$q$-Hypergeometric solutions of quantum differential equations, quantum Pieri rules, and Gamma theorem

TL;DR

This paper develops q-hypergeometric solutions for quantum differential equations of cotangent bundles of partial flag varieties, establishing mirror symmetry, Pieri rules, and a Gamma theorem linking asymptotics to Gamma classes.

Contribution

It introduces new q-hypergeometric solutions for quantum differential equations and proves Pieri rules and a Gamma theorem for cotangent bundles of partial flag varieties.

Findings

q-hypergeometric solutions exhibit Landau-Ginzburg mirror symmetry

Pieri rules for quantum equivariant cohomology are formulated and proved

Leading asymptotics relate to the equivariant Gamma class of the tangent bundle

Abstract

We describe \,$q$-hypergeometric solutions of the equivariant quantum differential equations and associated qKZ difference equations for the cotangent bundle $T^*F_\lambda$ of a partial flag variety \,$F_\lambda$\,. These \,$q$-hypergeometric solutions manifest a Landau-Ginzburg mirror symmetry for the cotangent bundle. We formulate and prove Pieri rules for quantum equivariant cohomology of the cotangent bundle. Our Gamma theorem for \,$T^*F_\lambda$ \,says that the leading term of the asymptotics of the \,$q$-hypergeometric solutions can be written as the equivariant Gamma class of the tangent bundle of $T^*F_\lambda$ multiplied by the exponentials of the equivariant first Chern classes of the associated vector bundles. That statement is analogous to the statement of the gamma conjecture by B.\,Dubrovin and by S.\,Galkin, V.\,Golyshev, and H.\,Iritani, see also the Gamma theorem for \,$F_\lambda$ \,in Appendix B.