Structure of Higher Genus Gromov–Witten Invariants of Quintic 3-folds
There is a set of remarkable physical predictions for the structure of BCOV’s higher genus B-model of mirror quintic 3-folds which can be viewed as conjectures for the Gromov–Witten theory of quintic 3-folds. They are (i) Yamaguchi–Yau’s finite generation, (ii) the holomorphic anomaly equation, (iii) the orbifold regularity and (iv) the conifold gap condition. Moreover, these properties are expected to be universal properties for all the Calabi–Yau 3-folds. This article is devoted to proving first three conjectures.
The main geometric input to our proof is a log GLSM moduli space and the comparison formula between its reduced virtual cycle (reproducing Gromov–Witten invariants of quintic 3-folds) and its nonreduced virtual cycle [CJR18P]. Our starting point is a Combinatorial Structural Theorem expressing the Gromov–Witten cohomological field theory as an action of a generalized -matrix in the sense of Givental. An -matrix computation implies a graded finite generation property. Our graded finite generation implies Yamaguchi–Yau’s (nongraded) finite generation, as well as the orbifold regularity. By differentiating the Combinatorial Structural Theorem carefully, we derive the holomorphic anomaly equations. Our technique is purely A-model theoretic and does not assume any knowledge of B-model. Finally, above structural theorems hold for a family of theories (the extended quintic family) including the theory of quintic as a special case.
The computation of the Gromov–Witten (GW) theory of compact Calabi–Yau 3-folds is a central and yet difficult problem in geometry and physics. During last twenty years, it has attracted a lot of attention from both physicists and mathematicians. In the early 90s, the physicist Candelas and his collaborators [CdOGP91] surprised the mathematical community by using mirror symmetry to derive a conjectural formula for a certain generating function (the -function) of genus zero Gromov–Witten invariants of a quintic 3-fold in terms of the period integral (or the -function) of its B-model mirror. The effort to prove the formula has directly lead to the birth of mirror symmetry as a mathematical subject. In a seminal work [BCOV93] in 1993, Bershadsky, Cecotti, Ooguri and Vafa (BCOV) introduced the higher genus B-model in physics in an effort to push mirror symmetry to higher genus. During the subsequent years, a series of conjectural formulae was proposed by physicists based on the BCOV B-model. Let be the generating function of genus GW-invariants of a quintic 3-fold. BCOV had already proposed a conjectural formula for and in their original paper (see also [YaYa04]). A conjectural formula for was proposed by Katz–Klemm–Vafa in 1999 [KKV99]. Afterwards, it became clear that we need a better understanding of the structure of . A fundamental physical prediction of Yamaguchi–Yau is that can be written as a polynomial of five generators constructed explicitly from the period integral of its mirror. Among the five generators, four of them are the holomorphic limit of certain non-holomorphic objects in the B-model. The famous holomorphic anomaly equation of BCOV can be recasted into equations determining the dependence on four of the generators. Abusing terminology, we still refer to them as holomorphic anomaly equations even though the four generators are holomorphic. The initial condition of the holomorphic anomaly equations is expected to be a degree polynomial of a single variable . Two other physical predictions regarding the structure of are the conifold gap condition and the orbifold regularity which determine the lower and upper parts of . Furthermore, the above structures of Gromov–Witten invariants are expected to be present in some fashion for all Calabi–Yau 3-folds, and hence can be considered as universal properties. Based on the above B-model structural predictions and an additional A-model conjecture called Castelnuovo bound, Klemm and his collaborators [HKQ09] derived a formula for for all !
