Workshop on Quivers, Calabi-Yau threefolds and DT Invariants

Liste des Contributions

8. 3d Coulomb Branch and Magnetic Quivers

Julius Grimminger (Imperial College London)

Three-dimensional N=4 theories admit a rich moduli space of vacua, with two distinct hyper-Kähler (HK) subspaces. While the Higgs branch is long understood as a HK quotient (1980s), the Coulomb branch (CB) has proven a much tougher nut to crack (2010s). It turns out it is a new HK construction in its own right.Understanding CBs is still an active area of research, and we will review the...

9. A universal theory of enumerative invariants and wall-crossing formulae.

Dominic Joyce (Oxford)

I outline a (very long and complicated, sorry) programme which gives a common universal structure to many theories of enumerative invariants counting semistable objects in abelian or derived categories in Algebraic Geometry, for example, counting coherent sheaves on curves, surfaces, Fano 3-folds, Calabi-Yau 3- or 4-folds, or representations of quivers (with relations). Write A for your...

4. BPS Lie algebras, from mathematics, for physicists

Ben Davison (Edinburgh)

Ten years ago, Kontsevich and Soibelman provided a mathematical construction, utilising quiver representations, vanishing cycles, and Hall algebras, of the algebra of BPS states associated to certain 4d N=2 supersymmetric gauge theories.  The starting data for the construction is a quiver with potential.  In the context of quantum groups, these algebras should be analogues/generalisations of...

13. Counting Lagrangian A-branes with networks

Pietro Longhi (Uppsala)

The framework of spectral networks was introduced in physics as a way to compute BPS states of 4d N=2 gauge theories. In this talk I will review a generalization, known as exponential networks, which produces enumerative invariants associated to special Lagrangians in certain Calabi-Yau threefolds. Applications include the computation of the exact spectrum for the mirror of the local...

3. D-branes on local P2 revisited

Pierrick Bousseau (CNRS and ETH)

I will describe a one-parameter family of scattering diagrams computing Donaldson-Thomas invariants of local P2 at any point of the physical slice in the space of Bridgeland stability conditions. The scattering diagrams are made of attractor flow trees and are also projections of special Lagrangian submanifolds in the universal family of mirror curves. I will also present a connection with the...

2. Donaldson--Thomas invariants of quivers with potentials from the flow tree formula

Hulya Arguz (IST Vienna)

A categorical notion of stability for objects in a triangulated category was introduced by Bridgeland. Donaldson--Thomas (DT) invariants are then defined as virtual counts of semistable objects. We will focus attention on a natural class of triangulated categories defined via the representation theory of quivers with potentials, and explain how to compute DT invariants in this case from a...

5. Donaldson-Thomas invariants of toric quivers

Pierre Descombes (LPTHE. Sorbonne U)

The category of sheaves on a toric threefold is derived equivalent to the category of representation of a quiver with potential obtained from a brane tiling of the torus. On this class of examples, the numerical DT invariants can be computed by toric localization with respect to an action scaling the arrows of the quiver, by enumerating the fixed points, described combinatorially by pyramid...

18. Exceptional Complex Structures and K-stability

David Tennyson (Imperial College London)

I will introduce a new geometric object called the Exceptional Complex Structure (ECS). This is an extension of the notion of complex structure to include all of the degrees of freedom of string backgrounds, much like the Generalised Complex Structure of Hitchin and Gualtieri was an extension that naturally included the B-field. In the first half of the talk I will define the ECS and provide...

12. From BPS crystals to BPS algebras: constructions, representations, and applications

Wei Li (CAS Beijing)

I will explain how to construct BPS algebras for string theory on general toric Calabi-Yau threefolds, based on the crystal melting description of the BPS sectors. The resulting quiver Yangians, together with their trigonometric and elliptic versions, unify various known results and generalize them to a much larger class. I will then explain how to describe their representations using...

15. Generalized Symmetries in String Compactifications

Sakura Schafer-Nameki (University of Oxford)

Generalized symmetries, such as higher-form and higher-group symmetries, will be discussed in Quantum Field Theories constructed in geometric engineering. The talk will provide an overview of recent, very lively, developments on this topic.

