Particles and Fields

? 0. The action (6.22) suffers from the contact term divergencies 211 ' 212 ' 213,214 which arise when a pair of X-s collides in a point. This sort of singularities appears already at the tree level. Really, the tree level graphs are generated by solving the classical equation of motion by perturbation theory. For Witten's action (6.22) the equation reads QBA

+ XA*A

= 0 .

Here QB = QNS- The first nontrivial iteration (4-point function) involving the pair of XA * A vertices produces the contact term singularity when two of X-s collide in a point.

6.4. Action in 0 Picture and Double Step Inverse Changing Operator

Picture

To overcome the troubles that we have discussed in the previous subsection it was proposed to change the picture of NS string fields from - 1 to 0, i.e. to replace |0)_i in (6.23) by |0) 140>141. States in the —1 picture can be obtained from the states in the 0 picture by the action of the inverse picture-changing operator Y 200

Y = Acd^e-^iw)

100

with X y = YX = 1. This identity holds outside the ranges kerX(w) and kery(«;) 2 1 9 . These kernels are: kerXiw)

= [o{w) | lira X{w)0{z)

= o)

kerYtw)

= \o(w)

= o) .

I lim Y(w)0(z)

and

This means that kerX(w) and ker Y(w) depend on a point w. In particular, the following statement is true: for all operators O(z), z ^ w the product X(w)0(z) ^ 0. The existence of these local kernels leads to the fact that at the 0 picture there are states that can not be obtained by applying the picture changing operators X and Y to the states at the —1 picture. The action for the NS string field in the 0 picture has the cubic form with the insertion of a double-step inverse picture-changing operator y_ 2 140>141 J; Ss

[Y-2A*QBA+?-

[Y-2A*A*A.

(6.28)

We discuss Y-2 in the next subsection. In the 0 picture there is a variety of auxiliary fields as compared with the —1 picture. These fields become zero due to the free equation of motion: QBA = 0, but they play a significant role in the off-shell calculations. For instance, a low level off-shell NS string field expands as A £* [dk{u{k)cl-\All{k)icla,i_l-\Bll(k)lh J

\

l

% + ^ ( t j d ^ ! ^ i 4

2

2

j

+ B(k)co + r(Jb)ci 7 i P-1 + • • • [ eifc'x(°'°) |0) .

2

2

(6.29)

Here u and r are just the auxiliary fields mentioned above. The SSFT based on the action (6.28) is free from the drawbacks of the Witten's action (6.22). The absence of contact singularities can be explained shortly. The action (6.28) yields the following equation Y_2(QBA

+ A*A)

=0.

(6.30)

Since operator YL2 is inserted in the mid point in the bulk and A is an operator in a point on the boundary we can drop out Y-2 and write the following equation QBA J

+ A*A = 0.

(6.31)

One can cast the action into the same form as the action (2.7) for the bosonic string if one modifies the NS string integral accounting the "measure" Y-2 : f = JY-2-

101

So the interaction vertex does not contain any insertion leading to singularity. The complete proofs of this fact can be found in 140 > 141 . 142 . To have well defined SSFT (6.28) the double step inverse picture changing operator must be restricted to be a) in accord with the identity k : F_ 2 X = X7_ 2 = Y ,

(6.32)

b) BRST invariant: [QB , F_ 2 ] = 0 , c) scale invariant conformal field, i.e. conformal weight of Y_2 is 0, d) Lorentz invariant conformal field, i.e. Y"_2 does not depend on momentum. The point a) provides the formal equivalence between the improved (6.28) and the original Witten (6.22) actions. The point b) allows us "to move" BRST charge from one string field to another. The point c) is necessary to make the insertion of Y_2 compatible with the ^-product. And the point d) is necessary to preserve unbroken Lorentz symmetry. As it was shown in paper 141 , there are two (up to BRST equivalence) possible choices for the operator Y_2 • The first operator, the chiral one 140 , is built from holomorphic fields in the upper half plane and is given by y_ 2 (z) = - 4 e - 2 * W - i l e-3*cd&

• dX(z) .

(6.33)

The identity (6.32) for this operator reads Y-2(z)X(z)

= X(z)Y.2(z)

= Y(z) .

This F_ 2 is uniquely defined by the constraints a) - d). The second operator, the nonchiral one 141 , is built from both holomorphic and antiholomorphic fields in the upper half plane and is of the form: Y-2(z,z)

= Y(z)Y(z).

(6.34)

Here by Y(~z) we denote the antiholomorphic field 4c(z)d^(z)e~ 2 < t > ^ z \ this choice of Y_2 the identity (6.32) takes the form: Y„2(z,z)X(z)

= X(z)Y_2(z,z)

= Y(z)

For

.

One can find the inverse operators W to YL 2 (z,z) and Y-2{z) W(z,z)Y„2{z,z) W(z)Y_2(z)

k

= 1,

= l,

W{z,z)=X{z)X{z) 2

;

W(z) = X (z) + c^Wiiz)

(6.35a) + a2W2(z)

We assume that this equation is true up to BRST exact operators.

+ a3W3(z) , (6.35b)

102

where Wi(z) = {Q,fi;(z)}, fij is given by fii = (db)d e2

= (02b) e24> ,

ft2

n3 = bd2e2*

(6.36)

and a, are the following numbers ai = _

a2 Q3 = _ (6 3?) 192' = ~48' 96The issue of equivalence between the theories based on chiral or nonchiral insertions still remains open. The first touch to the problem was performed in 210 . It was shown that the actions for low-level space-time fields are different depending on insertion being chosen.

6.5. SSFT in Conformal

Language

As for the bosonic string (see Sec. 5.2) for SSFT calculations it is convenient to employ the tools of CFT 2 0 8 . In the conformal language f and ^-product mentioned above are replaced by the odd bracket ((• • • | • • •)), defined as follows «K_ 2 |Ai,... ,An)) = (Pn o F_ 2 (0,0) A (n) o ^ ( 0 ) . . . / M o = (fjn)oY-2(i,i)fin)oA1(0)...fWoAn{0)) »

, '

An(0)\ n = 2,3.

(6.38)

Rn

Here r.h.s. contains SL(2, E) -invariant correlation function in CFT on string configuration surface Rn described in Section 5.2. Aj (j = 1 , . . . ,n) are vertex operators and {/• '} are given by eq. (5.27b). Y-2 is the double-step inverse picture changing operator (6.34) inserted in the center of the unit disc. This choice of the insertion point is very important, since all the functions fj maps the points i (the middle points of the individual strings) to the same point that is the origin. In other words, the origin is a unique common point for all strings (see Figure 11). The next important fact is the zero weight of the operator Y-2, so its conformal transformation is very simple foY-2(z,~z) = Y-2(f(z),f(~z)). Due to this property it can be inserted in any string. This note shows that the definition (6.38) is self-consistent and does not depend on a choice of a string on which we insert Y-2Due to the Neumann boundary conditions, there is a relation between holomorphic and antiholomorphic fields. So it is convenient to employ a doubling trick (see details in 1 9 8 ) . Therefore, Y-2(z,~z) can be rewritten in the following form1

Y-2{z,z) = Y(z)Y(z*). Because of this formula, the operator Y-2{ z,z) is sometimes called bilocal.

103

Here Y(z) is the holomorphicfieldand z* denotes the conjugated point of z with respect to a boundary, i.e. for the unit disc z* = 1/z and z* = ~z for the upper half plane. From now on we work only on the whole complex plane. Hence the odd bracket takes the form /1(n)oA1(0).../Wo4(0)) .

{{Y^2\AU . . . ,An)) = (Y(Pn(0))Y(Pn(oo))

(6.39) To summarize, the action we start with reads

S[A]

UY-2\A,QBA)) 9l L2

+

\({Y-2\A,A,A))

(6.40)

where g0 is a dimensionless coupling constant 6.6. Free Equation

of

Motion

The aim of this section is to illustrate that the free equation of motion (6.30) obtained from the action (6.40) and the free equation of motion (6.31) are the same1". To this purpose we consider the following GSO+ string field (in picture zero and ghost number one) A = J - ^ ^ L(k)ce2ikX^ - ^B^k^^e™-*^ + r-x(k)dce 2 i k X L ^

+

^A^cdX*e2ikX- (w)

+

^F^(k)cip^iljve2ik-XL^

+ r 2 {k)cde 2ikx ^^ i

(6.41)

and show that free equations of motion for the component fields are usual Maxwell equations for the massless vector field plus some relations between the other fields. The (6.41) contains all possible excitations in picture 0 on levels LQ = — 1 and LQ = 0. There are several unexpected properties of this formula: • First, we find that the level LQ = — 1 is not empty — it contains one field u(k); • Second, we find that there are two candidates for the Maxwell field: A^ and 1?M; • Third, we find that there are too many fields as compared with the massless spectrum of the first quantized string. m

In this section we use the convention a ' = 2.

104

As it will be shown in Section 6.8 the field B^ is physical one, while all other fields are auxiliary ones. Let us first examine the equations that follow from QB\A) = 0. We have d26k

u(k) (2k2 - 1) dec + crje^ik • -0 +

\k2dccdX»

+ A^k)

- 1 i ktid2cc +^cd

+ \ crje^ik • tpdX» + ^ d ^ e ^ S X " \k2dcj]e't'i)tl

+ B»(k) + liWd

8

•* ILV

-dm*?4.

02ik-XL(w)

(i/e V )

p2ik-XL(w)

- \cd (yje'V) - J c t / e V * * • dX

(9we 2 *) - ldr1r]e2ik • V>V>"02ifc-Xi(tu)

8

k2dcc^ij}v

4

- -cd(r] e^ik^ip,,} + -cqe^ik P

4

+ ri(fc)

2

^,/,^/,"

v

• ^ ^

2«*-Xx,(iu)

o

<92cc - dec 2iJfc • 3X + deqe^ik • ip + jd (drjrje2'1')

+ r2(k) -d2cc+2k2dccd(f> + ci]de't'2ik • tp + \d^de2^

8

- -crje^xp • dX - ce^d^ik + -bcdrjrje2 +

2

2

02ikXL(w)

• ip) 2ik-XL(w)

\d2r]r]^

From {QB, A} = Q one gets the following equations for component fields: u =0, Btl = All ,

r2 = 0 ,

(6.42a)

FM„ = - ih^Av] ,

r1 = --ikllAll

k2A„ = 4ifcMri .

