Murad Alim: Difference equations for Gromov-Witten potentials

Abstract: I will discuss a difference equation for the Gromov-Witten potential of the resolved conifold. Using the Gopakumar-Vafa resummation of the Gromov-Witten invariants of any Calabi-Yau threefold, I will further show that similar difference equations are satisfied by the part of the resummed potential containing the contribution of the genus zero GV invariants.

Todor Milanov: Matrix model for the total descendent potential of a simple singularity of type D

Abstract: I am planning to talk about my recent paper, joint with Alexander Alexandrov, with the same title. It was conjectured by Witten that the intersection theory on the moduli space of curves is governed by the KdV hierarchy. The conjecture was first proved by Kontsevich. He was able to express the intersection numbers in terms of the asymptotic expansion of a certain Hermitian matrix integral and to prove that the asymptotic expansion coincides with a tau-function of KdV given in the so-called Miwa parametrization. We were able to construct a Hermitian matrix integral whose asymptotic expansion coincides with the Miwa parametrization of the total descendent potential of a simple singularity of type D. Recalling a LG/LG mirror symmetry result of Fan--Jarvis--Ruan, we can also conclude that the asymptotic expansion of our integral gives the FJRW-invariants of the Berglund--Hubsch dual singularity D^T. 

Wei Gu: Nonabelian mirrors and Gromov-Witten invariants

Abstract: We propose Picard-Fuchs equations for periods of nonabelian mirrors [1] in this paper. Thenumber of parameters in our Picard-Fuchs equations is the rank of the gauge group in the nonabelian GLSM, although we reduce them to the actual number of Kähler parameters ultimately. These Picard-Fuchs equations are concise but novel, so we reproduce existing mathematical Picard-Fuchs equations of Gr(k,N) and Calabi-Yau manifolds as complete intersections in Grassmannians [2] to justify our proposal. Furthermore, our approach can be applied to other nonabelian GLSMs, so we compute mathematical Picard-Fuchs equations of some other Fano-spaces, which were not calculated in the literature before. Finally, the cohomology-valued generating functions of mirrors can be read off from our Picard-Fuchs equations. Using these generating functions, we compute Gromov-Witten invariants of various Calabi-Yau manifolds including complete intersection Calabi-Yau manifolds in Grassmannians and non-complete intersection Calabi-Yau examples such as Pfaffian Calabi- Yau threefold and Gulliksen-Negård Calabi-Yau threefold, and find agreement with existing results in the literature. However, the generating functions we propose for the non-complete intersection Calabi-Yau manifolds are genuinely new.

Jeremy Guere: Congruences on K-theoretic Gromov-Witten invariants

Abstract: K-theoretic Gromov-Witten invariants of smooth projective varieties have been introduced by YP Lee, using the Euler characteristic of a virtual structure sheaf. In particular, they are integers. In this talk, I look at these invariants for the quintic threefold and I will explain how to compute them modulo 41, using the virtual localization formula under a finite group action, up to genus 19 and degree 40.

Yaoxiong Wen: 3d Toric Mirror Symmetry

Abstract: In this talk, I will introduce a new version of 3d mirror symmetry for toric stacks, inspired by a 3d N=2 abelian mirror symmetry construction in physics introduced by Dorey-Tong. More precisely, for a short exact sequence                           0 -> Z^k -> Z^n -> Z^{n-k} -> 0,we consider the toric Artin stack $[C^n/(C^*)^k]$, and its mirror is given by the Gale dual of the above exact sequence, i.e., $[C^n/(C^*)^{n-k}]$. We introduce the modified equivariant K-theoretic I-functions for the mirror pair; they are defined by the contribution of fixed points. Under the mirror map, which switches the Kälher parameters and equivariant parameters and maps $q$ to $q^{-1}$, we see that modified I-functions with the effective level structure of mirror pair coincide. This talk is based on the joint work with Yongbin Ruan and Zijun Zhou.

Ming Zhang: Verlinde/Grassmannian Correspondence

Abstract: In the 90s', Witten gave a physical derivation of an isomorphism between the Verlinde algebra of $GL(n)$ of level $l$ and the quantum cohomology ring of the Grassmannian $\text{Gr}(n,n+l)$. In the joint work arXiv:1811.01377 with Yongbin Ruan, we proposed a K-theoretic generalization of Witten's work by relating the $\text{GL}_{n}$ Verlinde numbers to the level $l$ quantum K-invariants of the Grassmannian $\text{Gr}(n,n+l)$, and refer to it as the Verlinde/Grassmannian correspondence.  The correspondence was formulated precisely in the aforementioned paper, and we proved the rank 2 case (n=2) there. In this talk, I will first explain the background of this correspondence and its interpretation in physics. Then I will discuss the main idea of the proof for arbitrary rank. A new technical ingredient is the virtual nonabelian localization formula developed by Daniel Halpern-Leistner.

