Theoret ical and M a t h e m a t i c a l Phys ics , Vol. 104, No. 1, 1995

O N T H E A L G E B R A I C S T R U C T U R E O F T H E H O L O M O R P H I C

A N O M A L Y F O R 5 = 3 T O P O L O G I C A L S T R I N G S

E. Ldpez

1. I n t r o d u c t i o n . We begin this lecture with an introduction to topological field theories (TFT) and topo- logical strings, which will be necessary for the presentation of our results.

A TFT can be defined associated with an N = 2 supersymmetric theory. Let us consider N = 2 supercon- formal FT's. In the NS sector there are fields, called chiral (antichiral) fields, satisfying

+ . G+ ,h i _ _ G _ l / 2 ¢, = _1/2,e = 0 (G-1/2#9i = G-1/2dpi = 0 ) . (1)

We can restrict ourselves to the set of fields, finite in a unitary N = 2 SCFT, which are both chiral and primary. The OPE among them is, up to regular terms, closed and independent of their positions

¢~ ej k = c ~ j e k (2)

defining a ring structure, the chiral (antichiral) ring [1]. When acting in the Ramond sector, the states gen- erated by the chiral (antichiral) primary fields are in one-to-one correspondence with the Ramond vacua, ¢ , I o > R = li>n.

All this suggests that it is possible to build a TFT out of the N = 2 theory, by choosing O + ( G - ) as the BRST-current and whose physicM states are precisely the Ramond vacua. Indeed, this can be done by a process called twisting [2, 3], which consists in coupling the U(1)-current of the N = 2 superconformal algebra to 1/2 ( - 1 / 2 ) of the spin connection. We fix for the following the A-twist, in which G + and G+ will become BRST- currents. The energy-momentum tensor is then modified to

1 T (z) = T(z) + (3)

and the same for the right sector. The new T t is Q+-exact, as is necessary for defining a topologically invariant theory. From (3), we see that the spin s of any field is shifted to s - q /2 , where q is its U(1) charge. Therefore, in the twisted theory Q+ has spin i instead of 3/2, allowing one to define a BRST-current, and the chiral ring fields turn out to have conformal dimension zero in accordance with topological invaxiance. It is important to notice that the twist induces an anomaly in the U(1)-current

A J = A J = ~:(1 - g ) , c a = ~, (4)

c being the central extension of the untwisted theory. We can consider now the most general (untwisted) lagrangian

,c = Lo + f + E. (5)

where ¢!2)(z) = {Q-, [Q-, ¢,(z)]}

t q;!2)(z) = {Q+, [Q+, q;i(z)]). (6)

Instituto de Matem£ticas y F~sica Fundamental, Serrano 123, 28006 Madrid, Spain. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 43-54, July, 1995.

0040-5779/95/1041-0793512.50 ©1996 Plenum Publishing Corporation 793

The coupling constants (ti, ti) parametrize the space of N = 2 theories and the fields ~(2) 3!2) ,-i , ~, can be inter- preted as its tangent vectors. Therefore, the metric in this space, i.e., the Zamolodchikov metric Gi3 [4], will be given by

Gi. ~ ----(¢12)(1)(~2)(0))sphere . (7)

Let us reduce to marginal deformations, which maintain conformal invariance. They are characterized by h = ft = 1//2 and q -- ~ = =kl. For the case ~ = 3, which will be the case of interest in this lecture, the moduli space of N = 2 SCFT verifies very strong constraints, called a special geometry [5]:

(i) the metric G/j is restricted K£hler, namely, there exists a line bundle / : such that its first Chern class is 2rr times the K/ihler form;

(ii) there exists a holomorphic, integrable and completely symmetric tensor Cijk E t::2 in such a way that the curvature for the connection associated with G~3 satisfies

e2K~k-C = = + - 3 ( 8 )

where K is the Kghler potential and C3 = (Cj)*. Condition (ii) is a particular case of a more general structure, called tt-equations [6]. The t[-equations are

equivalent to the existence of an improved connection in the space of N = 2 theories with fiber the Ramond vacua, which is flat [7]. In this sense, these equations are always solvable. Let us remark that only in the case

= 3 do all the indices in (8) belong to marginal fields. The case ~ = 3 is interesting because it includes a- models in Calabi-Yau threefolds, which are the internal spaces for compactifying the superstring down to 4 dimensions. In this sense, the special geometry of the moduli space o f N = 2 ((2,2)) SCFT can give information about the low-energy effective lagrangian derived from superstring models [8].

