Quantum K-theory of flag varieties

Algebraic Geometry and Number Theory Seminar

The Gromov-Witten theory studies enumerative invariants of a projective variety $X$ through counting nodal curves, and gives a family of deformed multiplication structures on the cohomology of $X$, known as the quantum cohomology. For flag variety $X=G/B$, relations in its quantum cohomology may be obtained through studying the differential equations solved by the generating function known as the J-function, which in turn come from the Toda lattice system of the Langlands dual group of $G$. In the first part of this talk, we review the story above. In the second part of this talk, we discuss how much of it still holds in K-theoretic settings, where quantum K-theory will replace quantum cohomology, and difference equations will replace differential equations. Quantum K-theory arises naturally in the framework of 3D mirror symmetry, a duality between mirror holomorphic symplectic varieties.