On the Null Vectors in the Spectra of the 2D Integrable Hierarchies

TL;DR

This paper introduces a new method to describe the spectrum of local fields in classical integrable quantum field theories, systematically deriving null-vectors and providing explicit results for specific hierarchies.

Contribution

It offers an alternative, systematic approach to derive null-vectors in the spectra of 2D integrable hierarchies, connecting classical limits with geometric constructions.

Findings

Explicit null-vector derivations for A_1^{1}-(m)KdV and A_2^{2}-(m)KdV hierarchies.

Results align with classical limits of existing quantum constructions.

Provides insights into quantization and off-critical extensions.

Abstract

We propose an alternative description of the spectrum of local fields in the classical limit of the integrable quantum field theories. It is close to similar constructions used in the geometrical treatment of W-gravities. Our approach provides a systematic way of deriving the null-vectors that appear in this construction. We present explicit results for the case of the A_1^{1}-(m)KdV and the A_2^{2}-(m)KdV hierarchies, different classical limits of 2D CFT's. In the former case our results coincide with the classical limit of the construction of Babelon, Bernard and Smirnov.Some hints about quantization and off-critical treatment are also given.