Schedule for: 21w5002 - Geometry, Analysis, and Quantum Physics of Monopoles (Online)

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

This talk will be part pedagogical introduction and part survey. I will review some of the foundational tools used to construct and study monopoles, including the Nahm transform, spectral curves, and rational maps. I will go on to survey recent progress on the problems of explicitly constructing monopoles and classifying their dynamics, both on Euclidean R^3 and on other geometries.

It is well-known that the Riemannian geometry of the moduli space of Euclidean SU(2)-monopoles of charge $k$ is determined by spectral curves of monopoles. I had often wondered whether there is a purely differential-geometric explanation of this fact, i.e. whether there exists an infinitesimal object on the moduli space which makes it so. I shall show that the answer is yes, and that the object in question is what I call a hyper-Poisson bivector, i.e. a bivector which induces, for each complex structure, a Poisson structure on holomorphic functions, compatible with the respective complex-symplectic form.

I will review the basics of what is known about monopoles in higher dimensions. Time permitting, I will mention a few possible future directions regarding the use of monopoles in special holonomy.

The Hitchin-Simpson equations defined over a Kähler manifold are first order, non-linear equations for a pair of a connection on a Hermitian vector bundle and a 1-form with values in the endomorphism bundle. We will describe the behavior of solutions to the Hitchin–Simpson equations with norms of these 1-forms unbounded. We will also discuss the deformation problem of Taubes' Z2 harmonic 1-form.

For a semisimple group G and a smooth curve C, open zastava space Z(G,C) is a smooth variety, affine over a configuration space of C. In case C is the additive or multiplicative group, Z(G,C) is isomorphic to a moduli space of euclidean or periodic monopoles. It carries a natural symplectic form, and the projection to the configuration space is an integrable system (open Toda lattice for G=SL(2)). I will explain what happens when C is an elliptic curve. This is a joint work with Mykola Matviichuk and Alexander Polishchuk.

An embedded curve in a Poisson surface $\Sigma\subset X$ defines a smooth deformation space $\mathcal{B}$ of nearby embedded curves. In this talk we will describe a key idea of Kontsevich and Soibelman to equip the Poisson surface $X$ with a foliation in order to study the deformation space $\mathcal{B}$. For example, $X=TP^1\to P^1$ is a Poisson surface surface foliated by its fibres. The foliation, together with a vector space $V_\Sigma$ of meromorphic differentials on $\Sigma$, endows an embedded curve $\Sigma$ with the structure of the initial data of topological recursion, which defines a collection of symmetric tensors on $V_\Sigma$. These tensors produce a formal series, which turns out to be a formal Seiberg-Witten differential, that descends under a quotient to an analytic series.

The moduli spaces $M_k$ of $SU(2)$ monopoles on $R^3$ of charge $k$ are among the oldest studied objects in gauge theory, yet open questions still remain, such as Sen's conjecture for their $L^2$ cohomology. I will discuss compactifications of these moduli spaces as manifolds with corners, with respect to which their hyperKahler metrics are of "quasi fibered boundary" (QFB) type, a metric structure which generalizes the quasi asymptotically locally euclidean (QALE) and quasi asymptotically conic (QAC) structures introduced by Joyce and others. This geometric structure, which is best understood by comparison to the simpler moduli space of point clusters on $R^3$, systematically organizes the various asymptotic regions of the moduli space in which charge $k$ monopoles decompose into widely separated monopoles of charges summing to $k$. This is joint work with M. Singer and K. Fritzsch.

Magnetic monopoles and quantum theory have been joined at the hip from the beginning. Over the years monopoles have motivated, or at least had front row seats at, many of the most fundamental advances in quantum field theory. In recent times, monopoles have become a primary conduit through which quantum field theory directly impacts mathematics and vice versa. In this talk I will highlight some of these developments and connections

In this talk we will discuss some of the features of monopole-fermion interactions. In the low energy limit of SUSY gauge theories, monopole interactions can be described by dynamics on monopole moduli space. Coupling these theories to fermionic matter the theory on the moduli space M to a hyperholomorphic bundle on M. We will discuss how this bundle arises and how it can be understood in the rational map formulation of monopole moduli space.

We shall discuss equivalences between various types of monopoles and difference modules. They are variants of Kobayashi-Hitchin correspondences between algebro-geometric objects and differential geometric objects. We may also regard them as equivalences between monopoles and their underlying scattering data, which have been pursued in various contexts. Though it is difficult to construct monopoles explicitly, we would like to explain that some asymptotic properties can be easily understood through the corresponding difference modules.

Four-dimensional complete hyperkaehler manifolds can be classified into ALE, ALF, ALG, ALG*, ALH, ALH* families. It has been conjectured that every ALG or ALG* hyperkaehler metric can be realized as a 4d Hitchin moduli space. I will describe ongoing work with Rafe Mazzeo, Jan Swoboda, and Hartmut Weiss to prove a special case of the conjecture, and some consequences. The hyperkaehler metrics on Hitchin moduli spaces are of independent interest, as the physicists Gaiotto—Moore—Neitzke give an intricate conjectural description of their asymptotic geometry.

In this talk I will describe recent applications of Higgs bundles (including wild/irregular Higgs bundles) which arise within a wide range of applications in string compactifications over special holonomy manifolds --- including Calabi-Yau and G2 holonomy geometries. In string theory, much recent work has exploited links between Calabi-Yau and Hitchin Integrable systems. In particular, the work of Diaconescu, Donagi and Pantev brought to light a remarkable isomorphism between (non-compact) Calabi-Yau integrable systems and those of Hitchin. Extensions of this correspondence to compact Calabi-Yau geometry have recently been found in the context of string compactifications but many open questions remain. In particular, links between Hitchin and Calabi-Yau moduli spaces have the potential to shed light on a number of important physical questions regarding string vacuum spaces.

The advances of the last 20 years of nonperturbative supersymmetric gauge theory led to the double deformation of complex geometry of Hitchin systems and moduli spaces of periodic monopoles. I will review these results and sketch a speculative path towards construction of hyperkahler metrics on these moduli spaces.

I will discuss the effect of negative curvature and girth on singularity formation in Yang Mills and harmonic maps. This work is joint with Luca DiCerbo.

I will discuss my recent work constructing a non-conical singular Hermitian Yang-Mills connection on a homogeneous reflexive sheaf over $\mathbb{C}^3$, which is supposed to model the generic situation of bubbling phenomenon when the Fueter section has a zero. This example in particular shows that the uniqueness part of the Hitchin-Kobayashi correspondence does not extend naively to noncompact manifolds. A variant of this construction gives a sequence of HYM connections on the unit ball in $\mathbb{C}^3$ with uniformly bounded $L^2$ curvature, but the number of codimension 6 singularities tends to infinity along the sequence. This illustrates the substantial difficulty of the compactification problem in higher dimensional gauge theory.

The moduli space of the Bogomolny Equations on R3 with certain asymptotic conditions has been fully studied by S.Donaldson, based on an algebraic geometry approach developed by N.Hitchin. An alternative analytical approach towards studying the moduli space was established by C.Taubes. An adaption of C.Taubes' method can also be used to study the moduli space of the Bogomolny equations on R3 with a knot singularity. Moreover, it is natural to ask, whether it is possible to study the equations with a knot singularity by algebraic geometry method? If this is possible, then it may have the potential to bring knot theory into algebraic geometry.