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An Index for Quantum Integrability

by Shota Komatsu, Raghu Mahajan, Shu-Heng Shao

Submission summary

As Contributors: Shu-Heng Shao
Arxiv Link:
Date submitted: 2019-08-13
Submitted by: Shao, Shu-Heng
Submitted to: SciPost Physics
Domain(s): Theoretical
Subject area: High-Energy Physics - Theory


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

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