aut in the pointed case). In practice some groups appear naturally endowed with the profinite topology; main class of examples: the arithmetic Galois groups GkOthers can be seen naturally as completions of discrete groups; main class of examples: the geometric fundamental groups Trfeom(X). Here, under rather general assumptions on X, Trfeom(X) is the completion of the discrete group irl°p(Xan) where Xan is an analytification of X. Recall that 1) this depends on the choice of an embedding of the groundfield k into C, 2) again under rather general assumptions {e.g. X quasi-projective) the discrete group rr\op(Xan) is finitely generated. For the sake of clarity in §3.1 and §3.2 we will often use the letter V to denote a discrete finitely generated group and the letter G for profinite groups. It is useful to bear in mind the elementary fact that if G = T is the profinite completion of Y, V carries more information than G since G can be reconstructed from T but not vice versa: the topological fundamental group carries more information than the geometric fundamental group. In his Longue Marche, Grothendieck suggests that given G, say topologically finitely generated, it would be interesting to study the set of its discretifications, i.e. the set of discrete finitely generated T C G such that T = G. Such a F he calls a discretification of G, which is the nonlinear analog of

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an integral structure (say on a vector space). Given the importance of the integral lattices in Hodge theory, this seems like a natural object to study although nothing substantial seems to exist at present on the issue. Let us recall some very basic features of profinite topologies before returning to Galois actions, referring the reader to [47] for a recent thorough treatment with references. A profinite group G is an inverse limit of finite groups, which thus appear as finite quotients of G. If F is a discrete group, its profinite completion G = T is denned as the inverse limit of the system (F/JV) where N runs over the normal subgroups of finite index of F. If F is finitely generated, the sequence of characteristic subgroups N is cofinal so that one can restrict to these in the inverse system denning G. The pronilpotent quotient Gnd of G is obtained by restricting to the subsystem of nilpotent finite groups appearing in the definition of G. A first important feature is that primes do not interact in Gml, so that the adelic picture is enough to deal with the pronilpotent world. This rests on the following elementary (but not so easy!) lemma: Any nilpotent finite group is the direct product of its Sylow subgroups. One thus gets that any pronilpotent group is a direct product of pro-p groups: the situation can be decomposed along the primes and these do not interact. Starting from a discrete F, its pro-p completion G^ for p a given prime is of course defined as the inverse limit of the F/7V with F/7V" a finite p-group and its pronilpotent completion is the direct product of the G^ 's. It may be useful to add a word about the connection with the descending central series. Given a discrete group F one defines as usual the (lower) central series (F(n))n>i by: F(l) — F, F(n 4-1) = (F, F(n)), the group generated by the commutators of elements of F and F(n). Let F be the inverse limit of the quotients F/F(n). This group is a kind of partial pronilpotent completion; it is however in general not complete, and this is why we use the letter F. One can prove that the natural map F —» Gnli [Gml the pronilpotent completion of F) is injective. In particular, F is nilpotent residually finite, i.e. injects into Gml, if and only if the natural map F —> F is injective. This is the case for finitely generated free groups and for Teichmuller modular groups. These groups also inject into their pro-p completions for any odd prime p. Profinite groups and actions associated to them seem very difficult to study in general. For instance, letting again F2 = Z * Z denote the free group on two generators, Fi its profinite completion, its automorphism group Auti^F-i) (which contains Gq; see Section 1) seems rather intractable as such (contrast with the discrete case: Out(F2) — GL2CZ)). Essentially nothing is known about that group. For instance in the profinite case it

239 is extremely difficult, usually impossible to-date, to determine whether an explicitly given endomorphism is invertible; this is much more doable in the pronilpotent setting. In general, pronilpotent and prounipotent groups are much more amenable to computations for several reasons, so that it is tempting and useful to study nilpotent quotients first. Of course that entails a big loss of information: in terms of Galois groups, one can only hope to capture nilpotent finite extensions. We already mentioned that one is essentially reduced to studying the pro-p situation for varying prime p. The next big simplification comes from the fact that in the pro-p (or pronilpotent) world, there is no difference between projectivity and freeness; see for instance [52], §1.4, for the basic properties in that direction. In particular, if G is a pro-p group, not only can one test freeness cohomologically, but it is enough to do so with Z/p coefficients and trivial action: G is free if and only if H2(G, Z/p) = 0, in which case Hl{G, Z/p) describes generators (cf. op. cit.). This is exactly the strategy which we have seen in action above (cf. Section 2) in terms of mixed Tate motives, under slightly different circumstances. Compare with the full profinite situation and especially the famous Shafarevic conjecture; Shafarevic himself proved that Gqab is projective but it is a long standing conjecture that this group is profree. A third feature of the nilpotent framework is that one can use Lie algebras and the attending toolbox, in particular derivations, the infinitesimal form of automorphisms. We will detail this linearization for Galois actions and refer to [42] for more on pro-p groups. So we return to the action Gk -> Out(n9leom(X)), where G = 7rfeom(X) is topologically finitely generated and can be viewed as the profinite completion of F = TT1op(Xan). The following construction was first explored by Y.Ihara in [27]; see [28] for a first-hand survey and references. These studies concern the case k = Q, X = P* which remains of central interest; the information in other cases, especially higher dimensional ones, remains much more fragmentary and much less detailed although much of the 'general nonsense' adapts with no problem. Because G is topologically finitely generated, Aut(G) is naturally endowed with a profinite topology; because Gntl is a characteristic quotient, and indeed so are the pro-p quotients G^p\ one may study the pro-p actions <^> = G^p\ An important theorem of Grothendieck implies that the latter morphism factors through the group Gal(M/k) where M is the maximal extension of k unramified outside the primes lying over p. More precisely this is true when X is complete or a curve e.g. P*; the case of a regular higher dimensional noncomplete X is somewhat more subtle and there does not seem to exist a good account

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of it in the literature at present. So if M* denotes the fixed field of the kernel of cj>, one has M* C M where we drop the dependence on p from the notation. Next one forms the derived sequence (G(n))n>i and the quotient (G/G{n))n>i. Here G = G^ is now a pro-p group, e.g. the pro-p completion F2(p) of F2 in the case X = P*. Let <j>n : Gk -> Out(G/G(n)) denote the corresponding actions, which exist because G(n) is characteristic in G. Letting M* denote the fixed field of Ker((f>n), we get a nested sequence: k C Mi C ... C M* = D n >i Mn C M, and a corresponding filtration of G = Gal(M*/k) by the Gn = Gal(M*/k). The corresponding graduation is given by grm(G) = Gn/Gn+1 = Gal(MJMn^), gr(G) = @ngrn{G). Wefinallygot to the object which is most amenable to a detailed study. In particular gr(G) is a graded Lie algebra over Z p with the bracket inherited from the commutator in the group G, using that (Gm, Gn) C Gm+n to ensure that it is well-defined. In the case X = P*/Q, it was shown in [27] that each grn(G) is a free Zp-module of finite rank. Adding two conjectures to this tantalizing picture, it is believed that the rank of grn(G) is independent of p (an assertion of motivic flavor) and that M* = M. These results (resp. conjectures) are valid (resp. tantalizing) for more general X. At this point we refer the reader to [28] and references therein for further action: we have just reached the foot of this as yet largely unexplored range of mountains. The connection with the Grothendieck-Teichmiiller group, that is a pro-p, genus 0 version of it, is summarized in Lecture II of [28]. This process also has a lot in common with the motivic approach of section 2; indeed, after it has been rephrased in terms of linear representations and proalgebraic groups, it may in large part be viewed as the Galois side of the motivic picture, as already largely foreshadowed in [9]; see [14] for a careful description of the picture. To summarize, through this linearization process we have gained the possibility of using a lot of powerful tools, sometimes leading to concrete computations: Lie algebras, cohomology of pro-p groups etc. It also makes contact with the motivic picture, thus (in the case of P*) with multiple zeta values etc. On the other hand on the Galois side one has to restrict (at best) to the Galois group of the maximal extension unramified outside p (of course that should provide enough work for several generations...) and the corresponding quotient on the Grothendieck-Teichmiiller side. We finally remark that the importance of the geometry of the modular tower may well be one of the deepest intuitions in the Esquisse. Here it enters only (at least at present) through the structure of the tower of pro-p braid groups

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(see [28], Lecture II), that is through the geometry at infinity of the genus 0 case (see also the contribution of P.L. and L.Schneps in [11]). 4.2. Group actions versus linear

representations:

Taking things somewhat in the reverse order, a Tannakian category is equivalent to the category of the linear representations of a proalgebraic group, namely the fundamental group of the given category, after fixing a fiber functor. Grothendieck's classical (SGA 1) theory of the fundamental group uses a kind of nonlinear version: instead of linear representations, it establishes a dictionary between the category (Cov/X) of finite etale covers of a scheme X (or stack; see [43]) and the category of finite sets with TTI(X)action. Although one can in principle linearize by passing from the action of a group on a finite set to the associated linear permutation representation this is not an innocuous operation. Turning again to Grothendieck-Teichmuller theory, it deals first and foremost with the action of arithmetic Galois groups on geometric fundamental groups in conjunction - and here lies the charm of the situation with the geometry of the modular tower M. In particular one starts from the collection of stacks MgiTl and their discrete fundamental groups r*°£, which are finitely generated and residually finite discrete groups. It is of course quite natural to study the linear representations of these groups, associated monodromy operators etc. These and other topics are the subject matters of topological quantum field theory and related subjects; we refer to [5] for a survey with references. In particular conformal blocks in conformal field theory provide a linear form of lego, as first pointed out and explored in the classical 1989 paper of G.Moore and N.Seiberg. Of course this is only a form of the lego at infinity, to take up the terminology of Section 1 above. There also exists for instance an extended form of the Tannakian formalism for braided tensor categories. The point we want to make however in this subsection is that, when it comes to arithmetic fundamental groups and in particular Galois actions, it is quite hard to use linear representations, again beyond the prounipotent (motivic) setting. This can be seen as a variant of what has been pointed out above and one way of phrasing it more precisely is as follows. Let F be a discrete finitely generated and say residually finite group in order to fix ideas. By definition the proalgebraic completion (or envelope) of F over the field k (assume char(k) = 0) is thefc-proalgebraicgroup G°'9 whose category of finite dimensional linear representations over k coincides

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with that of F. In other words any morphism F —> GLn(k) (for some n) factors through Galg. There is a proalgebraic form of the Levi decomposition which asserts that Galg is a split extension of the proreductive group Gred by its prounipotent radical G u m , all groups being linear proalgebraic over k. The group Gred (resp. Gunl) can also be seen as the proreductive (resp. prounipotent) envelope of F and can be defined directly in the same way as Gal9, by considering reductive (resp. unipotent) linear representations of F. The prounipotent part Gunl can be fairly well understood; the point here is that Gred often looks intractable, or at least not easily amenable to computations. This last sentence can be illustrated by means of a theorem of J.Tits (see [56]) which implies the following: Let G be a semisimple algebraic group over k; then there exists a countable infinite set F inside G(k) such that any two elements of F generate a Zariski dense free subgroup of G(k). Now consider the case F = F2; by the above result one finds among the quotients of its reductive envelope F%ed a countably infinite number of copies of any linear semisimple algebraic group over k. It does seem hard to use such an object in a concrete way. The situation does not improve of course if one considers the proalgebraic completion of a profinite group such as GQ , although this group GQ9 occurs almost by definition if one studies Galois representations and is connected with the motivic Galois group of . The (needless to say temporary and provisional) conclusion seems to be that when studying of geometric Galois actions, one can either work in a profinite setting, or if one goes to the proalgebraic setting, one almost inevitably has to stick to a prounipotent or almost prounipotent framework. Here 'almost prounipotent' refers primarily to a series of works by R.Hain (see [25] and references therein) in which the author(s) investigate(s) extensions of reductive algebraic (as opposed to proalgebraic) groups by prounipotent groups. The case where the reductive quotient is simply the multiplicative group Gm is of particular interest and occurs naturally in various situations, as examplified in Section 2. It seems however difficult to tackle situations which involve proreductive envelopes of 'really' nonabelian groups in an essential way. Finally we note that in the pronilpotent and prounipotent cases, the dictionary between profinite and proalgebraic settings is well established. We refer in particular to [10], §9 and references therein. Let us just state some bare facts in a basic but representative situation; the proofs are essentially formal. Let F be a nilpotent torsionfree finitely generated group, G = F its profinite (equivalently pronilpotent) completion. On the algebraic side,

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one constructs the Q-prounipotent completion Gunl = Galg of V over Q. Let I be a prime and consider Guni(Q*), the group of Q^points of Guni. Then there is a natural injection T •—> Guni(Qe), the closure of the image of F is compact open and it is isomorphic to the pro-f completion G^ of F, that is the pro-^ part of G, which is the direct product of the G''''s. This statement provides the basic connection between the pronilpotent and prounipotent objects. 4.3. Good groups versus rigid ones: Let F be a finitely generated and residually finite discrete group, G = T its profinite completion and j : T •—> G the natural inclusion. Let M be a finite abelian group (in other words a finite Z-module). The inclusion j induces a map on the cohomology with constant coefficients in M (that is with trivial T and G actions) j * : H*(G,M) -> H*(T,M). Following J-P.Serre ([52], §2.6) the group T is called good if j * is an isomorphism for any finite M. By taking direct limits on the coefficients, this holds true for any torsion abelian group M. Consider the special case where V = ir-fp(X) and X is a classifying space, that is a K(T, 1) and is defined over a field embedded in C, so that G = 7rfeom(X). There is a natural isomorphism H*{X,M) ~ H*(T,M), where X = X(C) is viewed as a complex variety. If j * is surjective, any cohomology class in c £ H*(X, M) comes from a class in H*(G, M); since G is profinite any such class vanishes when restricted to some open subgroup. This translates into the fact that there exists a finite etale cover TT : Y —> X (here X = X^om = X ®k), such that 7r*(c) = 0. We thus find that X has many finite etale covers in the sense that given any cohomology class one can find a cover such that the pullback to it of the given class vanishes. The above considerations could apply to the moduli spaces of curves Mg^n with some qualifications; namely they are Q-stacks and thus orbifolds when viewed analytically. As a consequence, in the above one should consider only such M whose elements have orders prime to the cardinals of the automorphism groups, which here are the just the automorphism groups of the curves of type (g,n), whose order is bounded by the classical Hurwitz bound. However the main stumbling block here is that it is unknown whether or not the mapping class groups F*°£ are good for g > 2 (see note added in proof). We return to this important point below but note for now the connection between goodness and the abundance of etale covers. This was actually a motivating geometric property and can be used (SGA 4,

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§XI.4.6) in order to show that the etale topology is reasonable in the sense that any point on a scheme has a basis of acyclic neighborhoods. Let us go on exploring goodness a little further and give some basic examples. We refer to [41] for the precise statements and proofs. First goodness behaves well under extension: start from an exact sequence of discrete groups: l-+K^>E-4H^l.

(*)

Assume that H and K are good and K is of type FP (it has a finite projective resolution); then the sequence (*), obtained from (*) by profmite completing each group is exact and the extension group E is good too. Since the completion functor is right exact, only the left-hand injectivity is in question in the first assertion; the second assertion is a direct consequence of the Hochschild-Serre spectral sequence computing the cohomology of E. In particular a group F is good if it has a good subgroup of finite index (use the above and the fact that finite groups are good); in geometric parlance goodness can be thought of up to finite etale covers. This also implies that we can accommodate non trivial continuous actions in the cohomology, that is the case where F acts on M via a finite quotient and so its completion G = F acts continuously on M. So in fact if F is good H*(G, M) ~ H*(T, M) for any torsion module M acted on by F via a finite quotient. Next say that F isfc-goodif Hk(G,M) ~ Hk(T,M) for M as above (equivalently all M finite and trivial action). It is important to note that all (FPQO discrete) groups are 1-good; in fact a 1-cocycle for the trivial action is just a morphism and is determined by its values on the dense subgroup F of G. In dealing with the sequence (*) and its term-by-term completion (*) we actually had to require only that H and K be 2-good. As is often the case with cohomology, this condition on the #2-groups is the most often encountered in practice. Examples: Finite groups are good; (finitely generated )free groups are good because they are of cohomological dimension 1; braid groups and genus 0 mapping class groups are good because they are constructed as successive extensions of free and finite groups; 5Z,2(Z) is good because it has a free subgroup of finite index. Well-known and important conjecture: the mapping class groups (or Teichmiiller modular groups) F*op are good for all g. Note that F^i ~ SX2(Z) is good. Easy exercices: F2 is good and if Tg is good, the pointed group F Sjn is good for any n > 0 (one could also accommodate permutations of the marked points). As mentioned above, for classifying spaces, goodness does in some sense measure the abundance

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of etale covers, which is also connected with the richness of the Galois action and the attending anabelian phenomena. This is dramatically illustrated by the difference between curves and abelian varieties in this respect and this may well be the main point of the present subsection. To see this we first introduce a main class of counterexamples: higher rank arithmetic groups are not good in general. This is a consequence of the rigidity of such groups as we illustrate on examples leaving it to the reader to generalize. Specifically SLn(Z) (resp. Sp2g(Z)) is not good for n > 2 (resp. g > 1). A proof for SLn(Z) goes as follows and is easily adapted to the case of Sp2g(Z)- The first and crucial step consists in using the (MennickeBaas-Milnor-Serre) congruence subgroup property which translates as: SXn(Z) ~ SXn(Z); here and below n > 2. Next and essentially by definition, SLn(Z) ~ Y[pSLn(Zp) where the product runs over the prime integers. Finally SXn(Zp) has a nontrivial center whenever the n-th roots of unity belong to Z p , that is when p is congruent to 1 modulo n, which by Dirichlet theorem happens infinitely often. Conclusion: the profinite completion of SLn (Z) has an infinite torsion center and thus does not have virtually finite cohomological dimension. By contrast the discrete group SLn(Z) is virtually torsionfree and does have virtually finite cohomological dimension. Hence it cannot be good. It is plain from the above that we used first and foremost the fact that SLn(Z) or Sp2g(Z) has relatively 'few' open subgroups. Let us push this a little further, contrasting curves (the nonlinear side...) and abelian varieties (see also the contribution of Y.Ihara and H.Nakamura in [21]). On the side of Mg we have a classifying stack with conjecturally good topological fundamental group and many etale covers. In fact another topological conjecture, perhaps less widely believed than the goodness conjecture, predicts that a cofinal sequence is given by the geometric congruence subgroups, whose construction we do not recall here. In the case of Ag (g > 1), the classifying stack of principally polarized Abelian varieties, the topological fundamental group Sp2g(Z) is not good as indicated above, there are few etale covers and the associated action of the arithmetic Galois group does not carry much information. To wit, any finite etale cover of Ag is actually defined over Qab (the maximal abelian extension of Q) so that the action of GQ actually factors through Gal(Qa6/Q) ~ Z x and reduces to the cyclotomic character. By contrast, the moduli spaces of curves are conjecturally anabelian, which at any rate involves the fact that the Galois action is quite 'rich' (cf. loc. cit for a discussion of anabelianity' in the higher dimensional case). Note however that the Modular and the Abelian towers share the

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first level and that the arithmetic Galois action is already faithful there (i.e. on SL^Z)). The similarities and contrasts between mapping class groups and linear arithmetic groups have provided a rich theme in the literature since the late sixties and the above adds to the list of contrasts. We will not elaborate on the very embryonic theme of higher dimensional anabelian geometry but would like to mention that this phenomenon can also be partly connected (not only analogically) with all sorts of analytic and topological rigidity properties. We refer to [40] for very interesting results in this direction, which involve goodness, Artin good neighborhoods etc. FinaDy we mention the similarity with the purely topological long-standing Borel conjecture which asserts that a closed manifold which is a K(G, 1) is determined up to homeomorphism by that property (see [15]; note that it implies Poincare's conjecture). One can wonder for instance whether a Q-stack which is the classifying space of a finitely generated good and universally centerfree group is anabelian. Of course such a general and not even completely precise statement should not be taken too seriously but it may help making expectations a little more precise. 4.4. Amalgamation versus extension: Extensions of groups, modules or more generally of objects in abelian categories are one of the main tools which enable one to analyze a situation in terms of the associated 'simple' objects. For instance nilpotent (or solvable) finite groups are obtained by definition as successive extensions of finite abelian groups, which are easily classified. As another example the simple objects of a Tate category, that is the pure Tate motives, are by definition also completely classified and easy to list and analyze, including from the Galois viewpoint. The same holds true in principle for pure Hodge structures etc. Moving back to our favorite moduli stacks and their geometric fundamental groups, one finds that the genus 0 situation can be described in terms of extensions, although this may not always be the most revealing description. This rests on the fibration Mo,n+i —> Mo,n (n > 2) obtained by erasing a given labelled marked point and which can be viewed as the universal geometric monodromy fibration, since Mo,n+i ~ Co,n is the universal curve over Mo,n (which is a scheme since curves with labelled marked points have no automorphisms. So the situation is really simple here; no stacky niceties involved). Now one can consider the homotopy exact se-

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quence associated with that fibration and recall that Mo,n is a classifying space, so that all higher homotopy groups vanish. As a result, one finds that ro,n+i is an extension of Fo,n by the fundamental group of the fiber, which is nothing but .Fn_i, the free group on n — 1 generators. This unravels r o , n as a successive extension of free groups and rO|[n] is nothing but an extension of the permutation group Sn by Fo,n. Note that already here, free groups that is amalgamations of copies of Z have occurred. In fact F n -i came in as the fundamental group of the sphere with n points removed, which can be computed by induction using Van Kampen theorem, thus leading to amalgamation. In this subsection we would like to stress the importance of amalgamation in general as a way of describing the group theoretic Teichmuller tower from its constituents. Amalgamation occurs in terms of fundamental groups through various forms of the Van Kampen theorem, just as naturally as extensions occur in homological algebra through the various types of long exact and spectral sequences. Dealing with amalgamation and in order of increasing generality, one finds free products, amalgams of groups (and HNN extensions) , the Bass-Serre theory of the fundamental groups of graphs of groups (the standard reference being [53]), andfinallycomplexes of groups of which we will say but a few words at the end of this paragraph. We are especially interested here in comparing the discrete (or say abstract, that is paying no attention to topology) and the complete (or profinite) theory, the latter being actually fairly recent and still fairly incomplete. The first piece of relevant information is that any mapping class group r t o fi can be written as the fundamental group of a finite graph of groups involving only mapping class groups of strictly smaller modular dimensions (d(g,n) — dim(Mgt[n}) = 3g — 3 + n), provided d(g,n) > 2. This can be seen as a reflection of Grothendieck's 'two levels principle' (see [32]) and should perhaps be written up in detail. It goes roughly as follows (there are several possible variants): Consider the topological surface 5 g>n of finite hyperbolic type {g,n). Associated to it is the complex A = AgiTl based on the so-called cut-systems, that is multicurves 7 which are such that the result of cutting 5S]TJ open along 7 produces a sphere with holes. Such a 7 is a union of g simple loops, so one gets a (not locally finite) complex of dimension g — 1 (see [23] for detail and references), which is intimately connected with Schottky uniformization. Now F*°£ acts on Ag,n and it is easy to see that one can take one simplex of the first subdivision of Ag 2 ([23], Theorem 6.4; the full result

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is actually more precise). This makes it possible to apply classical results (especially [54]) in order to write a presentation of T*°£ as the fundamental group of a graph of groups. The groups which appear are the mapping class groups of surfaces obtained from S9,n by cutting along part of a cut-system, plus finite groups coming from symmetries of graphs and permutations of points. Concerning the latter one should in fact introduce the partially colored groups F*°£ where a C Sn records the sugroup of the 'allowed' permutations. One recovers Tt0?, by choosing a = Sn. All these groups are extensions of finite groups by F*°£ and here the real sticky point consists in understanding the bare groups Tg (n = 0) for g > 3. The T*°P simply occur naturally as vertex groups of the graphs, which it is a good exercise to write down explicitly in the case of F 3 and IV We insist that at a geometric level, this construction again reflects the geometry at infinity of the M5ira, giving rise to the stable stratification etc. Now what happens upon completion? Here we will discuss the relevant case (for us) of a finite graph and completion of the vertex and edge groups (see [58] and references therein). One can also treat the case of profinite graphs, where surprises do occur (see a later paper by the authors of [58]). So let C be a finite graph of groups, with vertex (resp. edge) groups Gv, v £ V = V(C) (resp. Ge, e € E = E(C)), and write G = TTI(C) for its fundamental group where we do not specify the choice of a base point, e.g. a maximal tree of the underlying combinatorial graph, which plays no interesting role here. Here the Gv, Ge and G are finitely generated discrete groups and we assume that the structure morphisms je : Ge —> Gv are injective; it is then part of the theory that the natural morphisms jv : Gv —> G are also injective. Consider now the term-by-term completed graph C, obtained by profinite completion of the vertex and edge groups, as well as the maps je connecting them (one can also use other sorts of completions, along admissible classes of finite groups). So C is defined by a finite projective system of maps je : Ge —> Gv. The fundamental group G — TTI(C') exists, with structure morphisms j v : Gv —> G. In particular one can view G as the limit of the finite system denned by C, that is any system of compatible maps from C to a profinite group H factors through G. But several very serious difficulties now arise: there is no guarantee that either the maps je or the maps jv are injective (even assuming as we do that the je are injective); G is a quotient of the full completion G but there is no guarantee that it coincides with it. In fact one can give a fairly explicit description of the topology of G: It is obtained by declaring open the subgroups H of G such that all the preimages jJ"1(G) C Gv have finite

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index in the Gv's. The standing condition 3.6.A. in [58] consists in assuming precisely that the je and jv are injective. Let us assume that it holds true and on top of it that G ~ G; we will discuss below the meaning of these assumptions in the case of mapping class groups. In this ideal situation, there is a lot that can be said. In particular, if the constituent groups Ge and Gv are good, then G is good (P.L. and V.Sergiescu; unpublished). In order to prove this, one compares a discrete and completed long exact sequence, much as one does with the relevant Hochschild-Serre spectral sequences in order to show that goodness behaves well under extension (see §3.3 above). Here the relevant long exact cohomology sequence is obtained in the completed case from the short exact sequence of complexes in [58] (3.8, bottom row of the diagram) which is the analog of the discrete sequence first emphasized by I.M.Chiswell. So in particular, given the above, if the assumptions above hold true, one easily derives the goodness of the groups r*°£ by induction on the modular dimension. More generally it would be a very significant step in trying to unravel the structure of the profinite Teichmiiller modular groups Vg [nj. We will now examine how these assumptions translate in our case into a nice and apparently difficult geometric problem which it may be interesting to state in some detail. We outlined above how the discrete modular groups (mapping class groups) can be viewed as fundamental groups of finite graphs built out of modular groups of strictly inferior dimensions and finite groups. Moreover the injective structure maps je and j v corresponding to edge and vertex groups are basically all of the same type: Letting as usual Sgtn denote the topological surface of type (g, n) and 7 a non disconnecting multicurve [i.e. a sub-cut-system), split Sgin along 7 and consider the injection of the modular group of the split surface into the original one. All in all, after using induction on the number of curves in a multicurve and a few simple manipulations on extensions involving finite groups etc. one shows that the only serious question is as follows. Let 7 be a (possibly separating) simple closed curve on the topological surface S9ln, and let Z1 be the centralizer in the discrete modular group of the Dehn twist r 7 along 7. We thus have an inclusion j : Zy <—* r*°P, which gives rise after completion to a map j : Z 7 <—> T9tn- The question is simply: Is j injective? We emphasize that modulo the results mentioned above a positive answer would vindicate all the assumptions about completed graphs, would imply the goodness of the discrete modular groups etc. Let us dwell a little more on the topological and geometric meanings of that question, by way of illustration of some of the ideas we discussed in the past items. Prom a

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topological viewpoint, we find first that the centralizer Z 7 is an extension of the modular group of the split surface (with the holes collapsed to marked points) by the cyclic group generated by the twist along 7. In other words we have a short exact sequence:

i^z-z^r^^-i.

