ICTS

This series of lectures is devoted to the geometry of moduli spaces of vortices on compact Riemann surfaces. Vortices can be regarded as solutions to some gauge theoretic equations involving a unitary connection on a fibre bundle and a Higgs field, or alternatively as pairs consisting of a holomorphic bundle and a holomorphic section of some associated bundle. To establish the link, a stability condition is required on the Higgs pair. Themes to be treated include abelian and non-abelian vortices, dimensional reduction of instantons and vortices, vortices and Hodge bundles, the Kähler-Yang-Mills equations and gravitating vortices.

Vortex moduli spaces support intrinsic geometry (their L^2 Kähler metric), which has been used to approximate the classical dynamics of vortices in low-energy field theory, e.g. via the associated geodesic flow. An extension of this idea is to use the same L^2 geometry to address the quantum mechanics of vortices in 2+1 dimensions via quantisation of their moduli spaces. There are various ways of doing this, depending on how the vortex dynamics is set up, and this series of lectures is meant to illustrate some of the possibilities. The holomorphic quantisation of the moduli space (relevant for first-order dynamics) will be discussed in the simplest example of vortices in line bundles. Some rudiments of the canonical quantisation of vortex moduli will also be presented, specifically the counting of states via L^2-Betti numbers, which is the viewpoint appropriate in a supersymmetric extension of second-order dynamics.

Vortex equations arise from supersymmetric quantum field theory (QFT) in describing certain BPS field configurations in three spacetime dimensions (or two spatial dimensions). In this minicourse, we will describe aspects of the corresponding QFTs and will extract from these vortex equations a category describing certain BPS line defects therein. Time permitting, we will use this physical setup to connect to notions in generalized affine Springer theory.

The low energy classical dynamics of topological solitons can often be modelled as geodesic motion in the space of static solitons with respect to a natural Riemannian metric called the L^2 metric. In this minicourse we will develop techniques to calculate this metric, and extract information about the resultant dynamics, for sigma model lumps and abelian vortices. A recurrent theme will be interesting phenomena arising due to noncompactness of the moduli space of static solitons.

Vortex equations arise from supersymmetric quantum field theory (QFT) in describing certain BPS field configurations in three spacetime dimensions (or two spatial dimensions). In this minicourse, we will describe aspects of the corresponding QFTs and will extract from these vortex equations a category describing certain BPS line defects therein. Time permitting, we will use this physical setup to connect to notions in generalized affine Springer theory.

Vortex moduli spaces support intrinsic geometry (their L^2 Kähler metric), which has been used to approximate the classical dynamics of vortices in low-energy field theory, e.g. via the associated geodesic flow. An extension of this idea is to use the same L^2 geometry to address the quantum mechanics of vortices in 2+1 dimensions via quantisation of their moduli spaces. There are various ways of doing this, depending on how the vortex dynamics is set up, and this series of lectures is meant to illustrate some of the possibilities. The holomorphic quantisation of the moduli space (relevant for first-order dynamics) will be discussed in the simplest example of vortices in line bundles. Some rudiments of the canonical quantisation of vortex moduli will also be presented, specifically the counting of states via L^2-Betti numbers, which is the viewpoint appropriate in a supersymmetric extension of second-order dynamics.

The low energy classical dynamics of topological solitons can often be modelled as geodesic motion in the space of static solitons with respect to a natural Riemannian metric called the L^2 metric. In this minicourse we will develop techniques to calculate this metric, and extract information about the resultant dynamics, for sigma model lumps and abelian vortices. A recurrent theme will be interesting phenomena arising due to noncompactness of the moduli space of static solitons.

This series of lectures is devoted to the geometry of moduli spaces of vortices on compact Riemann surfaces. Vortices can be regarded as solutions to some gauge theoretic equations involving a unitary connection on a fibre bundle and a Higgs field, or alternatively as pairs consisting of a holomorphic bundle and a holomorphic section of some associated bundle. To establish the link, a stability condition is required on the Higgs pair. Themes to be treated include abelian and non-abelian vortices, dimensional reduction of instantons and vortices, vortices and Hodge bundles, the Kähler-Yang-Mills equations and gravitating vortices.

3d N=4 gauge theories admit two topological twists, often called A and B, that are expected to lead mathematically to fully extended 3d topological quantum field theories (TQFTs). I will review some aspects of these putative TQFTs, some of their known and expected connections to representation theory, and (especially) connections to moduli spaces of vortices and their cohomology. I will then present some recent work on accessing these 3d TQFTs via boundary vertex algebras -- much as was done for Chern-Simons TQFT using the WZW model in the '80s and '90s. In particular, I will discuss using boundary vertex algebras to define braided tensor categories of line operators and to prove their equivalence under 3d mirror symmetry. (These developments in joint work with Andrew Ballin, Thomas Creutzig and Wenjun Niu.)

The enumerative geometry of vortex moduli spaces plays a key role in the study of supersymmetric gauge theories in three dimensions. For instance, in the presence of eight supercharges, field configurations contributing to the path integral can often be localised to moduli spaces of solutions of generalised vortex equations. In the presence of less supercharges, however, the path integral can receive additional contributions, which interplay in a remarkable way with those originating from vortex moduli spaces. In this talk, I will introduce some of these phenomena in simple, abelian examples, and comment on the expected relation to and potential implications for topics in mathematics such as wall-crossing.

