[论文解读] The complex Liouville string: the matrix integral

Scott Collier, Lorenz Eberhardt+2 more|arXiv (Cornell University)|Oct 9, 2024

Computational Physics and Python Applications被引用 6

一句话总结

这篇论文提出了 complex Liouville string 在 worldsheet 上的对偶性与一个 double-scaled two-matrix integral，并发展了 topological recursion 和一个 Feynman-diagrammatic 框架，以从 matrix model 计算 perturbative string amplitudes。它还分析了具有 infinitely many branch points 和 nodal singularities 的 spectral curve，并通过多种一致性检验来测试对偶性。

ABSTRACT

We propose a duality between the complex Liouville string and a two-matrix integral. The complex Liouville string is defined by coupling two Liouville theories with complex central charges $c = 13 \pm i λ$ on the worldsheet. The matrix integral is characterized by its spectral curve which allows us to compute the perturbative string amplitudes recursively via topological recursion. This duality constitutes a controllable instance of holographic duality. The leverage on the theory is provided by the rich analytic structure of the string amplitudes that we discussed in arXiv:2409.18759 and allows us to perform numerous tests on the duality.

研究动机与目标

• Motivate and establish a new string theory/matrix integral duality for the complex Liouville string.
• Provide a detailed two-matrix integral realization whose spectral curve encodes the string amplitudes.
• Show how topological recursion computes perturbative string amplitudes from the matrix model.
• Demonstrate analytic structures and consistency checks that support the duality.
• Outline connections to CohFT, TQFT, and potential non-perturbative extensions in follow-up work.

提出的方法

• Define the complex Liouville string as two Liouville CFTs with complex central charges coupled to bc ghosts.
• Propose a double-scaled two-matrix integral with a specific spectral curve x(z) = -2 cos(pi b^{-1} sqrt{z}), y(z) = 2 cos(pi b sqrt{z}).
• Develop loop equations and the genus expansion to obtain the spectral curve and resolvents.
• Apply topological recursion for multi-branch-point spectral curves to compute A_{g,n}^{(b)} from the matrix model observables omega_{g,n}^{(b)}.
• Relate matrix-model resolvents to string amplitudes via a branch-point sum and intersection-number structure, interpreted as a CohFT.
• Discuss tests including reproducing bootstrapped amplitudes, analytic continuation, and symmetry properties.

实验结果

研究问题

• RQ1 Can the complex Liouville string be captured by a double-scaled two-matrix integral with the given spectral curve?
• RQ2 How does topological recursion for this spectral curve reproduce perturbative string amplitudes A_{g,n}^{(b)}?
• RQ3 What is the precise dictionary between matrix-model resolvents and Liouville string amplitudes, including branch-point structures?
• RQ4 Does the construction realize a CohFT/TQFT framework and what are the implications for holography and dS/cosmological correlators?
• RQ5 What non-perturbative aspects and future directions follow from this duality?

主要发现

• A dual description of the complex Liouville string via a double-scaled two-matrix integral is established.
• The spectral curve exhibits infinitely many nodal singularities and branch points, leading to an extended topological recursion.
• A direct dictionary links matrix-model resolvents to string amplitudes through sums over branch points and stable-graph degenerations.
• The perturbative string amplitudes satisfy expected dilaton equations and symmetry properties, including an x-y exchange symmetry under b -> b^{-1}.
• The recursion matches a form analogous to Mirzakhani-type recursion for Weil-Petersson volumes and parallels with quantum volumes of the Virasoro minimal string.