,

(6.42b) (6.42c)

Note that we get Maxwell equation for B^ after excluding n in (6.42c) via (6.42b) k2 B^ = kf, ku Bv .

(6.43)

Eqs. (6.42a) exclude fields u and r2 and (6.42b) leave only one vector field.

105

Now let us present the result of computations of the action (6.40) on the string field (6.41):

f d™k 2

J

(2TT)26

u(-k)u(k)

- ^(-fc)AM(fc) -

ifi M (-fc)5 M (fc)

+ -A p (-fc)B M (fc) - iFM„(-fc)FM„(fc) + - 2ir2(-k)kliBfl(k)

ikllFliV{-k)B„{k)

- 10r2(-fc)r2(fc) - 4n(-fc)r 2 (fc)

One can check that the equations of motion following from this action are precisely the eqs. (6.42). Moreover, free equations of motion following from Qs|.4) = 0 and from Y-IQB\-A) — 0 are the same for any string field A. 6.7. Scattering

Amplitudes

6.7.1. Gauge Fixing Before proceeding with the investigation of tree graphs in the theory with the action (6.40), we should specify a gauge fixing procedure. It is convenient to choose the Siegel gauge 216 b0\A) = 0.

(6.44)

To find the propagator we have to invert the operator Y- 2 Q on the space of states satisfying (6.44). The propagator can be presented in a number of forms: A=

b

fWQ^

Li

= ^WLi

b

iwb-±Q

XJ

LI

+ Qb-^Wbi

= W^

LI

LI

LI

.

(6.45)

LI

One can prove that AY_ 2 Q = 1 in the space (6.44) by a simple calculation based on the following set of equalities: [L,Q} = 0,

6.7.2. ^Point

{b0,Q} = L,

[W,Q} = 0,

[y_ 2 ,Q] = 0 ,

Y_2W = 1 . (6.46)

On-Shell Amplitude

Now we turn to the calculation of the four-boson scattering amplitude in the cubic theory. We shall proceed as close as possible to the analogous calculation in the usual Witten theory. In the bracket representation for two- and threepoint vertices one can write h(i)

lii)

AA = 4ii(V3\ i23(V3\Y$Y$^WQ^\V2)ii\A\U\A1)1\A2)a\A3)3

. (6.47)

106

In comparison with the usual formula (5.61), we evidently find in (6.47) the additional midpoint insertions of Y_l , Y_^ a n d the new propagator (6.45). The string fields \Ar) = X\A) is a field in the —1 picture defined by the expression like (6.29). Taking into account the BRST invariance of the vertices and the on-shell condition for \Ar), we can rewrite eq. (6.47) in the form 1,(0 A4 = 41i(V3| i 23y| J 44>4|il>l|i 2 )2|i3)3 ,

(6-48)

where one of the two potentially dangerous Y-2's is cancelled by the operator W in the propagator. In the conformal formulation this expression corresponds to r°°

"Wo

dT

r dz dz

-

-

-

-

f^id-p{0^°^bY^-

(6 49)

-

where Z{p) is the Giddings map for N = 4. To be more rigorous we can consider first the regularized expression f dz

°°

/

dT

j2ridp

dz

-

-

-

-

(0i0203C46K.2) ,

(6.50)

with the regularization parameter T . We will prove that: i) the correlation function (6.50) is well-defined in the limit r —> 0; ii) in the limit r —> 0 the correlation function (6.49) after summation over all channels is equal to the Koba-Nielsen amplitude. We will proceed in the same way as described for Witten's diagram in Section 5.4 above. The first step is to move two of the X(0), say X^ and X^ , to the midpoint via the famous rule l " » ( 0 ) - I w ( i ) = {Q,(0)

(6.51)

107

and we find A$ in the form Ai = A° + A\ + A\

(6.52) In the RHS of eq. (6.52) the first term contributes to the Koba-Nielsen amplitude A°A= T d T ^ ^ -

{O^OzOib)

.

(6.53)

The last two terms in (6.52), A\ and A\ , are contact terms with cancelled propagator A\ = (YS^O^OsO^b)

,

(6.54)

and A\ = ( y _ 2 ^ ( 1 ) O i 6 2 6 3 0 4 6 ) •

(6.55) 142

In contrast to the usual calculation (see Appendix A in for details) we have not obtained any singularity in contact terms (see Appendix 6.9.3) so there is no need to look for a regularization to define A4 correctly. Therefore, the limit T —1 0 does exist . Finally note that the amplitudes A\ and A\ after summation over all channels are identically zero. For illustration we will prove this for A\. The explicit formula for A\ in the s-channel reads fOO

A\{s)

=1

C A

A

dTJ^.-^(01(z1)d2(z2)d3(z3)Oi(z4)b(z)mzi)-azoW(z0)). (6.56)

108 In the t-channel we get A\{t)

=J dTJ^-.-^(d2(z1)d3(Z2)0A(z3)01(zi)b(z)((az3)

-

&o))Y(z0)). (6.57)

Making a SL(2, C) transformation of the form z -> z' = Z—^- , z+1

(6.58) '

K

which gives (Zi, Z2,Zz,Zi)

-¥ (z2,Z3,Zi,Zi)

,

and using the fact that OT are anticommuting operators we find that and so the sum of (6.56) and (6.57) is equal to zero. In the oscillator language this corresponds to the cancellation of 4-point contact terms by cyclic symmetry arguments. The term (6.53) is also well-defined in the limit r -> 0 and gives the s-channel contribution in the total Koba-Nielson amplitude 215 ' 217 ' 218 . As in a usual gauge theory the issue of finiteness of 4-point off-shell amplitudes crucially depends on the choice of a gauge. One can use the alternative gauge bo(Y_2A) = 0, which leads to the propagator W(i) -±Y_2Q-± Li

W(i) = W(i) -°- - W(i) y- F_ 2 ^ W(i)Q . Li

LI

(6.59)

Li

LI

In this gauge the NS 4-point function (off-shell NS amplitude) is obviously finite. 6.7.3. N-Point On-Shell Amplitudes Let us start with the example of the 5-point amplitude. From now on we choose the propagator in the form (see eq. (6.45)) A =

hW(i)-bj-W(i)bj-Q,

(6.60)

then the 5-point function is given by the sum of two Feynman graphs 2

3

4

2

3

4

(6.61)

109 where the line with a dot stands for the second term in eq. (6.60). To analyze the graphs of this type, which also appear in N-point amplitudes, we adopt the following regularization

A -* ATur2

b

^e~^W(i)-b^e-^LW(i)e-^Q

= Li

Li

/»00

= bo

/*00

dre-TLW(i)

-b0

Li /»00

dTe-TLW{i)b0

/

dre~TLQ

.

(6.62)

This regularization is a lagrangian one and it's special case was used previously for the 4-point function. Adopting the regularization eq. (6.62) we find that in the limit n -> 0 the operator Q in the second term of (6.61) directly cancels the right-standing propagator thus leading to the diagram with the contact 4-vertex. 2 3 4

W (6.63) According to the Lemma in Appendix 6.9.3 we cannot remove the regularization parameter r 2 in this diagram. However, literally reproducing the arguments of the previous subsection, we conclude that after summation over all the channels these contact terms give zero. Hence we find that only the first graph in (6.61) contributes on-shell. Reproducing once again the arguments of the previous subsection, we finally conclude that the first graph in (6.61) gives the Koba-Nielsen formula. Following just the same device one can analyze any N-point on-shell amplitude and demonstrate the total agreement with the Koba-Nielsen formulae. The off-shell analysis of N-point functions is much more subtle. The key point is the behavior of multi-factor OPEs of the form y_ 2 • W • Y-2 • • • Y-i • 6.8. Restriction

on String

Fields

The aim of this section is to try to reduce the number of excitations appearing in the zero picture string field. Such reductions can be made if we have degenerate quadratic part of the action. But the reduction we describe is not of this sort. In this construction we use only the fact that the expectation value of the operator on the complex plane in super conformal two dimensional field theory is non-vanishing only if it compensates all background charges of the theory.

110

Let us decompose the string field A according to the (^charge q: J\. — y ^ J\q

7

J\q c Vq ,

qez where [jo , Ag] = q Ag

with

j0 = ^- j> d( d4>(() .

(6.64)

The BRST charge QB has also the natural decomposition over (^-charge: QB = QO + QI + Q2-

(6.65)

Since QB2 = 0 we get the identities: Qo2 = 0 , g2

2

=0

{Q0,Qi} and

= 0,

{Qi,Q2} = 0 , (6.66)

{<5o,<32} + <5i2 = 0 .

The non-chiral inverse double step picture changing operator Y-2 has ^-charge equal to —4. Therefore to be non zero the expression in the brackets ((Y^2 \ • • •)) must have (^-charge equal to +2. Hence the quadratic S2 and cubic S3 terms of the action (2.7) read: 5

2 = ~\ Y^((Y-2\A2-q , QoAq)) - \ Yjy-*\Al-* q€l

> QlAq))

q&

-\52((Y-2\A-q,Q2Aq)). 53

= ~\

E

(6.67a)

((y-2\A2-q-q' , Aql,Aq)) .

(6.67b)

q,q'eZ

We see that all the fields Aq , q ^ 0,1, give only linear contribution to the quadratic action (6.67a). We propose to exclude such fields. We will consider the action (6.67) with fields that belong to the space Vo© Vi only. To make this prescription meaningful we have to check that the restricted action is gauge invariant. The action restricted to subspaces Vo and V\ takes the form S2,restricted = -\{{Y.2\A0

, Q2Ao)) ~ ((Y-Mo

-^«r_2L4i,Q04i»,

^3, restricted

=

-{{Y^Ao^AuAx))

.

