Non-linear sigma model: Difference between revisions

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==Description==

The target manifold ''T'' is equipped with a [[Riemannian metric]] ''g''. {{mvar|Σ}} is a differentiable map from [[Minkowski space]] ''M'' (or some other space) to ''T''.

The [[Lagrangian density]] in contemporary chiral form<ref>{{cite doi|10.1007/BF02860276|noedit}}</ref> is given by:

:<math>\mathcal{L}={1\over 2}g(\partial^\mu\Sigma,\partial_\mu\Sigma)-V(\Sigma)</math>

where here, we have used a +&nbsp;−&nbsp;−&nbsp;− [[metric signature]] and the [[partial derivative]] <{{math>\partial\Sigma</math>| ''∂Σ''}} is given by a section of the [[jet bundle]] of ''T''&times;''M'' and ''{{mvar|V''}} is the potential.

In the coordinate notation, with the coordinates {{math|''Σ^''a''''}}, ''a''&nbsp;=&nbsp;1,&nbsp;...,&nbsp;''n'' where ''n'' is the dimension of&nbsp;''T'',

where here, we have used a +&nbsp;−&nbsp;−&nbsp;− [[metric signature]] and the [[partial derivative]] <math>\partial\Sigma</math> is given by a section of the [[jet bundle]] of ''T''&times;''M'' and ''V'' is the potential.

In the coordinate notation, with the coordinates Σ^''a'', ''a''&nbsp;=&nbsp;1,&nbsp;...,&nbsp;''n'' where ''n'' is the dimension of&nbsp;''T'',

:<math>\mathcal{L}={1\over 2}g_{ab}(\Sigma) (\partial^\mu \Sigma^a) (\partial_\mu \Sigma^b) - V(\Sigma).</math>

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This model proved to be relevant in string theory where the two-dimensional manifold is named '''worldsheet'''. Proof of renormalizability was given by [[Daniel Friedan]].<ref name="Frie80">

{{cite journal|last=Friedan|first=D.|authorlink=Daniel Friedan|title=Nonlinear models in 2+ε dimensions | journal = PRL | volume = 45 | issue = 13| pages = 1057 |publisher=|___location=| year = 1980 | url = http://link.aps.org/doi/10.1103/PhysRevLett.45.1057 |doi= 10.1103/PhysRevLett.45.1057 | bibcode=1980PhRvL..45.1057F}}</ref> He showed that the theory admits a renormalization group equation, at the leading order of perturbation theory, in the form

:<math>\lambda\frac{\partial g_{ab}}{\partial\lambda}=\beta_{ab}(T^{-1}g)=R_{ab}+O(T^2).~,</math>

<{{math|''R<sub>R_{ab}</mathsub>''}} being the [[Ricci tensor]] of the target manifold.

This represents a [[Ricci flow]], obeying [[Einstein field equations]] for the target manifold as a fixed point. The existence of such a fixed point is relevant, as it grants, at this order of perturbation theory, that [[conformal field theory|conformal invariance]] is not lost due to quantum corrections, so that the [[quantum field theory]] of this model is sensible (renormalizable).