## NS@1

• NS@1

Reminiscences by Narasimhan and Seshadri (as well as others) of the historical context out of which the Theorem was born.

## Courses

• C Sabbah

Twistor D-modules

Lecture 1: Overview from the Narasimhan-Seshadri theorem to the theorems of T. Mochizuki

Lecture 2: Introduction to twistor D-modules

Lecture 3: Tameness and wildness

• Biquard

Milnor-Wood inequality from the Higgs bundle viewpoint

Biquard: Milnor-Wood inequality from the Higgs bundle viewpoint

I will describe how the Milnor-Wood inequality follows from the Higgs bundle viewpoint, using some constructions on symmetric spaces of noncompact type, and I will describe some applications like the Cayley transform.

• Boalch

Connections on curves and wild character varieties

The character varieties of algebraic curves are spaces of representations of the fundamental group of the curve into a Lie group G. For smooth complex curves they thus arise as conjugacy classes of 2g tuples of group elements (a_i,b_i) satisfying a single relation of the form

(*) [a_1,b_1]...[a_g,b_g] = 1

where [a,b] is the group commutator. For irreducible unitary representations it has been known since 1965 that the character varieties have a Kahler structure and are isomorphic to the algebraic moduli space of stable rank n degree zero vector bundles. For geometers coming of age in Britain in the 90s almost every talk seemed to involve such spaces and the equation (*). The underlying symplectic structure is topological and may be approached algebraically by setting up a multiplicative version of Hamiltonian geometry and viewing the left of (*) as a group valued moment map, so the character variety is a "multiplicative" symplectic quotient (Alekseev-Malkin-Meinrenken 1997, reinterpreting work of Guruprasad-Huebschmann-Jeffrey-Weinstein 1995). For G=GL_n(C) the geometry is richer still and Hitchin showed the character variety is a hyperkahler manifold. The aim of this course is to describe a large class of multiplicative symplectic quotients generalising (*), which still admit hyperkahler metrics. They arise by considering meromorphic connections on curves, and so make contact with the classical theory of differential equations in the genus zero case. Amongst the new features that occur in this story are the wild mapping class groups, the global Weyl groups, and several gravitational instantons that had been missed by the physicists.

• Pantev

Foliations in derived geometry, symplectic structures, and potentials

I will explain how Lagrangian foliations in shifted symplectic geometry give rise to global potentials. I will give natural constructions of isotropic foliations on moduli spaces and will discuss the associated potentials. I will also give applications to the moduli of representations of fundamental groups and to non-abelian Hodge theory. This is based on joint works with Calaque, Katzarkov, Toen, Vaquie, and Vezzosi.

• Du Pei

A New TQFT from Equivariant Integration over Moduli Space of Higgs Bund les

## NS Reviews

• Balaji

The Narasimhan-Seshadri theorem and the geometry of principal bundles.

I will survey the development in the geometry of principal bundles and its generalizations from the standpoint of the Narasimhan-Seshadri theorem.

From Narasimhan-Seshadri to Uhlenbeck-Yau and beyond

A compelling measure of a theorem’s significance is the extent to which it can be generalized. To describe, even cursorily, the full range of generalizations of the Naraismhan-Seshadri theorem requires more than one talk. In this talk we will concentrate on just two directions: from closed Riemann surfaces to higher dimensional closed complex manifolds, and from single holomorphic bundles to more general collections of holomorphic bundles with additional structures.

Higgs bundles on Riemann surfaces

We give an introduction to the theory of G-Higgs bundles on Riemann surfaces, with special emphasis on the role of Higgs bundles as a generalizaion of the the Narasimhan-Seshadri theorem for non-compact reductive Lie groups (both complex and real).

• Nagaraj

Parabolic bundles and Parabolic Higgs bundles

• Nitsure

The Narasimhan-Seshadri theorem is closely tied with the theory of deformations and the construction of moduli spaces for representations, for simple bundles and for stable bundles. This lecture is designed to give a rapid introduction to the Narasimhan-Seshadri theorem, highlighting these aspects.

The Narasimhan-Seshadri Theorem and theoretical physics

I will describe how the Theorem relates to Gauge Theory and Conformal Field Theory.

