The crepant resolution conjecture for orbifolds

Let $\mathcal{X}$ be an orbifold satisfying the hard Lefschetz condition and admitting a crepant resolution $Y$. Let $F^Y$ and $F^\mathcal{X}$ denote their potential functions, with cohomological variables $y_0,\dotsc,y_a$ and $x_0,\dotsc,x_a$, respectively, and quantum parameters $q_1,\dotsc,q_r$ and $u_1,\dotsc,u_s$. Let $\pi:Y\to X$ be the resolution map, and write $L_i^j$ for the coefficients of a graded linear isomorphism $L:H^*(Y)\to H^*_{\mathit{orb}}(\mathcal{X})$. Crepant resolution conjecture. There exists a graded linear isomorphism

$$L:H^*(Y)\to H^*_{\mathit{orb}}(\mathcal{X})$$

and roots of unity $c_{s+1},\ldots,c_r$ such that: (1) the inverse of $L$ extends the map $\pi^*:H^*(\mathcal{X})\to H^*(Y)$; (2) regarding $F^Y$ as a power series in $y_0,\dotsc,y_a,q_1,\dotsc,q_s$, its coefficients admit analytic continuation from $(q_{s+1},\dotsc,q_r)=(0,\dotsc,0)$ to $(q_{s+1},\dotsc,q_r)=(c_{s+1},\dotsc,c_r)$; and (3) $F^\mathcal{X}$ and $F^Y$ are equal after the substitution

y_i=\sum_jL_i^jx_j,\qquad q_i=\begin{cases}c_i&\text{when }i>s,\u_i&\text{when }i\leq s. \end{cases}

The conjecture identifies orbifold quantum data with the analytically continued quantum data of a crepant resolution; its unstable coefficients are excluded from the asserted equality, and the conjecture predicts preservation of the (orbifold) Poincare pairing by $L$.