# Quivers, Clusters, Moduli and Stability

#### 9th–13th January 2023, University of Oxford

 Main Venue Participants Program Abstracts Getting Here

### Abstracts for “Quivers, Clusters, Moduli and Stability” talks:

• Luis Álvarez-Cónsul (ICMAT, Madrid): Embedding superconformal vertex algebras from Killing spinors and $$(0,2)$$ mirror symmetry

I will construct embeddings of the $$N=2$$ superconformal vertex algebra under appropriate conditions inspired by the Killing spinor equations in supergravity. Firstly, when these equations are formulated on a quadratic Lie algebra, they become purely algebraic conditions that induce an embedding in the corresponding superaffine vertex algebra. Secondly, when they are formulated on a suitable class of Courant algebroids, they induce an embedding in the vertex algebra of global sections of the corresponding chiral de Rham complex. As an application, I will present an example of $$(0,2)$$ mirror symmetry given by pairs of homogeneous Hopf surfaces equipped with a Bismut-flat pluriclosed metric. Joint work with Andoni De Arriba de La Hera and Mario Garcia-Fernandez (arXiv:2012.01851, to appear in IMRN, and further work in progress).

• Tom Bridgeland (Sheffield): Clusters and twistors

Given an ADE quiver $$Q$$ I will explain how to construct a complex manifold $$Z$$ with a map to $$\mathbb{P}^1$$ whose fibre over $$0$$ is the stability space of the CY3 Ginzburg algebra of $$Q$$, quotiented by spherical twists, and whose general fibre is an étale cover of the cluster Poisson variety of $$Q$$. This is joint work with Helge Ruddat.

• İlke Çanakçı (VU Amsterdam): A super CC-map

In cluster theory, establishing explicit formulae for cluster variables is an important problem. In the setting of surface cluster algebras this may be given combinatorially via snake graphs and homologically by the CC-map. Recently a combinatorial formula was introduced by Musiker, Ovenhouse and Zhang in an attempt to introduce super cluster algebras of type A (which computes super lambda-lengths in Penner–Zeitlin’s super-Teichmüller spaces). This formula is given in terms of double dimer covers on snake graphs. Motivated by this construction, we propose a representation theoretic interpretation of super lambda-lengths and introduce a super-CC formula which recovers the combinatorial formula. This is joint work in progress with Fedele, García Elsener and Serhiyenko.

• Joe Chuang (CU London): Rank functions on triangulated categories

A rank function is a nonnegative real-valued, additive, translation-invariant function on the objects of a triangulated category satisfying the triangle inequality on distinguished triangles. One way to obtain a rank function is as the mass of a Bridgeland stability condition, but rank functions may exist in the absence of t-structures. Rank functions on the perfect derived category of a ring are related to Sylvester rank functions on finitely presented modules, and therefore, via the work of Cohn and Schofield, to representations of the ring over skew fields. This is joint work with Andrey Lazarev.

• Alastair Craw (Bath): Birational geometry of quiver varieties

Alastair King’s description of quiver moduli spaces allows one to construct (Nakajima) quiver varieties as GIT quotients. I’ll describe joint with Gwyn Bellamy and Travis Schedler where we identify the movable cone of a quiver variety with a region in the corresponding GIT fan, and hence prove that every minimal model of a quiver variety is itself a quiver variety. The key tool is a new sufficient condition for a GIT quotient to be a relative Mori Dream Space.

A finite order automorphism of a complex semisimple Lie group determines a cyclic grading of its Lie algebra. Vinberg’s theory is concerned with the geometric invariant theory associated to this grading. Important examples include the case of involutions and representations of cyclic quivers. After reviewing some basic facts about Vinberg’s theory, in this talk I will discuss about its relation to the geometry of moduli spaces of Higgs bundles over a compact Riemann surface.

• Amihay Hanany (Imperial College London): Branes and Quivers

• Yang-Hui He (London Institute for Mathematical Sciences): Machine-learning algebraic geometry (Slides)

The search for the Theory of Everything has led to superstring theory, which then led physics, first to algebraic/differential geometry/topology, and then to computational geometry, and now to data science. With a concrete playground of the geometric landscape, accumulated by the collaboration of physicists, mathematicians and computer scientists over the last 4 decades, we show how the latest techniques in machine-learning can help explore problems of interest to theoretical physics and to pure mathematics. At the core of our programme is the question: how can AI help us with mathematics?

• Victoria Hoskins (RU Nijmegen): Projectivity of quiver moduli revisited

King used GIT to construct moduli spaces of semistable quiver representations, which are projective in the case of an acyclic quiver. I will present a modern GIT-free proof of this projectivity result using the existence theorem for stacks of Alper, Halpern-Leistner and Heinloth to moduli-theoretically produce a proper good moduli space and showing ampleness of a determinantal line bundle. Our approach also leads to new effective bounds for global generation of determinantal line bundles. This is joint work with Pieter Belmans, Chiara Damiolini, Hans Franzen, Svetlana Makarova and Tuomas Tajakk.

