Convergence and analytic decomposition of quantum cohomology of toric bundles. (English) Zbl 07794559
Summary: We prove that the equivariant big quantum cohomology \(Q H_T^\ast(E)\) of the total space \(E\) of a toric bundle \(E \to B\) converges provided that the big quantum cohomology \(Q H^\ast(B)\) converges. The proof is based on Brown’s mirror theorem for toric bundles [5]. It has been observed by Coates, Givental and Tseng that the quantum connection of \(E\) splits into copies of that of \(B\)[10]. Under the assumption that \(Q H^\ast(B)\) is convergent, we construct a decomposition of the quantum \(D\)-module of \(E\) into a direct sum of that of \(B\), which is analytic with respect to parameters of \(Q H_T^\ast(E)\). In particular, we obtain an analytic decomposition for the equivariant/non-equivariant big quantum cohomology of \(E\).
MSC:
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
Keywords:
Gromov-Witten theory; quantum cohomology; analytic quantum D-module; toric bundle; toric varietyReferences:
| [1] | Atiyah, M. F.; Bott, R., The moment map and equivariant cohomology. Topology, 1, 1-28 (1984) · Zbl 0521.58025 |
| [2] | Audin, M., Torus Actions on Symplectic Manifolds. Progress in Mathematics (2004), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 1062.57040 |
| [3] | Barannikov, S., Semi-infinite Hodge structures and mirror symmetry for projective spaces, available at · Zbl 1304.58003 |
| [4] | Berline, N.; Vergne, M., Classes caractéristiques équivariantes. Formule de localisation en cohomologie équivariante. C. R. Acad. Sci., Sér. 1 Math., 9, 539-541 (1982) · Zbl 0521.57020 |
| [5] | Brown, J., Gromov-Witten invariants of toric fibrations. Int. Math. Res. Not., 5437-5482 (2014) · Zbl 1307.14077 |
| [6] | Ciocan-Fontanine, I.; Kim, B., Big \(I\)-Functions, Development of Moduli Theory—Kyoto 2013. Adv. Stud. Pure Math., 323-347 (2016), Math. Soc. Japan: Math. Soc. Japan Tokyo · Zbl 1369.14018 |
| [7] | Coates, T.; Corti, A.; Iritani, H.; Tseng, H.-H., Computing genus-zero twisted Gromov-Witten invariants. Duke Math. J., 3, 377-438 (2009) · Zbl 1176.14009 |
| [8] | Coates, T.; Corti, A.; Iritani, H.; Tseng, H.-H., Hodge-theoretic mirror symmetry for toric stacks. J. Differ. Geom., 1, 41-115 (2020) · Zbl 1464.14044 |
| [9] | Coates, T.; Givental, A., Quantum Riemann-Roch, Lefschetz and Serre. Ann. Math. (2), 1, 15-53 (2007) · Zbl 1189.14063 |
| [10] | Coates, T.; Givental, A.; Tseng, H.-H., Virasoro constraints for toric bundles, available at · Zbl 1536.14049 |
| [11] | Givental, A. B., Homological geometry and mirror symmetry, 472-480 · Zbl 0863.14021 |
| [12] | Givental, A. B., Equivariant Gromov-Witten invariants. Int. Math. Res. Not., 613-663 (1996) · Zbl 0881.55006 |
| [13] | Givental, A. B., A mirror theorem for toric complete intersections, 141-175 · Zbl 0936.14031 |
| [14] | Givental, A. B., Elliptic Gromov-Witten invariants and the generalized mirror conjecture, 107-155 · Zbl 0961.14036 |
| [15] | Givental, A. B., Gromov-Witten invariants and quantization of quadratic Hamiltonians. Mosc. Math. J., 4, 551-568 (2001), 645, English, with English and Russian summaries, dedicated to the memory of I.G. Petrovskii on the occasion of his 100th anniversary · Zbl 1008.53072 |
| [16] | Givental, A. B., Symplectic Geometry of Frobenius Structures, Frobenius Manifolds. Aspects Math., 91-112 (2004), Friedr. Vieweg: Friedr. Vieweg Wiesbaden · Zbl 1075.53091 |
| [17] | Goresky, M.; Kottwitz, R.; MacPherson, R., Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math., 1, 25-83 (1998) · Zbl 0897.22009 |
| [18] | Iritani, H., Convergence of quantum cohomology by quantum Lefschetz. J. Reine Angew. Math., 29-69 (2007) · Zbl 1160.14044 |
| [19] | Iritani, H., Quantum \(D\)-modules and generalized mirror transformations. Topology, 4, 225-276 (2008) · Zbl 1170.53071 |
| [20] | Iritani, H., An integral structure in quantum cohomology and mirror symmetry for toric orbifolds. Adv. Math., 3, 1016-1079 (2009) · Zbl 1190.14054 |
| [21] | Kouchnirenko, A. G., Polyèdres de Newton et nombres de Milnor. Invent. Math., 1, 1-31 (1976) · Zbl 0328.32007 |
| [22] | Lee, Y.-P.; Lin, H.-W.; Wang, C.-L., Invariance of quantum rings under ordinary flops II: a quantum Leray-Hirsch theorem. Algebr. Geom., 5, 615-653 (2016) · Zbl 1373.14055 |
| [23] | Ruan, Y., The Cohomology Ring of Crepant Resolutions of Orbifolds, Gromov-Witten Theory of Spin Curves and Orbifolds. Contemp. Math., 117-126 (2006), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1105.14078 |
| [24] | Wasow, W., Asymptotic Expansions for Ordinary Differential Equations (1987), Dover Publications, Inc.: Dover Publications, Inc. New York, reprint of the 1976 edition · Zbl 0169.10903 |
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