(*)

Note that the action of T °Z\ n+2 on Z = (T 7 ) is the monodromy action and that the extension class c e H2(Tgopln+2,Z) corresponding to (*) has a clear geometric meaning. Indeed, moving to a more geometric and modular picture, consider the complex moduli space Mg,n(C), its completion Mg]7l(C) and Z = Z(^) the component of the divisor at infinity corresponding to 7 (for instance there is one component defined by all non separating 7's). Let Ze be a sufficiently thin tubular neighborhood of Z in Mg>n(C) and consider the intersection Ze \ Z of Ze with Mg
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next step in terms of considering amalgams of (pro)finite projective systems of groups consists in looking at complexes of groups, roughly speaking replacing graphs by higher dimensional CW complexes. We refer to [22], [6] and further papers by these and other authors. They consider only the discrete case and there does not seem to exist at present papers dealing with completion in this context (even for a finite two dimensional complex, say a triangle of groups). It may be that using the special loci in moduli spaces, that is the loci corresponding to curves with nontrivial automorphism groups, mapping class groups can be viewed as fundamental groups of complexes of groups in which only finite groups occur. Passing to the completed groups would then not involve any change in the constituents but might involve difficult questions on profinite spaces (see [48]). This circle of ideas, which perhaps deserves further investigation can be seen as the group theoretic facet of the notions briefly reviewed in §3.6 below. 4.5. All genera versus genus O.We will be brief on this contrast, as it has already been mentioned several times. The modular tower in genus 0 is clearly an extremely interesting object, which by now has appeared recurrently under several guises. The strong focus on this object can in large part be ascribed to V.Drinfeld and his series of groundbreaking articles culminating in [9]. Genus 0 objects are now legion as for instance quasi-Hopf algebras, braided categories and several types of operands (rooted trees and gravity operands). In some sense they describe the 'geometry of associativity' unearthed in the original papers of V.Drinfeld, and which categorically speaking is attached to the familiar McLane coherence relations (and Yang-Baxter equations). The theory of the multiple zeta values is also purely genus 0, at least to-date. Moreover from the viewpoint of Grothendieck-Teichmuller theory, if one takes the automorphisms of curves into account, as one should, the genus 0 setting in some sense tells the whole story: the outer automorphisms of the geometric fundamental group respecting both the inertia at infinity and the automorphism groups in genus 0 coincides with IT, the GrothendieckTeichmuller group at infinity in all genera; we refer to [51] for a precise statement and proof. In the next item, we will return to the emerging duality between automorphisms of curves and stable degeneration. Finally we have seen already that from a group theoretic viewpoint the genus 0 objects are also significantly simpler. For instance all groups occurring (braid groups, mapping class groups with or without permutations) are

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good for simple reasons and generally speaking, although they are far from well-understood (e.g. from a representation theoretic viewpoint), plane and sphere braid groups are much more amenable to computation than higher genera Teichmuller modular groups. Yet it seems that one should really consider the whole modular tower M, as is strongly suggested in Grothendieck's Esquisse, with all its geometric complexity and richness, not only that coming from the stable stratification (see also §3.6 below). And one should also take stacks seriously, that is not kill automorphisms too quickly by adding level structures. In essence, by considering curves as one dimensional Deligne-Mumford stacks, allowing for coverings, quotients by automorphisms, stable fibrations, point adding and erasing morphisms and the modular counterpart of these operations, one arrives at a structure governed by an enriched version of the modular tower which looks like a natural environment for Grothendieck-Teichmuller theory, and in which it can for instance make contact with Thurston's topological vision (see [33] for a preliminary exploration along these lines). 4.6. Stack inertia versus inertia at infinity: As already mentioned several times, the current version of the modular tower M essentially takes care of the structure at infinity (and partially of the universal geometric monodromy, via adding marked points). This can be phrased in many ways: it contains the Knudsen morphisms, it deals with the stable stratification, it is based on multicurves and pants decompositions, it stresses Dehn twists that is parabolic (reducible) diffeomorphisms of topological surfaces (in terms of the Thurston-Bers classification) or, in more algebraic parlance generators of the procyclic inertia subgroups associated with the irreducible components of the divisors at infinity of the moduli stacks of curves. On the other hand, the loci in the moduli spaces representing curves with nontrivial automorphisms have attracted attention quite early; see [7] for a nice modern reference with classical flavor. In the mid seventies H.Popp noticed that they build up stratifications of the moduli stacks in the sense of Zariski, reflecting the singularities of the associated coarse moduli spaces viewed as complex varieties. This viewpoint can be extended to general separated Deligne-Mumford stacks (see [35], Remarque 11.5.2, [43] and [38]), using the basic fact that geometric points of stacks come along with a group of automorphisms. Moreover, using the notion of inertia, one can give a unified treatment of the case of a divisor at infinity ([19]) and of

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these loci with nontrivial automorphisms. So one gets two stratifications of the completed moduli stacks of curves. The stable stratification ('at infinity') is defined over Q, whereas the other one, via automorphisms ('stack inertia') is not. Indeed it carries enormous arithmetic information which remains largely mysterious at present. In many respects the stable stratification is more familiar: it embodies, with a stacky grain of salt, the familiar setting of a regular quasiprojective scheme X which can be written as X = X \ Z with X regular projective and Z a divisor with strict normal crossings. So modulo some stacky nuances, one is in the framework of [19], with each component of the divisor at infinity Z corresponding to a conjugacy class of free procyclic groups Z with Tate twist 1: a topological generator up to conjugacy consists of a small loop 'around' the given divisor and the action of the arithmetic Galois group is that of the cyclotomic character on the roots of unity. The stratification via loci with nontrivial automorphisms (henceforth 'special loci' for short) is of a much more stacky nature; in group theoretic terms it is not about just (pro) cyclic groups but at the topological level reflects the boolean lattice of finite groups contained in the mapping class groups r t o f ,. Here one should recall that TtoF, is virtually torsionfree but apart from a few exception (see [32] for detail and references) it is also generated by its torsion elements. The contrast between these two stratifications deserves to figure in this list of nonlinear-versus-linear items partly because the stratification at infinity is intimately connected with (mixed) Hodge theory and has a distinct unipotent flavor, which can also be called parabolic in dynamical terms, thinking again of the Dehn twists in terms of diffeomorphisms of surfaces. This is however surely not the only relevant feature here and it seems that much remains to be unearthed. We will add a few remarks about the (discrete and profinite) group theoretic aspects and our embryonic knowledge of the Galois and Grothendieck-Teichmuller actions. The Grothendieck-Teichmuller group has been defined as the outer automorphism group of the modular tower respecting the inertia groups at infinity. In the topological setting a result of N.Ivanov asserts that the maximal free abelian groups inside the mapping class group F*°P are precisely the groups generated by the Dehn twists on the curves of a pants decomposition of the model surface of type (g, n). Is there an analog in the profinite setting? Note that one should allow here for conjugates of the groups mentioned above, which in the discrete case are of the same type, not so in the profinite case. Such results would help build up a group theoretic characterization of inertia at infinity. But again our current knowledge on these

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matters is poor; for instance it is likely but at present unknown (for g > 2) whether T9in is actually centerfree (this is now a consequence of goodness; see note added in proof); as a piece of warning recall the discussion for SLn(Z) in §3.3 above. This theme 'free abelian groups and completion' which is connected with inertia at infinity can be paralleled under the heading 'finite subgroups and completion'. Here again, for g > 2, what happens in terms of finite subgroups upon profinite completion of the modular groups is unknown. The finite subgroups of the (discrete) mapping class groups are exactly the automorphism groups of Riemann surfaces and they survive completion (i. e. map injectively into the completed groups) because the modular groups are residually finite. But is it true that any finite subgroup of the r 9 j n arises in this way, i.e. is conjugate to the image of a finite subgroup of F*°£ (again SLn(Z), n > 2 does not satisfy the analogous statement)? In other words is it true that the torsion of the profinite group Tg>n 'essentially' comes from its discrete part? This looks both interesting and hard... It is interesting because one might want to study the modular groups using the lattice of their finite subgroups and such tools as Farrell cohomology (see [4], [48] and [37]) or again complexes of groups as in §3.4. In fact even in the discrete case, and even maybe in genus 0, that is for the TQT,, little seems to have been done, as alluded to in the Esquisse. Note that we know that the mapping class group Fojn] is generated by its torsion, that it is virtually torsionfree ( To,n is torsionfree), that it is centerfree, that the corresponding discrete group is good etc. All this information is lacking in higher genera, and for good reasons. But it would still be interesting to better understand these groups in terms of their torsion. The baby example is PSL2CZ) or SL2CI'): PSL2CZ) contains a subgroup of finite index which is torsionfree and indeed free on two generators; it is just the principal subgroup of order 2, PT(2) ~ F2 = Z * Z, parameterizing elliptic curves rigidified by a level structure which 'kills' automorphisms (and not taking the generic involution into account as one in effect should). On the other hand PSX2(Z) = Z/2 * Z/3, reflecting the structures of the elliptic curves with automorphisms and, more accurately SLiffi) is the amalgamation of Z/4 and Z/6 over the involution Z/2. There does not seem to exist at present a careful description, using say complexes of groups, of the analogous situation in higher dimensions, that is for the spaces MQ^ (n > 4) and the attending discrete and profinite modular groups (see [50] for the elementary description of the automorphism groups in those cases). In principle one would like to have available descriptions of the discrete and

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profinite modular groups by means of -say- complexes of groups which use both kinds of stratifications, the stable one and the one associated with automorphism groups, in the style of the above presentations of PT(2) and PSL2{Z) (cf. §3.4 above). Such descriptions of the profinite modular groups, which seem quite hard to obtain in higher genera, would be relevant to the Galois and Grothendieck-Teichmuller viewpoint. At present, in the Galois case one can prove general stack theoretic results which for instance imply that for any hyperbolic type (g,n), any element of finite order in T3tn (or r g [ n ]) is mapped to a conjugate of itself by an open subgroup of GQ (see [38] for detail and more along these lines). The analogous result is known for the Grothendieck-Teichmuller group in genus 0; see [51] for this and other results on the Grothendieck-Teichmuller action on finite elements. Finally we mention [39] where the Galois action on some specific finite order elements in genus 0 is studied in detail. In closing, we note that in the above we have concentrated on the special loci, or loci of curves with nontrivial automorphisms. There are actually many other arithmetic loci inside the moduli stacks which are interesting from the Galois and Grothendieck-Teichmuller viewpoints, such as arithmetic geodesic curves and arithmetic eigenloci; these are actually used in a Galois context in [39]. We refer to [33] for a general geometric study and more references. Note added in Proof: M.Boggi recently showed (in [3]) that the Teichmuller groups are indeed good (see §3.3 above). This is an important new piece of information which hopefully will be put to use in a near future. Appendix: Belyi's theorem and 'dessins d'enfant' 'Dessins d'enfants' (at least the phrase) were introduced by Grothendieck in section 3 of his Esquisse; they are in line with Grothendieck-Teichmuller theory in the sense that they have to do with curves and somehow also with moduli spaces thereof. They offer the archetype of a situation where arithmetic is (unexpectedly?) encoded in topology. Note that other examples, starting with 'origamis' are discussed in [33]. In any case it is no wonder that this struck Grothendieck, and no chance that in the Esquisse, dessins are discussed after and as a kind of simple illustration of the first mathematical and most 'burning' section devoted to 'Grothendieck-Teichmuller theory'. We refer to [11] for material on dessins; to the best of our knowledge [57] is one of the very rare places, if not the only one, where the connection between dessins and moduli spaces of curves is effectively used.

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A feature of the situation that was most striking for Grothendieck is embodied in Belyi's theorem, of which we give a brief discussion, partly because it is relevant to the above and partly because it may be useful to emphasize the hyperbolic side of it, which is less commonly mentioned than the conformal viewpoint. As appears from the Esquisse and Grothendieck's correspondence, he had somehow anticipated the result but without being completely convinced of its validity nor of course being able to prove it. The fact is that around 1978 G.Belyi gave a surprisingly simple necessary and sufficient condition for a complex curve to be defined over a numberfield. We refer to the contribution of L.Schneps in [11] for the proof and the original reference. The statement and its amazingly simple proof should for instance be compared with the 'amazingly complicated' and roundabout proof of the Shimura-Tanyama-Weil conjecture which in analytic terms provides a very similar characterization of the elliptic curves which are defined over (Q> (not just Q). The original statement goes as follows: Theorem 4.1. (G.Belyi, 1978): A smooth complex curve X can be defined over a numberfield if and only if there exists a finite set S C X of points of X such that X' = X \ S can be realized as a finite unramified cover of P^O.l.oo}. The 'if part was actually known to A.Weil in the mid-fifties and was quite elementary for Grothendieck; see the contribution of J.Wolfart in [21] for a detailed discussion and proof in the spirit of SGA 1, Corollaire X.I.8. The 'only if part became central in Grothendieck's vision which partly materialized under the name 'Grothendieck-Teichmiiller theory'. Paraphrasing the 'only if part, the first message is that all smooth curves defined over numberfields look like P 1 \ {0,1, oo}, in the sense that they have a Zariski open dense set which can be realised as a finite etale covering of it. Such a cover is defined by a regular function 0 : X' —> P 1 \ {0, l,oo}, nowadays called a Belyi function. Equivalently a Belyi function is given purely analytically as a meromorphic function on the Riemann surface X which is ramified at most over the three points 0, 1 and oo of the Riemann sphere. Note also that the curve X' is always affine (S is not empty) and that the points which have to be deleted are Q-points ofX. Because of the rigidity of P 1 \ {0,1, oo}, namely the fact that its complex structure is unique and thus determined by the topology, an unramified finite covering can be given topologically and is completely encoded in a combinatorial graph looking just as candid as a 'dessin d'enfant' and yet

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carrying enough information to define for instance a numberfield (say its field of moduli). In terms of the associated Belyi function /?, the dessin is nothing but the preimage of the segment (0,1), viewed up to homotopy on the curve X considered as a topological surface. So dessins d'enfant parametrize unramified coverings of P 1 \ {0,1, oo} and can be thought of as a pictionary of the geometric fundamental group of P 1 \ {0,1, oo}, namely 2*2, or almost equivalently of the all important SL2CZ); see again §3 of the Esquisse for an inspired praise of this group. In connection with the ubiquity of P 1 \ {0,1,00}, we record the following suggestive Proposition 4.1. V\ \ {0,1,00} = Z \ {0,1} is the only smooth (marked) hyperbolic curve over Z.

For the proof, note first that if a marked curve of genus 0 has at least 4 marked points, then two of them must collide modulo some prime, which takes care of the genus 0 case. In genus 1, there must be at least 1 marked point to ensure hyperbolicity, so we get an elliptic curve, and it cannot have good reduction everywhere. In higher genera we can ignore marked points and use the fact (due to J.-M.Fontaine) that there does not exist an Abelian variety over Z, as would be given by the Jacobian of a smooth curve. We also notice, and this is a crucial remark of Grothendieck, that 1 P \ {0,1,00} is both a rigid hyperbolic curve and the first nontrivial moduli space (P1 \ {0, l,oo} ~ Mo,4) so that it lies at the crossroad of curves and their moduli, a bit as a nonlinear analog of the fact that elliptic curves provide a connection between curves and Abelian varieties. In any case, this elementary fact certainly has many deep consequences and is most relevant in the perspective of Grothendieck-Teichmiiller theory. Let us now move to the hyperbolic side of Belyi's result, refering in particular to [8]. First recall a classical notion: if G is a discrete or profinite group and H and H' are two subgroups of G, they are said to be commensurate if their intersection H D H' has finite index in both H and H'. They are commensurable if there exist g and g' in G such that the conjugate subgroups gHg~l and g'H'g^1 are commensurate; of course one can always take - say - g' = 1, replacing g with g'~lg, but we favored a symmetric definition. These are obviously equivalence relations and the notion of commensurability makes sense for conjugacy classes of subgroups. We can now state the following:

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Theorem 4.2. (Unramified hyperbolic version of Al): A smooth complex curve X can be defined over a numberfield if and only if there exists a finite set S C X such that the affine curve X' = X \ S is uniformized by a Fuchsian group T c PSL2(IR) (i.e. X' ~ T\H) with T commensurable to PSL2{Z). Next and last we revisit the same situation in an orbifold (or stacky) fashion, paying attention to ramification. Here triangle groups, which are almost never arithmetic, play a role which is completely parallel to that of the arithmetic groups in the last statement. Let Api<7)7. C PSL2(M.) be the Fuchsian triangle group denned by the triple of positive (finite or infinite) integers (p,q,r). We assume hyperbolicity as usual, namely that p~l + q~x + r~x < 1. As an abstract group APjqj?. is generated by three generators x,y,z with relations xyz = 1, xp = yq — zr = 1. Triangle groups are rigid objects in the sense that all Fuchsian groups inside PSX2OR) with this abstract presentation are conjugate. So Ap
259 situation of theorem A3, that is choose p = q = r = oo. In order to prove the proposition one notes that by puncturing X at the points lying over 0, 1 and oo, we get a curve which is dominated by a finite etale cover of P 1 \ {0,1, oo}. This cover X' can be written as X' ~ T'\H, where here the action is free, T' uniformizes X' and I" ~ fti(X') (topological fundamental group). We insist on the fact that T' is commensurable with, indeed is a finite index subgroup of PSL2(Z), whereas F in general is not commensurable with that group. So 'puncturing', that is in hyperbolic terms converting elliptic elements in a Fuchsian group into parabolic ones, is a very deep operation but somehow it does not alter the fact of being defined over a numberfield.

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17. A.B.Goncharov, Multiple Zeta Values, Galois groups, and geometry of modular varieties, ECM 2000, Progr. Math. 201, Vol. 1, 361-392, Birkhauser, 2001. 18. G.Gonzalez-Diez, P.Lochak, On the fundamental groups at infinity of the moduli spaces of compact Riemann surfaces, Michigan Math. J. 49 (2001), 493-500. 19. A.Grothendieck, J.P.Murre, The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme, Lecture Notes in Mathematics 208, Springer Verlag, 1971. 20. A.B.Goncharov, Y.I.Manin, Multiple ^-motives and moduli spaces Mo>n, preprint, 2002. 21. Geometric Galois Actions, L.Schneps and P.Lochak eds., London Math. Soc. Lect. Note Ser. 242, Cambridge University Press, 1997. 22. A.Haefiiger, Extension of complexes of groups, Ann. Inst. Fourier 42 (1992), 275-311. 23. J.L.Harer, The cohomology of the moduli spaces of curves, in Theory of Moduli, E.Sernesi ed., LN 1337 (1988), Springer, 138-221. 24. A.Hatcher, P.Lochak, L.Schneps, On the Teichmiiller tower of mapping class groups, J.reine angew. Math. 521 (2000), 1-24. 25. R.Hain, M.Matsumoto, Tannakian fundamental groups associated to Galois groups, in Galois groups and fundamental groups, MSRI Publ. 41, Cambridge Univ. Press., 2003. 26. A.Hatcher, W.Thurston, A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980), 221-237. 27. Y. Ihara, Profinite braid groups, Galois representations and complex multiplications, Ann. of Math. 123 (1986), 43-106. 28. Y.Ihara, Some arithmetic aspects of Galois actions on the pro-p fundamental group of P 1 \ {0, l,oo}, Proc. Symp. Pure Math. 70, 247-273, AMS Publ., 2002. 29. N.V.Ivanov, Subgroups of Teichmiiller modular groups, Translations of Mathematical Monographs 115, AMS Publ., 1992. 30. Y.Imayoshi, M.Taniguchi, An introduction to Teichmiiller spaces, Springer Verlag, 1992. 31. Y.Ihara, H.Nakamura, On deformation of maximally degenerate stable curves and Oda's problem, J. reine und angew. Math. 487 (1997), 125-151. 32. P.Lochak, The fundamental groups at infinity of the moduli spaces of curves, in [21], 139-158. 33. P.Lochak, On arithmetic curves in the moduli spaces of curves, to appear in JIMJ (Journal de l'lnstitut Mathematique de Jussieu). 34. M.Levine, Tate motives and the vanishing conjecture for algebraic /("-theory, in Algebraic K-theory and algebraic topology, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci 407, Kluwer Acad. Publ., 1993, 167-188. 35. G.Laumon, L.Moret-Bailly, Champs algebriques, Ergebnisse der Mathematik und ihrer Grenzgebiete 39, Springer Verlag, 2000. 36. P.Lochak et L.Schneps, The Grothendieck-Teichmiiller group and automorphisms of braid groups, in [11], 323-358.

261 37. P.Lochak et L.Schneps, A cohomologicaJ interpretation of the GrothendieckTeichmiiller group, Invent. Math. 127 (1997), 571-600. 38. P.Lochak, M.Vaquie, Inertie champetre, inertie singuliere et action galoisienne, in preparation. 39. P.Lochak, H.Nakamura, L.Schneps, Eigenloci in genus 0 and the parameters of the Grothendieck-Teichmiiller group, preprint, 2003. 40. H.Nakamura, Galois rigidity of algebraic mappings into some hyperbolic varieties, International J.Math. 4 (1993), 421-438. 41. H.Nakamura, Galois rigidity of pure sphere braid groups and profinite calculus, J. Math. Sci. Univ. Tokyo, 1 (1994), 71-136. 42. New Horizons in pro-p Groups, M. du Sautoy, D.Segal, A.Shalev Eds., Prog. Math. 184, Birkhauser, 2000. 43. B.Noohi, Fundamental groups of algebraic stacks, to appear in JIMJ (Journal de l'lnstitut Mathematique de Jussieu). 44. H.Nakamura, L.Schneps, On a subgroup of the Grothendieck-Teichmiiller group acting on the tower of profinite Teichmuller modular groups, Invent. Math. 141 (2000), 503-560. 45. H.Nakamura, A.Tamagawa, S.Mochizuki, Grothendieck's conjectures concerning fundamental groups of algebraic curves, Sugaku Expositions 14 (2001), 31-53; Japanese original in Sugaku 50 (1998), 113-129. 46. Espaces de modules des courbes, groupes modulaires et theorie des champs, Panoramas et Syntheses 7, SMF Publ., 1999; to appear in English, AMS Publ. 47. L.Ribes, P.Zalesskii, Profinite Groups, Springer Verlag, 2000. 48. C.Scheiderer, Farrell cohomology and Brown theorems for profinite groups, manuscripta math. 91 (1996), 247-281. 49. L.Schneps, The Grothendieck-Teichmller group GT: a survey, in [21], 183203. 50. L.Schneps, Special loci in moduli spaces of curves, in Galois groups and fundamental groups, MSRI Publ. 41, Cambridge Univ. Press., 2003. 51. L.Schneps, Automorphisms of curves and the Grothendieck-Teichmiiller group, preprint, 2002. 52. J-P. Serre, Cohomologie Galoisienne, LN 5, Springer Verlag, 1994 (cinquieme edition). 53. J-P. Serre, Arbres, Amalgames, 5L 2 , Asterisque 46 (1983), SMF Publ. (troisieme edition corrigee); available in English under the title Trees, Springer Verlag, 1980. 54. C.Soule, Groupes operant sur un complexe simplicial, C.R. Acad. Sciences 276 (1973), 607-609. 55. W.P.Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bulletin AMS 19 (1988), 417-431 (this paper was actually written around 1976). 56. J.Tits, Free subgroups in linear groups, J. of Algebra 20 (1972), 250-270. 57. L.Zapponi, Fleurs, arbres et cellules: un invariant galoisien pour une famille d'arbres, Compositio Mathematica 122 (2000), 113-133.

