Aswin Balasubramanian: Class S theories and the geometric Langlands correspondence
I will describe a physical way to think about the geometric Langlands correspondence using a class of 4d N = 2 theories (called class S theories). This point of view can be taken to be complementary to the approach of Kapustin–Witten and clarifies how their approach is related to the one based on 2d conformal field theories. The Hitchin integrable system plays a critical role in many of these considerations, so the talk can alternatively be thought of as an exploration of the geometry of the Hitchin system from various perspectives.
Indranil Biswas: Monopoles on Sasakian three-manifolds
We consider monopoles with singularities of Dirac type on quasiregular Sasakian three-manifolds fibering over a compact Riemann surface X, for example the Hopf fibration. We show that these correspond to holomorphic objects on X, which we call twisted bundle triples. These are somewhat similar to Murray’s bundle gerbes. A spectral curve construction allows us to classify these structures, and, conjecturally, monopoles.
Joint work with Jacques Hurtubise.
Facundo Camano: Monopolization of tesserons
Monopolization is a conjecture by Sergey Cherkis stating that all tesserons (hyperkähler gravitational instantons) may be realized as monopole moduli spaces. In the first half of the talk, we will go over the motivation for the conjecture, which tesserons have already been realized as monopole moduli spaces, the idea that all tesserons are related via Gromov–Hausdorff limits, and the moduli spaces proposed for the remaining tesserons. In the second half of the talk, I will present recent work investigating a particular Gromov–Hausdorff limit of moduli spaces of singular monopoles on ℝ3.
Benoit Charbonneau: Monopoles with arbitrary symmetry breaking: some further musings
The theory of SU(n)-monopoles is at its most developed in the case of maximal symmetry breaking. By contrast, monopoles with repeated Higgs-field eigenvalues inhabit a considerably rougher landscape. In this setting, familiar tools often require substantial reformulation: for instance, spectral curves alone do not determine the monopole, and the corresponding Nahm data exhibits phenomena with no analogue in the maximally broken case. I will explore several routes through this landscape, drawing on both classical constructions and more recent developments. Some of these paths are well travelled, while others remain only partially charted. The goal of the talk is not to present a definitive theory, but rather to offer a snapshot of a subject in development and to highlight some of the challenges that remain.
Aradhita Chattopadhyaya: Vafa–Witten theory and mock modularity
I will review aspects of Witten and Vafa’s 1994 work on S-duality. This is a supersymmetric, topologically twisted version of Yang–Mills theory on a 4-manifold M. In this talk I will focus on M = ℂℙ2 and gauge group SU(N). The instanton partition functions in this theory are given in terms of mock modular forms. I will also discuss my recent work with Jan Manschot, where we proved the structure of holomorphic anomaly as predicted in the original paper of Vafa and Witten. If time permits, I will briefly discuss my recent paper with Sergey Alexandrov where we comment on these invariants on the walls of marginal stability.
Josh Cork: Rotating calorons and delayed Nahm equations
Rotating calorons are screw-symmetric anti-self-dual connections on ℝ4, that is, instantons invariant under a glide rotation. In this talk, after a brief review of ordinary calorons (periodic instantons) and their relationship with monopoles, we shall formulate a Nahm transform for rotating calorons, identifying them with solutions of integrable delay-differential equations. These equations generalise the classical Nahm equations, and exhibit new phenomena such as non-constant rank 1 solutions. We shall describe these solutions in the simplest non-trivial case, and discuss some new directions and open problems.
Justin David: Dyonic black holes in N = 4 string theory and modular forms
We review the counting of the degeneracy of BPS states carrying electric and magnetic charges in N = 4 string theories. We consider states which preserve either half or quarter of the supersymmetries of the theory. For large charges, these states can be identified as extremal black holes. We discuss how modular forms transforming under SL(2,ℤ) and Sp(2,ℤ) appear as partition functions from which the BPS degeneracy can be extracted. We then show how properties of modular forms can be used to obtain the Hawking–Beckenstein entropy of the corresponding black hole. Surprisingly, recent studies have shown that properties of black holes can be used to predict certain positivity properties of the Fourier coefficients of the modular forms.
Daniel Fadel: Monopoles at large mass: concentration and abelianization
I will discuss the large mass behaviour of SU(2) monopoles on asymptotically conical manifolds in dimensions three, six, and seven. In dimension three, corrected and expanded joint work with Gon¸calo Oliveira shows that the mass-normalized energy concentrates at finitely many points, which coincide with the limiting Higgs-zero set. Each point carries a finite cluster of mass-one Euclidean monopoles whose total charge determines the concentration weight. In subsequent work, I prove exponential abelianization away from the concentration set and convergence, after translating the Higgs fields by their masses, to a reducible monopole with Dirac singularities. I will then turn to joint work with Gon¸calo Oliveira on G2 and Calabi–Yau monopoles with fixed asymptotic monopole class. We place the complementary compactness theories of Parise–Pigati– Stern and Yang Li in a unified asymptotically conical framework and identify the limiting currents they produce; this single current is a compactly supported calibrated integral cycle governing the mass-normalized energy concentration. Taking this cycle as our starting point, our main results compare its support with the limiting Higgs-zero set and with a limiting nonabelian locus defined from Li’s curvature concentration sets. We identify the excess of the nonabelian locus over the calibrated support with the obstruction to effective codimension-three monotonicity and prove abelianization away from the nonabelian locus.
