[Vecteurs de Whittaker à énergie finie, récurrence topologique et nombres de Hurwitz]
We upgrade the results of Borot–Bouchard–Chidambaram–Creutzig [BBCC24] to show that the Gaiotto vector in $4d$ $\mathcal{N} = 2$ pure supersymmetric gauge theory admits an analytic continuation with respect to the energy scale (which can therefore be taken to be finite, instead of infinitesimal), and is computed by topological recursion on the (ramified) half Seiberg–Witten spectral curve. This has a number of interesting consequences for the Gaiotto vector: relations to intersection theory on $\overline{\mathcal{M}}_{g,n}$ in at least two different ways, Hurwitz numbers, quantum curves, and (almost complete) description of the correlators as analytic functions of $\hslash $ (instead of formal series). The same method is used to establish analogous results for the more general Whittaker vector constructed in the recent work of Chidambaram–Dołęga–Osuga [CDO24].
Nous améliorons les résultats de Borot–Bouchard–Chidambaram–Creutzig [BBCC24] en montrant que le vecteur de Gaiotto pour la théorie de jauge pure $\mathcal{N} = 2$ super-symétrique en dimension $4$ admet un prolongement analytique dans la variable d’énergie — qui peut donc prendre une valeur finie plutôt qu’infinitésimale. Ce prolongement analytique se calcule par la récurrence topologique sur une courbe spectrale ramifiée admettant un revêtement double par la courbe spectrale de Seiberg–Witten. Plusieurs conséquences intéressantes pour le vecteur de Gaiotto s’ensuivent : des représentations multiples en termes d’intersections dans $\overline{\mathcal{M}}_{g,n}$ ainsi qu’en termes de nombres de Hurwitz, l’existence d’une courbe quantique, et une description complète (à une constante inconnue près) non-perturbative dans le paramètre quantique $\hslash $. La même méthode permet d’établir des résultats analogues pour des vecteurs de Whittaker plus généraux qui ont été introduits dans les travaux récents de Chidambaram–Dołęga–Osuga [CDO24].
Accepté le :
Publié le :
Keywords: Gaiotto state, Whittaker vector, AGT correspondence, topological recursion, W-algebras, $N=2$ gauge theory, Hurwitz theory, quantum curves
Mots-clés : État de Gaiotto, vecteur de Whittaker, correspondance AGT, récurrence topologique, W-algèbres, théorie de jauge $N=2$, théorie de Hurwitz, courbes quantiques
Gaëtan Borot 1 ; Nitin Kumar Chidambaram 2 ; Giacomo Umer 3
CC-BY 4.0
@article{JEP_2025__12__1503_0,
author = {Ga\"etan Borot and Nitin Kumar Chidambaram and Giacomo Umer},
title = {Whittaker vectors at finite energy scale, topological recursion and {Hurwitz} numbers},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1503--1564},
publisher = {\'Ecole polytechnique},
volume = {12},
year = {2025},
doi = {10.5802/jep.316},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.316/}
}
TY - JOUR AU - Gaëtan Borot AU - Nitin Kumar Chidambaram AU - Giacomo Umer TI - Whittaker vectors at finite energy scale, topological recursion and Hurwitz numbers JO - Journal de l’École polytechnique — Mathématiques PY - 2025 SP - 1503 EP - 1564 VL - 12 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.316/ DO - 10.5802/jep.316 LA - en ID - JEP_2025__12__1503_0 ER -
%0 Journal Article %A Gaëtan Borot %A Nitin Kumar Chidambaram %A Giacomo Umer %T Whittaker vectors at finite energy scale, topological recursion and Hurwitz numbers %J Journal de l’École polytechnique — Mathématiques %D 2025 %P 1503-1564 %V 12 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.316/ %R 10.5802/jep.316 %G en %F JEP_2025__12__1503_0
Gaëtan Borot; Nitin Kumar Chidambaram; Giacomo Umer. Whittaker vectors at finite energy scale, topological recursion and Hurwitz numbers. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 1503-1564. doi: 10.5802/jep.316
[ABC + 23] - “Topological recursion for Masur–Veech volumes”, J. London Math. Soc. (2) 107 (2023) no. 1, p. 254-332 | DOI | MR | Zbl
[ABCO24] - “The ABCD of topological recursion”, Adv. Math. 439 (2024), article ID 109473, 105 pages | MR | Zbl | DOI
