Tadpole operator for minimal models on arbitrary genus Riemann surfaces

Volume 237, number 3,4

PHYSICS LETTERS B

22 March 1990

TADPOLE OPERATOR FOR M I N I M A L M O D E L S O N ARBITRARY G E N U S R I E M A N N SURFACES

G. CRISTOFANO, G. MAIELLA, R. MUSTO and F. NICODEMI Dipartimento di Scienze Fisiche, Universith di NapolL and 1NFN, Sezione di Napoli, 1-80125 Naples, Italy Received 13 December 1989

Techniques developed for the analysis of bosonic and fermionic strings are used to obtain a tadpole operator for the Coulomb gas representation of minimal models. Correlators and conformal blocks on Riemann surfaces of any genus are then constructed.

1. In the last few years the Coulomb gas approach has provided a general tool for computing correlation functions for 2D conformal field theories with extended algebras, the simplest example being the so-called minimal models [ 1-8 ]. The basic idea is to replace the irreducible highest-weight representations of a Virasoro algebra with a simpler Fock space of a free scalar field [ 5,6 ]. In this paper we will exploit well-known techniques developed for the analysis ofbosonic and fermionic strings (see e.g. refs. [ 9,10] and references therein) and the action of the BRST charges introduced by Felder [ 1 1 ] to construct a 3-reggeon vertex and a tadpole operator for the Coulomb gas representation of minimal models. Then we give a prescription for sewing tadpoles and conformal blocks on the sphere so to build N-point correlators on Riemann surfaces of any genus. As an example we write the expression of a genus-2 partition function, giving all possible 2-loop characters of the minimal models; they agree with some results already given in the literature [ 12-14 ]. To build correlation functions of primary fields on higher genus Riemann surfaces the formalism of the tadpole operator is particularly suited. In CFT the one-loop tadpole operator, TI, corresponds to the physical states of an arbitrary Fock space, say F,,,n, circulating in the loop, while the free leg carries a momentum a,,,,,~ allowed by the fusion rules, after the insertion of screening charges necessary to enforce momentum conservation; note that one does not sum over internal m o m e n t u m otn, n. The advantage of this approach is that it easily generalizes to g loops. We recall that the explicit expression of the vertices associated with highest weight states is

Vn',n(Z) = :exp[ --w/2an,,.q~(z)]: ,

(1)

where the field ~0has the mode expansion ~0(z) = q - a o l n z +

-an -z -n,

(2)

n~:O n

and the "charge" (or " m o m e n t u m " ) is given by ~,,,n = ½(1 - n ' ) ~ _ + ½(1 - n)o~+.

(3)

Here 1 <~n' ~ 1 ) on the hws Iotn..). However these representations 0370-2693/90/$ 03.50 © ElsevierScience Publishers B.V. (North-Holland)

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are "larger" than the highest weight reprcscntations of CFT. Therefore one has to restrict himself to a subspace of F , , , corresponding to the Verma module, modding-outthe singular vectors [ 7]. In string theory, the similar problem of taking into account only "physical" states was solved by introducing ghost degrees of freedom, leading to a tadpole expressed as a product of matter and ghost field contributions. In CFT the reduction will be accomplishcd along the lines discussed in ref. [ 1 1 ], leading to a tadpole operator given by the difference of two terms. Felder introduces BRST charges Q, satisfying

Q.:F.,..~F.,,_.-Fp.+..,p_.,

Qp_. : F~.,_~--+F~. n_ 2~,,

(4)

or, in more mathematical language, a sequence ofcohomological charges

Q¢2,)_Q,, Q¢2j+,)_Op_,,

(5)

which verify the properties:

(I) Q,.Q~,_,=O. (II) Space of physical states--- ~ , , = Ker Q¢O)/Ira Q~ - 1). (Ill) Ker Qek)/Im Q~k-1)=0 Vk¢0. T h e Qek) operate on Fock spaces F ek) as F~k) _o ~k~,F~k+ ~) where F{k) -- F~2j) = Fn,,n_2j p , F(2d+ i ) _ Fn,,_n_2jk

.

