Mirror Symmetry and Differential Equations

Abstract: For a compact Lie group G, its massless Coulomb branch algebra is the G-equivariant Borel-Moore homology of its based loop space. This algebra is the same as the algebra of regular functions on the BFM space. In this talk, we will explain how this algebra acts on the equivariant symplectic cohomology of Hamiltonian G-manifolds when the symplectic manifolds are open and convex. This is a generalization of the closed case where symplectic cohomology is replaced with quantum cohomology. We also explain how it relates to the Coulomb branch algebra of cotangent-type representations. This is joint work with Eduardo González and Dan Pomerleano.

Abstract: Following a conjecture of C. Teleman, we discuss how to relate the quantum cohomology of a GIT quotient with the equivariant quantum cohomology via Fourier transformation. I will also discuss examples of the conjecture and applications to decomposition of quantum differential equations.

Abstract: To describe the asymptotic behavior of solutions of linear differential equation in one complex variable, Deligne introduced the notion of a Stokes filtered local system, or Stokes structure. We shall introduce the analogous notion for linear difference equations and formulate the Riemann-Hilbert correspondence. Possible application to mirror symmetry will also be discussed.

Abstract: Enumerative geometry sinks its roots many centuries back in time. In the last decades, ideas coming from physics brought innovation to this research area, with both new techniques and the emergence of new rich geometrical structures. As an example, Gromov--Witten theory, focusing on symplectic invariants defined as counting numbers of curves on a target space, led to the notion of quantum cohomology and quantum differential equations (qDEs). The qDEs define a class of ordinary differential equations in the complex domain, whose study represents a challenging active area in both contemporary geometry and mathematical physics. These equations, indeed, encapsulate information not only about the enumerative geometry of varieties but even (conjecturally) of their topology and complex geometry. The way to disclose such a huge amount of data is through the study of the asymptotics and monodromy of their solutions. In this talk, the speaker will address the problem of explicitly integrating the quantum differential equations of varieties. Focusing on the case of complete intersections and projectivizations of vector bundles, he will first introduce a family of integral transforms and special functions (the integral kernels). Then he will show how to use these tools to find explicit integral representations of solutions. Based on arXiv:2005.08262 (Memoirs of the EMS, Vol.2) and arXiv:2210.05445.

Abstract: Compound du Val (cDV) singularities are the appropriate threefold generalisation of the classically studied du Val surface singularities. Unlike du Val singularities, however, there is no finite (ADE) list of cDV singularities, the singularities can have moduli, and they need not even be isolated. Another interesting algebro-geometric phenomenon which first appears in (complex) dimension three is the notion of a small resolution -- a resolution of singularities whose exceptional set has codimension greater than one. Motivated by calculations which were made accessible by homological mirror symmetry, Lekili and Evans conjectured that the existence and behaviour of a small resolution of a given isolated cDV singularity should be witnessed by the symplectic cohomology of its Milnor fibre. In this talk, I will explain work-in-progress with Johan Asplund towards understanding the case of compound A_n singularities.

Abstract: Abouzaid--Ganatra--Iritani--Sheridan computed asymptotics of integrations of holomorphic volume forms on toric Calabi--Yau hypersurfaces over Lagrangian sections of SYZ fibrations by using tropical geometry. They gave a new proof of the gamma conjecture for ambient line bundles on Batyrev pairs of mirror Calabi--Yau hypersurfaces. In the talk, we review their work and discuss its generalization to the case of toric hypersurfaces which are not necessarily Calabi--Yau hypersurfaces.

Abstract: I will explain a proposal for the B-side categories appearing intoric homological mirror symmetry along the strata of the generalized discriminant locus due to Aspinwall, Plesser and Wang. A conjectural construction of the web of associated spherical functors and some K-theoretic supporting evidence will be discussed. This is joint work with Ludmil Katzarkov.