Classical or quantized statistical mechanics of critically-coupled Abelian Higgs vortices can be modelled by free dynamics on the N-vortex moduli space, with N large. Vortex interactions are captured by the non-trivial moduli space geometry. To avoid boundary effects and satisfy Bradlow’s constraint, the vortices are defined on a compact surface of large area A, with A/N > 4π. The classical partition function depends only on the moduli space volume, and the first quantum correction at high temperature T depends on the integrated scalar curvature. Using these known geometrical quantities, we deduce the high-T equation of state of the vortex gas. When A/N is only slightly larger than 4π, the moduli space simplifies to complex projective space with its Fubini–Study geometry. Here the quantum partition function and equation of state can be calculated for any temperature. (NSM thanks S. Nasir, J. Baptista, J.M. Speight and S. Wang for their collaboration and contributions.)

3d N=4 gauge theories admit two topological twists, often called A and B, that are expected to lead mathematically to fully extended 3d topological quantum field theories (TQFTs). I will review some aspects of these putative TQFTs, some of their known and expected connections to representation theory, and (especially) connections to moduli spaces of vortices and their cohomology. I will then present some recent work on accessing these 3d TQFTs via boundary vertex algebras -- much as was done for Chern-Simons TQFT using the WZW model in the '80s and '90s. In particular, I will discuss using boundary vertex algebras to define braided tensor categories of line operators and to prove their equivalence under 3d mirror symmetry. (These developments in joint work with Andrew Ballin, Thomas Creutzig and Wenjun Niu.)

In this talk I will give an overview of joint work with V. Pingali and C. Yao in arXiv:1911.09616, where we give a complete solution to the existence problem for gravitating vortices on acompact Riemann surface with non-negative topological constant c > 0.

Elliptic stable envelopes were introduced by Aganagic and Okounkov as a key tool in the study of K-theoretic quasimap invariants for Nakajima quiver varieties and other symplectic resolutions. They bear an interesting relation to symplectic duality: the stable envelopes of a dual pair of resolutions M and N should arise from a “duality interface” living on the product M × N. I will give a gentle introduction to some of these ideas. I will then describe joint work with Artan Sheshmani and Shing-Tung Yau which uniformizes the duality interface of a hypertoric dual pair, yielding a distinguished K-theory class on an affine analogue of M × N.

By now it is known that many interesting phenomena in geometry and representation theory can be understood as aspects of mirrorsymmetry of 3d N = 4 SUSY QFTs. Such a QFT is associated to a hyperk ̈ahler manifold X equipped with a hyperhamiltonian action of a compact Lie group G and admits two topological twists. The first twist, which is known as the 3d B-model or Rozansky–Witten theory, is a TQFT of algebro-geometric flavor and has been studied extensively by Kapustin, Rozansky and Saulina. The second twist, which is known as the 3d A-model or 3d Seiberg–Witten theory, is a more mysterious TQFT of symplecto-topological flavor. In this talk I will discuss what is known about the 2-categories of boundary conditions for these two TQFTs. They are expected to provide two distinct categorifications of category O for the hyperk ̈ahler quotient X///G and 3d mirror symmetry is expected to induce a categorification of the Koszul duality between categories O for mirror symplectic resolutions. For abelian gauge theories this picture is work in progress with Ben Gammage and Aaron Mazel-Gee. This generalizes works of Kapustin–Vyas–Setter and Teleman on pure gauge theory.

We will review the close relation between 3-dimensional gauge theory (N=4 SUSY 3d gauge theory, in physics language) and the 2-dimensional A-model, focusing on the role of the Toda integrable system. The latter plays the role, in 3d, that the classifying space BG of a compact Lie group G plays in lower dimension. Compact symplectic manifolds with Hamiltonian G-action give boundary conditions for the 3d theory. We will review the Floer-theoretic construction of the latter, as well as a key geometric application to the quantum cohomology of GIT quotients of Fano manifolds. Much of this material is joint work with Dan Pomerleano. In the last portion of the lectures, we will quickly review the GLSM construction of Coulomb branches for 3d gauge theory and their relation to Gromov-Witten theory, along with more speculative comments on the case of quaternionic matter.

I will discuss the problem of counting embedded holomorphic curves in Calabi–Yau manifolds of complex dimension three or, more generally, in symplectic six-manifolds. While the nave count does not typically lead to an interesting geometric invariant, I will outline an ongoing project with Thomas Walpuski whose goal is to define an invariant of symplectic six-manifolds by counting embedded holomorphic curves with weight given by the number of solutions to certain gauge-theoretic equations called the ADHM vortex equations.

We define and compute I-functions and central charges for abelian GLSMs using virtual factorizations of Favero and Kim. In the Calabi–Yau case we provide analytic continuation for central charges by explicit integral formulas. The integrals in question are called hemisphere partition functions and we call the integral representation Higgs–Coulomb correspondence. We then use it to prove GIT stability wall-crossing for central charges.

We will review the close relation between 3-dimensional gauge theory (N=4 SUSY 3d gauge theory, in physics language) and the 2-dimensional A-model, focusing on the role of the Toda integrable system. The latter plays the role, in 3d, that the classifying space BG of a compact Lie group G plays in lower dimension. Compact symplectic manifolds with Hamiltonian G-action give boundary conditions for the 3d theory. We will review the Floer-theoretic construction of the latter, as well as a key geometric application to the quantum cohomology of GIT quotients of Fano manifolds. Much of this material is joint work with Dan Pomerleano. In the last portion of the lectures, we will quickly review the GLSM construction of Coulomb branches for 3d gauge theory and their relation to Gromov-Witten theory, along with more speculative comments on the case of quaternionic matter.

Gaiotto introduced the notion of a conformal limit of a Higgs bundle and conjectured that these should identify the Hitchin component with the oper stratum in the de Rham moduli space. In the case of closed Riemann surfaces this result was proven by Dumitrescu et al., and the limits were shown to exist much more generally by Collier and the speaker. In this talk I will report on progress in the case of parabolic Higgs bundles, which were the context of Gaiotto’s original conjecture. (This is joint work with B. Collier and L. Fredrickson.)