An interesting development, over the past decade or so, has been the appearance of “multivalued” objects (such as functions, forms, spinors) in various branches of differential geometry. These include gauge theory, calibrated geometry and adiabatic limits of structures on manifolds. The developments extend more classical constructions in the case of Riemann surfaces. In the first part of the talk we review, in outline, some of the geometric and analytic aspects of these developments (related to other talks in this workshop). In the last part we focus in more detail on the adiabatic description of co-associative fibred G2 manifolds and research directions which these suggest.

I will explain a novel construction of K3 surfaces as moduli spaces of singular equivariant instantons on a 4-torus. This gives the first hyper-Kahler quotient construction of a compact non-toroidal manifold and yields explicit formulae for K3 metrics near torus orbifold limits. I will also describe a novel Fredholm theory — and its development using interesting ideas from microlocal analysis — for Laplacians and Dirac operators constructed out of these very singular connections which enables the construction. I will next introduce a variant of the Donaldson-Uhlenbeck-Yau theorem that operates in this setting, some of its consequences (such as non-emptiness of our moduli spaces), and the novel notion of stability for these singular connections that enters into the theorem. This theorem is proved by studying the gradient flow equation for the Yang-Mills functional, and I will describe the Fredholm theory for the heat operator that undergirds the proof of the short- and long-time existence and regularity of the flow. While these results are all proved for singular connections on a 4-torus, they are expected to generalize to enable the study of vast new classes of gauge-theoretic moduli spaces consisting of objects with severe singularities in codimension at least three. Based on joint work with Arnav Tripathy and Andras Vasy.

On a four-manifold underlying a complex smooth projective surface S, Tanaka-Thomas gave an algebro-geometric definition of the SU(r) Vafa-Witten partition function. When S has a non-zero holomorphic 2-form, the partition function is essentially topological with a rich modular structure. The algebro-geometric viewpoint provides new insights into Vafa-Witten theory such as (1) new mathematical verifications of S-duality, (2) K-theoretic refinement, (3) higher rank expressions with relations to the Rogers-Ramanujan continued fraction.

Holomorphic Floer theory is the analog of Floer theory for holomorphic symplectic manifolds. This topic has been recently explored, from several different perspectives, by Kontsevich-Soibelman and Doan-Rezchikov. In a different direction, I will in this talk present a conjectural picture relating holomorphic Floer theory of complex integrable systems to Donaldson-Thomas invariants. In physical terms, we will discuss a relation between the BPS spectrum of an N=2 4-dimensional field theory and the enumerative geometry of the corresponding Seiberg-Witten integrable system.

Instanton knot Floer homology admits an interesting deformation (unavailable without the knot). The deformation is a Floer homology with local coefficients. Though more complicated than the vanilla version they turn out to be easier to compute. We’ll describe some of computations of these modules. A key computation is the trivial n-strand braid in S1xS2, generalizing work of Ethan Street. This is joint work with Peter Kronheimer.

Given a compact 3-manifold $Y$ and a $\mathbb Z_2$-harmonic spinor $(\mathcal Z_0, A_0,\Phi_0)$ with singular set $\mathcal Z_0$, I will explain a construction of a family of local solutions to the two-spinor Seiberg-Witten equations parameterized by $\epsilon \in (0,\epsilon_0)$ on tubular neighborhoods of $\mathcal Z_0$. These solutions concentrate in the sense that the $L^2$-norm of the curvature near $\mathcal Z_0$ diverges as $\epsilon\to 0$, and after renormalization they converge locally to the original $\mathbb Z_2$-harmonic spinor. I will explain how these model solutions are used in a gluing construction showing a $\mathbb Z_2$-harmonic spinor satisfying mild assumptions necessarily arises as the limit of a sequence of two-spinor Seiberg-Witten solutions on Y.

The broad contours of mirror symmetry for Calabi-Yau manifolds are well-understood by now: classical complex-geometric information is exchanged for a symplectic-geometric expansion defined in terms of a mirror manifold. In the hyperkahler context, one may expect to make still stronger statements. In this talk, I will explain how to do so in the case of the K3 manifold (and some degenerations thereof). Namely, the exact Ricci-flat metric will have two expansions: (i) in terms of gauge theory on a four-torus, and (ii) as corrected by a list of enumerative invariants. In particular, these two constructions are the topics of the talks by M. Zimet and L. Fredrickson earlier in the workshop; I will review these constructions before explaining how they are related by this strong form of mirror symmetry. This talk surveys a series of joint works with L. Fredrickson, S. Kachru, A. Vasy, and M. Zimet.

I will discuss 5-dimensional N=1 super Yang-Mills compactified on X times S^1, with X a smooth, compact, oriented 4-manifold. After a partial topological twist along X, the theory is locally independent of the metric on X, while it does depend on the radius R of S^1. The coefficients of the R-expansion of the path integral correspond to the index of a Dirac operator on moduli spaces of instantons and monopoles, or more generally K-theoretic Donaldson invariants. I will evaluate path integrals using two methods: 1) the quantum mechanics of the theory reduced to S^1 and 2) the low energy effective theory reduced to X. Both methods reproduce the same wall-crossing formula for 4-manifolds with b_2^+=1. I will also discuss the evaluation of path integrals for 4-manifolds with b_2^+>1 using method 2. Our results agree with those for algebraic surfaces by Gottsche, Nakajima and Yoshioka (2006) and Gottsche, Kool and Williams (2021). This talk is based on work in progress with H. Kim, G. Moore, R. Tao and X. Zhang.

In this talk, we propose a monopole Fueter invariant of 3-manifolds, motivated by the Donaldson-Segal program.

The theory of Yang-Mills connections, and in particular instantons, revolutionized the study of 4-manifolds. Monopoles appear as the dimensional reduction of instantons to 3-manifolds. The moduli spaces of monopoles on R3 form ALF hyperkähler manifolds. An interesting feature of the monopole equation is that it can be generalized to certain higher-dimensional spaces. The most interesting examples appear on Calabi-Yau 3-folds and G2-manifolds. Donaldson and Segal hinted at the idea of defining invariants of Calabi-Yau 3-folds by counting monopoles on these manifolds. These monopole invariants, conjecturally, are related to the special Lagrangians, similar to the Taubes’ theorem, which relates the Seiberg-Witten and Gromov invariants of symplectic 4-manifolds.

Motivated by this conjecture, we propose numerical invariants of 3-manifolds by counting Fueter sections on hyperkähler bundles with fibers modeled on the moduli spaces of monopoles on R3. More ambitiously, one would hope this would result in a Floer theoretic invariant of 3-manifolds. A major difficulty in defining these invariants is related to the non-compactness problems. We prove partial results in this direction, examining the different sources of non-compactness, and proving some of them, in fact, do not occur.