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A quantum integrability index was proposed in cite{KMS}. It systematizes the Goldschmidt and Wittens operator counting argument cite{GW} by using the conformal symmetry. In this work we compute the quantum integrability indexes for the symmetric coset models ${SU(N)}/{SO(N)}$ and $SO(2N)/{SO(N)times SO(N)}$. The indexes of these theories are all non-positive except for the case of ${SO(4)}/{SO(2)times SO(2)}$. Moreover we extend the analysis to the theories with fermions and consider a concrete theory: the $mathbb{CP}^N$ model coupled with a massless Dirac fermion. We find that the indexes for this class of models are non-positive as well.
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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.
We investigate the quantum integrability of the Landau-Lifshitz model and solve the long-standing problem of finding the local quantum Hamiltonian for the arbitrary n-particle sector. The particular difficulty of the LL model quantization, which arises due to the ill-defined operator product, is dealt with by simultaneously regularizing the operator product, and constructing the self-adjoint extensions of a very particular structure. The diagonalizibility difficulties of the Hamiltonian of the LL model, due to the highly singular nature of the quantum-mechanical Hamiltonian, are also resolved in our method for the arbitrary n-particle sector. We explicitly demonstrate the consistency of our construction with the quantum inverse scattering method due to Sklyanin, and give a prescription to systematically construct the general solution, which explains and generalizes the puzzling results of Sklyanin for the particular two-particle sector case. Moreover, we demonstrate the S-matrix factorization and show that it is a consequence of the discontinuity conditions on the functions involved in the construction of the self-adjoint extensions.
It has been recently shown that every SCFT living on D3 branes at a toric Calabi-Yau singularity surprisingly also describes a complete integrable system. In this paper we use the Master Space as a bridge between the integrable system and the underlying field theory and we reinterpret the Poisson manifold of the integrable system in term of the geometry of the field theory moduli space.
Chiral anomalies give rise to dissipationless transport phenomena such as the chiral magnetic and vortical effects. In these notes I review the theory from a quantum field theoretic, hydrodynamic and holographic perspective. A physical interpretation of the otherwise somewhat obscure concepts of consistent and covariant anomalies will be given. Vanishing of the CME in strict equilibrium will be connected to the boundary conditions in momentum space imposed by the regularization. The role of the gravitational anomaly will be explained. That it contributes to transport in an unexpectedly low order in the derivative expansion can be easiest understood via holography. Anomalous transport is supposed to play also a key role in understanding the electronics of advanced materials, the Dirac- and Weyl (semi)metals. Anomaly related phenomena such as negative magnetoresistivity, anomalous Hall effect, thermal anomalous Hall effect and Fermi arcs can be understood via anomalous transport. Finally I briefly review a holographic model of Weyl semimetal which allows to infer a new phenomenon related to the gravitational anomaly: the presence of odd viscosity.
Higher-spin diffeomorphisms are to higher-order differential operators what diffeomorphisms are to vector fields. Their rigorous definition is a challenging mathematical problem which might predate a better understanding of higher-spin symmetries and interactions. Several yes-go and no-go results on higher-spin diffeomorphisms are collected from the mathematical literature in order to propose a generalisation of the algebra of differential operators on which higher-spin diffeomorphisms are well-defined.
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