Coulomb-gas representation on higher-genus surfaces

Nuclear Physics B330 (1990) 488-508 North-Holland

COULOMB-GAS R E P R E S E N T A T I O N ON HIGHER-GENUS SURFACES Jonathan BAGGER and Mark GOULIAN

lo'man Lahoratoly of Physics, Har~ard University, Cambridge, MA 02138, USA Received 16 June 1989

In this paper we use the Coulomb-gas approach to construct the minimal-model conformal blocks on higher-genus Riemann surfaces. We define the higher-genus blocks by sewing, and write them in terms of the rational blocks of a compactified scalar field. We show that spurious states decouple, which implies that the blocks degenerate correctly. As an example, we compute the genus-two partition function, and verify modular invariance for the subset of minimal models which only require one type of screening charge.

1. Introduction

The Coulomb-gas approach provides a powerful tool for computing correlation functions in the minimal series of two-dimensional conformal field theories. The basic idea is to replace the irreducible highest-weight representations of the Virasoro algebra with the simpler Fock space of a free scalar field. It has been known for some time [1] that on such Fock spaces one can construct representations of the Virasoro algebra with arbitrary central charge. In ref. [2] Feigin and Fuchs showed how to recover the minimal-model representations from these Fock spaces. Fateev and Dotsenko [3] then developed the Coulomb-gas approach, introducing screening operators to compute multipoint correlators on the plane. With this technique, they were able to calculate the minimal-model conformal blocks, their monodromy, the operator product coefficients, and thus the correlation functions; however, the detailed relation between their work and that of Feigin and Fuchs was unclear. More recently, Felder [4], using results from refs. [2, 5], explained this relation using a BRST-like construction. As a result of his analysis, he was able to find a Coulomb-gas construction for the conformal blocks on the torus. Similar expressions were found in refs. [6-8]. In this paper we will construct the minimal-model conformal blocks on highergenus surfaces. Our approach is to first represent the blocks by sewing zero- and one-loop amplitudes as in ref. [9]. These amplitudes are then represented in terms of a scalar field, following ref. [4]. Using the assumption that sewing scalar fields is 0550-3213/90/$03.50 ! Elsevier Science Publishers B.V. (North-Holland)

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equivalent to performing the functional integral on the sewn surface, we arrive at a representation of the higher-genus minimal-model blocks, in terms of the rational blocks of the higher-genus scalar field theory. We show that in the sewn amplitudes, only the BRST-invariant states propagate along the sewn lines, so the amplitudes degenerate correctly. As an example, we compute the genus-two partition function and verify modular invariance for the subset of minimal models which only require one type of screening charge. Throughout the paper we assume we are working with ratios of blocks in order to avoid subtleties connected with the conformal anomaly and holomorphic factorization. This is discussed in greater detail in appendix A.

2. Coulomb-gas representation In this section we review the Coulomb-gas representation of the minimal models. We start with a single scalar field, whose dynamics are described by the action s=

(1/4 )f

d2x~/h(h ~'" O/pO~g}-i%RO) + B.T.

(2.1)

Here h~,, is the metric on a Riemann surface of genus g, and R is its curvature, which satisfies

(1/8~')f d2x ~/hR = (1 - g).

(2.2)

B.T. denotes a boundary term which is necessary to define winding sectors on surfaces with g > 1 [10]. The action (2.1) differs from the usual action for a scalar field because of the term proportional to R. This curvature term introduces a "background charge" in the following way: Consider a correlator of vertex operators Vo - exp iadp on a Riemann surface of genus g,

(Uvo(zj))-f e e

(z,).

(2.3)

Under the change of variables g}--, {)+ c, with c a constant, the correlator transforms by a phase,

Since the functional integral should be invariant, the correlator (2.3) must vanish unless 2, o(1 - g ) = o. i

(2.4)

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If we think of the vertex operator V ~ ( z ) as inserting a "charge" a at the point z, we can interpret eq. (2.4) as ensuring charge neutrality on a genus-g surface. The curvature term in the action (2.1) gives rise to the "background charge" - 2%(1 - g). In the remainder of this section we will focus our attention on the simplest Riemann surface, the sphere S 2. On this surface, the neutrality condition reduces to ~aj-

2%= 0,

(2.5)

J which implies that the operator conjugate to V, is V2,° ,. The condition (2.5) has a simple interpretation, which is most easily seen by choosing h,, to be the singular metric on S 2 that is flat everywhere except at one point, " t h e point at infinity." This metric gives rise to a delta-function singularity in R, and the exponential of the curvature term in eq. (2.1) becomes a vertex operator at infinity. The charge of this vertex operator must be cancelled by the other fields in the correlation function. The energy momentum tensor is found by varying the action with respect to the metric h V In complex coordinates, we have T~. 1 oo = - aO.~OzeO + ia o O ~ .