, Q1A1)) (6.68a)

(6.68b)

111

The action (6.68) has a nice structure. One sees that since the charge Q 2 does not contain zero modes of stress energy tensor the fields A) play a role of auxiliary fields. On the contrary, all fields Ax are physical ones, i.e. they have non-zero kinetic terms. Let us now check that the action (6.68) has gauge invariance. The action (6.40) is gauge invariant under SA = QBA. + [A, A] .

(6.69)

But after the restriction to the space Vb © Vi it might be lost. Decomposing the gauge parameter A over ^-charge A = ^,Aq we rewrite the gauge Q

transformation (6.69) as: SAq = Q0Aq + QiAq-i + Q 2 A,_ 2 + ] T [Ag_,«, A,,] .

(6.70)

Assuming that Aq — 0 for q ^ 0,1 from (6.70) we get SA-2

= <2oA_2 ;

(6.71a)

6A-x

= Q 0 A - i + [ A ) , A _ i ] + QiA_2 + [>li > A_2];

(6.71b)

6Ao

= QoAQ + [Ao , A0] + QiA_i + [Ax, A_i] + Q 2 A_ 2 ;

(6.71c)

SAi

=Q0A1

(6.71d)

6A2

= QxAi + [Ax, Ax] + Q 2 A 0 ;

(6.71e)

SA3

= Q2A1 .

(6.71f)

+ [Ao,A1] + Q1AQ + [A1,A0]

+ Q2A-1]

To make the restriction consistent with transformations (6.71) the variations of the fields A-2 , A-x, A2 and A% must be zero. Since the gauge parameters cannot depend on string fields we must put A_ 2 = A-x = Ai = 0. So we are left with the single parameter Ao, but to have zero variation of A2 we must require in addition Q2A0 = 0. Therefore the gauge transformations take the form 6A0 = Q0A0 + [Ao , A0] , (6.72) 5Ax = QxA0 + [Ax, A0] ,

with

Q2A0 = 0 .

It is easy to check that (6.72) form a closed algebra. It is also worth to note that the restriction Q2A0 = 0 leaves the gauge transformation of the massless vector field unchanged.

112

6.9.

Appendix

6.9.1. Cyclicity Property The proof of the cyclicity property is very similar to the one given in 187 . But there is one specific point — insertion of the double step inverse picture changing operator Y_2 (6.34). So we repeat the proof with all necessary modifications. Let Tn and R denotes rotation by — ^ and — 2TT respectively: Tn(w) =e-^w

R(w) = e-2niw

,

.

These transformations have two fixed points namely 0 and oo. Let us apply the transformation Tn to the maps fjf1' (5.27b) and we get the identities: T„ o / W = / £ £ ,

k
T„o/W=i?o/W,

n = 2,3.

(6.73)

Since the weight of the operator Y_2 is zero and 0 and 00 are fixed points of Tn and R, the operator Yl 2 remains unchanged. Due to SL(2, M)-invariance of the correlation function we can write down a chain of equalities (Y_2 F[n) oA,...

F^

=
OV i ^ "

1

° Oh)

j n n \ o An^f^ o oh)

= (Y-2 Tn o f[n) oAx...Tno

f(nn\ o An^Tn

= e- 2 ; r i / l (y_ 2 F^ n ) o OhF2(n) oA1...

o /(») o Oh)

F | n ) o An.!)

.

(6.74)

In the last line we assume that Oh is a primary field of weight h and use the transformation law of primary fields under rotation: {R o Oh) (w) = e-2*ih Oh (e-^w)

.

Also we change the order of operators in correlation function without change of a sign, because the expression inside the brackets should be odd (otherwise it will be equal to zero) and therefore no matter whether $ odd or even. So the cyclicity property reads ((Y„2\AU...

,An))=e-2*ih»((Y„2\An,A1,...,An_1))

(6.75)

EXAMPLES. NOW we consider some applications of the cyclicity property (6.75). GSO+ sector consists of the fields with integer weights and therefore their exponential factor is equal to 1, while G S O - sector consists of the fields

113

with half integer weights and therefore their exponential factor is —1. Now we give few examples <(y_21.4+ , QB A+))

= «y_ 2 | Q B A+ ,A+)) ,

((Y_2\A-,QBA-))

=-((Y-2\QBA-,A-))

«X-2\A+,A-,A-))

= - ((Y-2\A-

(6.76a) ,

,A+,A-))

(6.76b) = ((Y.2\A-

,A- ,A+)) . (6.76c)

6.9.2. Twist Symmetry The proof of the twist symmetry is similar to the one given in 187 . But there is one specific point — insertion of the operator Y_ 2 . So we repeat this proof here with all necessary modifications. A twist symmetry is a relation between correlation functions of operators written in one order and in the inverse one: <(F_ 2 |0i,... , 0 „ » = ( - l ) ? « y _ 2 | O n ) . . . , Oi» .

(6.77)

We are interesting in this relation for n = 3. 1) Let us consider the following transformations M(w) = e~l*w and I(w) = el0/w. The transformation I has the following properties: J ( z i ) I ( z 2 ) = I(zlZ2)

l(z2'z)

and

= (/(*))^ •

The pair of points 0 and oo is not affected by M and I, therefore the doublestep inverse picture changing operator Y-2 remains unchanged. For the maps (5.27b) we have got the following composition laws /{3> o M = / o / | 3 ) ,

/ 2 (3) o M ^ / o / f

and

/f ) o M = I o f[3) . (6.78)

2) Since there is an identity M o O{0) = e _i7r/l 0(O) we can apply it to (6.77) ((YL2IO1,02 ,03)> = e " E fc' (F_ 2 A{3) o M o O ^ f o M o 02 / 3 (3) o M o 0 3 ) = eiv S ^

{Y_2

f

Q fW

o 0

J

0

f
/<3> o 0 3 )

= e " S h* (Y-2 /< 3 > o Oi / f ' o C?2 /< 3) o Os>

(6.79)

in the last line we use the invariance with respect to SL(2,R). Let N0dd and Neven be a number of odd or euen respectively fields in the set { 0 i , 0 2 , 0 3 } . After rearranging the fields one gets «y_ 2 |01 , 0 2 ,03» = J*^hl

(-l)N°dd{N*°dd~l)

«K- 2 |03 , 0 2 ,Ol» •

(6.80)

114

3) Since the correlation function is non zero only for odd expression, number N0dd is odd and N0dd — 2m + 1 for some integer m. Also we have an identity JVCTen + N0(id — 3. It's not difficult to check that (-l)"-"^"-1)

=

(_ir

= (

-i)^^ +^

+

^ " -

3

= (_i)JV^

+^ - . (6.81)

Combining (6.80) and (6.81) we get the twist property

«y_ 2 |Oi, o2,03» =fiin2 n3 «y_ 2 |o 3 , o2, 0$ , where

f(-1)^'+! , = < , , | (_l)^+i ,

fl,-

hj£Z

(i.e. GSO+)

(6.82)

1 hj€z+

(i.e. G S O - ) .

EXAMPLES

i) Let A+ = A\ + A2. Each term in A3+ = Al + {AiA\ + A\AX) + A2AXA2

+ (A^

+ AXA22) + A1A2Al

+ A\

should be twist invariant to be nonzero. Therefore we get fii — 1 and !72 = 1ii) Let we have fields A+ and A- = a\ + a2. Using cyclicity property (6.76) one gets 2(A+,ai +a2,ai

+ a2) = (A+aj + a\A+) + (A+a\ + a\A+) + (A+aia2 + a2aiA+)

+ {A+a2a\ + a\a2A+)

.

So we get il+ = 1 and Cl+fliCl2 = 1. If ill = 1 (tachyon's sector) then fl2 = 1 too. Therefore we can consider a sector with 0 = 1 only. 6.9.3. Power-Counting Lemma Lemma 1. Consider an off-shell N-point tree graph with V = N — 2 threestring vertices, and with LQ internal lines corresponding to the usual propagator Ao = i;W and Li internal lines corresponding to W-propagator term ^W^g-. In such a graph there is L = LQ + 2L\, V — LQ and L\ operators b(z), Z{z) and W(z) respectively. Let us denote the power of the leading singularity in the product (b(z))L(Z{z))y-L°(W(z))Ll by s, i.e. L

V-L0

L,

Y[b{zo + ate) J J Z {z0 + atf

J J W (z0 + afc) = - [O1 (z0) + o(e)} ,

n=l

k=l

j=\

where ai,a'j,a'^ are some constants. The amplitude is non-singular if s < 2L, it can contain a logarithmic singularity if s = 2L and it is divergent in the case s > 2L .

115

Proof. Note that we consider only the case when there are no cancelled propagators in the tree or, in other words, all the Q-s from the propagators ^W^Q sit on external legs. Going to the z-plane one can represent the amplitude in the form A=Y\[

^

/ poo

dTn )B(T1,...,TL)

,

(6.83)

where

-n(/£

6(T1,...,r1)=||(^s^

!

x

n=l

x (n °r M n ^ n (*°<) n ^) • (6-84) IN

L

b

V-L0

n=l

\r=l

z

Li

j=l

w

i=l

\

I

In the space of modular variables {Tn} we choose the spherical coordinates: Tn = Tan(il)

,

where r is the radius and Q, is the set of n — 1 angular variables. Then /*oo

T L _ 1 dTdniB(T,n) ,

A=

(6.85)

Jo where B(T, fi) = jB(rai,... , ran). To analyze the question of finiteness of the amplitude A let us estimate the behavior of the integrand TL~1B(T, fi) when r tends to zero. Let us denote by ZQ the common limit of the points zoi and z'oi when r —> 0. If we put e = zoi — ^o for some i then for every j one has: ZQJ = z0 + a^e. In this notation it is possible to write the OPE for the product of the operators b, Z, and W in eq. (6.84) as f[b(wn)

f[Z(zoi)f[W(z'oi)

n= 1

i= 1

= YtO'a(zo)ek°i[(u>n-zoy°-"

i=l

ct

.

n= 1

(6.86) By using eq. (6.86) one can write eq. (6.84) in the form

B{e) = ( ft Or W £ ^ (*) £fc° ft f ^ 7 5 (»» - *)'"" ) • (6-87) \r=l

a

n=l

"

/

116

Because the conformal map defined by equation (5.63) we have

/ | £ ^ ( « * - * ) ' " - ~ *-* + '"- +1 •

(6-88)

In summary we conclude that when e —>• 0, B(e) has a behavior:

\r=l

n=l

a

/

~ r ^ - O ^ B a e * ' J J c*"-.

