An Index for Quantum Integrability

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.

arXiv comment24 pages, 2 tables

References in corpus (31)

• An Index for 4 dimensional Super Conformal Theories
• Theta, Time Reversal, and Temperature
• Operator bases, $S$-matrices, and their partition functions
• Intrinsic and emergent anomalies at deconfined critical points
• Strings on Semisymmetric Superspaces
• Anomaly and global inconsistency matching: $\theta$-angles, $SU(3)/U(1{)}^2$ nonlinear sigma model, $SU(3)$ chains and its generalizations
• Anomalies and Bounds on Charged Operators
• Symbolic integration with respect to the Haar measure on the unitary group
• Coset Construction for Duals of Non-relativistic CFTs
• Generalization of the Haldane conjecture to SU(3) chains
• Integrable sigma models with theta=pi
• Classical Integrability of the Squashed Three-sphere, Warped AdS3 and Schroedinger Spacetime via T-Duality
• Haldane limits via Lagrangian embeddings
• Integrable sigma models and perturbed coset models
• Commuting charges and symmetric spaces
• Classical integrability of Schrodinger sigma models and q-deformed Poincare symmetry
• On the integrability of two-dimensional models with U(1)xSU(N) symmetry
• Sigma Models on Flags
• Integrable properties of sigma-models with non-symmetric target spaces
• Mixed global anomalies and boundary conformal field theories
• The geometry of antiferromagnetic spin chains
• Quantum, higher-spin, local charges in symmetric space sigma models
• Phase structure of the twisted $SU(3)/U(1{)}^2$ flag sigma model on $\mathbb{R}\times S^1$
• Non-local charges and quantum integrability of sigma models on the symmetric spaces SO(2n)/SO(n)xSO(n) and Sp(2n)/Sp(n)xSp(n)
• Condition on Ramond-Ramond fluxes for factorization of worldsheet scattering in anti-de Sitter space
• Classifying integrable symmetric space strings via factorized scattering
• Flag manifold sigma-models: the 1/N-expansion and the anomaly two-form
• Quantum integrability of sigma models on AII and CII symmetric spaces
• Constraining integrable AdS/CFT with factorized scattering
• Aspects of quantum integrability for pure spinor superstring in AdS(5)xS(5)
• Nonlocal Symmetries and Factorized Scattering

Cited by in corpus (1)

• Notes on index of quantum integrability