# Titles and abstracts

## Geometry and Physics of Mirror Symmetry

### The University of Sheffield, 2-6 September 2024

## Lectures

I will first give an introduction to the rich topics of mirror symmetry and enumerative geometry for toric Calabi-Yau 3-folds. I'll then provide a survey of recent results on an algebro-geometric construction of non-toric deformations of toric Calabi-Yau 3-folds. I will also discuss the significance of these non-toric deformations for the physics of 5-dimensional superconformal field theories.

In these lectures I will review some applications of geometric engineering correspondences between supersymmetric observables and enumerative invariants. Time permitting, I will discuss applications of 6d SCFTs to the higher DT theory of elliptically fibered CY singularities, some old and new features of 5d BPS quivers, and aspects of the theory of 5d conformal matter.

In these lectures I will provide a pedagogical overview of the resurgent structure of topological string theory on Calabi-Yau threefolds. I will first introduce the concept of resurgent structure associated to a factorially divergent series. Next I will review some basic ingredients of topological string theory, and I will proceed to analyze its resurgent structure, starting with solvable examples. I will finally discuss the conjectural results on the general case as well as illustrative examples.

An Enriques surface is the quotient of a K3 surface by a fixed point free involution. I will give an overview about known results and open questions regarding curve counting on Enriques surfaces. The first lecture will be about the Gromov-Witten invariants of the local Enriques surface, and in particular about how to prove the Klemm-Mariño formula which evaluates them all. In the second lecture we consider refinements of the Gromov-Witten invariants. This is best done on the sheaf side and there are two options: by K-theoretic invariants (following Nekrasov-Okounkov) or by a motivic refinement (following Kontsevich-Soibelman). We discuss both of them and their differences, and raise a couple of conjectures. In particular, we see a refinement of the Klemm-Mariño formula. The Enriques surface is a beautiful geometry, that serves as an interesting test case for enumerative theories beyond del Pezzo surfaces and local curves.

## Research talks

The KSBA moduli space, introduced by Kollár-Shepherd-Barron, and Alexeev, is a natural generalization of "the moduli space of stable curves" to higher dimensions. It parametrizes stable pairs (X,B), where X is a projective algebraic variety satisfying certain conditions and B is a divisor such that K_X+B is ample. For a polarised log Calabi-Yau variety (X,D) consisting of a projective variety X with an ample line bundle L, where B=D+\epsilon C, such that \epsilon is a small positive number, D is an anticanonical divisor and C is an ample divisor in the linear system of L, it was conjectured by Hacking-Keel-Yu that the KSBA moduli space is a toric variety (up to passing to a finite cover). We prove this conjecture for all log Calabi-Yau surfaces, using tools coming from the minimal model program, log smooth deformation theory and mirror symmetry. This is joint work with Valery Alexeev and Pierrick Bousseau.

Scattering diagram techniques allow to compute the BPS invariants of quiver with potential in terms of some initial data, which are expected to be simple in physically sensible examples (eg, they have been determined for class S theory).

We are interested in quiver with potential giving noncommutative resolution of CY3 singularities: in this case, the space of stability conditions of the quiver is divided into chambers, corresponding to different (commutative) resolutions of the singularity, related by Mori transformations. Using techniques from the theory of Bridgeland stability condition, we prove that the initial data of the quiver are supported on the wall between these chambers, hence can be determined from the birational geometry of the resolutions. It allows to compute the full BPS spectrum of such quiver with potentials, even in non-toric cases.

Alba Grassi: From mirror symmetry to black hole perturbation theory

One of the remarkable developments in recent years within the field of physical mathematics is the discovery of sharp connections between quantum spectral problems and topological string theory, as well as supersymmetric gauge theory. In this talk, I will review some aspects of this correspondence and show how it provides a new and useful perspective on black hole perturbation theory.

Spectral networks encode an awful lot of information about four-dimensional N=2 theories of class S, but are also an incredibly useful tool to study spectral problems that come up in this context. In this talk I will explain how to analyse spectral problems from the perspective of spectral networks, and how to formulate spectral determinants in terms of spectral coordinates. This is based on 1906.04271 with Andy Neitzke and work in progress with Alba Grassi and Qianyu Hao.

Amir-Kian Kashani-Poor: Non-perturbative topological string theory on compact Calabi-Yau manifolds (video)

We will apply resurgent techniques, as introduced in the lectures by Marcos Mariño, to the non-perturbative study of topological string theory on compact Calabi-Yau manifolds. Our approach will rely on explicitly constructing trans-series solutions to the holomorphic anomaly equations.

