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Monday, 06 February 2023
Time Speaker Title Resources
10:00 to 10:50 James Martin Speight (University of Leeds, UK) L^2 geometry of moduli spaces of vortices and lumps (Lecture 1)

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.

11:30 to 12:20 Nuno Romão (Augsburg University, Germany) Quantization of vortices (Lecture 1)

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.

14:30 to 15:20 Oscar García-Prada (ICMAT, Spain) Geometry of vortices on Riemann surfaces (Lecture 1)

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.

18:30 to 19:20 Chris Woodward (Rutgers University, USA) Symplectic vortices and the quantum Kirwan map (Lecture 1 online)

This minicourse is devoted to the relationship between gauged pseudoholomorphic maps to a symplectic manifold with Hamiltonian group actions and the pseudoholomorphic maps to its symplectic quotient. We will discuss the large-area Gaio-Salamon adiabatic limit of gauged maps as well as the small-area limit in which the moduli space essentially reduces to a classical symplectic quotient. In the Gaio-Salamon limit the Ziltener compactification of the moduli space of affine vortices naturally arises and the quantum Kirwan map is defined by integration over the Ziltener compactification.  We will explain the theory using the example of toric varieties as symplectic quotients of vector spaces.

Tuesday, 07 February 2023
Time Speaker Title Resources
10:00 to 10:50 Niklas Garner (University of Washington, USA) Categorical aspects of vortices (Lecture 1)

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.

11:30 to 12:20 James Martin Speight (University of Leeds, UK) L^2 geometry of moduli spaces of vortices and lumps (Lecture 2)

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.

14:30 to 15:20 Nuno Romão (Augsburg University, Germany) Quantization of vortices (Lecture 2)

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.

18:30 to 19:20 Chris Woodward (Rutgers University, USA) Symplectic vortices and the quantum Kirwan map (Lecture 2 online)

This minicourse is devoted to the relationship between gauged pseudoholomorphic maps to a symplectic manifold with Hamiltonian group actions and the pseudoholomorphic maps to its symplectic quotient. We will discuss the large-area Gaio-Salamon adiabatic limit of gauged maps as well as the small-area limit in which the moduli space essentially reduces to a classical symplectic quotient. In the Gaio-Salamon limit the Ziltener compactification of the moduli space of affine vortices naturally arises and the quantum Kirwan map is defined by integration over the Ziltener compactification.  We will explain the theory using the example of toric varieties as symplectic quotients of vector spaces.

Wednesday, 08 February 2023
Time Speaker Title Resources
10:00 to 10:50 Niklas Garner (University of Washington, USA) Categorical aspects of vortices (Lecture 2)

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.

11:30 to 12:20 Oscar García-Prada (ICMAT, Spain) Geometry of vortices on Riemann surfaces (Lecture 2)

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.

14:30 to 15:20 James Martin Speight (University of Leeds, UK) L^2 geometry of moduli spaces of vortices and lumps (Lecture 3)

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.

18:30 to 19:20 Chris Woodward (Rutgers University, USA) Symplectic vortices and the quantum Kirwan map (Lecture 3)

This minicourse is devoted to the relationship between gauged pseudoholomorphic maps to a symplectic manifold with Hamiltonian group actions and the pseudoholomorphic maps to its symplectic quotient. We will discuss the large-area Gaio-Salamon adiabatic limit of gauged maps as well as the small-area limit in which the moduli space essentially reduces to a classical symplectic quotient. In the Gaio-Salamon limit the Ziltener compactification of the moduli space of affine vortices naturally arises and the quantum Kirwan map is defined by integration over the Ziltener compactification.  We will explain the theory using the example of toric varieties as symplectic quotients of vector spaces.

Thursday, 09 February 2023
Time Speaker Title Resources
10:00 to 10:50 Niklas Garner (University of Washington, USA) Categorical aspects of vortices (Lecture 3)

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.

11:30 to 12:20 Oscar García-Prada (ICMAT, Spain) Geometry of vortices on Riemann surfaces (Lecture 3)

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.

14:30 to 15:20 Nuno Romão (Augsburg University, Germany) Quantization of vortices (Lecture 3)

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.

18:30 to 19:20 Chris Woodward (Rutgers University, USA) Symplectic vortices and the quantum Kirwan map (Lecture 4)

This minicourse is devoted to the relationship between gauged pseudoholomorphic maps to a symplectic manifold with Hamiltonian group actions and the pseudoholomorphic maps to its symplectic quotient. We will discuss the large-area Gaio-Salamon adiabatic limit of gauged maps as well as the small-area limit in which the moduli space essentially reduces to a classical symplectic quotient. In the Gaio-Salamon limit the Ziltener compactification of the moduli space of affine vortices naturally arises and the quantum Kirwan map is defined by integration over the Ziltener compactification.  We will explain the theory using the example of toric varieties as symplectic quotients of vector spaces.