(2.6)

Using the propagator ( Ojo( Z ) OweO( w ) ) = - 2 / ( z - w) 2,

it is not hard to show that eq. (2.6) describes a system with central charge c = 1 - 2 4 a 0 a. If we take a 2 = ( p' - p)2/4pp',

with p and p' positive coprime integers, the energy-momentum tensor (2.6) reproduces the central charges of the minimal models. In what follows, we compactify q~ on a circle of radius !/PP', so ff is identified with ff + 2~rl/~-p' . This is compatible with the curvature term in the action. In the canonical formalism, the Hilbert space then decomposes into a direct sum over products of Fock spaces F, ® F~, which are eigenstates of the charge operators

a 0 = ~ 2 ~ - 0z~ ,

do=~2 ~

0~ qS.

We will restrict our discussion to the holomorphic factor F~ since we are primarily interested in constructing conformal blocks.

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Let us now define the oscillator algebra (2.7)

[a,,, am] = 2n ~,,. _.,, where

T h e Fock space F~ is constructed from the state Ia ) - - e x p i a ~ ( 0 ) [ 0 ) by arbitrary applications of the a,,, with n < 0. The state [a) satisfies the conditions a0la ) = a l a ) , a,,la) = 0, n > 0. Although 0.4 is not a p r i m a r y field, a o still c o m m u t e s with the e n e r g y - m o m e n t u m tensor, so each Fock space F~ is a representation of the Virasoro algebra. T h e states in F~ are graded by their eigenvalues under L o. Since [L 0, a,,] = -na,,, the state [a) is the unique eigenvector with the minimal eigenvalue, h a. It satisfies Lol,x ) = ,x(a - 2 a o ) l a ) . F o r the radius V ~ ' ,

the allowed charges are

a = ll/2 ~ where l 1, l 2 ~ 71. For I 1 = ( n ' -

1)p + (1 - n)p' and 12 = 0, we have

a,,,,,=~(1-n')a where a + = ~

+ 12V~/2,

and a = - l / a + .

+½(1-n)a+, Thus the state l a , , , )

h,,.,, = [ ( n ' p - np') 2 - ( p - p')2]/4pp'.

(2.8) has dimension (2.9)

F o r 1 ~< n' ~

W e see that the Virasoro generators can be used to raise a l [ a k l ) to 1a1,1), but c a n n o t lower ]al,l) to a llal,1).

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To isolate the states that belong to the minimal models, we first restrict our attention to the subspace of F,,, ,, generated by the Virasoro algebra acting on la,,, ,,). In this subspace, IcL,, , ) is a highest-weight vector. We then mod out by the singular vectors to obtain the irreducible highest-weight representations of the minimal models. In refs. [2,4], this procedure was made precise. It was shown that there exist BRST-like operators* Q2i : F,,,,.+2/p

-+ F,,,2jp-,,,

Q2j+I : P,,,,2jp n -+ P,,, ,-~2j+l)p,

(2.10)

with j ~ 7/, which commute with the Virasoro algebra, and satisfy QjQ/+I = 0. It was also shown that the kernel of Q0 is the highest-weight representation generated by [cb,,, ), and that the image of Q-1 is the subspace of singular vectors. Thus the irreducible highest-weight representation of dimension h,,, ,, is given by J~,,,, = H ° - Ker Q 0 / I m Q 1, which is the cohomology in the middle degree of the complex:

The cohomology in all other degrees is zero. Thus, if we define

%F,

Y,

=eF,

,,+2jp

then we have Qeven "~odd

and o~, ,, _-- H ° ~.~,,, ....

(2.11)

which is an isomorphism of Virasoro modules. At this point, we should remark that the operators Qi are built out of vertex operators with charge ~+ [4]. One could instead work with operators QS, with the roles of c~+ and c~ reversed. It is easy to see that the same arguments carry through for Q~, with ,~,,,,. ,, replaced by , ~ ,,, ,,. The spaces o~,,,,, have a simple interpretation. Since the radius, ~ , has a rational square, we can organize the allowed charges into a finite number of families [11 ]:

* Our notation for Q differs from that in ref. [4].

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where fl(k, l) = k / 2 ~ f f + Iv~N/2, l ~ Z and k = 0 . . . . . N, with N = 2pp'. The ~k span representations of the chiral algebra generated by the currents J + = e +_iN~/~24. With the above definitions, we see that 0 % . = s k ° ,,

for

k,,,.,, = (n'p - np') + ( p " - p) mod N.

(2.12)

To compute the conformal blocks, we introduce chiral vertex operators. A chiral vertex operator ~ , . ,,rm is a primary field which maps J'~',,,,, to ..~F/,l:

normalized so that its matrix element between the highest-weight states is one. In ref. [4] it is shown that ¢~,,,,,,, may be represented by a BRST-invariant operator V,r',,. ~', n which maps the Fock space F m,, m to the Fock space Fr, l r',r,

V / , , , t . Fm, m - - - ) V [ , [ .