(6.89)

71=1

If s is the maximal power of singularity for the OPE (6.86) near the point ZQ (one must also put wn — ZQ ~ e in eq. (6.86) then L n=l

and eq. (6.89) takes the form B{e) ~

e-(£-i)£-*

.

(6.90)

Next we change the variables in (6.90) from e to r. As it is obvious from (5.63) when r -> 0 : r ~ e^+1 .

(6.91)

In terms of r eq. (6.90) looks like B ~

T-£+vT2(2£-s)

.

(692)

That is why the entire integrand in (6.85) behaves as Th-\

T-L+v^(2L-s)

= T-l+v^(2L-s)

^

when r ->• 0. From (6.93) we can derive the statement of our Lemma.

^g3^

117

6.9.4. Table of Notations, Correlation Functions and OPE

X£(z)X£(w)

~ -—rf a' 2

log(z - w)

a' 2 if"

dX>i(z)dXv(w)

1 z—w

c(z)b(w) ~ b{z)c(w)

•y(z)P(w)

P(z)y{w) •

(z)cp(w) ~ — log(z — to)

£(z)r)(w) ~ ??(z)£(iu) ~

(c(zi)c(z 2 )c(z 3 )) = - ( z x - z 2 )(z 2 - z 3 )(z 3 - zi)

(e-2*(»>> = 1

TB

i> =

2 —W 1 -6 — w

= ——dX

• dX - — dip • \j>

a'

a'

2{z-w)4

dx -v lpdl _ I a / 3 7

Tbc = -Ibdc - 96c

TB(z)rs(w).

+ ,

2

^TBH +

(z-u>)2

-^~arB(w)

TB(Z)7>(«O ~ , 3/2,,?>(™) + — L _ d r F ( » (z — u))'! z—w 5/2 1/2 m , . TP(Z)TF(«J) • ' iu) 3 +—zL— (z — — w TB{w) 7 = Jje*'

7> 7 = T4> + <j> — charge:

7 2 = J7S7je2*

T

ii 1 2iti

-JdCd
=

qAq

QB = ^ - 7 + ~Tbc) + ~-r4> • dX - ifc 7 2 2m J I 2 Q' 4 QB = ^~ i
1 (z-w)2

118

7. Cubic (Super)String Field Theory on Branes and Sen Conjectures 7.1. Sen's

Conjecture

As we have seen in Section 5 there is a tachyon mode in the spectrum of bosonic open string. There is also a tachyon mode in GSO— sector of fermionic string. Normally we impose GSO+ projection on superstring spectrum and obtain tachyon free spectrum. But there are objects in the string theory such as nonBPS branes or brane-anti-brane pairs, which necessarily contain both GSO+ and GSO— sector 144 . Therefore, it becomes important to explore the tachyon phenomenon. The existence of the tachyon mode does not necessarily signify a sickness of the theory, but may simply indicate an existence of a ground state with the energy density lower than in the starting configuration. One can compare this situation with Higgs phenomenon. The Higgs field has a potential looking like a Mexican hat. On the top of the potential the Higgs field has negative mass square, while at the bottom it has positive mass square. To find a new vacuum in the theory of strings one has to be able to calculate the effective tachyon potential, that is a functional on a constant field configuration. Since a non zero constant tachyon field is far away from its on-shell value one has to use an off-shell formulation of string theory to do this. As we know from sections 4-6 string filed theories give such off-shell formulations of the string theory. There are also so-called background independent string filed theories (BSFT) which are also used to calculate the effective tachyon potential 234 - 236 ' 237 . First attempts to find a stable vacuum in the bosonic string using SFT was made by Kostelecky and Samuel 138 . They have computed tachyon potential using level truncation scheme and shown that there is another vacuum, in which there is no tachyonic state. About two years ago A. Sen 233 suggested to interpret tachyon condensation as a decay of unstable D-branes. Let us consider one non-BPS D9-brane. As it is mentioned above the string attached to this brane has a tachyon in the spectrum. For this case, the Sen's conjecture says 144 : (1) The tachyon potential Vc(t) looks like Mexican hat with minimum at the point tc. The difference V(0) — V(tc) is precisely the tension fg of the non-BPS D9-brane. (2) The theory around the new vacuum does not contain any open string excitation. Therefore, this new vacuum can be also considered as a vacuum of a closed string. (3) All D-branes in type IIA (IIB) string theory can be regarded as classical

119 solutions in the open string field theory living on a system of non-BPS D9-branes (or D9-anti-D9-brane pair). In this section we will show how one can use SFT to check these conjectures. We will mainly concentrate on the first conjecture and will say only few words about others. Before proceed let us summarize what was obtained in the literature. The first Sen's conjectures was numerically verified almost in all SFTs listed in the Introduction. In Table below we collect results of the computations of the ratio ( V ( 0 ) - V ( £ C ) ) / T performed in different SFTs using the level truncation scheme. S F T Type Cubic SFT

V(0) -

V(tc)

level

number of the fields

10

102

0.9991

1

1 (exact) 0.944

BI SFT

T

Nonpolynomial SSFT

4

70

Modified Cubic SSFT

2

10

1.058

1

1 (exact)

BISSFT

These results were presented in 148,234,232,183 a n ( j 237 correspondingly. See also 187>188>232 about calculations of the tachyon potential in nonpolynomial SSFT 185 ' 186 and 189 about calculations in the Witten cubic SSFT. One sees that Background Independent SFTs give exact results for the minimum of the tachyon potential. The formulation of Background Independent SFT can be found in 220>234, and for further developments related to tachyon see 223,221,222,224,225,226,235,227,228,229,230,231

The

second

Sen's

conjecture

SFTs238,239,240,241,242,243,244,245,246,247

was also a n d

248

por

verified

almost

superstringS

in all

the Solution

representing, for example, a decay non-BPS D9 -> BPS D8 (D8) is a kink (anti-kink) solution. There are also solutions which represent a decay Dp —> Dq. Such solutions can be obtained by combining several kink solutions representing the decay Dp -> D(p - 1). This chain relations between the solutions are called the brane descend relations 144 ' 145 . In the case of the bosonic strings the solution representing a decay Dp —> Dq is called lump solution of codimension (p — q). Below we summarize some numeric results concerning lump solutions. The third conjecture implies that there are different solutions of the equations of motion of the SFT which can be identified with different objects ap-

120 pearing in the string theory. For example, the vortex and kink solutions in the Super SFT represent lower dimensional D-branes (of various types) in IIA (B) theory. Another example is lump solutions in the Bosonic SFT, which also represent lower dimensional D-branes. 7.2. CFT on

Branes

Dp-brane is incorporated into the string field theory by considering boundary CFT. This boundary CFT is constructed by p + 1 bosons Xa(z,z) (a = 0 , . . . ,p) with Neumann boundary conditions and 25 — p bosons X'l(z,z) (i = p+ 1 , . . . ,25) with Dirichlet boundary conditions. But to be able to use vertex operators we need to consider T-dual CFT. This CFT consist of 26 bosons Xfl(z,'z) with Neumann boundary conditions, but part of them is related to Xn(z,~z) by T-duality transformation: Xi(z,z)=XiL(z)+XR(z)

=U

X'i(z,z)=XiL(z)-XR(z).

7.3. String Field Theory on Pair of

(7.1)

Branes

To incorporate more than one brane into string field theory one can use ChanPaton factors. We will consider the string field theory for a pair of Dp-branes. To this end we need to introduce 2 x 2 Chan-Paton factors: 1 0' 0,

p 0

p CD 0

0 0' p 1 0 0' 1 0

denotes a string that begins and ends on the first brane; denotes a string that begins on the first and ends on the second brane; denotes a string that begins and ends on the second brane;

denotes a string that begins on the second and ends on the first brane.

For these matrices the following compositions are true:

'0 A /0 0\

(\

0

o o/ vi oy ~ vo oj '

(7 2a)

i o

(7 2b)

o o = (o i •

'

-

121

The string field is modified in the following way:

• - • " ' • ( J ! ! + -i'!!W.nw:;?

(7.3)

The action is of the form 1 $ Tr | « * , Q B # » + | « 4 , * , *»

9l

9l

\((*W,QB*{1)))

+ \W,QB
+ ^ ( ( * ( 1 ) , Q B $ ( 3 ) ) > + \w,QBci>))

+ i«* (1) ,* (1) ,$ (1) )) +

\{{^M2\^))

2 + \ ( ( ( f W . f ^ H c p . ) + \ ( « * < \ 0 , O + c.p.)

