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A σ-model, generically, is a spinless quantum field theory with an appropriate group symmetry structure. Normally, it serves as an effective theory of pseudoscalar mesons rising out of chiral symmetry breaking in QCD, and a scalar σ whose v.e.v. controls PCAC; this is the linear model. The nonlinear σ model has this σ field frozen to its v.e.v. and thus absent from the spectrum.

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Saying that a splitting varies over the moduli space is not completely well defined: you have to say how to identify the total spaces at different points of the moduli i.e. to specify a flat connectio …
answered Mar 6 '14 by user40227