Notes on index of quantum integrability

A quantum integrability index was proposed in Komatsu et al (2019 SciPost Phys. 7 065). It systematizes the Goldschmidt and Witten's operator counting argument (Goldschmidt and Witten 1980 Phys. Lett. B 91 392) by using the conformal symmetry. In this work we compute the quantum integrability indexes for the symmetric coset models SU(N) SO(N) and SO(2N) SO(N) x SO(N). The indexes of these theories are all non-positive except for the case of SO(4) SO(2) x SO(2). Moreover we extend the analysis to the theories with fermions and consider a concrete theory: the CPN model coupled with a massless Dirac fermion. We find that the indexes for this class of models are non-positive as well.