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Cell Decomposition and Compactification of Riemann's Moduli Space in Decorated Teichmiiller Theory R. C. Penner Departments of Mathematics and Physics/Astronomy University of Southern California Los Angeles, CA 90089

Abstract This survey covers earlier work of the author as well as recent work on Riemann's moduli space, its canonical cell decomposition and compactification, and the related operadic structure of arc complexes.

Introduction Riemann's moduli space M = M(F) of a surface F has an essentially canonical cell decomposition ([Ha]-[Ko]-[St] or [PI]) and admits various interesting compactifications, some of which are less or more compatible with the cell decomposition. There are furthermore several related operads based upon supersets of M, that is, there is a natural composition of appropriate (possibly degenerate) surfaces. These structures on moduli space are useful tools in several guises on the interfaces of mathematics and physics. For instance and essentially by definition, a conformal field theory leads to an algebra over the operad of uncompactified moduli space, where the operad structure is induced by gluing surfaces along boundary components, and algebras over this operad give Batalin-Vilkovisky structures [Ge]. In practice, one is furthermore interested in compactifications, or partial compactifications, of M; for instance, the Deligne-Mumford compactification [DM] and its operad in an appropriate setting correspond to Gromov-Witten theory

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and quantum cohomology. These remarks emphasize the importance of understanding compactifications of M. A basic fact about moduli space M = M(F) is that it admits a natural cell decomposition; there are several incarnations of cell decompositions depending upon the type of surface F, and we distinguish between "punctured " (surfaces without boundary and with at least one puncture) and "bordered" (surfaces with non-empty boundary which may also have punctures). As is well-known in the punctured case, M is a non-compact orbifold, which comes equipped with a canonical cell decomposition, where the cells in the decomposition are in one-to-one correspondence with suitable classes of graphs, or equivalently, with suitable classes of arc families in F; this cell decomposition does not extend to the DM compactification in any known way. In the setting of this paper in the case of a punctured surface F 1 ; upon choosing a distinguished puncture of i*\, there is a canonical compactification Arc(Fi) of M(Fi), which arises as the quotient of the simplicial completion of the natural cell decomposition. In the setting of this paper in the case of a bordered surface F2, there is a natural action of the positive reals R>o on M(F2) which is generated by scaling all of the hyperbolic lengths of the boundary geodesies, and we shall let M(_F2)/R.>o denote the quotient. There is a canonical compactification ylrc(jF2) of a space proper homotopy equivalent to M(_F"2)/R>o, where Arc(F2) comes equipped with a canonical cell decomposition. We had previously conjectured [P3] that in the case of a bordered surface F 2 , Arc(F2) is PL-homeomorphic to a sphere and shall discuss in §5 this "sphericity conjecture*". This generalizes to arbitrary bordered surfaces a classical combinatorial fact about triangulations of polygons as we shall see. In fact, this sphericity of Arc(F2) for any bordered surface F2, in turn, implies that the compactification Arc{F\) of moduli space is an orbifold, for any punctured surface Fi as we shall also see. In particular, Arc(F{) is conjecturally a new orbifold compactification of M(F\) for a punctured surface JF\, where the cell decomposition of

M(Fi) extends to Arc(F{).

Other aspects of the geometry and combinatorics of these new compactifications had already been described and will be recalled here as well. For instance: whereas the DM compactification is regarded in our formalism as "forgetting" the geometric structure on sub-annuli of F, our compactification allows one to "forget" geometric structure on any sub-surface embedded in F; a point in our compactification gives a canonical decomposition of * Note added in proof: In recent work with Dennis Sullivan, the general sphericity conjecture has been disproven, and a complete list is known of those arc complexes that are spherical. Nevertheless, most of the structure and all of the arguments described in this paper survive this unexpected development.

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F into respective geometric and topological sub-surfaces, and the topological part is non-empty if and only if the point is ideal; whereas Thurston's compactification (see [PH] for instance) of Teichmiiller space records only those hyperbolic lengths which diverge, our compactification records only those hyperbolic lengths which do not diverge. On the algebraic side, Sullivan [CS], [Su], [CJ], [Vo] has introduced a "string prop", which is intended to formalize in the language of traditional algebraic topology certain algebraic aspects of physical and mathematical theories involving colliding strings. This prop is intended to be "universal" in the sense that it should act on the loop space of any manifold. The precise definition of the appropriate compactification is apparently still being formulated. In our setting, there are natural operad structures on arc complexes in the bordered case which are closely connected to the string prop. These "arc operads" have been studied in [KLP] (and are recalled here in §7 with a more thorough survey given in Kaufmann's paper [K2] in this volume, which also surveys his independent work from [Kl] as well as more recent work on Deligne's Conjecture). Thus, this paper both surveys older results [P1-P3] (in §§1,2,6) and sketches (and even excerpts at points) the newer works [P4] on bordered surfaces (in §3), on sphericity (in §5), and [KLP] on the operad structure (in §7). We have attempted to include enough background material to provide a meaningful survey, for instance, including many of the principal identities of decorated Teichmiiller theory in §2 and sketches of proofs, where appropriate. The intention has been to emphasize the background material for the compactification and for the operad structure, and there are other aspects [P5] of decorated Teichmiiller theory (notably matrix models, universal constructions, homeomorphisms of the circle and Lie algebras, wavelets and Fourier transform) which are not discussed here. This paper is organized as follows. §1 gives basic definitions and compares the conformal treatment of the cell decomposition of moduli space in the punctured case with the hyperbolic treatment of the cell decomposition. §2 describes the several coordinates systems of decorated Teichmiiller theory and develops from first principles several deep results. As mentioned before, §§3-4 survey our newer work on bordered surfaces, and §5 gives all the details of a proof of the sphericity conjecture for punctured polygons. §6 contains older work that explains the degeneration of hyperbolic structure in our new compactification, and §7 surveys the arc operad and its relationship with the sphericity conjecture. A final appendix is intended to give a mathematical overview of folding problems in computational biology, employs the arc complex constructions to give new combinatorial models for certain biopolymers, and ends with a speculative remark.

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1. Definitions and cell decomposition for punctured surfaces Consider a smooth surface F = F* T of genus g > 0 with r > 0 labeled boundary components and s > 0 labeled punctures, where 6g — 7+4r+2s > 0. The surface F is said to be bordered if r ^ 0. The pure mapping class group PMC = PMC{F) of F is the group of all isotopy classes of orientation-preserving homeomorphisms of F which pointwise fix each boundary component and each puncture, where the isotopy is likewise required to pointwise fix these sets. In the unbordered case r = 0, define the classical Teichmuller space Tg of F to be the space of all complete finite-area metrics of constant Gauss curvature —1 ("hyperbolic metrics") on F modulo push-forward by diffeomorphisms fixing each puncture which are isotopic to the identity relative to the punctures. Tg is homeomorphic to an open ball of real dimension 6g—6+2s, the mapping class group PMC acts on Tg by push-forward of metric under representative diffeomorphism, and the quotient Mg = Tg/PMC is by definition Riemann's moduli space of F. In the unbordered gunctured case, where r = 0 and s > 0, there is a trivial R> 0 -bundle Tgs —^T°, where the fiber over a point is the set of all s-tuples of horocycles in F, one horocycle about^each puncture, and there is a corresponding trivial bundle Tgs/PMC = Mg —• Mg; the total spaces of these bundles are the decorated Teichmuller and decorated moduli spaces, respectively, and they are studied in [P1-P3]. In the bordered case r ^ 0, there are two geometric treatments of distinguished points in the boundary, which will be carefully described and compared in §3 leading to two closely related models of corresponding decorated moduli spaces. The remainder of this section is dedicated to a discussion of the well-known canonical cell decomposition of decorated moduli space in the punctured unbordered case r = 0 with s > 0 (as we tacitly assume in the remainder of this section), which may be described either in the spirit of [P1-P3] ("in the hyperbolic setting") or using quadratic differentials [St] ("in the conformal setting") as in [Ha] or [Ko]. Our discussion begins in the conformal setting, where moduli space is regarded as the space of all equivalence classes of conformal structures under push-forward by any orientation-preserving diffeomorphism. A "fatgraph" or "ribbon graph" G is a graph whose vertices have valence at least three plus the further structure of a cyclic ordering on the half-edges about each vertex. A fatgraph G may be "fattened" to a bordered surface as follows: start with disjoint neighborhoods in the plane of the vertices of G, where the cyclic ordering is determined by the orientation of the plane; glue orientation-preserving bands to these neighborhoods in the natural way, one band for each edge of G, to yield a topological surface FQ D G

267

with G a spine of FQ. The complement FQ — G is a union of topological annuli Ai C FG, i = I, • • •, s > I, where each annulus Ai has one boundary component d[ contained in G and the other di contained in 8FQ. Letting E{G) denote the set of edges of G, a "metric" on G is any function w € (R>o) B(G) - The curve d[ is a cycle on G, and we define the "length" of Ai to be £i(w) — ^2w(e), where the sum is over e £ &[ counted with multiplicity. More generally, for any simple cycle 7 on G, we similarly define t{y). We shall say that w is "positive" if £(7) > 0 for each simple closed cycle 7 and shall let 0, for i = l , . . . , s . We construct a metric surface homeomorphic to Fjf as follows, where 2 — 1g — s is the Euler characteristic of G. Give each Ai = Ai(w) the structure of a flat cylinder with circumference ti and height unity, and isometrically identify these cylinders in the natural way along common edges in U{c^}f as dictated by the metric and fatgraph. This yields a metric structure on FQ SO that the boundary component <9; of FQ is a standard circle of circumference I,; for each i = 1 , . . . , s, glue a standard flat disk of circumference ti to each di and take the center of the flat disk as a puncture *i to produce a conformal structure on a surface which is identified with An analytic fact is: Theorem A [St;§23.5] Suppose w € cr(G) is a positive metric on the fatgraph G, and let ti = ii(w) denote the corresponding lengths, for i = 1, . . . , s . Then there is a unique meromorphic quadratic differential q on Fg so that for each i = 1 , . . . , s, the non-critical horizontal trajectories of q in F foliate Ai(w) C Fg by curves homotopic to the cores, and the residue °f \/Q at *i is ti. Let nq denote the conformal structure on Fg determined by the quadratic differential q. Theorem A thus describes an effective construction

(G,»)4(i,x(
contracting each edge e 6 E(G) with w(e) = 0 to produce Gw. Identifying E{GW) with {e £ E(G) : w(e) ^ 0} in the natural way, we thus induce w' €
9=\

II ^(G)l/~.

268 where ]J denotes disjoint union, and (G 1 ,^ 1 ) ~ (G 2 ,w 2 ) if and only if G^ t C F agrees with G£,2 C F as members of C° and w[ € o^G1) agrees with w'2 € • F) >->• (/ o 0 : G —> F) for any diffeomorphism f : F —}• F, and the quotient "fatgraph complex" is G* = U°/PMC; let [G, w] € G* denote the class of (G,w)£U°. Theorem B [St;§25.6] and [HM] (G,w) i-> ng x (^)f induces real-analytic homeomorphisms U* -> T* x R^ o and G* -» M* x R> 0 By Theorem A, the mapping G* —¥ Mg x R5, o is well-defined and injective, and by Theorem B, this mapping is surjective (but its inverse requires solving the Beltrami equation and is highly non-computable). There is thus a PMG-invariant cell decomposition of T° x R> 0 induced by the cell structure of Gg, which is the main result in the conformal setting positing the existence of a cell decomposition of M°. In the hyperbolic setting, fix a (conjugacy class of) Fuchsian group F uniformizing a point of Tg, and specify a collection of horocycles, one horocycle about each puncture of F = F* (called a "decoration"); these data uniquely determine a point F e Tgs by definition. The main ingredient for the cell decomposition of moduli space in the hyperbolic setting is the "convex hull construction" [Pl;§4], which assigns to F G Tg a corresponding point (Gp,Wf) e f/g as follows: Via affine duality, we may identify the open positive light-cone L+ in Minkowski three-space with the collection of all horocycles in the hyperbolic plane (cf. §2) and thus produce a F-invariant set B C L+ corresponding to the decoration, where we regard F as acting via Minkowski isometries. In fact, B is discrete in L+, and we may consider the closed convex hull of B in the vector space structure underlying Minkowski space. The extreme edges of the resulting F-invariant convex body project to a collection of disjointly embedded arcs a r connecting punctures, and each component of F — U«f is simply connected; we shall say that such an arc family «f "fills" F. Given any arc family a filling F, we may define a subset C(a) = {F €Tg : Qf is homotopic to a}. The Poincare dual of the cell decomposition Ua r of F is a fatgraph G embedded as a spine of F. The explicit formula in terms of Minkowski geometry for the "simplicial coordinate" of an edge of G, gives a positive metric Wf on G; the simplicial coordinates are effectively the signed volume of the corresponding simplex with light-like vertices, as we shall describe in §2In the conformal setting, the effective construction maps Gsg -* ( M | x R ^ 0 ) , while in the hyperbolic setting, the effective construction (i.e., the convex

269 hull construction) maps M* ~> Gg in the opposite direction! Thus, each of the hyperbolic and conformal treatments has its difficult theorem: surjectivity of the effective construction. The proof of surjectivity in decorated Teichmuller theory devolves to showing that the putative cells C{a) C Tgs are in fact cells, and this cellularity is proven in [PI,'§5] (independent of the Jenkins-Strebel theory) as described further in §2. Just as the conformal setting has a non-computable inverse (M* x R> 0 ) -> Gsg, there is a non-computable (or at least, very difficult to compute) inverse Gg —> Mg in the hyperbolic setting. These "arithmetic problems" are studied in [P2] and will be discussed in §2; a solution to the arithmetic problem is thus a solution to an appropriate Beltrami equation. We believe that one may combine bounded distortion results of EpsteinMarden/Sullivan from the 1980's with arguments from [PI,§6] and prove that the hyperbolic construction followed by the conformal construction has bounded distortion on open moduli space (and the details of this will appear elsewhere). For instance, one can thus view the arithmetic problem as the calculation of hyperbolic geometry from conformal combinatorics, up to a bounded error. It is worth emphasizing that the conformal and hyperbolic versions of the cell decomposition of Tjf agree combinatorially but not pointwise. Since PMC acts more or less discretely on Tg, it is not surprising that a given cell decomposition can be "jiggled" to produce a combinatorially equivalent but pointwise distinct cell decomposition. (On the other hand, another hyperbolic version of the cell decomposition due to Bowditch-Epstein was shown in [P2] to agree pointwise with the cell decomposition based on the convex hull construction.) Finally, there is further structure in the hyperbolic setting without analogue in the conformal setting as discussed in §2-3 (from [PI] in the punctured and [P4] in the bordered case).

2. Coordinates on Teichmuller space for punctured surfaces There are several coordinatizations of the Teichmuller space Tgs = T{Fg) of a punctured unbordered surface which we shall next describe. All depend upon the Minkowski inner product < •, • > on R 3 whose quadratic form is given by x2 + y2 - z2 in the usual coordinates. As is well-known, the upper sheet H = {u = (x, y,z) G R 3 :< u,u >— - 1 and z > 0} of the twosheeted hyperboloid is isometric to the hyperbolic plane. Furthermore, the open positive light cone L+ = {u = (x, y, z) € R 3 :< u, u >= 0 and z > 0} is identified with the collection of all horocycles in H via the correspondence u H-> h(u) = {w e H :< w,u >= - 1 } . The first basic building block to be understood is a "decorated geodesic", by which we mean a pair of horocycles ho, hi in the hyperbolic plane with

270

distinct centers, so there is a well-defined geodesic connecting the centers of ho and hi; the two horocycles may or may not be disjoint, and there is a well-defined signed hyperbolic distance S between the horocycles (taken to be positive if and only if /i0 (~l hi = 0) as illustrated in the two cases of Figure 1. The lambda length of the pair of horocycles {ho,hi} is defined to be the transform A(/i0, hi) = y/2 exp S. Taking this particular transform renders the identification h geometrically natural in the sense that X(h(u0),h(ui)) = y/— < uOlUi >, for u 0 ,u x e L+ as one can check. h(ux

h(

)

U l

)

Figure 1 Decorated geodesies. By an arc family in F", we mean the isotopy class of a family of essential arcs disjointly embedded in F£ connecting punctures, where no two arcs in a family may be homotopic rel punctures. If a is an arc family so that each component of F — Ua is a polygon, then we say that a fills F, and in the extreme case that a is maximal so that each complementary component is a triangle, then a is called an ideal triangulation. Let us choose an ideal triangulation r of F£ once and for all and regard this as a "choice of basis" for the first global coordinatization: Theorem 1 [PI; Theorem 3.1] For any ideal triangulation T of F, the assignment of lambda lengths defines a real-analytic homeomorphism Ig ~> K > 0 -

For the proof, we refer the reader to [PI], which gives an effective construction of the underlying Fuchsian group from the lambda lengths on r (and is the decorated version of an easy case of Poincare's original construction of Fuchsian groups). The next basic building block to be understood is a "decorated triangle", by which we mean a triple ho,hi,ti2 of horocycles with distinct centers, so these centers determine an ideal triangle in the hyperbolic plane. We

271

shall refer to a neighborhood of a cusp of this triangle as a "sector" of the triangle, so there is one horocylic segment contained in each sector, and we define the h-length of the sector to be half the hyperbolic length of the corresponding horocyclic segment; see Figure 2, where we have identified an arc with its lambda length for convenience. One can think of the falength as a kind of angle measurement between geodesies asymptotic to a puncture.

h(u-, )

/

n

o

Figure 2 Decorated triangle. Lemma 2 Fix a decorated ideal triangle ho,hi,h,2, and define the lambda lengths Xi = X(hj, hk), where {i,j, k} = {0,1, 2}. Then a) [Pl;Proposition 2.8] As in Figure 2, the h-length of the sector opposite Xi is given by ^ - for [i, j , k} = {0,1, 2}. b) [P2;Propostion 2.3] There is a point in the hyperbolic plane equidistant to ho, hi,h,2 if and only if XQ, AI, A2 satisfy all three possible strict triangle inequalities, and this equidistant point is unique in this case as illustrated in Figure 3; furthermore, for a fixed ideal triangle, all points in the hyperbolic plane arise for some decoration on it. c) [P5;WP volumes;Theorem 3.3.6] The Weil-Petersson Kdhler two-form is given by - 2 £ ) rflog Ao A dlog Ai + dlog Ai A cflog A2 + dlog A2 A dlog Ao,

where the sum is over all triangles complementary to r with edges XQ, AI, A2 in this cyclic order in accordance with the orientation of F. The proofs of a) and b) follow from direct calculation, and c) follows from Wolpert's formula plus calculation.

272

inside triangle outside triangle Figure 3 Equidistant points to triples of horocycles. The last basic building block to be understood is a "decorated quadrilateral" by which we mean four horocycles with distinct centers, which thus determine an ideal quadrilateral. Let us furthermore triangulate this quadrilateral by adding one of its diagonals e and adopt the notation of Figure 4 for the nearby lambda lengths, where again we identify an arc with its lambda length for convenience. Furthermore, adopt the notation of Figure 4 for the nearby h-lengths. Define the simplicial coordinate of the edge e to be 2

E=

a

+ b2- e2 c2 + d2-e2 + abe cde

. . . ^ o r = (a + P - e) + 7 + 6 - 4> .

In fact, abcdE is the signed volume of the corresponding tetrahedron in Minkowski space, as one can check.

Figure 4 Decorated quadrilateral. Lemma 3 [Pl:§2] Adopting the notation of Figure 4, we have a) the coupling equation holds: a/3 = 76; b) the cross-ratio of the four ideal vertices is given by jjj; c) the diagonal f of the quadrilateral other than diagonal e has lambda length given by a Ptolemy transformation: / = s&k±d_

273

The proofs follow from easy calculation. The coupling equations give a variety determined by a collection of coupled quadric equations in the vector space spanned by the sectors, where the simpicial coordinates are given by linear constraints, and this variety is identified with decorated Teichmuller space. Fix an ideal triangulation r of Fg. Define a cycle of triangles (tj)™ to be a collection of triangles complementary to r so that tj fltj+i = ej, for each j = l , . . . , n , and the index j is cyclic (so that t n + i = t\). If the edges of tj are {ej, ej_i, bj}, for j = 1 , . . . , n, then the collection {6j}™ is called the boundary of the cycle, and the edges {e,,} are called the consecutive edges of the cycle. A function X : {edges of T} ->• R> 0 satisfies the no vanishing cycle condition if for any cycle of triangles with consecutive edges en+i = e i , . . . , e n , we have J™=i x(ej) > °A nifty little fact is the "telescoping property" of simplicial coordinates, as follows: If e i , . . . , en are the consecutive edges of a cycle of triangles with boundary b\, ...,&„, and the sector in t{ opposite bi has h-length hi, then X)"_x-Ei — 2 53™=i hi, where the simplicial coordinate of edge e^ is given by Ei. The proof follows immediately from the definition of simplicial coordinates. Suppose that T e Tg gives rise via the convex hull construction to the ideal cell decomposition otf. Let us complete ctf to an ideal triangulation r in any manner (triangulating the polygonal complementary regions). From the very definition of the convex hull construction and simplicial coordinate, we find that the simplicial coordinate of each edge is non-negative (by convexity) and satisfies the no vanishing cycle condition (since complementary regions are polygonal). Conversely, the main hard result of decorated Teichmuller theory is Theorem 4 [Pl;Theorem 5A]Given an ideal triangulation T and any Xe G R>o, for each e € T, with no vanishing cycles, there is a unique corresponding assignment of lambda lengths in R> 0 determining a decorated hyperbolic structure f £ Tgs so that Qf = r - {e G r : Xe — 0}, and the lambda lengths on r realize the putative simplicial coordinates Xe. The idea of the proof is to consider the vector space spanned by the sectors of an ideal triangulation r and introduce the "energy"

where the sum is over all edges e of r in the notation of Figure 4. In fact, solving for lambda lengths amounts to minimizing K subject to the constraints given by the simplicial coordinates, the constrained gradient

274 flow of K is dissipative with a unique minimum, this unique constrained minimum of K solves the coupling equations, and this gives the desired lambda lengths from a/3 = e~2 = j5. Thus, the explicit formulas for simplicial coordinates in terms of lambda lengths can be uniquely inverted provided the simplicial coordinates are non-negative and satisfy the no vanishing cycle condition. This explicit calculation is called the Arithmetic Problem [P2] Fix any ideal triangulation of any punctured surface, and calculate the lambda lengths from the simplicial coordinates. Lemma 5 [Pl;Lemma 5.2] The non-vanishing cycle condition on simplicial coordinates implies that the lambda lengths on any complementary triangle satisfy all three strict triangle inequalities; in particular, the product of the lambda length and the simplicial coordinate of an edge is bounded above by four. Proof Adopt the notation in the definition of simplicial coordinates for the lambda lengths near an edge e. If c+d < e, then c2 + d2 — e2 < —led, so the non-negativity of the simplicial coordinate E gives 0 < cd[(a — b)2 — e 2 ], and we find a second edge-triangle pair so that the triangle inequality fails. This is a basic algebraic fact about simplicial coordinates. It follows that if there is any such triangle t so that the triangle inequalities do fail for t, then there must be a cycle of triangles of such failures. This possibility is untenable since if ej+i > bj + ej, for j = 1 , . . . n, where we again identify an arc with its lambda length, then upon summing and canceling like terms, we find 0 > $2?=i bj, which is absurd since lambda lengths are positive. The second part follows from the elementary observation in the notation of Figure 4 that _, a 2 + b2 - e 2 c 2 + d2 - e2 eE = + , ab cd i.e., each summand is twice a cosine, and hence eE < 4.

q.e.d.

By taking Poincare duals of filling arc families, there is established an isomorphism between fatgraphs and ideal cell decompositions, and the simplicial coordinates are the analogues of the metric lengths of edges in the conformal treatment. Indeed, we may restate the main theorem of §1 in the hyperbolic setting using arc families as: Theorem 6 [Pl;Theorem 5.5] There is a PMC(F^)-invariant cell decomposition of Tg isomorphic to the geometric realization of the partially ordered set of all arc families filling Fg.