(7.4)

The dimensionless coupling constant g2, is the same as in the action (5.36). The expression (7.4) can be simplified if we employ cyclic symmetry of the odd bracket:

$( 1 ),$( 2 ), < A,«A*1=-4^(($ ( 1 ) ,Q B $ ( 1 ) )) + J(($ (2) J Q B * (2) )> (1) (1) (1) 2 2 2 + u\QB)) + | « * , * , * » + |«*< >,*< >,*< >»

(7.4') Let us consider the following string fields (7.5a)

* ^

=

/ (2TT)P+1 ^ f c ° ^ V f ( ^ ' ^ ' 2 ^ ) +iui(ka)Vlv

(w,ka,

2^rJ , (7.5b)

(7.5c) where V^ is a primary operator representing the vector field (5.47) and Vt is the tachyon vertex operator. And we introduce additional restrictions blUi(ka) — 0 and blu*(ka) = 0. We are interested in the following action S[6xi,t,Ui]

= —r- l((*(1),QB*{1))) 9o L

• (7-6) + ((f,QB)) + {(*{1},)}

122

The quadratic action can be easily read from (5.52) and it is of the form

<5 M

*

+1

1

1

f

dP dV '

/J.2 a'A;,

k

r )P+i

- [a'k2a+a'ml

•*Xi(-*)*x'(*)

+ [a'k2a + a'm2b] «,*(-*)«'(*)

- l] t*(-k)t(k)

(7.7)

where m2. = ( ^ ) . To compute the interaction terms we need to know the following correlation functions: ((«Xi(*i)V{,(fci,0), t*(k2)Vt {k2, -&.) =

i6xj(h)t*(k2)t(h)i x (cdX'Ux)

x8(k1

,ia'k\-l J2

c(/ 2 ) c(/ 3 ) e «*i-^(/»)

2 e

.ioc'k\ h

^ - ^ ( / 2 ) e 2i*s-x L (/ 3 )^

k2 + k3) (A - f 2 ) 2 a ' h k * (fl - f 3 ) 2 a ' k ^

+

~~bj

x (fi ~ h) (f2 - h) (f3 - fi) ibj6xj(ki) ria'kk x

J2

— 1 , I a' k\ — l i t

f3

-i a 2ira'

•

, ~lCt

fi — fi

(f2 -

> ^tr'tife / r

l/i - h)

f3)2a'k>k*

/ bj

2~™P

fi - h

(2irr+1a'-^S(ki+k2

t*(k2)t(k3)

2lT

—l

2 f,a'k lf^a'kl-lf,a'kl-lpiry+la,-2±l

Sxj(k1)t*(k2)t(k3)

=

r>a'k\ J\

, t(k3)Vt {k3, ^ T ) ) )

+

t \2a'k\ki. i r

(h - 73)

k3)f;a'kt

t \2a' k2k3+2

(h - h) (7.8)

Substitution of the maps (5.35) into (7.8) leads to the following expression

ibj6x3(ki) 2n x7

i*(A; 2 )i(A;3)(27r) p+1 a'-^J(fci + k2 + k3)

a'k\+a'k\+a'k\-2

(7.9)

123

Now we consider the correlation functions ( ( « * ( * i ) V V ( * i , 0 ) , iu*(k2)VJv {k2 , -^r)

, iuk(k3)Vkv

-8Xi{k1)u*Ak2)uk{h)}[alkh2,a'klfla'kl

=

x /cdX* e2ikl-XLUl)cdXj

{k3 , 5 ^ ) ) )

a'

e2ik2-XL{h)cdXk

e2ik3XL(-hA

.

(7.10)

Using the fact that Ui(ka)bl = u*{ka)bl — 0 we get the following expression 2_ a' x (/s -

/ 1 ' a ' f c ? / 2 ' a '* a 2 /3 a ' * ' (/i - / 2 ) (/ 2 - /s)

5xi(h)u*(k2)uk(kz) _2_ a1

h)

J*

n

(/2 - / 3 ) 5

(-ia')

X (/l - / 2 ) 2 a *1,la (/l - / 3 ) 2 Q ^ ^ (/2 -

2na'

•

/i — h

27ra'

h —h

hf

-^W6^u;{k2)u]{k3)f;a^f,a-klf,a^ X (/i - / 2 ) 2 Q ' * l f c a (/i - /a)2"'*1*8 (/2 -

(7.11)

/3)2Q'fc2fc3

Substitution of the maps (5.35) into (7.11) leads to the following expression ttMxi(fci)u*(k2)UJ(k3)(2ny+1

a'-^6(k1+k2+k3)r'k*+a'k>+a'kl

• (7.12)

Combining (7.9) and (7.12) we get the interaction term of the action (7.6) S3[5Xi,t,Ui] =

2a r v ' '

9o

x

r 2 1 - f dp+ik1dp+ik2dp+ik3

stu

?\

^+fc^+fc3)

J

(2^(2^+1

»6i^x'(fci) 7 a . f c ; + a > t g + a > t g [ 7 - 2 t . ( t 2 ) t ( f c 3 ) _ u*(fe2)ui(jb3)] ,

where 7 = -4= . Let us choose Sxl(x)

(7.13)

to be a constant

*X<(*) = (27r) lH - 1 «x i *(*) and let t, t* and u*, ttj be on mass shell. This simplifies the action (7.13). The

124 whole action S (7.6) is of the form p +l f ddP ~k

1

S[t Ui] =

>

^¥JW

a'kl-a'

)p+i

5

xt*{-k)t(k)

a'kl+a'

i bj S xl

2ita<

2TT

ibidx*

+ 1+

2ira

2TT

a'

<(-*)«'(*).

(7.14)

One sees t h a t the t e r m proportional to Sxi has n a t u r a l interpretation as a shift of mass of the fields t and Uj. B u t we want t o interpret the t e r m with S Xi as a shift of &;. Therefore, we have t o compare two shifts of the mass: one produced by Sxi and another produced by a shift of fy: , u

2

47r a

ibi6xi

2

2bi6V

a

/2

7

(7.15)

2-K

So we get 5

Sx'=i

6V

(7.16)

ira'

This expression can be obtained in a more simple way. Let us consider the following operator product expansion: e2i%j-X'L(z)

e2i£Xl(w)

(z _

= 1

/ Sbjb

= (z-w) z — w)

w^<*

(Sija-

i

e2i^-X'L(z)+2i^X'L

i

!>)

2

( ^ e2i^£h_X L(w)+2i^dX (w)(z-w)+0(z-w)

<2*> e

1 + (s - tu) —

0X*(u;) + 0(z -

wf

7T

(7.17) T h e state |0, b) is generated by the vertex operator Vt(0,

- ^ T ) , there-

fore, the change of this operator under a small shift of bi must be equal t o * X i V £ ( O , ^ ) . So we get ibj

c W e ^ ^ W

ibi

i<56

=

c(w)-^dXJ(w)e^x^

iva 2

-i a' 2

2

ifl>,- . 2 ' —7« 7ra' .a 7 .

2

c(itf)dJf'(tt;)

e^^

( t u )

i 2

7ra'

'

2w a1)

'

(7.18)

125 which is equal to the previously obtained expression up to the vertex operator of the auxiliary field. So we can now identify 6xl by using 5

i6Xl

=

Sbj_ ira'

(7.19)

One sees that up to the sign (that is easily explained) we get the same result as in (7.16). Now we substitute (7.16) or (7.19) into (7.7) and get for the field 8b1 the following action

r
_i

S2[Sbi] = 2TT

2 5

2Q"^ J

(7.20)

(2TT)H-I

2

As a consequence we get the brane's tension: _

1

(7.21)

2

2ir g„a'

7.4. Superstring

2

Field Theory on Non-BPS

D-Brane

To describe the open string states living on a single non-BPS D-brane one has to add G S O - states 144 . G S O - states are Grassmann even, while GSO+ states are Grassmann odd (see Table 6). Table 6.

Parity of string fields and gauge parameters in the 0 picture.

Name

Parity

GSO

Superghost number

•4+

odd

+

1

heZ,

A-

even

-

1

hei+

Weight (h)

Comments

/i > - 1

string

-,h>-2

A+

even

+

0

h e Z, h ^ 0

A_

odd

-

0

heZ+-,h>2

fields

2 gauge

2

parameters

The unique (up to rescaling of the fields) gauge invariant action unifying GSO+ and GSO— sectors is found to be

S[A+,A-] = ±

\{{Y^\A+

+ -({Y-2\A-

,QBA+))

, QBA-))

+ \{iX-2\A+

, A+ , ^ »

- ((Y-2\A+ , A- , A-)) . (7.22)

126

Here the factors before the odd brackets are fixed by the constraint of gauge invariance, that is specified below, and reality of the string fields A± . Variation of this action with respect to A+ , A- yields the following equations of motion" QBA+ + A+ * A+ - A- * AQBA-

= 0, (7.23)

+ A+ * A- - A- * A+ - 0 .

To derive these equations we used the cyclicity property of the odd bracket (6.76) (see Section 6.9.1). The action (7.22) is invariant under the gauge transformations SA+ = QBA+ + [A+, A+] + {A-, A_ } , SA-

= Q B A_ + [A-,A+] + {A+, A_} ,

where [, ] ({ , }) denotes ^-commutator (-anticommutator). To prove the gauge invariance, it is sufficient to check the covariance of the equations of motion (7.23) under the gauge transformations (7.24). A simple calculation leads to S {QBA+

+ A+

* A+ - A-

* A-)

= [QBA+

+ A+ * A+ - A-

* A-

- [QBA- + A+ * A- - A-*A+ 8 {QBA-

+ A+ * A- - A- * A+) =

[QBA-

+

, A+]

, A_] ,

+ A+ * A- - A- • A+ , A+]

[QBA+

+ A+*A+-A-*A-,

A_] .

