[论文解读] The complex Liouville string: the worldsheet

Scott Collier, Lorenz Eberhardt+2 more|arXiv (Cornell University)|Sep 27, 2024

Computational Physics and Python Applications被引用 5

一句话总结

引入一个通过在世界面上将两个具有复数中心荷的 Liouville CFT 耦合而成的新二维字符串理论，利用世界面自举分析其振幅，并讨论一个对偶的双矩阵积分以及与二维 de Sitter 引力的含义。

ABSTRACT

We introduce a new two-dimensional string theory defined by coupling two copies of Liouville CFT with complex central charge $c=13\pm i λ$ on the worldsheet. This string theory defines a novel, consistent and controllable model of two-dimensional quantum gravity. We use the exact solution of the worldsheet theory to derive stringent constraints on the analytic structure of the string amplitudes as a function of the vertex operator momenta. Together with other worldsheet constraints, this allows us to completely pin down the string amplitudes without explicitly computing the moduli space integrals. We focus on the case of the sphere four-point amplitude and torus one-point amplitude as worked examples. This is the first in a series of papers on the complex Liouville string: three subsequent papers will elucidate the holographic duality with a two-matrix integral, discuss worldsheet boundaries and non-perturbative effects, and connect the theory to de Sitter quantum gravity.

研究动机与目标

• Motivate and construct a two-dimensional string theory built from two complex-central-charge Liouville CFTs and bc-ghosts.
• Derive and constrain string amplitudes using exact worldsheet data and bootstrap principles.
• Show the analytic structure of amplitudes, including poles and discontinuities, to determine amplitudes without explicit moduli space integrals.
• Demonstrate a dual description of the theory via a double-scaled two-matrix integral.
• Illustrate how sphere four-point and torus one-point amplitudes can be solved and related to Virasoro minimal string quantities.

提出的方法

• Define the worldsheet theory as two coupled Liouville CFTs with c+ = 13 + iλ and c− = 13 − iλ, along with bc-ghosts.
• Use DOZZ structure constants C_b and the leg factor ρ_b to construct Liouville four-point and related correlators.
• Express string amplitudes A_{g,n}^{(b)} by integrating worldsheet correlators over moduli space with bc-ghosts and vertex operators.
• Exploit analytic structure, including an infinite series of Liouville-resonance poles and moduli-space discontinuities, to bootstrap amplitudes.
• Provide explicit formulas for sphere 3-point, torus 1-point, and sphere 4-point amplitudes (V_{1,1}^{(b)}, V_{0,4}^{(b)}, a_{0,3}^{(b)}, a_{1,1}^{(b)}, a_{0,4}^{(b)}).
• Outline the dual description via a double-scaled two-matrix integral and relate to topological recursion and Mirzakhani-type structures.

实验结果

研究问题

• RQ1 What is the analytic structure of amplitudes in the complex Liouville string as a function of external momenta?
• RQ2 Can one bootstrap and uniquely determine sphere four-point and torus one-point amplitudes from worldsheet constraints without performing explicit moduli-space integrals?
• RQ3 How do Liouville resonance poles and moduli-space discontinuities constrain the amplitudes and their factorization properties?
• RQ4 What is the role of the dual matrix model in encoding perturbative amplitudes and how does topological recursion arise in this context?
• RQ5 How do the sphere four-point and torus one-point amplitudes relate to the corresponding Virasoro minimal string quantities?

主要发现

• The complex Liouville string is defined by two complex-conjugate Liouville theories with c± = 13 ∓ iλ, yielding a real total theory with PSL(2,C) symmetry.
• The sphere four-point and torus one-point amplitudes can be expressed in terms of building blocks V_{1,1}^{(b)} and V_{0,4}^{(b)} that are analytic continuations of Virasoro minimal-string quantum volumes.
• The amplitudes display an infinite set of Liouville resonance poles and an infinite series of discontinuities from moduli-space integration, enabling a stringent bootstrap that uniquely fixes the amplitudes under mild growth assumptions.
• A dual description exists as a double-scaled two-matrix integral, whose perturbative expansion is governed by a spectral curve with nodal singularities, with a topological recursion closely related to Mirzakhani-type recursions.
• The sphere four-point amplitude and torus one-point amplitude coincide with proposed expressions involving sums over m of sine factors and Liouville volumes, and these proposals can be verified numerically against moduli-space integrals.