北京雁栖湖应用数学研究院

Genus one fibered Calabi-Yau threefolds appear in various places in string theory and algebraic geometry. We will present a class of such fibrations with 5-sections that arise as homologically projective dual pairs. Their topological string partition functions shows interesting modular behaviour. We will briefly comment on the case of 6-sections.

The main theme of this talk is Fukaya's program on SYZ mirror symmetry. Let X be a Calabi-Yau manifold equipped with an SYZ fibration to a base manifold B. Then Fukaya conjectured two correspondences: one between holomorphic disks in X with boundaries on Lagrangian torus fibers and Morse theory on the SYZ base B, and the other between deformations of complex structures on the SYZ mirror and the same Morse theory on B. I will explain joint works with Conan Leung and Ziming Ma where we proved a modified version of the latter correspondence. Along the way, we will see intimate relations between tropical geometry and smoothing of maximally degenerate Calabi-Yau varieties.

Generating series of enumerative invariants are often suprisingly connected to period integrals. In this talk I will explain a relation between Gromov-Witten invariants of elliptic curves and configuration space integrals of cohomology classes constructed from Poincare bundles.

Let $D$ be a smooth rational ample divisor in a smooth projective surface $X$. In this talk, we will present a simple uniform recursive formula for (primary) Gromov-Witten invariants of $O_X(-D)$. The recursive formula can be used to determine such invariants for all genera once some initial data is known. The proof relies on a correspondence between all-genus Gromov–Witten invariants and refined Donaldson–Thomas invariants of acyclic quivers. In particular, the corresponding BPS invariants are expressed in terms of Betti numbers of moduli spaces of quiver representations. This is a joint work with Pierrick Bousseau.

We report on a new development in asymptotic Hodge theory, arising from work of Golyshev-Zagier and Bloch-Vlasenko, and connected to the Gamma Conjectures in Fano/LG-model mirror symmetry. The talk will focus on aspects of this story related to periods and invariants of algebraic cycles, and their limits at a point of maximal unipotent monodromy. In particular, we will explain why the leading Taylor coefficients at positive integers of (a period of) the Mellin transform of certain Calabi-Yau VHS are given by special values of (higher) normal functions. That is, their Picard-Fuchs equations already “know about” the relevant algebraic cycles on the total space of the family.

In 1992, Cecotti and Vafa introduced a new index for $N=2$ supersymmetric theories, which generalizes the Witten index. For $N=2$ supersymmetric theories from Calabi-Yau manifolds, this new index is known as BCOV (Bershadsky, Ceccoti, Ooguri and Vafa) formula. In particular, for Calabi-Yau threefolds, this BCOV formula is well-known as the generating function of Gromov-Witten invariants of genus one. In this talk, I will consider the BCOV formula for lattice polarized K3 surfaces. There is no Gromov-Witten invariants in the BCOV formula for $K3$ surfaces, however, we will find some nice cusp forms (which we call BCOV cusp forms) on the relevant period domains. As by-products, we also find K3 differential operators for all the genus zero groups of type $\Gamma(n)_{+}$. This is a joint work with Atsushi Kanazawa (mathAG: arXiv:2303.04383).

One of the most challenging problems in geometry and physics is to compute higher genus Gromov-Witten invariants of compact Calabi-Yau 3-folds, such as the famous quintic 3-fold. I will briefly describe how physicists compute Gromov-Witten invariants of the quintic 3-fold up to genus 53, using five mathematical conjectures. Three of them have been already proven, and one of the remaining two conjectures has been solved in some genus. I will explain how to prove the last open one, called the Castelnuovo bound, which predicts the vanishing of Gopakumar-Vafa invariants for a given degree at sufficiently high genus. This talk is based on the joint work with Yongbin Ruan.

Using ideas from matrix model theory, given a sequence of numbers one can construct some tau-functions of the KP hierarchy. There are many sequences of integers which have rich combinatorial meanings and have connections with various branches of mathematics. For examples, the Catalan numbers and the Narayana numbers are related to cluster algebras, asociahedra, and they can be defined for Coxeter groups. They are also related to Wigner's semicircle law and arcsine law, and they arise in various lattice path counting problems. Our considerations of the expansions of the tau-function in Schur functions reveal some surprising mirror symmetry phenomena, in particular, an unexpected mirror symmetry between Lie algebras of tyep A and type B emerges in this setting.

