Mirror symmetry for extended affine Weyl groups
Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 907-957.

We give a uniform, Lie-theoretic mirror symmetry construction for the Frobenius manifolds defined by Dubrovin–Zhang in [21] on the orbit spaces of extended affine Weyl groups, including exceptional Dynkin types. The B-model mirror is given by a one-dimensional Landau–Ginzburg superpotential constructed from a suitable degeneration of the family of spectral curves of the affine relativistic Toda chain for the corresponding affine Poisson–Lie group. As applications of our mirror theorem we give closed-form expressions for the flat coordinates of the Saito metric and the Frobenius prepotentials in all Dynkin types, compute the topological degree of the Lyashko–Looijenga mapping for certain higher genus Hurwitz space strata, and construct hydrodynamic bihamiltonian hierarchies (in both Lax–Sato and Hamiltonian form) that are root-theoretic generalisations of the long-wave limit of the extended Toda hierarchy.

Nous donnons une construction de symétrie miroir, de façon uniforme et par des méthodes de théorie de Lie, pour les variétés de Frobenius définies par Dubrovin-Zhang sur les orbites des groupes de Weyl affines étendus, y compris les types de Dynkin exceptionnels. Le modèle miroir est donné par un superpotentiel de Landau-Ginzburg construit à partir d’une dégénérescence convenable des courbes spectrales de la chaîne de Toda affine relativiste pour le groupe de Lie-Poisson affine correspondant. Nous fournissons également plusieurs applications de ce théorème miroir. Celles-ci incluent des expressions explicites pour les coordonnées plates pour la métrique de Saito et le prépotentiel de Frobenius en tout type de Dynkin ; le calcul du degré topologique de l’application de Lyashko-Looijenga pour certaines strates des espaces d’Hurwitz en genre supérieur ; et la construction de hiérarchies hydrodynamiques bi-hamiltoniennes (à la fois dans le formalisme de Lax-Sato et hamiltonien) qui donnent des généralisations de la limite de dispersion nulle de la hiérarchie de Toda étendue.

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Published online:
DOI: 10.5802/jep.197
Classification: 53D45, 14B07, 20H15
Keywords: Frobenius manifolds, mirror symmetry, integrable systems
Mot clés : Variétés de Frobenius, symétrie miroir, systèmes intégrables
Andrea Brini 1; Karoline van Gemst 2

1 School of Mathematics and Statistics, University of Sheffield S11 9DW, Sheffield, United Kingdom On leave from CNRS, DR 13, Montpellier, France
2 School of Mathematics and Statistics, University of Sheffield S11 9DW, Sheffield, United Kingdom
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Andrea Brini; Karoline van Gemst. Mirror symmetry for extended affine Weyl groups. Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 907-957. doi : 10.5802/jep.197. https://jep.centre-mersenne.org/articles/10.5802/jep.197/

[1] G. Antoniou, M. Feigin & I. A. B. Strachan - “The Saito determinant for Coxeter discriminant strata”, 2020 | arXiv

[2] V. I. Arnold - “Topological classification of complex trigonometric polynomials and the combinatorics of graphs with an identical number of vertices and edges”, Funktsional. Anal. i Prilozhen. 30 (1996) no. 1, p. 1-17, 96 | DOI | MR | Zbl

[3] V. I. Arnold, S. M. Gusein-Zade & A. N. Varchenko - Singularities of differentiable maps. Vol. I, Monographs in Math., vol. 82, Birkhäuser Boston, Inc., Boston, MA, 1985 | DOI

[4] G. Borot & A. Brini - “Chern–Simons theory on spherical Seifert manifolds, topological strings and integrable systems”, Adv. Theo. Math. Phys. 22 (2018) no. 2, p. 305-394 | DOI | MR | Zbl

[5] V. Bouchard & M. Marino - “Hurwitz numbers, matrix models and enumerative geometry”, in From Hodge theory to integrability and TQFT tt*-geometry, Proc. Sympos. Pure Math., vol. 78, American Mathematical Society, Providence, RI, 2008, p. 263-283 | DOI | MR | Zbl

[6] A. Brini - “E 8 spectral curves”, Proc. London Math. Soc. (3) 121 (2020) no. 4, p. 954-1032 | DOI | MR

[7] A. Brini - “Exterior powers of the adjoint representation and the Weyl ring of E 8 , J. Algebra 551 (2020), p. 301-341 | DOI | MR

