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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.

MSC:
 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)
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