# An Index for Quantum Integrability

## TL;DR

This paper introduces an integrability index to determine the presence of quantum conserved currents in two-dimensional sigma models, providing a systematic way to assess quantum integrability across different models.

## Contribution

It defines a new integrability index for each spin, derives explicit formulas for coset models, and applies these to identify quantum conserved currents in specific sigma models.

## Key findings

- Established the existence of a spin-6 quantum conserved current in the O(N) model.
- Indices for CP^{N-1} models are non-positive, indicating non-integrability.
- Indices for flag sigma models are negative, suggesting these models are not integrable.

## Abstract

The existence of higher-spin quantum conserved currents in two dimensions guarantees quantum integrability. We revisit the question of whether classically-conserved local higher-spin currents in two-dimensional sigma models survive quantization. We define an integrability index $\mathcal{I}(J)$ for each spin $J$, with the property that $\mathcal{I}(J)$ is a lower bound on the number of quantum conserved currents of spin $J$. In particular, a positive value for the index establishes the existence of quantum conserved currents. For a general coset model, with or without extra discrete symmetries, we derive an explicit formula for a generating function that encodes the indices for all spins. We apply our techniques to the $\mathbb{CP}^{N-1}$ model, the $O(N)$ model, and the flag sigma model $\frac{U(N)}{U(1)^{N}}$. For the $O(N)$ model, we establish the existence of a spin-6 quantum conserved current, in addition to the well-known spin-4 current. The indices for the $\mathbb{CP}^{N-1}$ model for $N>2$ are all non-positive, consistent with the fact that these models are not integrable. The indices for the flag sigma model $\frac{U(N)}{U(1)^{N}}$ for $N>2$ are all negative. Thus, it is unlikely that the flag sigma models are integrable.