Editing Polyakov action (section)

{{Short description|2D conformal field theory used in string theory}}
{{String theory}}
In [[physics]], the '''Polyakov action''' is an [[action (physics)|action]] of the [[two-dimensional conformal field theory]] describing the [[worldsheet]] of a string in [[string theory]]. It was introduced by [[Stanley Deser]] and [[Bruno Zumino]] and independently by [[Lars Brink|L. Brink]], [[Paolo Di Vecchia|P. Di Vecchia]] and P. S. Howe in 1976,<ref>{{cite journal |last1=Deser |first1=S. |authorlink1=Stanley Deser |last2=Zumino |first2=B. |authorlink2=Bruno Zumino |date=1976 |title=A Complete Action for the Spinning String |url=https://cds.cern.ch/record/201648|journal=Phys. Lett. B |volume=65 |issue= 4|pages=369–373 |doi=10.1016/0370-2693(76)90245-8}}</ref><ref>{{cite journal |last1=Brink |first1=L. |authorlink1=Lars Brink |last2=Di Vecchia |first2=P. |authorlink2=Paolo Di Vecchia |last3=Howe |first3=P. |date=1976 |title=A locally supersymmetric and reparametrization invariant action for the spinning string |url=https://dx.doi.org/10.1016/0370-2693%2876%2990445-7 |journal=Physics Letters B |volume=65 |issue=5 |pages=471–474 |doi=10.1016/0370-2693(76)90445-7|url-access=subscription }}</ref> and has become associated with [[Alexander Markovich Polyakov|Alexander Polyakov]] after he made use of it in quantizing the string in 1981.<ref>{{cite journal |last1=Polyakov |first1=A. M. |authorlink1=Alexander Markovich Polyakov |date=1981 |title=Quantum geometry of bosonic strings |url=https://dx.doi.org/10.1016/0370-2693%2881%2990743-7 |journal=Physics Letters B |volume=103 |issue=3 |pages=207–210 |doi=10.1016/0370-2693(81)90743-7|url-access=subscription }}</ref> The action reads:
: <math>\mathcal{S} = \frac{T}{2} \int\mathrm{d}^2\sigma\, \sqrt{-h}\,h^{ab} g_{\mu\nu}(X) \partial_a X^\mu(\sigma) \partial_b X^\nu(\sigma),</math>

where <math>T</math> is the string [[Tension (mechanics)|tension]], <math>g_{\mu\nu}</math> is the metric of the [[target manifold]], <math>h_{ab}</math> is the worldsheet metric, <math>h^{ab}</math> its inverse, and <math>h</math> is the determinant of <math>h_{ab}</math>. The [[metric signature]] is chosen such that timelike directions are + and the spacelike directions are −. The spacelike worldsheet coordinate is called <math>\sigma</math>, whereas the timelike worldsheet coordinate is called <math>\tau</math>. This is also known as the [[nonlinear sigma model]].<ref name="Frie80">{{cite journal |last=Friedan |first=D. |author-link=Daniel Friedan |title=Nonlinear Models in 2+ε Dimensions |journal=[[Physical Review Letters]] |volume=45 |pages=1057–1060 |date=1980 |issue=13 |url=http://www.physics.rutgers.edu/~friedan/papers/PRL_45_1980_1057.pdf |doi=10.1103/PhysRevLett.45.1057 |bibcode=1980PhRvL..45.1057F}}</ref>

The Polyakov action must be supplemented by the [[Liouville field theory|Liouville action]] to describe string fluctuations.