Non-linear sigma model: Difference between revisions

Revision as of 17:55, 28 August 2015

Revision as of 21:43, 28 August 2015

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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>{{citeCite journal doi| last1 = Gürsey | first1 = F. | title = On the symmetries of strong and weak interactions | doi = 10.1007/BF02860276 |noedit journal = Il Nuovo Cimento | volume = 16 | issue = 2 | pages = 230–240 | year = 1960 | pmid = | pmc = }}</ref> is given by

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

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

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Further adding nonlinear interactions representing flavor-chiral anomalies results in the [[Wess–Zumino–Witten model]],<ref>{{cite journal |first=E. |last=Witten |title=Non-abelian bosonization in two dimensions |journal=[[Communications in Mathematical Physics]] |volume= 92| issue= 4 |year=1984 | pages= 455–472 | doi= 10.1007/BF01215276|bibcode = 1984CMaPh..92..455W }}</ref> which

augments the geometry of the flow to include [[Torsion tensor|torsion]], preserving renormalizability and leading to an [[infrared fixed point]] as well, on account of [[teleparallelism]] ("geometrostasis").<ref>{{citeCite journal doi| last1 = Braaten | first1 = E. | last2 = Curtright | first2 = T. L. | last3 = Zachos | first3 = C. K. | doi = 10.1016/0550-3213(85)90053-7 |noedit title = Torsion and geometrostasis in nonlinear sigma models | journal = Nuclear Physics B | volume = 260 | issue = 3–4 | pages = 630 | year = 1985 | pmid = | pmc = |bibcode = 1985NuPhB.260..630B }}</ref>

{{further|Ricci flow}}