275

3. Bordered surfaces We next turn our attention to bordered surfaces. Not only does much of the decorated Teichmuller theory extend naturally to this setting, but there are also new aspects of the theory to be described. In this section, we shall assume only that r ^ O (and allow both cases s = 0 or s > 0). Enumerate the (smooth) boundary components of F as di, where i = 1 , . . . , r, and set d = U{di}l. Let Hyp(F) be the space of all hyperbolic metrics with geodesic boundary on F. Define the moduli space to be r

M = M(F) = [Hyp(F) x (JJft)]/ ~, I

where ~ is the equivalence relation generated by push-forward of metric under orientation-preserving diffeomorphism

/.(r,(&)D) = (/*(r),(/(&)a where F >-¥ /*(F) is the usual push-forward of metric on Hyp(F). Let ^i(F) denote the hyperbolic length of di for F £ Hyp(F), and define the first model M = M(F) of decorated moduli space to be { (I\ (6)i, (*0i) • r € Hyp(F), & € du 0 < U < ti(r), t = 1 , . . . , r } / ~, where ~ denotes push-forward by diffeomorphisms on (F, (£i)i), as before, extended by the trivial action on (ti)\. A point of M is thus represented by F € Hyp(F) together with a pair of points & / pi in each component di, where Pi is the point at hyperbolic distance t; along di from & in the natural orientation. (In the special case g = 0 = s = r — 2 where F is an annulus, we define M(F) to be the collection of all configurations of two distinct labeled points in an abstract circle of some radius.) Turning to the second model for decorated moduli space, begin with a smooth surface F with smooth boundary, choose one distinguished point di € di in each boundary component, and set D = {di}^. Define a quasi hyperbolic metric on F to be a hyperbolic metric on Fx = F — D so that d? = di — {di} is totally geodesic, for each i = l , . . . , r , and set dx = U{d*}\. To illuminate the definition, take a hyperbolic metric on a once-punctured annulus A with geodesic boundary and the simple essential geodesic arc a in A which is asymptotic in both directions to the puncture; the induced metric on a component of A — a gives a model for the quasi hyperbolic metric near a component of dx as in Figure 5.

276

Figure 5 The model for F*. In this second model, there is a version of decorated Teichmuller space as the space T = T{F) of all quasi hyperbolic metrics on F — D, where we furthermore specify for each di a horocyclic segment centered at di (again called a "decoration"), modulo push-forward by diffeomorphisms of F — D which are isotopic to the identity, where^ the action is trivial on hyperbolic lengths of horocylic segments. In fact, T is homeomorphic to an open ball of dimension 6gj- 6 + 5r + 2s. The second model for the decorated moduli space isM = M(F) = f{F)/PMC{F). For a fixed quasi hyperbolic metric on F, there is a unique separating geodesic d* C F in the homotopy class of di . Excising from F - L>{d*}\ the components containing points of dx yields a surface F*, which inherits a hyperbolic metric with geodesic boundary. (In the special case that F is an annulus, the surface F* collapses to a circle.) F 6 T has its underlying hyperbolic metric given by a conjugacy class of Fuchsian group F for F*. Define an arc in F to be a smooth path o embedded in F whose endpoints lie in D and which meets OF transversely, where a is not isotopic rel its endpoints to a path lying in dF — D. Two arcs are parallel if there is an isotopy between them fixing D pointwise. An arc family in F is the isotopy class of a set of disjointly embedded arcs in F, no two components of which are parallel. A collection a of arcs representing an arc family in F is said to meet efficiently an arc or curve C embedded in F if there are no bigons in F complementary to a U C. If a is an arc family in F where each component of F — Ua is a polygon or a once-punctured polygon, then we say that a quasi fills F. In the extreme case that each component is either a triangle or a once-punctured monogon, then a is called a quasi triangulation. Theorem 7 [P4;Theorem 1] For any quasi triangulation r of F, the

277 assignment of lambda lengths defines a real-analytic homeomorphism lg,r -> K > 0

•

The proof is verbatim that of [PI; Theorem 3.1] briefly mentioned before. In fact, much of the number-theoretic utility of lambda lengths in the punctured case [P1-P3] extends without great difficulty to the bordered case as discussed in further detail in the Side-Remark after Theorem 1 of [P4]. One introduces simplicial coordinates in the bordered case in analogy to the previous discussion, where if e € dx, then it bounds a triangle on only one side, say with edges o, 6, e, and we define its simplicial coordinate to be abe

Fix a quasi triangulation r of F, and define the subspace C(T) = {(y-,x) € R 9

x (R> 0 ) T : there are no vanishing cycles or arcs}

CRTuaX, where the coordinate functions are the simplicial coordinates (rather than the lambda lengths as in the previous theorem). We say that there are "no vanishing cycles" as before for a point of C(T) if there is no essential simple closed curve C C F meeting a representative r efficiently so that

o= Y, x*» peCnu-r

where p £ C f~l o, for a e r, contributes to the sum the coordinate xp of o. We say that there are "no vanishing arcs" for a point of C(T) if there is no essential simple arc A C F meeting r efficiently and properly embedded in F with endpoints disjoint from D so that

°= E peAnd*

y? +

E *i» PEAHT

where again xp and yp denote the coordinate on a at an intersection point p = j 4 n a or p = C n a for a € r . This is a convex constraint on y for fixed x. Just as in the case of punctured unbordered surfaces, F gives rise to a quasi filling arc family a^ via the convex hull construction; fixing the topological type of Qp and varying T then gives a cell in a natural decomposition of T. T h e o r e m 8 [P4; Theorem 3] A point F E T gives rise to a quasi filling arc family ctf via the convex hull construction as well as a tuple of simplicial coordinates {y,x) € C(T) for any quasi triangulation T D a r , where

278

x vanishes on r - ap. Furthermore, this gives a real-analytic homeomorphism of the decorated Teichmuller space of F with [|JT C(T)] / ~, where (TI,VI,XI) ~ (r2,y2,x2) if xx agrees with x2 on n - {o € n : i 1 = 0} = T2 — {a e T2 : x2 = 0}, where x? denotes the x coordinate on a, for j = 1,2, andyi = y2. Define a variant of the decorated Teichmiiller space

T=f(F)=[{JC(r)]/~, T

where ~ is as in Theorem 8, and for membership in C(T) we demand that there are no vanishing cycles or arcs and for any triangle t C F complementary to T, and the lambda lengths on the edges of t satisfy all three possible strict triangle inequalities. Using cycles of triangles, the quotient M. = T/PMC can be shown [P4;Lemma 5] to be a strong deformation retract of M. Using the existence of equidistant points to suitable triples of horocycles as in Lemma 2b in order to determine distinguished points, one can then prove Theorem 9 [P4; Theorem 8] There is a real-analytic homeomorphism between M. and M. Thus, our two models M and M. of decorated moduli space are homotopy equivalent (and indeed are homeomorphic except for the required passage

to

MCM).

4. The arc complex of a bordered surface Let us inductively build a simplicial complex Arc'(F), where there is one p-simplex a (a) for each arc family a in F of cardinality p+1; the simplicial structure of cr (a) is the natural one, where faces of
Arc{F) = Arc'(F)/PMC (F). If a is an arc family in F with corresponding simplex cr(a) in Arc'(F), then we shall let [a] denote the PMC(.F)-orbit of a and a[a] denote the

279 quotient of
Arc#(F) = {a[a\ : a quasi - fills F}. Let M(F) /R>o denote the quotient of the moduli space of the bordered surface F by the R>o-action by homothety on the tuple of hyperbolic lengths of the geodesic boundary components. A further analysis of the homeomorphism in Theorem 9 shows Theorem 10 [P4;Theorem 14] For any bordered surface F ^ F$2, is proper homotopy equivalent to M(F)/H>o.

Arc#(F)

Furthermore, each of M ( F ) / R > 0 and Arc#(F) admit natural (S'1)r-actions (moving distinguished points in the boundaries for the former and "twisting" arc families around the boundary components for the latter); the proper homotopy equivalence in Theorem 10 is in fact a map of (5 1 ) r spaces. This is the main result of [P4]. It follows from Theorem 10 plus the sphericity conjecture 12 discussed in the next section that M ( F ) / R > 0 compactifies to a sphere. Notice that the space of all projectivized laminations VC{F) is itself a sphere (cf. [PH]) which in particular contains the complex Arc'(F). It is well-known that VC{F)/PMC(F) is a space whose largest Hausdorff quotient is a singleton, yet it contains as a dense open set the quotient Arc{F), whose further study we undertake in the next section.

5. Sphericity Extending the notation from before, let F = Fs - denote a fixed smooth surface with genus g > 0, s > 0 punctures, and r > 1 boundary components, where the ith boundary component comes equipped with 5i > 1 distinguished points d; as well, for i = 1 , . . . , Si and 6 = (51, <S2, • • • Sr). Let Arc(F) = Arc1 (F)/PMC(F) denote the arc complex of F as denned before, and recall the following combinatorial fact. Classical Fact 11 For F = F°{ ., Arc(F) = Arc#{F) = Arc'(F) is PL-homeomorphic to the sphere of dimension n — 4 provided n > 4. Gian-Carlo Rota told us that sphericity for polygons was almost certainly known to Whitney. This is a special case of the following general conjecture.

280 Sphericity Conjecture 12* [P3;Conj.B] The arc complex Arc(F) is PLhomeomorphic to the sphere of dimension 6g — 7+3r + 2s + 6i + 62 H h 6r provided this number is non-negative. This result has been a long-standing goal for us. We had proved this conjecture for g = 0 = r - 1 in [P3] and shall recall this proof and add all the details here. New tools have arisen to address the general sphericity conjecture in collaboration with Dennis Sullivan. Let D = D(F) = U{di:i

=

l,2,...,r}

denote the set of all the distinguished points in the boundary of F, A = A ( F ) = S1 + 52 + --- + Sr, be the cardinality of D, and set N = N{F) = 65 - 7 + 3r + 2s + A to be the dimension of the arc complex. Furthermore, in the special case that A = r so 6 is an r-dimensional vector each of whose entries is unity, then we shall also sometimes write F% _ = Fs ?. Notice that N > 0 except for the triangle F° ,3-. and once-punctured monogon FQ^ — FQ ,J>. It is convenient in the sequel to imagine weighted arcs in the spirit of train tracks [PH] as follows. If a is an arc family, then a weighting on a is the assignment of a non-negative real number, the weight, to each arc comprising a, and the weighting is positive if each weight is positive. A protective weighting is the projective class of a weighting on an arc family. Since the points of a simplex are described by "projective non-negative weightings on its vertices", we may identify a point of a (a) with a projective weighting on a. The projective positive weightings correspond to the interior Int a(a) of cr(a), and Arc'(F) itself is thus identified with the collection of all projective positively weighted arc families in F. Suppose that a = {ao,-..,a p } is an arc family. A useful realization of a positive weighting w = (wi)^ on the respective components of a is as a collection of p + 1 "bands" & disjointly embedded in the interior of F with the width of /?8 given by u>i, where at traverses the length of /%, for i = 0 , . . . ,p. The bands coalesce near the boundary of F and are attached one-to-the-next in the natural manner to form "one big band" for each of the A distinguished points as illustrated in Figure 6. (More precisely, we are constructing here a "partial measured foliation from a weighted arc * Note added in proof: In recent work with Dennis Sullivan, the general sphericity conjecture has been disproven, and a complete list is known of those arc complexes that are spherical.

281

family regarded as a train track with stops"; for further details, see [PH] or [KLP;§1].)

Figure 6 The band picture of a weighted arc family. Given an arc family a in F, consider the surface F — Ua. The non-smooth points or cusps on the frontiers of the components of F — Ua give rise to distinguished points in the boundary of F — Ua in the natural way, and this surface together with these distinguished points is denoted Fa. Our basic approach to sphericity is by induction on N(F) > N(Fa), and more will subsequently be said on this point. We begin the discussion of sphericity with a number of examples. Example 1 If g = s = 0 and r = 1, then F is a polygon with <5i vertices, N = S\ — 4, PMC(F) is trivial, and we are in the setting of the Classical Fact. When S\ = 4 with F a quadrilateral, we have N = 0, and there are exactly two chords, which cannot be simultaneously disjointly embedded. Thus, Arc(F) = Arc'(F) consists of two vertices, namely, Arc(F) is the 0-sphere. (The reader may likewise directly investigate the case N = 1 where F is a pentagon, and the usual "pentagon relation" shows that Arc(F) itself is also a pentagon.) More generally, in order to handle all polygons inductively, we recall Lemma 13 If F' arises from F by specifying one extra distinguished point in the boundary of F, then Arc(F') is PL-homeomorphic to the suspension of Arc(F) provided Arc(F) ^ 0. Proof If p is the extra distinguished point of F' in the boundary of F, let Po,Pi,P,P2 be the consecutive distinguished points lying in a single boundary component of F' in its induced orientation, where the pj may not be distinct. There is a natural mapping of Arc'(F) x [0,1] to Arc'{F') using partial measured foliations gotten by sliding proportion t € [0,1] of the band that hits p\ over to p, which descends to an embedding of (Arc(F) x [0,1])/ ~) to Arc(F'); the equivalence relation ~ collapses the interval [0,1] to a point for any arc family that does not hit p\. There is an

282

arc ai near the boundary which connects p\ to P2 and another arc oo near the boundary which connects po and p, each of which is essential in F' but either non-essential or non-existent in F. The surface Frai\ has two components, one of which is a triangle, and the other of which is homeomorphic to F. The collection Cj of projectively weighted arc families in F' which either contain or to which we may add at is thus homeomorphic to the topological join Ci « [a*] * Arc(F) of Arc(F) with a point, namely, the vertex <x[{ai}] corresponding to a;, for i = 0,1. Identifying the copy of Arc(F) lying in Cj with

[(Arc(F) x {i})/ ~] C [(Arc(F) x {0,1})/ ~] in the natural way for i = 0,1, we find a mapping of the suspension of Arc(F) onto ylrc^i*1') which is evidently a PL-homeomorphism. q.e.d. Proof of the Classical Fact Induct on the number Jj > 4 of distinguished points using Lemma 13 starting from the basis step Si = 4 treated in Example 1 (using the fact that the suspension of a PL-sphere is again a PL-sphere of one greater dimension). q. e. d. Example 2 If g = 0, r = s = 1, and 5\ — 2, then F is a once-punctured bigon. Any arc in F with distinct endpoints must be inessential, and any arc whose endpoints coincide must be separating. There are two complementary components to a, one of which has one cusp in its frontier, and the other of which has two cusps. The component of the former type must contain the puncture and of the latter type must be a triangle. It follows that Arc(F) = Arc'(F) is in this case again a zero-dimensional sphere, and Lemma 13 applies as before to prove sphericity whenever g = 0 and r = s = 1. As in this example, if a is a separating arc whose endpoints coincide with the point p in any surface, then its two complementary components are distinguished by the fact that one component has exactly one cusp arising from p, and the other component has exactly two cusps arising from p, and we shall refer to these respective complementary regions as the "onecusped" and "two-cusped" components. In particular, any arc in a planar surface whose endpoints coincide is necessarily separating. Example 3 If g = s = 0, r = 2, and <5X = <$2 = 1, then F is an annulus with one distinguished point on each boundary component. It is elementary that any essential arc must have distinct endpoints and its isotopy class is classified by an integral "twisting number", namely, the number of times it twists around the core of the annulus; furthermore, two non-parallel essential arcs can be disjointly embedded in F if and

283

only if their twisting numbers differ by one. Thus, Arc'{F) is in this case isomorphic to the real line, the mapping class group PMC{F) is infinite cyclic and generated by the Dehn twist along the core of the annulus, which acts by translation by one on the real line, and the quotient is a circle. Again, Lemma 13 extends this to any surface with g — s = 0 and r = 2. Example 4 If g = 0, s = 2, and r = 1 = Si, then F is a twice-punctured monogon. If a is an essential arc in F, then its one-cusped component contains a single puncture, which may be either of the two punctures in F, and the two-cusped component contains the other puncture of F. Labeling the punctures of F by 0 and 1, choose disjointly embedded arcs ai so that a* contains puncture i in its one-cusped complementary component, for i = 0,1. The Dehn twist r along the boundary generates PMC{F), and the 0-skeleton of Arc'(F) can again be identified with the integers, where the vertex cr({r^ (ai)}) is identified with the integer 2j + i for any integer j and i = 0,1. Applying Example 2 to the two-cusped component of F{ a .j, for i = 0,1, we find exactly two one-simplices incident on cr({«i}), namely, there are two ways to add an arc disjoint from m corresponding to the two cusps in the two-cusped component, and these arcs are identified with the integers i ± 1. Thus, Arc'(F) is again identified with the real line, PMC(F) acts this time as translation by two, and Arc(F) is again a circle. Lemma 13 then proves sphericity of Arc(F) for any planar surface with r = 1 and s = 2 The typical aspect illustrated in this example is that the arcs in an arc family come in a natural linear ordering which is invariant under the action of the pure mapping class group. Specifically, enumerate once and for all the distinguished points pi,p2, • • • ,PA in the boundary of F in any manner. The boundary of a regular neighborhood of pi in F, for i = 1,2, . . . , A contains an arc Ai in the interior of F, and Ai comes equipped with a natural orientation (lying in the boundary of the component of F—Ai which contains pi). If a is an arc family in F, then there is a first intersection of a component of a with the Ai of least index, and this is the first arc Oi in the linear ordering. By induction, the (j + l ) s t arc (if such there be) in the putative linear ordering is the first intersection (if any) of a component of a - {ai,..., Oj_i} with the Ai of least index. Since we take the pure mapping class group which fixes each distinguished point and preserves the orientation of F, this linear ordering descends to a well-defined linear ordering on the PMC(F)-orbits in Arc'(F). It follows immediately that the quotient map Arc'(F) —»• Arc(F) is injective on the interior of any simplex in Arc'(F), so that the simplicial complex Arc'(F) descends to a CW decomposition on Arc(F).

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Furthermore, in the action of PMC\F) on Arc'(F), there can be no finite isotropy, and the isotropy subgroup of a simplex a (a) in PMC(F) is either trivial or infinite. The former case occurs if and only if a quasi fills the surface F. One fundamental inductive tool is: Proposition 14 [P3;Lemma Z]Fix a bordered surface F, and suppose that Arc(Fa) is PL-homeomorphic to a sphere of dimension N(Fa) for each non-empty arc family a in F. Then Arc(F) is an N(F)-dimensional PLmanifold. Idea of Proof The idea is that the link of a simplex a[a] in Arc(F) in the first barycentric subdivision is canonically isomorphic to Arc(Fa); this is a general fact about geometric realizations of partially ordered sets. Thus, any point x G Arc(F) admits a neighborhood which is PL-homeomorphic to an open ball of dimension N = N(F). Of course, x lies in the interior Int a[a] of a[a\, for some arc family a in F, and we suppose that a is comprised of p + 1 component arcs. We have the identity N + 1 = (N(Fa) + 1) + (p + 1) since both sides of the equation give the number of arcs in a quasi-triangulation of F. Since Arc(Fa) is PL-homeomorphic to an N(Fa)-dimensional sphere by hypothesis, Int o~[ct] * Arc(Fa) is PLhomeomorphic to an open ball of dimension p + N(Fa) + 1 = N. This gives the required neighborhood of x in Arc(F) and proves that Arc(F) is q.e.d. inductively a manifold. As was mentioned before, our basic approach to sphericity is inductive, and the proof depends upon the inductive hypothesis that the sphericity conjecture holds for Fa for any non-trivial arc family a in F, i.e., by induction on N(F). In order to avoid the Poincare Conjecture in dimensions less than four, our basis step involves an explicit analysis of all arc complexes of dimension at most four. In the special case g = 0 = r — l o f multiply punctured polygons, we have already analyzed in the examples above (in combination with Lemma 13) all of the arc complexes of dimension at most two. In this case, the only remaining complex in dimension three is for the surface FQX and in dimension four is for Fjf ,2y which follows from F = Fgtl by Lemma 13. To handle the arc complexes of F = Ffi l5 for s > 3, label the various punctures of F with distinct members of the set S = {1,2,..., s}. If a is an arc family in F and a is a component arc of a, then a has a corresponding one-cusped component containing some collection of punctures labeled by a proper non-empty subset S(a) C S, which we regard as the "label" of a itself. More generally, define a "tableaux" r labeled by S to be a rooted tree

285

embedded in the plane where: the (not necessarily univalent) root of r is an unlabeled vertex, and the other vertices of r are labeled by proper nonempty subsets of S; for any n > 1, the vertices of r at distance n from the root are pairwise disjoint subsets of S; and, if a simple path in r from the root passes consecutively through the vertices labeled Si and S2, then S2 is a proper subset of S\. Given an arc family a in F, inductively define the corresponding tableaux T — r(a) in the natural way: For the basis step, choose as root some point in the component of Fa which contains the boundary of JF; for the inductive step, given a vertex of T lying in a complementary region R of Fa, enumerate the component arcs ao,ai,...,am of a in the frontier of R, where we assume that these arcs occur in this order in the canonical linear ordering described in §2 and ao separates R from the root. Each arc at separates R from another component Ri of Fa, and we adjoin to r one vertex in each such component Ri with the label S(a;) together with a one-simplex connecting R to Ri, for each i — 1,2,..., m, where the onesimplices are disjointly embedded in F. It follows immediately from the topological classification of surfaces that PMC(F)-orbits of cells in Arc(F) are in one-to-one correspondence with isomorphism class of tableaux labeled by S. Furthermore, since the edges of T{Q) are in one-to-one correspondence with the component arcs of a, a point in Jnt(a(a)) is uniquely determined by a projective positive weight on the edges of r(a). It follows that Arc(F) itself is identified with the collection of all such projective weightings on all isomorphism classes of tableaux labeled by S.

I. I. A- -X Ak

Bk

K(j

Ljj

A ' A A c

u

D

y

Ek

F k

Figure 7 The tableaux for F$A.

286

Example 5 The surface FQX. Let us adopt the convention that given an ordered pair ij, where i,j € {1,2,3}, we shall let k = k(i,j) = {1,2,3} — {i, j}, so k actually depends only upon the unordered pair i, j . The various tableaux for FQ^ are enumerated, labeled, and indexed in Figure 7, where in each case, ij varies over all ordered pairs of distinct members of {1, 2, 3}, k — 1, 2, 3, and the bullet represents the root. In this notation and letting d denote the boundary mapping in Arc = ATC(FQ{), one may directly compute incidences of cells summarized as follows. dCij =M- Aj, dDij = Aj - Bk, dEk = Ak - Bk, 8Fk = Bk- Ak, dGij = Cij — Dji + Dij,

dHu = Dij -Cjk + Fk, din = Di:j -Ek + Ckj, dJij = Cij - Cik + Cjk, dKij = dj - Hij + Hji - Jy, dLij — Iij — Gij + Jki — Iji.

We may symmetrize and define sub complexes Xk = Xij U Xji, for X = K,L, and furthermore set Mk = KkULk, for A; = 1,2,3. Inspection of the incidences of cells shows that each of Kk and Lk is a 3-dimensional ball embedded in Arc, as illustrated in Figures 8a and 8b, respectively, with Kij on the top in part a) and Lji on the top in part b). Gluing together Lk and Kk along their common faces Gij,Gji, we discover that Mk is almost a 3-dimensional ball embedded in Arc except that two points in its boundary are identified to the single point Ak in Arc, as illustrated in Figure 8c.

/.""D«\T""H""*\

Figure 8a Kk.

{•••""D*V£"'"*»\

A**"&r^"*""»\

Figure 8b Lk.

Figure 8 Mk and the balls

Figure 8c Mk. Kk,Lk.

Each Mk, for k = 1, 2,3, has its boundary entirely contained in the sub complex J of Arc spanned by {Ak : k = 1, 2,3} U {Qj, J{j :i,je

{1, 2,3} are distinct}.