The above operators Q, are explicitly given by [ 11,12 ]

(exp(2nina2 ) - l ) l ~

Q " - \ exp(2nioF+)-1 J n J dvo

"~i' i V~,+(vo),=~

dvi

V,~+(v,),

(6)

V0

and commute with the L, since the final integrand is single-valued, while the v~are integrated along the contours defined in ref. [ 11 ]. The properties fo the Q allow one to express the trace over physical states for any operator ~Fas Tr...,.~=

~

( - - l y T r r ~ , ~ e~) ,

(7)

where the ~u~ are defined on F tj} recursively by ~O)Q o - ,) = Q o-,~ y - o - ~ with ~-~o) _= ~. Therefore the trace becomes an alternatingsum of Fock space traces. We are interested in extending "off-shell" this approach in order to apply it in the framework of the sewing procedure used to construct to higher loops. 2. Our starting point will be a 3-reggeon vertex operator for which we will use the notation r'lrl r'2r2 r'3r3 w{ I.;., I.~.21.~.3 I} ,

(8)

which yields thc three point function relative to the screened primary fields [ 11 ]

,,~.,~-j

Vm,,,(z)

dUb V~_(ua) 2

380

h=l

dr1 V,~+(Vt) , l=1

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once it is saturated with the "incoming" state l 13= ~ [ an~ni ), satisfying the fusion rules. Analytically this corresponds to the tree-level vertex operator VN;o,( N = r' + r + 3, r' = Zi r~, r = )',ri), for a field ~0compactified on a circle, as given for instance in ref. [ 15 ], saturated with r' "tachyonic" states of m o m e n t u m a _ and r states of m o m e n t u m a + at the positions of the screenings, and with the " m o m e n t a " of the three remaining legs fixed at ant, n, ( i = 1,2, 3 ). In the notation of ref. [ 10 ] one has (V.~,;ol = l-I ( o ~ , 0 , 1 6 i~l

t=l

a t - ° 2x/'~a °

19,v.o.,

(9)

where f%:o=exp

i~

(a°+2x/2Oto) ~ ~ O : i n [ V ~ ( z ) ] ~ = o

i=1

×exp

½ ~..

i#j=l

n~0

~..

n,m=O

n~

m ~ ~O~ln[Vi(z)-Vj(w)] . . . . o •

(10)

For definiteness sake we are going to sew leg 1 to leg 3, so we start from the vertex W ,f Io,o "

r',r

o,o

where the tree-level m o m e n t u m "conservation", enforced by the fi-function, fixes (r', r) to be r ' = ( m ' - 1 )/2, r = ( m - 1 ) / 2 . This is possible because one can show that if the fusion rules are verified, the relation hw,,=h v,_~.p_, and the fact that either p o r p ' is odd, allow one to choose ( m ' , m ) both odd. From this 3-reggeon vertex one builds the one-loop tadpole T In'.n) by employing the sewing technique (see, e.g. rcfs. [9, l 0,16 ] ), i.e. reversing one leg, which corresponds to

(n',nl--,In',n),

ao-,a~=2x/2ao-a~,,

a,,--,a_,,=at,,,

inserting a suitable propagator and summing over all possible intermediate states, i.e. taking the trace over Fn'.n, since each one-loop conformal block corresponds to a fixed m o m e n t u m flowing around the loop. Then one has to subtract the contribution of null states. This procedure in the end will reconstitute the O function associated with the Riemann surface, which in string theory was built by the sum over the discrete values of the loop m o m e n t u m . The cohomological properties of the BRST charges enable us to have a well defined method for modding out the "null states". To this end one has to determine the covariance properties of the 3-regge0n vertex trader a BRST transformation. We have obtained the action of the Q's on the 3-reggeon vertex starting from a formulalion of this operator in terms of Sciuto-Della Selva-Saito vertices (rcf. [17 ] ), and using the action of Q on these vertices constructed in ref. [ 12 ]. The result can be expressed by the following relation: W{[O,O

n',--n

r',r m',m

o,o

n',n

"~()(1)

J~¢..n

~"

W{ In',,, o,o . . . . .

m',--m

+r

0,0

n',n [ } Q ~ ) + 6 .

w{ o,o n',n r,,. m ' , m o.o n ' , - - n l iQ~3)

(11)

where the factors y and fi have been calculated explicitly and we have chosen a configuration appropriate to take care of the contribution of the first null state in the loop. However the central point is that, if one is interested in writing correlators among primary fields, the first term on the fight-hand side ofeq. ( 1 1 ) is irrelevant, since the BRST charge Q(2), attached to the free leg of the tadpole, will always end up acting either on another loop, in which, by construction, only "physical" states circulate, or on a highest weight state, and in both cases it gives zero. Therefore one can mod out the null states contribution independently in each loop. The only coefficient of interest is then ~, which turns out to be just a phase factor, i.e. fi=exp(-i2~z0) ,

0=nam,ma+ .

