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 -model comes from the original theory, which described QCD phenomenology, and contained a pion triplet field and a scalar, the particle; It was a harmonic map with target space S³ and fields ^A, with the constraint _k ^{k k} + ² = f ² = const; 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 -model is a theory of a spacetime scalar field , described by an action

S =  tr(J_a J^a) dv =  h_{AB a}^{A a B} dv ,

where the metric h_AB is defined by ds² = h_AB d^A d^B = tr[(d) ^–1]ⁿ, n being the dimension of spacetime, and J_a:= (_a) ^–1.
* Field equations: They are of the form

J^a_{, a} = 0 .

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

S = (R – tr J_a J^a) dv   and   R_ab = tr(J_a J_b) .

Examples, Special Types > s.a. solution methods for the Einstein equation.
* On a symmetric space G/H: A non-linear -model on G/H is an example of a harmonic map between two Riemannian spaces, as can be seen from the action; Non-linear -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, R)/SO(j, k–j+1) -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-in; Batalin & Marnelius PLB(01) [generalized]; Cattaneo m.QA/07 [and deformation quantization].
@ SU(2)-sigma model and gravity: Bizon & Wasserman PRD(00)gq [self-similar spherical].
@ Supersymmetric: Barnes PLB(99)ht; Imbiriba ht/99; Higashijima & Nitta ht/00-in [as gauge theories]; Albertsson et al NPB(04) [with bdries].
@ 2+1: Fujii & Suzuki LMP(98) [conserved currents].

Specific Features and Effects > s.a. duality; soliton.
@ 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; 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).
@ Textbooks and reviews: Percacci 86.
@ Quantization: Isham & Klauder JMP(90).
@ Renormalization: Bonneau ht/99 [and BRS]; Ferrari & Quadri IJTP(06)ht/05 [weak power-counting theorem], JHEP(06)ht/05 [4D, two-loop].

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 25 may 2008