We call this operator a screened vertex operator. It is given by

g,:;i',;(z ) =~dUl...du,.,dvt...dvrV~,r.,,(z)V ~ (ul)... V~ (ur,)V~+(vl)... V~(vr)

-- vo (:)v"

(2.13)

where r ' = ½ ( / ' + l - m ' - n ' ) a n d r=~(l+l-m-n). In eq. (2.13), the operators V~ and V,+ are dimension-one screening operators which ensure that the screened vertex operator has the correct charge, without changing the conformal properties of V,~,,,,. The contours begin and end at the point z, as described in ref. [4]. By deforming the contours, one can show that the correlators computed with eq. (2.13) agree with those computed by Fateev and Dotsenko [3]. Finally, note that the above operator exists only if r' and r are integers. In fact, the only nonvanishing screened vertex operators are those which satisfy the fusion rules of the minimal models [4]. Using the results of this section, any computation with the minimal models that can be described in terms of the representation theory of irreducible highest-weight

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modules has an easy translation into a computation in the corresponding scalar field theory. This is the case for genus zero and one; correlation functions on the sphere become inner products, while correlation functions on the torus are simply traces [4]. To extend these results to higher genus, we introduce the notion of sewing, and define the higher-genus correlators in terms of the sewing procedure.

3. Sewing conformal field theories 3.1. M I N I M A L

MODELS

Sewing is a very intuitive method for constructing a genus-g surface out of two surfaces of lower genus. Formally, we sew two Riemann surfaces X 1 and X 2 of genus gl and g2 as follows [12]: Remove points Pa and P2 from X 1 and Z2, and choose coordinates z~ and z 2 which identify neighborhoods of Pl and P2 with the open unit disk in the complex plane. For each t ~ C, with It] < 1, we construct a new surface Z = Z~ooX 2 of genus g~ + g2 by identifying points on Z 1 and ~2 which satisfy z 1 = t/z2. A correlator (+1 . . - ~ . ) ~ on the sewn surface is then defined by [9] t - a ' . k ( ~ 1 ... i, j.

+,,~e_,xk(Pl))Z~(~jy(k(p2)~,,+l...

+,,)z2~f-k:,j,

(3.1)

k

where we have suppressed the antiholomorphic dependence. In (3.1), the sum over ~ ° i X k denotes a sum over the full set of primary fields and their Virasoro descendants. The ~ ° j ~ k are the corresponding conjugate fields, and Ai, k is the dimension of ~-iXk- The matrix Jgk, q is defined in terms of the two-point functions on the sphere,

Eq. (3.1) tells us how to construct a genus-g correlator out of two correlators defined on surfaces of lower genus. In a similar fashion, we can attach a handle to a single Riemann surface X by removing two points from the surface and identifying disks as above. In this case the sewn correlator is of the form ~-~ t - a " k ( ' ~ P - j 2 k ( P 2 ) + l i, .j, k

...

~bn~-iXk(Pl)).~

,/~-1

k,ij

.

(3.2)

Attaching a handle promotes a correlator from genus g to genus g + 1. To isolate a particular block in the sewn correlator, we choose blocks in the original correlators and then restrict the sum over k to a single field X- Since we are interested in representing this block in terms of a free scalar field, we must also discuss sewing with the action (2.1).

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3.2. SCALAR FIELDS

For scalar fields, we can define higher-genus correlation functions by performing a functional integral on the sewn surface

(~1

"'" l~n)X

=

f

,e-%a ...

q~.,

(3.3)

where q~: Z = ~ 1 ~ 2 - - - > S 1. Alternatively we can define the correlators by sewing, with the sum running over all Fock-space descendents. Thus, instead of (3.1), we write

t-n"*(~l... ~,,S~' iXk(Pl))X~(s~C_j2k(p2)+m+l... ~,)X2J/g,jl,

(3.4)

i' l] "

where the sum over k now runs over all fields Xk which are highest weight with respect to the oscillator algebra (2.7), and ~'-iXk denotes an arbitrary Fock descendent. In what follows, we shall assume that (3.3) and (3.4) are equal. At first sight, there appears to be a problem with this construction, because the background charge in eq. (3.3) is greater by 2 % than the total background charge of the two correlators in (3.4) (see (2.4)). However, in (3.4), one must not forget the charge carried by the fields Xk and Xk- Since Xk is defined to be the primary field conjugate to X~, the total charge carried by Xk and )(k is 2a 0. The difference in background charge is compensated by the charge carried by the fields Xk and Xk-

4. Coulomb-gas representation on higher-genus surfaces 4.1. CONSTRUCTING THE BLOCKS

In this subsection we compute the conformal blocks for the minimal-model correlators on a higher-genus Riemann surface. Let us first consider a block constructed by (3.1), in which the sum over k is restricted to a fixed field X of dimension h,,,,,. We proceed by induction, and assume that we already have scalar field representations of the c o r r e l a t o r s (~l...~m.~P'iX(Pl))Xl and ( ~ J X ( P 2 ) ~ b - , +1 "'" +,,)x2" This is a perfectly safe assumption, since it certainly holds if X 1 and X 2 are the sphere or the torus [4], and from these we can build all other surfaces. As in sect. 2, we represent the field X by a screened vertex operator r', r

,,(pl) =

~+ •

(4.1)

However, there is an important difference, which follows from the fact that the vertex operator V~,,,,(pl ) does not appear on the sewn surface, even though its

496

J. Bagger. M. Goulian / Coulomb-gas representation

Fig. 1. For the case of one screening operator, the contour winds around the point Pl.

