ORBIFOLD QUANTUM COHOMOLOGY OF THE CLASSIFYING SPACE OF A FINITE GROUP

By Tyler Jarvis, T. Kimura. 2001

78 citations recorded by the source corpus.

We work through, in detail, the quantum cohomology, with gravi- tational descendants, of the orbifold BG, the point with action of a finite group G. We provide a simple description of algebraic structures on the state space of this theory. As a consequence, we find that multiple copies of commuting Virasoro algebras appear which completely determine the correlators of the theory. 1. Introduction W. Chen and Y. Ruan (CR1, CR2) introduced the notion of the Gromov-Witten invariants of an orbifold V. Their construction reduces to the usual Gromov-Witten invariants when V is a smooth, projective variety. When V = (Y/G), where Y is a smooth, projective variety, and G is a finite group, the state space of this theory is generally larger than the G-invariant part of the cohomology of Y ; indeed, it has additional direct summands associated to loci in Y with nontrivial isotropy. These loci are called the twisted sectors of the theory, and their presence should be part of the proper notion of the cohomology of an orbifold. The purpose of this paper is to provide a detailed treatment of this theory for the simplest case, namely, when V is the classifying stack BG of a finite group G. Many of the features and subtleties of the Gromov-Witten invariants of orbifolds are present even here. The state space of this theory contains twisted sectors and the correlators in this theory are intersection numbers on Mg,n(BG), the moduli space of genus-g, n-pointed orbifold stable maps into BG. The correlators in this theory can be described in purely group theoretic terms, and we recover the result (Ru) that the algebraic structure on the state space H is isomorphic to the center of the group algebra, ZC(G), together with an invariant metric. Furthermore, we show that on the large phase space of this theory, there are r commuting copies of "half" the Virasoro algebra in this theory, where r is the dimension of H, all of which annihilate, and completely determine, the exponential of the large phase space potential function. We obtain a proof of the usual Vira- soro conjecture (EHX) as the special case where a diagonal action (after a variable rescaling) of these Virasoro algebras is considered. Similarly, the relevant integrable hierarchy consists of r commuting copies of the KdV hierarchy. Finally, it is worth pointing out that the moduli spaces Mg,n(BG) have many

Subfields: Geometry and Topology, Mathematical Physics

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Citations in the corpus

Symplectic geometry of Frobenius structures (2003, methodology): “In this section, we collect some results from [23, 24, 7, 25, 26, 29, 30] illustrating the following general observation: the higher genus theory is quantization of the genus 0 theory and inherits the twisted loop group of symmetries.”
The spaces of Laurent polynomials, Gromov-Witten theory of ℙ1-orbifolds and integrable hierarchies (2008, background): “— Quantum cohomology of a point (which coincides with the miniversal deformation of A1 singularity) and the KdV hierarchy ([35], [25]).”
Orbifold quantum Riemann–Roch, Lefschetz and Serre (2005, background): “In [44], explicit formulas expressing descendant orbifold Gromov-Witten invariants of BG in terms of descendant integrals on moduli stacks of stable curves have been proven.”
The Hirzebruch--Riemann--Roch theorem in true genus-0 quantum K-theory (2011, methodology): “The paper [16] by Jarvis–Kimura describes stable maps to the orbifold point=Zm D BZm in a way that can be easily adjusted to our case X=Zm D X BZm.”
LOCALIZATION IN GROMOV-WITTEN THEORY AND ORBIFOLD GROMOV-WITTEN THEORY (2011, background): “Jarvis-Kimura relate the descendant integrals on Mg,~c(BG) to those on Mg,n [29].”
Higher genus Gromov–Witten invariants as genus zero invariants of symmetric products (2003, background): “However, all we ever need to do with Artin stacks is intersect regularly embedded smooth divisors with regularly embedded smooth closed substacks, and also take first Chern classes of line bundles.”