BKP-affine coordinates and emergent geometry of generalized Brézin-Gross-Witten tau-functions.(English)Zbl 07972363
The article deals with Gromov-Witten type theories and mirror symmetry, in the context of the so-called “emergent geometry”. A key question is: How can one find the associated B-model geometry which plays the role of a mirror? Using the Sato Grassmannian, an infinite-dimensional homogeneous space representing the set of all tau-functions of the KP hierarchy, the authors explore the duality among different tau-functions through the action of the infinite-dimensional Lie group \(\mathrm{GL}(\infty )\). They provide an example of the emergent geometry associated with a tau-function of the integrable BKP hierarchy, introduced by the Kyoto School, which shares many characteristics with the KP hierarchy. In particular, tau-functions of the BKP hierarchy can be expressed as sums of Schur \(Q\)-functions and are uniquely characterized by their BKP-affine coordinates on the big cell of the isotropic Sato Grassmannian. Focusing on generalized Brezin-Gross-Witten (BGW) models, the authors demonstrate that the spectral curve, along with the Eynard-Orantin topological recursion, naturally arises from the Virasoro constraints governing these tau-functions. Furthermore, they provide explicit expressions for the BKP-affine coordinates of these tau-functions and show that the quantum spectral curve, associated with the spectral curve, emerges naturally from the BKP-affine coordinates and the Eynard-Orantin topological recursion.
Reviewer: Anatoliy K. Prykarpatsky (Kraków)
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37K25 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry |
| 53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |
| 53D37 | Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category |
| 14J33 | Mirror symmetry (algebro-geometric aspects) |
| 57R56 | Topological quantum field theories (aspects of differential topology) |
Keywords:
generalized BGW model;emergent geometry;BKP-affine coordinates;topological recursion;quantum spectral curveCite
References:
| [1] | Alexandrov, A., Cut-and-join description of generalized Brézin-Gross-Witten model, Adv. Theor. Math. Phys., 22, 6, 2018; Alexandrov, A., Cut-and-join description of generalized Brézin-Gross-Witten model, Adv. Theor. Math. Phys., 22, 6, 2018 ·Zbl 07430950 |
| [2] | Alexandrov, A., KdV solves BKP, Proc. Natl. Acad. Sci., 118, 25, Article e2101917118 pp., 2021; Alexandrov, A., KdV solves BKP, Proc. Natl. Acad. Sci., 118, 25, Article e2101917118 pp., 2021 |
| [3] | Alexandrov, A., Generalized Brézin-Gross-Witten tau-function as a hypergeometric solution of the BKP hierarchy, Adv. Math., 412, Article 108809 pp., 2023; Alexandrov, A., Generalized Brézin-Gross-Witten tau-function as a hypergeometric solution of the BKP hierarchy, Adv. Math., 412, Article 108809 pp., 2023 ·Zbl 1509.14105 |
| [4] | Alexandrov, A., Intersection numbers on \(\overline{\mathcal{M}}_{g , n}\) and BKP hierarchy, J. High Energy Phys., 2021, 9, Article 013 pp., 2021; Alexandrov, A., Intersection numbers on \(\overline{\mathcal{M}}_{g , n}\) and BKP hierarchy, J. High Energy Phys., 2021, 9, Article 013 pp., 2021 ·Zbl 1472.83028 |
| [5] | Alexandrov, A.; Shadrin, A., Elements of spin Hurwitz theory: closed algebraic formulas, blobbed topological recursion, and a proof of the Giacchetto-Kramer-Lewański conjecture, Sel. Math. New Ser., 29, 26, 2023; Alexandrov, A.; Shadrin, A., Elements of spin Hurwitz theory: closed algebraic formulas, blobbed topological recursion, and a proof of the Giacchetto-Kramer-Lewański conjecture, Sel. Math. New Ser., 29, 26, 2023 ·Zbl 1519.81282 |