In order to understand J, we again symmetrize and define Jk = Jij U
287 refer to the 1-dimensional simplices Cjk,Cjk as the "generators" and to Cij,Cji as the "lips" of
Figure 9 The cone Jfc and the one-skeleton of J. In order to finally recognize the 3-dimensional sphere, it is best to take a regular neighborhood of J in Arc, whose complement is a disjoint union of three 3-dimensional balls. Each 3-dimensional ball is naturally identified with the standard "truncated" 3-simplex, where a polyhedral neighborhood of the 1-skeleton of the standard 3-simplex has been excised. These truncated simplices are identified pairwise along pairs of hexagonal faces to produce the 3-dimensional sphere in the natural way. We next prove that the arc complex Arc(F) minus a point is contractible, for any surface F — Fos - . . Let a denote an arc labeled by {s}, and let b denote an arc labeled by S — {s}. We shall show that A = Arc(F) — {[a]} strong deformation retracts onto {[&]}. The retraction proceeds in four steps, as follows: A retracts to B — {[a] £ A : no vertex of r(a) is labeled by {s}}, B retracts to C = {[a] e B : s is not a member of the label of any vertex of r ( a ) } , C retracts to V = {[a] £ C : 3 ! vertex of r{a) adjacent to the root labeled by 5 — {s}},

288 and finally V retracts to the point [b]. In order to coherently describe these retractions, let us introduce a natural linear ordering, the so-called "pre-ordering", on the vertices of any tableaux r. Begin with the root of r as the first element in the putative pre-ordering. Given a vertex v at distance n > 0 from the root, there is a (possibly empty) family vi, V2, • • •, vm of vertices of r at distance n + 1 from the root which are adjacent to v, and this family of vertices comes in the right-to-left linear ordering from the definition of a tableaux. Furthermore, each Vj is the root of a (possibly empty) sub tableaux TJ of T, for j = 1,2,..., m, whose other vertices (if any) are at distance at least n + 2 from the root in r. Take the pre-ordering on T\, then on T 2 , and so on up to r m , to complete the recursive definition of the pre-ordering on the vertices of r itself. For the first retraction from A to B, simply decrease the projective weights, one at a time, on the one-simplices of r ( a ) , for [a] € A—B, which terminate at vertices labeled by {s} beginning with the greatest such vertex in the pre-ordering. For the second retraction from B to C, consider the greatest vertex v of r(a), for [a] £ B — C, in the pre-ordering whose label S contains s, and let e be the edge of T(Q) terminating at v, where e has a projective weight w. Insert a new bi-valent vertex u into e labeled by S and re-label v by S — {s}. The edge e is thus decomposed into two edges, and the one incident on v is given projective weight tw, while the other is given projective weight (1 — t)w, for 0 < t < 1. The corresponding deformation as t goes from 0 to 1 effectively replaces the label S on v with the label S — {s} and thus reduces the number of vertices whose label contains s. Iteratively removing s from the label of the greatest vertex in this manner defines the retraction of B to C. For the third retraction, suppose that [a] lies in C — T>, and let v denote the root of r(a). Adjoin to r(a) a new edge e incident on v, and let the other endpoint of e be the new root of the resulting tableaux, where v is labeled by S — {s}. Let w denote the projective class of the sum of all of the weights on r(a), and define a projective weight tw on e, for 0 < t < 1. The corresponding deformation as t goes from 0 to 1 effectively adds a new vertex with label S — {s}. Finally, simply decrease the projective weights on all of the edges of r(a) except the edge incident on the root in order to contract V to [6], completing the proof that the complement of a point in the arc complex Arc{F^ ,^)\s contractible. We shall say that a surface F "satisfies the inductive hypothesis" provided Arc(Fa) is PL-homeomorphic to a sphere of dimension N(Fa) for each non-empty arc family a in F. (For the perspicacious reader, notice that we tacitly assumed this inductive hypothesis in the proof of Lemma 13.)

289

The inductive hypothesis thus implies that Arc(F) is a manifold by Proposition 14, and this manifold is simply connected and a homology sphere since the complement of a point is contractible by the previous discussion. By the Poincare Conjecture in high dimensions, it follows inductively that Arc(F) is a sphere for F = FQ (1)' an<^ hence for F = F£ ^ for any n > 1 by Lemma 13. This completes the promised proof of the sphericity theorem for punctured polygons.

6. Punctured surfaces and fatgraphs We take this opportunity to point out that in [P3] we claimed that our compactification maps continuously to the Deligne-Mumford compactification, and this may not be true; the precise connection between the two compactifications will be taken up elsewhere. Secondly, Theorem 5 of [P3] is slightly corrected by Theorem 17 below. We return to the setting F = F* 0 of punctured surfaces with s > 1 and r = 0. Let us choose from among the punctures of F a distinguished one, to be denoted n. Build as before the geometric realization A'n (F) of the partially ordered set of all isotopy classes of arc families of simple essential arcs which are asymptotic to TT in both directions (referred to subsequently as arcs "based" at n). Consider the punctured arc complex of F = Fg Notice that given an arc family a in given by A^{F) = A'V{F)/PMC(F). F, each component of Fa inherits a labeling as a bordered surface, where the ends of Fa are taken to be the distinguished points in the boundary; this is the basic connection between bordered and punctured arc complexes. Several of the main results of [P3] are summarized in the following theorem. Theorem 15 For any choice of puncture •n, the Teichmuller space Tjf of F — Fg is naturally isomorphic to the subspace of A'v = A'V(F) corresponding to arc families based at ix which quasi-fill F, so An is a compactification of M = Mg. The third barycentric subdivision of A'v descends to an honest simplicial complex on the compactification An of M. Degenerate structures (i.e., "ideal" points of An - M) will be fully described presently as will a graphical description of cells together with the corresponding matrix-model. For the moment, though, we turn to sphericity. In fact, orbifoldicity of A* follows from the sphericity conjecture. Indeed, as in the proof of Proposition 14, where the link of a simplex a[a] in the second barycentric subdivision of Arc(F) is isomorphic to Arc(Fa), this time there is a finite group Qa acting on a[a] * Arc'(Fa), and the natural mapping Int(cr[a\* Arc'{Fa)) -t A'(F) induces a homeomorphism

290 between its image and the quotient Int(a[a] * Arc'(Fa))/Qa. Thus, the links of simplices are virtually arc complexes of bordered surfaces (i.e., spheres), so A^ is indeed an orbifold. The local groups Qa in the orbifold structure of An are easily described and recognized as follows. Consider the stabilizer Sa < PMC(F) of Fa in F (which may permute the labeling of Fa as a bordered surface), and notice that the quotient Qa = SjPMCiF) is a finite group. For instance, a component of Fa might be an annulus and there may be some € PMC(F) which cyclically of type F°,lny permutes the n distinguished points on a common boundary component; the resulting coset corresponds to a "fractional Dehn twist". The starting point for the sphericity conjecture was the request/challenge from Ed Witten in 1989 to find an orbifold compactification of moduli space together with a cell decomposition which restricts to the usual one on moduli space itself. The sphericity theorem for multiply punctured disks (proved in §5) already proves orbifoldicity for punctured spheres as noted in [P3]. The sphericity conjecture then gives orbifoldicity of our new compactification for any punctured surface. Given a quasi triangulation r of F = F* based at TT, we may formally add in a choice-free manner an edge to T in each punctured component of FT to produce an ideal triangulation r + of F. There is a fatgraph G in F which is Poincare dual to T+ in F. On this level of dual fatgraphs, there is one loop of G for each puncture of F other than vr; these loops are taken formally here in the sense that we associate a lambda length to each such loop, where these lambda lengths are taken to be large relative to the non-loop lambda lengths and are otherwise unrestricted. The promised description of degenerate structures is given by the next theorem. Theorem 16 [P3; Theorem 7] Fix a quasi triangulation T of F£ with associated ideal triangulation r + . Let Ej(t) be a degeneration of simplicial coordinates on T+, and let (3 denote the collection of arcs in T+ SO that the corresponding lambda lengths ei do not diverge. Fixing a horocycle about 7T, each component of Fp comes equipped with a canonical decoration at the cusps on the boundary. The polygons and once-punctured polygons of Fp combine to give a decorated hyperbolic structure on a bordered subsurface GF, and the complement TF = F — GF is regarded as a topological bordered surface (with no hyperbolic structure) together with a decoration coming from the specified horocycle. Thus, a point in An gives rise to a geometric part GF C F together with a topological part TF C F, and the topological part is non-empty if and

291 only if the point is ideal. One is thus permitted to "forget" the hyperbolic structure on a subsurface TF. In particular, the Deligne-Mumford compactification corresponds to "forgetting" the structure only on disjoint collections of annuli TF C F. An elementary observation is that any graph G admits a canonical decomposition into its recurrent part RG and its non-recurrent part NG, as follows. We shall say that a closed edge-path P = ei,e2,---,en on G, thought of as a concatenation of edges, is "edge-simple" provided: i) e; 7^ e I + i for any i (where the index i is taken modulo n + 1); ii) P never twice traverses the same oriented edge of G. The decomposition is given by: RG = {e £ G : 3 edge — simple curve through e},

NG = G- RG. The next result improves and slightly corrects an earlier result. Theorem 17 [P3; Theorem 5] Consider a degeneration of hyperbolic structure, where simplicial coordinates Ei{t) —> 0 for the indices i € I and lambda lengths ej(t) —>• oo for indices j € J. Then J C. I. Furthermore, letting Gi = {e € G : e 6 / } and likewise Gj, we have R{Gj) = Gj. Indeed, that J C I follows directly from the second part of Lemma 5, and the first part of Lemma 5 is applied to prove the remaining assertions about recurrent subgraphs. The combinatorics of our compactification is clear on the level of arc families based at TT in a surface F = F?. Cells simply correspond to arc families which may or may not quasi fill, and the topological part of the decomposition of F, if any, arises from the complementary components which are different from (perhaps once-punctured) polygons. The graphical formulation is somewhat complicated but interesting nonetheless, and we finally describe it. Given the quasi triangulation r of F, complete it to an ideal triangulation r + as before with corresponding dual marked fatgraph G' = G'T. In contrast to the previous discussion, we here simply remove all the loops to produce a marked fatgraph G = GT, where G has s - 1 external nodes, and we label these external nodes with the indices 2 , . . . , s of the corresponding punctures. One considers Whitehead moves (i.e., contraction/expansion of edges of the fatgraphs) as usual, and now also another elementary move supported on a surface of type F^ , 2 .. This new move interchanges the two possible quasi triangulations of this surface, so on the level of dual fatgraphs, given

292 a puncture i, the new move just flips the edge containing the terminal node labeled i by altering the fattening at its other vertex. In fact, this new elementary move is just the composition of two Whitehead moves on ideal triangulations of F02,2y We now two-color the edges of G by specifying "regular" and "ghost" edges, where we thing of ghost edges as "missing", and there must be at least one regular edge. Take equivalence classes of these two-colored fatgraphs with labeled terminal nodes under Whitehead moves and the new elementary move, where we are only allowed to perform moves along the ghost edges. Furthermore, once an edge is a ghost edge, it remains a ghost ("once you're dead, you're dead"), and otherwise the color of an edge is unaffected. For instance, in order to collapse a single edge with distinct vertices in a cubic fatgraph to a quartic vertex, instead, put a ghost icon on the edge to be collapsed; collapse and expand it to produce another cubic fatgraph with a ghost icon on the new edge, and identify these two ghostly fatgraphs. Theorem 18 [P3; Theorem 9] The dual of the simplicial complex A'n is naturally isomorphic to the geometric realization of these marked twocolored labeled fatgraph equivalence classes. An obvious project is to compute the virtual Euler characteristics using this two-colored matrix model. It is also worth mentioning that the graphical formulation of Arc(F) for F a bordered surface can be described in analogy to this discussion, where each distinguished point in the boundary gives rise to a "hair" in the natural way. The dual of Arc'(F) is naturally isomorphic to the set of two-colored fatgraphs with s external nodes and A hairs with external nodes, both labeled, modulo Whitehead moves on the ghost edges, where each boundary component has at least one hair and hairs are never allowed to be ghosts (which has a macabre corporeal interpretation).

7. Operads Several topological and homological operads based on families of projectively weighted arcs in bordered surfaces F* T are introduced and studied in [KLP]. This work as well as further material [Kl] due to Kaufmann are surveyed in his paper [K2] in this volume, so our treatment here will be brief and contextual. The spaces underlying the basic operad are identified with open subsets of Arc(F) which contain Arc#(F) for suitable families of bordered surfaces F. Specifically, say that an arc family a in F is exhaustive if for every boundary component of F, there is some arc in a which is incident on this boundary component, and define Arcsg{n) = {[a] € Arc{Fgn+1)

: a is exhaustive}.

293 We shall see that "gluing in the spirit of train tracks" imbues Arc = {Arc^(n)}n>i with a cyclic operadic structure which restricts to a suboperadic structure on Arccp = {ArcQ(n)}n>i; there are also natural twistings Arcg{n) — Arcg(n) x ( 5 1 ) n + 1 with related operadic structures for which we refer the reader to [KLP]; this trivial Cartesian product is interpreted as a family of coordinatizations of the boundary of F which are twisted by Arcag{ri).

Already algebras over Arccp < Arc are shown in [KLP] to be BatalinVilkovisky algebras where the entire BV structure is realized simplicially. An example of this simplicial realization is given by the symmetric diagram which describes the BV equation itself, as described in [K2]. Furthermore, this basic operad contains the Voronov cacti operad [Vo] up to homotopy. Our operad composition on Arc depends upon an explicit method of combining families of projectively weighted arcs in surfaces, that is, each component arc of the family is assigned the projective class of a positive real number; we next briefly describe this composition. Suppose that F1, F2 are surfaces with distinguished boundary components d1 C Fx,d2 C F2. Suppose further that each surface comes equipped with a properly embedded family of arcs, and let a\,..., a'{ denote the arcs in Fl which are incident on <9e, for i = 1,2, as illustrated in part I of Figure 10. Identify d1 with d2 to produce a surface F. We wish to furthermore combine the arc families in F*,F2 to produce a corresponding arc family in F, and there is evidently no well-defined way to achieve this without making further choices or imposing further conditions on the arc families (such as p1 = p2). The additional data required for gluing is given by an assignment of the projective class of one real number, a weight, to each arc in each of the arc families. The weight uij on aj is interpreted geometrically as the height of a rectangular band J?j = [0,1] x [-u>j/2,u>j/2] whose core [0,1] x {0} is identified with o*-, for i — 1,2 and j = l,...,p*. If we assume that Y^=i wj = £ j = i wj f° r simplicity, so that the total height of all the bands incident on dl agrees with that of d2; in light of this assumption, the bands in F1 can be sensibly attached along d1 to the bands in F2 along d2 to produce a collection of bands in the surface F as illustrated in part II of Figure 10; notice that the horizontal edges of the rectangles {-Rj}? de2

compose the rectangles {-Rf}? into sub-rectangles and conversely. The resulting family of sub-bands, in turn, determines a weighted family of arcs in F, one axe for each sub-band with a weight given by the width of the sub-band; thus, the weighted arc family in F so produced depends upon the weights in a non-trivial but combinatorially explicit way. This describes the basis of our gluing operation on families of weighted arcs, which is derived from the theory of train tracks and partial measured foliations [PH]. In fact, the simplifying assumption that the total heights agree is obviated by

294

considering not weighted families of axes, but rather projectively weighted families of arcs since we may de-projectivize in order to arrange that the simplifying assumption is in force (since plp2 ^ 0 by exhaustiveness), perform the construction just described, and finally re-projectivize. The gluing operation just described may give rise to bands which are embedded annuli foliated by simple curves, and these are simply discarded to finally produce a resulting projectively weighted arc family which is again exhaustive.

i

II

Figure 10 Gluing weighted arc families as train tracks Enumerate the boundary components of F% as dQ,d\,..., dlni once and for all for i = 1,2; the gluing operation applied to boundary components 3} and 3Q induces the usual o, operation for i = 1 , . . . , n\. Letting S p denote the pth symmetric group, there is a natural S n +i-action on the labeling of boundary components which restricts to a natural S n -action on the boundary components labeled {1,2,..., n}. Thus, S n and S n + i act on ArCg(n), and extending by the diagonal action of S n + i on (S1)n+1, the symmetric groups S n and S n + i likewise act on Arcg(n), where S n by definition acts trivially on the first coordinate in (51)""1"1. Theorem 19 [KLP; Theorem 1.5.2-3] The compositions ot are well-defined and imbue Arccp with the structure of a topological operad under the natural Sn-action on labels. The operad has a unit and is cyclic for the natural Sn+i-action. Arc likewise inherits the structure of a cyclic operad with unit. An algebra over the homology of this operad has the structure of a Batalin- Vilkovisky algebra with a product that is compatible with the induced Gerstenhaber bracket. Just to give the flavor, we illustrate as one-parameter families of weighted arc families in Figure 11 the various operations (the identity, BV operator, dot product, and star product), where the parameter s satisfies 0 < s < 1 and the outermost boundary component is labeled zero by convention. In this same spirit, the BV equation itself is given by an explicit and symmetric chain homotopy of weighted arc families as explained in [KLP] or [K2].

295

Figure 11 The identity, BV operator, dot product, and star operator Just as for cacti, Arccp acts on the loop space of any manifold; does this action extend to an action of Arc or Arc itself? Does our formalism allow for a coherent description and calculation of the full string prop or its compactification? Does the BV structure of Arc or Arccp survive injectively on the level of homology? As to the calculation of the homology of Arccp, the planar sphericity conjecture is of utility since the complement of Arccp(n) in Arc{F$)n+1) is a finite union of spheres Arc(F^r), where r + s = n + 1. This arrangement of spheres should be analyzed using a Mayer-Vietoris argument, and the homology of Arccp itself could then be computed using Alexander duality in the sphere Arc{F$jn+i); the twistings Arccp could also then be computed using the Kunneth formula. Of course, the full sphericity theorem is likewise useful^ in calculating the homology operad of Arc itself, of twistings such as Arc, and presumably also of the string prop.

Appendix. Biological Applications The prediction of "macromolecular folding" is a central problem in contemporary biology. The "primary structure" (i.e., the sequence of nucleotides C, G, A, U for RNA, or of 20 amino acids for a general protein) can be determined experimentally, and one wishes to predict from it the full threedimensional structure or "folding" of the macromolecule. We have glossed over interesting and important mathematics here, the problem of "sequence alignment", one aspect of which is fundamental to the problem of reconstructing the entire primary structure of the macromolecule from empirically known snippets. The basic model of the "denatured macromolecule" is / = Im = [0, m] C Z C R for some m. A general "bonding" on Im is simply a collection B of unordered pairs {i,j}, where i,j € Z, \i— j \ > 1, and 0 < i, j' < m. Various pairs of "sites" i,j E / are "bonded" if there is a corresponding element {i,j} € B, and we may consider the semi-circle CV,- C R x R > 0 Q R 2 with endpoints (i,0), (j, 0). One thinks of Im C R as the "backbone" of the macromolecule, a tensile rod which bends to allow the bonding, where each Cij is collapsed to a point; in particular, the backbone is sufficiently

296 rigid to prohibit any bond {i, j} with \i - j \ = 1. We say that a bonding B is a "secondary structure" [Wa] if Cj, n Cu C R = R x {0} C R 2 for all {i,j},{k,l} € B, whence {Qj : {i,j} G B} is an arc family in R x R> 0 5 Z x R> 0 D Im x R> 0 . The secondary structure is said to be "binary" provided djOCke ¥= 0 implies {i,j} = {Jfc.i}. A "helix" of length £ is a collection {i, j}, {i + 1, j - 1 } , . . . {i + £, j - £} e B, for j < i, whence it is a sequence of consecutive bonds. There is a natural binary secondary structure associated with an arbitrary one as illustrated in Figure 12.

Figure 12 Binary from arbitrary foldings An "RNA secondary structure" is a binary secondary structure on Im for some m. This evidently depends upon a plane of projection, but in practice, there seems to be a more or less well-defined plane of projection for the arc family. (This is the plane the biologists draw for RNA which contains most of the bonding.) In any case, the definition of an RNA secondary structure is mathematically complete. In practice in R 3 , there is further bonding between sites which does not lie in the plane of projection, and this is the "tertiary structure" or full three-dimensional folding, of which we shall give mathematical formulations later. The prediction of tertiary from primary structure for RNA is the full folding problem. The prediction of the biologically correct' secondary structure from the primary structure for RNA is another example of the macromolecular folding problem, which we shall discuss further below. For a general protein, the "secondary structure" is a decomposition of it into two standard motifs: the a-helix and the /3-sheet. The a-helix is an (essentially always right-handed) helix, and the /3-sheet is a motif where the macromolecule repeatedly folds back upon itself in a plane. (There are other basic motifs for general proteins which can be added to the definition of its secondary structure, but we shall not take this up here.) Again in any case, the full three-dimensional folding of a general protein is its "tertiary stucture", and the folding problem is to predict secondary or tertiary from primary structure. In the context of general proteins, most effective methods of folding prediction are based upon motif searches and matches with a database of primary structures whose foldings have been experimentally determined by X-ray crystallography or other methods (and these experiments are delicate and relatively few). This is obviously disappointing to a theorist seeking ab initio methods based on physical or mathematical principles. Possibly, such a model should incorporate what some argue is a driving force of "hydo-

297 phobicity/hydrophillicity" (the tendency of individual nucleotides, amino acids or sequences of them to shun/seek water in the biological environment), and more generally, other "solvents" (i.e., the liquids in which the molecules exist) are also considered in many studies. Structural considerations determined by the stereochemistry of the primary structure may also play a role. Furthermore, various models have been purported to capture the "essential" energetics between atoms in the sequences. It is safe to say that the definitive model that simply captures the essential features of atomic interactions is still unknown, and in fact, effective prediction for biologically interesting primary sequences of length 103 and larger may just be too complex for any ab initio methods. For RNA secondary structures, however, there are rigorous and reasonably reliable thermodynamic models, the "standard" one being Mfold [ZMT,ZS] which is defined as follows. The input is the primary structure (that is, a long sequence of letters C, G, A, U); the output is not only the optimal (i.e., global energy minimizing) folding or foldings but also the "sub-optimal" band of the local minima whose values are within, say, five percent of the global minimum. Experience seems to indicate that the correct foldings often lie in such a sub-optimal band. The thermodynamics is governed by "nearest neighbor bonds", that is, Boltzmann weights are assigned to bonds and to pairs and triples of bonds in a helix at adjacent sites; the energies of these configurations at fixed temperature have been determined experimentally, and the thermodynamic model can be solved (for instance stochastically) on the computer. This model apparently very accurately predicts melting points of RNA in most known examples. Bondings are naturally partially ordered by inclusion, and the restriction to secondary structures on a fixed number of sites thus has a geometric realization given by an arc complex for a polygon as was observed in [PW]. By the Classical Fact 11, this space of secondary structures is a sphere. Statically, this gives combinatorial information about the space of secondary structures, and dynamically, one imagines the evolution of a secondary structure in time as a dynamical system on a suitable sphere. Also proved in [PW] is the fact that an appropriate space of all RNA secondary structures is likewise homotopy equivalent to an arc complex (for instance, the space of all RNA secondary structures with no non-trivial helices and a fixed number of unbonded sites is again homeomorphic to the arc complex of a polygon), and a novel constructive proof (based on train tracks) of the Classical Fact 11 is presented. In this context, the planar sphericity conjecture has the immediate interpretation as capturing the combinatorics of several interacting secondary structures on macromolecules in both the general and binary contexts. Any model of full three-dimensional folding of systems of macromolecules should be explicable in terms of arc families, and towards this end, we next describe several recent models of "pseudo-knotting" of secondary structures.

298 Among the many elaborations and extensions of the basic thermodynamical model Mfold described before, [IS] and especially [XBTI] emphasize a related notion of pseudo-knotting among other innovations. The first model we propose is the three-dimensional folding of a single binary macromolecule. Binary Chiral Model F Given a binary bonding B on J m , construct a bordered surface F' C R 3 as follows. Attach to Im x [0,1] C R 2 C R 3 a collection of bands in the natural way, one band 0k for each bond {ii,jk} € B, where the core of the band /3k agrees with Cikjk; the ^-coordinates are adjusted in any convenient way to arrange that the bands are all disjointly embedded. Consider a boundary component K of F'; as illustrated in Figure 13, there are two cases for K: either the projection of K bounds a polygon, or it does not, and we adjoin to F' an abstract polygon along each component of the former type to produce a surface F. The boundary components of F are called "pseudo-knots". Notice that altering the lengths of helices or breaking helices by adding free sites leaves invariant the topological type of F, and F is planar if and only if the folding is a secondary structure. ^^^»™^^^^

/y'

pseudo-knot

^ty£

cap

this

Figure 13 Boundary components of F'. The "achiral model" for binary folding allows each endpoint of a band to be attached either above the a;-axis or below the x-axis (so there are four ways to attach a band, and more data is required for each band). In the same manner as before, we may build a surface F' and cap off certain of its boundary components to produce another surface F, whose boundary components are the pseudo-knots. The chiral and achiral models for arbitrary macromolecules are defined similarly. It is worth saying explicitly that there are further elaborations of these four basic models (chiral/achiral for binary /arbitrary) tailored to other natural biophysical circumstances. We had already studied in [P2] a thermodynamic model with Boltzmann weights given by simplicial coordinates; this physics project was undertaken as a pure mathematician since the structure and attendant calculations were pleasing. The achiral model in the binary case is closely related to the thermodynamic model described in [P2]. (The two cyclic orderings on a set with three elements in [P2] corresponds to

299

the bond being attached above or below the x-axis.) There is no doubt that the combinatorics of suitable models of macromolecules are described by arc complexes. Whether protein folding will be effectively predicted by ab inifcio or by refined empirical motif recognition, a better understanding of this combinatorics should be biologically useful. Speculative Remark As we have seen in §2, the simplicial coordinates are effectively volumes in Minkowski space, so in the spirit of [Kh] and [AS], the thermodynamic model in [P2] has "the Boltzmann weights as volumes". Might the convex hull construction itself give a method of predicting RNA or protein folding? The key question is how to impose the primary structure of the macromolecule on the geometry. Perhaps of utility in this regard, [KV] describes a hyperbolic version of conformal field theory, which is especially interesting to us because the authors argue that it models suitable polymers; furthermore (without realizing it!), the authors employ the special and unnatural case of decorated Teichmuller theory in which all of the horocycles have a common Euclidean radius e in a model of upper half space, and then take the limit e —> 0. Essentially the entire analysis of [KV] applies in our setting and suggests an explicit model of "hyperbolic field theory", where the two-point function is the lambda length.

