On equivariant Gromov–Witten invariants of resolved conifold with diagonal and anti-diagonal actions
Tóm tắt
We propose two conjectural relationships between the equivariant Gromov–Witten invariants of the resolved conifold under diagonal and anti-diagonal actions and the Gromov–Witten invariants of
$$\mathbb {P}^1$$
and verify their validity in genus zero approximation. We also provide evidences to support the validity of these relationships in genus one and genus two.
Tài liệu tham khảo
Belorousski, P., Pandharipande, R.: A descendent relation in genus 2. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 29, 171–191 (2000)
Behrend, K., Fantechi, B.: The intrinsic normal cone. Invent. Math. 128, 45–88 (1997)
Brini, A.: The local Gromov-Witten theory of \(\mathbb{P} ^{1}\) and integrable hierarchies. Commun. Math. Phys. 313, 571–605 (2012)
Brini, A., Carlet, G., Rossi, P.: Integrable hierarchies and the mirror model of local \(\mathbb{P} ^{1}\). Phys. D 241, 2156–2167 (2012)
Bryan, J., Pandharipande, R.: The local Gromov-Witten theory of curves. J. Am. Math. Soc. 21, 101–136 (2008)
Carlet, G., Dubrovin, B., Zhang, Y.: The extended Toda hierarchy. Mosc. Math. J. 4, 313–332 (2004)
Coates, T., Givental, A.: Quantum Riemann-Roch, Lefschetz and Serre. Ann. Math. 165, 15–53 (2007)
Coates, T., Givental, A., Tseng, H.-H.: Virasoro constraints for toric bundles, arXiv:1508.06282
Dijkgraaf, R., Witten, E.: Mean field theory, topological field theory, and multi-matrix models. Nucl. Phys. B 342, 486–522 (1990)
Dubrovin, B.: Integrable systems in topological field theory. Nucl. Phys. B 379, 627–689 (1992)
Dubrovin, B.: Integrable systems and classification of 2D topological field theories. In: Babelon, O., Cartier, P., Kosmann-Schwarzbach, Y. (eds.) “Integrable Systems", The J.-L. Verdier Memorial Conference, Actes du Colloque International de Luminy, pp. 313–359. Birkhäuser (1993)
Dubrovin B.: Geometry of 2D topological field theories. In: Francaviglia M., Greco S. (eds.) “Integrable Systems and Quantum Groups" (Montecatini Terme, 1993), pp. 120–348. Springer (1996)
Dubrovin, B.: On almost duality for Frobenius manifolds, in geometry, topology, and mathematical physics. Am. Math. Soc. Transl. Ser. 212, 75–132 (2004)
Dubrovin, B.: On universality of critical behaviour in Hamiltonian PDEs. Am. Math. Soc. Transl. 224, 59–109 (2008)
Dubrovin, B.: Hamiltonian perturbations of hyperbolic PDEs: from classification results to the properties of solutions. In: Sidoravičius, V. (ed.) New trends in mathematical physics, pp. 231–276. Springer, Dordrecht (2009)
Dubrovin, B., Liu, S.-Q., Yang, D., Zhang, Y.: Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs. Adv. Math. 293, 382–435 (2016)
Dubrovin, B., Liu, S.-Q., Yang, D., Zhang, Y.: Hodge-GUE correspondence and the discrete KdV equation. Commun. Math. Phys. 379, 461–490 (2020)
Dubrovin, B., Liu, S.-Q., Zhang, Y.: On the genus two free energies for semisimple Frobenius manifold. Russ. J. Math. Phys. 19, 273–298 (2012)
Dubrovin, B., Yang, D.: Generating series for GUE correlators. Lett. Math. Phys. 107, 1971–2012 (2017)
Dubrovin, B., Yang, D.: On cubic Hodge integrals and random matrices. Commun. Number Theory Phys. 11, 311–336 (2017)
Dubrovin, B., Yang, D., Zagier, D.: Gromov-Witten invariants of the Riemann sphere. Pure Appl. Math. Q. 16, 153–190 (2020)
Dubrovin, B., Zhang, Y.: Frobenius manifold and Virasoro constraints. Sel. Math. 5, 423–466 (1999)
Dubrovin, B., Zhang, Y.: Bi-hamiltonian hierarchies in 2D topological field theory at one-loop approximation. Commun. Math. Phys. 198, 311–361 (1998)
Dubrovin, B., Zhang, Y.: Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov–Witten invariants, arXiv:math/0108160
Dubrovin, B., Zhang, Y.: Virasoro symmetries of the extended Toda hierarchy. Commun. Math. Phys. 250, 161–193 (2004)
Eguchi, T., Getzler, E., Xiong, C.-S.: Topological gravity in genus 2 with two primary fields. Adv. Theor. Math. Phys. 4, 981–1000 (2000)
Eguchi, T., Yamada, Y., Yang, S.-K.: On the genus expansion in the topological string theory. Rev. Math. Phys. 7, 279–309 (1995)
Eguchi, T., Yang, S.-K.: The topological CP1 model and the large-N matrix integral. Mod. Phys. Lett. A 9, 2893–2902 (1994)
Getzler, E.: The Toda conjecture. In: Symplectic geometry and mirror symmetry, pp. 51–79. World Science Publishing, NJ (2001)
Getzler, E.: Intersection theory on \(\overline{{\cal{M} }}_{1,4}\) and elliptic Gromov-Witten invariants. J. Am. Math. Soc. 10, 973–998 (1997)
Getzler, E.: Topological recursion relations in genus 2. In: Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), pp. 73–106. World Science Publishing, NJ (1998)
Givental, A.B.: Gromov-Witten invariants and quantization of quadratic Hamiltonians. Mosc. Math. J. 1, 551–568 (2001)
Kontsevich, M., Manin, Yu.: Gromov-Witten classes, quantum cohomology and enumerative geometry. Commun. Math. Phys. 164, 525–562 (1994)
Li, J., Tian, G.: Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties. J. Am. Math. Soc. 11, 119–174 (1998)
Liu, S.-Q., Yang, D., Zhang, Y., Zhou, C.: The Hodge-FVH correspondence. J. Reine Angew. Math. 775, 259–300 (2021)
Mumford, A.: Towards an enumerative geometry of the moduli space of curves. In: Arith. Geom., pp. 271–328. Birkhäuser, Boston (1983)
Okounkov, A., Pandharipande, R.: Gromov-Witten theory, Hurwitz theory and completed cycles. Ann. Math. 163, 517–560 (2006)
Okounkov, A., Pandharipande, R.: The equivariant Gromov-Witten theory of \({\mathbb{P} }^1\). Ann. Math. 163, 561–605 (2006)
Ruan, Y., Tian, G.: A mathematical theory of quantum cohomology. J. Diff. Geom. 42, 259–367 (1995)
Strachan, I.A.B., Stedman, R.: Generalized Legendre transformations and symmetries of the WDVV equations. J. Phys. A 50, 17 (2017)
Vekslerchik, V.E.: Functional representation of the Ablowitz-Ladik hierarchy. J. Phys. A 31, 1087–1099 (1998)
Vekslerchik V. E.: Universality of the Ablowitz–Ladik hierarchy, arXiv:solv-int/9807005v1
Witten, E.: Two dimensional gravity and intersection theory on moduli space. Surv. Diff. Geom. 1, 243–310 (1991)
Zhang, Y.: On the \(CP^1\) topological sigma model and the Toda lattice hierarchy. J. Geom. Phys. 40, 215–232 (2002)