Abstract: Moduli of r-spin curves, rth roots of the canonical bundle, have interesting applications to mirror symmetry. In turn, these applications shed new light on classical problems in the geometry of moduli of curves (e.g. tautological relations). We look at them from a purely geometric point of view. For r=2, their rank gives rise to a Z/2-quadratic form and a cohomological field theory. We show that the quadratic form generalizes to tropical hyperelliptic curves. For any r, the direct image is related to the so-called double ramification divisor and to the Witten top Chern class.

Abstract: 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)

Abstract: In this talk I recall some basic facts concerning monodromy of Gel’fand-Kapranov-Zelevinski hypergeometric functions that are considered as periods of a Calabi-Yau hypersurface. Our principal concern will be the monodromy behaviour of GKZ hypergometric functions as they are analytically continued along loops avoiding discriminantal loci. We shall use the language of amoeba to describe the analytic continuation process. We recall a monodromy theorem by P. Horja that can be interpreted as a variation of Picard-Lefschetz formula in our special setting. We show that in certain cases, our monodromy formula gives supporting evidence for the homological mirror symmetry conjecture proposed by Kontsevich. Namely, the monodromy representation of the period of a C.Y. variety is expressed by means of multiplication by Todd class of the mirror symmetric variety. In a similar vein, we discuss the case of Delsarte hypersurface.

Abstract: We briefly review the origin in physics of attractor points on the moduli space of Calabi-Yau threefolds. We turn to their mathematical interpretation as special cases of Hodge loci. This leads to fascinating conjectures on the modularity of the Calabi-Yau threefolds at these points in terms of their periods and L-functions. For hypergeometric one-parameter families of Calabi-Yau threefolds, these conjectures can be verified at least numerically to very high precision. This involves analytic continuation of the solutions of the corresponding hypergeometric differential equations.

Abstract: We define the stable Fukaya category of a Liouville domainas the quotient of the wrapped Fukaya category by the full subcategory consisting of compact Lagrangians, and discuss the relation between the stable Fukaya categories of affine Fermat hypersurfaces and the Fukaya categories of projective hypersurfaces. We also discuss homological mirror symmetry for Milnor fibers of Brieskorn-Pham singularities along the way. This is a joint work in progress with Yanki Lekili.

Abstract: In this talk, we will discuss the reconstruction problem of the mirror manifold, starting with Fukaya’s conjectural construction using counting gradient flow trees/counting holomorphic disks. We will relate it to the famous Gross-Siebert construction, deformation theory and construction of B-model Frobenius manifold, using a newly constructed dgBV algebra. This is a joint work with Kwokwai Chan and Naichung Conan Leung.

Abstract: This talk will report on recent progress on obtaining (open) Gromov-Witten invariants from the Fukaya category. A crucial ingredient is showing that the cyclic open-closed map, which maps the cyclic homology of the Fukaya category of X to its S1-equivariant quantum cohomology, respects connections. Along the way we will encounter R-matrices, which were used in the Givental-Teleman classification of semisimple cohomological field theories, and allow one to determine higher genus Gromov-Witten invariants from genus 0 invariants. I will then present some evidence that this approach might extend beyond the semisimple case. Time permitting, I will explain how one can extend these results to obtain open Gromov-Witten invariants from the Fukaya category.

Abstract: Let X be a Fano variety with G action. The quantum GIT conjecture predicts a formula for the quantum cohomology of "anti-canonical" GIT quotients X//G in terms of the equivariant quantum cohomology of X. The formula is motivated by ideas from 3- dimensional gauge theory ("Coulomb branches") and provides a vast generalization of Batyrev's formula for the quantum cohomology of a toric Fano variety. I will explain the conjecture and then describe joint work in progress with C. Teleman towards proving it. The strategy of proof involves ideas from Varolgunes' theory of relative symplectic cohomology.

Abstract: I will recall the notion of an involutive cover of a symplectic manifold and the local-to-global principle which shows the existence of a spectral sequence starting from the relative symplectic cohomology of the cover and ending at quantum cohomology. I will then discuss the compatibility of this spectral sequence with natural algebraic structures that are present. This leads to a research program that aims to explain genus 0 closed string CY mirror symmetry in a purely symplectic manner, which I will sketch at the end of my talk.