screening operators V2' and V2~ are still present. This requires us to modify the contours in eq. (4.1) to run around the point Pv For the case of one screening operator, such a contour is shown in fig. 1. The vertex operators with the closed contours are related to those with the standard contours by a change of normalization. In a similar fashion, we represent ;~ by the vertex operator V~"9,,, p_,,(P2). The fields q~ which remain on the sewn surface are also represented by screened vertex operators (or their Virasoro descendents). The screening operators associated with these fields are chosen to have the same contours that they had on the original surfaces 2;t and ,7,2. With this said, we represent (3.1) by

F. ~t-',(,#, ... ,l,°,v2' U ff %,,.,,( p,))~, t, I

x (.z yo,, ,,

,,(p2)V~i ~v-++,,,+, ... +,,)~:~';:

(4.2)

and (3.2) by

F.f>t ~,(~/v% ,,, ,,(p~)v:'V£+~I... ~ ~,V£" V,~" ~iV,~,,.,, (pl))~JJ/{,.-/I (4.3) I,]

where, as in (3.1) and (3.2), the sums run over all Virasoro descendents. 4.2. D E C O U P L I N G OF SPURIOUS STATES

The sums in (4.2) and (4.3) are not particularly natural for a scalar field theory; we would rather sum over all Fock descendents. This would allow us to compute the blocks using a functional integral over the sewn surface. However, this requires that the extra states decouple. To see how this works, let us first consider (4.2), which becomes

~ ~'(+1...+.,U"v2:e Y~,,,(pl)>., l, /

x (d,v%

,,,..,(::2:vo ~+¢..+, q.,,)~2~,/ -

k T1~'

V

~

. . .

We must show that the extra states in { sg iV, --

--

1

.

(4.4)

) do not contribute to the sum. We n'.

n

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497

start by choosing a splitting for the Fock space F,,,,,, and its conjugate Fp, ,,, p_,,:

KerQo = ImQ l~Dh,

F,,, ,, = Im Q_l ~) h G C,

Ker00=Im0

Fp.

~e13,

,,,p

,,=ImQ

l~Ta~(~.

(4.5)

In (4.5), the spaces h and T3correspond to irreducible representations of the minimal models, while C, (~, Im Q z and Im Q-1 contain the spurious states. To see that these spurious states decouple, note that the matrix elements of Im Q ~ with Ker Q0 and Im ~? 1 with Ker Q0 are zero. Therefore, we can choose the splitting (4.5) so that ~ is of the form C h ~,=

ImQ

*

*

0

0

*

0

1 ),

(4.6)

C ImQ

1

where * denotes a nonzero element. Inverting, we find

./g-l=

*

0

*)

o

o

•

•

(4.7)

Now, all the fields ~ in (4.4) are BRST invariant, so the states in I m Q 1 and Im Q z do not contribute to the sum. From (4.4), we see that the only nonvanishing contributions come from the elements of h and ta, which are precisely the Virasoro descendents in (3.1). Therefore, extending the sum over the full Fock space does not change the correlator because none of the extra states actually propagate across the neck. We now consider the extension of (4.3) to include all Fock-space descendents ,,. . . . . ,

(p2)Vf

r"

,

.

(4.8)

i,/

In this case, the spurious states do not automatically decouple since they appear inside the same correlator. For example, Im Q_ 1 appears in a correlator with (~, and (~ is not annihilated by Q. These states must be removed explicitly. This can be done by first extending the sum over F,,, n to include the larger space ~ , ,,, and then, using (2.11), subtracting a sum over ~ , , _,,. The subtraction removes all spurious states from the correlator, as shown in fig. 2.

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> 0

>-Q >-(3

Fig. 2. The n', n minimal-model block is given by the difference of the n', n and n', - n rational blocks.

4.3. G E N U S O N E

N o w that we have extended the sums in the sewing formulae to include all Fock space descendents, we can replace (4.2) and (4.3) by expressions involving rational blocks on the sewn surface. In this subsection we illustrate this procedure by computing the blocks of the genus-one partition and one-point functions. Felder was able to derive these results quite easily, using the H o p f trace theorem [4]. Our argument here is really equivalent; only the words are different. Let us first compute the block in the partition function corresponding to the dimension h,,,.,, field. We start with the following two-point function on the sphere:


. ,p

n

. ,n

)s2.

As described above, we shall sew this two-point function via (3.2), and extend the sum to include the entire space o~, ,, where k,,, ,, satisfies (2.12). We then subtract a sum over . ~ , _,, to remove the spurious states. Finally, we represent the sums by rational blocks on the torus. The metric we use to evaluate the blocks depends on the value of t, and also on the initial metric on the sphere [9]. However, we are free to choose any conformal factor because we know the transformations of the blocks (see appendix A). We therefore choose the flat metric with Teichmuller parameter ~-. For a flat metric, the action (2.1) coincides with the free action with c = 1. Thus the one-loop path integrals are the same. (Of course if we change the conformal factor, they no longer agree since the two theories have different values of c.) The full path integral is easily evaluated to give [10]

X Z=

t~

(2(p--p')AIN'c)

