document: Classical, two-dimensional sigma models on compact symmetric spaces G/H are integrable by virtue of conserved quantities which can arise as integrals of local or non-local functions of the underlying fields (the accounts in [1–5] contain references to the extensive literature). Since these models are asymptotically free and strongly coupled in the infrared, their quantum properties are not straightforward to determine. Nevertheless, following Lüscher [6], Abdalla, Forger and Gomes showed [7] that, in a G/H sigma model with H simple,11Here, and throughout this Letter, we shall use ‘simple’ to mean that the corresponding Lie algebra has no non-trivial ideals. Hence U(1) is simple in our terminology, in addition to the usual non-Abelian simple groups of the Cartan–Killing classification [13]. the first conserved non-local charge survives quantization (after an appropriate renormalization [6–8]), which suffices to ensure quantum integrability of the theory. By contrast, calculations using the 1/N expansion reveal anomalies that spoil the conservation of the quantum non-local charges in the CPN−1=SU(N)/SU(N−1)×U(1) models for N>2, and in the wider class of theories based on the complex Grassmannians SU(N)/SU(n)×SU(N−n)×U(1) for N>n>1 [9].
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extractive_keyphrases: quantization quantum integrability renormalization simple
abstractive_keyphrases: calculations using the 1/n expansion complex grassmannians su(n)/su(n)×su(n−n)×u(1) for n>n>1 conservation of the quantum non-local charges corresponding lie algebra has no non-trivial ideals cpn−1=su(n)/su(n−1)×u(1) models for n>2 g/h sigma model with h simple integrals of local or non-local functions two-dimensional sigma models on compact symmetric spaces g/h