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 S^{3} and fields
Φ^{A},
with the constraint ∑_{k} *π*^{k }*π*^{k}
+ *σ*^{2} = *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(*J _{a}*

where the metric *h** _{AB}* is
defined by d

*

*J*^{a}_{,
}* _{a}* = 0 .

* __Coupled to gravity__:
The action and the equations of motion become

*S* = ∫ (*R* – \({1\over4}\)tr *J _{a} J*

**Examples, Special Types** > s.a. canonical quantum theory;
solution methods for the einstein equation.

* __On a symmetric space G/H__:
A non-linear

- In the study of

- In dimensionally reduced supergravity.

@

@

@

@

@

**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).

@ __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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sep 2016