300

Bibliography [AS] M. Atiyah and P. Sutcliffe, "The geometry of point particles"; hep-th/0105179. [CS] M. Chas and D. Sullivan. "String topology", math.GT/9911159; to appear Ann. Math. [CJ] R. L. Cohen and J.D.S. Jones, "A homotopy theoretic realization of string topology", Preprint math.GT/0107187. [DM] P. Deligne and D. Mumford, "Irreducibility of the space of curves of a given genus", Publ. IHES 36 (1979), 75-110. [Ge] E. Getzler, "Two-dimensional topological gravity and equivariant cohomology", Comm. Math. Phys. 163 (1994), 473-489. [Ha] J. L. Harer, "The virtual cohomological dimension of the mapping class group of an orientable surface, Inv. Math. 84 (1986), 157-176. [HM] J. H. Hubbard and H. Masur, "Quadratic differentials and foliations", Ada Math.142 (1979), 221-274. [Kl] R. M. Kaufmann, "On several varieties of cacti and their relations", preprint MPI-2002-113, math.QA/0209131. [K2] —, this volume. [KLP] Ralph L. Kaufmann, Muriel Livernet, R. C. Penner, "Arc operads and arc algebras", preprint math.GT/0209132. [Kh] A. Kholodenko, "Kontsevich-Witten model from 2+1 gravity: new exact combinatorial solution", Jour. Geom. and Phys. 43 (2002), 4591. [Ko] M. Kontsevich, "Intersection theory on the moduli space of curves and the matrix Airy function", Comm. Math. Phys. 147 (1992), 1-23. [KV] P. Kleban and I. Vassileva, "Conformal field theory and hyperbolic geometry", Phys. Rev. Lett. 72 (1994) , 3929-3932. [IS] H. Isambert and E. Siggia, Proc. NAS USA 97 (2002), 6515-6520. [PI] R. C. Penner, "The decorated Teichmuller space of punctured surfaces", Comm. Math. Phys. 113 (1987), 299-339. [P2] —, "An arithmetic problem in surface geometry", The Moduli Space of Curves, Birk-hauser (1995), eds. R. Dijgraaf, C. Faber, G. van der Geer, 427-466. [P3] —, "The simplicial compactification of Riemann's moduli space", Proceedings of the 37th Taniguchi Symposium, World Scientific (1996),

301 237-252.

[P4] —, "Decorated Teichmiiller theory of bordered surfaces", preprint math.GT/0210326. [P5] —, "Perturbative series and the moduli space of Riemann surfaces" Jour. Diff. Geom. 27 (1988), 35-53;"Weil-Petersson volumes", Jour. Diff. Geom. 35 (1992), 559-608; "Universal constructions in Teichmuller theory" Adv. Math. 98 (1993), 143-215; (with F. Malikov) "The Lie algebra of homeomorphisms of the circle", Adv. Math. 140 (1998), 282-322; "On Hilbert, Fourier, and wavelet transforms", Comm. Pure Appl. Math. 55 (2002), 772-814. [PH] R. C. Penner with J. L. Harer, Combinatorics of Train Tracks, Annals of Mathematical Studies 125, Princeton Univ. Press (1992, 2001). [PW] R. C. Penner and M. S. Waterman "Spaces of RNA secondary structures", Advances in Mathematics 101 (1993), 31-49. [St] K. Strebel, Quadratic Differentials, Ergebnisse der Math. 3:5, Springer-Verlag, Heidelberg (1984). [Su] D. Sullivan, "Moduli of closed string operators and the homology action" (2002); to appear in Abel Bicentennial Proceedings. [Vo] A. A. Voronov, "Notes on universal algebra", math.QA/0111009. [Wa] M. S. Waterman, Introduction to Computational Biology: Maps, Sequences and Genomes, Chapmann and Hall-CRC (1995). [XBTI] A. Xayaphoummine, T. Bucher, F. Thalmann, and H. Isambert, "Prediction and statistics of pseudoknots in RNA structures using exactly clustered stochastic simulations", pre-print (2003). [ZMT] M. Zuker, D. H. Mathews, and D. H. Turner, "Algorithms and Thermodynamics for RNA secondary structure prediction: a practical guide", In RNA Biochemistry and Biotechnology, Kluwer (1999), 11-43. [ZS] M. Zuker and D. Sankoff, "RNA secondary structures and their prediction", Bull. Math. Biol. 46 (1984), 591-621.

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SPATIAL INTERMITTENCY IN TWO-DIMENSIONAL TURBULENCE: A WAVELET APPROACH KAI SCHNEIDER L3M-CNRS & CMI, Universite de Provence, Marseille, Prance Email: [email protected] MARIE FARGE LMD-IPSL-CNRS, Ecole Normale Superieure, Paris, France Email: [email protected] NICHOLAS KEVLAHAN Department of Mathematics and Statistics, McMaster University, Hamilton, Canada Email: [email protected] Turbulence is characterized by its intermittency, which is defined classically as localized bursts of small-scale activity in the observed quantity (e.g. velocity, vorticity, dissipation). Consequently, the simultaneous space and scale localization of the wavelet representation makes it a natural choice for studying intermittency. We propose different measures of intermittency, based on orthogonal wavelets, which avoid some problems associated with classical measures. We then apply them to study the intermittency of two-dimensional turbulence computed by direct numerical simulation.

1. Introduction Turbulence is considered fully-developed when nonlinear effects, due to the advection terms of the Navier-Stokes equation, strongly dominate linear effects due to the dissipative terms. Intermittency, namely isolated bursts of activity in the measured quantity, has long been recognized as an essential characteristic of fully-developed turbulent flows. When introducing the energy spectrum as the Fourier transform of the two-point correlation Taylor 33, noted that dissipation is distributed unevenly: ... the fact that small quantities of very high frequency distur-

303

bances appear, and increase as the speed increases, seems to confirm the view frequently put forward by the author that the dissipation of energy is due chiefly to the formation of very small regions where the vorticity is very high. Apart from these very small regions the turbulence behind a grid is similar at all speeds. Taylor already had the intuition that bursts of high frequency vorticity are responsible for dissipation. This is what we now refer to as spatial intermittency. Soon afterwards Kolmogorov 17 and Obukhov 26 introduced their theory of homogeneous isotropic turbulence, where energy is transferred inviscidly in the inertial range from large to small scales until it is finally dissipated at the smallest scales of motion. In their theory Kolmogorov and Obukhov assumed that the dissipation of energy is space-filling (i.e. non-intermittent). However, in 1944 Landau noticed that the spectral energy transfer rate cannot be constant in space, and thus the fluctuation of energy dissipation must be intermittent at small scales. This is consistent with the physical intuition of Taylor 33. To correct his original theory for intermittent dissipation, Kolmogorov 18 supposed that the dissipation at small scales is distributed log-normally in space. This correction leads to a small increase in the slope of the energy spectrum in the inertial range. The first quantitative estimation of intermittency was presented by Townsend 34. He pictured turbulence as a sequence of active bursts separated by quiescent regions. To quantify this intermittency he introduced an 'intermittency factor' 7 which measures the ratio of active regions to quiescent regions ( 7 = 1 corresponds to an entirely active signal, 7 = 0 corresponds to an entirely quiescent signal). He showed that 7 is inverse proportional to flatness 2~7'

(1) [ )

where / is the signal and (•) denotes the average. The flatness is now one of the standard measures of intermittency. Batchelor & Townsend 2 used this tool to study the intermittency of the first three derivatives of the velocity, finding that flatness increases with the order of derivatives for isotropic turbulence as well as for the wake behind a cylinder. This increase in flatness was attributed to the presence of coherent vortices produced by nonlinear instabilities. Because intermittency is associated with the small scales, many studies involved signal filtering to extract either the small scales, or a range

304

of scales. This filtering is done in Fourier space, and the filtered signal is transformed back to physical space before computing its flatness using (1). Sandborn 29 employed a constant relative band-width filter to measure flatness of the longitudinal velocity fluctuations as a function of wavenumber in channel flow boundary layers. He found that the flatness increases strongly with wavenumber, i. e. the smaller the scale the more intermittent the flow, independently of the distance from the wall. Note that the constant relative band-width filter (Ak/k constant) used by Sandborn has the same spectral properties as the wavelet filters we consider later. It is important at this point to mention that, although these experiments attempt to study spatial intermittency, they use time series data (from a fixed probe in a mean flow) which are interpreted in terms of spatial series using Taylor's hypothesis 33

±--±1

(2)

[) dx~ udt' where the mean velocity U is supposed to be much larger than the fluctuation velocity. Note that this hypothesis is valid only on average. However, some intense turbulent fluctuations may exceed U, which would then cause Taylor's hypothesis to fail. In addition, Lin 20 showed that Taylor's hypothesis is not strictly valid forflowswith mean gradients (e.g. shear flows, boundary layers). Thus, experimental measurements of intermittency may not always be reliable estimates of spatial intermittency. In the case of numerical simulations (as analyzed below), Taylor's hypothesis is not required to study spatial intermittency. Following the method proposed by Sandborn 29, Kennedy & Corrsin 16 used a band-pass filter with constant relative bandwidth to study intermittency in a free jet. They compared the flatness of the turbulent velocity fluctuations with the flatness of a squared Gaussian process as a function of scale, in order to see whether nonlinear processes might be responsible for turbulence intermittency. They found that squared Gaussian noise is more intermittent at all scales than the turbulent fluctuations, although flatness increases with decreasing scale in both cases. They also remarked that averaging tends to make a non-Gaussian signal appear Gaussian (i.e. flatness converges to 3). The intermittent character of turbulence affects other diagnostics used for its study:

• the slope of the inertial range energy spectrum is steepened,

305

• the exponent £p of the p—th order structure function,

{\u{x + l)-u{x)\p)<xfc,

(3)

increases more slowly than linearly with p, • ratios of subsequent moments grow with the order of the moments, i.e. M4/M2 < M6/Mf < ..., where the moment of order p is denned as Mp = (up). This implies that the probability density function (PDF) decays slower than a Gaussian at large values (this characteristic is often called 'heavy tails'), • as suggested by Batchelor & Townsend 2 and others, intermittency is linked to the presence of coherent vortices in the flow. Hence, any physically sound model of turbulence must take into account intermittency. Currently, the most physically accurate turbulence model is large eddy simulation (LES), but it takes into account only a weak intermittency limited to the resolved scales of the flow, namely the large eddies. Let us mention here that from our point of view large eddies are not the same as coherent vortices. We have shown previously 12>10>9 that coherent vortices, i.e. localized concentrations of energy and enstrophy which survive on times much longer than the eddy turn-over time, are multiscale structures. Consequently, the low-pass niters used in LES remove the small-scale part of coherent vortices. The goal of this paper is to point out the limitations of classical measures of intermittency, and to present a unified set of wavelet-based alternatives (many of which have been introduced separately elsewhere). We show how the classical measures can be thought of as a special case of wavelet filtering using a singular wavelet. It is this lack of regularity that limits the usefulness of classical measures for sufficiently smooth signals. In the following section we review classical methods for studying intermittency and note their drawbacks which lead to incorrect results in certain cases. In §3 we present wavelet-based methods that overcome these limitations, and produce accurate results in all cases. We also show precisely how the wavelet methods relate to the classical methods reviewed in §2. The differences between the two approaches are then illustrated by applying them to a direct numerical simulation (DNS) of two-dimensional turbulence in §4. In particular, we see that, in this cas, the classical structure function gives the wrong result when applied to the velocity field, whereas the appropriate wavelet equivalent works correctly. Finally, in §5 we summarize the main results of this paper.

306

2. Classical methods for studying intermittency As we noted in the introduction, intermittency is defined as localized bursts of high frequency activity. This means that intermittency is a phenomenon localized in both physical space and spectral space, and thus a suitable basis for representing intermittency should reflect this dual localization. The Fourier basis is perfectly localized in spectral space, but fully delocalized in physical space. Therefore when a turbulence signal is filtered using a high-pass Fourier filter and then reconstructed in physical space, e.g. to calculate its flatness, some spatial information is lost. This leads to smoothing of strong gradients and the appearance of spurious oscillations in the background, which come from the fact that the modulus and phase of the discarded high wavenumber Fourier modes have been lost. The spatial errors introduced by such a filtering lead to errors in estimating the flatness, and hence intermittency, of the signal. When a quantity (e.g. velocity derivative) is intermittent it contains rare but strong events (i.e. bursts of intense activity), which correspond to large deviations reflected in the 'heavy tails' of the PDF. Second-order statistics (e.g. energy spectrum, second-order structure function) are relatively insensitive to such rare events because their time or space support is very small and thus do not dominate the integral. However, these events become increasingly important for higher-order statistics, and finally come to dominate. High-order statistics therefore characterize intermittency. Of course, intermittency is not essential for all problems: second-order statistics will suffice to measure dispersion (dominated by energy-containing scales), but not to calculate drag (dominated by vorticity production in thin boundary layers). Classical examples of high-order statistics are the pth-order structure functions. In §3.5 we will see that structure functions correspond to the Lp-norm of the wavelet coefficients using the wavelet difference of Diracs (DOD) wavelet 4 , which has only a single zero moment (the minimum required for a wavelet). We will show that this limits the usefulness of the structure functions for analyzing sufficiently smooth fields. The drawback of higher-order statistics, however, is that the number of samples required for an accurate estimation increases with order p. For instance, the number of samples required to estimate the moments of order 12 is about 109, and thus estimation of high-order moments quickly becomes impractical. To circumvent this difficulty we have proposed a different approach: namely to separate the rare and extreme events from the dense and weak

307

events, and then calculate the statistics for each independently. In turbulent flows the rare events are the coherent vortices and the dense events correspond to the incoherent background flow. A major difficulty in turbulence theory is that there is no clear scale separation between these two kinds of events. This lack of a 'spectral gap' excludes Fourier filtering. Since the rare events are well localized in physical space one might try using an on-off filter in physical space to extract them, but in this case you need an a priori model of the coherent structures. However, this approach changes the spectral properties by introducing spurious discontinuities (adding an artificial k~2 component to the energy spectrum). To avoid these two problems we propose using the wavelet representation, which combines both physical and spectral localization (bounded from below by Heisenberg's uncertainty principle). We have shown 10 that nonlinear wavelet filtering can be used to separate the coherent vortices from the background flow. Since we have extensively discussed the use of wavelets for conditional averaging in previous work {e.g. 10 ), in this paper we focus on the use of wavelets in the context of Lp-norms. The most basic Lp-norms are the pth-order moments. They are defined for a quantity / with PDF P(f) as

MP(f) = I P(f)fpdf.

(4)

One can then calculate the ratios of moments of different orders

o

m-

Mp(/)

a)

(5j QvM) - {Mq(f))r/oThe QPtq{f) measure the shape of the distribution P(f). For example, if q = 2 we can define the following quantities:

• • • •

skewness S = Qz^U), flatness F = Q4,2(/), hyperskewness Sh = Qs^tf), hyperflatness Fh = Q&aU)-

The departure of the PDF from Gaussianity can then be measured by comparing the above quantities to their values in the Gaussian case {e.g. S = 0, F = 3 for a Gaussian distribution). The p-th order structure function is used extensively to study homogeneous turbulence since it is translation invariant, characterizes the selfsimilar structure of the flow and is easy to measure experimentally. The

308

p-th order structure function of a random scalar field / is denned as SPj(l)

= {\f{x + l)-f{x)\>).

(6)

Note that under assumptions of self-similarity and using the (exact) Karman-Howarth expression for the third-order structure function Sp(l) ~ lp/3, i.e. a straight line on a log-log graph. However, experimental measurements fall below this straight line prediction. A related second-order statistic is the spectrum

E(k) = ±\f(k)\2

with f{k) = ±Jf(x)exp(-ikx)dx

, (7)

which is related to the second-order structure function Szj(l) and the autocorrelation function R(l) in the following way,

R(l) = (f(x + l)f(x)) = 2 r Jo

and hence we get

cos(kl)E(k)dk,

S2,f(l) = (\f(x + I) - f{x)\2) = 2R(0) - 2R(l) = 2/

(1 - cos(kl))E(k)dk.

(8) (9) (10)

The above relation shows that the structure function corresponds to a filtered spectrum, and the corresponding filter is insensitive to sufficiently smooth fields. In §3.5 we will propose wavelet tools to improve the filter selectivity. 3. Wavelet methods for studying intermittency 3.1. Orthogonal wavelet

transform

In this section we describe some statistical tools based on the orthogonal wavelet transform. The wavelet approach avoids the limitations of structure functions and allows moment ratios to be defined as a function of scale. We present them considering, as example, a one-dimensional scalar field f(x) which has vanishing mean and is periodic (the extension to higher dimensions and vector fields is straightforward). Hence we employ a periodic multi-resolution analysis (MRA) 6>n>23 and develop the signal /, sampled on N — 2J points, as an orthonormal wavelet series from the largest scale lmax = 2° to the smallest scale lmin = 2~ J :

(11) j=0

i=0

309 where ipjti is the 27r-periodic wavelet. Due to orthogonality the coefficients are given by /,,, = (/, ipj,i) where (•,•) denotes the L2-inner product. The wavelet coefficients are indexed in terms of scale j , position i. 3.2.

Wavelet spectra

We define the scale distribution of energy, also called scalogram, as 2J-1

Ej = £

\ki\2-

(12)

i=0

To be able to relate the scale distribution to the Fourier spectrum, we introduce the mean wavenumberfcoof the wavelet I/J, defined by

Ji wwi*

(13)

Thus each scale 2~i of the wavelet tpj is inversely proportional to the mean wavenumber kj — k^lK The discrete local wavelet spectrum 8>28 is then denned as

E(kj,Xi) = \fjti\2£r.>

(14)

where Afcj = -^/fcjfcj+i - y/kjkj~[ is the mean wavenumber. By measuring E(kj,Xi) at different positions Xi in a turbulent flow one can study how the energy spectrum depends on local flow conditions, and estimate the contribution to the overall Fourier energy spectrum of different components of the flow. For example, one can determine the scaling of the energy spectrum contributed by coherent structures, such as isolated vortices, and the scaling of the energy spectrum contributed by the incoherent background flow. The spatial variability of the local energy spectrum E(kj,Xi) measures the flow's intermittency. This quantity also allows us to study the global spectral behaviour of / by summing the local energy spectrum over all positions,

E(kj)=J2E(kj,xi).

(15)

i=0

The relationship between the global wavelet spectrum E(kj) and the usual Fourier energy spectrum E(k) is described in the following section.

310 3.3. Relation between wavelet and Fourier spectra First note that due to the orthogonality of the wavelet decomposition, the total energy is preserved and we have E = J2. Ej. The global wavelet spectrum is related to the Fourier energy spectrum according to 2 8 ' n ~

m

=

1

r°°

c^J0

E k

( ')M*ok'/k)\2dkf.

(16)

dk.

(17)

where

CV = J

k

The wavelet spectrum is therefore a smoothed Fourier spectrum weighted by the modulus of the Fourier transform of the analyzing wavelet 28 . Note that, as the wavenumber increases, the smoothing interval becomes larger u . A sufficient condition guaranteeing that the global wavelet spectrum is able to detect the same power-law behaviour k~a as the Fourier spectrum is that ip should have enough vanishing moments m 2 8 , i. e. f+°° a- 1 / xmip{x) dx = 0 for 0 < m < .

(18)

If this condition is not satisfied the global wavelet spectrum saturates at the critical cancellation order m. In this case it only shows a power-law behaviour with a slope not steeper than a = 2 (TO + 1). Since large a corresponds to smooth functions, when the analyzed function is smoother than the wavelet, then ip(x) should have more vanishing moments m in order to correctly detect the slope of the signal's spectrum. If the wavelet does not have enough zero moments we simply measure the spectral scaling of the wavelet and not of the signal! 3.4. Wavelet intermittency measures In this section we use the space-scale information contained in the wavelet coefficients to define scale-dependent moments and moment ratios. Useful diagnostics to quantify the intermittency of a field are the moments of its wavelet coefficients at different scales j 3 0 , 2 J —1

MPAf) = ^ E i / ^ r j=0

Note that (12) implies Ej = 2jM2,j.

(19)

311 The sparsity of the wavelet coefficients at each scale is a measure of intermittency, and it can be quantified using ratios of moments at different scales,

o

m

Mpj(/)

r2oi

l - (Mqii(f))p/i ' ' which may be interpreted as quotient norms between different Lp- and Lqspaces. Classically, one chooses q = 2 to define typical statistical quantities as a function of scale. Recall that for p = 4 we obtain the scale dependent flatness Fj = QA,2,JI which is equal to 3 at all scales j for a Gaussian white noise (non intermittent signal). The scale dependent skewness, hyperflatness and hyperskewness are obtained for p = 3,5 and 6, respectively. For intermittent signals Qp,q,j increases with j .

VP,
3.5. Relation to structure functions In this section we link the scale dependent moments of wavelet coefficients with structure functions. In the case of second order statistics, we show that global wavelet spectra correspond to second order structure functions. Furthermore, we give a rigorous bound for the maximum exponent of the structure functions and propose a way to overcome this limitation. The increments of a signal, also called the modulus of continuity, can be seen as its wavelet coefficients using the DOD wavelet mentioned earlier, i.e. ip{x)=5{x + l)-5{x).

(21)

We thus obtain u(x + l)-u(x)

= ux,i = (u,ipXil)

(22)

with ipXti(y) = j[(5(Jif£ + 1) - Si^)]. Note that the wavelet is normalized with respect to the L1-norm. The p-th order structure function Sp(l) therefore corresponds to the p-th order moment of the wavelet coefficients at scale /, Sp(l) = I\ux,i\pdx.

(23)

As the DOD wavelet has only one vanishing moment (its mean), the exponent of the p-th order structure function in the case of a self-similar behaviour is limited by p, i.e. if Sp(l) oc Zc(p) then Cip) < P- This 'saturation' behaviour was originally observed by Babiano et al. 1 for DNS of

312 two-dimensional flows. To be able to detect larger exponents one has to use increments with a larger stencil, or wavelets with more vanishing moments, i.e. J xmip(x)dx = 0 for m = 0,1, . . . , M - 1. This will become clearer below in the context of Besov regularity of functions. We now concentrate on the case p = 2, i.e. the energy norm. Equation (16) gives the relation between the global wavelet spectrum E(k) and the Fourier spectrum E(k) for an arbitrary wavelet ip. For the DOD wavelet we find, since ^(A;) = eik - 1 = eikl2{eik'2 - eik'2) and hence |^(fe)|2 = 2(1 -cosfc), that 2

^ik)=C^JQ

r°°

k k'

£( f c ')(l-cos(^))*'.

(24)

Setting I = ko/k and comparing with (10) we see that the wavelet spectrum corresponds to the second order structure function, i.e. E(k) = -^S2(l).

