# Timetable, titles and abstracts

## BPS states, mirror symmetry, and exact WKB II

### The University of Sheffield, 5-9 September 2022

## Lectures

Sergey Mozgovoy (Dublin):

*Some aspects of wall-crossing structures,*notes

I will survey different ways to encode and to manipulate wall-crossing structures (stability data). I will also discuss some examples related to quivers with potentials.

Boris Pioline (Jussieu):

*The global scattering diagram for local P^2,*slides

Given a triangulated category $C$, the set of Donaldson-Thomas invariants $\Omega_Z(\gamma)$ is usefully encoded in a family of consistent scattering diagrams $\{D_\psi(C),\psi\in\IR/2\pi\IR\}$, defined as the union of real-codimension 1 loci in the space $S$ of Bridgeland stability conditions where the central charge $Z(\gamma)$ has argument $\psi$ for some semi-stable object $E$ of charge $\gamma$. I shall construct the scattering diagrams $D_\psi(C)$ for the derived category of coherent sheaves on one of the simplest examples of a Calabi-Yau, namely the total space of the canonical bundle over the projective plane. In this model, the reduced space of stability conditions $\tilde S=Aut(C)\S/GL(2,\R)^+$ coincides with the modular curve $H/\Gamma_1(3)$, where $H$ is the Poincaré upper half-plane. The diagram $D_\psi(C)$ on $H$ interpolates between the scattering diagram associated to a quiver with potential near the orbifold point, and the scattering diagram for coherent sheaves on $\IP^2$ constructed in a recent work by Pierrick Bousseau near the large volume point. Physically, the scattering diagram encodes the attractor flow trees and provides a global view on the spectrum of BPS bound states. Based on work with Pierrick Bousseau, Pierre Descombes, Bruno Le Floch, to appear soon.

Johannes Walcher (Heidelberg):

*Extended Mirror Symmetry and Arithmetic (of BPS states?),*slides

Dimitri Zvonkine (Versailles):

*Introduction to cohomological field theories and Givental's group action*

We will start with the definition of a topological field theory = Frobenius algebra = fusion algebra, follow with the upgrade to cohomological field theories, define Givental's group action on cohomological field theories, and give as many examples as possible.

## Research talks

Sergey Alexandrov (Montpellier):

*Joyce structures, twistors and topological strings,*slides

As shown by T. Bridgeland, a Riemann-Hilbert problem determined by Donaldson-Thomas invariants naturally gives rise to the so-called Joyce structure. It can be characterized by a function known as Plebanski potential, or its close cousin Joyce potential. I'll show that a twistorial solution to the RH problem provides a simple integral expressions for both potentials. Then I'll explain the relation of this solution to the conformal limit of the twistor spaces appearing in gauge and string theories, and physical interpretation acquired by the two potentials in these setups. For the case of the resolved conifold, I'll present a recipe to make the twistorial solution well-defined despite an infinite BPS spectrum, and trace out the emergence of a tau-function, its relation to topological strings and its behavior under S-duality to the twistor space implementation of instantons in string theory.

Michele Cirafici (Trieste):

*(Refined) BPS States from HEll,*slides

K3 surfaces play a prominent role in string theory and algebraic geometry. The properties of their enumerative invariants have important consequences in black hole physics and in number theory. To a K3 surface string theory associates an Elliptic genus, a certain partition function directly related to the theory of Jacobi modular forms. A certain multiplicative lift of the Elliptic genus produces another modular object, an Igusa cusp form, conjectured to be the generating function of Donaldson-Thomas invariants of K3xT2. For certain values of the K3 moduli, such a generating function computes microstates of supersymmetric black holes. In this talk I will discuss some work in progress concerning the refinement of this chain of conjectures. The Elliptic genus can be generalized into a so called Hodge-Elliptic genus which is then related to the refined Donaldson-Thomas invariants of K3xT2.

Ben Davison (Edinburgh):

*BPS cohomology and nonabelian Hodge theory,*slides

BPS cohomology is a cohomology theory that assigns to a 3-Calabi-Yau category (along with a choice of numerical class, and stability condition) a vector space, whose dimension recovers the associated BPS invariants. The theory extends naturally to 2CY categories, by taking 3CY completions (in the sense of Keller). In particular one can study the BPS cohomology of the category of representations of the fundamental group of a Riemann surface, or the BPS cohomology of the category of semistable Higgs bundles on a smooth complex projective curve of genus g. In this talk I'll explain why these BPS cohomologies turn out to be isomorphic, and how this enables us to offer a partial answer an old question question of Simpson: to what extent can the classical nonabelian Hodge correspondence be extended to stacks of strictly semistable objects on both sides of the correspondence?

