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# Timetable (MAMW01)

## Metric and Analytic Aspects of Moduli Spaces

Monday 27th July 2015 to Friday 31st July 2015

 09:00 to 09:50 Registration INI 1 09:50 to 10:00 Welcome from John Toland (INI Director) INI 1 10:00 to 11:00 Folded hyperKähler metrics Nigel Hitchin recently introduced folded hyperKähler metrics in his study of the limit of higher Teichmüller spaces. I will explain a construction of such metrics. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 H Auvray (Université Paris-Sud)An analytic construction of dihedral ALF gravitational instantons Gravitational instantons are 4-dimensional complete non-compact hyperkähler manifolds with some curvature decay at infinity. The asymptotic geometry of these spaces plays an important role in a conjectural classification; for example, instantons of euclidean, i.e. quartic, large ball volume growth, are completely classified by Kronheimer, whereas the cubic regime, i.e. the {\it ALF (Asymptotically Locally Flat)} case, is not fully understood yet. More precisely, ALF instantons with {\it cyclic topology at infinity} are classified by Minerbe; by contrast, a classification in the {\it dihedral} case at infinity is still unknown. A wide, conjecturally exhaustive, range of dihedral ALF instantons were constructed by Cherkis-Kapustin, adopting the moduli space point of view, and studied explicitly by Cherkis-Hitchin. I shall explain in this talk another construction of such spaces, based on the resolution of a Monge-Ampère equation in ALF geometry. INI 1 12:30 to 13:30 Lunch at Wolfson Court 13:30 to 14:30 Discussion INI 1 14:30 to 15:30 P Boalch (Université Paris-Sud)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 were 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"). INI 1 15:30 to 16:00 Afternoon Tea 16:00 to 17:00 Kähler metrics and Chern forms on the moduli space of punctured Riemann surfaces I will review known properties of the WP and TZ metrics on the moduli space of punctured Riemann surfaces, discuss Chern forms of the associated line bundles and some open problems. This is a joint work with J. Park and L.P. Teo. INI 1 17:00 to 18:00 Welcome Wine Reception
 09:00 to 10:00 On the geometry of some Hyperkaehler manifolds I will discuss the geometry of some hyperkaehler manifolds : the QALE geometry of the Hilbert scheme of n-points in the complex plane or the QAC geometry of the cotangent bundle of Grassmannian. INI 1 10:00 to 11:00 X Zhu (Massachusetts Institute of Technology)Nodal degeneration of hyperbolic metrics and application to Weil-Petersson metric on the moduli space This is joint work with Richard Melrose. We analyze the behavior of the Laplacian on the fibres of a Lefschetz fibration and use it to describe the behavior of the constant curvature metric on a Riemann surface of genus $>1$ undergoing nodal degeneration. We apply this to deduce the asymptotics of the Weil-Petersson metric on the moduli space $\mathcal{M}_g$. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Asymptotics of hyperboilic, Weil-Peterssen and Takhtajan-Zograf metrics This will be a continuation of the talk by Xuwen Zhu on our joint work concerning the regularity of the fibre hyperbolic metrics up to the singular fibres for Lefschetz fibrations. In particular this applies to the universal curve over moduli space. I will discuss the marked case with the moduli space $\mathcal{M}_{g,n}$ of surfaces of genus $g$ with $n$ ordered distinct points in the stable range, $2g+n\ge3.$ As in the unmarked case the description of the regularity of the fibre hyperbolic metrics, up to the divisors forming the `boundary' of the Knudsen-Deligne-Mumford compactification, implies boundary regularity for the Weil-Peterssen metric. In this case it also leads to an asymptotic description of the Takhtajan-Zograf metric which contributes to the Chern form of the determinant bundle for $\bar\partial$ on the fibres of the universal curve. INI 1 12:30 to 13:30 Lunch at Wolfson Court 13:30 to 14:30 Discussion INI 1 14:30 to 15:30 Renormalized volume on the Teichmuller space of punctured Riemann surfaces We define and study the renormalized volume for geometrically finite hyperbolic 3-manifolds that may have rank-1 cusps. We prove a variation formula, and show that for certain families of convex co-compact hyperbolic metrics degenerating to a geometrically