**To connect to the HET Seminar via Zoom, please click the following link:**

https://theias.zoom.us/j/83432499354?pwd=TkhMZWFJK2E1UzRyeUppU01pZVQwUT09

Abstract: This is a continuation of a series of talks I have given in recent years reporting on work with Jan Manschot, et. al.

There will be some repetition, and some updates and new results. (See below for some relevant links.)

Manschot and I have revisited the derivation of topological invariants of four-manifolds (smooth, compact, oriented, no boundary) using N=2 supersymmetric field theory.

For the N=2* theory topological twisting introduces a dependence on an ``ultraviolet'' spin-c structure. The dependence on the ultraviolet spin-c structure is thoroughly investigated in this work in the case where the gauge group is SU(2) or SO(3).

A key step in the derivation of the invariants involves an integral over the Coulomb branch of the theory. It should be possible to write a Coulomb branch integral for any class S theory. However, the present example illustrates some subtleties, which were not appreciated when the subject was first studied in the 1990's, in forming a well-defined measure. There has also been recent progress in evaluating Coulomb branch integrals. This is due to an improved understanding of the relation of the integrand to Jacobi-Maass forms, indefinite theta functions, and mock modular forms.

In the N=2* case the partition functions are are (mock) modular in the ultraviolet coupling constant and are (suitably) S-duality covariant in that parameter. For b2+>1 the path integral can be explicitly expressed as a finite sum over (infrared) spin-c structures of modular expressions in the ultraviolet coupling.

The SU(2) N=2* theory also has a mass parameter. Using the large mass limit one can rederive the renowned Witten conjecture expressing the Donaldson invariants in terms of the Seiberg-Witten invariants. When the ultraviolet spin-c structure is the canonical one determined by an almost complex structure, the mass to zero limit reproduces (somewhat surprisingly) the results of Vafa and Witten for topologically twisted N=4 SYM. The expressions are also closely related to recent results in enumerative algebraic geometry due to Gottsche, Kool, Nakajima, and Williams.

**Previous talks closely related to this one:**

StringMath 2018: http://www.physics.rutgers.edu/~gmoore/StringMath2018-Final.pdf

JMM January 2020: http://www.physics.rutgers.edu/~gmoore/JMM-Talk-Jan18-2020.pdf

WHCGP July 2020: http://www.physics.rutgers.edu/~gmoore/WHCGP-July20-2020.pdf

KITP December 2020: https://online.kitp.ucsb.edu/online/mod20/moore/