Y'~

(4.9)

where the theta function is given by

z I1E7

and the sum over k is a sum over rational blocks. We choose the a cycle on the

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499

torus to be the one induced by the nontrivial cycle on the twice-punctured sphere in (3.2). In eq. (4.9), ~i is the Riemann class of the torus [10,12], which we can represent by ~-/2 + 1/2. Although we could shift away the Riemann class by shifting k, we include it because it will be important when we consider higher-genus surfaces where there is a nonzero background charge. Another reason to explicitly include A is because degeneration of the genus-two amplitudes produces one-loop amplitudes with the Riemann class appearing as in eq. (4.9). Eq. (4.9) contains contributions from all the rational blocks. The block corresponding to the sum over ~ , , ~ is easily extracted; it is given by

l lkn ]

~(~.)v~ N - - ( 2 ( p - p ' ) A l N , r ) , 0 which is just Tr 5

q L0 c/24. n', n

Using this result, the h,,, ,, block in the minimal-model partition function is given by

1 Irk]

ik I (2(p-p')Z~lN'~),1

"0('~) O[ N (2(p-P')ZXlN'~)-~ L N

in accord with refs. [13,14]. In a similar way, we can construct the one-point functions at genus one from the three-point functions on the sphere. Consider, for example, the block shown in fig. 3, where the pairs of integers m', m and n', n denote the minimal-model representations with dimensions h,,, ,, and h,, ,,. From the fusion rules, we know that this block is nonvanishing if (I)

m', m odd, ½(m + 1) ~
5' ( m ' + 1 ) ~ , , ' l' ~ p ' - ½ ( m ' + 1 ) ,

m',m 0 n',n Fig. 3. The n', n block of the m', m one-point function on the torus.

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or

(II)

p'-m',p-modd 1-m<~p-2n, 1-

p-

2l~m-1,

(4.10)

m ' <~p ' - 2 n ' , p ' - 2 l ' <~ m ' - l .

Since h,,,m = hp, ,,,',p-m, we could have equally well chosen the pair of integers p ' - m', p - m to represent the field of dimension kin, ' m" Therefore we m a y assume without loss of generality that m' and m are both odd. T o c o m p u t e this block, we represent the chiral vertex operator

by a screened vertex operator. There are two possibilities: r', r . V.~, m- F . , . --. F.,,.

with r ' =

(4.11 /

~(rn' - 1) and r = l ( m - 1); and r*,

r

.

,

,

V p , _ , , , p _ m. F,, ,,--* Fp, ,, p ,1

(4.12)

with r ' = ~ ( 2 n ' - m ' - 1) and r = ½ ( 2 n - m - 1). Note, however, the operator in (4.12) does not map F,,, n to itself, but rather to its conjugate Fp .... ,, p_,,. Therefore, it has a vanishing expectation value on the torus. This can be fixed by composing V;r', with the dimension-zero operator P r.q~P, p m r', r

V ; ' - l. p

(4.131

1" F p . . . . , p _ . --+ F~, . ,

with r ' = p ' - n' - 1 and r = p - n - 1. F r o m ref. [4], we know that this operator is equivalent to the identity in the minimal models. T h e sewing now goes through essentially in the same way as for the partition function. As above, the correlators are given by the difference of two terms, one which involves a sum over o~, ,, and the other a sum over o~, , (see fig. 4). However, because of the details of the BRST invariance, the second term picks up a

m',m

=

m',m

O

- m',m

O

v

n',n

Fn',n

Fw,-n

Fig. 4. The n', n minimal-model block is given by the difference of two rational blocks. The phases, screening operators and insertion of e 2~'~'~are suppressed.

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501

phase relative to the first [4]. If we use the vertex operator in (4.11), the one-point function is given by Tr~,,,.,,q t.oV ,r'. ~ ,rm -

e2rri°l(~ y Tr y . ,

r'. ,,q L0Vr,,m, r

(4.14)

where r ' = 7(m ~ ' - 1 ) , r = ½ ( m - 1 ) , and the phase 01 = a,,,,mO~+n. On the other hand, if we use the vertex operators (4.12) and (4.13), we find Try,, ,,q CoV~'l':'p li P-"-1v/r'" ~m',p ,,,

' -- e 2=i02 Tr,~,,, _,,qLOVpP,' - l~,pli ~ 1vr', p ' - np / , n-<,,+l>/2 p-m ,

(4.15)

where r ' = ~(2n' - m ' - 1), r = ½(2n - m - 1), and 0z = ap . . . . , p _ m a + ( p - n) + 2(p'-p)n/p. All of the contours of the screening operators start and end at the locations of their respective vertex operators, and are homologous to the a cycle of the torus. 4.4. H I G H E R - G E N U S SURFACES

We will now show how to compute blocks on higher-genus surfaces. Recall that we can increase the genus by one unit either by attaching a torus and using (3.1) or by attaching a handle via (3.2). We shall use the former approach, which allows us to borrow results from subsection 4.3. Therefore we represent the genus-g blocks by q,3 diagrams, as shown in fig. 5. F o r simplicity, we illustrate the procedure by sewing two genus-one blocks to find the genus-two blocks for the minimal-model partition functions. (The generalization to higher genus and to other correlation functions is straightforward.) Let us start by considering the two-loop minimal-model block shown in fig. 6. This block is obtained by sewing the one-loop blocks of fig. 7. Since we sew a field with its conjugate, we must represent one of the blocks in fig. 7 by the vertex operator with charge a,,, .... and the other by the operator of charge a p , m',p-m" The next step is to write the sewn block in terms of rational blocks on a genus-two surface. Since each of the one-point functions contains two rational blocks (fig. 4), the sewn block contains four rational blocks (fig. 8). In the figure, we have suppressed the phases, screening operators and the insertion of e 2i%ee.