[ABDB + 25] - “KP integrability through the - swap relation”, Selecta Math. (N.S.) 31 (2025) no. 2, article ID 42, 37 pages | MR | Zbl | DOI
[ACEH20] - “Weighted Hurwitz numbers and topological recursion”, Comm. Math. Phys. 375 (2020), p. 237-305 | DOI | MR | Zbl
[ACM92] - “Higher genus correlators from the hermitian -matrix model”, Phys. Lett. B 282 (1992), p. 341-348 | DOI | MR
[AGT10] - “Liouville correlation functions from four-dimensional gauge theories”, Lett. Math. Phys. 91 (2010), p. 167-197 | DOI | MR | Zbl
[Ale18] - “Cut-and-join description of generalized Brezin-Gross-Witten model”, Adv. Theo. Math. Phys. 22 (2018) no. 6, p. 1347-1399 | MR | Zbl | DOI
[AN24] - “Super volumes and KdV tau functions”, 2024 | arXiv | Zbl
[BBC + 24] - Higher Airy structure, -algebras and topological recursion, Mem. Amer. Math. Soc., vol. 296, no. 1476, American Mathematical Society, Providence, RI, 2024 | Zbl
[BBC + 25] - “Taking limits in topological recursion”, J. London Math. Soc. (2) 112 (2025) no. 3, article ID e70286, 104 pages | DOI | MR | Zbl
[BBCC24] - “Whittaker vectors for -algebras from topological recursion”, Selecta Math. (N.S.) 30 (2024) no. 2, article ID 33, 91 pages | MR | Zbl | DOI
[BDBKS24] - “Topological recursion for Kadomtsev–Petviashvili tau functions”, J. London Math. Soc. (2) 109 (2024) no. 6, article ID e12946, 57 pages | DOI
[BDK + 23] - “Double Hurwitz numbers: polynomiality, topological recursion and intersection theory”, Math. Ann. 387 (2023), p. 197-243 | DOI | MR | Zbl
[BE12] - “Geometry of spectral curves and all order dispersive integrable systems”, SIGMA 8 (2012), article ID 100, 53 pages | Zbl | DOI
[BE17] - “Reconstructing WKB from topological recursion”, J. Éc. polytech. Math. 4 (2017), p. 845-908 | MR | Numdam | Zbl | DOI
[BEO15] - “Abstract loop equations, topological recursion, and applications”, Commun. Number Theory Phys. 9 (2015) no. 1, p. 51-187 | DOI | MR | Zbl
[BHSLM14] - “Mirror symmetry for orbifold Hurwitz numbers”, J. Differential Geom. 98 (2014) no. 3, p. 375-423 | MR | Zbl
[BKL + 21] - “Special cases of the orbifold version of Zvonkine’s -ELSV formula”, Michigan Math. J. 70 (2021) no. 2, p. 369-402 | MR | Zbl
[BKS24] - “Higher Airy structures and topological recursion for singular spectral curves”, Ann. Inst. H. Poincaré D 11 (2024) no. 1, p. 1-146 | Zbl | MR | DOI
[BN19] - “Loop equations for Gromov–Witten invariants of ”, SIGMA 15 (2019), article ID 061, 29 pages | DOI | MR | Zbl
[BS17] - “Blobbed topological recursion: properties and applications”, Math. Proc. Cambridge Philos. Soc. 162 (2017) no. 1, p. 39-87 | MR | Zbl | DOI
[CD22] - “Non-orientable branched coverings, -Hurwitz numbers, and positivity for multiparametric Jack expansions”, Adv. Math. 409 (2022), article ID 108645, 72 pages | MR | Zbl | DOI
[CDO24] - “-Hurwitz numbers from Whittaker vectors for -algebras”, 2024 | arXiv
[CGFG25] - “Relations on and the negative -spin Witten conjecture”, Invent. Math. 241 (2025) no. 3, p. 929-997 | MR | Zbl | DOI
[Chi08] - “Towards an enumerative geometry of the moduli space of twisted curves and th roots”, Compositio Math. 144 (2008) no. 6, p. 1461-1496 | DOI | MR | Zbl
[DBKPS23] - “Loop equations and a proof of Zvonkine’s -ELSV formula”, Ann. Sci. École Norm. Sup. (4) 56 (2023) no. 4, p. 1199-1229 | DOI | MR | Zbl
[DHS09] - “Quantum curves and -modules”, J. High Energy Phys. (2009) no. 11, article ID 047, 59 pages | MR | Zbl | DOI
[DK18] - “Towards the topological recursion for double Hurwitz numbers”, in Topological recursion and its influence in analysis, geometry and topology (C. Liu & M. Mulase, eds.), Proc. Symposia Pure Math., vol. 100, American Mathematical Society, Providence, RI, 2018, p. 151-178 | Zbl
[DLM] - “NIST Digital Library of Mathematical Functions”, Release 1.1.4 of 2022-01-15, Ed. by F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. B. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, M. A. McClain, http://dlmf.nist.gov/
[DOSS14] - “Identification of the Givental formula with the spectral curve topological recursion procedure”, Comm. Math. Phys. 328 (2014) no. 2, p. 669-700 | DOI | MR | Zbl
