An index for quantum integrability

The existence of higher-spin quantum conserved currents in two dimensions guarantees quantum integrability. We revisit the question [1] of whether classically-conserved local higher-spin currents in two-dimensional sigma models survive quantization. We define an integrability index IpJq for each spin J, with the property that IpJq 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 CP^N´1 model, the OpNq model, and the flag sigma model _U^U_p^p₁^N_qN^q . For the OpNq 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 CP^N´1 model for N ą 2 are all non-positive, consistent with the fact that these models are not integrable. The indices for the flag sigma model _U^U_p^p₁^N_qN^q for N ą 2 are all negative. Thus, it is unlikely that the flag sigma models are integrable.