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SIGMA 15 (2019), 061, 29 pages arXiv:1905.01890
https://doi.org/10.3842/SIGMA.2019.061
Contribution to the Special Issue on Integrability, Geometry, Moduli in honor of Motohico Mulase for his 65th birthday
Loop Equations for Gromov-Witten Invariant of $\mathbb{P}^1$
Gaëtan Borot a and Paul Norbury b
a) Max Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
b) School of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia
Received May 16, 2019, in final form August 14, 2019; Published online August 23, 2019
Abstract
We show that non-stationary Gromov-Witten invariants of $\mathbb{P}^1$ can be extracted from open periods of the Eynard-Orantin topological recursion correlators $\omega_{g,n}$ whose Laurent
series expansion at $\infty$ compute the stationary invariants. To do so, we overcome the technical difficulties to global loop equations for the spectral $x(z) = z + 1/z$ and $y(z) = \ln z$ from the local loop
equations satisfied by the $\omega_{g,n}$, and check these global loop equations are equivalent to the Virasoro constraints that are known to govern the full Gromov-Witten theory of $\mathbb{P}^1$.
Key words: Virasoro constraints; topological recursion; Gromov-Witten theory; mirror symmetry.
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References
- Behrend K., Fantechi B., The intrinsic normal cone, Invent. Math. 128 (1997), 45-88, arXiv:alg-geom/9601010.
- Borot G., Eynard B., Orantin N., Abstract loop equations, topological recursion and new applications, Commun. Number Theory Phys. 9 (2015), 51-187, arXiv:1303.5808.
- Dubrovin B., Geometry of $2$D topological field theories, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Editors M. Francaviglia, S. Greco, Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348, arXiv:hep-th/9407018.
- Dunin-Barkowski P., Norbury P., Orantin N., Popolitov A., Shadrin S., Primary invariants of Hurwitz Frobenius manifolds, in Topological Recursion and its Influence in Analysis, Geometry, and Topology, Editors C.C.M. Liu, M. Mulase, Proc. Sympos. Pure Math., Vol. 100, Amer. Math. Soc., Providence, RI, 2018, 297-331, arXiv:1605.07644.
- Dunin-Barkowski P., Orantin N., Shadrin S., Spitz L., Identification of the Givental formula with the spectral curve topological recursion procedure, Comm. Math. Phys. 328 (2014), 669-700, arXiv:1211.4021.
- Eguchi T., Hori K., Xiong C.-S., Quantum cohomology and Virasoro algebra, Phys. Lett. B 402 (1997), 71-80, arXiv:hep-th/9703086.
- Eynard B., Orantin N., Topological recursion in enumerative geometry and random matrices, J. Phys. A: Math. Theor. 42 (2009), 293001, 117 pages, arXiv:0811.3531.
- Givental A.B., Gromov-Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J. 1 (2001), 551-568, arXiv:math.AG/0108100.
- Kontsevich M., Manin Yu., Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), 525-562, arXiv:hep-th/9402147.
- Kontsevich M., Manin Yu., Relations between the correlators of the topological sigma-model coupled to gravity, Comm. Math. Phys. 196 (1998), 385-398, arXiv:alg-geom/9708024.
- McClure J.P., Wong R., Explicit error terms for asymptotic expansions of Stieltjes transforms, J. Inst. Math. Appl. 22 (1978), 129-145.
- Norbury P., Stationary Gromov-Witten invariants of projective spaces, Acta Math. Sin. (Engl. Ser.) 33 (2017), 1163-1183, arXiv:1112.6400.
- Norbury P., Scott N., Gromov-Witten invariants of $\mathbb{P}^1$ and Eynard-Orantin invariants, Geom. Topol. 18 (2014), 1865-1910, arXiv:1106.1337.
- Okounkov A., Pandharipande R., Gromov-Witten theory, Hurwitz theory, and completed cycles, Ann. of Math. 163 (2006), 517-560, arXiv:math.AG/0204305.
- Okounkov A., Pandharipande R., Virasoro constraints for target curves, Invent. Math. 163 (2006), 47-108, arXiv:math.AG/0308097.
- Orantin N., Symplectic invariants, Virasoro constraints and Givental decomposition, arXiv:0808.0635.
- Teleman C., The structure of 2D semi-simple field theories, Invent. Math. 188 (2012), 525-588, arXiv:0712.0160.
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