A proof of loop equations in 2d topological gravity

TL;DR

This paper provides a detailed, explicit proof of the loop equations for multi-boundary correlators in 2d topological gravity, clarifying previous derivations and verifying them through new examples including JT gravity and Airy models.

Contribution

It offers the first fully explicit proof of the loop equations in 2d topological gravity with general couplings, including detailed treatment of genus zero correlators.

Findings

Loop equations are verified in several new examples.

The proof clarifies the genus zero part of the one-boundary correlator.

Comparison of correlator representations reveals similarities and differences.

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.