The progress for mathematicians to prove these conjectures has been slow. The genus zero conjecture was proved by Givental [Gi98b] and Lian–Liu–Yau [LLY97], and it was considered to be a major event in mathematics during the 90s. It took another ten years for Zinger to prove BCOV’s conjecture in genus one [Zi08]. It took yet another ten years for the authors’ recent proof of the genus two BCOV conjecture [GJR17P]. The geometric input to our work on genus two is a construction of a certain reduced virtual cycle on an appropriate log compactification of the GLSM moduli space [CJR18P] (see also [CJRS18P]). Its localization formula expresses the Gromov–Witten invariants of quintic 3-folds in terms of a graph sum of (rather mysterious) effective invariants and (rather well-understood) twisted GW-invariants of . This formula works in arbitrary genus as long as one can compute the effective invariants. In particular, there is only one effective invariant for , and it can be computed from the , degree zero GW-invariant. So in principle, we can push our technique to to prove the conjectural formula of Katz–Klemm–Vafa. The main difficulty is directly related to the physical prediction of Yamaguchi–Yau that should be a polynomial of five generators. A direct generalization of our argument in [GJR17P] only implies that is a polynomial of nine generators. The appearance of extra generators is similar to the simpler case of the Gromov–Witten theory of an elliptic curve, where is a quasi-modular form of . On the other hand, the corresponding twisted theory of on is a quasi-modular form for . The ring of quasi-modular forms of has more generators than that of ! In the case of the quintic, the presence of the extra four generators increases the computational complexity significantly. We could try to prove the genus three formula by brute force but the proof would not be illuminating and is unlikely to generalize to higher genus. Our eventual goal is to reach to and beyond. It is clear to us that we should first attack the set of physical predictions for the structures of .
To describe the main idea, recall that the general twisted theory naturally depends on six equivariant parameters, five for the base and one for the twist. It is complicated to study the general twisted theory, and therefore Zagier–Zinger [ZaZi08] specialize the equivariant parameter to , where is a primitive fifth root of unity. This theory is referred as formal quintic (see [LhPa18]). They show that the twisted theory is generated by the five generators predicted by physicists. Unfortunately, in our work, the natural specialization of equivariant parameters is . The bulk of [GJR17P] is to show that the corresponding twisted theory for equivariant parameter has four extra generators. The main input of the current article is a comparison formula expressing the reduced virtual cycle of the Log GLSM moduli space in terms of its canonical (non-reduced) virtual cycle. The advantage of non-reduced theory is that it admits a action with six equivariant parameters which we can specialize to as for the formal quintic.
To state our precise results, we need to setup some notation. Let be a quintic 3-fold. Fix such that , and fix ambient classes .111By a dimension consideration, insertions of primitive classes are not very interesting. Let
where is the forgetful map, be the generating series of Gromov–Witten classes defined by . For , let
be the corresponding numerical generating series. In the cases , and to avoid unstable terms, we need markings (see below).
Let be the GLSM moduli space for a quintic 3-fold, that is the moduli space of stable maps to with a -field [ChLi12]. In [CJR18P], we construct a certain logarithmic compactification (see Section 2 for more details) where is a partition representing the contact order (or relative condition). Traditionally, we call the case the holomorphic theory, and the case the meromorphic theory. The moduli space is a proper Deligne–Mumford stack with a two-term perfect obstruction theory. Hence, it admits a virtual fundamental cycle . The cycle is equivariant for both the action of and the action on the -field. The holomorphic theory is very special in the sense, that in [CJR18P] we construct a reduced virtual cycle . The main result of [CJR18P] is that computes the GW-invariants of quintic 3-folds. The boundary of contains rubber moduli spaces with their own virtual fundamental cycles. The key new higher genus information for quintic 3-folds are the effective invariants
for integral such that . In particular, the invariants are only defined when , and when . Furthermore, the are determined by the corresponding low degree Gromov–Witten invariants.
The comparison formula of [CJR18P] between the reduced and non-reduced cycle yields a formula of the form (the precise result is in Theorem LABEL:thm:compare)
where is a decorated bipartite graph, the are effective invariants associated to vertices and is the remaining contributions which can be expressed in terms of non-reduced virtual cycles. We can replace in the above formula by formal parameters and denote the resulting generating series by . We call the extended quintic family. We can then obtain the theory of by setting . By setting , we obtain the theory of holomorphic GLSM.
1.1. Graded finite generation and orbifold regularity
Let us consider the -function (or period integral of its mirror) of the quintic 3-fold
where is a formal variable satisfying . We separate into components:
The genus zero mirror symmetry conjecture of quintic 3-folds can be phrased as a relationship
between the - and -function, involving the mirror map , where
Now we introduce the following degree “basic” generators
where , and . We often abbreviate , and .
