N=2* Supersymmetric Yang-Mills Theory, Four-Manifold Invariants, And Mock Modular Forms

Duration: 54 mins 16 secs

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Description:	Moore, G Wednesday 26th May 2021 - 15:30 to 16:30


Created:	2021-05-27 15:39
Collection:	New connections in number theory and physics
Publisher:	Isaac Newton Institute for Mathematical Sciences
Copyright:	Moore, G
Language:	eng (English)


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 revisted the derivation of invariants of smooth (compact, oriented) four-manifolds without boundary using twisted N=2 supersymmetric field theory. A key step in the derivation involves an integral expression for the invariants, valid for manifolds with b2+=1. The integral is known as the Coulomb branch, or u-plane, integral. While a Coulomb branch integral can, in principle, be written down for any class S theory we will focus on the case of pure SU(2) SYM and the N=2* SU(2) (and SO(3)) SYM. The recent progress is due in large part to an improved understanding of the relation of the integrand to Jacobi-Maass forms and indefinite theta functions. When handled with care, the integral can be done using integration by parts. In the N=2* case the forms are bimodular, and the result of the integral is (mock) modular in the ``UV'' modular parameter τuv and is (suitably) S-duality covariant in that parameter. For the N=2* theory topological twisting introduces a dependence on an ``ultraviolet'' spin-c structure cuv, which has not previously been investigated in detail. For b2+>1 the path integral can be explicitly expressed as a finite sum over (infrared) spin-c structures of modular expressions in τuv. The SU(2) N=2* theory also has a mass parameter. In the limit m→∞, and for any cuv, we recover the renowned ``Witten conjecture'' expressing the Donaldson invariants in terms of the Seiberg-Witten invariants. In the case that cuv is associated to an almost complex structure, the m→0 limit reproduces the results of Vafa and Witten for topologically twisted N=4 SYM. The expressions are also closely related to recent results on generating functions in enumerative algebraic geometry due to Gottsche, Kool, Nakajima, and Williams. This talk will be very similar to a talk I gave at the IAS, March 2021, as well as: WHCGP July 2020: http://www.physics.rutgers.edu/~gmoore/WHCGP-July20-2020.pdf

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 revisted the derivation of invariants of smooth (compact, oriented) four-manifolds without boundary using twisted N=2 supersymmetric field theory. A key step in the derivation involves an integral expression for the invariants, valid for manifolds with b2+=1. The integral is known as the Coulomb branch, or u-plane, integral. While a Coulomb branch integral can, in principle, be written down for any class S theory we will focus on the case of pure SU(2) SYM and the N=2* SU(2) (and SO(3)) SYM. The recent progress is due in large part to an improved understanding of the relation of the integrand to Jacobi-Maass forms and indefinite theta functions. When handled with care, the integral can be done using integration by parts. In the N=2* case the forms are bimodular, and the result of the integral is (mock) modular in the ``UV'' modular parameter τuv and is (suitably) S-duality covariant in that parameter. For the N=2* theory topological twisting introduces a dependence on an ``ultraviolet'' spin-c structure cuv, which has not previously been investigated in detail. For b2+>1 the path integral can be explicitly expressed as a finite sum over (infrared) spin-c structures of modular expressions in τuv. The SU(2) N=2* theory also has a mass parameter. In the limit m→∞, and for any cuv, we recover the renowned ``Witten conjecture'' expressing the Donaldson invariants in terms of the Seiberg-Witten invariants. In the case that cuv is associated to an almost complex structure, the m→0 limit reproduces the results of Vafa and Witten for topologically twisted N=4 SYM. The expressions are also closely related to recent results on generating functions in enumerative algebraic geometry due to Gottsche, Kool, Nakajima, and Williams.

This talk will be very similar to a talk I gave at the IAS, March 2021, as well as:

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

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