(12) 381

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As it could be expected this phase coincides with the one found by Felder when the free leg is on the mass shell. Subtraction of the contribution of the first null state from the trace over F~,n amounts then to subtract exp (2hi0) times the trace over F,,,,. Iteration o f this procedure yields the alternating sum over Fock spaces, eq. (7). Now, since F(2J)=F, ',~-2sp and a,,,,,,_z~p=ot,,,,,+jx/~, the sum over j e Z corresponds to the sum over all possible values o f the quantized loop m o m e n t u m in the case of a free scalar field coupled to a background charge [ 15 ]. Therefore this contribution is just given by the l-loop vertex VR;I for the scalar field case, with R = r' + r + 1, once the legs corresponding to the screenings operators are saturated with r' "tachyonic" states o f m o m e n t u m a _ , and r of m o m e n t u m a + while the free leg has m o m e n t u m a,,,,m. Explicitly, following the notation o f ref. [ 10 ] we have

l~(Otg'_m',p-m,O~l

•

n=l

h(a_,O.I

z(a+,O~ld

h=l

R. . . . . o x / ~n a~i X 12,v.o'exp ( ½i,jY, •

a°

I=1

v/-~a~ m~

i

078'~ lnE[Vi(z)' V,(w) ] z - w - o )

,

V,(z)-

YAw)

_

_

v,(~) X ~ exp j~Z

Xexp

(

(

izE(jx/~+x/~ot~,,~-2x/~ao)z(j.v/N+x/~et~,,~)+i(j.v/N+x/r2a~.,.) f

~ ~

i= 1 n=O

f o z:o)

zO

i2x/~

~ ~ - - . ) 071ntriV,(z)],=o "=

n=0

)

,

(13)

where I?'n.;ois given in eq. (10), N = 2p'p and the variables relative to the screenings should be integrated over. The sum over F (2j+t) can be carried out in the same way and yields an analogous contribution, except that one has to make the substitution a,. , - , a , , . _ , . The'one-loop tadpole operator T~ "''") (fig. 1 ) is then given by the difference between these two contributions with a relative phase factor exp (i2zr0). To obtain now the contribution of a conformal block (n', n) to a l-loop M-point correlator o f primary fields, one has to attach to TI "',") a vertex operator V,v-;o,including the required number of screening vertices, and saturate its legs with tachyonic states o f suitable momentum. The sum over F (2s) leads to the tachyonic l-loop amplitude:

AM+R;,(Z,,...,Z~+g)=(~

ai \

X 0[%,

i=l

( 1 - k " ) - ' I - I [E(zi, zJ)] 2'~"~ 17 [ a ( z , ) ] -4~°~' n=[

I
i=l

~'m. ,~, x/r2ot,iog+2ao(A+½) ,N0.

(14)

z0

Here 2,,~ = x/~(ot#,,, -(Xo) = n'p-np' and the z~ indicate the positions of conformal vertices for i = 1.... , M, and of the screening vertices for i=M+ 1, ..., M+R, R being the total number o f screenings. Introducing the torus variables w~= ( 1/2rd )In zi, specializing a, A and 09 to the toms and taking into account the conformal weights of the fields, the alternating sum over F (2j) and F (2s+ I ) gives as final answer for the general 1-100p M-point corrclator:

( V,~,(wt )...V,~u(wM) )g=l = ~%;1 (Wl, ..., wM; ,;t~.~) - e x p ( i 2 r c 0 ) ~ ; l (wl, ..., wM; 2~,,_,) ,

.

(15)

(m' ,n0

Fig. I. Symbolic representation of the l-loop tadpole operator. The screening operators are not shown. 382

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where O=

PHYSICS LETTERS B

22 March 1990

no6,,,,a+,and q,./12

'~+~ 2 " [ ~9, ( w/: 1T) 72'~x~

×,Lexp[inNz(m+~)2+2nix/~(m+~)~yv/2aiw,]

(16,

Here w~}= w~- ~), q= exp(iz~r), and we have indicated explicitly the integration variablcs. For the screenings associated with the tadpole the integration contour is a linear combination of a contour around the annulus of the loop, i.e. along an a-cycle, with coefficient one and a contour around the neck of the tadpole, while for those associated with external legs the contour starts and ends at the position of the corresponding puncture going around the tadpole's neck (see refs. [ 11,18,19] ) *t. If one takes into account the difference in configuration, this expression coincides with the one obtained in refs. [ 11,20]. In particular, if one sets all the a~= 0 one recovers the well known characters for the conformal blocks [ 21,4 ]. We notice that the properties of these characters under modular transformation depend crucially on the rational characteristics ).,,,, of the O-functions and on the subtraction procedure which leads to the alternating sum. 3. To evaluate correlators on higher gcnus surfaces it is quite simple in this formalism. In fact one can construct a g-loop tadpole 7~ by sewing g l-loop tadpoles to ( g - 1 ) 3-vertex operators, obeying the fusion rules. This allows to specify the configuration of primary fields propagating in each loop and on the lines connecting them, as indicated pictorially in fig. 2. The correct g-loop m o m e n t u m conservation, i.e. ~ a ~ = - ( g - 1 )2~Xo, will automatically be taken care of because of the relation 5"3=, o4 = 2O~oat each 3-vertex and of the scalar product rule = 6 ( o q +o¢2-2o~o) .