We finish here the introduction to TFT, and continue with a brief summary about topological string models. We will begin with the simplest case, i.e., topological gravity.

Topological gravity is the T F T associated with the moduti space of Riemann surfaces, Adg,~. Its aim is to obtain a topological field representation of the Mumford-Mori ta cohomology classes [9]. An important difference from the TFT's presented above is the presence of local symmetries, namely diffeomorphisms and local WeyI transformations. The set of physical observables is again given by the cohomology of a certain BRST charge, but now this cohomology should be defined on gauge-invariant objects. This allows Q-exact operators ¢ = {Q, A} to become nontrivial when A is Q-close but not gauge invariant. All the observables in topological gravity, the gravitational descendents cry, n > 0, are of this kind (except the puncture operator P = or0).

Owing to this, the amplitudes (a,~ ...a,~)g give rise to recursion relations among the amplitudes at genus g and g' < g. These relations will allow one to exactly solve the theory. Indeed, the recursion relations are equivalent to the KdV equations, showing that topological gravity coincides with one-matrix models [9, 10].

The same result can be derived in terms of a contact term algebra between the gravitational fields [11]. The advantage of this formulation is that the precise form of the recursion relations is completely determined fromthe consistency conditions of the contact term algebra, namely, the requirement that the value of the correlators is independent of the order in which the fields are integrated over the Riemann surface E. This fact will be strongly used in what follows.

Let us recall the functional integral representation for topological gravity correlators [11]

where X~ )(~ are Bettrami differentials and G, G are the superpartners of the energy-momentum tensor. The 1-139-~ G(xa)G()~ ) comes from the integration over the superpartners of the moduli parameters. ~ This factor l l a , 8 : 1

can be done independently of the moduli because in topological string models the supersymmetry is rigid and

1The integration over the moduli space A4a.~ includes the necessary factors to project out the zero modes of the b, b ghosts and their superpartners fl, ft.

794

therefore the supermoduli split. The external insertions are represented in the functional integral by 2-forms o .(2) , given by

~BRSTO'(n 1} = d o . n , (~BRSTo. (2) = do" (1) ( 1 0 )

of ghost number n - 1 and which are analogous to expressions (6). From (9), we can derive the following selection rule for nonvanishing correlators:

s

~ ] ] ( n i - 1) = 39 - 3. (11) i----1

Topological strings consist in the coupling of topological matter systems to topological gravity [9]. The physical observables are

'-- n _> 0, (1_9)

where ~bi runs over the chiral primary fields. The correlators admit the same functional integral representation (9). The only new ingredient is the anomaly of the U(1) current of the matter sector (4), which modifies the section rule to $

~ ( q i + n i -- 1) = 3 9 - 3 + ~(1 - g) (13) i = 1

for an N = 2 TFT, and where qi is the U(1) charge of the corresponding operator. In the case ~ = 3, (9) reduces to the special rule

3

E(qi +n.i- 1 ) = 0 . (14) i----1

Let us notice, therefore, that we can have at any genus nonvanishing correlators of any number of marginal fields (q = 1, n = 0). Subsequently, we will restrict ourselves to fi = 3 topological strings.

The complete set of marginal perturbations is generated by both chiral (or topoIogical) fields ~bl 2) and an-

tichiral (or antitopological) fields ~ ) (see (5), (6)). After twisting, q;~2) becomes BRST exact, and therefore naively would be expected to decouple from any string correlator. In [12], an anomaly was found in the de- coupling of these operators, i.e., the holomorphic anomaly. The holomorphic anomaly is a recursive equation relating correlators at genus 9 and f < 9. It can be integrated in terms of a set of Feynman rules and allows one to solve any correlator, except for a holomorphic part, which in many cases is very constrained by modular invariance and boundary condition arguments. A surprising fact is the appearance in these Feynman rules of a new field with the defining properties of the dilaton field.

In this lecture, we propose to understand the holomorphic anomaly in terms of a contact term algebra that fuses the topological and antitopological sectors, and in which the diIaton is included in a natural way.