To simplify later formulae, we introduce following basis:
and the normalized Gromov–Witten classes
where , , [ZaZi08].
One can apply localization formula to the non-reduced virtual cycles in the comparison formula to compute , and specialize the equivariant parameter to the formal quintic parameter . The first main result of the article is a combinatorial structural theorem that packages the localization contributions into a Givental-style -matrix action.
(Combinatorial Structural Theorem)
Here, is the set of genus , -marked stable graphs with the decorations:
for each vertex , we assign a label or ;
for each flag or where labeled by , we assign a degree .
For each , the contribution is defined from the - and -matrices of the formal quintic theory in an explicit formula similar to that of Givental’s -matrix action (we refer to Section LABEL:NewRaction for the precise formula).
We cannot directly apply the structural results of formal quintic proven in [LhPa18P]. Nevertheless, after much computation, we can prove Yamaguchi–Yau’s prediction.
We introduce the ring of -generators
the degree defined as follows
We define Yamaguchi–Yau’s finite generation ring222From the definitions one can check that it is a subring of . by
We denote by the linear subspace of degree elements.
This ring admits an additional structure: there exists a derivation acting on this ring, such that the ring is closed under and that increases the degree by . The explicit definition of is given in Lemma LABEL:lem:du.
By setting , we can regard as a subring of . Then we will lose the degree information.
The following “graded finite generation properties” hold for the extended quintic family
Our graded finite generation theorem is much stronger than Yamaguchi–Yau’s original non-graded finite generation. For example, a direct consequence is another key structural prediction:
(Orbifold regularity) Suppose that . Then we can write
where for .
represents the initial condition of the holomorphic anomaly equations. By using the boundary behavior of the large complex structure limit point, the conifold singularity and the orbifold regularity, it is expected to be a polynomial of of degree . At this moment, it is beyond our ability to calculate directly even at . There are two additional B-model structural predictions for . The orbifold regularity claims the vanishing of the lower part of . The other is the conifold gap condition, which determines the upper coefficients of of . As we see, for each there are of initial conditions. By using the orbifold regularity and the degree zero Gromov-Witten invariants, the number of initial conditions is reduced by . By using the conifold gap condition, it is further reduced to many, the same as the number of effective invariants. It is natural to speculate that our approach also implies the conifold gap condition. We leave this for future research.
There is a different approach to the higher genus theory of quintic 3-folds by Chang–Guo–Li–Li–Liu. Recently, they posted a series of articles [CGLL18P, CGL18Pa, CGL18Pb]. Among other things, they proved the original Yamaguchi–Yau (non-graded) prediction independently.
It is nontrivial to show [Mo11] that the five generators are algebraic independent and hence the expression of is unique. This is important for the statement of the holomorphic anomaly equation for which we need to consider derivatives with respect to generators. On the other hand, our proof of Theorem 1.4 gives a canonical expression of and in terms of the generators. Hence, we do not need the algebraic independence of five generators.
1.2. Holomorphic anomaly equations
We now consider the holomorphic anomaly equations (HAE). It is clear that the choice (2) of the generators of is not canonical. Let us make a choice.
Suppose be a finite set that generates . We pick a subset of such that
Let be the subring generated by . We call a choice of non-holomorphic subring, and we call the elements in the ring of the derivations of
the vector fields in the non-holomorphic direction.
A natural choice of the non-holomorphic subring is
Motivated by [YaYa04], we can pick another set of generators
of defined in (3), such that . Here .
We introduce the derivations to be
Then we have the following holomorphic anomaly equations conjectured in [YaYa04]
Note that we could consider as polynomials of either the generators or the generators .
The above holomorphic anomaly equation should be viewed as a higher genus generalization of the Picard–Fuchs equation. For example, for , it determines (see Remark LABEL:rmk:YYHAE) inductively up to a holomorphic function in , which is referred to as the holomorphic anomaly in the physics literature. Our Graded Finite Generation Theorem implies that the holomorphic anomaly is a polynomial of degree , whose lower part vanishes (orbifold regularity).
