## An index for quantum integrability

Shota Komatsu, Raghu Mahajan, Shu-Heng Shao

SciPost Phys. 7, 065 (2019) · published 26 November 2019

- doi: 10.21468/SciPostPhys.7.5.065
- Submissions/Reports

### 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.

### Authors / Affiliations: mappings to Contributors and Organizations

See all Organizations.-
^{1 }Shota Komatsu, -
^{1 }^{2 }Raghu Mahajan, -
^{1 }Shu-Heng Shao

- Instituto Nazionale di Fisica Nucleare (INFN) (through Organization: Istituto Nazionale di Fisica Nucleare / National Institute for Nuclear Physics [INFN])
- National Science Foundation [NSF]
- Simons Foundation
- United States Department of Energy [DOE]