[8] A. Brini, R. Cavalieri & D. Ross - “Crepant resolutions and open strings”, J. reine angew. Math. 755 (2019), p. 191-245 | DOI | MR | Zbl

[9] A. Brini & K. Osuga - “Five-dimensional gauge theories and the local B-model”, Lett. Math. Phys. 112 (2022), article ID 44, 48 pages | DOI | MR

[10] N. Caporaso, L. Griguolo, M. Marino, S. Pasquetti & D. Seminara - “Phase transitions, double-scaling limit, and topological strings”, Phys. Rev. D 75 (2007) no. 4, article ID 046004, 24 pages | DOI | MR

[11] G. Carlet - “The extended bigraded Toda hierarchy”, J. Phys. A 39 (2006) no. 30, p. 9411-9435 | DOI | MR | Zbl

[12] J. Cheng & T. Milanov - “The extended D-Toda hierarchy”, Selecta Math. (N.S.) 27 (2021) no. 2, article ID 24, 85 pages | DOI | MR | Zbl

[13] M. J. Crescimanno & W. Taylor - “Large N phases of chiral QCD in two-dimensions”, Nuclear Phys. B 437 (1995), p. 3-24 | DOI

[14] B. Dubrovin - “Hamiltonian formalism of Whitham type hierarchies and topological Landau-Ginsburg models”, Comm. Math. Phys. 145 (1992), p. 195-207 | DOI | MR | Zbl

[15] B. Dubrovin - “Integrable systems in topological field theory”, Nuclear Phys. B 379 (1992) no. 3, p. 627-689 | DOI | MR

[16] B. Dubrovin - “Geometry of 2D topological field theory”, in Integrable systems and quantum groups (M. Francaviglia & S. Greco, eds.), Lect. Notes in Math., vol. 1620, Springer-Verlag, Berlin, 1996, p. 120-348 | DOI

[17] B. Dubrovin - “Differential geometry of the space of orbits of a Coxeter group”, in Integrable systems, Surv. Differ. Geom., vol. 4, Int. Press, Boston, MA, 1998, p. 181-211 | DOI | MR | Zbl

[18] B. Dubrovin - “Painlevé transcendents and two-dimensional topological field theory”, in The Painlevé property, CRM Ser. Math. Phys., Springer, New York, 1999, p. 287-412 | DOI | Zbl

[19] B. Dubrovin - “On almost duality for Frobenius manifolds”, in Geometry, topology, and mathematical physics, Amer. Math. Soc. Transl. Ser. 2, vol. 212, American Mathematical Society, Providence, RI, 2004, p. 75-132 | MR | Zbl

[20] B. Dubrovin, I. Strachan, Y. Zhang & D. Zuo - “Extended affine Weyl groups of BCD-type: Their Frobenius manifolds and Landau–Ginzburg superpotentials”, Adv. Math. 351 (2019), p. 897-946 | DOI | MR | Zbl

[21] B. Dubrovin & Y. Zhang - “Extended affine Weyl groups and Frobenius manifolds”, Compositio Math. 111 (1998) no. 2, p. 167-219 | DOI | MR | Zbl

[22] P. Dunin-Barkowski, N. Orantin, S. Shadrin & L. Spitz - “Identification of the Givental formula with the spectral curve topological recursion procedure”, Comm. Math. Phys. 328 (2014), p. 669-700 | DOI | MR | Zbl

[23] T. Ekedahl, S. Lando, M. Shapiro & A. Vainshtein - “Hurwitz numbers and intersections on moduli spaces of curves”, Invent. Math. 146 (2001), p. 297-327 | DOI | MR | Zbl

[24] V. V. Fock & A. Marshakov - “Loop groups, clusters, dimers and integrable systems”, in Geometry and quantization of moduli spaces, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Cham, 2016, p. 1-66 | Zbl

[25] W. Fulton - “Hurwitz schemes and irreducibility of moduli of algebraic curves”, Ann. of Math. (2) 90 (1969), p. 542-575 | DOI | MR | Zbl

[26] W. Fulton & J. Harris - Representation theory, Graduate Texts in Math., vol. 129, Springer-Verlag, New York, 1991 | DOI

