Reconciling CRW with RW by functorially encoding geometric data

Develop a functorial encoding of the geometric data from Kapustin–Rozansky (2010) into the push–pull local systems so that the (∞,3)-category CRW reproduces the noncommutative 2-morphism categories (e.g., matrix factorizations) predicted for RW. Determine a coherent modification of the local systems Q^⊗ that overcomes the current discrepancy between CRW and RW.

Background

The authors compare the predicted RW hom 2-categories (often involving matrix factorizations and noncommutative dg-algebras) with the commutative Z2-graded quasicoherent sheaf categories that arise in their CRW. They identify a structural gap: CRW’s morphism categories are modeled by dg-modules over commutative dg-algebras, while RW’s are expected, in key examples, to be modules over noncommutative dg-algebras.

They suggest that the gap may be bridged by enhancing the geometric data carried by the local systems used in their push–pull construction, but acknowledge that how to do this is presently unclear. A functorial mechanism that incorporates the KR2010 geometric inputs into Q^⊗ could yield a closer approximation—or even a realization—of RW.