Geometry of quiver moduli

workshop at Utrecht University, February 23 and 24, 2026

The conference takes place in room HFG 7.07, on the Utrecht Science Park campus.

Capacity in the room is limited.

Schedule

Monday, February 23

10:00–11:00
Dubrovin conjecture for the quiver moduli space $K_{(2,3)}^3$
Junyu Meng (Université de Toulouse)
11:00–11:30
break
11:30–12:30
Interactions of Gromov-Witten theory and GKM theory
Daniel Holmes (Institute of Science and Technology Austria)
12:30–13:45
lunch
13:45–14:15
Homotopy path algebras and blob representations
Raf Bocklandt (Universiteit van Amsterdam)
14:30–15:00
Hochschild cohomology of homotopy path algebras coming from surface polygonalizations
Shin Komatsu (Universiteit van Amsterdam)
15:00–15:30
break
15:30–16:00
Resolutions of near-thin quiver moduli
Matthijs Holstege (Utrecht University)
16:15–16:45
Degree of a Fano quiver moduli space
Pengcheng Zhang (Universität Wuppertal)

Tuesday, February 24

10:00–11:00
Teleman quantisation applied to quiver moduli
Gianni Petrella (University of Luxembourg)
11:00–11:30
break
11:30–12:00
Motives of nullcones of quiver representations
Lydia Gösmann (Ruhr-Universität Bochum)
12:15–12:45
Counting real sheaves on toric surfaces
Tom Manopulo (Utrecht University)
12:45–14:00
lunch
14:00–15:00
Categorifying knot operations
Okke van Garderen (SISSA)

Abstracts

Homotopy path algebras and blob representations

Raf Bocklandt (Universiteit van Amsterdam)

To a quiver embedded in a topological space one can associate its homotopy path algebra. For certain representations that correspond to a contractible blob of vertices and arrows, we show how one can calculate the Ext groups in terms of relative simplicial cohomology. We use this to study the relation between homotopy path algebras coming from different quivers that are related by contracting arrows in a blob. Joint work with Wenjun Huang
Motives of nullcones of quiver representations

Lydia Gösmann (Ruhr-Universität Bochum)

For a finite-dimensional complex representation of a complex reductive group, its nullcone already introduced by Hilbert, plays an important role in the study of the invariant ring, the orbit structure and the quotient map for the group action. A particularly interesting and accessible class of representations are the representation spaces of quivers with their base change action of a product of general linear groups. It is thus natural, both from the invariant-theoretic and from the quiver-theoretic point of view, to study the nullcones of representation spaces of quivers, which parametrize nilpotent representations. A qualitative study of the nullcone was accomplished by Le Bruyn adapting the Hesselink stratification of the nullcone and describing it in terms of loci of semistable representations for so-called level quivers.

Building upon these results, we have studied these nullcones quantitatively, by describing their motives, that is their classes in the Grothendieck ring of complex varieties. We interprete the Hesselink stratification as an identity of motives, first yielding a positive summation formula for the nullcone, and culminating in a product formula relating the generating functions motives of nullcones to generating functions of semistable representations of level quivers, reminiscent of wall-crossing formulas. This is a joint work with Markus Reineke.

Interactions of Gromov-Witten theory and GKM theory

Daniel Holmes (Institute of Science and Technology Austria)

We explore the two-way interaction between Gromov-Witten theory and GKM theory established through equivariant localization. In one direction, GKM theory provides a setting where Gromov-Witten invariants and quantum products are nicely computable. We have implemented this in the Julia package GKMtools (joint with Giosuè Muratore). In the other direction, a well-behaved Gromov-Witten theory enforces structural properties of Hamiltonian GKM spaces. We will survey some recent results in both directions.
Resolutions of near-thin quiver moduli

Matthijs Holstege (Utrecht University)

In this talk, we translate Seshadri's resolution of the moduli space of vector bundles of rank $2$ and degree $0$ on a smooth projective curve to the setting of quiver moduli. This results in a resolution of the moduli space of $\theta$-polystable representations of $Q$ with dimension vector $\boldsymbol{d}$ in the case where $Q$ is any quiver, $d_i=2$ for all $i\in Q_0$, $\theta$ satisfies a genericity condition, and a $\theta$-stable representation of $Q$ with dimension vector $\boldsymbol{d}$ exists.
Hochschild cohomology of homotopy path algebras coming from surface polygonalizations

Shin Komatsu (Universiteit van Amsterdam)

In this talk, we study homotopy path algebras coming from polygonalizations of surfaces and investigate their Hochschild cohomology. We compare the Hochschild cohomology of this algebra with the topological cohomology of the surface. We show that the Hochschild cohomology has larger dimension, and that equality holds when the polygonalization is nondegenerate. If time permits, I will sketch the proof that if the dimension of the Hochschild cohomology is higher in degree 1, the surface must be a torus.
Counting real sheaves on toric surfaces

Tom Manopulo (Utrecht University)

In 1991, Klyachko proved a beautiful formula calculating the Euler characteristic of the moduli space of rank-two stable torsion-free sheaves on the complex projective plane with given Chern class. His method uses toric geometry and has been expanded and applied to prove different versions of the same computation for other surfaces, giving insight also to wall-crossing phenomena which aren't visible on the projective plane. After motivating the problem and recalling the basic constructions involved, we describe a readaptation of these methods for the real projective plane which generalizes to real models of other toric surfaces.
Dubrovin conjecture for the quiver moduli space $K_{(2,3)}^3$

Junyu Meng (Université de Toulouse)

We investigate the moduli space $K_{(2,3)}^3$ of quiver representations for the 3-Kronecker quiver with dimension vector $(2,3)$. It is a blow-down of $\operatorname{Hilb}^3\mathbb{P}^2$ and comes with interesting embeddings into Grassmannians, giving rise to the universal quiver representation. We construct an Aut-invariant full exceptional sequence for its derived category and calculate its small quantum cohomology ring. These allow us to verify the (refined) Dubrovin conjecture relating full Lefschetz collection of derived category to the generic semisimplicity of the big quantum cohomology ring. This is based on a joint work with Svetlana Makarova.
Teleman quantisation applied to quiver moduli

Gianni Petrella (University of Luxembourg)

We apply Teleman quantization to the GIT construction of moduli spaces of representations of quivers. This allows us to prove that the universal bundle of fine quiver moduli spaces is a partial tilting object, confirming a conjecture of Schofield, and to give several constructions of strongly exceptional collections. Our results are effective and are implemented in the SageMath and Oscar computer algebra systems.

This talk is based on joint work with Pieter Belmans, Ana-Maria Brecan, Hans Franzen and Markus Reineke.

Categorifying knot operations

Okke van Garderen (SISSA)

The knots-quivers correspondence relates certain invariants of knots and quivers, motivated by their meaning in string theory. Because knots are related by surgeries, this has lead to interesting new relations between generating series of different quivers. In this talk I will show how these relations can be understood in terms of the geometry of quiver moduli, leading to a categorification via the cohomological Hall algebras.
Degree of a Fano quiver moduli space

Pengcheng Zhang (Universität Wuppertal)

Given a quiver $Q$ and a dimension vector $\mathbf{d}$, the moduli space $X$ of $\theta$-stable representations of $Q$ with dimension vector $\mathbf{d}$ for some $\theta$ is a Fano variety if it satisfies a numerical condition. In this talk, we will explore how to compute the degree of $X$ in its Chow ring. One key technique is to reduce the calculation to the toric case where the dimension vector assigned to a quiver is 1. In the end, we will see a degree formula in the case of the bipartite quiver $K_{(2,q)}^m$.