## Talks

• Andersen

Quantization of the moduli spaces of flat SU(n) and SL(n,C) connections

We will discuss the geometric quantisation of both the moduli spaces of flat SU(n) and SL(n,C) connections. The main issue in both cases is to demonstrate invariant of the chosen polarisation need in geometric quantisation. This is done by construction of the Hitchin-Witten connection in the bundle over Teichmüller space, which the quantisation procedure poses and by proving that this connection is projectively flat. This results in projective representations of mapping class groups, which respectively fit into SU(n) quantum Chern-Simons theory or equivalently the Witten-Reshetikhin-Turaev TQFT and into the SL(n,C) quantum Chern-Simons theory, which we have very recently constructed jointly with Rinat Kashaev by more combinatorial means.

• Berczi

Tautologial integrals on curvilinear Hilbert schemes

The punctual Hilbert scheme of k points on a smooth projective variety X parametrises zero dimensional subschemes of X of length k supported at one point. A point of the punctual Hilbert scheme is called curvilinear if it sits in the germ of a smooth curve on X. The irreducible component of the punctual Hilbert scheme containing these points is called the curvilinear component. We give a description of the curvilinear Hilbert scheme as a projective completion of the non-reductive quotient of holomorphic map germs from the complex line into X by holomorphic polynomial reparametrisations. Using an algebraic model of this quotient and equivariant localisation we develop an iterated residue formula for tautological integrals over the curvilinear component. We discuss possible generalisations for other non-reductive moduli problems.

• Biquard

The Hitchin component for SL(\infty)

Nigel Hitchin recently introduced a notion of Higgs bundle for the group SL(\infty). I shall explain a construction of the corresponding Hitchin component

• Boalch

Non-perturbative hyperkahler manifolds

Following Kronheimer's construction of the ALE spaces from the ADE affine Dynkin graphs, and Kronheimer-Nakajima's subsequent extension of the ADHM construction, a large class of hyperkahler manifolds attached to graphs emerged, known as "quiver varieties". Nakajima has shown they play a central role in representation theory. If the underlying graph is of a special type it turns out that the corresponding quiver varieties have natural partial compactifications, which also admit complete hyperkahler metrics. They arise as spaces of solutions to Hitchin's equations on Riemann surfaces, with wild boundary conditions. (They are particular cases of the moduli spaces constructed in work with Biquard published in 2004). The class of graphs for which this works are known as "supernova" graphs and includes all the complete multipartite graphs. In particular the square and the triangle are supernova graphs, and so some gravitational instantons arise in this way. In this talk I will review this story focussing on specific examples and recent developments such as the algebraic construction of the underlying holomorphic symplectic manifolds (the "wild character varieties").

• Brambila-Paz

Coherent Higgs systems

The Hecke correspondence have been an important tool to study moduli spaces of stable vector bundles and of Higgs bundles. In this talk we generalize the Hecke correspondence and define the coherent Higgs systems. The aim is to study the moduli space of such systems. The work is in progress and is a joint work with Steve Bradlow, Peter Gothen and Oscar Garcia-Prada.

• Collier

Maximal SO(2,3) surface group representations and Labourie's conjecture

The nonabelian Hodge correspondence provides a homeomorphism between the character variety of surface group representations into a real Lie group G and the moduli space of G-Higgs bundles. This homeomorphism however breaks the natural mapping class group action on the character variety. Generalizing techniques and conjectures of Labourie for Hitchin representations, we restore the mapping class group symmetry for all maximal SO(2, 3) = PSp(4, R) surface group representations. More precisely, we show that for each maximal SO(2, 3) representation there is a unique conformal structure in which the corresponding equivariant harmonic map to the symmetric space is a conformal immersion, or, equivalently, a minimal immersion. This is done by exploiting finite order fixed point properties of the associated maximal Higgs bundles.

• Fernandez

Gravitating vortices, cosmic strings and the Kähler-Yang-Mills equations.

In joint work with Luis Álvarez-Cónsul and Oscar García-Prada, we study equations describing Abelian vortices on a Riemann surface with back reaction of the metric. The gravitating vortex equations are derived by dimensional reduction of the Kahler-Yang-Mills equations on the product of the complex projective line with a Riemann surface, and inherit their moment map interpretation. As a particular case of the gravitating vortex equations on P^1 we find the Einstein-Bogomol'nyi equations, whose solutions correspond to Nielsen-Olesen cosmic strings in the Bogomol'nyi phase. Applying the general theory for the Kahler-Yang-Mills equations, in this lecture we give evidence of an analogue of the Narasimhan-Seshadri Theorem for gravitating vortices. Using an existence theorem by Yisong Yang, our main result implies a Hitchin-Kobayashi correspondence for the particular case of the Einstein-Bogomol'nyi equations, and also a solution to a conjecture by Yang about the non-existence of cosmic strings on P^1 superimposed at a single point.

