Title and Abstract (workshop home)

• Martijn Kool
  Title: Proof of Magnificent Four
  Abstract: Motivated by super-Yang–Mills theory on a Calabi–Yau 4-fold, Nekrasov assigned weights to solid partitions (4-dimensional piles of boxes) and conjectured a formula for their weighted generating function. We define K-theoretic virtual invariants viaHilbert schemes of points on affine 4-space by realizing them as zero loci of isotropic sections of orthogonal bundles. Using the Oh–Thomas localization formula, we recover Nekrasov’s weights. Applying ideas from Okounkov in the 3-dimensional case, we deduce Nekrasov’s formula. Joint work with J. Rennemo.


• Yuki Koto
  Title: Convergence and analytic decomposition of quantum cohomology of toric bundles
  Abstract: By taking the GIT quotient of a direct sum of line bundles on a complex projective variety, we can obtain a fiber bundle with fiber toric variety. The Gromov-Witten invariants and quantum cohomology of such toric bundles can be calculated using the I-functions constructed by Brown. Using Brown's result, we can see that the formal quantum D-modules of the toric bundles are decomposed into direct sums of the quantum D-modules of the base space. In this talk, I will explain that when the quantum cohomology of the base space converges, the quantum cohomology of the toric bundles also converges, and the direct sum decompositions of the quantum D-modules are analytic.


• Navid Nabijou
  Title: A tale of four theories
  Abstract: I will explain the web of relationships connecting four genus-zero Gromov-Witten theories of snc pairs: logarithmic, orbifold, naive and local. We will see that the orbifold, naive and local theories all coincide, but that their relationship to the logarithmic theory is complicated, involving delicate geometry and combinatorics. Our proofs hinge on a technique - “rank reduction” - which reduces questions about snc divisors to questions about smooth divisors, where the situation is much-better understood. This technique originates from some basic observations on the geometry of the moduli space of logarithmic stable maps, which I will present. This talk is based on joint works with Ranganathan and Battistella-Tseng-You, and on upcoming joint work with Battistella-Ranganathan.


• Helge Ruddat
  Title: The proper Landau-Ginzburg potential is the open mirror map
  Abstract: In a joint work with Gräfnitz and Zaslow, we study the Landau-Ginzburg (LG) model that is mirror to a smooth del Pezzo surface with smooth anticanonical divisor. The mirror LG potential $W$ is proper in this situation. We compute the broken line expansion near infinity of the generalized theta function which gives $W$ in the Carl-Pumperla-Gross-Siebert degeneration. It turns out that it coincides with a well known series: the mirror map of the corresponding local Calabi-Yau. In the case of $\mathbb{P}^2$: $W = 1 + 2Q + 5Q^2 + 32Q^3 + 286Q^4 +\cdots $. It has been known that the mutations of the LG mirror of a toric del Pezzo cover only a bounded subset of the affine manifold that is the SYZ base of the del Pezzo with smooth elliptic curve, e.g. by Thomas Prince or cluster theory. We venture into the unbounded part where the LG potential receives infinitely many terms. The potential can be studied via log Gromov-Witten theory and tropical geometry and I will try to explain the connecting relationships.


• Fumihiko Sanda
  Title: Mirror symmetry of Fano manifolds via toric degenerations
  Abstract: Let $X$ be a Fano manifold and $L$ be a monotone Lagrangian in $X$. Then (a chart of) a Landau-Ginzburg mirror of $X$ is a Laurent polynomial $f$ which is computed by counting holomorphic disks bounded by $L$. Suppose that $X$ admits a toric degeneration to a normal toric Fano variety $X'$. In this talk, I will explain that the Newton polytope of $f$ is equal to the fan polytope of $X'$.


• Nobuyoshi Takahashi
  Title: Log BPS numbers and contributions of degenerate log maps
  Abstract: The relative (or log) Gromov-Witten theory gives a moduli theoretic formulation of enumerative problems on log varieties, but the relation between those invariants and the actual counts is quite subtle. In this talk, I will talk about the log BPS number, a conjecturally integer-valued invariant related to the enumeration of $\mathbb{A}^1$-curves. Then I will look at the contribution of an $\mathbb{A}^1$-curve to these invariants, which shows an intriguing analogy with the case of a rational curve on a K3 surface. The talk is based on joint work with Jinwon Choi, Michel van Garrel, and Sheldon Katz.


• Umut Varolgunes
  Title: Locality for relative symplectic cohomology
  Abstract: I will start by introducing relative symplectic cohomology with a focus on its relevance to mirror symmetry. This will involve some computations in the context of Lagrangian torus fibrations with non-compact complete integral affine bases. I will highlight the role of the locality statement that recently appeared in a preprint with Yoel Groman for such computations.


• Yaoxiong Wen
  Titile: $3d$ $\mathcal{N}=2$ mirror symmetry
  Abstract: In this talk, I will introduce a new version of $3d$ mirror symmetry for toric stacks, inspired by a $3d$ $\mathcal{N}=2$ abelian mirror symmetry construction in physics introduced by Dorey-Tong. More precisely, 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.


• Xiaomeng Xu
  Title: Stokes matrices via the method of isomonodromy deformation
  Abstract: In this talk, we give an introduction to the Stokes matrices, as well as the isomonodromy deformation, of meromorphic linear systems of ordinary differential equations with Poncare rank 1. We then give an explicit expression of the Stokes matrices via the asymptotics of solutions of isomonodromy deformation equations. In the end, we apply the explicit expression to the study of the WKB approximation of the differential equations.