# An Index for Quantum Integrability

### Submission summary

 As Contributors: Shu-Heng Shao Arxiv Link: https://arxiv.org/abs/1907.07186v1 Date submitted: 2019-08-13 Submitted by: Shao, Shu-Heng Submitted to: SciPost Physics Discipline: Physics Subject area: High-Energy Physics - Theory Approach: Theoretical

### 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.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission 1907.07186v1 on 13 August 2019

## Reports on this Submission

### Strengths

1) Systematic construction of an index giving a lower bound on the number of quantum conserved currents of some classically integrable sigma modelsa
2) Article technically sound, interesting and well written

### Weaknesses

1) The limit of the method is that it does not allow to conclude that a model is not quantum integrable.

### Report

The authors give a systematic construction of an index which is a lower bound on the number of quantum conserved currents of a given spin in some classically integrable sigma models. This is a generalization of a method originally developed by Y. Goldschmidt and E. Witten. Whereas if the index is strictly positive one can conclude that there exists quantum conserved currents, no definite conclusion can be reached when the index is negative or zero. However, the authors check that in many cases where the models are believed not to be quantum integrable, all indices that they compute are negative or zero

• validity: high
• significance: good
• originality: good
• clarity: high
• formatting: excellent
• grammar: excellent