finite hyperbolic metric with rank-1 cusps, the renormalized volume converges to the renormalized volume of the limiting metric. INI 1 15:30 to 16:00 Afternoon Tea 16:00 to 17:00 LD Saper (Duke University)Perverse sheaves on compactifications of locally symmetric spaces Perverse sheave have important applications to representation theory, number theory, and algebraic geometry. I will discuss work in progress to understand the category of perverse sheaves on the Baily-Borel compactification of a Hermitian locally symmetric space; the method is to first work on the less singular reductive Borel-Serre compactification and then push down. Along the way I will introduce these various compactifications and give examples. INI 1
 09:00 to 10:00 R Bielawski (Leibniz Universität Hannover)Asymptotics and compactification of monopole moduli space I shall give several (equivalent) descriptions of asymptotic metrics on various regions of SU(2)-monopole moduli spaces, as well as of their gluing which provides an asymptotic description of SU(2)-monopole metric. I shall also describe the common compactification of the monopole moduli space and of the above asymptotic approximation. INI 1 10:00 to 11:00 Coulomb branches of 3-dimensional $\mathcal N=4$ gauge theories Let $M$ be a quaternionic representation of a compact Lie group $G$. Physicists study the Coulomb branch of the 3-dimensional supersymmetric gauge theory associated with $(G,M)$, which is a hyper-Kaehler manifold, but have no rigorous mathematical definition. When $M$ is of a form $N\oplus N^*$, we introduce a variant of the affine Grassmannian Steinberg variety, define convolution product on its equivariant Borel-Moore homology group, and show that it is commutative. We propose that it gives a mathematical definition of the coordinate ring of the Coulomb branch. If time permits, we will discuss examples arising from quiver gauge theories. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 ALG and the SU($\infty$) Toda equation The purpose of the talk is to pose a question about 4-dimensional moduli spaces of Higgs bundles: what are the minimal assumptions one needs to prove that the moduli space is ALG? INI 1 12:30 to 13:30 Lunch at Wolfson Court 16:00 to 17:00 'Breakout' Session Format: 2 x (10 min introduction + 5-10min discussion) M. Singer - Spectral Curves: What are they good for? B. Schroers - Monopole Clouds: What are they? Problems: For those who have a problem that they can clearly state in less than 3mins and explain why it is interesting in less than (a further) 2mins INI 1 19:30 to 22:00 Conference Dinner at Emmanuel College
 09:00 to 10:00 T Hausel (EPFL - Ecole Polytechnique Fédérale de Lausanne)Hyperkähler toy models Motivated by visualizing hyperkähler moduli spaces, we present here examples, typically four dimensional, which can serve as toy models understanding their shape. INI 1 10:00 to 11:00 Quantization of integrable systems of periodic monopoles I will review what we learned over the recent years on the quantization of the periodic monopole moduli spaces appearing as the phase spaces of algebraic integrable systems in connection with the supersymmetric gauge theories (Seiberg-Witten integrable systems). INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Schiffer variations and Abelian differentials Given a Riemann surface and an Abelian differential, we consider Cech style deformations based at zeros of the differential. Deformations are given in terms of slit mappings, degenerate Schwarz Christoffel mappings. We describe the associated deformation cocycles valued in vector fields. Schiffer gave an exact formula for the change in the double pole Green's function corresponding to his conformal gluing deformation. We follow his approach and develop the second order variation formula for the double pole Green's function. Consequences are second order variation formulas for Abelian differentials and for the Riemann period matrix. The second variation of the period matrix is in the style of Rauch's celebrated formula and is given in terms of the 2-jet of the corresponding differentials at the base point zero. Applications may include the Teichmuller geodesic flow on the space of Abelian differentials and the curvature of the Siegel upper half space metric on the image of Teichmuller space by the period matrix mapping. INI 1 12:30 to 13:30 Lunch at Wolfson