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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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[1] Bertram, A., Ciocan-Fontanine, I., Kim, B.: Two proofs of a conjecture of Hori and Vafa. Duke Math. J. 126(1), 101–136 (2005) · Zbl 1082.14055 · doi:10.1215/S0012-7094-04-12613-2
[2] Bertram, A., Ciocan-Fontanine, I., Kim, B.: Gromov–Witten invariants for nonabelian and abelian quotients. J. Algebr. Geom. to appear, math.AG/0407254 · Zbl 1166.14035
[3] Coates, T., Givental, A.: Quantum Riemann-Roch, Lefschetz, and Serre. Ann. Math. (2) 165(1), 15–53 (2007) · Zbl 1189.14063 · doi:10.4007/annals.2007.165.15
[4] Dubrovin, B.: Geometry of 2D topological field theories. In: Integrable Systems and Quantum Groups (Montecatini Terme, 1993). Lect. Notes Math., vol. 1620, pp. 120–348. Springer, Berlin (1996)
[5] Dubrovin, B.: Painlevé transcendents in two-dimensional topological field theory. In: The Painlevé Property. CRM Ser. Math. Phys., pp. 287–412. Springer, New York (1999) · Zbl 1026.34095
[6] Dubrovin, B.: Geometry and analytic theory of Frobenius manifolds. In: Proceedings of the International Congress of Mathematicians, vol. II, Berlin, 1998. Doc. Math., Extra vol. II, pp. 315–326 (1998) · Zbl 0916.32018
[7] Ellingsrud, G., Strømme, S.A.: On the Chow ring of a geometric quotient. Ann. Math. 130, 159–187 (1989) · Zbl 0716.14002 · doi:10.2307/1971479
[8] Ginzburg, V.A.: Equivariant cohomology and Kähler geometry (Russian). Funkts. Anal. Prilozh. 21(4), 19–34, 96 (1987) · Zbl 0625.33006 · doi:10.1007/BF01077982
[9] Givental, A.: Equivariant Gromov–Witten invariants. Int. Math. Res. Not. 1996(13), 613–663 (1996) · Zbl 0881.55006 · doi:10.1155/S1073792896000414
[10] Givental, A.: Elliptic Gromov–Witten invariants and the generalized mirror conjecture. In: Integrable Systems and Algebraic Geometry (Kobe/Kyoto, 1997), pp. 107–155. World Sci. Publ., River Edge, NJ (1998) · Zbl 0961.14036
[11] Givental, A.: Symplectic geometry of Frobenius structures. In: Frobenius Manifolds. Aspects Math., E36, pp. 91–112. Vieweg, Wiesbaden (2004) · Zbl 1075.53091
[12] Goulden, I.P., Jackson, D.M.: Combinatorial Enumeration. In: Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons, Inc., New York (1983) · Zbl 0519.05001
[13] Kim, B., Kresch, A., Pantev, T.: Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee. J. Pure Appl. Algebra 179, 127–136 (2003) · Zbl 1078.14535 · doi:10.1016/S0022-4049(02)00293-1
[14] Kirwan, F.: Refinements of the Morse stratification of the normsquare of the moment map. In: The Breadth of Symplectic and Poisson Geometry. Prog. Math., vol. 232, pp. 327–362. Birkhäuser Boston, Boston, MA (2005) · Zbl 1073.32015
[15] Kontsevich, M., Manin, Y.I.: Gromov–Witten classes, quantum cohomology, and enumerative geometry. Commun. Math. Phys. 164(3), 525–562 (1994) · Zbl 0853.14020 · doi:10.1007/BF02101490
[16] Lee, Y.P., Pandharipande, R.: A reconstruction theorem in quantum cohomology and quantum K-theory. Am. J. Math. 126(6), 1367–1379 (2004) · Zbl 1080.14065 · doi:10.1353/ajm.2004.0049
[17] Lee, Y.P., Pandharipande, R.: Frobenius manifolds, Gromov–Witten theory, and Virasoro constraints, Part I. available at www.math.utah.edu/plee and/or www.math.princeton.edu/ahulp
[18] Manin, Y.I.: Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces. Am. Math. Soc. Colloq. Publ., vol. 47. Am. Math. Soc., Providence, RI (1999) · Zbl 0952.14032
[19] Martin, S.: Symplectic quotients by a nonabelian group and by its maximal torus. math.SG/0001002
[20] Sabbah, C.: Frobenius manifolds: Isomonodromic deformations and infinitesimal period mappings. Expo. Math. 16, 1–58 (1998) · Zbl 0904.32009
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