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Mathematics, Algebraic Geometry, Mathematical Physics, Quantum Algebra
Abstract:
In the previous paper by Tarasov and Varchenko 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 the paper by Tarasov and Varchenko.
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.