<7< Fig. 5. A genus-g block represented by a ~3 diagram.

J. Bagger, M. Goulian / Coulomb-gasrepresentation

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n',n

Y,1

Fig. 6. A minimal-model block that contributes to the genus-two partition function.

m',m

O n',n

p'-m', p-m

O l',l

Fig. 7. The one-point blocks that give rise to fig. 6.

n',n

1',1

-- Q - - Q Fn,,n

FI,,1

- O - Q Fn,,n

C>-O Fn',-n

FI',I

FI,_1


FI'- 1

Fig. 8. A genus-two minimal-model block is given by the difference of four rational blocks. The phases, screening operators and insertion of e 2i"c~¢' a r e suppressed.

E a c h of the r a t i o n a l blocks in fig. 8 can be e x t r a c t e d from the following t w o - l o o p c o r r e l a t i o n function:

{Vat,, l.... (z)Va ( U l ) " " Va (Up'-2)Va+ (U1)''" Va+(Up-2)}"

F r o m (4.11), (4.12), and (4.13), the total n u m b e r of a a n d c~+ screening o p e r a t o r s is p ' - 2 a n d p - 2, respectively. The total charge of this correlator is therefore

ap, t,p_ l + ( p ' - 2 ) a

+(p-2)c~+=-2a0,

w h i c h precisely cancels the genus-two b a c k g r o u n d charge.

503

J. Bagger, M. Goulian / Coulomb-gas representation

Yk

y~,

Fig. 9. A genus-two rational block.

Following ref. [10], we now evaluate this correlator and extract the relevant rational blocks. A typical block is shown in fig. 9; it is given by

Zk~ = 0

p' 2

Ik

~

0

0

f

2( p - p ' ) ( A - z ) - 2 p

p 2

u,+2p'

i=1

p' 2 i~ 1

E

E ":l N~? j=l

p- 2 /: /=I

p-2 p'-2

×

i~al

i~J"

H e ( . , . . j ) -2

i=l j=l

p'-2 p-2 XO(Z)--4~00~p ' l.p 1 U O(~.~i ) 4a0a U O(Uj) i=1 j=l

4a0a+.

(4.16)

In eq. (4.16), A is the Riemann class, ~2 is the period matrix for the genus-two surface, E is the prime form and o gives the contraction of the vertex operators with the curvature term in the action (see ref. [10] for details). We have dropped overall factors which cancel in ratios of blocks (see appendix A). The theta function is defined by

ncZ×Z

We choose the a and b cycles to be those induced by the a and b cycles on the two sewn tori. Combining these results, we find that the genus-two minimal-model block in fig. 6 is given by

Zk,,.,,kc.j- e2~riOt:zk,:. ,kr. I

-- e2~i°~-M#:Zk,,, ,,k,. , + e2~ri(°' +°2)Jf'~Zk,,,. ,,k,, ,,

(4.18)

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J. Bagger, M. Goulian /

Coulomb-gas representation

where the phases follow from (4.14) and (4.15). The normalization sV" arises when converting the open contours in (4.15) to closed contours, as discussed above, It appears in (4.18) because the two terms in (4.15) have different numbers of screening operators associated with Vp, m,,p_.,. If we degenerate the surface by pinching the coordinate t in moduli space, then by applying the appropriate formulae for the degeneration of 0, E, o, and A [10], we recover the original genus-one amplitudes. Of course, this is guaranteed by the sewing procedure, and by the fact that the spurious states decouple.

4.5. M O D U L A R C O V A R I A N C E

We would now like to check the modular covariance of the genus-two blocks. The modular group at genus two is generated by Dehn twists about the cycles a x, a 2, b 1, b 2, a~-la 2 [15]. The resulting transformations on the homology bases are:

Oal ~

al

a2

bl

b2

tl 1

1

0

0

0

~12 h1

0 1 0

1 0 0

0 1 0

0 0 1

b2

Do=

-

0

1

0

0

0

1

0

1

0

0 0

o 1 0

Y~

"-%'

0

'

'

D~; 1~2 =

Ob2

1 0 0 0

1

0

0

0

0 -1 1

1 1 -1

0 1 0

0 0 1

0 1 0 0

0 0 1 0

0 1 0 1

(4.19)

These matrices generate Sp(4, 7/). Under a modular transformation U ~ Sp(4, 7/), the period matrix transforms as follows,

B The Riemann class also transforms [10]:

A~A(CA+D)-I+½

diag(CD t ) ) diag(ABt) ,

where diag(X) denotes the two component column vector with entries equal to the

505

J. Bagger, M. Goulian / Coulomb-gas representation

diagonal entries of X. Finally, E and o transform as

E(z, w) --->exp[ivr(w-z)(Ct2 + D) XC(w- z)]E(z, w), o(z)-*exp

~_ l (A-(g-1)z)(Ct2+ D) 1C(k-(g-1)z) o(z).