Fabrizio del Monte (Montreal):

*Five-dimensional BPS spectra and quantum periods from q-Painleve' equations,*slides

While the study of BPS states of four-dimensional supersymmetric QFTs is by now a well-established field with connections to various branches of mathematical and theoretical physics, the results on the case of five-dimensional theories, related to Donaldson-Thomas invariants of local Calabi-Yau threefolds, are still scarce. Stimulated by the four-dimensional results, there has been a recent surge of studies on the topic, but traditional methods have mostly been able to tackle cases without wall-crossing, corresponding to threefolds without compact divisors such as the resolved conifold. In this talk I will show how to solve the BPS spectral problem for local threefolds originating from del Pezzo and Hirzebruch surfaces, by exploiting their underlying affine symmetries. This perspective leads to classes of stability conditions where both the BPS spectrum and the Kontsevich-Sobeilman wall-crossing invariant can be computed exactly, and allows to identify solutions of the corresponding set of TBA equations as special cases of q-Painleve' functions. The case of the q-Painleve' "algebraic solutions" in turn describes WKB quantum periods of a corresponding difference equation, the quantum mirror curve of the threefold, at a special point in its moduli space.

Veronica Fantini (IHES):

*2d-4d wall crossing and deformations of holomorphic pairs,*slides

2d-4d coupled systems were introduced by Gaiotto, Moore and Neitzke as supersymmetric 4d gauge theories coupled to supersymmetric surface defects. Their BPS spectrum can be determined using wall crossing formulas which are a generalization of Kontsevich--Soibelman wall crossing formula for pure 4d theories. The aim of this talk is to discuss the equivalence between the 2d--4d wall crossing formulas and consistent scattering diagrams in the extended tropical group $\tilde{\mathbb{V}}$. The latter has been defined by studying asymptotics of deformations of holomorphic pairs, relaying on Fukaya’s approach to mirror symmetry.

Tim Graefnitz (Cambridge):

*The proper Landau-Ginzburg potential is the open mirror map*

The mirror dual of a smooth toric Fano surface X equipped with an anticanonical divisor E is a Landau–Ginzburg model with superpotential W. Carl-Pumperla-Siebert give a definition of the the superpotential in terms of tropical disks using a toric degeneration of the pair (X,E). When E is smooth, the superpotential is proper. We show that this proper superpotential equals the open mirror map for outer Aganagic–Vafa branes in the canonical bundle K_X, in framing zero. As a consequence, the proper Landau–Ginzburg potential is a solution to the Lerche-Mayr Picard-Fuchs equation. This is joint work with Helge Ruddat and Eric Zaslow.

Qianyu Hao (UT Austin):

*Exact WKB, supersymmetric field theory and topological string,*slides

It has been shown from different perspectives that 4d N=2 theories are closely related to differential equations, e.g. Schrodinger equations. In the exact WKB method, we study both the solutions and the Voros symbols. The Voros symbols, also known as quantum periods, contain information about the monodromy of the differential equations. In the dictionary between the two sides, singularities of Borel transforms of asymptotic series for local solutions or quantum periods correspond to 2d-4d or 4d BPS central charges. Analogously, 5d N=1 theories correspond to difference equations. I will show some generalization of the 4d results to 5d, where I will talk about simple examples corresponding to taking the toric Calabi-Yau manifold to be $\mathbb{C}^3$ or the resolved conifold for the geometric engineering. The local solutions correspond to open topological string partition functions. I will show that singularities in the Borel plane of local solutions are related to 3d-5d BPS KK modes and a direct connection between exponential networks and exact WKB for difference equations. And I will also talk about closed topological string partition functions, which can be thought of as playing the role of quantum periods in the examples we study. The singularities in the Borel transform are related to central charges of 5d BPS KK modes. This talk is based on joint work with Alba Grassi and Andrew Neitzke.

Oliver Leigh (Stockholm):

*The Blowup Formula for the Instanton Part of Vafa-Witten Invariants on Projective Surfaces*

In this talk I will present a blow-up formula for the generating series of virtual χ_y-genera for moduli spaces of sheaves on projective surfaces. The formula is related to a conjectured formula for topological χ_y-genera of Göttsche, and is a refinement of a formula of Vafa-Witten relating to S-duality. I will also discuss the proof of the formula, which is based on the blow-up algorithm of Nakajima-Yoshioka for framed sheaves on ℙ^2. This talk is based on joint work with Nikolas Kuhn and Yuuji Tanaka.

Alexander Shapiro (Edinburgh):

*DT-transformations for K-theoretic Coulomb branches*

As conjectured by Gaiotto, K-theoretic Coulomb branches of 4d N=2 quiver gauge theories possess cluster structure. If the gauge quiver has no loops, the cluster structure can be used to write out the Donaldson-Thomas transformation, which enumerates the BPS states of the theory. In this talk I will describe the cluster structure, the Donaldson-Thomas transformation, and its relation to the bi-fundamental Baxter operators. The talk will be based on our joint work with Gus Schrader.

Joerg Teschner (Hamburg):

*Clusters, line bundles, and topological partition functions,*slides

The goal of my talk will be to review some aspects of a program inspired by work of Tom Bridgeland aiming at a non-perturbative characterisation of topological string partition functions. The program is based on two main ingredients: The complex geometry of the underlying moduli spaces on the one hand, and cluster algebra structures defined by BPS- or DT-invariants on the other hand. Based on evidence gathered from some examples we will propose a conjectural characterisation of the partition functions for local Calabi-Yau manifolds, generalising earlier proposals by Marino and collaborators, and related to earlier proposals by Alexandrov, Pioline and collaborators based on the geometry of hypermultiplet moduli spaces. If time permits we will discuss the role of S-duality, and some steps towards a possible derivation of our proposal.