The t~-amplitudes that originate this anomaly are given by

s 3g--3

n g , s z i=1 z i a = l

C-(xo)d-(2o) ). (15)

The string interpretation of this amplitude as a measure on the corresponding moduli space .M g,n + 1 [13] is by no means direct. It is precisely in this point where the holomorphic anomaly differs from the standard BRST- anomaly of the bosonic string. In fact, the integration of the moduli parameter associated with the insertion of the antitopological field is performed using the SUSY current G +, instead of G - , which in these theories plays a role analogous to the b-ghost (see (15)). Searching for a standard string interpretation of the t [ amplitude (15), we can formally try to represent it as follows:

C-G- ¢,(z,) I-[ (161 g , s + l z = z i a = l ~ g , s + l

795

where we have introduced new operators q~7 with positive ghost number equal tp one [14]. In order to make ex- plicit the ghost number counting, we interpret the operators ¢; as having implicitly a gravitational descendent index n, such that

gh($;) = n + ~ , (17)

with q; its U(1) charge. According to this heuristic argument, the operators q$~ should be associated with n = 2. The philosophy underlying (16) is as follows. First, we interpret the antitopological insertion as representing

a pure BRST but nontrivial object, in analogy with what happens for gravitational descendents. Then we integrate over the position of the corresponding puncture in the standard topological way, namely by using G- as the b-ghost.

Another reason supporting (16) comes from the subleading divergences in the operator product

3(z) ¢,(0) = (zs) Izl2'

where G G is the Zarnolodchikov metric (7). These subleading singularities [15], which depend linearly on the curvature of the surface, determine the "dilaton" contribution to the holomorphic anomaly [12]. The formal

^

operators ¢~, based on the ghost number counting (17), allow a natural translation of them into the definition of the contact term

iD¢7(z)l¢i) = 1<~), (19) G,j

with O" 1 representing the dilaton field. The dynamics of the dilaton field will be later determined by its contact terms with the rest of the fields. The domain of integration D in (19) is an infinitesimal neighborhood of the point where the field ~, is inserted.

The previous arguments should be interpreted only as providing some heuristic support for the rule defined in Eq. (16). The testing of this idea will consist in finding a contact term algebra from which to derive, as consistency conditions, all the dynamics of the t-insertions, namely, the holomorphic anomaly and, as a par- ticular case of it, the special geometry of the moduli space of N = 2, ~ = 3 SCFT. We believe these ideas can be extended to the moduli space of N = 2, theories for any ~.

2. t t -Equa t ions as c o n s i s t e n c y cond i t ions o f a con tac t t e r m a lgebra . Let us consider the algebra of operators generated by ¢i, ¢7 and the dilaton field al with i = 1, .., n for n the number of marginal deforma- tions. We define the following contact term algebra [14]:

- - - -

(20)

In order to take into account the contribution of the curvature and the twist, we introduce the operator

(21)

for ~ = ~o + 2zr, where • is the operator that bosonizes the/./(1) current of the N = 2 SCFT and zr is the conjugate of the Liouville field. The contact term algebra for this operator is defined as follows:

~ ¢i[e½'N~)l = Ai[e½~(z)>, ~ ¢ T l e ½ ~ ( z ) ) = O, ~ al,e½~(~)) = a,e½'~(~)). (22)

796

The undetermined constants appearing in (20) and (22) will be now fixed by imposing consistency conditions

/Da/Db,C) = /Db/Da,C ) (23)

for three arbitrary operators. 2 To solve the consistency conditions we will assume that: (i) Gi, 3 is invertible; (ii) the value of a equals 1 (this condition is based on the way the dilaton field measures the curvature); (iii) the derivation rules