1.3. Plan of the paper
The paper is organized as follows. In Section 2, we introduce the logarithmic compactification of the GLSM moduli space and the comparison formula between its reduced and nonreduced virtual cycle. In Section LABEL:sec:rewrite, we prove a geometric comparison between the formal quintic and the true quintic theory. The combinatorial structural theorem is proved in Section LABEL:sec:Rmatrixaction. Graded finite generation and orbifold regularity are proven in Section LABEL:sec:fgen. The holomorphic anomaly equation is proven in Section LABEL:sec:HAE. In the final section LABEL:sec:orbifoldregularity, we prove a technical result of the formal quintic theory which we used in the proof of graded finite generation. It implies a conjecture of Zagier–Zinger [ZaZi08].
Our program (including the results of the current paper) to the higher genus theory was first announced in a workshop in Zurich in January 2018. A special thanks to R. Pandharipande for the opportunity to present our program and his constant support. We thank D. Maulik, A. Pixton, G. Oberdieck and X. Wang for interesting discussions. The third author would like to thank S. Donaldson, H. Ogouri and C. Vafa for valuable discussions regarding the program.
The first author was partially supported by supported by NSFC grants 11431001 and 11501013. The second author was partially supported by an AMS/Simons Travel Grant. The third author was partially supported by NSF grant DMS 1405245 and NSF FRG grant DMS 1159265. Part of the work was done during last two authors’ visit to the MSRI, which was supported by NSF grant DMS-1440140.
2. Logarithmic GLSM Moduli Space and its Virtual Cycles
In this Section, we collect some results that will appear in [CJR18P, CJR19P].
2.1. Moduli space
Let be a smooth projective variety with a line bundle admitting a section cutting out a smooth hypersurface . We are mainly interested in the case that and .
Given a map from a log-smooth curve , let be the projective bundle equipped with the logarithmic structure pulled back from and the divisorial log structure from the infinity section. Let be the moduli space of maps of degree together with an aligned333This imposes that the partial order on the degeneracies of the irreducible components is actually a total order. This is a variant of the logarithmic moduli space that behaves well for the localization. log-section with contact order along the infinity section, and with all markings mapping to the zero section, and such that , where is the zero section of , is ample for all sufficiently large . For the computation of the Gromov–Witten theory of it suffices to consider the case when is the empty partition but in the proof of the Combinatorial Structural Theorem, we will also need to consider more general .
Theorem 2.1 ([Cjr18p]).
The space is a proper Deligne–Mumford stack.
There are evaluation maps , and there is a forgetful map .
2.2. Virtual cycles
The moduli space has a canonical perfect obstruction theory defining a virtual cycle .
The maximal degeneracy defines a line bundle on . In the case that , in [CJR18P], a surjective homomorphism of sheaves
is constructed using the section cutting out the hypersurface . Using this cosection-like homomorphism, an alternative reduced virtual cycle is constructed, which agrees with on the locus where the log-section does not map any components of the source to the infinity section. In particular, the reduced virtual dimension is the same as the ordinary virtual dimension. The crucial property of the reduced virtual cycle is the following:
Theorem 2.2 ([Cjr18p]).
where on both sides denotes the corresponding forgetful map to .
2.3. Localization formula
The moduli space admits a -action defined via scaling the log-section . Both the ordinary and reduced perfect obstruction theories are equivariant with respect to this torus action. Hence, we can apply the virtual localization theorem [GrPa99] to compute their virtual classes.
We first consider the fixed loci of . They are classified by a set of decorated bipartite graphs , which we describe now. Let and be the sets of vertices and edges of , respectively. There are various decorations:
the bipartite structure
a genus mapping
a curve class mapping
a fiber class mapping
a distribution of markings
The valence of a vertex is the sum of the number of edges at and the number of preimages under . We set for all that and that is the partition formed by for all edges at .
We require the decorated graphs to statisfy the following conditions:
If for , we have , then the unique incident edge has .
In the case that , , Figure 1 and Figure 2 show all localization graphs for and which do not have any vertices with . These are all localization graphs for the analogous moduli space defined using stable quotients. The full set of stable maps localization graphs is obtained by attaching additional genus zero vertices via degree-one edges to any and distributing among them and the original stable vertices in .
Given a decorated graph , consider the fiber product