Computing A Categorical Gromov Witten Invariant

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arXiv:1706.09912v2 [math.AG] 11 Jul 2017Computing a categorical Gromov-Witten invariantANDREI CĂLDĂRARU1 AND JUNWU TUABSTRACT: We compute the g 1, n 1 B-model Gromov-Witten invariant ofan elliptic curve E directly from the derived category Dbcoh (E). More precisely, wecarry out the computation of the categorical Gromov-Witten invariant defined byCostello using as target a cyclic A model of Dbcoh (E) described by Polishchuk.This is the first non-trivial computation of a positive genus categorical GromovWitten invariant, and the result agrees with the prediction of mirror symmetry: itmatches the classical (non-categorical) Gromov-Witten invariants of a symplectic2-torus computed by Dijkgraaf.Contents1 Introduction . . . . . . . . . . . . . . . . . . . . . . .2 The classical invariants . . . . . . . . . . . . . . . . .3 Geometric mirror symmetry . . . . . . . . . . . . . .4 Quasi-modular forms and Kaneko-Zagier theory . . . .5 Polishchuk’s algebra and its holomorphic modification6 Costello’s formalism . . . . . . . . . . . . . . . . . .7 A roadmap to the computation . . . . . . . . . . . . .8 String vertices . . . . . . . . . . . . . . . . . . . . . .9 The computer calculation . . . . . . . . . . . . . . . .10 Proof of the main theorem . . . . . . . . . . . . . . .Bibliography . . . . . . . . . . . . . . . . . . . . . . . .16812131822273034421. Introduction1.1. The initial form of mirror symmetry, as described in 1991 by Candelas-de laOssa-Green-Parkes [COGP91], centered on the surprising prediction that the genus1Partially supported by the National Science Foundation through grant number DMS1200721.

2Căldăraru and Tuzero Gromov-Witten invariants of a quintic threefold X̌ could be computed by solvinga differential equation governing the variation of Hodge structure associated to anotherspace, the so-called mirror quintic X . Many other such mirror pairs (X, X̌) were laterfound in physics, satisfying similar relationships between the genus zero GromovWitten invariants of X̌ and the variation of Hodge structure of X .1.2. A far-reaching generalization of mirror symmetry was proposed several yearslater by Kontsevich [Kon95] in his address to the 1994 International Congress ofMathematicians. He conjectured that the more fundamental relationship between thespaces X and X̌ in a mirror pair should be the existence of a derived equivalence betweenthe derived category Dbcoh (X) of coherent sheaves on X and the Fukaya category Fuk(X̌)of X̌ . This statement became known as the homological mirror symmetry conjecture.1.3. Implicit in Kontsevich’s proposal was the idea that the equality of numericalinvariants predicted by the original version of mirror symmetry should follow tautologically from the homological mirror symmetry conjecture. To achieve this oneneeds to construct categorical Gromov-Witten invariants: invariants associated to an(enhanced) triangulated category C , with the property that they recover the classicalGromov-Witten invariants of the space X̌ when the target category C is taken to beFuk(X̌). Once one has such invariants, evaluating them on Dbcoh (X) yields new invariants of X , the so-called B-model Gromov-Witten invariants of X . These invariants aredefined for any genus, not just for genus zero. (The genus zero B-model invariantsare expected to match the data of the variation of Hodge structures used before.) Thecategorical nature of the construction automatically implies, for a pair of spaces (X, X̌)which satisfies homological mirror symmetry, that the B-model invariants of X matchthe Gromov-Witten invariants of X̌ .1.4. Genus zero categorical Gromov-Witten invariants satisfying the desired properties were defined in 2015 by Ganatra-Perutz-Sheridan [GPS15] following ideas ofSaito [Sai83, Sai83] and Barannikov [Bar01]. However, according to the authors, thisapproach does not extend to positive genus.For arbitrary genus Costello [Cos09] proposed a definition of categorical invariantsassociated to a cyclic A algebra (or category), following ideas of Kontsevich andSoibelman [KonSoi09]. Unfortunately many details of [Cos09] were left open, andcomputing explicit examples turned out to be a difficult task. Costello (unpublished)computed one example where the target algebra is the ground field (correspondingto the case where the target space X is a point). No other explicit computations ofCostello’s invariants exist. Costello-Li [CosLi12] wrote in 2010:

