Floer Cohomology with Gerbes

Abstract

This is a written account of expository lectures delivered at the summer school on “Enumerative invariants in algebraic geometry and string theory” of the Centro Internazionale Matematico Estivo, held in Cetraro in June 2005. However, it differs considerably from the lectures as they were actually given. Three of the lectures in the series were devoted to the recent work of Donaldson–Thomas, Maulik–Nekrasov–Okounkov–Pandharipande, and Nakajima–Yoshioka. Since this is well documented in the literature, it seemed needless to write it up again. Instead, what follows is a greatly expanded version of the other lectures, which were a little more speculative and the least strictly confined to algebraic geometry. However, they should interest algebraic geometers who have been contemplating orbifold cohomology and its close relative, the so-called Fantechi–Gottsche ring, which are discussed in the final portion of these notes. Indeed, we intend to argue that orbifold cohomology is essentially the same as a symplectic cohomology theory, namely Floer cohomology. More specifically, the quantum product structures on Floer cohomology and on the Fantechi–Gottsche ring should coincide. None of this should come as a surprise, since orbifold cohomology arose chiefly from the work of Chen–Ruan in the symplectic setting, and since the differentials in both theories involve the counting of holomorphic curves. Nevertheless, the links between the two theories are worth spelling out. To illustrate this theme further, we will explain how both the Floer and orbifold theories can be enriched by introducing a flat U(1)-gerbe. Such a gerbe on a manifold (or orbifold) induces flat line bundles on its loop space and on its inertia stack, leading to Floer and orbifold cohomology theories with local coefficients. We will again argue that these two theories correspond. To explain all of this properly, an extended digression on the basic definitions and properties of gerbes is needed; it comprises the second of the three lectures.