A Donaldson-Thomas crepant resolution conjecture on Calabi-Yau 4-folds

Abstract

Let G G be a finite subgroup of S U ( 4 ) \mathrm {SU}(4) such that its elements have age at most one. In the first part of this paper, we define K K -theoretic stable pair invariants on a crepant resolution of the affine quotient C 4 / G {\mathbb {C}}^4/G , and conjecture a closed formula for their generating series in terms of the root system of G G . In the second part, we define degree zero Donaldson-Thomas invariants of Calabi-Yau 4-orbifolds, develop a vertex formalism that computes the invariants in the toric case, and conjecture closed formulae for their generating series for the quotient stacks [ C 4 / Z r ] [{\mathbb {C}}^4/{\mathbb {Z}}_r] , [ C 4 / Z 2 × Z 2 ] [{\mathbb {C}}^4/{\mathbb {Z}}_2\times {\mathbb {Z}}_2] . Combining these two parts, we formulate a crepant resolution correspondence which relates the above two theories.