A McKay-like correspondence for $(0,2)$-deformations

Abstract

We present a local computation of deformations of the tangent bundle for a resolved orbifold singularity $\mathbb{C}^d/G$. These correspond to $(0, 2)$-deformations of $(2, 2)$-theories. A McKay-like correspondence is found predicting the dimension of the space of first-order deformations from simple calculations involving the group. This is confirmed in two dimensions using the Kronheimer-Nakajima quiver construction. In higher dimensions such a computation is subject to nontrivial worldsheet instanton corrections and some examples are given where this happens. However, we conjecture that the special crepant resolution given by the $G$-Hilbert scheme is never subject to such corrections, and show this is true in an infinite number of cases. Amusingly, for three-dimensional examples where $G$ is abelian, the moduli space is associated to a quiver given by the toric fan of the blow-up. It is shown that an orbifold of the form $\mathbb{C}^3 / \mathbb{Z}_7$ has a nontrivial superpotential and thus an obstructed moduli space.