Court 13:30 to 14:30 Discussion 14:30 to 15:30 The charge density of a monopole and its asymptotic tail Co-author: Daniel Nogradi Magnetic monopoles are finite-energy solutions of the Yang-Mills-Higgs equations in non-abelian gauge theory. From afar, they resemble sources of magnetic charge in Maxwell electromagnetism (hence the name). It is a long-standing problem to identify a smooth magnetic charge density which induces the asymptotic magnetic field of a monopole. In this talk I will present a novel solution to this problem. I define a charge density by summing the squared norms of an L^2-orthonormal basis for the kernel of a Dirac operator associated with the monopole -- this is the analog for monopoles of the Bergman kernel in Kaehler geometry. I will show that the expansion of its induced magnetic field agrees with the asymptotic field of the monopole, to all orders in 1/r. I will also discuss the explicit evaluation of this asymptotic field. INI 1 15:30 to 16:00 Afternoon Tea 16:00 to 17:00 The elliptic genus - a view from conformal field theory The elliptic genus of a compact Calabi-Yau manifold Y is a weak Jacobi form of weight zero. It is a topological invariant of Y which allows a reinterpretation as part of the partition function of so-called associated conformal field theories. As such, the coefficients of the elliptic genus count generic dimensions of spaces of states across the moduli space of such theories. On the other hand, if the theories have extended supersymmetry, then this conformal field theoretic view on the elliptic genus implies a novel decomposition of the underlying virtual bundle. This phenomenon occurs, for example, if Y is a hyperKaehler manifold. The talk will review the elliptic genus from this perspective in terms of geometric data. INI 1
 09:00 to 10:00 Polynomial Pick forms for affine spheres, real projective polygons, and surface group representations in PSL(3,R). Abstract: (Joint work with David Dumas.) Discrete surface group representations into PSL(3, R) correspond geometrically to convex real projective structures on surfaces; in turn, these may be studied by considering the affine spheres (an interpretation of the Hitchin system of equations in this case) which project to the convex hull of their universal covers. As a sequence of convex projective structures leaves all compacta in its deformation space, a subclass of the limits is described by polynomial cubic differentials on affine spheres which are conformally the complex plane. We show that those particular affine spheres project to polygons; along the way, a strong estimate on asymptotics is found, which translates to a version of Stokes data. We begin by describing the basic objects and context and conclude with a sketc h of some of the useful technique and an application. INI 1 10:00 to 11:00 Mass in Kaehler Geometry Given an ALE (asymptotically locally Euclidean) Riemannian manifold, one can define a real number called its mass that measures an important feature of the asymptotic geometry. In this lecture, I will describe a new result that offers a reinterpretation of the mass of ALE Kaehler manifolds. In the AE (asymptotically Euclidean) case, this not only implies the positive mass theorem for Kaehler manifolds, but also yields a Penrose-type inequality for the mass. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Coulomb Branch and the Moduli Space of Instantons The moduli space of k G instantons on C^2, where G is a classical gauge group, has a well known HyperKahler quotient formulation known as the ADHM construction. The extension to exceptional groups is an open problem. In string theory this is realized using a system of branes, and the moduli space of instantons is identified with the Higgs branch of a particular supersymmetric gauge theory with 8 supercharges. A less known, and less studied aspect of moduli spaces of instantons is that they can be realized as the Coulomb branch of a supersymmetric gauge theory in 2+1 dimensions. Recent developments on the understanding of the Coulomb branch gives us a nice solution to the problem where G is an exceptional group, thus allowing a systematic study of these moduli spaces. I will discuss these developments, and present the corresponding quivers, and the Coulomb branch Hilbert Series - the main tool which lead to the recent progress. INI 1 12:30 to 13:30 Lunch at Wolfson Court