Note that to obtain this result the associativity of ^-product and Leibnitz rule for QB must be employed. These properties follow from the cyclicity property of the odd bracket. The formulae above show that the gauge transformations define a Lie algebra. 7.5. Computation

of Tachyon Potential

in Cubic

SSFT

Here we explore the tachyon condensation on the non-BPS D-brane. In the first subsection, we describe the expansion of the string field relevant to the tachyon condensation and the level expansion of the action. In the second subsection we calculate the tachyon potential up to levels 1 and 4, and find its minimum. 7.5.1. Tachyon String Field in Cubic SSFT The useful devices for computation of the tachyon potential were elaborated j n 147,146,186 -^ye e m p i 0 y these devices without additional references. "We assume that r.h.s. is zero modulo ker Y_2 •

127

Denote by Hi the subset of vertex operators of ghost number 1 and picture 0, created by the matter stress tensor Tg, matter supercurrent 2> and the ghost fields b, c, d£, 77 and cf>. We restrict the string fields A+ and A- to be in this subspace Hi. We also restrict ourselves by Lorentz scalars and put the momentum in vertex operators equal to zero. Next we expand A± in a basis of LQ eigenstates, and write the action (2.7) in terms of space-time component fields. The string field is now a series with each term being a vertex operator from Hi multiplied by a space-time component field. We define the level K of string field's component Ai to be h + 1, where h is the conformal dimension of the vertex operator multiplied by Ai, i.e. by convention the tachyon is taken to have level 1/2. To compose the action truncated at level (K, L) we select all the quadratic and cubic terms of total level not more than L for the space-time fields of levels not more than K. Since our action is cubic, L < SK. To calculate the action up to level (2,6) we have a collection of vertex operators listed in Table 7. Note that there are extra fields in the 0 picture as Table 7. Vertex operators in pictures —1 and 0 . Level L0 + l

Weight Lo

GSO

0

-1

+

even

u

—

c

1/2

-1/2

-

even

t

ce_*

e*r7

1

0

+

odd

r;

cdcdt,e-2,t>

dc, cd<j>

3/2

1/2

-

odd

Si

cdc/>e~*

Twist

Name

n

Picture — 1 Picture 0 Vertex operators

cTF ,

d {ne't')

bcrje'f', Tide*' 2

1

+

even

Vi

T),

TF<XT*

2

d c,

cTB,

cTiv

2

c T 0 , cd 4>, d2^cdce~2'tl

TF^

bcdc, dcd(f>

dicdcde-2

compared with the picture —1 (see Section 6.4). Surprisingly the level LQ = — 1 is not empty, it contains the field u. One can check that this field is auxiliary. In the following analysis it plays a significant role. Only due to this field in the next subsection we get a nontrivial tachyon potential (as compared with one given in 189 ) already at level ( 1 / 2 , 1). As it is shown in Appendix 6.9.2 the string field theory action in the restricted subspace Hi has Z2 twist symmetry. Since the tachyon vertex operator

128 has even twist we can consider a further truncation of the string field by restricting A± to be twist even. Therefore, the fields ri,Si can be dropped out. Moreover, we impose one more restriction and require our fields to have the (/•-charge (see 6.64) equal to 0 and 1. String fields (in GSO± sectors) up to level 2 take the form0 A+(z) = uc(z) + v1 d2 c(z) + v2 cTB(z) + v3cT^(z)

+ vA

+ v5cd2(j>(z)+v6TFrje'l'(z)+v7bcdc(z)+

cT^z)

v8dcd(j>(z) ,

(725)

D 0(z ]

A-(z) =

v(z) •

7.5.2. Tachyon Potential Here we give expressions for the action and the potential by truncating them up to level (2, 6). Since the field (7.25) expands over the levels 0, \ , and 2 we can truncate the action at levels (1/2, 1) and (2, 6) only. All the calculations have been performed on a specially written program on Maple. All we need is to give to the program the string fields (7.25) and we get the following lagrangians

£(i.D =

u2 + -t2 + 2-ut 4 37

9l*>^ £(2,6) _

9 +

4w

2

(7.26)

u2 + -t2 + (4wi - 2v3 - 8v4 + 8v5 + 2v7) u ^

15 77 i + y u 2 + vl + Yv*

+ 22v

%

+ l0v

e +

8viV3

~~ Mvxvi

+ 24viw5 + 4vx v7 — I6U3V4 + 4vsv5 — 2v$v7 + 12u3i>8 — 52V4U5 — 8V4V7 - 20w4w8 + 8v5v7 + 8v5v8 + (-30v 4 + 20w5 + 30v2) v6 + 4v7v8

+

1 U 372

+

-V\

/40

I —7 u + 4 5 73 V 215 7 3

8O73

25 V2 32

9_ V3 16

4573

^-v2

.

59 32^

+

43 2AV5

+

ZV7]t

4573 29573 -V4 -v3 -

2

"The stringfieldsare presented without any gaugefixingconditions.

(7.27)

129

where 7 = -^ choice

. To simplify the suscceeding analysis we use a special gauge

3V2 - 3^4 + 2«5 = 0

(7.28)

This gauge eliminates the terms linear in VQ and drastically simplifies the calculation of the effective potential for the tachyon field. We will discuss an issue of validity of this gauge in Section 8. The effective tachyon potential is defined as V(t) = —C(t,u(t),Vi(t)), where u(t) and Vi(t) are solution to equations of motion duC = 0 and dViC = 0. In our gauge the equation dV6C = 0 admits a solution i>6 = 0 and, therefore, the tachyon potential computed at levels (2, 4) and (2, 6) is the same. The potential at levels (1/2, 1) and (2, 6) has the following form:

vi|'1)(*) = 9oOi'

1

81 A 2 t -\t 1024 (7.29) 5053 4 t -^2 69120

6)

vil' (t) =

9i<x-

One sees that the potential has two global minima, which are reached at points tc = ± 1.257 at level (1/2, 1) and at points tc = ± 1.308 at level (2, 6) (see also Figure 14 and Table 8).

7.6. Tension of Non-BPS Conjecture

Hp-Brane

in Cubic SSFT and Sen's

To find a tension of Dp-brane following 187 one considers the SFT describing a pair of Dp-branes and calculates the string field action on a special string field. This string field contains a field describing a displacement of one of the S7\ * ~ *• branes and a field describing arbitrary V-' /R\ excitations of the strings stretched between the two branes. For simplicity one can use low-energy excitations of the strings stretched between the branes. The cubic SSFT describing a pair of non-BPS Dp-branes includes 2 x 2 ChanPaton (CP) factors 187 - 198 and has the Figure 13. The system of two non-BPS Dp-branes and strings attached to them.

f 0 u o w i n g form

130

S[A\,A-} = ± \{(X-2\A+ , QBA+)) + l((Y„2\A+ , A+ , A+l JO

+ i((Y-_2|i- , QBA-)) - ((Y-2\A+

,A-,A.

(7.30)

Here g0 is a dimensionless coupling constant. The hatted BRST charge QB and double step inverse picture changing operator Y-2 are QB and Y-2 tensored by 2 x 2 unit matrix. The string fields are also 2 x 2 matrices

(7.31) and the odd bracket includes the trace over matrices. The action is invariant under the following gauge transformations: SA+ = QBA+ + [A+ , A+] + {A- , A_} , (7.32) 5A- = QBA-

+ [A- , A+] + {A+ , A_} .

The fields A± describe excitations of the string attached to the first brane, (2) while A± describe excitations of the string attached to the second one. Excitations of the stretched strings are represented by the fields B± and B*± (see Figure 13). The action for a single non-BPS D-brane (2.7) that we have used above is derived from the universal action (7.30) by setting A± , B± and B± to zero. Note also that we have not changed the value of the coupling constant g0. The "±" subscript specify the GSO sector. Let us take the following string fields A± : A+ = A^

+ B*+ + B+ ,

A- = B*_+B_

where

r dp+lk

/

-> \

,

(7.33)

131

*--=/^«-wv. (*.-*)« (21)Here bi is a distance between the branes, a = 0 , . . . ,p and i = p + 1 , . . . ,9 and VJ, and Vt are vertex operators of a massless vector and tachyon fields respectively defined by V„(A;a,fcj) — -

' 2'

1/2

V.

cax" +

Vt(fca,fci) = - c2ik-ip-

O2ik-XL(0)

(7.34) O2ifc-XL(0)

-r/e*

These vertex operators are written in the 0 picture and can be obtained by applying the picture changing operator (6.27) to the corresponding operators in picture — 1. The Fourier transform of Ai(ka) has an interpretation of the Dp-brane's coordinate up to an overall normalization factor 198 . Further we will assume that blBi(ka) = 0. The action for the field (7.33) depending on the local fields Bi(ka), B,*(fcQ), t(ka), t*(ka) and Ai(ka) is given by S[Ai,Bi,t]

= -i ^((Y^lA^UV1')) 9o L^ 2\AV,B*+ + ((Y({Y-2

,B+))

+ ((Y-2\K,B+)) + ({Y-2\B*_,B-)) - ((r_ 2 |4 1 ) ,s*,£_»]

x [*«+TAA + **(-*)*(*) [ki+(A* dP+1p r dp-

-M

2a'(*2+p2+P.fc+?^7^)

+ I (2*) p + l x [ B ? ( - p - * ) # ( * ) H ^ f f - P "*)*(*)]

MJ'(P) 27ra'v / 2a 7 (7.35)

where 7 = -z^ • Let us now consider the constant field Ai(£) = const, where £Q are coordinates on the brane. Its Fourier transform is of the form Ai(p) = (2ny+1Ai6(p). Let also B^k), B*{k) and t(p), t*(p) be on-shell, i.e.

kl +

6? A2 (27TQ')

= 0

132 and b2

PI +

1

(27ra')

2

= 0.

2a'

In this case the action (7.35) is simplified and takes the form: dp+1k .shell

9o'

*2 +

(2TT)H-1

6?

1

(27ra') 2

2a'

biA*

+

27ra'v'2a'

t*(-k)t(k)

rfP+1fc

+ s2

,E±±

mass shell (2TT)IH-I

6?

k A

ki + (27ra') :

+ 27ra'V2a'

B*(-k)Bj(k)

.

(7.36)

T h e terms proportional to Aj have n a t u r a l interpretation as shifts of masses of the fields t and Bi. We want to interpret the term with Ai as a shift of brane coordinate &,. Therefore, we have t o compare two shifts of the mass: one produced by Ai and another produced by a shift of bi 6(m2)

=

biAi

2bi5V 2

2

a'

A-K

(7.37)

27ra'V2a'

So one gets A% =

-

(7.38)

J6*

7T

This formula determines the normalization of the field Ai. Therefore, we we can introduce the profile of the first non-BPS Dp-brane as Xi (£) = 7r

MO-

Substitution of t h e Aj(£) into t h e first t e r m of t h e action (7.35) yields 5

° = ~

2

/

,E±I /

^

^ ( 0 ^ ( 0

•

(7-39)

T h e coefficient before the integral is t h e n o n - B P S Dp-brane tension fp: f

p =

2

2 2 )£±i 71 a 2

•

(7-40)

ffo "

As compared with t h e expression for t h e tension given in 1 8 7 we have t h e addition factor 4. T h e origin of this factor is the difference in the normalization of the superghosts /?, 7.