Computing a categorical Gromov-Witten invariant3A candidate for the B-model partition function associated to a CalabiYau category was proposed in [Cos07], [Cos07], [KonSoi09] based ona classification of a class of 2-dimensional topological field theories.Unfortunately, it is extremely difficult to compute this B-model partitionfunction.1.5. In this paper we compute the g 1, n 1 B-model categorical invariant ofan elliptic curve Eτ , starting from Costello’s definition and using as input an A model of the derived category Dbcoh (Eτ ) proposed by Polishchuk [Pol11]. It is thefirst computation of a categorical Gromov-Witten invariant with non-trivial target andpositive genus.More precisely, for a complex number τ in the upper-half plane H let Eτ denote theelliptic curve of modular parameter τ , Eτ C/Z Zτ . For each such τ we compute aB (τ ), the corresponding B-model categorical invariant. Regardingcomplex number F1,1the result as a function of τ we obtain the so-called B-model Gromov-Witten potential,a complex-valued function on the upper half plane.1.6. Mirror symmetry predicts the result of the above computation. There is a standardway to collect the classical g 1, n 1 Gromov-Witten invariants of an elliptic curveA (q). The resultin a generating power series, the A-model Gromov-Witten potential F1,1is known through work of Dijkgraaf [Dij95]:AF1,1(q) 1E2 (q).24Here E2 is the standard Eisenstein holomorphic, quasi-modular form of weight 2,expanded at q exp(2πiτ ). For this computation to give a non-trivial answer weneed to insert at the one puncture the class [pt]PD which is Poincaré dual to a point.The prediction of mirror symmetry is that the A- and B-model potentials should matchafter the Kähler and complex moduli spaces are identified via the mirror map, whichin the case of elliptic curves takes the formq exp(2πiτ ).Thus the prediction of mirror symmetry is that the B-model potential should equalBF1,1(τ ) 1E2 (τ ).24

4Căldăraru and Tu1.7. To get our computation off the ground we need a cyclic A -algebra model ofthe derived category Dbcoh (Eτ ). Such an algebra was described by Polishchuk [Pol11],using structure constants that are modular, almost holomorphic forms. We will use bothPolishchuk’s original algebra, and a gauge-equivalent modification of it whose structureconstants are quasi-modular, holomorphic forms. The interplay between calculationsin these two models, via the Kaneko-Zagier theory of quasi-modular forms, will forma central part of our final computation.1.8. Like in the classical Gromov-Witten calculation, in order to get a non-trivialanswer in the B-model computation we need to insert a certain Hochschild class[ξ] HH 1 (Aτ ) at the puncture, mirror dual to [pt]PD . This class will be representedby the Hochschild chain in A 1τ1dz̄.ξ τ τ̄(The identification of Eτ with C/Z Zτ yields a well-defined class dz̄ in H 1 (Eτ , OEτ ).This group is a direct summand of Aτ . Therefore ξ is a well-defined element ofhomological degree ( 1) of the algebra Aτ , and as such it gives rise to a class inHH 1 (Aτ ).)The following theorem is the main result of this paper.1.9. Theorem. With insertion the class [ξ] , Costello’s categorical Gromov-Witteninvariant of Aτ at g 1 , n 1 equals1BF1,1(τ ) E2 (τ ).241.10. We interpret this result in two ways. On one hand we think of it as confirmationof the mirror symmetry prediction at g 1 as in (1.6). On the other hand, through theprism of homological mirror symmetry we can view our result as a statement about theFukaya category of the family Ěρ which is mirror to the family Eτ of elliptic curves.Indeed, by work of Polishchuk-Zaslow [PolZas98] we know that homological mirrorsymmetry holds for elliptic curves. The authors construct an equivalenceDb (Eτ ) Fuk(Ěρ ),cohwhere Ě is the 2-torus mirror to Eτ , endowed with a certain complexified Kähler classρ. Therefore our computation, which is a priori about Dbcoh (Eτ ), can be reinterpretedas a calculation about Fuk(Ěρ ). From this perspective we regard Theorem 1.9 asverification of the prediction that Costello’s categorical Gromov-Witten invariants ofthe Fukaya category agree (in this case) with the classical ones of the underlying space,as computed by Dijkgraaf.

