GL Verlinde numbers and the Grassmann TQFT

Abstract

These notes are concerned with moduli spaces of bundles on a smooth projective curve. Over them we consider determinant line bundles and their holomorphic Euler characteristics, the Verlinde numbers. The goal is to give a brief exposition of the two-dimensional topological quantum field theory that captures the structure of the GL Verlinde numbers, associated with spaces of bundles with varying determinant. Our point of view is to emphasize the close connection with another TQFT, the quantum cohomology of the Grassmannian. Two different geometries are related here, the moduli of bundles on a curve C and the space of maps from C to a suitable Grassmannian. The connection between them was established in the classic paper [W] where the open and closed invariants of the GL Verlinde TQFT, in all genera, were exhaustively written in both geometries. On the mathematical side, it was shown [A] that the underlying algebras of the two TQFTs are isomorphic, as the genus zero three-point invariants match. The TQFTs turn up different invariants overall, due to a discrepancy in the metrics of the associated Frobenius algebras. Moreover, the higher genus GL Verlinde invariants, open or closed, have not been systematically written down in the mathematics literature although they were shown in [W] to have compelling closed-form geometric expressions. We found it useful therefore to render the results of [W] in standard mathematical language, also with a view toward future studies of q-deformations of ordinary two-dimensional Yang Mills theory. The exposition is organized as follows. After briefly recalling the notion of a two-dimensional TQFT in the next section, we introduce in our context, on a smooth projective curve C, the two spaces of interest: the ancestor of all moduli spaces of sheaves, the Grothendieck Quot scheme, and the moduli space of semistable bundles. We present the former here primarily as compactifying the space of maps from the curve to a Grassmannian. Relevant aspects of the geometry and intersection theory of the two spaces are discussed. The last section studies the relation between them, in the form of the GL Verlinde TQFT, which we also refer to as the Grassmann TQFT.