Stringy and Orbiforld Cohomology of Wreath Product Orbifolds

Abstract

Let $[X/\sfG]$ be an orbifold which is a global quotient of a compact almost complex manifold $X$ by a finite group $\sfG$. Let $\Sigma_n$ be the symmetric group on $n$ letters. Their semidirect product $\sfG^n \rtimes \Sigma_n$ is called the {\em 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 $\calH(X^n, \sfG^n\rtimes \Sigma_n)$ be the stringy cohomology ~\cite{FG, JKK1} of the $(\sfG^n\rtimes \Sigma_n)$-space $X^n$. We prove that the space $\sfG^n$-invariants of $\calH(X^n, \sfG^n\rtimes \Sigma_n)$ is isomorphic to the algebra $H_{orb}([X/\sfG])\{\Sigma_n\}$ introduced by Lehn and Sorger ~\cite{LS}, where $H_{orb}([X/\sfG])$ is the Chen-Ruan orbifold cohomology of $[X/\sfG]$. We also prove that, if $X$ is a projective surface with trivial canonical class and $Y$ is a crepant resolution of $X/\sfG$, then the Hilbert scheme of $n$ points on $Y$, denoted by $Y^{[n]}$, is a crepant resolution of $X^n/(\sfG^n\rtimes \Sigma_n)$. Furthermore, if $H^{\ast}(Y)$ is isomorphic to $H_{orb}([X/\sfG])$ as Frobenius algebras, then $H^{*}(Y^{[n]})$ is isomorphic to $H^{\ast}_{orb}([X^n/(\sfG^n \rtimes \Sigma_n)])$ as rings. Thus we verify a special case of the cohomological hyper-K\"{a}hler resolution conjecture due to Ruan ~\cite{R}.