Schedule for: 17w5149 - The Analysis of Gauge-Theoretic Moduli Spaces

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Taub-NUT manifolds are possibly the simplest examples of four dimensional ALF manifolds; as the Taub-NUT are hyperkahler , their instantons admit a treatment in terms of algebraic geometry. We explain this geometry, in parallel with that of instantons on the flat R^3xS^1 model. Joint work with Sergey Cherkis.

By my joint work with Y.Takayama, bow varieties are Coulomb branches of affine quiver gauge theories of type A. The corresponding Higgs branches are quiver varieties of affine type A, and I constructed integrable highest weight representations of the corresponding affine Lie algebra on top degree homology groups of their lagrangian subvarieties many years ago. In this talk, I will explain a construction of the same representations on top degree homology groups of certain lagrangian in bow varieties (work in progress). This is closely related to geometric Satake correspondence for affine Kac-Moody groups, proposed by Braverman-Finkelberg.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

We will explain how to construct complete Calabi-Yau metrics on $\mathbb{C}^n$ for $n\ge 3$ by smoothing singular Calabi-Yau cones and using suitable compactifications by manifolds with corners. Our examples are of Euclidean volume growth, but with tangent cone at infinity having a singular cross section. This is a joint work with Ronan J. Conlon.

A sequence of the Seiberg-Witten monopoles with two spinors can degenerate to a $\mathbb{Z}/2$ harmonic spinor. I will show that there are obstractions for $\mathbb{Z}/2$ harmonic spinors to be realizable as degenerations of the Seiberg-Witten monopoles.

In joint work with Mark Haskins and Johannes Nordström we develop a powerful new analytic method to construct complete non-compact G2-manifolds, that is, Riemannian 7-manifolds with holonomy the exceptional Lie group G2. Our construction starts with an asymptotically conical Calabi-Yau 3-fold B and a Hermitian Yang-Mills connection on a circle bundle M over B satisfying a necessary topological condition. The construction then produces a 1-parameter family of complete circle-invariant G2-holonomy metrics on the total space of M that collapses with bounded curvature to the original Calabi-Yau metric on B. The geometry at infinity of the G2-metrics we construct is that of a circle bundle over a Calabi-Yau cone with fibres of fixed finite length and it is therefore the 7-dimensional analogue of the asymptotic geometry of 4-dimensional ALF hyperkähler metrics. We illustrate the power of our construction by constructing infinitely many diffeomorphism-types of complete non-compact G2-manifolds (only a handful of such manifolds are currently known) and families of complete G2-holonomy metrics of arbitrarily high dimension (only rigid and 1-parameter families of such metrics are currently known).

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I will explain recent joint work with Rafe Mazzeo, Hartmut Wei{\ss} and Frederik Witt on the asymptotics of the natural $L^2$-metric $G_{L^2}$ on the moduli space $\mathcal M$ of rank-$2$ Higgs bundles over a Riemann surface $X$ as given by the set of solutions to the self-duality equations $$ \begin{cases} 0=\bar\partial_A\Phi\\ 0=F_A+[\Phi\wedge\Phi^{\ast}] \end{cases} $$ for a unitary connection $A$ and a Higgs field $\Phi$ on $X$. I will show that on the regular part of the Hitchin fibration $(A,\Phi)\mapsto\det\Phi$ this metric is well-approximated by the semiflat metric $G_{\operatorname{sf}}$ coming from the completely integrable system on $\mathcal M$. This also reveals the asymptotically conic structure of $G_{L^2}$, with the (generic) fibres of the above fibration being asymptotically flat tori. This result confirms some aspects of a more general conjectural picture made by Gaiotto, Moore and Neitzke. Its proof is based on a detailed understanding of the ends structure of $\mathcal M$. The analytic methods used here in addition yield a complete asymptotic expansion of the difference $G_{L^2}-G_{\operatorname{sf}}$ between the two metrics, with leading order term decaying at some polynomial rate as $\|\Phi\|\to\infty$.

Hitchin's equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichm\"uller theory, and the geometric Langlands correspondence. In this talk, I'll describe what solutions of $SL(n,\mathbb{C})$-Hitchin's equations ``near the ends'' of the moduli space look like, and the resulting compactification of the Hitchin moduli space. Wild Hitchin moduli spaces are an important ingredient in this construction. This construction generalizes Mazzeo-Swoboda-Weiss-Witt's construction of $SL(2,\mathbb{C})$-solutions of Hitchin's equations where the Higgs field is ``simple.''

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I will discuss the application of techniques initially developed to study singularities of Yang Mill's fields and harmonic maps to obtain Betti number bounds, especially for negatively curved manifolds.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Bow moduli spaces have various applications. They are isomorphic to moduli spaces of Yang-Mills instantons on Asymptotically Locally Flat spaces. We begin by defining the instantons topological charges and identifying the relevant balanced and cobalanced bow representations. Bow moduli spaces can also be identified with moduli spaces of vacua of quantum supersymmetric gauge theories. We identify gauge theories for which bows deliver both their Higgs and Coulomb branches. Moreover, for these theories the gauge theory mirror symmetry acts as isometry interchanging the types of the branches. Two independent approaches are presented: 1. the generalizations of the ADHM-Nahm transform, that we call Up and Down transforms (developed in collaboration with Mark Stern and Andres Larrain-Hubach) and 2. the monad construction approach (developed in collaboration with Jacques Hurtubise).

