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Math-Physics Joint Seminar

Thursday, March 14, 2019 - 4:30pm

Anton Zeitlin



University of Pennsylvania


The correspondence between integrable systems and enumerative geometry
started roughly 25 years ago in the works of Givental and his collaborators,
studying quantum cohomology and quantum K-theory. Around 10 years ago,
physicists Nekrasov and Shatashvili proposed an unexpected relation between
quantum K-theory and quantum integrable systems based on quantum groups
within their studies of 3-dimensional gauge theories. Their bold proposal
led to a lot of interesting developments in mathematics, bringing a new life
to older ideas of Givental, and enriching it with flavors of geometric
representation theory via the results of Braverman, Maulik, Nakajima,
Okounkov and many others. In this talk I will focus on recent breakthroughs
in the subject, leading to the proper mathematical understanding of
Nekrasov-Shatashvili original papers as well as some other subsequent
conjectures made by physicists. Our main illustration of such a relation is
an interplay between equivariant quantum K-theory of the cotangent bundles
to Grassmanians and the Heisenberg XXZ spin chain. We will also discuss
relation of equivariant quantum K-theory of cotangent bundles of flag
varieties and many-body integrable systems of Ruijsenaars-Schneider and