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Boundary Compactified Imaginary Liouville Theory

Yang Xiao, Yuxiao Xie

TL;DR

This work constructs Boundary Compactified Imaginary Liouville Theory (BCILT), extending the nonunitary, logarithmic CILT to manifolds with boundary by incorporating Neumann boundary conditions, curvature couplings, and imaginary Gaussian Multiplicative Chaos (GMC) on both bulk and boundary. The authors develop a rigorous probabilistic framework, starting from a compactified Gaussian Free Field, employing separating families to define curvature terms, and leveraging imaginary GMC to define vertex insertions, while ensuring invariance under translations and conformal changes. A central achievement is proving Segal's axioms for BCILT: well-defined correlation functions, Segal amplitudes, Weyl covariance, spin, and a gluing mechanism that preserves the amplitude structure under all allowed surgeries. The curvature-anomaly analysis, Neumann doubling, and GMC estimates adapted to boundary settings establish a robust path-integral construction with explicit structure constants expressed via Coulomb gas and related integrals, laying groundwork for BCILT and its connections to loop models and boundary CFTs. This framework provides rigorous tools for future bootstrap analyses and potential ties to SLE/CLE-type boundary phenomena in statistical physics.

Abstract

We generalize the construction of Compactified Imaginary Liouville Theory (CILT), a non-unitary logarithmic Conformal Field Theory (CFT) defined on closed surfaces, to surfaces with boundary. Starting from a compactified Gaussian Free Field (GFF) with Neumann boundary condition, we perturb it by adding in curvature terms and exponential potentials on both the bulk and the boundary. In physics, this theory is conjectured to describe the scaling limit of loop models such as the Potts and $O(n)$ models. To define it mathematically, the curvature terms require a detailed analysis of the topology, and the potential terms are defined using the imaginary Guassian Multiplicative Chaos (GMC). We prove that the resulting probabilistic path integral satisfies the axioms of CFT, including Segal's gluing axioms. This work provides the foundation for future studies of boundary CILT and will also help with the understanding of CILT.

Boundary Compactified Imaginary Liouville Theory

TL;DR

This work constructs Boundary Compactified Imaginary Liouville Theory (BCILT), extending the nonunitary, logarithmic CILT to manifolds with boundary by incorporating Neumann boundary conditions, curvature couplings, and imaginary Gaussian Multiplicative Chaos (GMC) on both bulk and boundary. The authors develop a rigorous probabilistic framework, starting from a compactified Gaussian Free Field, employing separating families to define curvature terms, and leveraging imaginary GMC to define vertex insertions, while ensuring invariance under translations and conformal changes. A central achievement is proving Segal's axioms for BCILT: well-defined correlation functions, Segal amplitudes, Weyl covariance, spin, and a gluing mechanism that preserves the amplitude structure under all allowed surgeries. The curvature-anomaly analysis, Neumann doubling, and GMC estimates adapted to boundary settings establish a robust path-integral construction with explicit structure constants expressed via Coulomb gas and related integrals, laying groundwork for BCILT and its connections to loop models and boundary CFTs. This framework provides rigorous tools for future bootstrap analyses and potential ties to SLE/CLE-type boundary phenomena in statistical physics.

Abstract

We generalize the construction of Compactified Imaginary Liouville Theory (CILT), a non-unitary logarithmic Conformal Field Theory (CFT) defined on closed surfaces, to surfaces with boundary. Starting from a compactified Gaussian Free Field (GFF) with Neumann boundary condition, we perturb it by adding in curvature terms and exponential potentials on both the bulk and the boundary. In physics, this theory is conjectured to describe the scaling limit of loop models such as the Potts and models. To define it mathematically, the curvature terms require a detailed analysis of the topology, and the potential terms are defined using the imaginary Guassian Multiplicative Chaos (GMC). We prove that the resulting probabilistic path integral satisfies the axioms of CFT, including Segal's gluing axioms. This work provides the foundation for future studies of boundary CILT and will also help with the understanding of CILT.

Paper Structure

This paper contains 42 sections, 35 theorems, 207 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Sketch of the construction
  3. Statement of the main results
  4. Open questions and future works
  5. Organization of the paper
  6. Acknowledgments
  7. Geometric preliminaries
  8. Neumann doubling
  9. Distribution spaces
  10. Laplacians and Green's functions
  11. Poisson and Dirichlet-to-Neumann operators
  12. Separating families
  13. Cohomology
  14. Gluing
  15. Gluing two surfaces along a circle
  16. ...and 27 more sections

Key Result

Theorem 1.2

This defines a Conformal Field Theory in the following sense:

Figures (11)

  • Figure 1: On the left is a possible choice of a separating family $(d_1,d_2,d_3)$ on the annulus $\mathbb A$ with two Neumann boundary circles $\partial_\pm{\mathbb{A}}$ and two insertions $(z_1,z_2)$. Note that the orientation of the $d_j$ does not matter. On the right, we obtain a new separating family by replacing $d_1$ with the blue curve between $z_1$ and $z_2$. Then the value of $\int_{x_0}^z\omega$ changes by $m_1$, the winding number around $z_1$.
  • Figure 2: An extended surface of type $(1,1,2,1,2,2,2)$ and its double
  • Figure 3: A separating family of an extended surface
  • Figure 4: Gluing two surfaces along a Dirichlet circle
  • Figure 5: Gluing two surfaces along a Dirichlet semicircle
  • ...and 6 more figures

Theorems & Definitions (81)

  • Definition 1.1
  • Theorem 1.2
  • Remark
  • Remark
  • Definition 2.1
  • Remark
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • ...and 71 more