Computing a categorical Gromov-Witten invariant51.11. There is one important aspect of Costello’s work that we have suppressed in theabove discussion. In order to extract an actual Gromov-Witten potential from a cyclicA -algebra A (as opposed to a line in a certain Fock space) we need to choose asplitting of the Hodge filtration on the periodic cyclic homology of A. The correctsplitting is forced on us by mirror symmetry. The Hochschild and cyclic homology ofAτ agree with those of Eτ , as they are derived invariants. Under this identification, asplitting of the Hodge filtration is the choice of a splitting of the natural projection1HdR(Eτ ) H 1 (Eτ , OEτ ) HH 1 (Aτ ).Mirror symmetry imposes the requirement that the lift of [ξ] must be invariant undermonodromy around the cusp, which in turn uniquely determines the lifting. It is withthis choice that we carry out the computations in Theorem 1.9. See Section 9 for moredetails.1.12. There is another approach to higher genus invariants in the B-model, due toCostello and Li [CosLi12, Li11, Li12, Li16], inspired by the BCOV construction inphysics [BCOV94].These other invariants also depend on a choice of splitting of the Hodge filtration.In their works Costello and Li analyzed the BCOV-type invariants of elliptic curvesobtained from arbitrary splittings of the Hodge filtration and they showed that theseinvariants satisfy the Virasoro constraints. Moreover, they studied a family of splittingsdepending on a parameter σ H and they proved the modularity of the correspondingBCOV potentials. The monodromy invariant splitting that we consider corresponds tothe limiting splitting σ i . We have learned the idea that this is the correct onefor mirror symmetry from conversations with Costello and Li.1.13. The BCOV-type invariants have the advantage that they give a more geometricdefinition of B-model Gromov-Witten invariants for Calabi-Yau spaces, and are alsomore easily computed than the original categorical ones of Costello [Cos09]. In fact,for elliptic curves Li was able to establish mirror symmetry at arbitrary genus for BCOVB-model invariants, and to directly compute the potential functions in any genus andfor arbitrary insertions.However, the BCOV-type invariants are fundamentally different from the ones westudy in this paper, in that they are not a priori categorical: knowing homologicalmirror symmetry does not allow one to conclude the equality of the A- and B-modelinvariants. Moreover, the BCOV approach does not immediately generalize to othernon-geometric situations wherein one only has a category, and not an underlying space.

6Căldăraru and Tu1.14. Outline of the paper. Section 2 outlines Dijkgraaf’s computation in the classicalsetting. Section 3 discusses mirror symmetry in the geometric setting. The next twosections review modular forms, Kaneko-Zagier theory, and Polishchuk’s A algebra.Costello’s general formalism is outlined in Section 6, and the next section contains aroadmap to the computation for elliptic curves. Section 8 describes a computation,essentially due to Costello, of the string vertices for χ 1. The last two sectionsB that wepresent two different ways to compute the Gromov-Witten invariant F1,1want. The first method involves reducing the problem to a very large linear algebracomputation, which is then solved by computer. The second method, presented inSection 10, gives a purely mathematical deduction of the result, using a comparisonbetween computations in the holomorphic and modular gauges, respectively.1.15. Standing assumptions. We work over the field of complex numbers C.Throughout the paper we will need to use various comparison results between algebraichomology theories (Hochschild, cyclic) and geometric ones (Hodge, de Rham). Mostof these comparison results are in the literature; however, some appear to be knownonly to the specialists but are not published. In particular we have not been able to findin the literature a comparison between the algebraic Getzler-Gauss-Manin connectionand the classical geometric one. We tacitly assume that they agree, but this should beconsidered a conjectural result. (A similar assumption is made in [GPS15].)1.16. Acknowledgments. We would like to thank Nick Sheridan, Kevin Costello,Si Li, Dima Arinkin, Alexander Polishchuk, and Jie Zhou for patiently listening tothe various problems we ran into at different stages of the project, and for providinginsight. Stephen Wright’s explanation of the use of L1 -optimization techniques hasbeen extremely helpful for computing a small solution to the lifting problem. Some ofthe larger computations were carried out on the SBEL supercomputer at the Universityof Wisconsin, whose graceful support we acknowledge.2. The classical invariantsIn this section we outline Dijkgraaf’s computation [Dij95] of the classical g 1,n 1 Gromov-Witten invariants of elliptic curves.2.1. Let Ě R2 /Z2 denote the two dimensional torus, endowed with any complexstructure making it into an elliptic curve. (The specific choice of complex structure

7Computing a categorical Gromov-Witten invariantwill not matter.)