Probabilistic construction of Toda Conformal Field Theories
Annales Henri Lebesgue, Volume 6 (2023), pp. 31-64.


Keywords Gaussian Free Field, Gaussian Multiplicative Chaos, Two-dimensional Conformal Field Theory, Random geometry, W-algebras


Following the 1984 seminal work of Belavin, Polyakov and Zamolodchikov on two-dimensional conformal field theories, Toda conformal field theories were introduced in the physics literature as a family of two-dimensional conformal field theories that enjoy, in addition to conformal symmetry, an extended level of symmetry usually referred to as W-symmetry or higher-spin symmetry. More precisely Toda conformal field theories provide a natural way to associate to a finite-dimensional simple and complex Lie algebra a conformal field theory for which the algebra of symmetry contains the Virasoro algebra. In this document we use the path integral formulation of these models to provide a rigorous mathematical construction of Toda conformal field theories based on probability theory. By doing so we recover expected properties of the theory such as the Weyl anomaly formula with respect to the change of background metric by a conformal factor and the existence of Seiberg bounds for the correlation functions.


[Ara17] Arakawa, Tomoyuki Introduction to W-Algebras and Their Representation Theory, Perspectives in Lie Theory (Callegaro, Filippo; Carnovale, Giovanna; Caselli, Fabrizio; De Concini, Corrado; De Sole, Alberto, eds.), Springer, 2017, pp. 179-250 | DOI | Zbl

[Bor86] Borcherds, Richard Vertex algebras, Kac–Moody algebras, and the Monster, Proceedings of the National Academy of Sciences of the United States of America, Volume 83 (1986), pp. 3068-3071 | DOI | MR | Zbl

[BPZ84] Belavin, Aleksandr A.; Polyakov, Aleksandr M.; Zamolodchikov, Aleksandr B. Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys., B, Volume 241 (1984) no. 2, pp. 333-380 | DOI | MR | Zbl

[CH22] Cerclé, Baptiste; Huang, Yichao Ward identities in the 𝔰𝔩 3 Toda field theory, Commun. Math. Phys., Volume 393 (2022) no. 1, pp. 419-475 | DOI | Zbl

[DDDF20] Ding, Jian; Dubédat, Julien; Dunlap, Alexander; Falconet, Hugo Tightness of Liouville first passage percolation for γ(0,2), Publ. Math., Inst. Hautes Étud. Sci., Volume 132 (2020), pp. 353-403 | DOI | MR | Zbl

[DFG + 20] Dubédat, Julien; Falconet, Hugo; Gwynne, Ewain; Pfeffer, Joshua; Sun, Xin Weak LQG metrics and Liouville first passage percolation, Probab. Theory Relat. Fields, Volume 178 (2020), pp. 369-436 | DOI | MR | Zbl

[DKRV16] David, François; Kupiainen, Antti; Rhodes, Rémi; Vargas, Vincent Liouville Quantum Gravity on the Riemann Sphere, Commun. Math. Phys., Volume 342 (2016), pp. 869-907 | DOI | MR | Zbl

[DM21] Duplantier, Bertrand; Miller, Scott Jason Sheffield Liouville quantum gravity as a mating of trees, Astérisque, 427, Société Mathématique de France, 2021 | Zbl

[DO94] Dorn, Harald; Otto, Hans-Jörg Two- and three-point functions in Liouville theory, Nucl. Phys., B, Volume 429 (1994) no. 2, pp. 375-388 | DOI | MR | Zbl

[DRV16] David, François; Rhodes, Rémi; Vargas, Vincent Liouville quantum gravity on complex tori, J. Math. Phys., Volume 57 (2016) no. 2, 022302 | DOI | MR | Zbl

[Dub09] Dubédat, Julien SLE and the Free Field: partition functions and couplings, J. Am. Math. Soc., Volume 22 (2009) no. 4, pp. 995-1054 | DOI | MR | Zbl

[FdV69] Freudenthal, Hans; de Vries, Hendrik Linear lie groups, Pure and Applied Mathematics, 35, Academic Press Inc., 1969 | Zbl

[FL88] Fateev, Vladimir A.; Lukyanov, Sergei L. The Models of Two-Dimensional Conformal Quantum Field Theory with Z(n) Symmetry, Int. J. Mod. Phys. A, Volume 3 (1988), p. 507 | DOI | MR

[FL05] Fateev, Vladimir A.; Litvinov, Alexey V. On differential equation on four-point correlation function in the Conformal Toda Field Theory, Jetp Lett., Volume 81 (2005), pp. 594-598 | DOI

[FLM89] Frenkel, Igor; Lepowsky, James; Meurman, Arne Vertex Operator Algebras and the Monster, Academic Press Inc., 1989 | Zbl

[FOR + 92] Fehér, László; O’Raifeartaigh, Lochlainn; Ruelle, Philippe; Tsutsui, Izumi; Wipf, Andreas On Hamiltonian reductions of the Wess–Zumino–Novikov–Witten theories, Phys. Rep., Volume 222 (1992) no. 1, pp. 1-64 | DOI | MR

[FZ85] Fateev, Vladimir A.; Zamolodchikov, Aleksandr B. Parafermionic Currents in the Two-Dimensional Conformal Quantum Field Theory and Selfdual Critical Points in Z(n) Invariant Statistical Systems, Sov. Phys., JETP, Volume 62 (1985), pp. 215-225

