Ruan cohomologies of the compactifications of resolved orbifold conifolds

Abstract

In this paper, we study the Ruan cohomologies of $X^s$ and $X^{sf}$, the natural compactifications of $V^s$ and $V^{sf}$, where $V^s$ and $V^{sf}$ are the two small resolutions of \[ V=\{(x,y,z,w)\mid xy-zw=0\}/\mu _r(1,-1,0,0),\quad r>1, \] the finite group quotient of the singular conifold. There is an additive isomorphism between the Chen-Ruan cohomologies $\phi :H^*_{CR}(X^s)\to H^*_{CR}(X^{sf})$. We study the three-point orbifold Gromov-Witten invariants of the exceptional curves $\Gamma ^s$ on $X^s$ and $\Gamma ^{sf}$ on $X^{sf}$ and show that the corresponding Ruan cohomology ring structures on the Chen-Ruan cohomologies of $X^s$ and $X^{sf}$, defined by these three-point functions, are isomorphic to each other under the map $\phi $ and the identification $[\Gamma ^s]\leftrightarrow -[\Gamma ^{sf}]$.