The partition function of topological string theory is an asymptotic series in the topological string coupling and provides in a certain limit a generating function of Gromov-Witten (GW) invariants of an underlying Calabi-Yau threefold X. I will discuss how the resurgence analysis of the partition function for a class of non-compact geometries allows one to extract the Donaldson-Thomas (DT) or BPS invariants of the same underlying geometry X. I will further discuss how the analytic functions in the topological string coupling obtained by Borel summation admit a dual expansion in the inverse of the topological string coupling leading to another asymptotic series at strong coupling and to the notion of topological string S-duality. I will further discuss how this S-duality leads to a new modular structure in the topological string coupling which is captured in terms of elliptic gamma functions. I will also discuss relations to refined topological string theory, difference equations and the exact WKB analysis of the mirror geometry. This is based on various joint works with Lotte Hollands, Arpan Saha, Iván Tulli and Jörg Teschner as well as on work in progress.

I will discuss a road map to study higher genus Gromov-Witten invariants of the Calabi-Yau quintic threefolds. I will concentrate on the geometric aspect of the method used, which is called N-mixed spin P-fields. The method and its variations are developed by Huai-Liang Li, Guo Shui, Jun Li, Melissa Liu, Wei-Ping LI and Zhou Yang.

We study a novel relation between quantum periods and TBA(Thermo-dynamic Bethe Ansatz)-like difference equations, generalize previous works to a large class of Calabi-Yau geometries described by three-term quantum operators. We give two methods to derive the TBA-like equations. One method uses only elementary functions while the other method uses Faddeev’s quantum dilogarithm function. The two approaches provide different realizations of TBA-like equations which are nevertheless related to the same quantum period.

High genus GW invariants of the quintic Calabi Yau threefold are governed by a BCOV Feynman structure. The structure was proved in 2018 by enumeratiing rational curve chains in an N-dimensional conic extension of quintic threefold, in the sense of Landau Ginzburg theory. The proof used the pacakge of two point NMSP correlations to approach the BCOV B model propagators. In this talk we will exibit an alternate proof using the package of the R matrix factorization of NMSP theory. The method is more efficient and physics oriented than the 2018 A-model geometric approach. This is a joint work with Shuai Guo and Jun Li.

We investigate a two-dimensional chiral analogue of the algebraic index theorem via the theory of chiral algebras developed by Beilinson and Drinfeld. We construct a trace map on the elliptic chiral homology of the free beta gamma-bc system, and explain the corresponding holomorphic anomaly equation.

Topological string theory has (spacetime) instanton sectors, which the resurgence theory predicts to be completely controlled by the perturbative free energy via Stokes transformations. Recent results also suggest the Stokes constants are related to BPS/DT invariants. To make this picture concrete, one needs to first solve the instanton amplitudes and then calculate the Stokes constants. We demonstrate that the first problem can be solved exactly and completely through a trans-series extension of the BCOV holomorphic anomaly equations. We also show that valuable information on BPS invariants can be obtained through the calculation of Stokes constants. We will demonstrate our results with the example of the famous quintic manifold.

I give a survey of the recent progress on Nakajima-Yoshioka's blowup formulas. I will discuss the generalizations of blowup formulas to various supersymmetric gauge theories in 4d, 5d, 6d that are cohomological, K-theoretic, elliptic respectively, and to refined topological string theory on local Calabi-Yau threefolds. I also discuss the relation between blowup equations and (refined) holomorphic anomaly equations which are two universal approaches to solve enumerative invariants of local Calabi-Yau threefolds.

Assuming certain comparison between non-commutative Hodge structures with classical Hodge structures, we prove the categorical enumerative invariants associated with a smooth projective family of Calabi-Yau 3-folds satisfy the holomorphic anomaly equation. (This is a joint work with Yefeng Shen).

The finite generation and Yamaguchi-Yau type functional equations are structural properties predicted by physicists for the Gromov-Witten theory of Calabi-Yau manifolds. In this talk I will introduce the Givental-Teleman classifi- cation for semisimple cohomological field theories, and explain how to obtain the properties for formal Calabi-Yau complete intersections. This is a joint work with Shuai Guo.

Quantum Lefschetz property relates GW invariants of a complete intersection a toric variety and and GW invariants of the toric variety. In genus 0, Givental used this to prove mirror symmetry conjectures. For genus one, Zinger introduced a refinement of GW invariant, reduced GW invariant and succeeded to extend Quantum Lefschetz property in genus one. In this talk, I will briefly review the past researches on quantum Lefschetz property of GW invariants, and introduce my recent works joint with Jeongseok Oh and Mu-lin Li. We used the results of desingularization and local equation of the stable map space when genus one and two, given by Hu-Li and Hu-Li-Niu. Using this, we could describe intrinsic normal cones locally and study the decomposition of the virtual cycles. This gives another proof of Zinger's comparison formula comparing GW invariants and reduced GW invariants, and also gives a new comparison formula in genus two. This gives us a way to compute genus two GW invariants for CY3 complete intersections in projective spaces.