| [6] | Anderson, P. W., More is different, Science, 177, 4047, 393-396, 1972; Anderson, P. W., More is different, Science, 177, 4047, 393-396, 1972 |
| [7] | Balogh, F.; Yang, D., Geometric interpretation of Zhou’s explicit formula for the Witten-Kontsevich tau function, Lett. Math. Phys., 107, 10, 1837-1857, 2017; Balogh, F.; Yang, D., Geometric interpretation of Zhou’s explicit formula for the Witten-Kontsevich tau function, Lett. Math. Phys., 107, 10, 1837-1857, 2017 ·Zbl 1381.37088 |
| [8] | Bertola, M.; Dubrovin, B.; Yang, D., Correlation functions of the KdV hierarchy and applications to intersection numbers over \(\overline{\mathcal{M}}_{g , n} \), Physica D, 327, 30-57, 2016; Bertola, M.; Dubrovin, B.; Yang, D., Correlation functions of the KdV hierarchy and applications to intersection numbers over \(\overline{\mathcal{M}}_{g , n} \), Physica D, 327, 30-57, 2016 ·Zbl 1373.37153 |
| [9] | Bertola, M.; Ruzza, G., Brézin-Gross-Witten tau function and isomonodromic deformations, Commun. Number Theory Phys., 13, 827-883, 2019; Bertola, M.; Ruzza, G., Brézin-Gross-Witten tau function and isomonodromic deformations, Commun. Number Theory Phys., 13, 827-883, 2019 ·Zbl 1427.14030 |
| [10] | Brézin, E.; Gross, D. J., The external field problem in the large N limit of QCD, Phys. Lett. B, 97, 1, 120-124, 1980; Brézin, E.; Gross, D. J., The external field problem in the large N limit of QCD, Phys. Lett. B, 97, 1, 120-124, 1980 |
| [11] | Chidambaram, N. K.; Garcia-Failde, E.; Giacchetto, A., Relations on \(\overline{\mathcal{M}}_{g , n}\) and the negative r-spin Witten conjecture, 2022, arXiv preprint; Chidambaram, N. K.; Garcia-Failde, E.; Giacchetto, A., Relations on \(\overline{\mathcal{M}}_{g , n}\) and the negative r-spin Witten conjecture, 2022, arXiv preprint |
| [12] | Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T., Transformation groups for soliton equations IV. A new hierarchy of soliton equations of KP-type, Physica D, 4, 3, 343-365, 1982; Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T., Transformation groups for soliton equations IV. A new hierarchy of soliton equations of KP-type, Physica D, 4, 3, 343-365, 1982 ·Zbl 0571.35100 |
| [13] | Date, E.; Jimbo, M.; Miwa, T., Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras, 2000, Cambridge University Press; Date, E.; Jimbo, M.; Miwa, T., Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras, 2000, Cambridge University Press ·Zbl 0986.37068 |
| [14] | Deligne, P.; Mumford, D., The irreducibility of the space of curves of given genus, Publ. Math. l’IHÉS, 36, 1, 75-109, 1969; Deligne, P.; Mumford, D., The irreducibility of the space of curves of given genus, Publ. Math. l’IHÉS, 36, 1, 75-109, 1969 ·Zbl 0181.48803 |
| [15] | Dijkgraaf, R.; Verlinde, H.; Verlinde, E., Loop equations and Virasoro constraints in nonperturbative two-dimensional quantum gravity, Nucl. Phys. B, 348, 3, 435-456, 1991; Dijkgraaf, R.; Verlinde, H.; Verlinde, E., Loop equations and Virasoro constraints in nonperturbative two-dimensional quantum gravity, Nucl. Phys. B, 348, 3, 435-456, 1991 |
| [16] | Do, N.; Norbury, P., Topological recursion on the Bessel curve, Commun. Number Theory Phys., 12, 1, 53-73, 2018; Do, N.; Norbury, P., Topological recursion on the Bessel curve, Commun. Number Theory Phys., 12, 1, 53-73, 2018 ·Zbl 1419.14035 |