• Dominic Joyce (Oxford): The structure of invariants counting coherent sheaves on complex surfaces (Slides)

Let $$X$$ be a complex projective surface with geometric genus $$p_g > 0$$. We can form moduli spaces $$\mathcal{M}_{(r,a,k)}^{\mathrm{st}} \subset \mathcal{M}_{(r,a,k)}^{\mathrm{ss}}$$ of Gieseker (semi)stable coherent sheaves on $$X$$ with Chern character $$(r,a,k)$$, where we take the rank $$r$$ to be positive. In the case in which stable $$=$$ semistable, there is a (reduced) perfect obstruction theory on $$\mathcal{M}_{(r,a,k)}^{\mathrm{ss}}$$, giving a virtual class $$[\mathcal{M}_{(r,a,k)}^{\mathrm{ss}}]_{\mathrm{virt}}$$ in homology.

By integrating universal cohomology classes over this virtual class, one can define enumerative invariants counting semistable coherent sheaves on $$X$$. These have been studied by many authors, and include Donaldson invariants, K-theoretic Donaldson invariants, Segre and Verlinde invariants, part of Vafa–Witten invariants, and so on.

In my paper “Enumerative invariants and wall-crossing formulae in abelian categories”, in a more general context, I extended the definition of the virtual class $$[\mathcal{M}_{(r,a,k)}^{\mathrm{ss}}]_{\mathrm{virt}}$$ to allow strictly semistables, proved wall-crossing formulae for these classes and associated “pair invariants”, and gave an algorithm to compute the invariants $$[\mathcal{M}_{(r,a,k)}^{\mathrm{ss}}]_{\mathrm{virt}}$$ by induction on the rank $$r$$, starting from data in rank $$1$$, which is the Seiberg–Witten invariants of $$X$$ and fundamental classes of Hilbert schemes of points on $$X$$. This is an algebro-geometric version of the construction of Donaldson invariants from Seiberg–Witten invariants; it builds on work of Mochizuki 2008.

This talk will report on a project to implement this algorithm, and actually compute the invariants $$[\mathcal{M}_{(r,a,k)}^{\mathrm{ss}}]_{\mathrm{virt}}$$ for all ranks $$r > 0$$. I prove that the $$[\mathcal{M}_{(r,a,k)}^{\mathrm{ss}}]_{\mathrm{virt}}$$ for fixed $$r$$ and all $$a,k$$ with $$a$$ fixed mod $$r$$ can be encoded in a generating function involving the Seiberg–Witten invariants and universal functions in infinitely many variables. I will spend most of the talk explaining the structure of this generating function, and what we can say about the universal functions, the Galois theory and algebraic numbers involved, and so on. This proves several conjectures in the literature by Lothar Göttsche, Martijn Kool, and others, and tells us, for example, the structure of $$\mathrm{U}(r)$$ and $$\mathrm{SU}(r)$$ Donaldson invariants of surfaces with $$b^2_+ > 1$$ for any rank $$r \ge 2$$.

• Bernhard Keller (Paris): The blue vs. red game and applications

We will present the blue vs. red game where mutations at frozen vertices are counterbalanced by mutations at non frozen vertices. As an application, we will obtain group actions (e.g. braid group actions) on cluster categories and cluster algebras.

• Miles Reid (Warwick): Remarks on the Tate–Oort group

The Tate–Oort group scheme $$\mathbb{TO}_p$$ is a good reduction of $$\mathbb{Z}/p$$ or $$\mu_p$$ in characteristic $$0$$ to $$p$$. I have recently written a very simple and useful treatment based on a splitting base change involving the variables $$t,S$$ with $$S*t^{p-1} + p = 0$$. I will explain the simple construction, and why I refer to $$t^{p-1}$$ as a splitting pullback.

If I make any progress on the problem, I will also talk about my current game of trying to make this work with prime $$p$$ replaced by a composite number, starting from the most basic cases of $$p_1*p_2$$ or $$p^2$$.

• Markus Reineke (Bochum): Some questions on quiver moduli

We discuss recent results and formulate open questions on quiver moduli, related to stability and expansion, cell decompositions, Fano varieties, motivic invariants, and Cohomological Hall algebras.

• Konstanze Rietsch (King’s College London): Tropical Edrei theorem

The classical Edrei theorem from the 1950’s gives a parametrisation of the infinite upper-triangular totally positive real Toeplitz matrices by positive real parameters with finite sum. These matrices (and their parameters) are central for understanding characters of the infinite symmetric group, as was discovered by Thoma who reproved Edrei’s theorem in the 1960’s. A totally different theorem, related to quantum cohomology of flag varieties and mirror symmetry, gives inverse parametrisations of finite totally positive Toeplitz matrices. In this talk I will present an analogue of the finite theorem over the field of Puiseaux series that comes out of work of Judd and Judd–R, as well as a tropical version of this theorem combining results of Judd and Ludenbach. Finally, I will explain a new ‘tropical’ version of the infinite (Edrei) theorem and connect the finite and infinite theories.

• Gregory Sankaran (Bath): Abelian surfaces, a famous list of primes and no group theory

The fifteen primes that divide the order of the Monster turn up in several places in number theory: we still don’t quite know why. A few years ago they also turned up in algebraic geometry. I will explain why we shouldn’t have been surprised.

