Workshop on Higgs bundles
Dates: 10-13 November 2026
Venue: ICMAT, Madrid
Titles and abstracts
Anna Borri (Essen)
Title: Untwisting the twisted character variety
Abstract: On a smooth projective complex curve X, Simpson's non abelian Hodge correspondence establishes a homeomorphism between the moduli space of degree 0 Higgs bundles on X and the character variety, parametrising representations of the fundamental group of X. For degree d Higgs bundles, the character variety is replaced by a twisted version. In this talk, we explain an alternative construction of this twisted version. The key insight is that the twist appearing in the variety of representations can be thought of as monodromy around an orbifold point. This insight is formalized by adding a stacky point to the curve X and then showing that the moduli space of degree d Higgs bundles on X can be realized as a component of the moduli space of degree 0 Higgs bundles on the stacky curve. Following a result by Shende, we use this construction to compute the Hodge weights of the tautological classes on the twisted character variety, by replacing the stacky curve with a simplicial scheme constructed from a triangulation.
Mischa Elkner (Vienna)
Title: Models of Hitchin maps, medium algebras, and automorphisms
Abstract: I will recall conjectures of Hausel about Hecke modifications of the Hitchin section in the moduli space M(G) of G-Higgs bundles. In particular, the conjectures model the Hitchin map in terms of so-called medium algebras. In a proven special case, González and Hausel have extended this model to the fixed points of an involution on M(G). I will show how to implement some further automorphisms of M(G) studied by García-Prada and Ramanan. Time permitting, I will also connect this to an interesting duality of real reductive groups.
Camilla Felisetti (Modena)
Title: A Parabolic approach to intersection theory on singular moduli spaces of vector bundles
Abstract: Moduli spaces of semistable vector bundles of fixed rank and degree on curves are fundamental objects in algebraic geometry. When rank and degree are coprime. the stability and semistability conditions coincide and the corresponding moduli space is a smooth projective variety whose geometry, topology and intersection theory have been studied extensively over the past several decades. When we drop the coprimality assumption, singularities appear and this has historically posed significant challenges for computing intersection pairings. In this talk we present a recent work in collaboration with Olga Trapeznikova, which introduces a novel and more accessible methodology for tackling this problem. The core strategy involves realizing the intersection cohomology of the singular moduli space as a canonical subspace of the cohomology of a smooth moduli space of parabolic bundles. We will show how this parabolic approach enables the calculation of intersection pairings by leveraging the Hecke correspondence and the Jeffrey-Kirwan residue formula. By comparing this methodology to the standard Jeffrey-Kirwan-Kiem-Woolf blow-up construction, we will discuss how this framework not only provides a computationally efficient alternative but also offers a clear geometric interpretation for arbitrary rank r.
Emilio Franco (Madrid)
Title: Quasi-algebraic quantization for the B-twist Langlands TQFT
Abstract: We will begin this brief talk providing insight on the Moore-Tachikawa category and its relation with the Relative Langlands program of BenZvi-Sakellaridis-Venkatesh. Then we will explain how we are able to generalize the later in the context of non-abelian Hodge theory by means of quasi-algebraic sheaves on the Deligne moduli stack. Time permitting, we will discuss the next steps of our program towards the construction of a B-twist quantization of the Langlands TQFT.
This is joint work with Eric Y.-Y. Chen (EPFL).
Alexander Fruh (Birmingham)
Title: The U(p,q)-Hitchin fibration via a central gerbe extension
Abstract: U(p,q)-Higgs bundles provide simple examples of both Higgs bundles associated to real groups and twisted quiver bundles; however, for (p-q) > 1, the corresponding Hitchin fibration does not have an abelian structure. Nevertheless, the work of Hameister and Morrissey shows that the fibration is governed by a non-abelian gerbe. We show how this can be described as a central gerbe extension using the U(q+1,q)-Hitchin fibration (which has an abelianisation by work of García-Prada and Peón-Nieto); and we show how this relates to a Picard stack action on the U(p,q)-Hitchin fibration.
Guillermo Gallego (Madrid)
Title: On the q-nonabelian Hodge correspondence (after Mochizuki)
Abstract: Mochizuki formulated an analogue of the nonabelian Hodge correspondence, where the role of the "Dolbeaut side" is played by multiplicative Higgs bundles on C^*, the role of lambda-connections is played by q-difference modules, and the "Betti side" concerns bundles on an elliptic curve with two "anti-Harder-Narasimhan" filtrations (the so-called "boomerangs"). The role of the Hitchin equations is played by the Bogomolny equations on C^* x S^1 (equivalently, on the product of an elliptic curve and a real line); thus we are concerned with the doubly-periodic monopoles or "monowalls" from Cherkis and Ward. I will review the main aspects of this correspondence, and explain the connection with other related problems, like the study of bundles whose structure group is a parabolic subgroup of a complex reductive group.
Miguel Gonzalez (Vienna)
Title: Introduction to Hodge bundles
Abstract: Hodge bundles arise naturally as fixed points of the natural C*-action on moduli spaces of Higgs bundles. We will introduce these objects and discuss some of the ingredients that appear in their study, such as Vinberg C*-pairs, the Toledo invariant, Arakelov-Milnor inequalities or the Cayley correspondence, following the work of Biquard, Collier, García-Prada and Toledo. We will illustrate these concepts with examples throughout.
Enya Hsiao (Leipzig)
Title: Odd magical triples and maximal Higgs bundles
Abstract: Higher Teichmüller theory is the study of connected components in the G^R-character variety consisting entirely of discrete and faithful surface group representations. Under the nonabelian Hodge correspondence, various problems in higher Teichmüller theory can be approached by using topological methods on the G^R-Higgs bundle moduli space. Following recent developments of Theta-positivity on the character variety side, it has been proposed by Bradlow, Collier, Garcia-Prada, Gothen and Oliveira that the corresponding components on the Higgs bundle moduli space are characterized by Slodowy slices of magical sl2-triples. In this talk, I will explain how the above framework can be extended to include the case of maximal components of a nontube type Hermitian Lie group by introducing the notion of odd magical triples, further supporting the expectation that all higher Teichmüller components arise via a Cayley correspondence.
Jia Choon Lie (Shanghai)
Title: Spectral correspondence for cyclic Higgs bundles
Abstract: The classical spectral correspondence describes Higgs bundles in terms of sheaves on spectral curves. In this talk, I will discuss the generalization to cyclic Higgs bundles from the viewpoint of quiver bundles: the spectral data can be expressed naturally in terms of sheaves on a suitable noncommutative surface. I will also discuss how this perspective reveals a connection between U(p,q)-Higgs bundles and modules over the sheaf of even Clifford algebras associated to a conic fibration. This is joint work with Ana Peón-Nieto.
Alfonso Zamora (Madrid)
Title: Stringy invariants of abelian character varieties
Abstract: We will discuss the computation of stringy invariants for $G$-character varieties of free abelian groups (with analogous results for moduli of $G$-Higgs bundles on abelian varieties) through the study of the orbifold cohomology of $[T^r/W]$, where $T)$ is a maximal torus of $G$ and the Weyl group $W$ acts diagonally on $T^r$. This is joint work with Carlos Florentino and Ángel González-Prieto.