Friday, 10 February 2023
Time Speaker Title Resources
10:00 to 10:50 James Martin Speight (University of Leeds, UK) L^2 geometry of moduli spaces of vortices and lumps (Lecture 4)

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.

11:30 to 12:20 Nuno Romão (Augsburg University, Germany) Quantization of vortices (Lecture 4)

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.

14:30 to 15:20 Oscar García-Prada (ICMAT, Spain) Geometry of vortices on Riemann surfaces (Lecture 4)

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.

16:30 to 17:20 Niklas Garner (University of Washington, USA) Categorical aspects of vortices (Lecture 4)

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.

Monday, 13 February 2023
Time Speaker Title Resources
10:00 to 10:50 Tudor Dimofte (University of Edinburgh, UK) 3d gauge theories: vortices and vertex algebras (Lecture 1)

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.)

11:30 to 12:20 Jeongseok Oh (Imperial College, UK) Quasimaps, their wall-crossings and mirror symmetry (Lecture 1)

Inspired by the role of loop spaces in the proof of the mirror theorem by his advisor (Alexander Givental), Bumsig Kim and his collaborators developed the theory of spaces of quasimaps. Interestingly, the change of their generating functions of invariants (defined by integration over these spaces) according to their stability conditions, known as a wall-crossing formula, has exactly the same form as mirror symmetry for certain potential functions predicted by physics. So it gives a geometric interpretation of mirror symmetry. A huge computational advantage is that the moduli spaces of quasimaps have less boundary components than those of stable maps, so that they become easier to work with. Together with the wall-crossing formula, we can compute genus g Gromov-Witten invariants for quintics, up to a few low degree invariants which remain unknown.
In this tribute talk to Bumsig Kim, I shall review his achievements in mirror symmetry through quasimaps.

14:30 to 15:20 Andrea Ferrari (Durham University, UK) Vortices, other saddles, and wall-crossing

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.

16:30 to 17:20 Mathew Bullimore (Durham University, UK) Vortices and generalised symmetry

There is much recent interest in generalised or categorical symmetries that go beyond the paradigm of groups and into the realm of higher groups and fusion categories. I will discuss aspects of such symmetries in the context of three-dimensional abelian GLSMs and moduli spaces of vortices. I will then speculate on generalised notions of equivariance in quasimap K-theory of toric stacks.

17:30 to 18:20 Nick Manton (University of Cambridge, UK) Statistical mechanics of vortices

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.)

Tuesday, 14 February 2023
Time Speaker Title Resources
10:00 to 10:50 Tudor Dimofte (University of Edinburgh, UK) 3d gauge theories: vortices and vertex algebras (Lecture 2)

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.)

11:30 to 12:20 Luis Álvarez-Cónsul (ICMAT, Spain) Obstructions to the existence of gravitating vortices

In this talk, I will start explaining some background about the self-dual Einstein–Maxwell–Higgs equations on a compact surface, and then analyze obstructions to the existence of solutions, expressed in terms of the multiplicities of the zeroes of the Higgs field and the vortex number. Joint work with Mario García-Fernandez, Oscar García-Prada, Vamsi Pritham Pingali and Chengjian Yao.

14:30 to 15:20 Mario García-Fernandez (ICMAT, Spain) Gravitating vortices with positive curvature

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.

17:30 to 18:20 Jeongseok Oh (Imperial College, UK) Quasimaps, their wall-crossings and mirror symmetry (Lecture 2)

Inspired by the role of loop spaces in the proof of the mirror theorem by his advisor (Alexander Givental), Bumsig Kim and his collaborators developed the theory of spaces of quasimaps. Interestingly, the change of their generating functions of invariants (defined by integration over these spaces) according to their stability conditions, known as a wall-crossing formula, has exactly the same form as mirror symmetry for certain potential functions predicted by physics. So it gives a geometric interpretation of mirror symmetry. A huge computational advantage is that the moduli spaces of quasimaps have less boundary components than those of stable maps, so that they become easier to work with. Together with the wall-crossing formula, we can compute genus g Gromov-Witten invariants for quintics, up to a few low degree invariants which remain unknown.
In this tribute talk to Bumsig Kim, I shall review his achievements in mirror symmetry through quasimaps.