[EMS11] - “The Laplace transform of the cut-and-join equation and the Bouchard-Mariño conjecture on Hurwitz numbers”, Publ. RIMS, Kyoto Univ. 47 (2011), p. 629-670 | DOI | Zbl
[EO07] - “Invariants of algebraic curves and topological expansion”, Commun. Number Theory Phys. 1 (2007) no. 2, p. 347-452 | MR | Zbl | DOI
[EO09] - “Topological recursion in random matrices and enumerative geometry”, J. Phys. A 42 (2009) no. 29, article ID 293001, 117 pages | Zbl | DOI
[Eyn04] - “All genus correlation functions for the hermitian -matrix model”, J. High Energy Phys. (2004) no. 11, article ID 031, 35 pages | DOI
[Eyn14] - “Invariants of spectral curves and intersection theory of moduli spaces of complex curves”, Commun. Number Theory Phys. 8 (2014) no. 3, p. 541-588 | MR | Zbl | DOI
[Giv01] - “Gromov–Witten invariants and quantization of quadratic Hamiltonians”, Moscow Math. J. 1 (2001), p. 551-568 | DOI | MR | Zbl
[GPH17] - “Generating functions for weighted Hurwitz numbers”, J. Math. Phys. 58 (2017) no. 8, article ID 083503 | DOI | MR | Zbl
[GT93a] - “Twists and Wilson loops in the string theory of two-dimensional QCD”, Nuclear Phys. B 403 (1993) no. 1–2, p. 395-449 | DOI | MR | Zbl
[GT93b] - “Two-dimensional QCD is a string theory”, Nuclear Phys. B 400 (1993) no. 1–3, p. 181-208 | DOI | MR | Zbl
[HK10] - “Holomorphicity and modularity in Seiberg-Witten theories with matter”, J. High Energy Phys. (2010) no. 7, article ID 083, 49 pages | MR | Zbl | DOI
[Hoc25] - “Symplectic (non-)invariance of the free energy in topological recursion”, 2025 | arXiv
[IKT19] - “Voros coefficients for the hypergeometric differential equations and Eynard–Orantin’s topological recursion: Part II: For confluent family of hypergeometric equations”, J. Integrable Syst. 4 (2019) no. 1, article ID xyz004, 46 pages | MR | Zbl | DOI
[KN24] - “Polynomial relations among kappa classes on the moduli space of curves”, Internat. Math. Res. Notices (2024) no. 3, p. 1825-1867 | MR | Zbl | DOI
[KS18] - “Airy structures and symplectic geometry of topological recursion”, in Topological recursion and its influence in analysis, geometry, and topology, Proc. Symposia Pure Math., vol. 100, American Mathematical Society, Providence, RI, 2018, p. 433-490 | Zbl
[MMO05] - “Large N limit of D Yang–Mills theory and instanton counting”, J. High Energy Phys. 03 (2005), p. 027 | DOI | Zbl
[MO19] - Quantum groups and quantum cohomology, Astérisque, vol. 408, Société Mathématique de France, Paris, 2019
[Mum83] - “Towards an enumerative geometry of the moduli space of curves”, in Arithmetic and geometry, Vol. II, Progress in Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, p. 271-328 | DOI | Zbl
[NO06] - “Seiberg-Witten theory and random partitions”, in The unity of mathematics, Progress in Math., vol. 244, Birkhäuser Boston, Boston, MA, 2006, p. 525-596 | Zbl | DOI
[Nor23] - “A new cohomology class on the moduli space of curves”, Geom. Topol. 27 (2023), p. 2695-2761 | DOI | MR | Zbl
[Nov24] - “On the D Yang–Mills/Hurwitz correspondence”, 2024 | arXiv | Zbl
[NY04] - “Lectures on instanton counting”, in Algebraic structures and moduli spaces, CRM Proc. Lecture Notes, vol. 38, American Mathematical Society, Providence, RI, 2004, p. 31-101 | Zbl | DOI
[OP06] - “Virasoro constraints for target curves”, Invent. Math. 163 (2006) no. 1, p. 47-108 | MR | Zbl | DOI
[SV13] - “Cherednik algebras, -algebras and the equivariant cohomology of the moduli space of instantons on ”, Publ. Math. Inst. Hautes Études Sci. 118 (2013), p. 213-342 | DOI | MR | Zbl
[Tac15] - supersymmetric dynamics for pedestrians, Lect. Notes in Physics, vol. 890, Springer/Hindustan Book Agency, Cham/New Delhi, 2015 | MR | Zbl | DOI
[VW14] - “A note on the asymptotic expansion of generalized hypergeometric functions”, Anal. Appl. (Singapore) 12 (2014) no. 1, p. 107-115 | DOI | Zbl
[YZ24] - “On the Hodge-BGW correspondence”, Commun. Number Theory Phys. 18 (2024) no. 3, p. 611-651 | MR | Zbl | DOI
[Zho15] - “Emergent geometry and mirror symmetry of a point”, 2015 | arXiv | Zbl
Cité par Sources :