133

_2

-1

\l

i t

1

1

\\ 2 i.i

i

fi.2 /-0.4: /

-0.6

/

-0.8 ~1

y

Legenc1 level 1 level 4

Figure 14. Graphics of the tachyon potential at the levels (1/2, 1) and (2, 6). "—1" is equal to the minus tension of non BPS Dp-brane fp = —=TIJ- . 7r29?a'Hi-

Now we can express the coupling constant gl in terms of the tension fp. Hence the potential (7.29) at levels (1/2 1) and (2, 6) takes the form (see also Figure 14)

•UV ( 2i l l )

•eff

( * ) = ^

V £ . . )\t), * _

81 1024 5053 69120

1 , 4

4

4

1 4

(7.41) 2

The critical points of these functions are collected in Table 8. One sees that the potential has a global minimum and the value of this minimum is 97.5% at level (1/2, 1) and 105.8% at level (2, 6) of the tension fp of the non-BPS Dp-brane.

134 Table 8. The critical points of the tachyon potential at levels (1/2, 1) and (2, 6). Potential (4.D •eft

*c=0

V

Vc = 0

4C = ± - 5 — v(

Critical values

Critical points

2 6

. )

Vc fa -0.975 fp

w ±1.257

tc-0

Vc = 0 94

tc = ±

x/75795 «

Vc « -1.058 fp

±1.308

5053

7.7. Test o / A b s e n c e o / Kinetic Vacuum

Terms

Around

Tachyon

In this section we perform calculations of the effective potential at level 0, i.e. we take only low levels fields: dp+lk

/

2ik-XL(w)

— — ^ u(k) U(w, k)\w=Q JP+1 k

,

/

(2^r*(*)TKt)U ,

V(w, k) = c(w)e 1 c(w)2ik

T(«,,A) = --T,(w)e+M

• ip(w) e2ik-XL(w)

_

(7.42) Note t h a t the expression for the tachyon field can be obtained by applying rasing picture changing operator X to the tachyon vertex operator in picture — 1. T h e action in the m o m e n t u m representation is of the form p+1

f dv~ k

1

1

f

t(-k)u(k) - h(-k)t(k)

dP+lp

-^/^'Hfe-p)'(p)«(*>Mte'

[a'kl - 1/2] (7.43)

(P2+k2+kP)

This expression can b e shortly rewritten in t h e ^-representation

S[«.*] = V ^ 5 i r [dP+h 1 37 :2

u(x)u(x)

u{x) t(x) t(x)

~\t{x)

(-a'dada

- 1/2) t(x)

(7.44)

135

where 0(a;) = exp ( - a ' log 7 dada) {x) .

(7.45)

The operation " has the following properties • to = *o if to = const, •

/ dx i(x) =

dx t(x),

•

I dx a(x)b(x) = I dx a(x) b(x).

Using the second property we can move " from u(x) on the term i(x)i(x). this one can integrate over u(x) and gets an effective action SeffM =

*

9

1}

(-a'dada

fd^x\-\t{x)

After

- 1/2) t ( x ) - - 4 - 4 f ^ ) 2 (*) • (7.46)

Expanding t(a;) = to + *i(a;) and keeping only terms that are quadratic in t\ and linear in a' we get ^2,shifted\P\

— 2 /

X

'1

10

2

*b,

ti(a;)a'aaaati(x) .

(7.47)

So to vanish the kinetic terms for t\ we have to have to — 1.099, which is not far from the critical value t0 = 1.257 for the minimum of the tachyon potential at level ( | , 1). 7.8. Lump

Solutions

Here we give a brief review of the numeric computations of lump solutions (see review 240 and 2 4 1 ~ 2 4 8 for more details'3). First, let us define what is the lump solution. Let us denote by <J>0 the string field representing the solution of momentum independent equations of motion. In other words, $ 0 is a tachyon condensate. We want to find a lump solution $ of codimension one, which represents a decay D25 —> D24. This means that string field $ is a static configuration depending on one space coordinate, say x25 = x. This configuration has nontrivial boundary conditions $(a;) ->• $0

as

x -> ± 0 0 .

Physically this condition means that we want to have true string vacuum at plus and minus infinity. We expect that the solution is concentrated near the PSee

249 250

'

for relations with p-adic

strings

14 251 252 253 254

'

'

'

'

.

136 origin and, therefore, it distinctly differs from $ 0 only in small domain around the origin. Second, to be able to find a solution numerically we have to have finite number of fields. To this purpose we use level truncation scheme for the computations of the tachyon potential. To find the lump solution authors of the paper 241 proposed to use a modified level truncation scheme. Their idea consists of two steps: (1) We compactify the direction a; on a circle of radius Ry/a'. This allows us to represent component fields (j>i(x) of the string field $(a;) by Fourier series rather than Fourier integral:
^2i,ne^

(7.48)

(2) We modify the level grading: lvlij(0i,») = - ^ + i V i + l ,

(7.49)

where JVj is oscillator number operator evaluated on the vertex operator corresponding to the field 4>i a n d the normalization of the grading was chosen in such a way that it is equal to 1 on the zero momentum tachyon field. Using this level grading we can define (M, N)-\evel approximation of the action. This means that we get all fields with lvljj ^ M from the string field $ and keep only the terms in the action for which lvl^ ^ N. Third, we have to choose the Hilbert space % in which we are going to find our solution. The analysis performed in 241 shows that without loss of generality we can consider the following reduced space: (7.50)

H ~ ^ m a t t e r ® rlghost © rlmatter (x)

where ^^"atter i s a universal (background independent) Hilbert space'241 gen,24 erated by matter Virasoro generators restricted to the directions x1,... , x" ^ghost is a Hilbert space spanned by all ghost operators and ^matter (x) is a Hilbert space generated by all oscillators a 2 5 . Now consider a simple example of the computations. Let us choose R = \/3 and assume that we have only tachyon field t(x). Since it has to be even we can write t(x) = y2tncos

( -^L== 1

(7.51)

137

The terms tn have the following grading: n2 l v l ^ tn = 1 + — . Now substitution into the cubic action (5.52), (5.59') yields the following expressions at levels (0,0) and ( | , | ) : (7.52a)

0

(7.52b)

27 "3"

where 7 = -4= and V is related to S by formula _ 27r-RvVVol 2 4

(7.53) 2 Tni " 9oa The solution of the equations of motion obtained from (7.52) is presented on the left pane on Figure 15. —

*

W 0.5

0.6

~=

0.4

\

0.4

0.3

\

0.2

\ -3

0.2

01

-2

Vo.1

/

x

2

3

-2

-4

\

/

2

4 X

Figure 15. Plot of t(x) for R = V3 on different levels. On the left pane the solid line represents t(x) at level ( | , | ) . On the right pane the dashed line shows a plot of t(x) at level (^ , ^4-) and solid line shows a plot of t(x) at level (3,6).

One can also compute the ratio of tension of the lump solution and the tension of original D25-brane. One expect that the tension of the lump solution of codimension 1 is equal to the tension of D24-brane. Therefore, we expect that r (lump) TD24

= 1

where

T(lump) =

2

^ ^ f (V($)-V(g 0 )) •

9Za

(7.54)

138

There are several ways to check this equality numerically. We can express the tension of D25 brane through the coupling constant g0 as 147 1 TU25

=

27r2ff2a/13

Substitution in (7.54) yields _ r (lump) _ 7rj25 TD24

2-KRVO1

2TT2 (V($) - V($ 0 ))

TD24

= R (2TT2 V(*)

- 2TT2 V ( * O ) )

•

(7.55)

So, now we can compute the value of r for our approximate solution and check how accurately it approaches unity. Since we know that 27r2V($o) = —1> we can write two expressions for r: rW=ii[27r2V^JV)(*) + l ] )

(7.56a)

r<2> = 2TT 2 J R[V( M ^($) - V ^ J ^ o ) ] •

(7.56b)

By adding more fields one can increase the level and make computations of the solution more and more accurate. Now we present the results of such computations, which were obtained in 241 for radius R = \ / 3 . On the right pane of Figure 15 we present a plot of t(x) at levels ( | , y-) and (3,6). In the Table below one can find the values of r' 1 ) and A2^ depending on the level. Level 11

1)

\3

' 3/

r(D

r (2)

1.32002

0.77377

1.25373

0.707471

(2,4)

1.11278

1.02368

11

1.07358

0.984467

1.06421

0.993855

(1 !} \3

\3

' 3)

'

I!) 3 /

(3,6)

So, it seems that r' 1 ' and A2^ converge to unity from top and bottom correspondingly. 8. Level Truncation and Gauge Invariance 8.1. Gauge Symmetry

on Constant

Fields

In this section we restrict our attention to scalar fields at zero momentum, which are relevant for calculations of a Lorentz-invariant vacuum. The zero-

139

momentum scalar string fields ,4+ and A- can be expanded as oo

oo

A+ = ^2f^i

.4_=J]£ Q T a ,

and

i=0

(8.1)

a=0

where conformal operators $ ; and T a are taken at zero momentum and % and ta are constant scalar fields. The action (2.7) for the component fields %, ta is a cubic polynomial of the following form

i,j

a,b

i,j,k

(8.2) where Mij = « y _ 2 | * i , Q B *,-» ,

Sijfc = « r _ 2 | * i , *,-, **)> .

(8.3a)

?ab = ((Y_ 2 |T a , Q B T 6 » ,

SM = « ^ - 2 | * i , T a , T 6 » .

(8.3b)

For the sake of simplicity we consider the gauge transformations with GSO— parameter A_ equal to zero. The scalar constant gauge parameters {<5Aa} are the components of a ghost number zero GSO+ string field A+ = ^ < 5 A a A + , a .

