Stringy and orbiforld cohomology of wreath product orbifolds

Abstract

Let [X/G] be an orbifold which is a global quotient of a compact almost complex manifold X by a finite group G. Let $Sigma_n$ be the symmetric group on n letters. Their semidirect product $G^n rtimes Sigma_n$ is called the wreath product of G and it naturally acts on the n-fold product $X^n$, yielding the orbifold $[X^n /(G^n rtimes Sigma_n)]$. Let $frak{H} (X^n,G^n rtimes Sigma_n)$ be the stringy cohomology [7, 10] of the $(G^n rtimes Sigma_ n)$-space $X^n$. We prove that the space $G^n$-invariants of $frak{H} (X^n,G^n rtimes Sigma_n)$ is isomorphic to the algebra $H_{orb}([X/G]){Sigma_ n}$ introduced by Lehn and Sorger [14], where $H_{orb}([X/G])$ is the Chen-Ruan orbifold cohomology of [X/G]. We also prove that, if X is a projective surface with trivial canonical class and Y is a crepant resolution of X/G, then the Hilbert scheme of n points on Y , denoted by $Y^{[n]}$, is a crepant resolution of $X^n/(G^n rtimes Sigma_n)$. Furthermore, if H∗(Y ) is isomorphic to $H_{or b}([X/G])$ as Frobenius algebras, then $H^∗(Y^{[n]})$ is isomorphic to $H^∗_{orb}([X^n/(G^n rtimes Sigma_ n)])$ as rings. Thus we verify a special case of the cohomological hyper-Kähler resolution conjecture due to Ruan [22].