Quantum cohomology of partial flag manifolds

TL;DR

This paper computes the quantum cohomology rings of partial flag manifolds and introduces a new concept of vertical quantum cohomology for algebraic bundles, advancing understanding in geometric representation theory.

Contribution

It provides explicit computations of quantum cohomology rings for partial flag manifolds and defines the vertical quantum cohomology ring for associated algebraic bundles.

Findings

Quantum cohomology rings of partial flag manifolds are explicitly computed.

A new notion of vertical quantum cohomology ring is introduced.

The vertical quantum cohomology ring for flag bundles is determined.

Abstract

We compute the quantum cohomology rings of the partial flag manifolds F_{n_1\cdots n_k}=U(n)/(U(n_1)\times \cdots \times U(n_k)). The inductive computation uses the idea of Givental and Kim. Also we define a notion of the vertical quantum cohomology ring of the algebraic bundle. For the flag bundle F_{n_1\cdots n_k}(E) associated with the vector bundle E this ring is found.