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Boundary Toda Conformal Field Theory from the path integral

Baptiste Cerclé, Nathan Huguenin

TL;DR

The paper develops a rigorous probabilistic construction of Boundary Toda Conformal Field Theory on hyperbolic surfaces with boundary by extending the Davids-DKRV framework to the Toda setting. It employs a vector-valued Gaussian Free Field and Gaussian Multiplicative Chaos to define Toda fields, vertex operators, and correlation functions, while carefully handling boundary conditions that arise from outer automorphisms of the underlying Lie algebra. The main results establish existence (via Seiberg bounds), Weyl anomaly, and diffeomorphism invariance of correlation functions, and reveal that the boundary theory carries a twisted $W$-algebra symmetry whose currents transform predictably under outer automorphisms; Neumann and Cardy boundary conditions are distinguished and connected to the invariant subspaces of $\mathfrak a$. The work also outlines future directions toward solvability, exact structure constants in the $\mathfrak{sl}_3$ case, and connections to classical Toda geometry in the semi-classical limit, thereby providing a solid probabilistic foundation for boundary Toda CFT and its rich symmetry structure.

Abstract

Toda Conformal Field Theories (CFTs hereafter) are generalizations of Liouville CFT where the underlying field is no longer scalar but takes values in a finite-dimensional vector space induced by a complex simple Lie algebra. The goal of this document is to provide a probabilistic construction of such models on all compact hyperbolic Riemann surfaces with or without boundary. To do so, we rely on a probabilistic framework based on Gaussian Free Fields and Gaussian Multiplicative Chaos. In the presence of a boundary, one major difference with Liouville CFT is the existence of non-trivial outer automorphisms of the underlying Lie algebra. The main novelty of our construction is to associate to such an outer automorphism a type of boundary conditions for the field of the theory, leading to the definition of two different classes of models, with either Neumann or Cardy boundary conditions. This in turn has implications on the algebra of symmetry of the models, which are given by $W$-algebras.

Boundary Toda Conformal Field Theory from the path integral

TL;DR

The paper develops a rigorous probabilistic construction of Boundary Toda Conformal Field Theory on hyperbolic surfaces with boundary by extending the Davids-DKRV framework to the Toda setting. It employs a vector-valued Gaussian Free Field and Gaussian Multiplicative Chaos to define Toda fields, vertex operators, and correlation functions, while carefully handling boundary conditions that arise from outer automorphisms of the underlying Lie algebra. The main results establish existence (via Seiberg bounds), Weyl anomaly, and diffeomorphism invariance of correlation functions, and reveal that the boundary theory carries a twisted -algebra symmetry whose currents transform predictably under outer automorphisms; Neumann and Cardy boundary conditions are distinguished and connected to the invariant subspaces of . The work also outlines future directions toward solvability, exact structure constants in the case, and connections to classical Toda geometry in the semi-classical limit, thereby providing a solid probabilistic foundation for boundary Toda CFT and its rich symmetry structure.

Abstract

Toda Conformal Field Theories (CFTs hereafter) are generalizations of Liouville CFT where the underlying field is no longer scalar but takes values in a finite-dimensional vector space induced by a complex simple Lie algebra. The goal of this document is to provide a probabilistic construction of such models on all compact hyperbolic Riemann surfaces with or without boundary. To do so, we rely on a probabilistic framework based on Gaussian Free Fields and Gaussian Multiplicative Chaos. In the presence of a boundary, one major difference with Liouville CFT is the existence of non-trivial outer automorphisms of the underlying Lie algebra. The main novelty of our construction is to associate to such an outer automorphism a type of boundary conditions for the field of the theory, leading to the definition of two different classes of models, with either Neumann or Cardy boundary conditions. This in turn has implications on the algebra of symmetry of the models, which are given by -algebras.
Paper Structure (42 sections, 22 theorems, 94 equations)

This paper contains 42 sections, 22 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Toda Conformal Field Theories
  3. Liouville theory on closed surfaces
  4. Toda CFTs on closed Riemann surfaces
  5. Boundary Toda CFT
  6. Statements of the main results and perspectives
  7. Existence of the correlation functions
  8. Conformal and diffeomorphism invariance
  9. Implications on the algebra of symmetry
  10. Further perspectives
  11. Background and notations
  12. Compact Riemann surfaces, Laplacians and Green's functions
  13. Uniform metrics
  14. Green's functions
  15. Regularized determinants
  16. ...and 27 more sections

Key Result

Theorem 1.1

For $F$ bounded and continuous over $H^{-1}(\Sigma\to\mathfrak a,g)$, the correlation function $\langle F(\Phi)\prod_{k=1}^NV_{\alpha_k}(z_k)\prod_{l=1}^MV_{\beta_l}(s_l) \rangle_{g,\tau}$ exists and is non-trivial if and only if for $i=1,\cdots, r$:

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Definition 3.1
  • Definition 3.2
  • ...and 40 more