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Quantum cohomology via \(D\)-modules. (English) Zbl 1081.53077
Author’s summary: We propose a new point of view on quantum cohomology, motivated by the work of Givental and Dubrovin, but closer to differential geometry than the existing approaches. The central object is a \(D\)-module which “quantizes” a commutative algebra associated to the (uncompactified) space of rational curves. Under appropriate conditions, we show that the associated flat connection may be gauged to the flat connection underlying quantum cohomology. This method clarifies the role of the Birkhoff factorization in the “mirror transformation”, and it gives a new algorithm (requiring construction of a Gröbner basis and solution of a system of o.d.e.) for computation of the quantum product.

MSC:
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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