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
Toda.
Math-Physics Joint Seminar
Thursday, March 14, 2019 - 4:30pm
Anton Zeitlin
LSU