[27] I. P. Goulden & D. M. Jackson - “The combinatorial relationship between trees, cacti and certain connection coefficients for the symmetric group”, European J. Combin. 13 (1992) no. 5, p. 357-365 | DOI | MR | Zbl

[28] I. P. Goulden, D. M. Jackson & A. Vainshtein - “The number of ramified coverings of the sphere by the torus and surfaces of higher genera”, Ann. Comb. 4 (2000) no. 1, p. 27-46 | DOI | MR | Zbl

[29] A. Goupil & G. Schaeffer - “Factoring n-cycles and counting maps of given genus”, European J. Combin. 19 (1998) no. 7, p. 819-834 | DOI | MR | Zbl

[30] K. Ito & S.-K. Yang - “Flat coordinates, topological Landau-Ginzburg models and the Seiberg-Witten period integrals”, Phys. Lett. B 415 (1997), p. 45-53 | DOI | MR

[31] I. M. Krichever - “The tau function of the universal Whitham hierarchy, matrix models and topological field theories”, Comm. Pure Appl. Math. 47 (1994), p. 437 | DOI | Zbl

[32] S. K. Lando - “Ramified coverings of the two-dimensional sphere and intersection theory in spaces of meromorphic functions on algebraic curves”, Uspehi Mat. Nauk 57 (2002) no. 3(345), p. 29-98 | DOI | MR

[33] S. K. Lando & D. Zvonkine - “On multiplicities of the Lyashko–Looijenga mapping on strata of the discriminant”, Funct. Anal. Appl. 33 (1999) no. 3, p. 178-188 | DOI | MR | Zbl

[34] S. K. Lando & D. Zvonkine - “Counting ramified converings and intersection theory on spaces of rational functions. I. Cohomology of Hurwitz spaces”, Moscow Math. J. 7 (2007) no. 1, p. 85-107 | DOI | MR | Zbl

[35] W. Lerche & N. P. Warner - “Exceptional SW geometry from ALE fibrations”, Phys. Lett. B 423 (1998) no. 1-2, p. 79-86 | DOI | MR

[36] E. Looijenga - “The complement of the bifurcation variety of a simple singularity”, Invent. Math. 23 (1974), p. 105-116 | DOI | MR | Zbl

[37] O. V. Lyashko - “The geometry of bifurcation diagrams”, Uspehi Mat. Nauk 34 (1979) no. 3(207), p. 205-206 | Zbl

[38] A. McDaniel & L. Smolinsky - “A Lie-theoretic Galois theory for the spectral curves of an integrable system. I”, Comm. Math. Phys. 149 (1992) no. 1, p. 127-148 | DOI | MR | Zbl

[39] A. McDaniel & L. Smolinsky - “A Lie-theoretic Galois theory for the spectral curves of an integrable system. II”, Trans. Amer. Math. Soc. 349 (1997) no. 2, p. 713-746 | DOI | MR | Zbl

[40] T. Milanov, Y. Shen & H.-H. Tseng - “Gromov–Witten theory of Fano orbifold curves and ADE-Toda Hierarchies”, Geom. Topol. 20 (2016) no. 4, p. 2135-2218 | DOI | Zbl

[41] N. Nekrasov - “Five dimensional gauge theories and relativistic integrable systems”, Nuclear Phys. B 531 (1998) no. 1-3, p. 323-344 | DOI | MR | Zbl

[42] P. Norbury & N. Scott - “Gromov–Witten invariants of 1 and Eynard–Orantin invariants”, Geom. Topol. 18 (2014) no. 4, p. 1865-1910 | DOI | MR | Zbl

[43] D. Panov & D. Zvonkine - “Counting meromorphic functions with critical points of large multiplicities”, J. Math. Sci. 126 (2005) no. 2, p. 1095-1110 | DOI | Zbl

[44] P. Rossi - “Gromov–Witten theory of orbicurves, the space of tri-polynomials and symplectic field theory of Seifert fibrations”, Math. Ann. 348 (2010) no. 2, p. 265-287 | DOI | MR | Zbl

[45] B. M. Szablikowski & M. Blaszak - “Dispersionful analog of the Whitham hierarchy”, J. Math. Phys. 49 (2008) no. 8, p. 082701, 20 | DOI | MR | Zbl

[46] H. Williams - “Double Bruhat cells in Kac–Moody groups and integrable systems”, Lett. Math. Phys. 103 (2013), p. 389-419 | DOI | MR | Zbl

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