Among the set of all finite dimensional HyperKahler cones, there is a special family, constructed using quivers, which we term "minimally unbalanced" quivers. This family turns out to solve a problem in super symmetric gauge theories in five dimensions. After an introduction to HyperKahler cones and Coulomb branches, I will present the family and discuss its properties, perhaps going over some physics of five dimensions.

For a compact Riemann surface of genus $g\ge 2$, the components of the moduli space of $\text{Sp(4}\text{,}\mathbb{R}\text{)}$-Higgs bundles are partially labeled by an integer $d$, known as the Toledo invariant. Of particular interest is the subspace for which this integer attains a maximum, and this subspace has been shown to have $3\cdot {{2}^{2g}}+2g-4$ connected components. In the search of model Higgs bundles parametrizing these components, a gluing construction between parabolic $\text{Sp(4}\text{,}\mathbb{R}\text{)}$-Higgs bundles plays an important role. This construction is coined in terms of solutions to Hitchin's equations, where existing analytic techniques have been quite effective.

I will describe a Kobayashi-Hitchin correspondence for the extended Bogomolny equations on the product of a Riemann surface and $R^+$ with Nahm pole boundary conditions and knot singularities at $\Sigma \times \{0\}$. This is joint work with Rafe Mazzeo

A Hodge elliptic genus is a refinement of the complex elliptic genus of a compact Calabi-Yau manifold. While the complex elliptic genus is an invariant which Calabi-Yau manifolds share with certain superconformal field theories, there are several competing versions of Hodge elliptic genera, the first of which were introduced by Kachru and Tripathy, motivated through string theory. For K3 surfaces, some of these Hodge elliptic genera yield invariants, whose properties are still under investigation. In my talk, I will discuss these new invariants, emphasizing that not all of the elliptic genera should be expected to have an impact in superconformal field theory.

The Basu-Harvey-Terashima equations are a lift of Nahm's equations to a branched double cover of a spectral curve. I'll explain how these equations arise, their relation with Lie superalgebras and discuss their interpretation in terms of flows on Jacobians. Time permitting, I"ll mention putative applications to hyperkaehler geometry.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The monopole moduli spaces ${\mathcal M}_k$ have a number of `asymptotic regions'. In the compactification of ${\mathcal M}_k$ as a manifold with corners, the intersections of these regions correspond to corners of codimension $>1$. A neighbourhood in the moduli space of a corner of codimension $d$ parameterizes families of monopoles obtained by gluing monopoles at a configuration of points characterized by $d$ different length-scales. This talk, which is based on joint work with Karsten Fritzsch and Chris Kottke, will report on the uniform construction of such families of monopoles.

In this talk I will describe the beginnings of a new approach to boundary value problems for generalized self-duality equations, such as the Haydys--Witten and extended Bogomolny equations, via the holographic correspondence in string theory. This is based on work done in collaboration with Sophia Domokos.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Abelian Higgs Vortex Equations on a Riemann surface S with smooth metric g are fascinating to study. The existence of N-vortex solutions is well established for a certain range of parameters, and the moduli space is the symmetrised Nth power of S. For exotic parameter values, results are less clear, even though for some g with constant curvature the equations are integrable, and solutions are explicitly known. A vortex solution can be used to define a non-smooth metric on S, the Baptista metric. This metric has conical singularities, with angular excess an integer multiple of $2 \pi$. There is a natural gauge theoretic metric on the moduli space of vortices, but it remains unclear if this is a standard metric on the moduli space of conical (Baptista) metrics.

Abstract: The moduli space of hyperpolygons is a finite-dimensional hyperkaehler quotient that extends polygon space and serves as a distinguished locus within the moduli space of parabolic Higgs bundles on the punctured Riemann sphere. In the spirit of the work of Mazzeo-Swoboda-Weiss-Witt on ordinary Higgs bundles, we use properties of the hyperkaehler moment map for hyperpolygon space to construct a limiting sequence that terminates in a moduli space of degenerate hyperpolygons. We use this partial compactification to determine and speculate upon some of the global geometric properties of hyperpolygon space. This is joint work with Hartmut Weiss.

In this joint work with Rafe Mazzeo, we would like to understand the deformation theory of constant curvature metrics with prescribed conical singularities on a compact Riemann surface. We construct a resolution of the configuration space, and prove a new regularity result that the family of constant curvature conical metrics has a nice compactification as the cone points coalesce. This is one key ingredient to understand the full moduli space of such metrics with positive curvature and cone angles bigger than $2\pi$.

The Hitchin fibration is a natural tool through which one can understand the moduli space of Higgs bundles and its interesting subspaces (branes). After reviewing the type of questions and methods considered in the area, we shall dedicate this talk to the study of certain branes which lie completely inside the singular fibres of the Hitchin fibrations. Through Cayley and Langlands type correspondences, we shall provide a geometric description of these objects, and consider the implications of our methods in the context of representation theory, Langlands duality, and within a more generic study of symmetries on moduli spaces.

Moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. For representations into complex semisimple Lie groups, the components of these spaces are labeled by obvious topological invariants. This is no longer true if one restricts to real forms of the complex groups, in which case factors other than the obvious invariants lead to the existence of extra `exotic' components which can have special significance. Formerly, all known instances of such exotic components were attributable to one of two distinct mechanisms. Recent Higgs bundle results for the groups SO(p,q) shed new light on this dichotomy and reveal new examples outside the scope of the two known mechanisms. This talk will survey what is known about the exotic components and describe the new SO(p,q) results.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.