¢ ~ r ~ ( ~ , O = 0~r%z(t ,~ , ~ r % ~ ( t , ~ = (-1)F(r%~ ~ @ r ~ ( t , O , F(F%z) = q~ - q~ - q~, (24)

are satisfied, where F ~ stands for a generic contact term tensor, and q~ for the U(1) charge associated with the corresponding field; these rules define the way in which the operators act on the coefficients appearing in the contact term algebra. Notice that in general these coefficients witl depend on the moduli parameters (t, t-). The logic for this rule is the equivalence between the insertion of a marginal field and the derivation with respect to the corresponding moduli parameter. For this reason we will not associate any derivative with the dilaton field. The derivation rule (24) is forced by the topological interpretation of the { insertions we are using. Once we decide to work with the operators ¢~ and to define the measure using only G - insertions, we must accommodate to this picture the coupling of the spin connection to the U(1) current. Since the derivation c%//corresponds to the insertion of an antitopological field, we need to change, in the neighborhood of the insertion, the sign of the coupling of the U(1) current to the background gauge field defined by the spin connection. This fact gives rise to the factor ( -1 ) F(r) in (24).

Using (i), (ii), and (iii), let us start by analyzing the following consistency condition:

Applying the contact term algebra (20), we get

b r~j ICk> - r~j I¢~ > -- - 2 r~j ICk > (26)

which, for a nonvanishing ri~. , implies that b = - I . (27)

From the condition

together with Eq. (27) and the derivation rules (24), we obtain

0i~t~l> - e l¢d = l¢i>,

which is solved by e = - l .

To continue the study, we take the condition

which leads to

(29)

(30)

(31)

2 Notice that there are two contributions to each side of (23): the successive contact terms of the operators a and b with the bracketed c, and the contact term between a and b first, then carried over lc).

797

Using that Gij is invertible, and for a general number of marginal deformations, we get from the above equation

d = 0. (33)

Moreover, the consistency condition

and Eq. (33) imply that C--~0.

From (27), (30), (33) and the consistency condition

we get easily

(34)

(35)

(36)

Condition (43), together with fD 41 fD 4~14£) term for antitopological operators.

To conclude the study of the consistency conditions, we will consider now the relation

3The symmetry of Ftj will assure that fD ~i fo Cjl¢i> = fD ¢d fD ¢il¢~> is satisfied.

798

0 i F ~ = 0 . (43)

= fD ¢} fD q$714~>, allows us to impose a vanishing contact

(44)

(42)

we obtain that I ' ~ is the only function of the antitopological variables

Using now

Gi3 = O. (37)

The next conditions that we will analyze involve the curvature operator e½ @(*)

from which we get, assuming that F/kj is symmetric in the lower indices a

G,~ = @Ai, OiAj = OjAi. (39)

Equations (39) imply that the metric Gi), is Kghler for a certain potential K(t, t~

Gi~ = OiO~tt'. (40)

With this information, we can return to (32) and deduce that the tensor f'~. is symmetric in the lower indices: U

-k ~ . (41) FT~= ~,.

Using Eqs. (27) and (37), we obtain

D¢i J,~ ¢.~1¢k> = 1¢~> - ISi> - I¢~c> + fact P

O~Gk3 Gky Gi3 terms,

/o/o ¢3 ¢i1¢k) = - ~ r ~ k let) + rikGt3l¢t). (45)

In order to justify the inclusion of factorization terms in (45), let us notice two facts. The necessity of in- cluding factorization terms at the level of consistency conditions (23) is already present in the simplest case of contact term algebra, i.e., in pure gravity. The reason for that is an asymmetry in the contact term coefficients [11]. Second, the heuristic argument (17) seems to indicate a hidden gravitational descendent index in the op- erators ¢;. Therefore, and due to the nonvanishing correlation function Cijk at genus zero for three marginal

fields, we should consider the possible existence of factorization terms associated with the 6j insertions. We can write them generically as follows:

fact terms = B/-- n Cikn t ¢ t ) " (46) J

From Eqs. (44)-(46), we obtain that the coefficient P~j is the connection for the metric G~3, which we already know is Kghler

k = (OiGfi)ark (47)

and a (t, t-) type equation OaFkij = Gi~,5~ + Gja6ki _ BamkCijm. (48)

The tensor B/~ can be derived from the contact term algebra by the following argument. Let us consider 2

the consistency condition on a general string amplitude

f l s (6;6; ¢t}9 = (6~6; I ] Cz)g; (49) / = 1 l = l

from (20) we get

fI - ,, )(6~ ¢~ )~ + (50) 1=1 1=1 n o d e s

where ~¢~.Dl denotes the commutator of the contact terms of q~; and 63 with Ct, and Rzx the commutator of

those at the nodes. Using now the symmetry of I'~3 in the lower indices (41), we can conclude

s = = s = 0. (51)

l = 1 n o d e s

The contribution at a node associated with the factorization of the surface will be defined by the tensor Bg ~ as follows: 3