Guido Franchetti: The L2 metric for hyperbolic 2-monopoles
While it has been known for a long time that the L2 metric on the moduli space of hyperbolic monopoles is divergent, it has been recently shown that it can be made finite by modifying the gauge fixing condition. In this talk, based on joint work with Derek Harland, I will show how applying the modified gauge fixing condition to the moduli space of charge 2 mass 1/2 monopoles results in a complex non-degenerate bilinear form which can be explicitly computed. This bilinear form restricts to a Riemannian metric on the subspace of inversion-symmetric 2-monopoles.
Tom Galvin: A new metric for hyperbolic SU(2) 2-monopoles
There is significant interest in the L2 metric on the moduli space of SU(2) Euclidean monopoles, both because it is hyperkähler, and because it models monopole dynamics. No direct analogue of such a metric exists for hyperbolic monopoles, though alternatives have been suggested. I will discuss a new metric on the moduli space of hyperbolic SU(2) 2-monopoles based on methods of O. Nash. While this metric is not hyperkähler, it does possess a more general geometry known as a pluricomplex structure. I will introduce this concept in the talk before examining some of the metric’s curvature properties. This is joint work with Derek Harland and Linden Disney-Hogg.
Amihay Hanany: Quotient quiver subtraction and the moduli space of monopoles
The moduli spaces of classical BPS monopoles on ℝ3— together with variants carrying singular framing or non-trivial asymptotic boundary conditions — are complete hyperkähler manifolds with an underlying integrable structure. In the physics description, these same spaces arise as Coulomb branches of 3d N = 4 quiver gauge theories: each monopole moduli space is encoded combinatorially by a quiver, a graph whose nodes and edges record gauge and matter content. This dictionary is powerful, but computing explicit metrics — or evaluating hyperkähler quotients by isometry subgroups — remains difficult once the group action fails to be free and singular loci appear. In this talk we introduce quotient quiver subtraction, a graph-theoretic algorithm that performs SU(n) hyperkähler quotients directly on the quiver associated to a monopole moduli space, without first knowing an explicit metric. Rather than assuming the quotient is a smooth geometric slice, the procedure handles singular group actions by producing a finite set of resulting quivers, whose Coulomb branches together reconstruct the metric and global structure of the quotient. We illustrate the method on quivers built from finite A4-type root systems — that is, linear-chain quiver diagrams.
Christopher Lang: Monopoles, co-Higgs bundles, and topological recursion
In this talk, we discuss the relationship between the moduli spaces of SU(2) monopoles and stable co-Higgs bundles. We focus on charge two monopoles, showing that the normal bundle of the monopole moduli space in the Hitchin base carries a natural metric, which can be described using topological recursion. This talk is based on ongoing joint work with Linden Disney-Hogg.
Yang Li: Large mass limit of G2 and Calabi–Yau monopoles
I will discuss some recent progress on the Donaldson–Segal programme, which concerns the limiting behaviour of gauge theory equations known as G2-monopoles (resp. Calabi–Yau monopoles) on asymptotically conical G2-manifolds (resp. Calabi–Yau 3-folds). We develop a structure theory for the limit of SU(2) G2-monopoles (resp. Calabi–Yau monopoles) as the mass parameter tends to infinity, while the topological data for the bundle stays fixed. We show how to extract a singular abelian G2-monopole (resp. Calabi–Yau monopole) with Dirac singularity along a calibrated cycle in the large mass limit, and we prove an energy identity for monopole bubbles.
Richard Melrose: Compactification, monopoles, moduli and configuration spaces
TBA
Pranav Pandit: Kähler geometry of categories
A Bridgeland stability condition may be thought of as a categorical analogue of a Kähler class. I will discuss the problem of refining it to a notion of Kähler metric, with corresponding notions of mass, harmonic objects, minimizing flows, and a Kobayashi–Hitchin-type picture. Part of the framework I will describe is closely related to a noncommutative-geometric formalism encompassing, among other examples, the Bogomolny equations.
Joint work with F. Haiden, L. Katzarkov, and M. Kontsevich.