| [17] | Dubrovin, B.; Yang, D.; Zagier, D., On tau-functions for the KdV hierarchy, Sel. Math., 27, 1-47, 2021; Dubrovin, B.; Yang, D.; Zagier, D., On tau-functions for the KdV hierarchy, Sel. Math., 27, 1-47, 2021 ·Zbl 1458.14061 |
| [18] | Eskin, A.; Okounkov, A.; Pandharipande, R., The theta characteristic of a branched covering, Adv. Math., 217, 873-888, 2008; Eskin, A.; Okounkov, A.; Pandharipande, R., The theta characteristic of a branched covering, Adv. Math., 217, 873-888, 2008 ·Zbl 1157.14014 |
| [19] | Eynard, B.; Orantin, N., Invariants of algebraic curves and topological expansion, Commun. Number Theory Phys., 1, 2, 347-452, 2007; Eynard, B.; Orantin, N., Invariants of algebraic curves and topological expansion, Commun. Number Theory Phys., 1, 2, 347-452, 2007 ·Zbl 1161.14026 |
| [20] | Fukuma, M.; Kawai, H.; Nakayama, R., Continuum Schwinger-Dyson equations and universal structures in two-dimensional quantum gravity, Int. J. Mod. Phys. A, 6, 08, 1385-1406, 1991; Fukuma, M.; Kawai, H.; Nakayama, R., Continuum Schwinger-Dyson equations and universal structures in two-dimensional quantum gravity, Int. J. Mod. Phys. A, 6, 08, 1385-1406, 1991 |
| [21] | Giacchetto, A.; Kramer, R.; Lewański, D., A new spin on Hurwitz theory and ELSV via theta characteristics, 2021, arXiv preprint; Giacchetto, A.; Kramer, R.; Lewański, D., A new spin on Hurwitz theory and ELSV via theta characteristics, 2021, arXiv preprint |
| [22] | Gross, D. J.; Witten, E., Possible third-order phase transition in the large-N lattice gauge theory, Phys. Rev. D, 21, 2, 1980; Gross, D. J.; Witten, E., Possible third-order phase transition in the large-N lattice gauge theory, Phys. Rev. D, 21, 2, 1980 |
| [23] | Gukov, S.; Sułkowski, P., A-polynomial, B-model, and quantization, J. High Energy Phys., 2012, 2, Article 70 pp., 2012; Gukov, S.; Sułkowski, P., A-polynomial, B-model, and quantization, J. High Energy Phys., 2012, 2, Article 70 pp., 2012 ·Zbl 1309.81220 |
| [24] | Harnad, J.; Tau, B. F., Functions and Their Applications, Cambridge Monographs on Mathematical Physics, 2021, Cambridge University Press; Harnad, J.; Tau, B. F., Functions and Their Applications, Cambridge Monographs on Mathematical Physics, 2021, Cambridge University Press ·Zbl 1482.37001 |
| [25] | Hoffman, P. N.; Humphreys, J. F., Projective Representations of the Symmetric Groups: Q-Functions and Shifted Tableaux, Oxford Mathematical Monographs, 1992, Clarendon Press; Hoffman, P. N.; Humphreys, J. F., Projective Representations of the Symmetric Groups: Q-Functions and Shifted Tableaux, Oxford Mathematical Monographs, 1992, Clarendon Press ·Zbl 0777.20005 |
| [26] | Ji, C.; Wang, Z.; Kac-Schwarz, Y. C., Operators of type B, quantum spectral curves, and spin Hurwitz numbers, J. Geom. Phys., 189, Article 104831 pp., 2023; Ji, C.; Wang, Z.; Kac-Schwarz, Y. C., Operators of type B, quantum spectral curves, and spin Hurwitz numbers, J. Geom. Phys., 189, Article 104831 pp., 2023 ·Zbl 1521.37072 |
| [27] | Jimbo, M.; Miwa, T., Solitons and infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci., 1983, 19, 943-1001, 1983; Jimbo, M.; Miwa, T., Solitons and infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci., 1983, 19, 943-1001, 1983 ·Zbl 0557.35091 |
| [28] | Kac, V.; Schwarz, A., Geometric interpretation of the partition function of 2D gravity, Phys. Lett. B, 257, 3-4, 329-334, 1991; Kac, V.; Schwarz, A., Geometric interpretation of the partition function of 2D gravity, Phys. Lett. B, 257, 3-4, 329-334, 1991 |
| [29] | Kac, V.; van de Leur, J., Polynomial tau-functions of BKP and DKP hierarchies, J. Math. Phys., 60, 7, Article 071702 pp., 2019; Kac, V.; van de Leur, J., Polynomial tau-functions of BKP and DKP hierarchies, J. Math. Phys., 60, 7, Article 071702 pp., 2019 ·Zbl 1421.37028 |