[ II = E oE_ ,,co; <s, II II lES r = 0 X --S lEX h E Y

IEX h E Y

(52)

where S refers to the set of all punctures, X and Y is a partition of it, and the tensor B can be chosen to be symmetric in the upper indices. Using now (51) we get

B-~God = Bff~G~,;,, &iB] f3 = ~ B ~ ~. (53)

799

By an analogous argument, we find from condition (44) and for a general string amplitude

B~#F ~ _ OiB] # + B~'rr~3 "Y + 3 i.v 20'KB~#3 = 0. (54)

Let us define B2 # = B3~#e2KGe'°~G~#. Then Eqs. (53) and (54) imply that BUS is proportional to the three- 3

point correlation function for the antitopological fields. Substituting this information into Eq. (48), we obtain the (t, {)-equation

o~r~j = a , , ~ + a j ~ - O;'~c~j,,. (55) Notice that in order to get the special geometry relation (55) from the contact term algebra, it was necessary to ma~e use of the derivation rule (24). From (55) we can conclude that the metric G G is the Zamolodehikov metric for the marginal deformations. With this result we finish the derivation of the (t, t-) equations as consis- tency conditions for the contact term algebra (20). Our next objective will be the derivation of the holomorpbSc anomaly.

3. The h o l o m o r p h i c a n o m a l y . We will derive the holomorphic anomaly for correlators at any genus 9 using the contact term algebra introduced in (20), (22). Let us recall the expression of the t{-amplitudes with the help of the formal operators ¢3 :

s 3g--3

O{C~i...i = L < /C G - G - ~ { H / c G-G-¢i H g,s+l z i----1 zi a,~-----t

i6S

a-(xo)O-(2~) }~ ,,+,

(56)

where S denotes the set of all punctures, and we have introduced the last equality to simplify the notation. The contributions to (56) can be written as

= + E i6S i6S nodes

(57)

where RDi is the contact term of q~ with the ¢i insertion, and RA is the contact term contribution that fac- torizes the surface through a node. Let us start by analyzing the R D i boundaries:

i6S {6S j6S i6S j~i i6S j~i

(5s)

The internal nodes A are associated with the two types of boundaries of a Riemann surface of genus g and s punctures. The first one (we will denote it as A 1), comes from pinching a handle, leading to a surface of genus g - l :

1 B ~ # ( ¢ ~ ¢ # H ¢i )g-1 (59) (3 v i e ~ i6S i6S

where the factor ½ should be added to reflect the equivalency between the order in which the two new insertions 4~ are integrated. The factorization tensor B ' satisfies the same set of Eqs. (53) and (54) as the tensor B; thus, it is also proportional to the three-point correlation function. With an appropriate choice of normalization of the string amplitudes, the proportionality constant between both factorization tensors can be set equal to one [16].

The second ones, denoted A2, come from the factorization of the surface into two surfaces of genus r and punctures in the subset X, and genus 9 - r and punctures in Y respectively:

1 g ( II = E E (,o II II (60)

i6S r----O XUY----S j6X k6Y

800

Collecting now Eqs. (58)-(60), we obtain the equation for the t-dependence of any string amplitude:

1 9

i E S i E S r = 0 X u Y = S j E X k E Y

+ 2g- + i)< H CJ (61) iES j ¢ i

Notice that in our derivation of the holomorphic anom^aly from the contact term algebra we have only consid- ered the contact terms of the antitopological operator ¢{ with the rest of the operators ¢i, but not the contact terms among the operators ¢i themselves. This is equivalent to defining the correlators (I-I ¢i) by covariant derivatives of the generating functional. There are, however, some aspects of the previous derivation that should be stressed at this point.

1. The correlators <I] ¢i> for topological operators cannot be determined by the contact term algebra, in contrast to what happens in topological gravity. In fact, from the contact term algebra we can only get relations of the type

(rL - rL )< H > = E + E (62) lES IES nodes

which does not imply (F~j - F~i = 0) anything about the surface contribution. Moreover, they are compatible with making all contact terms RDI equal to zero by covariantization.