(4.20)

Substituting these transformations into the expression for the theta functions (4.17), we find

k2]

kl

-~ (z+ 2(p-p')al~2), 0

0 Do:

~exp

i~(kj+p-p')2-if(p-p')2

kl

×0

0 D,, ~,:

--* exp

kl ×0

0 D., 1Dt71 DO 1 1.

k2 1(z + 2(p_p,)al~2) ' -~0

w

i~(k,

-

k2) 2 -

1

×exp

I

2w

~-i(p

- p ' ) ( k 1 + k2)

]

k2 ] N (z+Z(p-p')A[~2), 0

( _ i£2ll/N )1/2

xO -N 0

]

N /=1

~ exp( -

2rrikll/N )

k2] N (z + 2(p-p')AI~2) 0

iw[z+2(p-p')A]

(1:) ~11 0

1

[z+2(p-p')A] ,

(4.21)

and similarly for D~IDhD~1. Using these results, it is not hard to show that blocks of the form (4.18) transform correctly under D~ and Do2. These twists do not change the contours for the

506

J. Bagger, M. Goulian / Coulomb-gas representation

screening operators. Other twists, however, do change the contours. To prove covariance, it is necessary to relate the new contours to the old. For genus one, this is done in refs. [7,16]. This problem can be avoided by writing the partition function with the contour integrals replaced by surface integrals, as in refs. [3, 6,17]. Unfortunately, the integrands do not have the correct periodicity to give sensible surface integrals whenever there are insertions of both types of screening operators, V~+ and V~. In what follows, we will therefore restrict our attention to models with p ' = 2, in which only one type of screening operator contributes. Following refs. [6,18], for this restricted class of models, we can write the partition function as N

f d2Ul .. d2up "- 2 d2Vl --- d2vp - 2

N

~

IZk~ -- Zw(k)~ -- Z~w(~)+ Zw(k)w(~)l 2 ,

k=Z T,=a (4.22) where w(k) = (ap + bp')[k - (p'-p)]

+ ( p ' - p ) mod2pp',

with a and b integers satisfying a p - b p ' = 1. The only terms in the sum that contribute are those with k = k , , , or k , , ,, and k = kr, I or k r _ l , where 1 ~< n', l' < ~ p ' - l, 1 < ~ n , l < ~ p - 1 . It is now straightforward to check that (4.22) is modular invariant under all the above modular transformations. 5. Conclusions We have shown that it is possible to extend the Coulomb-gas construction to higher-genus surfaces. We use a sewing technique, and have demonstrated that only BRST-invariant states propagate along the sewn lines. This allows us to write our higher-genus expressions in terms of the rational blocks of a compactified scalar field. Our construction ensures that when the surface is pinched, the amplitudes degenerate correctly, splitting into a sum over lower-genus amplitudes. However, we were unable to verify modular covariance of the blocks directly, because of our inability to manipulate the higher-genus contours. Nevertheless, on passing to a surface integral representation, we have checked that the partition function is indeed modular invariant when p' = 2. Our treatment is incomplete. As the number of screening charges associated with internal states increases, the resulting contours become quite complex and we have not found a general expression for them. Furthermore, even if we were able to find the general contours, we would be unable to evaluate the integrals explicitly. Indeed, even in the simplest examples on the toms the integrals appear intractable [4]. However, in these examples it is known that closed form expressions exist. It would

J. Bagger, M. Goulian / Coulomb-gas representation

507

be interesting to derive these expressions directly from the integral representations and find analogous expressions on an arbitrary Riemann surface. There have been a number of applications of the Coulomb-gas technique to conformal theories with larger chiral algebras. In refs. [19,20], for example, the representations of the W,, algebras are constructed. A number of groups have worked out the construction of zero- and one-loop correlators for SU(2) WZW theories and parafermionic theories [21]. There have even been discussions of defining conformal field theories directly in terms of the Coulomb gas [22]. It appears that our treatment can be extended to these models as well. We would like to thank G. Felder and H. Sonoda for helpful comments. After this work was completed, we received papers by Foda and by Frau, Lerda, McCarthy and Sciuto [23] in which similar results are discussed. This work was supported by the Alfred P. Sloan Foundation, and by the National Science Foundation, grants PHY87-14654 and PHY86-57291.

