A proof of loop equations in 2d topological gravity

Abstract

We study multi-boundary correlators in 2d Witten-Kontsevich topological gravity. We present a proof of the loop equations obeyed by the correlators. While the loop equations were derived a long time ago, our proof is fully explicit in the presence of general couplings $t_k$. We clarify all the details, in particular the treatment of the genus zero part of the one-boundary correlator. The loop equations are verified by several new examples, including the correlators of Jackiw-Teitelboim gravity in the genus expansion and the exact correlators in the Airy case. We also discuss the free boson/fermion representation of the correlators and compare it with the formulation of Marolf and Maxfield and the string field theory of Ishibashi and Kawai. We find similarities but also some differences.

33 pages, v2: some explanations added, published version

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• JT gravity as a matrix integral
• Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models
• JT gravity, KdV equations and macroscopic loop operators
• Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy
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• CFT and topological recursion
• Density matrices in quantum gravity
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Cited by in corpus (1)

• Deformations of JT Gravity via Topological Gravity and Applications