Sigma Models  

In General
* Motivation: Non-linear models are useful in treating spontaneous symmetry breaking, where the absence of an invariant ground state is described in terms of constraints on the fields, equivalent to non-linear submanifolds of vector spaces on which the group acts.
* History: The name σ-model comes from the original theory, which described QCD phenomenology, and contained a pion triplet field and a scalar, the σ particle; It was a harmonic map with target space S3 and fields ΦA, with the constraint ∑k \(\pi^k \pi^k\) + σ2 = f 2 = constant; Notice that, with the constraint, the values of the fields do not form a vector space, but they have a Riemannian structure; Later the name has been extended to other kinds of theories, other kinds of harmonic maps.
$ Def: A σ-model is a theory of a spacetime scalar field χ, described by the action

S = \({1\over4}\)∫ tr(Ja Ja) dv = \({1\over4}\)∫ hAB ∂aΦA ∂aΦB dv ,

where the metric hAB is defined by ds2 = hAB dΦA dΦB = tr[(dχ) \(\chi^{-1}\)]n, n being the dimension of spacetime, and Ja:= (∂aχ) \(\chi^{-1}\).
* Field equations: They are of the form

Ja, a = 0 .

* Coupled to gravity: The action and the equations of motion become

S = ∫ (R − \({1\over4}\)tr Ja Ja) dv   and   Rab = \({1\over2}\)tr(Ja Jb) .

Examples, Special Types > s.a. canonical quantum theory; solution methods for the einstein equation.
* On a symmetric space G/H: A non-linear σ-model on G/H is an example of a harmonic map between two Riemannian spaces, as can be seen from the action; Non-linear σ-models on symmetric spaces appear naturally
- In the study of n-dimensional solutions of Einstein's equation with a set of k commuting Killing vector fields; The theory reduces to (d−k)-dimensional gravity coupled to a SL(k+1, \(\mathbb R\))/SO(j, k−j+1) σ-model on the space of orbits of the Killing vector fields, where j is the number of timelike Killing vector fields.
- In dimensionally reduced supergravity.
@ Topological: Witten CMP(88).
@ Poisson-sigma-models: Schaller & Strobl MPLA(94), LNP(94)gq, ht/94, LNP(96)ht/95 [intro]; Bandos & Kummer IJMPA(99)ht/97; Hirshfeld & Schwarzweller ht/00-proc; Batalin & Marnelius PLB(01) [generalized]; Cattaneo m.QA/07 [and deformation quantization]; Bonechi et al JHEP(12)-a1110.
@ Sigma-models and gravity: Bizoń & Wasserman PRD(00)gq [self-similar spherical]; Clément a0812-conf [higher-dimensional, and supergravity]; > s.a. massive gravity.
@ Supersymmetric σ-models: Barnes PLB(99)ht; Imbiriba ht/99; Higashijima & Nitta ht/00-conf [as gauge theories]; Albertsson et al NPB(04) [with boundaries].
@ Other types: Fujii & Suzuki LMP(98) [(2+1)-dimensional, conserved currents]; Percacci & Zanusso PRD(10)-a0910 [with higher-derivative terms].

Specific Features and Effects
> s.a. duality; Ricci Flow [renormalization-group flow]; solitons.
@ Critical phenomena: Liebling et al JMP(00)mp/99 [singular/non-singular solutions].
@ Global identity: Mazur PLA(84) [generalization of Green identities].
@ As deformed topological field theories: Fendley PRL(99)ht.
@ Related topics: Bastianelli et al PLB(00) [dimensional regularization]; > s.a. critical collapse; lie algebras [algebroids]; phase transitions.

References
@ Original model: Gell-Mann & Levy NC(60).
@ General: Coleman, Wess & Zumino PR(69); Salam & Strathdee PR(69), PR(69); Duff & Isham PRD(77); Lindström a1803-conf [uses, rev]; Ang & Prakash a1810 [solitonic sectors, and ordered phases of systems with spontaneously broken symmetries].
@ Textbooks and reviews: Percacci 86.
@ Quantization: Isham & Klauder JMP(90); Nguyen JMP-a1408 [2D, perturbative].
@ Renormalization: Bonneau ht/99-in [and BRS symmetry]; Ferrari JHEP(05)ht [flat connection structure]; Ferrari & Quadri IJTP(06)ht/05 [weak power-counting theorem], JHEP(06)ht/05 [4D, two-loop]; Codello & Percacci PLB(09)-a0810 [fixed points].
> Online resources: see Wikipedia page.

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