Ruan's crepant resolution conjecture for quantum D-modules

Let cmathcalXcmathcal{X} and YY be the orbifold and its crepant resolution, with cmathrmQDM(cmathcalX)cmathrm{QDM}(cmathcal{X}) and cmathrmQDM(Y)cmathrm{QDM}(Y) their quantum DD-modules. A global quantum DD-module is a tuple (cmathcalMA,F,cnabla,H(,)F)(cmathcal{M}_A,F,cnabla,H(,)_F) consisting of a connected complex analytic space, a holomorphic vector bundle with flat connection, and a non-degenerate flat inner product. Ruan's crepant resolution conjecture. There should exist a global quantum DD-module (cmathcalMA,F,cnabla,H(,)F)(cmathcal{M}_A,F,cnabla,H(,)_F) and open subsets VcmathcalX,VY\fesubsetcmathcalMAV_cmathcal{X},V_Y\fesubsetcmathcal{M}_A such that

(cmathcalMA,F,cnabla,H(,)F)VcmathcalXcsimeqcmathrmQDM(cmathcalX),(cmathcal{M}_A,F,cnabla,H(,)_F)|_{V_cmathcal{X}}csimeqcmathrm{QDM}(cmathcal{X}), (cmathcalMA,F,cnabla,H(,)F)VYcsimeqcmathrmQDM(Y).(cmathcal{M}_A,F,cnabla,H(,)_F)|_{V_Y}csimeqcmathrm{QDM}(Y).

This is a formulation of the crepant resolution conjecture asserting that the quantum theories of a crepant resolution and its orbifold are local manifestations of one global quantum DD-module; the source gives no resolution status.

Sources & referencesView supporting material

Primary source

Andrea Brini, Renzo Cavalieri and Dustin Ross, “Crepant resolutions and open strings”, arXiv:1309.4438 (2019).

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