• Franco

Moduli spaces of Lambda-modules on abelian varieties

Let \Lambda be a D-algebra in the sense of Bernstein and Beilinson. Higgs bundles, vector bundles with flat connections, co-Higgs bundles... are examples of \Lambda-modules for particular choices of \Lambda. Simpson studied the moduli problem for the classification of \Lambda-modules over Kahler varieties, proving the existence of a moduli space Lambda-modules. Using the Polishchuck-Rothstein transform for modules of D-algebras over abelian varieties, we obtain a description of the moduli spaces of \Lambda-modules of rank 1. We also proof that polystable \Lambda module decompose as a direct sum of rank 1 \Lambda-modules. This allow us to describe the module spaces of arbitrary rank.

• Gomez

Torelli theorem for the parabolic Deligne-Hitchin moduli space

The Deligne-Hitchin moduli space of a complex algebraic curve is a partial compactification of the moduli space of $\lambda$-connections. It includes as closed subvarieties the moduli spaces of Hitchin bundles ($\lambda=0$), the moduli of holomorphic connections ($\lambda=1$), and the moduli of Hitchin bundles on the complex conjugate curve ($\lambda=\infty$). It can also be interpreted as the twistor space for the moduli space of Hitchin bundles. We show a Torelli theorem for a parabolic verion of this moduli space (joint work with David Alfaya).

• Heinloth

Coarse moduli spaces for parahoric bundles using a stack theoretic GIT condition

One of the applications of GIT is that it allows to single out parts of moduli problems that admit separated coarse moduli spaces. Similar to the methods developed (independently) by Daniel Halpern-Leistner one can also find stack theoretic criteria allowing to find separated substacks of moduli stacks, which sometimes allow one to avoid explicit GIT constructions. To illustrate the criterion we apply it to some generalizations of principal bundles, known as parahoric group schemes on curves. In characteristic 0 many of these spaces were constructed by Balaji and Seshadri in order to obtain an equivariant version of the Narasimhan -Seshadri theorem.

• Hurtubise

Monopoles on circle bundles.

Following Witten and Kapustin, one can view monopoles with Dirac type singularities as mediating a Hecke transform. We consider the case of self-Hecke transform, given by monopoles on the product of a Riemann surface and a circle; more generally, we look at the case of a monopole on a circle bundle over the Riemann surface. Here the relevant geometry on the base is Sasakian, and the Narasimhan-Seshadri correspondence is with some rather unusual gerbe-like objects. (Joint with Benoit Charbonneau, then Indranil Biswas.)

• Jaya Iyer

Higher-order Chern-Cheeger-Simons invariants

In this talk, we will present a construction of higher-order Chern-Simons invariants, associated to a variation of flat connections. We will indicate a proof to show that this provides a map on homology of moduli space of flat connections, and taking values in appropriate degree C/Z-cohomology of the underlying manifold.

• Oliveira

A quadratic pair on a compact Riemann surface $X$ is a generalisation of an orthogonal bundle over $X$. One motivation for the study of the moduli spaces of these objects comes from Higgs bundles for the real symplectic group $\mathrm{Sp}(2n,\mathbb{R})$, hence from representations of $\pi_1X$ in $\mathrm{Sp}(2n,\mathbb{R})$. We will explain this motivation and present some basic results on the geometry and topology of the moduli spaces of rank $2$ quadratic pairs.

• Parameswaran

Parabolic curves in positive characteristic

• Du Pei

Branes and mirror symmetry in Hitchin moduli space

• Peon

TBA

• Pym

Meromorphic connections and the Stokes groupoids.

To any effective divisor on a smooth complex curve, one can associate, in a canonical way, a Lie groupoid. This groupoid is a complex analytic surface that serves as the natural domain for the parallel transport of meromorphic connections on the curve with poles bounded by the divisor. I will describe the local and global structure of these groupoids, and their application to some classical topics in the analysis of ODEs: the "resummation" of divergent series solutions, and the theory of isomonodromic deformations. This talk is based on joint work with Marco Gualtieri and Songhao Li.

