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The abelian/nonabelian correspondence and Frobenius manifolds. (English) Zbl 1164.14012
While genus-zero Gromov-Witten invariants (quantum cohomology) of toric varieties are relatively well understood the same can not be said about other varieties. A. Bertram and two of the current authors conjectured [J. Algebr. Geom. 17, No. 2, 275–294 (2008; Zbl 1166.14035)] a correspondence between Gromov-Witten potentials of nonabelian and abelian GIT quotients \(X//G\) and \(X//T\) of a projective variety \(X\). Here \(G\) is a reductive Lie group acting holomorphically on \(X\) with a maximal torus \(T\). This paper extends the conjecture to the equivariant setting and restates it in terms of Frobenius structures on big quantum cohomology rings \(QH^*(X//G)\) and \(QH^*(X//T)\).
The authors also describe a formulation of the correspondence in terms of big \(J\)-functions of Givental and show that it can often be reduced to a relation between small \(J\)-functions via the Gromov-Witten reconstruction theorems. They give a proof of the conjecture for partial flag manifolds \(X//G\) based on this reduction.
The main idea is as follows. It is known that any \(\sigma\in H^*(X//G)\) admits a (non-unique) lift \(\widetilde{\sigma}\) to the Weyl invariant subspace of \(H^*(X//T)\) and this lifting respects cup-products with the Weyl anti-invariant class \(\omega\), i.e. \[ \widetilde{(\sigma\cup_{X//G}\sigma')}\cup\omega =\widetilde{\sigma}\cup_{X//T}(\widetilde{\sigma}'\cup\omega). \] This fails for big quantum products \(*\), but the authors conjecture that the equality can be saved by replacing \(\widetilde{\sigma}\) with \(\xi\) satifying \(\xi\,*_{X//T}\,\omega=\widetilde{\sigma}\,\cup\,\omega\). Thus we get \[ (\widetilde{(\sigma*_{X//G}\sigma')}\cup\omega)(t) =(\xi*_{X//T}(\widetilde{\sigma}'\cup\omega))(\widetilde{t}), \] after an explicit change of coordinates \(\widetilde{t}(t)\). At the level of Gromov-Witten invariants this means that the naive expression for \(\langle\sigma_1,\dots,\sigma_n\rangle_{0,n,\beta}\) receives correction terms from sums of products of invariants of \(X//T\) of the same type.
The appearence of \(\xi\) is most naturally explaind in terms of Frobenius structures. The above lifting provides a canonical Frobenius structure on the Novikov ring \(N(X//G)\). However, the new flat coordinates are not the same as for the standard structure and therefore \(\widetilde{\sigma}\) are not horizontal. The horizontal fields are exactly \(\xi\) and \(\widetilde{t}(t)\) is the corresponding change of coordinates.

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
14L30 Group actions on varieties or schemes (quotients)
Full Text: DOI arXiv
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