18:30 to 19:20 David Favero (University of Alberta, Canada and University of Minnesota, USA) Cohomological field theories from GLSMs (Lecture 1)
Wednesday, 15 February 2023
Time Speaker Title Resources
10:00 to 10:50 Michael McBreen (CUHK, China) Uniformizing the elliptic stable envelopes of a hypertoric variety

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.

11:30 to 12:20 Mykola Dedushenko (Simons Center for Geometry and Physics, USA) Elliptic stable envelopes as interfaces in a 3d QFT

I will review the physical realization of elliptic stable envelopes within 3d N = 4 gauge theories. This consists of two steps: (1) explaining why the (Higgs phase) SUSY vacua on a two-torus are captured by the elliptic cohomology of the Higgs branch; (2) observing that a natural SUSY interface exists that interpolates between the massive and massless regimes of the theory.

16:30 to 17:20 Justin Hilburn (PITP, Canada) 2-categorical aspects of 3d mirror symmetry

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.

17:30 to 18:20 Dan Halpern-Leistner (Cornell University, USA) Infinite-dimensional geometric invariant theory and gauged Gromov–Witten theory

Harder–Narasimhan (HN) theory gives a structure theorem for vector bundles on a smooth projective curve. A bundle is either semistable, or it admits a canonical filtration whose associated graded bundle is semistable in a graded sense. After reviewing recent advances in extending HN theory to arbitrary algebraic stacks, I will discuss work in progress with Andres Fernandez Herrero to apply this general machinery to the stack of ”gauged” maps from a curve C to a G-scheme X, where G is a reductive group and X is projective over an affine scheme. Our main application is to use HN theory for gauged maps to compute generating functions for K-theoretic enumerative invariants known as gauged Gromov–Witten invariants. This problem is interesting more broadly because it can be formulated as an example of an infinite dimensional analog of the usual setup of geometric invariant theory, which has applications to other moduli problems.

18:30 to 19:20 David Favero (University of Alberta, Canada and University of Minnesota, USA) Cohomological field theories from GLSMs (Lecture 2)
Thursday, 16 February 2023
Time Speaker Title Resources
10:00 to 10:50 Constantin Teleman (UC Berkeley, USA) Gromov-Witten theory and gauge theory (Lecture 1)

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.

11:30 to 12:20 Sushmita Venugopalan (IMSc, India) Vortices on non-compact Riemann surfaces

Symplectic vortices on punctured Riemann surfaces are related by a Hitchin–Kobayashi correspondence togauged maps satisfying a semistability condition. These objects show up in various settings in gauged Gromov–Witten theory, such as in the definition of the quantum Kirwan map by Ziltener and Woodward using affine vortices; and in quasimaps to GIT quotients defined by Kim, Ciocan-Fontanine and Maulik.

14:30 to 15:20 Aleksander Doan (University College London, UK) Holomorphic curves and the ADHM vortex equations

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.

17:30 to 18:20 Mykola Dedushenko (Simons Center for Geometry and Physics, USA) Symplectic duality and the vortex partition function

Elliptic stable envelopes are known to appear in the 3d mirror symmetry of the “vertex function” (which encapsulates the K-theoretic count of vortices). Looking instead at the Higgs–Coulomb phase transition in a given 3d N = 4 theory (also known as symplectic duality), we explain this via the properties of the cigar partition function (or half-index) and the Janus interfaces from talk (i).

18:30 to 19:20 Chiu-Chu Melissa Liu (Columbia University, USA) Higgs–Coulomb correspondence and wall-crossing in abelian GLSMs

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.

Friday, 17 February 2023
Time Speaker Title Resources
10:00 to 10:50 Constantin Teleman (UC Berkeley, USA) Gromov-Witten theory and gauge theory (Lecture 2)

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.

11:30 to 12:20 Vamsi Pingali (IISc, India) The vortex ansatz as a fertile testing ground for certain systems of PDEs

I shall review the vortex bundle construction of García-Prada. Then I shall proceed to discuss the vortex ansatz as a way to dimensionally reduce several interesting systems of PDEs like the Kähler–Yang–Mills equations (and its offshoots) and the vector bundle Monge–Amprère equation. I shall discuss the latter in detail and end with an advertisement for studying fully nonlinear systems of PDEs arising from vector bundles (and the fact that the vortex ansatz can prove to be a powerfultesting ground for them).

14:30 to 15:20 Richard Wentworth (University of Maryland, USA) Conformal limits of parabolic Higgs bundles

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.)