| [30] | Knudsen, F. F., The projectivity of the moduli space of stable curves, II: The stacks \(M_{g , n}\), Math. Scand., 52, 2, 161-199, 1983; Knudsen, F. F., The projectivity of the moduli space of stable curves, II: The stacks \(M_{g , n}\), Math. Scand., 52, 2, 161-199, 1983 ·Zbl 0544.14020 |
| [31] | Kontsevich, M., Intersection theory on the moduli space of curves and the matrix Airy function, Commun. Math. Phys., 147, 1, 1-23, 1992; Kontsevich, M., Intersection theory on the moduli space of curves and the matrix Airy function, Commun. Math. Phys., 147, 1, 1-23, 1992 ·Zbl 0756.35081 |
| [32] | Laughlin, R. B., A Different Universe: Reinventing Physics from the Bottom Down, 2005, Basic Books: Basic Books New York; Laughlin, R. B., A Different Universe: Reinventing Physics from the Bottom Down, 2005, Basic Books: Basic Books New York |
| [33] | Liu, X.; Yang, C., Q-Polynomial expansion for Brézin-Gross-Witten tau-function, Adv. Math., 404, Article 108456 pp., 2022; Liu, X.; Yang, C., Q-Polynomial expansion for Brézin-Gross-Witten tau-function, Adv. Math., 404, Article 108456 pp., 2022 ·Zbl 1515.33014 |
| [34] | MacDonald, I. G., Symmetric Functions and Hall Polynomials, 1995, Clarendon Press; MacDonald, I. G., Symmetric Functions and Hall Polynomials, 1995, Clarendon Press ·Zbl 0899.05068 |
| [35] | Mironov, A.; Morozov, A., Superintegrability of Kontsevich matrix model, Eur. Phys. J. C, 81, 3, 2021; Mironov, A.; Morozov, A., Superintegrability of Kontsevich matrix model, Eur. Phys. J. C, 81, 3, 2021 ·Zbl 07408732 |
| [36] | Mironov, A.; Morozov, A.; Semenoff, G., Unitary matrix integrals in the framework of generalized Kontsevich model. I. Brézin-Gross-Witten model, Int. J. Mod. Phys. A, 11, 28, 5031-5080, 1996; Mironov, A.; Morozov, A.; Semenoff, G., Unitary matrix integrals in the framework of generalized Kontsevich model. I. Brézin-Gross-Witten model, Int. J. Mod. Phys. A, 11, 28, 5031-5080, 1996 ·Zbl 1044.81723 |
| [37] | Norbury, P., A new cohomology class on the moduli space of curves, Geom. Topol., 27, 7, 2695-2761, 2023; Norbury, P., A new cohomology class on the moduli space of curves, Geom. Topol., 27, 7, 2695-2761, 2023 ·Zbl 1553.14016 |
| [38] | Orlov, A. Y., Hypergeometric functions related to Schur Q-polynomials and BKP equation, Theor. Math. Phys., 137, 2, 1574-1589, 2003; Orlov, A. Y., Hypergeometric functions related to Schur Q-polynomials and BKP equation, Theor. Math. Phys., 137, 2, 1574-1589, 2003 ·Zbl 1178.33015 |
| [39] | Sato, M., Soliton Equations as Dynamical Systems on an Infinite Dimensional Grassmann Manifold, vol. 439, 30-46, 1981, RIMS Kokyuroku; Sato, M., Soliton Equations as Dynamical Systems on an Infinite Dimensional Grassmann Manifold, vol. 439, 30-46, 1981, RIMS Kokyuroku ·Zbl 0507.58029 |
| [40] | Schur, J., Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math., 1911, 139, 155-250, 1911; Schur, J., Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math., 1911, 139, 155-250, 1911 ·JFM 42.0154.02 |
| [41] | Schwarz, A., On the solutions to the string equation, Mod. Phys. Lett. A, 06, 29, 2713-2725, 2011; Schwarz, A., On the solutions to the string equation, Mod. Phys. Lett. A, 06, 29, 2713-2725, 2011 ·Zbl 1020.37579 |
| [42] | Segal, G.; Loop, W. G., Groups and equations of KdV type, Publ. Math. l’IHÉS, 61, 1, 5-65, 1985; Segal, G.; Loop, W. G., Groups and equations of KdV type, Publ. Math. l’IHÉS, 61, 1, 5-65, 1985 ·Zbl 0592.35112 |
