Stringy and Orbiforld Cohomology of Wreath Product Orbifolds

Abstract

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