(8.4)

a

Assuming that the basis {$.,, T^} is complete we write the following identities: QBA+,a = ^ V * Q * i )

(8.5a)

* j * A + l a - A+, a * $j = ^2 Wja$i ,

(8.5b)

i

T6*A+,a-A+,0*T,, = ^ 3 V T

a

.

(8.5c)

a

The variations of the component fields <j>1 and ta with respect to the gauge transformations (7.24) generated by 6Xa can be expressed in terms of the "structure constants" 80 a

6t

= 80P + Sift = (K + S'jaP) a

a

b

= 6xt = d bat 5\

a

SXa ,

.

(8.6a) (8.6b)

The constants V„ solve the zero vector equation for the matrix My: M«V£ = 0

(8.7)

140

and therefore the quadratic action is always invariant with respect to free gauge transformations. In the bosonic case one deals only with the gauge transformations of the form (8.6a) and finds Slja 258 using an explicit form of ^-product in terms of the Neumann functions 204 . In our case it is more suitable to employ the conformal field theory calculations by using the following identity: «y_2|*i, $ 2 * $ 3 » = «y-2|*i, * 2 , *3» •

(8.8)

To this end it is helpful to use a notion of dual conformal operator. Conformal operators {$*, T a } are called dual to the operators { $ j , Tj} if the following equalities hold «r_2|*\*,->>=<^

and

«r_2|f°,T6»=(JV

(8.9)

Using (8.8), (8.9) and (8.5) we can express the structure constants 2lja and 3aba in terms of the correlation functions: fja

= « r _ 2 | $ \ $,-, A+, a » - ( ( r - 2 | $ \ A + , Q , $,-» ,

daba = « K - 2 | f a , Tb, A+, a » - « F _ 2 | f \ A+,a, T 6 » .

(8.10a) (8.10b)

In the next section these formulae are used to write down gauge transformations explicitly.

8.2. Calculations

of Structure

Constants

In Section 7.5.2 we have computed 184 the restricted action (6.68a) up to level (2,6). The relevant conformal fields with 0-charge 1 and 0 are $0 = U = c

^3 = V3= cTni

$ 6 = Vi, = TFT)e+

(8.11a)

$1 = Vi = d2c

$4 = V4 = cT>

$ 7 = V7 = bcdc

(8.11b)

$ 2 = V2 = cTB

2

$ 8 = V8 = dcd(j>

(8.11c)

*S = V6 = cd 4> T0 = ^e*r)

(8.11d)

141

with <j>% — {u, vi,...,

vs} and ta — {t}. For this set of fields we have got

S(2'4) = u2 + i* 2 + {Avx - 2v3 - 8v4 + 8v5 + 2v7) u + Av\ + ^-v\ 77 + v3 + YV*

+ 22V

5

+

10U

8v y

6 +

l 3 ~ 32vlVi + 24^5

+ 4viV7

- I6113W4 + 4:V3v5 — 2v3v7 + 12t; 3 ii 8 — 52t;4?;5 — 81*4^7 - 20u4V8

+ 8v5v7 + 8v5vs + (-30u 4 + 20i;5 + 30u2) v6 + 4v7vs , 0(2,6) 53

M

9 U +

= VW

25 Vl

V2

8 ~V

9

59 3

~ Irf ~ 3 2 *

( 407 „ , 457 3 + I — j p u - 457^1 + -J-V2 215 7 3 j-vh

43 +

2

\

(8.12) 2

V7 l

+

24^ 3 )

457 3 2957 3 + -y-"3 + ^ ^ u 4

807 3 \ 2 - - y - v r j v2 .

,„„„. (8.13)

Here 7 = -4= . There is no gauge transformation at level zero. At level 2 the gauge parameters are zero picture conformal fields with ghost number 0 and the weight h = 1, see Table 7. There are two such conformal fields with 0 (/(-charge be and d
+ 5\2d(l) .

(8.14)

The zero order gauge transformation (6.69) of level 2 fields has the form 60A+(w) = QBk+{w)

= (-6X2 + p\i

+ SXi cT^w)

J d2c(w) + <5Ai cTB(w) + SXi cTin{w)

+ 6X2 cd2(j)(w) - SX2 rie^TF(w) + SXX bcdc(w)

+ SX2 dcd<j)(w) + 1 (<5Ai - 2<5A2) brjdrie2^

.

(8.15)

We see that in accordance with (6.71e) one gets the field $9 = br)dr)e2^ from the sector with q — 2. To exclude this field from the consideration we impose the condition <22A+ — 0, which links parameters SX\ and JA2 appearing in (8.14): Q 2 A+ = i (5\i - 2SX2) br)dr)e2*{w) = 0 ,

<JAi = 2<5A2 = 2SX .

(8.16)

142

We are left with the following zero order gauge transformations of the restricted action on level 2: S0vi = 25X ,

S0V4 = 26X ,

Sovj = 2SX ,

60v2 = 26X ,

S0v5 = 8X ,

80v8 = SX ,

S0V3 = 2<5A ,

S0v6 = —SX ,

5QU — 0 .

(8.17)

Transformations (8.17) give the vector V\ = V1 in (8.6a) in the form V = {0,2,2,2,2,1,-1,2,1}.

(8.18)

One can check that the quadratic action at level (2,4) (8.12) is invariant with respect to this transformation, 9

AC

<5052 = < 5 A ^ ^ V

i

= 0,

(8.19)

or in other words 9-component vector V1 (8.18) is the zero vector of the matrix Mtj defined by (8.12). Now we would like to find the nonlinear terms in the transformations (8.6). At level 2 we have 3)\ = 3] • The dual operators (8.9) to the operators (8.11) are the following $1 = ^-r)dr][l + dbc]e2'l' ,

$ 5 = - J L ^ [ 4 - d2 + 2 d^e2*

,

(8.20a) #2 =

** =

6^9r?e20TB'

I 3 = j-[dr]d2T] 48

- &r)dr)]e2^ ,

$ i = -lr]dridde2't>

,

~cTFdr)e+,

(8.20b)

<|7 = lr]dr][l + bdc\e2't' , 8

(8.20c)

<£8 = \ndr)bcd^>e2^ ,

(8.20d)

8

o

$0 = ^r]dT][6-d(bc)-2d2]e2'i>

+ l-drld2rie2

,

T ° = \ce4>dr)

.

(8.20e) It is straightforward to check that ((Y-2\&,*i))=&i

and

« r _ 2 | T ° , T„» = 1 .

(8.21)

We find the coefficients 3) in (8.6) up to level (2,4) and this gives the

143

following gauge transformations Sxu = (—-73

+ 327 j vi - - 7 3 w 4 + (-197 3 + 167)^5 3

+ (-y7 it *.

(8.22a)

- 3 2 7 ) v8 SX ,

(8.22b)

= | t SX , - 3 V

27 ,

|-y7

S1V1 =

3

8 \

/

11 ,

4 \

/

17

4 \

3

+ 37J v, + ( - - ^ + 777J -5 + ( - ^ 7 + 37J

(lb3 - b)V8.

+ Siv2

3

+ I 6 7 ) v7 + f ^ 7

O 0

d o

6X ,

oO

(8.22c)

V 0

O

0

Q

- 3 7 ^ 1 - 7 777 7 7 ^"5- -7 7777 «7 + r A <5A , 0 O 3

-

<17 y7

6iv3 =

3

16 \ /17 - y 7 J -1 + ( T 7 16 \

-?73 \ 3

+

r

+

3

(8.22d)

8 \ /17 - 737 ) ^5 + ( y 7

3

8 \ - 737 ) ^7 (8.22e)

<5A ,

y 7 j ^8

K

(8.22f)

777 Vl + -77-7 "4 - -77-7 ^5 - 777 ^7 + 777 «8 SX , 3 3 D D O

"4 $lV5

=

/61 , 7

+

32 \ 25 , 7 Wl + 7 W2

3 -3™ IT ~ y J

/

31 3

16 \

y

" i2~ "

/ 14 ±<± , —7

5 , 481 , 7 3+ 7 U4

16 \ - — 7 ) "7 +

v

IT

52 7

3

+

T3

32 \ SX, T 7 « 8 (8.22g) (8.22h)

<^i«6 = - 777^6 ^A , Ji«7

16 7 u + / 3 87 33 + 16 \

7 j n - y-7 " 2 + 7J7 v U ^ +^v T

+ ( f 73 + § 7 )

«B

3

+ ( 8 7 3 + §77 )J

v

7 + h1673

- — 7 u4 - -3-7] "8 SX, (8.22i)

I"4 dlW8 =

4

3 3

25

w

5

3

3

95

v

3

11

V

777" + -77-7 l - -77-7 2 + — 7 V3 + W7T 4 + T l

3

o

8

12

24

0

3

A V

, v

5 + ^T 7

1 „ SX ,

J (8-22J)

144

where 7 = j±= « 0.770. 8.3. Breaking

of Gauge Invariance

by Level

Truncation

As it has been mentioned above (see (8.7)) the quadratic restricted action (6.68a) is invariant with respect to the free gauge transformation (8.17). In contrast to the bosonic case 258 already the first order gauge invariance is broken by the level truncation scheme. Explicit calculation shows that = — -t26\ + {quadratic terms in Vi} S\ .

#5|first order = $iS2 ' + SoS3 '

(8.23) Note that the terms in the braces belong to level 6 and therefore we neglect them. The origin of this breaking is in the presence of non-diagonal terms in the quadratic action (8.12). More precisely, in the bosonic case the operators with different weights are orthogonal to each other, while in the fermionic string due to the presence of Y-2 this orthogonality is broken. Indeed, the substitution of {1} = {u,Vi, w1} and {ta} = {t, TA} (here by w1 and TA we denote higher level fields) into the action (8.2) yields 5

2 = -/ ~ E "*** * TA - \ E 3^ TA TB - 1 Y, M « ^