Equivariant quantum differential equation and $qKZ$ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and B-Theorem

Abstract

In arXiv:1901.02990v1 the equivariant quantum differential equation ($qDE$) for a projective space was considered and a compatible system of difference $qKZ$ equations was introduced; the space of solutions to the joint system of the $qDE$ and $qKZ$ equations was identified with the space of the equivariant $K$-theory algebra of the projective space; Stokes bases in the space of solutions were identified with exceptional bases in the equivariant $K$-theory algebra. This paper is a continuation of arXiv:1901.02990v1. We describe the relation between solutions to the joint system of the $qDE$ and $qKZ$ equations and the topological-enumerative solution to the $qDE$ only, defined as a generating function of equivariant descendant Gromov-Witten invariants. The relation is in terms of the equivariant graded Chern character on the equivariant $K$-theory algebra, the equivariant Gamma class of the projective space, and the equivariant first Chern class of the tangent bundle of the projective space. We consider a Stokes basis, the associated exceptional basis in the equivariant $K$-theory algebra, and the associated Stokes matrix. We show that the Stokes matrix equals the Gram matrix of the equivariant Grothendieck-Euler-Poincar\'{e} pairing wrt to the basis, which is the left dual to the associated exceptional basis. We identify the Stokes bases in the space of solutions with explicit full exceptional collections in the equivariant derived category of coherent sheaves on the projective space, where the elements of those exceptional collections are just line bundles on the projective space and exterior powers of the tangent bundle of the projective space. These statements are equivariant analogs of results of G. Cotti, B. Dubrovin, D. Guzzetti, and S. Galkin, V. Golyshev, H. Iritani.