Frederic Rochon: Geometry at infinity of the moduli space of centered monopoles of type (1, . . . , 1)
The moduli space of centered monopoles of type (1, . . . , 1) is a Taub–NUT deformation of maximal order of the Euclidean metric. We will use this fact to show that the natural metric on the moduli space is a quasi-fibered boundary metric. This will allow us to identify the tangent cone at infinity, to compute the reduced L2-cohomology and to prove that Sen’s S-duality conjecture holds for centered monopoles of type (1, . . . , 1).
Andy Royston: Interfaces and boundaries for the extended Bogomolny equations: a holographic perspective
I review my work with Sophia Domokos over the past eight years exploring aspects of the intersecting D3/D5 system in string theory. This system gives rise to a form of the holographic correspondence known as AdS/dCFT. We find that the extended Bogomolny equations (EBEs) play a role in the description of BPS states on both sides of the correspondence. On one side, the EBEs are defined on ℝ3 with an ℝ2 interface on which jumping data lives. On the other side, the EBEs are defined on a Euclidean half-space ℝ2 × [0,∞). The holographic correspondence implies relationships for each of these systems to others that may be of interest. I describe our work in progress towards understanding these relationships.
Filippo Sotgiu: Instantons on the Atiyah–Hitchin manifold
We construct SU(2)-instantons on the Atiyah–Hitchin manifold as superpositions of “fundamental constituents”. We classify the admissible boundary conditions and predict the dimension of the moduli space for each one. The constituents are the monopoles and “rotated monopoles” already studied in the case of ℝ3 × S1 by Foscolo and Ross, plus a rigid, SO(3)-invariant instanton which is specific to the Atiyah–Hitchin manifold. We explain how this result can be used as a stepping stone towards the construction of large families of Yang–Mills instantons over all ALF hyperkähler manifolds from similar constituents.
This talk is based on joint work with Lorenzo Foscolo, Calum Ross and Jakob Stein.
Weifeng Sun: Gauge theory with knot singularities
Singular solutions of gauge-theoretic equations provide a bridge between geometric analysis and knot theory. I will begin with a brief introduction to Kronheimer–Mrowka’s construction of instanton Floer homology for knots. Motivated by this picture, I will discuss several other gauge theories with singularities that may have connections to knot theory. One example is the Bogomolny equation on ℝ3 with singularities along a knot. I will describe my work on local singularity models and the analysis of the associated moduli spaces. Another direction is Witten’s proposal to recover the Jones polynomial and Khovanov homology through gauge theory. I will explain this proposal and how it leads to the extended Bogomolny equations.
Arya Yae: Metric convergence from parabolic Higgs bundle moduli spaces to hyperpolygon spaces
The moduli space of parabolic Higgs bundles on a Riemann sphere with n marked points is a hyperkähler manifold with a rich geometric structure. Points in the moduli space correspond to solutions to Hitchin’s equations, which arise via dimensional reduction of the Yang–Mills equations. These solutions are generally very hard to find, and we cannot describe the moduli space metric explicitly. Thus, in order to prove geometric results, we rely on gauge-theoretic techniques. We show that under a fine-tuned degeneration of the moduli parameters, the hyperkähler metric on the moduli space converges pointwise to that of an embedded Nakajima quiver variety called hyperpolygon space. The proof uses a construction of model solutions to Hitchin’s equations near the marked points, a delicate gluing procedure, and an application of the analytic implicit function theorem in the degenerate limit to perturb the approximate solution to an actual solution.
This talk is based on my recent paper arXiv:2601.10656 with Laura Fredrickson.
Jiajun Yan: A gauge-theoretic construction of the 4-cimensional hyperkähler ALE spaces
Non-compact hyperkähler spaces arise frequently in gauge theory. The 4-dimensional hyperkähler ALE spaces are a special class of complete non-compact hyperkähler spaces. They are in one-to-one correspondence with the finite subgroups of SU(2) and have interesting connections with representation theory and singularity theory, captured by the McKay correspondence. In this talk, we give a gauge-theoretic construction of these spaces, inspired by Kronheimer’s original construction via a finite-dimensional hyperkähler reduction. In the gauge-theoretic construction, we realize each ALE space as a moduli space of solutions to a system of equations for a pair consisting of a connection and a section of a vector bundle over an orbifold Riemann surface, modulo a hyperkähler gauge group action.
Yehao Zhou: Generalized chiral Potts models and hyperbolic monopoles from holomorphictopological Chern–Simons theories
The chiral Potts model is an integrable model which is exceptional in the sense that its spectral parameter lives on a surface of genus greater than one. Atiyah observed that the same spectral data appear in the study of SU(2) hyperbolic monopoles. In this talk, I will revisit and generalize the correspondence and explore the origin of the correspondence in the holomorphic-topological Chern–Simons theories.
This talk is based on joint work arXiv:2502.17545 with Moosavian and Yamazaki.