(25)

The above results show that, if the Fourier spectrum behaves like k~a (for k —> oo), E(k) ex k~a if a < 2m + 1, where m is the number of vanishing moments of the wavelets. Consequently we find for 52(0 that S2(l) ex ^ ( p ) = (^) C < P ) (for I -> 0) if C(2) < 2M. In the present case we have M = 1, i.e. the second order structure function can only detect slopes smaller than 2, corresponding to an energy spectrum with slopes shallower than —3. Thus we find that the usual structure function gives spurious results for sufficiently smooth signals. In the appendix we generalize the relation between structure functions and wavelet coefficients by introducing Besov spaces which are typically used in nonlinear approximation theory 7 . 4. Application to two-dimensional turbulence 4.1. Classical statistical analysis We now analyze quantitatively the emergence of intermittency in a twodimensional homogeneous isotropic decaying flow. The two-dimensional Navier-Stokes equations dtuj + u • Vw = — V2w

(26)

where u is the velocity, u = V x u is the vorticity, and Re is the Reynolds number (based on the size of the domain), are computed for periodic boundary conditions using a pseudo-spectral direct numerical simulation at res-

313 olution 2562. The Navier-Stokes dynamics rapidly organize the initial homogeneous flow into isolated coherent vortices which contain most of the vorticity, and this process results in an intermittent distribution of vorticity (figure 1). The idea that intermittency arises from instabilities which generate coherent vortices was already inferred by Batchelor & Townsend 2 from experimental data, when they wrote: It is suggested that the spatial inhomogeneity is produced early in the history of the turbulence by an intrinsic instability, in the way that a vortex sheet quickly rolls up into a number of strong discrete vortices. Note that in our case the initial vorticity field, although homogeneous, contains extreme values associated with the tails of the Gaussian PDF (cf. figure 2) which act as seeds for the formation of coherent vortices. We now analyze quantitatively the emergence of intermittency in a homogeneous two-dimensional turbulentflowat Reynolds number Re = 1000. This evolution may be divided into three different stages: • At early times, from t = 0 to t = 4, the vorticity dynamics are dominated by strong dissipation of enstrophy Z = 1/2 / |w|2cfa; and palinstrophy P = 1/2 / |Vw|2da;: 89% of the initial enstrophy ZQ and 98% of the initial palinstrophy Po are dissipated between t = 0 to t = 4 (cf. table 1). This stage corresponds to the formation of coherent vortices which emerge from the random initial vorticity distribution. • At intermediate times, from t — 4 to t = 10, both enstrophy and palinstrophy decay more slowly: 60%Z0 and 88%Po are dissipated between t = 4 and t = 10 (cf. table 1). During this stage the flow dynamics is dominated by strong nonlinear vortex interactions. • At later times, from t = 10 to t = 100, the energy decreases very slowly (cf. table 1), and theflowevolves towards a quasi-stationary state where only two opposite-sign coherent vortices remain. After the last stage the turbulence is effectively dead, as the nonlinear dynamics are zero (i.e. u- Va> w 0), and the flow evolves exclusively due to the diffusion of vorticity. We now analyze these three stages in more detail using the measures introduced earlier. The random initial distribution of vorticity (figure 1 a) is characterized by a Gaussian PDF (figure 2 a) and a large scale correlation (figure 4 a). At early times (from t = 0 to t = 4) the Navier-Stokes

314

-30

-20

-10

0

10

20

30

-10

0

10

-2

0

2

Figure 1. Vorticity field at t = 0,4,100. Note the emergence of coherent vortices from an initially homogeneous vorticity distribution.

nonlinear dynamics leads to a self-organization of the vorticity field into isolated coherent vortices (figure 1 b). The formation of coherent vortices is reflected in the following quantities: • Vorticity in physical space changes from homogeneous at t = 0 (figure la) to inhomogeneous (figure lb) at later times. • The PDF of vorticity changes from a Gaussian at t = 0 (figure 2a) to a stretched exponential (figure 2b) at later times. • The scatter-plot of w versus \t changes from decorrelated at t = 0 (figure 3a) to correlated as a superposition of several functional relationships u> — F{^) (figure 3b), each corresponding to a coherent vortex. • Wavelet coefficients of vorticity u> change from dense for the Gaussian distribution at t = 0 (not shown here) to sparse (figure 5d) at later times. At the intermediate and late stages for t = 4 to t = 100 the flow dynamics is dominated by the nonlinear interactions between isolated coherent vortices. Each vortex is advected and strained by the velocity resulting from all vortex motions; this evolution is conservative (without dissipation) as long as the vortices are far apart. If two vortices of the same sign move close together the interaction is no longer conservative and leads to the fusion of the vortices accompanied by a strong (although very intermittent in space and time) dissipation of enstrophy and palinstrophy. By t = 100 the flow has reached a quasi-stationary state where all same-sign vortices have merged and only two opposite-sign vortices remain. This final stage is characterized by two distributions predicted by analytical theories, as we

315 will now explain. As shown in figure 2, the initial Gaussian distribution evolves via a stretched exponential to a quasi-stationary final state approximating a Cauchy distribution. This Cauchy distribution agrees with the prediction of Min, Mezic & Leonard 24 based on a system of point vortices. For such a distribution the variance and all higher order moments diverge, showing that the Navier-Stokes equations can generate extremely non-Gaussian distributions with coherent vortices. This evolution of the PDF from Gaussian to Cauchy can be dynamically explained. Due to Biot-Savart's law the flow organizes itself around initial extreme values of the vorticity. The gradients formed between the coherent vortices by this process tend to dissipate weak vorticity and therefore isolate the vortices. The coherent vortices then merge, which results in further dissipation of weaker vorticity. As a result, the strong values of the vorticity decay more slowly than the weak values, which results in a steepening of the PDF. Note that the velocity remains quasi-Gaussian for all times, as is the case for three-dimensional turbulence.

Figure 2. PDF of vorticity field at t = 0, 4,100. The PDF is initially Gaussian and changes to a Cauchy distribution via a stretched exponential. The dotted curves show the ideal distribution (Gaussian, stretched exponential or Cauchy).

The coherence scatter plot (pointwise correlation between vorticity and stream function) illustrates the self-organization of the flow (see figure 3). Initially there is no correlation between the stream function $> and the vorticity w, while in the final state a functional relation ui = F(^f), with F(*) = asinh(|/?|#) where a = 1/5 and /? = - 2 , emerges. The functional relationship between u> and ^ implies that the nonlinearity has been depleted, and that the flow has reached a quasi-stationary state. This sinh functional relationship was predicted by Joyce & Montgomery 15 and verified numerically by Montgomery et al. 25. At intermediate times the co-

316

t

E

Z

0 0.5000 37.38 4 0.3949 4.146 10 0.3453 1.739 100 0.2546 0.3076

P M3/M?2 28312.6 529.2 61.97 0.5831

-0.01028 0.3436 0.8891 0.04206

M4/Mj 3.061 5.676 8.147 5.766

M5/Ml/2 -0.1926 8.325 21.81 0.4563

M6/Mj 15.83 76.36 150.4 46.23

herence scatter plot appears to be a superposition of many sinh curves with different a and f3 corresponding to the many individual coherent vortices (the thickening of the curves is due to the relative motion of the vortices). The coherence plot measures the organization of the flow, and shows that as the flow becomes more organized the statistics become less Gaussian. We consider the conjunction of these two effects to be characteristic for intermittency in incompressible turbulent flows.

:

f

h • y* 1 i * _ 1 i/\

Figure 3. Coherence scatter plot at t = 0,4,100. Note the transition from an uncorrelated state towards a functional relationship w = F(*l>), characteristic of quasi-stationary coherent vortices.

In figure 4(a), we show the time evolution of the energy spectrum at t — 0,4 and 100. It follows a power law in the inertial range, namely from k = 1 up to the dissipative wavenumbers larger than k = 64, where the slope changes from —3 at t = 0, to —4 at t — 4 and finally to —6 at t = 100. These negative slopes reveal the long range correlation of the energy spectrum which increases in time, i.e. the velocity field becomes increasingly correlated and smooth. Note that the statistical theory of two-dimensional homogeneous turbulence 19 predicts a A;~3 power-law behaviour. The steepening of the slope we observe as time evolves is attributed to the intermittency resulting from the emergence of coherent vortices 22 . In table 1, we show the time evolution of the ratio between the subsequent moments of vorticity Mp/M^ • At time t = 0 the behaviour of

317 Mp/M% versus p is consistent with the Gaussianity of the initial vorticity distribution. At later times, the ratio increases with p faster than for the Gaussian distribution, which confirms the fact that the vorticity field becomes intermittent, as we have already seen from the vorticity (cf. figure 2) 4.2. Wavelet statistical analysis We now apply the wavelet diagnostics introduced in §3 to analyze the intermittency of the freely decaying two-dimensional turbulent flow described in the previous section. In figure 4 (b) we show the scale dependence of enstrophy (15) at early, intermediate and late times. The scale I of maximum enstrophy increases from I = 2~5 at t = 0 to I = 2~2 at t = 100. Therefore this correlation scale of the vorticity field increases in time, which is due to the formation and subsequent merging of coherent vortices, as illustrated in figure 1. The scale dependent flatness of vorticity Fj is shown in figure 4(c). It evolves from Gaussian (i.e. Fj « 3 for all j) at t = 0 to non-Gaussian (characterized by the fact that Fj strongly increases with scale j) as time increases. io>j

ltf

,—,—„—„

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: > <

, — , — , — — ^ T

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Figure 4. (a) Energy spectra E(k). (b) Scale distribution of enstrophy Zj. (c) Scale dependent flatness Fj at t = 0,4,100 for the decaying case.

We focus now on the instant t = 4 (cf. figure lb), which is typical of the regime where coherent vortices have already formed and are interacting strongly. To study the dynamics of this flow regime we analyze the vorticity field u>, the linear dissipation term L = I/Re V2u; and the nonlinear advection term N = —u • Vw of the governing vorticity transport equation dtw = L + N

(27)

at time t = 4. We recall that, since V • u — 0, the velocity can be re-

318

constructed from the vorticity using Biot-Savart's relation, u = V^V""2^, where V1- = (—dy,dx) and V~2 denotes the Green's function of the Laplacian. We plot vorticity, dissipation and advection at t = 4 in both physical and wavelet space in figure 5. Figure 5b shows that dissipation is localized in the sheared regions between interacting vortices. Figure 5c) shows that the advection term is also well-localized in sheet-like regions. The wavelet coefficients of the three fields have similar intermittent structure. Note that the wavelet coefficients become increasingly sparse at smaller scales (figure 5d, e, f), which is a good indication of intermittency. It is interesting to note that the wavelet coefficients that are active for vorticity are also active for dissipation and advection, i.e. the same wavelet coefficients represent all three quantities. The wavelet coefficients u>, L and N reveal that vorticity, dissipation and advection are strongly intermittent, i.e. for these 3 fields the spatial support decreases with the scale, likewise their wavelet coefficients become sparser when scale decreases. This intermittency is quantified by computing the scale dependent flatness Fj (c.f. figure 6c). The moments Mv and the flatness Fj strongly increase with p and j , respectively, with the same scaling law for the three fields u>, L and N. This confirms the fact that they have the same type of intermittency. In figure 6a we display the scale distribution (in L2-norm) of vorticity, dissipation and advection. They all are multiscale, but have different distributions: vorticity is most active around scale I = 2~ 2 5 , dissipation around scale I = 2~6 and advection around scale / = 2~5. The fact that dissipation has its maximum at small scales agrees with the classical phenomenology. However, its multiscale distribution contradicts the assumption of a nondissipative inertial range (assumed by the statistical theory of turbulence), but agrees with the hypothesis of progressive dissipation throughout the inertial range as proposed by Castaing 5 , and Frisch & Vergassola 13, for three-dimensional turbulence. Now we take the vorticity field at t = 4 and randomize the phase of its Fourier coefficients, in order to construct a fractal field with a Gaussian PDF, retaining the spatial correlation of the original vorticity field. The randomized vorticity field wr is defined as wr(f) = ^ | C ( f c ) | e i V ^ l

(28)

k

where w(k) = 1/(2TT) J IJJ(X) exp(—ik • x)dx denotes the Fourier transform of the original vorticity field and 6 are uniformly distributed random num-

319

^ ^ • - ^ - w ;

-10

0

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10

-10

•• < **^-

-5

\

0

.,

5

10

„••

-100

•t*,,^..

-SO

-

0

• -

SO

100

"Sr^'v

Figure 5. Top: vorticity field w, linear dissipation term L = — I/Re V2w and nonlinear advection term N = u- Vw at t = 4. Bottom: the corresponding wavelet coefficients , L and iV. We use the Mallat representation 21 for the wavelet coefficients. .,

.•-•-*••••*••••%

'

\
j

A

^

S

"

I

" •

_

w

i:

Figure 6. Scale distribution of enstrophy Zj, normalized PDFs P(fn) for fn = //||/||oo and scale-dependent flatness Fj for vorticity, dissipation and advection terms at t = 4.

bers, i.e. 6 £ U(0,1). Although the resulting field has the same spectral behaviour (cf. figure 7b) and the same scale distribution (cf. figure 7c) as vorticity u>, it has neither coherent vortices in physical space (figure 7a), nor intermittency, i. e. its wavelet coefficients are not sparse at small scales (figure 7d). The phase-scrambled field also has no significant increase of flatness Fj with scale (figure 7f). Its PDF is Gaussian with flatness F = 2.9

320

(cf. figure 7e), compared with F = 5.7 and stretched exponential PDF for the original field (figure 2b). We have thus shown that a fractal field with the same long-range dependence as a turbulent field (i.e. same energy spectrum) is not necessarily intermittent. This also demonstrates that intermittency in turbulence is due the presence of coherent vortices.

*if$B'
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Figure 7. Top: vorticity with randomized phases, u)T at t = 4. Corresponding energy spectra E(k) and scale distribution of enstrophy Zj. Bottom: corresponding wavelet coefficients cDr, PDF of vorticity u>r and flatness Fj.

To illustrate the relation between structure functions and scale dependent moments of the wavelet coefficients we consider a typical statistically stationary two-dimensional turbulent flow field at resolution N — 2562, extensively studied in 10. In fig. 8 (top, left) we plotted its energy and enstrophy spectra exhibiting a k~5 and a k~z power law behaviour, respectively. Figure 8 (top, right) shows the enstrophy Fourier spectrum together with the global wavelet spectrum using quintic spline wavelets. We find perfect agreement between both Fourier and wavelet spectra as predicted by the theory (16) since the wavelet used here has 5 vanishing moments. Furthermore we plot the global wavelet spectrum, plus its standard deviation at different scales, to illustrate the fact that the fluctuations of the spectrum in physical space increase with the wavenumber. All these diag-

321

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Figure 8. Top: Isotropic Fourier energy and enstrophy spectra (left). Fourier and global wavelet enstrophy spectra and the standard deviation of the wavelet spectrum in physical space. Middle: Classical longitudinal structure functions 5|| p (/) for p = 2, ...,6 of velocity (left) and corresponding wavelet based longitudinal structure functions (right), both averaged over 256 lines. Bottom: Structure functions versus third order structure function, left classical case, right wavelet case.

322

nostics indicate the presence of intermittency. In figure 8 (middle) we plot the longitudinal structure functions S\\tP(l) of the velocity for p = 1 to 6. On figure 8 (middle, left) we use the DOD wavelet, i.e. the classical structure function, and on figure 8 (middle, right) we use quintic spline wavelets. For the classical structure functions (figure 8 (middle, left)) we observe that the slope is limited by the number of vanishing moments of the wavelet. For example at p = 2 we observe that S2{1) oc I2, whereas one should find for S^W °c I4 since E(k) oc k~5. This limitation is due to the fact that the DOD wavelet has vanishing mean only, and therefore the structure functions show the scaling of the wavelet (CO5) = P) a n d not that of the velocity field! Using the quintic spline wavelets instead of the DOD wavelets, we find the correct slope of Z4 for 52, since the quintic spline wavelets have 5 vanishing moments. For the higher order structure functions we find £(p) « 2p, which is the expected scaling of a two-dimensional velocity field. 4.3. Extended

self-similarity

To extend the scaling behaviour of structure functions one typically uses rescaled structure functions 3 , i.e. one considers ratios of structure functions of different order Sp(l)/Sq(l). One then studies the scaling behaviour of the p-th order structure function as a function of the q-th order structure function, i.e. Sp(l) oc S,(Z)«™>

(29)

with Sq(l) oc l^q). Typically, q = 3 due to the fact that S3(l) is known exactly from the Karman-Howarth equation 14. This approach is called extended self-similarity (ESS). It greatly increases the range over which one observes a well-defined power-law, even at moderate Reynolds numbers. Because the scale-dependent moments of the wavelet coefficients are equivalent to the structure functions using the L1 normalization of the wavelets, i.e. SpVL(2^) = 2^2MpJ(f) = 2^/2/2-> J^LQ1 \fj,i\p, the ratios of the moments at different scales Qp,q,j(f) (20) correspond to a generalized extended self similarity in wavelet space. This allows us to detect the selfsimilar behaviour of functions with steeper slopes: if tp has m vanishing moments then £(p) is bounded from above by rap. Finally, in figure 8 (bottom) we plot the structure functions versus the third order structure function, as used in the ESS approach. In both cases we observe that the functions are less curved than without using ESS. For

323

the classical structure functions based on the DOD wavelet (figure 8 bottom, left), we find slopes of 2/3,4/3,5/3 and 2 for p = 2,4,5 and 6, respectively. When we use the improved structure function based on the quintic spline wavelet, we find the same slopes of 2/3,4/3,5/3 and 2 for p = 2,4,5 and 6, respectively. This is due to the fact the we plot the structure functions versus the third order structure function and hence only information about the relative slope is obtained. The above results show that the slope of the classical structure functions is limited by the regularity of the underlying DOD wavelet. The scaling behaviour of smoother fields can only be detected using structure functions based on wavelets with a sufficient number of vanishing moments. We have also shown that the ESS approach may be misleading, as it only yields information about the relative slopes. These relative slopes might be the same, even if the slopes of the original structure functions are wrong. The relations summarized in this section have been presented for the one-dimensional case only, but they can be generalized easily to higher dimensions using tensor product constructions of wavelets 6 and Besov spaces in Mn 32. 5. Conclusion In this paper we have reviewed the usual ways of quantifying turbulence intermittency and its effects. In particular, we have emphasised the fact that structure functions can be interpreted as wavelet transformed quantities using a difference of Diracs (DOD) wavelet. Because this wavelet is singular, it is insensitive to sufficiently smooth signals. This means that for signals with an energy spectrum steeper than —3 the classical structure function gives spurious results. In order to overcome this limitation we propose using a sufficiently smooth wavelet (i.e. one with more zero moments). This point has been illustrated using the velocity field from a two-dimensional DNS which has an energy spectrum with a slope —5. We have found that the classical structure functions merely measure the properties of the DOD wavelet, whereas the structure functions based on a smoother (quintic spline) wavelet provide accurate scalings. We have also presented several other wavelet-based diagnostics to quantify intermittency. Each of them exploits the space—scale localization properties of the wavelet representation, that reflects the space-scale localization of intermittency itself. The characteristics of intermittent fields were highlighted by applying these wavelet tools to a decaying two-dimensional

324

turbulence, and to a non-intermittent Gaussian random field with the same energy spectrum. These wavelet diagnostics also showed that the nonlinear advection term and the linear dissipation term of Navier-Stokes equations are highly intermittent. This justifies the use of adaptive wavelet-based algorithms 31, that exploit this intermittency to reduce the number of degrees of freedom necessary to compute the evolution of turbulent flows. We have also studied the appearance of intermittency in a turbulent flow computed from non-intermittent initial conditions until it reaches a final quasi-stationary state. We showed how the nonlinear dynamics of the Navier-Stokes equations produces a highly intermittent vorticity distribution due to the formation of coherent vortices. Finally, we would like to emphasize that the wavelet measures of intermittency presented here are not specific to turbulence, and therefore can be applied to other intermittent signals. Acknowledgements We gratefully acknowledge financial support from McMaster University, the programme pluri-formation 'Modelisation et simulation numerique en mecanique des fluides' of Ecole Normale Superieure Paris, the European Program TMR on 'Wavelets in Numerical Simulation' (contract FMRXCT 98-0184) and the French-German Program Procope (contract 99090). N. Kevlahan would like to thank NSERC and CNRS for financial support, and he would especially like to thank M. Farge for the chance to visit LMD during part of this work. Appendix The relation between structure functions and wavelet coefficients can be generalized by using appropriate function spaces. For this we introduce Besov spaces 32 ' 7 , which can be characterized using wavelet coefficients and are related to structure functions 27. For q < oo we define the Besov space

v B'Ptq = {/ G W(R) ; r\j\f{x with 0 < s < 1 ,

+ l)-f{x)\>dxYI» G L"(R*+,j)} (30)

p,q>l.

This means that / € B*>q if and only if / £ Lp, and

rsq(j

\f(x + I) - f(x)fdx)^

jJ

< oo .

(31)

325

Using the p-th order structure function SP{1), this is equivalent to

(jf + ~r"5 p (0'j)'
(32)

This means that the p-th order structure function is related to Besov norms via the modulus of continuity. The corresponding norm is given by

11/11*.,, = II/HLP + I/IBJ,, where the semi-norm \J\B'

(33)

is defined as:

[l-{j\f(x

+ l)-f{x)\'>dx)1"']

-(r'-*""f)*

-(jf [•-^-•'H'f)'-

T

J

(34)

<»>36 <»

This shows that the Besov norms (q < oo) are intimately related to the structure functions Sp(l) and the wavelet coefficients f(a, •) =

(f(xUaib(x)).

In the case q = oo, we obtain

B'PtOO = {/ G W(R) ; r'(j \f(x +1) - fixWdx)1* G L°°(JR*+)} (37) and for the semi-norm we get

I/IBS,- = H r ' ( / 1/^ + 0 " fWW'Woo

= ||/-S5P(/)1/P||OO. (38)

In the case of a self-similar behaviour of the type Sp(l) ~ l ^

(39)

/ G B«5>/*.

(40)

it follows that

This implies that ^(p) < p as we restricted ourself to the case s < 1. To overcome this limitation the Besov spaces can be generalized for s > 1, where s is no longer an integer. For s > 1, s not an integer, we decompose s into s = [s] + a, ([s] being the integer part of s) and we introduce

B'Piq{M) = {/ G Lp(iR) ; / ( m ) G ££ g (JR), 0 < m < [s]}

(41)

326

where /("*) denotes the m-th derivative of / . The corresponding norm is defined as

"'Ik. = £

m=0

n/ (m) ik. •

(42)

In order to have norm equivalence with the wavelet coefficients, the wavelet tp has to have at least [s] + 1 vanishing moments. Let us mention that in the case where s is an integer, the modulus of continuity should be modified 32 to ||/(x+/)-2/(a;)-/(a;-Z)||/,p. Note that this second-order stencil is no longer equivalent to the structure function. Finally, when p = q = 2 we obtain the Sobolev space H" and for p — q = oo the Holder space Cs 32 . To summarize, structure functions of order p correspond to Besov norms of functions which can be characterized by means of weighted sums of wavelet coefficients due to norm equivalences. This remark completes the link between structure functions, wavelet coefficients and Besov norms. It also suggests that the limitations of classical structure functions may be overcome by using structure functions based on wavelets with more vanishing moments than the DOD wavelet. References 1. Babiano, B., Basdevant, C. & Sadourny, R. 1985 Structure functions and dispersion laws in two-dimensional turbulence. J. Atm. 5ci.42(9), 941-949. 2. Batchelor, G. K. & Townsend, A. A. 1949 The nature of turbulent motion at large wave-number. Proc. Roy. Soc. A 199, 238-255. 3. Benzi, R., Ciliberto, S., Tripicione, R., Baudet, C. & Massaioli, F. 1993 Extended self similarity in turbulent flows. Phys. Rev. E 48, R29. 4. Chainais, P., Abry, P. & Pinton, J. 1999 Intermittency and coherent structures in a swirling flow: A wavelet analysis of joint pressure and velocity measurements. Phys. Fluids 11(11), 3524-3539. 5. Castaing, B. 1989 Consequence d'un principe d'extremum en turbulence. J. Physique 59, 147-156. 6. Daubechies, I. 1992 Ten lectures on wavelets. SIAM. 7. DeVore, R. 1999 Nonlinear approximation. Ada Numerica 8, Cambridge University Press. 8. Do-Khac M., Basdevant C, Perrier V. and Dang-Tran K. 1990 Wavelet analysis of 2d turbulent flows. Physica D, 76, 252-277. 9. Farge M., Pellegrino G. and Schneider K. 2001 Coherent Vortex Extraction in 3D Turbulent Flows using orthogonal wavelets. Phys. Rev. Lett., 87(5), 054501-1-054501-4.

327

10. Farge M., Schneider K. and Kevlahan N. 1999 Non-Gaussianity and Coherent Vortex Simulation for two-dimensional turbulence using an adaptive orthonormal wavelet basis. Phys. Fluids, 11(8), 2187-2201. 117-200. 11. Farge, M. 1992 Wavelet transforms and their applications to turbulence. Ann. Rev. Fluid Mech. 24, 395-457. 12. Farge, M. & Rabreau, G. 1988 Transformed en ondelettes pour detecter et analyser les structures coherentes dans les ecoulements turbulents bidimensionnels. C. R. Acad. Sci. Paris Ser. II307, 433-462. 13. Frisch, U. & Vergassola, M. 1991 A prediction of the multifractal model: the intermediate dissipation range. Europhys. Lett. 14, 439-444. 14. Frisch, U. 1995 Turbulence. The legacy of A.N. Kolmogorov. Cambridge University Press. 15. Joyce, G. & Montgomery, D. 1973 Negative temperature states for the twodimensional guiding-center plasma. J. Plasma Phys. 10, 107. 16. Kennedy, D. A. &; Corrsin, S. 1961 Spectral flatness factor and 'intermittency' in turbulence and in non-linear noise. J. Fluid Mech. 10, 366-370. 17. Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR. 30(4), 301305. 18. Kolmogorov, A. N. 1962 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13, 82-85. 19. Kraichnan, R.H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417-1423. 20. Lin, C. C. 1953 On Taylor's hypothesis and the acceleration terms in the Navier-Stokes equations. Quart. Appl. Math. 10(4), 295-306. 21. Mallat, S. 1998 A wavelet tour of signal processing. Academic Press. 22. McWilliams, J.C. 1984 The emergence of isolated coherent vortices in turbulent flows. J. Fluid Mech. ,146, 21-43. 23. Meneveau, C. 1991 Analysis of turbulence in the orthonormal wavelet representation. J. Fluid Mech. ,232, 469-520. 24. Min, I. A., Mezic, I. & Leonard, A. 1996 Levy stable distributions for velocity and velocity difference in systems of vortex elements. Phys. Fluids 8(5), 11691180. 25. Montgomery, D., Matthaeus, W. H., Stribling, W. T., Martinez, D. & Oughton S. 1992 Relaxation in two dimensions and the "sinh-Poisson" equation. Phys. Fluids A, 4, 3-6. 26. Obukhov, A. M. 1941 On the distribution of energy in the spectrum of turbulent flow. Dokl. Akad. Nauk SSSR. 32(1), 22-24. 27. Perrier, V. & Basdevant, C. 1996 Besov norms in terms of the continuous wavelet transform. Applications to structure functions. Math. Mod. Meth. Appl. Sci. 6, 649-664. 28. Perrier, V., Philipovitch, T. & Basdevant, C. 1995 Wavelet spectra compared to Fourier spectra. J. Math. Phys., 36(3), 1506-1519. 29. Sandborn, V. A. 1959 Measurements of intermittency of turbulent motion in a boundary layer. J. Fluid Mech. 6, 221-240.