19. Holography, Quivers and Geometry

Alessandro Tomasiello (Università di Milano-Bicocca)

In string theory, a spacetime with an anti-de Sitter (AdS) factor is related to a conformal field theory (CFT) by a famous correspondence. Many such spacetimes are obtained by considering D-branes on conical Calabi–Yau singularities, which can in turn be obtained by K-stability techniques. The corresponding CFT is often, but not always, suggested by matrix factorization techniques.

1. Non-perturbative quantum geometry, resurgence and BPS structures

Murad Alim (University of Hamburg)

BPS invariants of certain physical theories correspond to Donaldson-Thomas (DT) invariants of an associated Calabi-Yau geometry. BPS structures refer to the data of the DT invariants together with their wall-crossing structure. On the same Calabi-Yau geometry another set of invariants are the Gromov-Witten (GW) invariants. These are organized in the GW potential, which is an asymptotic series...

7. Rank r DT theory from rank 1

Soheyla Feyzbakhsh (Imperial College London)

Fix a Calabi-Yau 3-fold X satisfying the Bogomolov-Gieseker conjecture of Bayer-Macr`i-Toda, such as the quintic 3-fold. After a brief introduction of weak Bridgeland stability conditions, I will describe work with Richard Thomas which expresses Joyce’s generalised DT invariants counting Gieseker semistable sheaves of any rank r on X in terms of those counting sheaves of rank 1. By the MNOP...

6. Remarks on Correspondences from Geometric Engineering

Michele Del Zotto (Uppsala)

I will briefly review a strategy to obtain correspondences between supersymmetric quantum field theories in various dimensions building upon geometric engineering techniques. Several new applications and examples will be presented, highlighting the interconnections with the enumerative geometry of backgrounds with special holonomy. In particular, we will include some results about the higher...

11. Stability Structures in Holomorphic Morse-Novikov Theory

Maxim Kontsevich (IHES)

I will talk about an elementary example illustrating the general program (by Y.Soibelman and myself) relating Floer theory for complex symplectic manifolds, quantization and resurgence. The specific question is about the space of morphisms between two specific branes in the cotangent bundle to a complex manifold. The first brane is the zero section endowed with a generic rank 1 local system,...

17. Surface defects, tt* Toda equations and BPS spectra

Alessandro Tanzini (SISSA)

We show that the partition functions of 4d supersymmetric gauge theories with 8 supercharges in presence of surface defects obey tt* equations for a suitable isomonodromic deformation problem, and we comment on its M-theory origin. The solution to these equations provides new recursion relations for instanton counting for all simple groups from A to E. The uplift to 5d is a discrete flow...

16. Three-dimensional Calabi-Yau cones with 2-torus action

Hendrick Suess (Jena University)

There are two main constructions of Calabi-Yau cones in dimension 3. Firstly, the anti canonical cones over (log) del Pezzo surfaces and secondly via Gorenstein toric singularities. The toric construction automatically comes with the action of a 3-dimensional torus and the Calabi-Yau condition is automatically fulfilled. For the construction from del Pezzo surfaces we only obtain a...

10. Topological String on Non-Commutative Resolutions

Albrecht Klemm (Bonn)

Based on a project with Sheldon Katz, Thorsten Schimannek and Eric Sharpe  we describe a simple example of a non-commutative resolution  namely the one of a singular double cover of $P^3$. This exhibits 84 nodes whose small blow ups give rise to torsion classes in $H_2(\hat M,Z)$. The torsion classes support a non-trivial B-field and can be described in terms of non-commutative geometry. We...

14. Wall-crossing structures arising from surfaces

Sergey Mozgovoy (Trinity College Dublin)

Families of Bridgeland stability conditions induce families of stability data (DT invariants), wall-crossing structures and scattering diagrams on the motivic Hall algebra. These structures can be transferred to the quantum torus if the stability conditions of the family have global dimension at most 2. I will discuss geometric stability conditions on a surface with nef anticanonical bundle....