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} \(\pi^k \pi^k\)
+ *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*s*^{2} = *h*_{AB}
dΦ^{A} dΦ^{B}
= tr[(d*χ*) \(\chi^{-1}\)]^{n},
*n* being the dimension of spacetime, and *J _{a}*:=
(∂

*

*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);
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.

main page
– abbreviations
– journals – comments
– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 31 oct 2018