Beyond descendants: integrable observables for cohomological field theories

TL;DR

The paper introduces integrable observables as a new class of objects in cohomological field theories, preserving integrability and connecting to known hierarchies, while providing new proofs and structural insights.

Contribution

It proposes integrable observables as alternatives to psi classes, demonstrating their role in unifying and extending existing hierarchies and offering new proofs of key conjectures.

Findings

Recover the Dubrovin-Zhang and double ramification hierarchies

Establish Miura equivalence of new hierarchies to known ones

Provide a short proof of Witten's conjecture

Abstract

We introduce the concept of integrable observables and propose them as alternatives to the standard Witten's psi classes (a.k.a. descendants in $2D$ quantum gravity) to be coupled with cohomological field theories and their generalisations. The main property of integrable observables is that they retain the integrability properties. We present three examples of integrable observables. The first two recover the Dubrovin-Zhang and double ramification hierarchies, while revealing new structural features in this framework. The third, a new example, builds on recently established properties of the so-called $\mathbb{\Pi}$-class, extending them and placing this class naturally within the theory of integrable systems. Notably, our integrable observables framework yields a proof that the new $\mathbb{\Pi}$-hierarchies are Miura equivalent both to the Dubrovin-Zhang hierarchies and to the double ramification hierarchies. A new very short proof of Witten's conjecture is also provided.