• Ramanan

Hyperelliptic curves and the Hitchin map

• Schaffhauser

Real Seifert manifolds and the Narasimhan-Seshadri correspondence.

We extend a result of Biswas, Huisman and Hurtubise on representations of the orbifold fundamental group of a compact connected Real Riemann surface of genus greater or equal to 2 into the group of unitary and anti-unitary transformations of a Hermitian vector space.

• Schaposnik

Higgs bundles, branes, and applications.

Higgs bundles (introduced by N. Hitchin in 1987) are pairs of holomorphic vector bundles and holomorphic 1-forms taking values in the endomorphisms of the bundle. The moduli space of Higgs bundles carries a natural Hyperkahler structure, through which we can study A-branes (Lagrangian subspaces) or B-branes (holomorphic subspaces) with respect to each structure.

We shall begin the talk by first introducing Higgs bundles for complex Lie groups and the associated Hitchin fibration, and recalling how to realize Langlands duality through spectral data. We will then look at a natural construction of families of subspaces which give different types of branes, and explain how the topology of some of these branes can be completely determined via the monodromy action of the Hitchin system. Finally, we shall give some applications of the above approaches in relation to Langlands duality and the study of character varieties. Some of the work presented during the talk is in collaboration with David Baraglia (Adelaide).

• Szabo

Spectral data for irregular Higgs bundles

We outline a general construction of spectral data for Higgs bundles with irregular singularities on curves. We then turn to specific low-dimensional examples. In some cases we carry out stability analysis.

• Thompson

Chern-Simons Theory with a Complex Gauge Group on a Seifert Rational Homology Sphere

• Venugopalan

Hitchin-Kobayashi correspondence for vortices on non-compact Riemann surfaces.

A Hitchin-Kobayashi correspondence relates stable holomorphic pairs (a holomorphic section on a Hermitian vector bundle and a holomorphic section on it) with the zeros of the vortex equation. This set up can be generalized - the vector bundle can be replaced by a fiber bundle, whose fibers are a Kaehler manifold with Hamiltonian action of a compact Lie group K. In this talk I present a Hitchin-Kobayashi correspondence for K-vortices on certain Riemann surfaces with inifinite volume. In the particular case of affine vortices - i.e. when the Riemann surface is the complex line, our result has applications in gauged Gromov-Witten theory.

• Weitsman

Recursion and vanishing relations in the cohmology ring of the moduli of parabolic vector bundles

We study a tautological subring of the moduli space of parabolic vector bundles, first in rank 2 and then in higher rank. By finding explicit geometric cycles Poincare dual to the generators of this ring, we show that, in rank 2, these classes satisfy recursion relations in the genus and number of marked points, analogous to the KdV and Virasoro relations in the cohomology of the moduli space of curves. The simplest of these relations gives a vanishing relation equivalent to the Newstead conjecture. We then discuss a generalization to higher rank in joint work with Adina Gamse.

• Wentworth

Towards a strange duality for odd orthogonal bundles

Strange duality is an isomorphism between spaces of nonabelian theta functions on Riemann surfaces. The motivation comes from the level-rank duality for conformal blocks that is associated to conformal embeddings of affine Lie algebras. The classical version of strange duality for vector bundles was proven independently by Belkale and Marian-Oprea, but there are many other examples of conformal embeddings for which one might expect dualities. In this talk I will discuss the case of moduli spaces of odd orthogonal bundles. In particular, I will explain the proof of a Verlinde type formula, conjectured by Oxbury-Wilson, for the space of twisted spin bundles, and the construction of a Hitchin connection. This is joint work with Swarnava Mukhopadhyay.

• Wilkin

Classification of Yang-Mills flow lines

The theorem of Narasimhan and Seshadri has an analytic interpretation (due to Donaldson) in terms of the Yang-Mills flow: a holomorphic bundle on a compact Riemann surface is polystable if it flows to a minimum of the Yang-Mills functional. A generalisation of this to unstable bundles (due to Daskalopoulos and Rade) says that the limit of the Yang-Mills flow is isomorphic to the graded object of the Harder-Narasimhan-Seshadri double filtration of the initial holomorphic structure, i.e. the limit of the flow is determined by the algebraic geometry of the initial condition. In this talk I will extend this analytic-algebraic correspondence further by showing that the pairs of Yang-Mills critical points which are connected by Yang-Mills flow lines have an algebraic description in terms of Narasimhan and Ramanan's Hecke correspondence.