We discuss techniques to calculate symplectic invariants on CY 3-folds M, namely Gromov-Witten (GW) invariants, Pandharipande-Thomas (PT) invariants, and Donaldson-Thomas (DT) invariants. Physically the latter are closely related to BPS brane bound states in type IIA string compactifications on M. We focus on the rank r_6=1 DT invariants that count anti-D6-D2-D0 brane bound states related to PT and high genus GW invariants, which are calculable by mirror symmetry and topological string B-model methods modulo certain boundary conditions, and the rank zero DT invariants that count rank r_4=1 D4-D2-D0 brane bound states. It has been conjectured by Maldacena, Strominger, Witten and Yin that the latter are governed by an index that has modularity properties to due S-duality in string theory and extends to a mock modularity index of higher depth for r_4>1. Again the modularity allows to fix the at least the r_4=1 index up to boundary conditions fixing their polar terms. Using Wall crossing formulas obtained by Feyzbakhsh certain PT invariants close to the Castelnuovo bound can be related to the r_4=1,2 D4-D2-D0 invariants. This provides further boundary conditions for topological string B-model approach as well as for the D4-D2-D0 brane indices. The approach allows to prove the Castenouvo bound and calculate the r_6=1 DT- invariants or the GW invariants to higher genus than hitherto possible.

The affine Grassmannian of a finite-dimensional simple complex Lie algebra \g plays an important role in representation theory. It is infinite-dimensional, but is the union of countably many finite-dimensional symplectic leaves (indexed by the dominant coweights for \g). The combinatorics of the closure order and the geometry of slices between symplectic leaves are well understood. The analogue for an affine Kac-Moody Lie algebra, the double affine Grassmannian, was (somewhat conjecturally) defined by Braverman-Finkelberg in 2007, and subsequently related to Coulomb branches of quiver gauge theories in their joint work with Nakajima. For affine type A, Nakajima-Takayama have described slices in the double affine Grassmannian as Cherkis bow varieties and have classified the symplectic leaves. An interesting phenomenon is the appearance of additional strata (not indexed solely by the dominant coweights). The analogous decomposition is expected to hold in other affine types. In this talk I will outline some known properties of the affine Grassmannian and the expected properties of the double affine Grassmannian, and will explore (from the perspective of a mathematician) a relationship between the double affine Grassmannian of type E8 and the Higgs branch of (A-type) orbi-instantons. If time permits, I will discuss possible extensions of this framework to D-type and E-type orbi-instantons.

Oh and Thomas have defined a K-theoretic sheaf counting invariant for moduli spaces of sheaves on a Calabi-Yau 4-fold. One of the simplest examples of such a moduli scheme is the Hilbert scheme of n points on C^4. The topic of this talk is a proof of a formula for the generating functions of invariants of these Hilbert schemes, confirming a conjecture of Nekrasov (as well a generalisation to Quot schemes of C^4, conjectured by Nekrasov and Piazzalunga).

I will start by introducing a class of nodal CY 3-folds that arise as symmetric determinantal double covers of P1-bundles on P2. The geometries are torus fibered over P2 and I will argue that the associated Tate-Shafarevich group is Z2 x Z2. Under mild assumptions one can show that the exceptional curves in any small resolution are 2-torsion in homology. This implies that the geometries do not admit any Kaehler small resolution. However, they conjecturally support a flat but topologically non-trivial B-field, corresponding to an element of the Brauer group of a non-Kaehler small resolution. From the string theory perspective, turning on the B-field stabilizes the singularities and leads to a well-defined topological string partition function. At the hand of these examples, I will then discuss the general vector valued modular properties of the topological string partition function on torus fibered CY 3-folds. This leads me to conjecture several new examples of twisted derived equivalences.

It is known that A-model topological string theory on a Calabi-Yau threefold X can be mathematically understood as Gromov-Witten theory. However, a similar interpretation for the so called refined topological string has, until now, been missing. Inspired by the anticipated connection to M-theory, Andrea Brini and I aim to close this gap by proposing a formulation of the refined theory in terms of equivariant Gromov-Witten theory of the extended target geometry X x C^2 given that X admits a non-trivial torus action. To convince you of our construction I will mention several precision checks our proposal passes and I will outline how it conjecturally relates to other refined curve counting theories.