328

30. Schneider, K. Sz Farge, M. 1998 Wavelet approach for modelling and computing turbulence. In Advances in turbulence modelling, von Karman Institute for Fluid Dynamics Lecture Series 1998-05. 31. Schneider, K., Kevlahan, N. K.-R. &: Farge, M. 1997 A comparison of an adaptive wavelet method and nonlinearly filtered pseudo-spectral methods for two-dimensional turbulence. Theoret. Comput. Fluid Dynamics 9, 191. 32. Stein, E. M. 1970 Singular integrals and differentiability properties of functions. Princeton University Press. 33. Taylor, G. I. 1938 The spectrum of turbulence. Proc. Roy. Soc. Lond. A 164, 476-490. 34. Townsend, A. A. 1948 Local isotropy in the turbulent wake of a cylinder. Austr. J. Sci. Res. 1, 161.

329

AN ELEMENTARY DEFINITION OF BROWNIAN MOTION IN HILBERT SPACE

NILS TONGRING WOODS HOLE OCEANOGRAPHIC INSTITUTION

1. Introduction Some years ago Kakutani and I considered the simple definition below of Brownian motion in Hilbert space, using a complete orthonormal basis and an associated sequence of independent 1-dimensional Brownian motions. Earlier Hasegawa, in a series of papers 1>2, constructed Brownian motion in an infinite dimensional space, recasting in a probabilistic way the potential theory from Paul Levy's monograph on functional analysis 3 . In Levy's account, mathematical objects in infinite dimensional space are limits of sequences of objects in finite dimensional Euclidean spaces R n as n —> oo, his "methode du passage du fini a l'infini". Hasegawa considers the space E of sequences {x = (xi, ...,xn,...) £ R°° : suptfl/N5Zn=i xn2 < 00} with the topology given by the seminorms MN = ijjT!l=^n2)l'2,N = 1,2,... and ||a:||oo = l i m s u p , ^ \\x\\N, with Brownian motion in the state space E a family of independent 1dimensional Brownian motions. In 2 Hasegawa compares his approach with other well-known efforts by Gross, Daletskii, and Hida. Our brief attempt in the summer of 1984 stopped with showing the existence almost everywhere of infinite dimensional Brownian motion for a fixed time (cf. page 3 below), but not for all time, a familiar problem in the study of stochastic processes, and which seems also a difficulty for the approach of Hasegawa. A question of Dennis Sullivan revived my interest hi the problem, and the following is a continuation of the work Kakutani and I had begun in the summer of 1984. This summer will mark Kakutani's 93rd birthday.

330

Consider a (real) separable Hilbert space H with a complete orthonormal system {ej}?^. Suppose {Xi}^ is a collection of independent Brownian motions on the real line R with the common probability space ft, the collection of measurable sets £ in fi, and the probability measure \i with the standard properties, each Brownian motion having continuous paths for a.a. u € fi, and each starting from the origin: Xi(0,w) = 0 for every i. Given the element x = YliLi a*e« °f H, where {fli}£i is a collection of (real) numbers such that Ei=i a ? < °°! the formal expression oo

(1)

X(t,w) = Y^aiXi(t,u)ei j=i

is a possible candidate for Brownian motion in Hilbert space (assume that the coefficients {aj} are non-zero). The first problem is to establish a meaning for (1), for almost all w £ f i . Denote the partial sum by S n : n

Sn = ^2aiXi(t,w)ei.

Lemma 1: The sequence {S n } is Cauchy.

Proof: 2

2

||Sn-Sm|| =

Y,

aiXi{t,cj)ei

i=m+l

= ^

\aiXi(t,u>)\2.

i=m+l

Is the sequence {Sn = Y^i=i di2Xi2(t, w)}n for a fixed t convergent? By the 0-1 law, this sequence either converges a.s. or diverges a.s.. Apply the "three series criterion" of Kolmogorov to the series X)i=i \alXi(t,Lj)\2 : (TSC) The series ^ Yi of independent summands converges a.s. to a random variable if and only if for any fixed c > 0, the three series (i) £ Pr^Yi|| > c] , (ii) E f f 2 ^ C , and (iii) £EKi C c o n v e r g e . I n (ii), the superscript "c" denotes the truncated variable: Yc is Y or 0 according as \Y\ < c or \Y\ > c. These three conditions are easily verified:

331

(i)ZPr[aiV>c}

= Y:Prl\Xi\>g-l}.

Now Pr[lXil

>^ M

=

2 /« e _ ^ , u < J ^ - ^ ) * V^iriJ^ ~ y/2-Kct

=

2ja^ e -^, Velvet

using the inequality P°°

2

1

2

f°°

1

2

/ e~^du < - / ue~^du =X-e~^. Jx ~ * Jx The sum of such terms converges, and condition (i) follows. (ii) The variance a2(a?X?)c of the truncated a^Xf is E[(a]X?y - E(a?X?yf

= E[K2X,2)^]2 - 2(E[(a2X2)c])2 + E[E(a 2 X 2 ) c ] 2 = E[a 2 (X 2 ) c ] 2 -2(E[a 2 (X 2 ) c ]) 2 + (E[a 2 X 2 ] c ) 2


= O(t2aj). Therefore J2 a2(afXf)c = O(t2 J2 aj) < oo for any t. (iii) The sum ^E[a 2 (X,(t, W )) 2 ] c < ^ E a 2 ( ^ ( t , a ; ) ) 2 = t ^ a 2 < oo. i

i

i

By the three series criterion then, the sum £^ a2(Xi(t, w))2 converges a.s.; therefore the sequence {S n } is Cauchy, and for a given t, X(£,w) exists for a.a UJ e fl, proving the lemma. There still remains the question of the existence of X for all t for a.a. LJ, and whether X is continuous in t.

Take the domain of the variable t to be the unit interval (this case can be transformed to the case [0, oo)).

332

Introduce the polynomial approximation to X denned by Y«(t,u/)=X(t,u;)

(2)

if t = k/2l , and interpolate linearly for the other tin [k — 1/21, k/21} , for k — 1,2,..., 2*. By excluding a set in ft of measure zero, Y ^ is well-defined for all i. We want to show that lim Y~«(*,a;) = Y(t,w)

(3)

z—>oo

exists uniformly in t £ [0,1] for a.a. LJ, and is therefore necessarily continuous. First, establish that lim Y' l '(t, w) exists in the H-norm sense, uniformly for t £ [0,1], for a.a. u. This will follow if the sequence {^||Y«(tlW)-Y(<+1)(t,a;)||}n »=i

converges uniformly in t; for then {Y^(t,u)} limi_Kx> YW(t,u) exists uniformly for t € [0,1]. Let

is Cauchy, and

Ei = {w € n : max ||Y*(*,W) - Y (<+1) (*,w)| > e j ,

(4)

and Pi

- Pr(Ei).

(5)

Assume that (A) there exists a sequence of positive real numbers {ej} with ][] e^ < oo, and that the associated {p*} satisfy ^2pi < oo. Let N = lim sup Ej. If w S fi\N, there exists an integer m such that u> £ Ej for i > m; i.e., max \\Y^(t,(j) - y( i+1 )(t,a;)|| < et for all i > m. Then the sequence

{^IIYW^O/I-Y^MIIK above does converge uniformly in t, and the limit of {Y^l\t, u>)} exists, uniformly for t in [0,1], for ui £ N.

333 Also Pr(N) < lim Pr( M m—>oo

^s

Em)

< lim Y^Pr{Em) n>m

= lim V p n m—>oo

z

—'

ri

Therefore the limit Y(t,uj) exists uniformly in t for a.a. w. Now Y(t,u)) = ~X.(t,w) if i is a dyadic rational. As in the proof of continuity for scalar Brownian motion, we have the diagram

Y(ti,w)

Y(t,w)

I X(tj,u)

X{t,u)

for arbitrary t, with t = limj-Kx, tj for some sequence of increasing dyadic rationals; convergence in the top line of the diagram is with respect to the H-norm, and the convergence in the bottom line is in probability (to be shown). As in the scalar case, convergence with respect to the H-norm implies convergence in probability, and limits in probability are unique up to equivalence, where equivalence means that Pr{u) : X(£,a>) ^ Y(t,u>)} = 0 for any t. X is equivalent to Y is written X ~ Y (a wider equivalence will be useful later). To prove convergence in probability for the bottom line of the diagram,

334

let Vj = Pr{u : || X(t,w) - Xfo.w) ||2 > e2} = Pr{ W : £ a n 2 ( X n ( i , u , ) -Xn(td,u>))2

> e 2 }.

(6)

n

Assume t,- < t for all t. Take a sequence {bn}, bn > 0, such that £) n ^n2 < !• Let snj- = Pr{w : a n 2 [X n (t,o;) - ^(tj-.w)] 2 > 6 n 2 e 2 }.

(7)

Claim: oo

oo

VJ < Y, Pri" • an2[Xn(t,w) - Xnitj^)}2

> bn2e2} = ^ snj, (8)

n=l

n=l

for since «i < Pr{^an2[Xn(t)

-Xn(tj)}2

n

> Y,bn2€2} n

< Pr( \J {an2(Xn(t) - X^tj))2

> bn2e2})

n=l

< J2 Pr{an2(Xn(t)

- Xn{ti))2 > bn2e2},

(9)

n

it follows that oo

Vj
( 10 )

Now ^ • = Pr{||X4i)-X n (i,)||>^i} =

.

/

e ^^du

^(t-tj)-/^

<2Kpe-=fei

(11)

and the probabilities

^'SffcJ,-^.

(12)

335

Assume that the sequence {6n} is chosen with the further requirement that un = ^J- —> 0. Such a sequence can always be found. The right side of the inequality (12) for Vj is cct~l^2Y^=i une~"™^> where c is a constant and a = a(e,j) = 2(t-ty *et ^n ~ ~£?- ^ o r a gi y e n e > 0, take j0 sufficiently large so that a(e,j0) > 1. The series Yl^Li e~XnCl converges for all j > jo if the An is a monotone increasing sequence of positive numbers such that j — ^ —» oo (by comparison with the Riemann zeta function). For a given e > 0, as tj —> t the sum in (12) decreases. The right side of the inequality Vj < C^YTJ X^^Li A~1/2e~A"Q is bounded by a number decreasing to zero as j —» oo. Since this result holds for any e > 0, X(tj) —» X(i) in probability. Consider some particular cases. (i) If an = 1/n for every n, take (properly normalized)6n2 = 2 r Gn / n-i 1/ ' 2 ' where the remainder r n _i = an2 + an+\2 + ... = 1/n2 + With this choice of the sequence {bn}, un = |a n |/6 n —> 0, and 0 < An < An+i < ..., An —> oo, while \n/logn > (n — I) 1 / 2 /logn —> oo. (ii) Next, consider an = 1/n"/2, 1 < a < 2, so that ^2nan2 = Enl/na < oo- Again using bn2 = a n 2 /r n _ 1 1 /2 ) An = bn2/an2 = [i/n°+i/(n+i)°+---]1/2 < ^"+i! t n e n • • • < ^n < An+i < .... To see that

T^ = S ^ f c = r-W'iogn ^ °°' t a k e

rn l

~

<

!

1

1

n so t h a t

2fc < n < 2fc-1; then

1

1

- ^2(fc-i)« + (2fe-1 + l ) Q + ' ' ' + 2 ^ ^ + ^2fc + l ) Q + ' ' '+2(fc+i)Q^ + ' ' ' '

The number of terms in the first bracket above is 2k~1 + 1, and an upper bound for the first bracket is therefore ^(t-itii a n d s 0 o n f° r the other terms, giving the estimate rn-i < (2

1 1 + 1 ) 2 (fc-i)a+ 2 ^

+ 2

1

+1

2(*+i)« + '''

=

1 2^~^ 2( f c - 1 ) a + M a 2( f c - 1 ) Q '

where Ma is a constant depending on a. This expression tends to zero as k -> oo so that rn _ l( | ogn yi > [i/2(*-.).+Mg2*iV2<»-D,](iogn)2 ^ oo, and (iii) The general case: with the same choice of the sequence {bn} as in example (i) above, namely bn2 = an2jrn-\xl2, we have An = l/u n 2 = l/rn-i 1 / 2 = (an2 + ...) —> oo as n —» oo. Monotonicity of {r n } implies that ... < An < An+i < — To achieve Xn/logn —> oo, and therefore convergence of the series, assume that (logn) r n _j —> 0 (a stronger condition would be that ^2n an2(logn)2 < oo, satisfied by exs. (i) and (ii)).

336

The procedure can be generalized still further: take the sequence bn2 to be of the form f(n)an2,n = 1,2,..., where f(n) is a sequence of positive numbers so that E n bn2 < oo and f(n)/logn —» oo (for example, the case (iii) above is f(n) = l/r n _ 1 1 / 2 ), the same argument works; the notation /(n) will be used below for such auxiliary sequences. Now pointwise convergence or convergence with respect to the H norm implies convergence in probability, and limits in probability are unique up to equivalence. Therefore X ~ Y. The proof of the existence and continuity of X is complete, except for verifying that (A) holds; this problem we consider next. Condition (A) requires the existence of a sequence of positive numbers {tj} satisfying (l)£<=i * < ° ° (2) EZiMEi) = E,~i^{a;: maxo u) = First observe that maxo e,}, replace (2) by (2')Ei=i« pi for every i. Let Pr{\\X(^,")

- X ( ^ , u ; ) | | > £i} = Si

(si is independent of j). Then Pr{u : H X ^ . u ) - X ( ^ i , W)|| < £*} = 1 - sit

337

and by independence Pr{\\X(^,u>)-X(J-^±,u,)\\<ei,j

= l,2,...,2i+1}

=

(l-sif+\

Therefore Pr{u : maxj\\X(^u>) - X ( ^ » | | > ej = 1 - (1 - Sif+\ and gi = l-(l- S i ) 2 i + 1 . Now 1 — (1 — Si)2'

< 2i+1Si. We can then replace condition (2') by

(2")ES12i+1Si : ||X(2?TT,W)|| > €i} (since X(0,w) = 0). Condition (A) will be satisfied if (1") there is a sequence of positive numbers {ei}

with YZLi €« < °°

and (2")E»" 1 2 i + 1 Si £»}. Next, calculate 2 Si=Pr{u>:\\X(^,uj)\\ >e*}

= Pr{u;:f^al[Xn(^,^>e2}.

(13)

Repeat the argument used to derive eq.(8): take a sequence {bn}, with bn > 0, such that £ ~ = 1 b\ < 1. Let s' ni = P r { o ; : a 2 [ X n ( ^ r ) u ; ] 2 > 6 ^ } . Claim: Si < X ) S'm

(14)

338

since Si

= Pr{ W :f>2 [ X n ( _L)]2 > e 2 } n=l oo

..

oo

^Pr^:^^-^)] 2 ^^ 2 } n=l

n=l

< Pr U«=1 {W : a*[X n (-L)] a > ^c?}

<£iM":^[*n(-^)]2>fc2},

(15)

proving (15). To satisfy condition (2") on Sj, it is sufficient that oo

oo

££2 i+ V ni
(16)

j=l n=l

where 2

s' 2

f

f°°

9

ft

a

< _£=

«2 (•n 2 «i 2 2 i

^^e—Tt^-

(17)

Inequality (17), that YT=i E^=i 2 i + l s 'ni < oo, will then be satisfied if „

oo oo

9

i±i

t^efa-

(18)

with 6n > 0, E^=i ^n < 1, £i > 0, XIi=i ei < °°) with aj real and £ ~ i ° « 2 < oo. Consider first the case a n = l/n. Then take 6n = riy~1, where 0 < 7 < 1/2, and et — 2~4/4. The double sum in the left side of the inequality (19) is then

(19)

339

where c is a constant. To estimate in the above expression the sum on the index i, write it in the form J2 2 3V4 e -n 2 ^/ 2

= c y>( 2 (i+l)/2)l/2 e -n^2(

i

i

+ 1 )/ 2 2- I / 2 ( 2 (i+l)/2 _ 2 i/2)

i


= c(n 2 T)- 3 / 2 r(3/2)

using the fact that the function is eventually monotone decreasing. Then

S < cJ2n"rn~3"1 n

= c5>- 4 ^.

(20)

n

Take 1 < 47 < 2, or 1/4 < 7 < 1/2; for 7 in this range, S is bounded, and condition (A) holds. The general case: the sequence {an} is square summable; take {bn} such that 2T^Li ^«2 = 1 a n d §*V —» 0, , and e^ = 2~s. Apart from a constant the double sum in (19) is then

where again un = '~^. Let A = un~2 > 0. Using the Dirichlet integral

[°°xh-Xxdx = \-§T(h,

Jo

*

= exactly as in the case above, write the double sum as c'}2nunun3T(^) cJ2nun4- If bn = al/rl/-i, where r n _i is the remainder a2 + a 2 + 1 + ..., then the series J^ n 62 converges more slowly than the series Yln°%i- Also = E n r n - i = Yunnar?• If this series converges, Z)n u " 4 = Hn(al/bl)2 inequalities (19) and (17) follow, and finally condition (A). Since this condition on the remainder implies a property (cf. p. 7 (iii)) giving convergence in probability, namely that [logn)2rn-i —> 0 (the sequence {rn} being monotone decreasing with positive terms, so that nrn-\ —> 0), expression (1) converges in H for a.a. ui e Q, for all t £ R+.

340

More generally, given a sequence of positive numbers g(n) with X)n g(n)an2 < oo and J^ n l/g(n) < oo, the above argument can be carried out for bn2 = g(n)an2. Summarizing the above, Lemma 2: For {an} S I2, the expression (1) for X(£, u>) converges a.e. for all t G R+ if the following two conditions are satisfied: (a) there exists a monotone increasing sequence of positive numbers {/(n)} with 5Zn f{n)an2 < °° a n ( l such that f(n)/logn —• oo; (/3) there exists a sequence of positive numbers {g(n)} such that E n 9 ( n K 2 < oo and £ n l/ff(n)2 < oo. Now if the sequence {g(n)} in (/?) is monotone increasing, then n/g(n) —> 0 as n —> oo, so that logn/g(n) —> 0 (the example above (p.ll) is the case /(n) = g(n) = l/rn1/2). Then only condition (/3) is needed. Compare Lemma 2 with the classical Menshov-Rademacher theorem: a sufficient condition for the convergence a.e. of the deterministic series Efclian<£n(a0 for an arbitrary orthonormal system = {(/>n(x) • n > l,x £ (0,1)} of functions is the inequality oo

^a n 2 (Zo f f n) 2
(21)

n=l

The Menshov theorem is sharp in the following sense: there exists an orthonormal system {n}"=i° o n (0> 1) s u c n t n a t t o e a c h monotone increasing sequence of positive numbers {u>(n)} with u>(n) = o((logn)2), there is a series J2T=i an<)>n{x) that diverges a.e. and whose coefficients satisfy z n = 1 an u(n) < oo. The sequence {(logn)2} is an example of a "Weyl sequence" for the property of almost everywhere convergence. In the Brownian motion case then, the monotone increasing sequence sequence {
341 We are now able to make the following definition Def.l: If the expression (1) converges a.e. for all t > 0, X is called Brownian Motion in H with weights {an}. To remove the dependence on the orthonormal system, the equivalence class of the vector-valued stochastic process can be extended by using the equivalence of all (normalized) unconditional bases of Hilbert space. Then the Brownian motion in H is essentially unique and independent of the complete orthonormal basis used. Also, the system of weights {an} can be normalized so that ^ ^ L j an2 = 1. The final result is Prop.l: If the conditions of lemma 1 are satisfied, Brownian motion exists and has continuous paths. Cor.: If there exists a monotone increasing sequence of positive numbers { Brownian motion exists. Brownian motion in H can be considered as a function mapping R+ into the space of measurable functions M(Q,H) taking t into X(t, •), a measurable function on Q with the natural norm and inner product, giving the identity ((X(t) - X(s), X(t) - X(s)}) = ||X(*) - X( S )|| 2 L2

= JdPr(u)\\X(t)-X(s)fH OO

n=l =

t~S

for t > s, as in the scalar case. Also it follows that ((X(t),X(s))> = min(t, s) for t, s e R + .

342

References 1. Yoshihei Hasegawa. Levy's Functional Analysis in Terms of an Infinite Dimensional Brownian Motion I, II, Proc. Japan Acad. 56 Ser. A (1980) 109-113, 114-118. 2. Yoshihei Hasegawa. Levy's Functional Analysis in Terms of an Infinitedimensional Brownian Motion I, Osaka J. Math. 19 (1982) 405-428. 3. Paul Levy. Problemes concrets d'analyse fonctionelle, Gauthier-Villars (1951). 4. K. Tandori. Uber die Konvergenz der Orthogonalreihen II, Ada Sci. Math. (Szeged) 25 (1964) 219-232. 5. F.Moricz and K. Tandori. An improved Menshov-Rademacher theorem, Proc. Amer. Math. Soc. 124, no.3, 877-885 (1996).

343 SERIES ON KNOTS AND EVERYTHING Editor-in-charge: Louis H. Kauffman (Univ. of Illinois, Chicago) The Series on Knots and Everything: is a book series polarized around the theory of knots. Volume 1 in the series is Louis H Kauffman's Knots and Physics. One purpose of this series is to continue the exploration of many of the themes indicated in Volume 1. These themes reach out beyond knot theory into physics, mathematics, logic, linguistics, philosophy, biology and practical experience. All of these outreaches have relations with knot theory when knot theory is regarded as a pivot or meeting place for apparently separate ideas. Knots act as such a pivotal place. We do not fully understand why this is so. The series represents stages in the exploration of this nexus. Details of the titles in this series to date give a picture of the enterprise. Published: Vol. 1:

Knots and Physics (3rd Edition) by L. H. Kauffman

Vol. 2:

How Surfaces Intersect in Space — An Introduction to Topology (2nd Edition) by J. S. Carter

Vol.3:

Quantum Topology edited by L. H. Kauffman & R. A. Baadhio

Vol. 4:

Gauge Fields, Knots and Gravity by J. Baez & J. P. Muniain

Vol. 5:

Gems, Computers and Attractors for 3-Manifolds by S. Lins

Vol. 6:

Knots and Applications edited by L. H. Kauffman

Vol. 7:

Random Knotting and Linking edited by K. C. Millett & D. W. Sumners

Vol. 8:

Symmetric Bends: How to Join Two Lengths of Cord by R. E. Miles

Vol. 9:

Combinatorial Physics by T. Bastin & C. W. Kilmister

Vol. 10: Nonstandard Logics and Nonstandard Metrics in Physics by W. M. Honig

344 Vol. 11: History and Science of Knots edited by J. C. Turner & P. van de Griend Vol. 12: Relativistic Reality: A Modern View edited by J. D. Edmonds, Jr. Vol.13: Entropic SpacetimeTheory by J. Armel Vol.14: Diamond — A Paradox Logic by N. S. Hellerstein Vol. 15: Lectures at KNOTS '96 by S. Suzuki Vol. 16: Delta — A Paradox Logic by N. S. Hellerstein Vol. 17: Hypercomplex Iterations — Distance Estimation and Higher Dimensional Fractals by Y. Dang, L. H. Kauffinan & D. Sandin Vol. 19: Ideal Knots by A. Stasiak, V. Katritch & L. H. Kauffinan Vol. 20: The Mystery of Knots — Computer Programming for Knot Tabulation by C. N. Aneziris Vol. 24: Knots in HELLAS '98 —• Proceedings of the International Conference on Knot Theory and Its Ramifications edited by C. McA Gordon, V. F. R. Jones, L Kauffinan, S. Lambropoulou & J. H. Pnytycki Vol. 25: Connections — The Geometric Bridge between Art and Science (2nd Edition) by J. Kappraff Vol. 26: Functorial Knot Theory — Categories of Tangles, Coherence, Categorical Deformations, and Topological Invariants by David N. Yetter Vol. 27: Bit-String Physics: A Finite and Discrete Approach to Natural Philosophy by H. Pierre Noyes; edited by J. C. van den Berg Vol. 28: Beyond Measure: A Guided Tour Through Nature, Myth, and Number by J. Kappraff Vol. 29: Quantum Invariants — A Study of Knots, 3-Manifolds, and Their Sets by T. Ohtsuki Vol. 30: Symmetry, Ornament and Modularity by S. V. Jablan Vol. 31: Mindsteps to the Cosmos by G. S. Hawkins Vol. 32: Algebraic Invariants of Links by J. A. Hillman Vol. 33: Energy of Knots and Conformal Geometry byJ. O'Hara Vol. 34: Woods Hole Mathematics — Perspectives in Mathematics and Physics edited by N. Tongring & R. C. Penner