Mathematics of gauge fields: October 10, 2016 – April 28, 2017

Organized by: Simon Donaldson, Kenji Fukaya, and John Morgan

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The program will focus on various mathematical aspects of gauge theory, including applications to topology and geometry. This area of study began when Donaldson showed how to use the moduli space of ASD connections on auxiliary SU(2)-bundles of charge one on a Riemannian 4-manifold to study the underlying smooth topology of the manifold. Eventually, this led Donaldson to what are now called the Donaldson polynomial invariants of a 4-manifold constructed from cohomlogy ring of compactifications of the moduli spaces of SU(2) ASD connections of all charges. These moduli spaces have interesting geometry and topology and natural compactifications.

Mathematical studies of gauge theories changed significantly when Seiberg-Witten (SW) theory was introduced. The SW moduli spaces tend to be zero dimensional, i.e., finite sets of points. Thus, their internal structure is not nearly as rich as the ASD moduli spaces. Nevertheless, they are easier to work with in many applications, and the emphasis of the field moved to studying these rather than the Donaldson moduli spaces. One motivating theme of this program is that the ASD moduli spaces are interesting and deserved to be revisited.

There has been a long-conjectured relationship, established to physicist’s satisfaction, between the SW invariants and the Donaldson polynominal invariants. One goal of the program is to clarify the present status of various classical problems including the state of this and related conjectures. This problem is closely related to so-called blow-up formulas and wall crossing formulas for the Donaldson polynomial invariants and equivariant versions of the invariants on the $4$-sphere.

In another direction, there is question of the analogue of this conjecture for the Floer homology which should relate the SW Floer homology and the ASD Floer homology of 3-manifolds. There is also a circle of questions concerning the extensions of these theories to 3-manifolds with boundary.

All of this work concerns the case of gauge group SU(2). There has been some, limited, work for gauge groups SU(n), and one can of course consider even the general compact, semi-simple Lie Group G.

In a different direction, there is the recent work of Taubes on the study of the compactification of the space of flat SL(2,C) connections on 3-manifolds and an analogue of Uhlenbeck’s compactness theorem for SL(2,C) ASD connections on 4-manifolds. This work has some connections with various higher dimensional theories, including gauge theories on Calabi-Yau threefolds and manifolds of exceptional holonomy and equations in 5 dimensions studied by Haydys and Witten, related to Khovanov homology.

In recent years the equivariant Donaldson invariants and their generalizations (to other gauge groups and matter content) were connected to representation theory of infinite-dimensional algebras. This, in turn, led to the new questions on moduli spaces of gauge fields living on singular four dimensional manifolds, embedded into higher dimensional spaces, e.g. Calabi-Yau fourfolds. The proper definition of these gauge field problems is still lacking.

The goal of the program is to bring experts to the Center to work on these and related questions.