# The Analysis of Gauge-Theoretic Moduli Spaces (17w5149)

Arriving in Banff, Alberta Sunday, August 27 and departing Friday September 1, 2017

## Organizers

Sergey Cherkis (University of Arizona)

Michael Singer (University College London)

Rafe Mazzeo (Stanford University)

## Objectives

There are a large number of basic questions in this area:

a) Moduli spaces arising in gauge-theoretic problems carry $L^2$- (or Weil-Petersson type) metrics which in many cases have interesting special structure. For example, they may be Kaehler or even hyperKaehler. We obtain, in this way, remarkable families of complete hyperKaehler (and hence Ricci-flat) metrics. The asymptotic structure of these moduli space metrics seems to fall into some generalization of the ALE/ALF/ALG/ALH classes that are well-known in the four-dimensional setting in the context of gravitational instantons. In fact, these higher dimensional moduli spaces appear to typically have the structure of a singular torus fibration over a quasi-asymptotically conic space with a generalized `fiber-boundary’ type metric. Proving that this is really the case, not only in special cases but perhaps as a general structural feature, is a guiding question. As a special subcase of this, we mention the moduli space of monopoles on $R^3$, studied initially by Taubes, Atiyah, Gibbons-Manton and many others. Dramatic new progress in understanding its metric structure at infinity, phrased in terms of a compactification as a manifold with corners, is the subject of intense ongoing work.

b) One application coming from the understanding of these metric asymptotics is that it then becomes possible to study the space of $L^2$ harmonic forms, which has direct physical meaning. For the monopole moduli spaces, these spaces of harmonic forms are the subject of the famous Sen conjectures. While no mathematical progress has been made on this question for the past decade, new techniques from geometric microlocal analysis and elsewhere have now matured to the point where these types of problems may soon become tractable. The quantum field theory and string theory interpretation of these moduli spaces suggests predictions for the dimensions of the spaces of their $L^2$ harmonic forms. The mathematical theory should lead to an interpretation of these dimensions in terms of intersection cohomology groups of certain compactifications of these moduli spaces. The key problems involve first establishing the necessary mathematical theory, and then verifying these predictions.

c) Since the work of Seiberg and Witten, the intriguing isometry between the moduli spaces of classical gauge theory solitons, such as monopoles or instantons, and moduli spaces of vacua of quantum gauge theories was intensely studied in physics. For example, moduli spaces of singular monopoles are isometric to Coulomb branches of supersymmetric Quantum Chromodynamics. Currently, a mathematical framework for understanding this relation between quantum and classical moduli spaces is just beginning to emerge. It is a propitious time to develop a much better understanding of this promising relation.

d) The theory of quiver varieties leads to many examples of complete hyperKaehler metrics with interesting asymptotics. The newer and far-reaching theory of bow moduli spaces provides a vast generalization which leads to an understanding of the moduli space of instantons over a 4-dimensional ALF space. These descriptions are very algebraic in a sense, and it is a high priority to understand them in more specifically geometric analytic terms.

e) The Kapustin-Witten equations on a four-manifold with boundary, where the boundary contains a knot, are conjectured to provide a new gauge-theoretic interpretation of the Jones polynomial of that knot; the Haydys-Witten theory in one higher dimension should lead to a similar interpretation of the Khovanov homology. These are conjectured to be related to G\_2 monopoles in seven dimensions. The development of the necessary analytic theory is still in its infancy, and it is expected that these new gauge theories will lead to very interesting new moduli spaces and interesting topological invariants.

a) Moduli spaces arising in gauge-theoretic problems carry $L^2$- (or Weil-Petersson type) metrics which in many cases have interesting special structure. For example, they may be Kaehler or even hyperKaehler. We obtain, in this way, remarkable families of complete hyperKaehler (and hence Ricci-flat) metrics. The asymptotic structure of these moduli space metrics seems to fall into some generalization of the ALE/ALF/ALG/ALH classes that are well-known in the four-dimensional setting in the context of gravitational instantons. In fact, these higher dimensional moduli spaces appear to typically have the structure of a singular torus fibration over a quasi-asymptotically conic space with a generalized `fiber-boundary’ type metric. Proving that this is really the case, not only in special cases but perhaps as a general structural feature, is a guiding question. As a special subcase of this, we mention the moduli space of monopoles on $R^3$, studied initially by Taubes, Atiyah, Gibbons-Manton and many others. Dramatic new progress in understanding its metric structure at infinity, phrased in terms of a compactification as a manifold with corners, is the subject of intense ongoing work.

b) One application coming from the understanding of these metric asymptotics is that it then becomes possible to study the space of $L^2$ harmonic forms, which has direct physical meaning. For the monopole moduli spaces, these spaces of harmonic forms are the subject of the famous Sen conjectures. While no mathematical progress has been made on this question for the past decade, new techniques from geometric microlocal analysis and elsewhere have now matured to the point where these types of problems may soon become tractable. The quantum field theory and string theory interpretation of these moduli spaces suggests predictions for the dimensions of the spaces of their $L^2$ harmonic forms. The mathematical theory should lead to an interpretation of these dimensions in terms of intersection cohomology groups of certain compactifications of these moduli spaces. The key problems involve first establishing the necessary mathematical theory, and then verifying these predictions.

c) Since the work of Seiberg and Witten, the intriguing isometry between the moduli spaces of classical gauge theory solitons, such as monopoles or instantons, and moduli spaces of vacua of quantum gauge theories was intensely studied in physics. For example, moduli spaces of singular monopoles are isometric to Coulomb branches of supersymmetric Quantum Chromodynamics. Currently, a mathematical framework for understanding this relation between quantum and classical moduli spaces is just beginning to emerge. It is a propitious time to develop a much better understanding of this promising relation.

d) The theory of quiver varieties leads to many examples of complete hyperKaehler metrics with interesting asymptotics. The newer and far-reaching theory of bow moduli spaces provides a vast generalization which leads to an understanding of the moduli space of instantons over a 4-dimensional ALF space. These descriptions are very algebraic in a sense, and it is a high priority to understand them in more specifically geometric analytic terms.

e) The Kapustin-Witten equations on a four-manifold with boundary, where the boundary contains a knot, are conjectured to provide a new gauge-theoretic interpretation of the Jones polynomial of that knot; the Haydys-Witten theory in one higher dimension should lead to a similar interpretation of the Khovanov homology. These are conjectured to be related to G\_2 monopoles in seven dimensions. The development of the necessary analytic theory is still in its infancy, and it is expected that these new gauge theories will lead to very interesting new moduli spaces and interesting topological invariants.