2. If, in the computation of < q~E Hie s ¢i >, we take into account all contact terms, i.e., contact terms between the ¢i operators, we will find, as a consequence of the derivation rules (24) and the (t, t-) equations (55), that the holomorphic anomaly is cancelled, reflecting the commutativity of ordinary derivatives [~, Oj] --- O.

3. We should say that from the contact term algebra we cannot prove, at least directly, that the correlators ( ¢~ l I i e s ¢i > are saturated by contact terms. The fact that we have proved is that the contact term contribu- tion dictated by the contact term algebra (20), (22) is precisely the holomorphic anomaly.

4. The curvature of the initial surface is augmented by two units in both processes of pinching a handle or factorizing the surface. In order to take this into account, the two insertions ¢~, ¢~ generated in these processes should include, in addition, an extra unit of curvature. Therefore, the total balance of curvature for the new insertions is zero. This can be seen as the reason for the zero contact term between the dilaton field al and the antitopological operators q~i (see Eqs. (33) and (35)).

4. F ina l c o m m e n t s . In this section, we will collect some concrete questions which we believe would be worth considering in more detail.

i. A direct derivation, using cancel propagator arguments [11], of the tt-contact term algebra (20). ii. To extend the holomorphic anomaly for correlators involving gravitational descendents and to massive

topological field theories. iii. The antitopological operators generate changes in the string background. Because of this, it is relevant

to determine whether it is possible to give an explicit expression for the operators ¢~. iv. The N = 2 couple (Q+, Q - ) is used to define string measures in (16). The main difference with respect

to the bosonic string is that G - , which plays the role of b-ghost, has nontrivial cohomology. It would be inter- esting to study in a more systematic way the properties of strings defined for a pair (Q, b) satisfying the Hedge relations, i.e., strings with nontrivial b-cohomology.

Acknowledgments

The authoI would like to thank the organizers of the Fifth International Conference on Mathematical Physics, String Theory, and Quantum Gravity, Alushta, June 10-20, 1994 for giving me the opportunity to present this work and for a very pleasant and interesting stay. This work was supported by M.E.C. fellowship AP9134090983.

801

R e f e r e n c e s

1. W. Lerche, C. Vafa, and N. Warner, Nucl. Phys., B324, 427 (1989). 2. E. Witten, Commun. Math. Phys., 118,411 (1988). 3. T. Eguchi and S. K. Yang, Mod. Phys. Left., A5, 1693 (1990). 4. A.B. Zamolodchikov, JETP Left., 43, 730 (1986). 5. B. de Witt and A. van Proeyen, Nucl. Phys., B245, 89 (1984); A. Strominger, Commun. Math. Phys.,

133, 163 (1990); L. Castellarfi, R. D'Auria, and S. Ferrara, Phys. Left., B241, 57 (1990); Class. Quant. Gray., 1,317 (1990); R. D'Auria, S. Ferrara, and P. Fr~, NucI. Phys., B359, 705 (1991).

6. S. Cecotti and C. Vafa, Nucl. Phys., B367, 359 (1991). 7. B. Dubrovin, Commun. Math. Phys., 152,539 (1993). 8. J. Dixon, V. S. Kaplunovsky and J. Louis, Nucl. Phys., B329, 27 (1990). 9. E. Witten, Nucl. Phys., B340, 281 (1990).

10. R. Dijkgra~and E. Witten, Nucl. Phys., B342, 486 (1990). 11. H. Verlinde and E. Verlinde, Nucl. Phys., B348, 457 (1991). 12. M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, "Holomorphic anomaly in topologicM field theories,"

Preprint HUTP93/A008; "Kodaira-Spencer theory of gravity, and exact results for quantum string am- plitudes," Preprint HUTP93/A025.

13. L. Alvarez-Gaume, C. Gomez, G. Moore, and C. Vafa, Nucl. Phys., B303, 455 (1988). 14. C. GSmez and E. Ldpez, Phys. Left., B334, 323 (1994). 15. D. Kutasov, Phys. Left., B220, 153 (1989). 16. K. Li, Nucl. Phys., B354, 711 (1991); Nucl. Phys., B354, 725 (1991).

802