| [43] | van de Leur, J., The Adler-Shiota-van Moerbeke formula for the BKP hierarchy, J. Math. Phys., 36, 4940-4951, 1995; van de Leur, J., The Adler-Shiota-van Moerbeke formula for the BKP hierarchy, J. Math. Phys., 36, 4940-4951, 1995 ·Zbl 0844.35109 |
| [44] | Wang, Z.; Yang, C., BKP hierarchy, affine coordinates, and a formula for connected bosonic N-point functions, Lett. Math. Phys., 112, 62, 2022; Wang, Z.; Yang, C., BKP hierarchy, affine coordinates, and a formula for connected bosonic N-point functions, Lett. Math. Phys., 112, 62, 2022 ·Zbl 1521.37076 |
| [45] | Wang, Z.; Zhou, J., A formalism of abstract quantum field theory of summation of fat graphs, 2021, arXiv preprint; Wang, Z.; Zhou, J., A formalism of abstract quantum field theory of summation of fat graphs, 2021, arXiv preprint |
| [46] | Wang, Z.; Zhou, J., Topological 1D gravity, KP hierarchy, and orbifold Euler characteristics of \(\overline{\mathcal{M}}_{g , n} \), 2021; Wang, Z.; Zhou, J., Topological 1D gravity, KP hierarchy, and orbifold Euler characteristics of \(\overline{\mathcal{M}}_{g , n} \), 2021 |
| [47] | Witten, E., Two-dimensional gravity and intersection theory on moduli space, Surv. Differ. Geom., 1, 1, 243-310, 1990; Witten, E., Two-dimensional gravity and intersection theory on moduli space, Surv. Differ. Geom., 1, 1, 243-310, 1990 ·Zbl 0757.53049 |
| [48] | Yang, D.; Zhang, Q., On the Hodge-BGW correspondence, Commun. Number Theory Phys., 18, 3, 611-651, 2024; Yang, D.; Zhang, Q., On the Hodge-BGW correspondence, Commun. Number Theory Phys., 18, 3, 611-651, 2024 |
| [49] | Y. You, Polynomial solutions of the BKP hierarchy and projective representations of symmetric groups. Infinite-dimensional Lie algebras and groups, Luminy-Marseille, Adv. Ser. Math. Phys. 7, 449-464. ·Zbl 0744.35052 |
| [50] | Zhou, J., Topological recursions of Eynard-Orantin type for intersection numbers on moduli spaces of curves, Lett. Math. Phys., 103, 11, 1191-1206, 2013; Zhou, J., Topological recursions of Eynard-Orantin type for intersection numbers on moduli spaces of curves, Lett. Math. Phys., 103, 11, 1191-1206, 2013 ·Zbl 1323.14028 |
| [51] | Zhou, J., Quantum deformation theory of the Airy curve and mirror symmetry of a point, 2014, arXiv preprint; Zhou, J., Quantum deformation theory of the Airy curve and mirror symmetry of a point, 2014, arXiv preprint |
| [52] | Zhou, J., Emergent geometry and mirror symmetry of a point, 2015, arXiv preprint; Zhou, J., Emergent geometry and mirror symmetry of a point, 2015, arXiv preprint |
| [53] | Zhou, J., K-Theory of Hilbert schemes as a formal quantum field theory, 2018, arXiv preprint; Zhou, J., K-Theory of Hilbert schemes as a formal quantum field theory, 2018, arXiv preprint |
| [54] | Zhou, J., Fat and thin emergent geometries of Hermitian one-matrix models, 2018, arXiv preprint; Zhou, J., Fat and thin emergent geometries of Hermitian one-matrix models, 2018, arXiv preprint |
| [55] | Zhou, J., Emergent geometry of matrix models with even couplings, 2019, arXiv preprint; Zhou, J., Emergent geometry of matrix models with even couplings, 2019, arXiv preprint |
| [56] | Zhou, J., Grothendieck’s Dessins d’Enfants in a web of dualities, 2019, arXiv preprint; Zhou, J., Grothendieck’s Dessins d’Enfants in a web of dualities, 2019, arXiv preprint ·Zbl 1524.37066 |
| [57] | Zhou, J., Grothendieck’s Dessins d’Enfants in a web of dualities. II, 2019, arXiv preprint; Zhou, J., Grothendieck’s Dessins d’Enfants in a web of dualities. II, 2019, arXiv preprint |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