Appendix A In this appendix we briefly review the mathematical framework for higher-genus conformal blocks. The discussion is taken directly from ref. [24]. The holomorphic blocks in the partition function define a holomorphic section ~p of a vector bundle E, ® Wg over the genus-g moduli space. E C is a projective line bundle and Wg is a projective holomorphic vector bundle with dimension equal to the number of blocks. Choosing a projectively flat hermitian metric h on Wg, we obtain the partition function as a section of E~ ® E,. over moduli space:

One can locally choose a basis of holomorphic sections of Wg, ~,, so that h takes the form 0 , e f hab where h°h is a constant matrix and f is a locally defined function. The ambiguity in the choice of h is encoded in f. Since for the minimal models we work with the A series [18], we will choose a basis so that hOt, = 6~t, . In trying to describe partition functions, we are faced with two problems: Z is a section of a nontrivial line bundle and it does not exhibit holomorphic factorization (since f will not in general be a holomorphic function). If we have chosen a particular basis as above then Z will be of the form z = Ez°,

z° =

gZ

Thus, ratios of the form

zo/z,

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J. Bagger, M. Goulian / Coulomb-gas representation

will b e h o l o m o r p h i c s q u a r e s :

za/z~ - -

I'~J~,bl

2

W e c a n s i m i l a r l y c o n s i d e r ratios o f a n y c o n f o r m a l b l o c k s in t h e theory. In fact, s i n c e t h e p r o p e r t i e s of f a n d E Cd e p e n d o n l y o n t h e c e n t r a l charge, we c o u l d c h o o s e t h e b l o c k s to b e f r o m d i f f e r e n t t h e o r i e s w i t h the s a m e v a l u e of c, such as a m i n i m a l m o d e l a n d a scalar field t h e o r y w i t h a c t i o n (2.1). T h i s is t h e c o n t e x t in w h i c h w e will b e c o m p u t i n g

the b l o c k s a n d p a r t i t i o n

f u n c t i o n s o f the m i n i m a l m o d e l s . W e will n o t e x p l i c i t l y w r i t e ratios of blocks, b u t a l w a y s a s s u m e t h e y are i m p l i e d . A s a result, w e will d r o p f a c t o r s w h i c h will c a n c e l in t h e s e ratios.

References [1] D.B. Fairlie, unpublished: A. Chodos and C. Thorn, Nucl. Phys. B72 (1974) 509 [2] B.E. Fcigin and D.B. Fuchs, Funct. Anal. Appl. 13, N4 (1979) 91; 16, N2 (1982) 47; Representations of the Virasoro algebra m Representations of infinite-dimensional Lie groups and Lie algebras (Gordon and Breach, New York, to appear) [3] V. Dotsenko and V.A. Fateev, Nucl. Phys. B224 (1984) 312; B251 (1985) 691 [4] G. Felder, Nucl. Phys. B317 (1989) 215 [5] C. Thorn, Nucl. Phys. B248 (1984) 551 [6] J. Bagger, D. Nemeschansky and J.-B. Zuber, Phys. Lett. B216 (1988) 320 [7] T. Jayaraman and K.S. Narain, ICTP preprint IC/88/306 (1988) [8] O. Foda and B. Nienhuis, Utrecht preprint THU-88-34 (1988) [9] H. Sonoda, Nucl. Phys. B311 (i988) 401; B311 (1988) 417 [10] E. Verlinde and H. Verlinde, Nucl. Phys. B288 (1987) 357 [11] R. Dijkgraaf, E. Verlinde and H. Verlinde, in Nonperturbative quantum field theory, ed. G. 't Hooft, A. Jaffe, P. Mitter and R. Stora (Plenum, New York, 1988): Commun. Math. Phys. 115 (1988) 649 [12] J.D. Fay, Theta functions on Riemann surfaces, Lecture notes in mathematics, vol. 352 (Springer, Berlin, 1973) [13] A. Rocha-Caridi, in Vertex operators in mathematics and physics, ed. J. gepowsky, S. Mandelstam and I.M. Singer (Springer, Berlin, 1984) [14] P. Di Francesco, H. Saleur and J.-B. Zuber, Nucl. Phys. B300 (1988) 393 [15] L. Alvarez-Gaum~, G. Moore, C. Vafa, Commun. Math. Phys. 106 (1986) 1 [16] G. Felder and R. Silvotti, ETH preprint ETH-TH/89-38 (1988) [17] V. Dotsenko, Kyoto preprint RIMS-559 (1986) [18] A. Capelli, C. Itzykson and J.-B. Zuber, Nucl. Phys. B280 (1987) 445 [19] V. Fateev and A. Zamolodchikov, Nucl. Phys. B280 (1987) 644 [20] V. Fateev and S. Lykanov, Int. J. Mod. Phys. A3 (1988) 507 [21] A.B. Zamolodchikov, Montreal conference (1988); V. Dotsenko, Harvard seminar (1989); D. Nemeschansky, Phys. Lett. B224 (1989) 121; J. Distler and Z. Qiu, Cornell preprint (1989); D. Bernard and G. Felder, Saclay preprint SPhT/89/113: A. Gerasimov, A. Marshakov, A. Morozov, M. Olshanetsky and S. Shatashvili, Moscow preprint (1989) [22] M. Caselle and K.S. Narain, Nucl. Phys. B323 (1989) 673 [23] O. Foda, Nijmegen preprint (1989); M. Frau, A. Lerda, J,G. McCarthy and S. Sciuto, Phys. Lett. B228 (1989) 205 [24] D~ Friedan and S. Shenker, Nucl. Phys. B281 (1987) 509