Schedule

Titles & Abstracts

Minicourses

Markus Reineke (Ruhr-Universität Bochum)
Donaldson–Thomas invariants and Cohomological Hall algebras of quivers
Abstract. We will first work through the definition of Donaldson-Thomas invariants of quivers (with potential and/or stability), as a mathematical precision of string-theoretic BPS counts. We will then discuss some tools (representation-theoretic and geometric) to compute these invariants. We will work through the "categorification" of DT invariants via Cohomological Hall algebras, analyze their structure, and compute some examples.

Katrin Wendland (Trinity College Dublin)
Invariants obtained by 'counting' BPS states in two dimensions

Abstract. This mini-course will focus on so-called BPS states in two dimensional quantum field theories and their geometric interpretations. We will begin by discussing the BPS conditions, and we will determine the BPS spectrum in some standard examples. Then we will introduce and discuss 'counting' functions, which in particular 'count' BPS states. This will allow us to read off some interesting invariants that are shared by geometry and quantum field theory.

Talks

Hülya Argüz (University of Georgia)
Fock–Goncharov dual cluster varieties and Gross–Siebert mirrors
Abstract. Cluster varieties come in pairs: for any X-cluster variety there is an associated Fock–Goncharov dual A-cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. I will explain how to bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry, and show that the mirror to the X-cluster variety is a degeneration of the Fock–Goncharov dual A-cluster variety. To do this, we investigate how the cluster scattering diagram of Gross–Hacking–Keel–Kontsevich compares with the canonical scattering diagram defined by Gross–Siebert to construct mirror duals in arbitrary dimensions. This is joint work with Pierrick Bousseau.

Dmitry Chicherin (Université Savoie Mont Blanc)

Cluster algebras for scattering alphabets

Abstract. The study of scattering amplitudes in gauge field theories is a thriving area of research with direct applications to collider physics and gravitational waves. Many interesting scattering amplitudes and Feynman integrals evaluate to multiple polylogarithmic functions, and the notion of the symbol alphabet provides a sharper description of the relevant function space. The knowledge of the symbol alphabet simplifies perturbative calculations, and in some cases, it helps to bootstrap multi-loop amplitudes without doing any loop integrations. For some scattering processes, it has been observed that the symbol alphabet allows for a cluster algebra description. This provides insight into the structure of loop corrections of the scattering amplitudes. I will review recent progress in the application of the cluster algebras to the Feynman integrals, and in particular to high-multiplicity scattering amplitudes in the maximally-supersymmetric Yang–Mills theory.

Bea de Laporte (Universität zu Köln)

Landau-Gizburg potentials via projective representations

Abstract. Many interesting spaces arise as partial compactifications of Fock-Goncharov's cluster varieties, among them (affine cones over) flag varieties which are important objects in representation theory of algebraic groups. Due to a construction of Gross-Hacking-Keel-Kontsevich those partial compactifications give rise to Landau-Ginzburg potentials on the dual cluster varieties whose tropicalizations define interesting polyhedral cones parametrizing the theta basis on the ring of regular functions on the cluster varieties.

In this talk, after explaining the background, we give an interpretations of these Landau-Ginzburg potentials as F-polynomials of projective representations of Jacobian algebras. This is joint work with Daniel Labardini-Fragoso.

Michele Del Zotto (Uppsala Universitet)

Supersymmetric Spectral Problems via Quiver Representations

Abstract. I will present a short review of applications of representation theory of quivers with potentials to the spectral problem of supersymmetric quantum fields in four and five dimensions. The dictionaries which arises are a fruitful source of interesting conjectures informing both mathematics from physics, as well as physics from mathematics. I will conclude by presenting an interesting parallel between the construction of the cluster category in representation theory and the 't Hooft screening mechanism in quantum field theory.

Fabian Haiden (Syddansk Universitet)

Surfaces, quivers, and derived categories

Abstract. Polyangulations — decompositions of a surface into n-gons for varying n — appear naturally in the study of quadratic differentials. I will explain how to encode them in terms of quivers with potential and how this leads to a construction of stability conditions on d-Calabi–Yau categories, generalizing results of Bridgeland–Smith. Based on work in progress with Merlin Christ and Yu Qiu.

Amihay Hanany (Imperial College London)

Quivers and Branes

Abstract. I will explain the natural connection between branes and quivers and how the study of branes leads to interesting findings in quiver gauge theories.

Motohico Mulase (University of California Davis)

Complex Lagrangian geometry and quantization

Abstract. Quantum theories of physics in real 2, 3, and 4 dimensions produce amazing duality relations in geometry. In each context, complex Lagrangian subvarieties play different key roles. In this talk I start with an elementary example of quantization of a complex Lagrangian submanifold, and propose a possible complex Lagrangian geometry. The talk is based on a collaboration in progress with John Alex Cruz Morales and Olivia Dumitrescu.

Pierre-Guy Plamondon (Université de Versailles Saint-Quentin)

Some configuration spaces via categorification

Abstract. The categorification of cluster algebras using representations of quivers is a powerful tool that has led to advances both in the theory of cluster algebras an in the representation theory of algebras. In this talk, we will study a variety defined from the representation theory of associative algebras. For quivers of type A, this variety recovers a partial compactification of the configuration space of n points in P¹. This generalizes the cluster configurations spaces of Arkani-Hamed, He and Lam for acyclic Dynkin quivers. This is a report on ongoing work with N. Arkani-Hamed, H. Frost, G. Salvatori and H. Thomas.

Koushik Ray (Indian Association for the Cultivation of Science)

Learning scattering amplitudes by heart

Abstract. I shall report on the categorification of scattering amplitudes for planar Feynman diagrams in scalar field theories with a polynomial potential. Amplitudes for cubic theories are written down in terms of projectives of hearts of intermediate t-structures of the derived category of A-type quivers. I shall indicate the correspondence with cluster categories as well. I shall briefly discuss the results for higher order polynomial potentials in terms of higher cluster categories and higher intermediate t-structures.

Johannes Walcher (Universität Heidelberg)

Exponential networks and representations of quivers

Abstract. Quivers have been a useful tool to understand both the dynamics of gauge theories and the spectrum of BPS states in string theory. I will report on (partly unfinished) work with R. Eager and S. Selmani on the problem of “counting” special Lagrangian submanifolds in local Calabi–Yau threefolds, in its dimensional reduction to a spectral curve.