Will Donovan: Windows on the Pfaffian-Grassmannian correspondence

Abstract: The Pfaffian-Grassmannian correspondence relates certain pairs of non-birational Calabi-Yau threefolds which can be proved to be derived equivalent. I construct a family of derived equivalences using mutations of an exceptional collection on the relevant Grassmannian, and explain a mirror symmetry interpretation. This follows a physical analysis of Landau-Ginzburg B-models by Eager, Hori, Knapp, and Romo, and builds on work with Addington and Segal.

Albrecht Klemm: Analytic Structure of all Loop Banana Amplitudes

Abstract: Using the Gelfand-Kapranov-Zelevinsk\uı system for the primitive cohomology of an infinite series of complete intersection Calabi-Yau manifolds, whose dimension is the loop order minus one, we completely clarify the analytic structure of all banana amplitudes with arbitrary masses. In particular, we find that the leading logarithmic structure in the high energy regime, which corresponds to the point of maximal unipotent monodromy, is determined by a novel Γˆ-class evaluation in the ambient spaces of the mirror, while the imaginary part of the amplitude in this regime is determined by the Γˆ-class of the mirror Calabi-Yau manifold itself. We provide simple closed all loop formulas for the former as well as for the Frobenius κ-constants, which determine the behaviour of the amplitudes, when the momentum square equals the sum of the masses squared, in terms of zeta values. We extend our previous work from three to four loops by providing for the latter case a complete set of (inhomogenous) Picard-Fuchs differential equations for arbitrary masses. This allows to evaluate the amplitude as well as other master integrals with raised powers of the propagators in very short time to very high numerical precision for all values of the physical parameters. Using a recent p-adic analysis of the periods we determine the value of the maximal cut equal mass four-loop amplitude at the attractor points in terms of periods of modular weight two and four Hecke eigenforms and the quasiperiods of their meromorphic cousins.

Tseng, Hsian-Hua: Relative Gromov-Witten theory without log geometry

Abstract: We describe a new Gromov-Witten theory of a space relative to a simple normal-crossing divisor constructed using multi-root stacks

Hiraku Nakajima：Quiver gauge theories with symmetrizers

Abstract: We generalize the mathematical definition of Coulomb branches of 3-dimensional N=4 SUSY quiver gauge theories in arXiv:1503.03676, arXiv:1601.03686, arXiv:1604.03625 to the cases with symmetrizers. We obtain generalized affine Grassmannian slices of type BCFG as examples of the construction, and their deformation quantizations via truncated shifted Yangians. Finally, we study modules over these quantizations and relate them to the lower triangular part of the quantized enveloping algebra of type ADE. (Based on the joint work with Alex Weekes, arXiv:1907.06522.)

Richard Thomas: Square root Euler classes and counting sheaves on Calabi-Yau 4-folds

Abstract: I will explain a nice characteristic class of SO(2n,C) bundles in both Chow cohomology and K-theory, and how to localise it to the zeros of an isotropic section. This builds on work of Edidin-Graham, Polishchuk-Vaintrob, Anderson and many others.This can be used to construct an algebraic virtual cycle (and virtual structure sheaf) on moduli spaces of stable sheaves on Calabi-Yau 4-folds.It recovers the real derived differential geometry virtual cycle of Borisov-Joyce but has nicer properties, like a torus localisation formula. Joint work with Jeongseok Oh (KIAS).Seminar slides:  

Kentaro Hori: D-brane central charges

Abstract: I will describe what are known as D-brane central charges in various contexts including string compactifications, tt^* geometry, hemisphere partition function, Gromov-Witten theory, and old matrix models. I will then discuss their relationships.

Shuai Guo：The Landau-Ginzburg/Calabi-Yau correspondence for the quintic threefold

Abstract: In this talk, we will first introduce the physical and mathematical versions of the Landau-Ginzburg/Calabi-Yau correspondence conjecture for the Calabi-Yau threefolds. Then we will explain our approach to prove this conjecture for the most simple Calabi-Yau threefold - the quintic threefold. This is a work in progress joint with Felix Janda and Yongbin Ruan. 

Huai-Liang Chang: BCOV Feynman structure in A side

Abstract:  Gromov Witten invariants Fg encodes the numbers of genus g curves in Calabi Yau threefolds. They play a role in enumerative geometry and are not easy to be determined. In 1993 Bershadsky, Cecotti, Ooguri, Vafa solved holomorphic anomaly equations for Fg's in B side and their solution exhibited a hidden ``Feynman structure” governing all Fg’s at once. The method was via path integral while its counterpart in mathematic has been missing for decades. In 2018, a large N topological string theory is developed in math and provides the wanted ``Feynman structure”. New features are (i) dynamically quantizing the Kaehler moduli parameterin Witten’s GLSM, and (ii) enhancing the Calabi Yau target to a large N bulk with Calabi Yau boundary. Both are achieved within a Landau Ginzburg type construction (P fields and cosections in math terms). This theory, called “N-Mixed Spin P field” in math, is thus responsible for BCOV theory in A side. In it the A model theory of the large N bulk encodes the B model propagators of the boundary’s mirror. This is a new phenomenon intertwining bulk-boundary correspondence, mirror symmetry, and Gauge-String(B-A) duality.In this talk we will see genuine ideas behind these features, along a new angle to realize NMSP from people familiar with Givental theory.