[FZ87] Fateev, Vladimir A.; Zamolodchikov, Aleksandr B. Conformal quantum field theory models in two dimensions having Z3 symmetry, Nucl. Phys., B, Volume 280 (1987), pp. 644-660 | DOI

[GKRV20] Guillarmou, Colin; Kupiainen, Antti; Rhodes, Rémi; Vargas, Vincent Conformal bootstrap in Liouville Theory (2020) (to appear in Acta Mathematica,

[GM92] Gervais, Jean-Loup; Matsuo, Yutaka W geometries, Phys. Lett., B, Volume 274 (1992) no. 3-4, pp. 309-316 | DOI

[GM21] Gwynne, Ewain; Miller, Jason Existence and uniqueness of the Liouville quantum gravity metric for γ(0,2), Invent. Math., Volume 223 (2021) no. 1, pp. 213-333 | DOI | MR | Zbl

[GRV19] Guillarmou, Colin; Rhodes, Rémi; Vargas, Vincent Polyakov’s formulation of 2d bosonic string theory, Publ. Math., Inst. Hautes Étud. Sci., Volume 130 (2019), pp. 111-185 | DOI | MR | Zbl

[HRV18] Huang, Yichao; Rhodes, Rémi; Vargas, Vincent Liouville quantum gravity on the unit disk, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 54 (2018) no. 3, pp. 1694-1730 | DOI | MR | Zbl

[Hum72] Humphreys, James Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, 9, Springer, 1972 | DOI | MR | Zbl

[JMO88] Jimbo, Michio; Miwa, Tetsuji; Okado, Masato Solvable lattice models related to the vector representation of classical simple Lie algebras, Commun. Math. Phys., Volume 116 (1988) no. 3, pp. 507-525 | DOI | MR | Zbl

[Kah85] Kahane, Jean-Pierre Sur le chaos multiplicatif, Ann. Sci. Math. Qué. (1985), pp. 105-150 | Zbl

[KRV19] Kupiainen, Antti; Rhodes, Rémi; Vargas, Vincent Local Conformal Structure of Liouville Quantum Gravity, Commun. Math. Phys., Volume 371 (2019), pp. 1005-1069 | DOI | MR | Zbl

[KRV20] Kupiainen, Antti; Rhodes, Rémi; Vargas, Vincent Integrability of Liouville theory: proof of the DOZZ formula, Ann. Math., Volume 191 (2020) no. 1, pp. 81-166 | DOI | MR | Zbl

[LG13] Le Gall, Jean-François Uniqueness and universality of the Brownian map, Ann. Probab., Volume 41 (2013) no. 4, pp. 2880-2960 | DOI | MR | Zbl

[LRV22] Lacoin, Hubert; Rhodes, Rémi; Vargas, Vincent The semiclassical limit of Liouville conformal field theory, Ann. Fac. Sci. Toulouse, Math. (6), Volume 31 (2022) no. 4, pp. 1031-1083 | DOI | MR | Zbl

[LS79] Leznov, Andreĭ N.; Saveliev, Mikhail V. Representation of zero curvature for the system of nonlinear partial differential equations x α,zz ¯ =exp(kx) α and its integrability, Lett. Math. Phys., Volume 3 (1979), pp. 489-494 | DOI | Zbl

[Mie13] Miermont, Grégory The Brownian map is the scaling limit of uniform random plane quadrangulations, Acta Math., Volume 210 (2013) no. 2, pp. 319-401 | DOI | MR | Zbl

[OPS88] Osgood, Brad; Phillips, Ralph; Sarnak, Peter Extremals of determinants of Laplacians, J. Funct. Anal., Volume 80 (1988) no. 1, pp. 148-211 | DOI | MR | Zbl

[Pol81] Polyakov, Aleksandr M. Quantum Geometry of bosonic strings, Phys. Lett., B, Volume 103 (1981), pp. 20-210 | DOI | MR

[RV14] Rhodes, Rémi; Vargas, Vincent Gaussian multiplicative chaos and applications: A review, Probab. Surv., Volume 11 (2014), pp. 315-392 | DOI | MR | Zbl

[RY91] Revuz, Daniel; Yor, Marc Continuous Martingales and Brownian Motion, Springer, 1991 | DOI

[Sch11] Schramm, Oded Scaling limits of loop-erased random walks and uniform spanning trees, Selected works of Oded Schramm. Volumes 1, 2 (Selected Works in Probability and Statistics), Springer, 2011, pp. 791-858 | DOI | MR

[Sei90] Seiberg, Nathan Notes on Quantum Liouville Theory and Quantum Gravity, Common trends in mathematics and quantum field theories (Progress of Theoretical Physics Supplement), Volume 102, Yukawa Institute for Theoretical Physics, 1990, pp. 319-349 | DOI | Zbl

[She07] Sheffield, Scott Gaussian free field for mathematicians, Probab. Theory Relat. Fields, Volume 139 (2007), pp. 521-541 | DOI | MR | Zbl

[Zam85] Zamolodchikov, Alexeĭ B. Infinite additional symmetries in two-dimensional conformal quantum field theory, Theor. Math. Phys., Volume 65 (1985) no. 3, pp. 1205-1213 | DOI

[ZZ96] Zamolodchikov, Aleksandr B.; Zamolodchikov, Alexeĭ B. Conformal bootstrap in Liouville field theory, Nucl. Phys., B, Volume 477 (1996) no. 2, pp. 577-605 | DOI | MR | Zbl