(17)

To obtain an M-point corretator < V~I(z,)... V~M(ZM)>g or a partition function at g loops, one has just to sew the free leg of the g-loop tadpole to a vertex operator saturated with "tachyonic" states of momentum ~xi at the positions z, of the primary fields, and of momentum a + at the positions of the screening operators which then should be integrated over. However, any time one has to contract a bra with a ket carrying the same momentum one has to insert a suitably screened "identity" operator to map the momentum into its dual (to have a nonvanishing scalar product). ~' We thank S. Sciuto for an enlightening discussion on this point.

(%,%)

,~-"~x

(m',m) I (,~. £i)

I

......

Fig. 2. Symbolic representation ot~the g-loop tadpole.

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As an example we are going to construct the contribution to a 2-loop partition function of two conformal blocks (n~, n~ ) and (n~, n2) when they exchange the states belonging to Fin',m- One has to notice that it is not possible, in general, simply to join the tadpole corresponding to (n'~, n~) to the conjugate of the tadpole corresponding to (n~, n2). Indeed to join the two blocks one has to insert the 3-vertex

WSt io.o Im',m i<,# Ip'--

l,p-- 1

i~o,. l}

(18)

containing a screened identity, with r' = p ' + m ' - 1 and r=p+ m - 1 , which transforms states of momentum am,,,, into states of momentum a v. -m,,v-,., as shown in fig. 3. It is easy to check that the total momentum flowing into the correlator is - 2ao, as it should. Therefore this contribution to the 2-loop "partition function" or conformal block is given by the 2-loop tachyonic amplitude of ( p ' - 2 ) state l a _ ) at v~, of ( p - 2 ) states lot+) at u~ and one "identify" state 1%, - ~,v- ~= 2 % ) at z. It can then be written as X(2)[n'l,nl " l _

£ p-2

p'--2

p--2

p'--2

Ln"rt2") -- ~ i~l~l d~i i=H' dr, i=l~Ii.:E(Z, Ui) 4Of+Ct'OimHl E(Z, V,) 4~-Ot0 p--2

p'--2

p-2p'--2

× I-[ E(u,, u:) ~"~+ H E(v,, v,) ~'~- I-I Fl E(u,, ~j)-~ iv~j=l

i~j=l

xa(z) - ~ I-I o ( u i ) - ~ + i=1

×

~/~-2ai

i=l j=l

I-[ a(v~) -~°~+0 ~;" /N z.,~. /N

i=1

~ou + 2~/2O
,

(19)

zo after integration over the positions of the screenings. The conformal properties of the identity operator, i.e. h2,,0 = 0, insure that this expression does not depend on z. Then the character associated to the two-loop conformal block will be Z(2)

f n'hnI "~ - tn~,n2J--

~

[Z(2){~',~n~2}--exp(2~ziO,)z(2){~"-m~,,2}--exp(27t102)Z' (2){n,2_n2 Jn'l,n,~

(m',m)

+exp[2ni(01 .a_t~ ~ v 2 1 J~A1~(2)r,'~,--. "(n~,--n2 J~] , where Oi=n,c~+a,,,,m. We should stress that in this approach all the phases are the same as at the l-loop level, being only a consequence of the BRST mechanism for modding out null states separately in each loop. The difference in the phase factors with respect to other authors [ 12,13,19 ] may be ascribed to the different choice of integration contours, which are forced by the configuration chosen in building the correlators. Analogous results have also been derived by quite different approaches [ 14,22 ]. The operatorial method presented in this paper to evaluate correlators and "partition functions" or conformal blocks at any loop order for minimal models can also be applied to other CFT's described by a coset space representation, e.g. N = 1 and N = 2 models, which enjoy a "Coulomb gas" like description.

p'-l,p-1 (m' , m) -/%, ->.

(m' ,m) Fig. 3. Symbolic representation of a 2-loop conformal block requiring the insertion of a screened identity.

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While this work was being completed we became aware of ref. [ 23 ] in which similar results are obtained.

Acknowledgement It is a pleasure to thank P. Di Vecchia and S. Sciuto for many interesting and clarifying discussions, and R. Pettorino for collaboration in the early stages of this work.

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