• Wolfgang Schief (New South Wales): Clifford lattices and a conformal generalization of Desargues’ theorem

The integrability of a discrete equation quite often has its origin in a classical (or novel) geometric incidence theorem. The Hirota equation constitutes a master equation of integrable systems theory and is applicable in a variety of physical and mathematical contexts, including the study of cluster algebras. We demonstrate that the integrability of its Schwarzian avatar is the consequence of a conformal generalisation of the classical Desargues theorem of projective geometry. The talk is based on the paper A.D. King and W.K. Schief, Clifford lattices and a conformal generalization of Desargues’ theorem, J. Geom. Phys. 62 (2012) 1088–1096. The talk is self-contained and mostly of a pictorial nature.

• Sibylle Schroll (Cologne): Braid group actions on branched coverings and full exceptional sequences

The existence of full exceptional sequences in triangulated categories is often difficult to determine. In case full exceptional sequences exist, there is a well-known action of the braid group and the question of whether this action is transitive arises. In this talk, on joint work with Wen Chang and Fabian Haiden, we consider this braid group action on full exceptional sequences in topological Fukaya categories of graded marked surfaces or equivalently in bounded derived categories of graded gentle algebras.

• Xiuping Su (Bath): Grassmannian cluster category and mirror symmetry

In this talk I will discuss joint work (old and new) with Bernt Tore Jensen and Alastair King on the Grassmannian cluster category $${\mathrm{CM}}(C)$$. I will first recall its construction and some key properties. Then I will explain how it provides a categorification of cluster algebra structures on the coordinate ring $$\mathbb{C}[\mathrm{Gr}(k, n)]$$ and the quantum coordinate ring $$\mathbb{C}_q[\mathrm{Gr}(k, n)]$$. Finally, I will discuss our work in progress on applying $${\mathrm{CM}}(C)$$ to give a categorical version of Rietsch–Williams’ mirror symmetry for Grassmannians.

• Richard Thomas (Imperial): Counting sheaves and representations of CY4 quivers (Slides)

Borisov–Joyce found a way to define a count of sheaves on Calabi–Yau $$4$$-folds, using real derived differential geometry. I will talk about joint work with Jeongseok Oh which gives a definition within algebraic geometry. To make things more interesting for Alastair I’ll say something about the quivery version.

• Michael Wemyss (Glasgow): Spherical objects in dimension 2 and 3

I will explain how to classify spherical objects in various geometric settings (Kleinian singularities, 3-fold flops) using simple modules for the corresponding noncommutative resolution. The main result is much more general, and also more surprising: in the null category $$\mathcal{C}$$, (1) all objects with no negative Ext groups belong to the heart of a bounded t-structure, and (2) the objects with no negative Ext groups, and whose self-$${\rm Hom}$$ is one-dimensional, are precisely the simples. We can even classify all bounded t-structures on $$\mathcal{C}$$. There are various geometric corollaries. The main technique also works for finite dimensional algebras, where in the derived category of a silting discrete algebra, every semibrick complex can be completed to a simple minded collection. This is all joint work with Wahei Hara.

• Katrin Wendland (Trinity College Dublin): An application of folding ADE to BCFG

We consider families of Calabi–Yau threefolds which are obtained from the deformation spaces of ADE type surface singularities. For these non-compact Calabi–Yau threefolds, Diaconescu, Donagi and Pantev discovered in 2007 that the associated Calabi–Yau integrable systems agree with the ADE type Hitchin integrable systems. In joint work with Beck and Donagi we show that these integrable systems allow ‘folding’ by automorphisms of the underlying ADE root systems, and we investigate the corresponding orbifoldings of Calabi–Yau threefolds.

• Alexandra Zvonareva (Stuttgart): Derived equivalences of Brauer graph algebras and the action of spherical twists

Brauer graph algebras first appeared in representation theory of finite groups and can be defined for any suitably decorated graph on an oriented surface. These algebras may be considered as a natural framework to study actions induced by certain configurations of spherical objects. In this talk I will explain how starting from a Brauer graph on a graded surface one can construct a graded $$n$$-Calabi–Yau version of a Brauer graph algebra whose category of perfect complexes depends only on the underlying graded surface. I will also describe the action induced by the corresponding configuration of spherical objects. This is based on joint work in progress with Wassilij Gnedin and Sebastian Opper.

#### Related UK activities:

• COW Seminar, a long-running UK-wide algebraic geometry seminar.
• The Calf, a graduate student oriented version of COW.
• GLEN, a northern COW.
• EmSG, Eastern Midlands Algebraic Geometry Seminar.
• 3CinG, an EPSRC-funded research collaboration between Imperial, Warwick, and Cambridge.
• Enhancing RNG, an EPSRC-funded research collaboration between Edinburgh, Glasgow, and Sheffield.
• Combinatorial Representation Theory, an EPSRC-funded research collaboration based in Leeds

@ Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford, OX2 6GG
Funded by the London Mathematical Society (LMS), Foundation Compositio Mathematica, and the Engineering and Physical Sciences Research Council (EPSRC)