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Quantum cohomology of hypertoric varieties. (English) Zbl 1322.14087

The quantum cohomology ring of an algebraic variety is a certain deformation of its cohomology ring. Roughly speaking, in such a deformation two subvarieties are considering intersecting if they are connected by one or more rational curves. The virtual numbers of such curves, counted by Gromov-Witten invariants, appear as coefficients in the decomposition of the quantum cohomology cup product. These ideas feature prominently in mathematical physics, especially in string theory, where Gromov-Witten invariants appear as quantum corrections to classical notions of geometry, hence the name quantum cohomology.
In this paper the authors study the equivariant quantum cohomology ring of hypertoric varieties. Hypertoric varieties are a version of toric varieties, which can be obtained as an hyperKähler quotient. Well known examples are crepant resolutions of \(A_n\) singularities. One can study such varieties using combinatorial techniques from toric geometry, and as a result their geometry is captured by certain hyperplane arrangements. The paper contains two results. The first one is an explicit presentation of the equivariant cohomology ring of such varieties, given in terms of generators and relations. This result follows from an explicit formula for the quantum multiplication by a divisor.
The second result concerns a mirror formula for the quantum connection on such varieties. The quantum connection depends on the equivariant parameters, and for fixed equivariant parameters is a deformation of an ordinary connection, which involves the quantum product. This results identifies such a quantum connection on a hypertoric variety with the Gauss-Manin connection of a certain mirror family of complex manifolds (with a local system). Such a family is defined in term of the toric data and hyperplanes of the original variety. The proof follows by identifying a certain quantum differential equation for the quantum connection, with a Picard-Fuchs equation for the periods of a specific cohomology class of the mirror family.
The results of the paper are quite explicit and can be useful for people working on hypertoric varieties. The authors have also put some effort in making the paper self-contained, and the relevant concepts of hypertoric geometry and quantum cohomology are reviewed, albeit rather concisely.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14J33 Mirror symmetry (algebro-geometric aspects)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
14T05 Tropical geometry (MSC2010)
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References:

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