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Derivation of all structure constants for boundary Liouville CFT

Morris Ang, Guillaume Remy, Xin Sun, Tunan Zhu

Abstract

We prove that the probabilistic definition of the most general boundary three-point and bulk-boundary structure constants in Liouville conformal field theory (LCFT) agree respectively with the formula proposed by Ponsot-Techsner (2002) and by Hosomichi (2001). These formulas also respectively describe the fusion kernel and modular kernel of the Virasoro conformal blocks, which are important functions in various contexts of mathematical physics. As an intermediate step, we obtain the formula for the boundary reflection coefficient of LCFT proposed by Fateev-Zamolodchikov-Zamolodchikov (2000). Our proof relies on the boundary Belavin-Polyakov-Zamolodchikov differential equation recently proved by the first named author, and inputs from the coupling theory of Liouville quantum gravity (LQG) and Schramm Loewner evolution. Our results supply all the structure constants needed to perform the conformal bootstrap for boundary LCFT. They also yield exact descriptions for the joint law of the area and boundary lengths of basic LQG surfaces, including quantum triangles and two-pointed quantum disks.

Derivation of all structure constants for boundary Liouville CFT

Abstract

We prove that the probabilistic definition of the most general boundary three-point and bulk-boundary structure constants in Liouville conformal field theory (LCFT) agree respectively with the formula proposed by Ponsot-Techsner (2002) and by Hosomichi (2001). These formulas also respectively describe the fusion kernel and modular kernel of the Virasoro conformal blocks, which are important functions in various contexts of mathematical physics. As an intermediate step, we obtain the formula for the boundary reflection coefficient of LCFT proposed by Fateev-Zamolodchikov-Zamolodchikov (2000). Our proof relies on the boundary Belavin-Polyakov-Zamolodchikov differential equation recently proved by the first named author, and inputs from the coupling theory of Liouville quantum gravity (LQG) and Schramm Loewner evolution. Our results supply all the structure constants needed to perform the conformal bootstrap for boundary LCFT. They also yield exact descriptions for the joint law of the area and boundary lengths of basic LQG surfaces, including quantum triangles and two-pointed quantum disks.
Paper Structure (35 sections, 52 theorems, 312 equations, 2 figures)

This paper contains 35 sections, 52 theorems, 312 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Overview of the proof
  4. Connection with the fusion and modular kernels of Virasoro conformal blocks
  5. Outlook and perspective
  6. Conformal bootstrap for boundary LCFT
  7. Implications for GMC measures
  8. Interaction between the integrability of LCFT and the mating-of-trees theory.
  9. Outline of the paper
  10. Definitions and meromorphicity of ${H_{\mathrm{PT}}}$, $G_{\mathrm{Hos}}$, $H$, $G$, and $R$
  11. The Ponsot-Teschner and Hosimichi functions
  12. Definition of $H$ and $G$ under the Seiberg bound
  13. Meromorphic extension of $H$
  14. Probabilistic definition of $R$ via truncations
  15. Analytic continuation of $H$: proof of Proposition \ref{['prop:H-def']}
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $\gamma \in (0, 2)$, $\sum_{j=1}^3 \beta_j > 2Q$, $\beta_j < Q$, and $\mu_j=\mu_B(\sigma_j)>0$ for $j=1,2,3$. Then

Figures (2)

  • Figure 1: Structure constants of boundary Liouville CFT.
  • Figure 2: The contour of integration $\mathcal{C}$ in the function ${H_{\mathrm{PT}}}$.

Theorems & Definitions (114)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4: Liouville field
  • ...and 104 more