Polyakov action

In physics, the Polyakov action is the two-dimensional action of a conformal field theory describing the worldsheet of a string in string theory. It was introduced by S.Deser and B.Zumino and independently by L.Brink, P.Di Vecchia and P.S.Howe (in Physics Letters B65, pgs 369 and 471 respectively), and has become associated with Alexander Polyakov after he made use of it in quantizing the string. The action reads

:$mathcal\{S\}\; =\; \{T\; over\; 2\}int\; mathrm\{d\}^2\; sigma\; sqrt\{-h\}\; h^\{ab\}\; g\_\{mu\; u\}\; (X)\; partial\_a\; X^mu\; (sigma)\; partial\_b\; X^\; u(sigma)$

where $T$ is the string tension, $g\_\{mu\; u\}$ is the metric of the target manifold, $h^\{ab\}$ is the worldsheet metric and $h$ is the determinant of $h\_\{ab\}$. The metric signature is chosen such that timelike directions are + and the spacelike directions are -. The spacelike worldsheet coordinate is called $sigma$ wheareas the timelike worldsheet coordinate is called $au$.

Global symmetries

The action is invariant under spacetime translations and infinitesimal Lorentz transformations::(i) $X^alpha\; ightarrow\; X^alpha\; +\; b^alpha$:(ii) where $omega\_\{mu\; u\}\; =\; -\; omega\_\{\; u\; mu\}$ and $b^alpha$ is a constant. This forms the Poincaré symmetry of the target manifold.

The invariance under (i) follows since the action $mathcal\{S\}$ depends only on the first derivative of $X^alpha$. The proof of the invariance under (ii) is as follows:

::

Local symmetries

The action is invariant under worldsheet diffeomorphisms (or coordinates transformations) and Weyl transformations.

Diffeomorphisms

Assume the following transformation:::$sigma^alpha\; ightarrow\; ilde\{sigma\}^alphaleft(sigma,\; au\; ight)$It transforms the Metric tensor in the following way:::$h^\{ab\}\; ightarrow\; ilde\{h\}^\{ab\}\; =\; h^\{cd\}\; frac\{partial\; ilde\{sigma\}^a\}\{partial\; sigma^c\}\; frac\{partial\; ilde\{sigma\}^b\}\{partial\; sigma^d\}$One can see that:::$ilde\{h\}^\{ab\}\; frac\{partial\}\{partial\; ilde\{sigma\}^a\}\; X^mu\; frac\{partial\}\{partial\; ilde\{sigma\}^b\}\; X^\; u\; =\; h^\{cd\}\; frac\{partial\; ilde\{sigma\}^a\}\{partial\; sigma^c\}\; frac\{partial\; ilde\{sigma\}^b\}\{partial\; sigma^d\}\; frac\{partial\}\{partial\; ilde\{sigma\}^a\}\; X^mu\; frac\{partial\}\{partial\; ilde\{sigma\}^b\}\; X^\; u\; =\; h^\{ab\}\; partial\_a\; X^mu\; partial\_b\; X^\; u$ One knows that the Jacobian of this transformation is given by:::which leads to:::$mathrm\{d\}^2\; sigma\; ightarrow\; mathrm\{d\}^2\; ilde\{sigma\}\; =\; mathrm\{J\}\; mathrm\{d\}^2\; sigma\; ,$::$h\; =\; mathrm\{det\}\; left(\; h\_\{ab\}\; ight)\; ightarrow\; ilde\{h\}\; =\; mathrm\{J\}^\{-2\}\; h\; ,$and one sees that:::$sqrt\{-\; ilde\{h\; mathrm\{d\}^2\; ilde\{sigma\}\; =\; sqrt\{-h\}\; mathrm\{d\}^2\; sigma$summing up this transformation leaves the action invariant.

Weyl transformation

Assume the Weyl transformation:::$h\_\{ab\}\; ightarrow\; ilde\{h\}\_\{ab\}\; =\; Lambda(sigma)\; h\_\{ab\}$then:::$ilde\{h\}^\{ab\}\; =\; Lambda^\{-1\}(sigma)\; h^\{ab\}$::$mathrm\{det\}\; (\; ilde\{h\}\_\{ab\}\; )\; =\; Lambda^2(sigma)\; h\_\{ab\}$And finally:::And one can see that the action is invariant under Weyl transformation. If we consider n-dimensional (spatially) extended objects whose action is proportional to their worldsheet area/hyperarea, unless n=1, the corresponding Polyakov action would contain another term breaking Weyl symmetry.

One can define the Stress-energy tensor:::$T\_\{ab\}\; =\; frac\{2\}\{sqrt\{-h\; frac\{delta\; S\}\{delta\; h^\{ab$Let's define:::$h\_\{ab\}\; =\; expleft(phi(sigma)\; ight)\; hat\{h\}\_\{ab\}$Because of Weyl symmetry the action does not depend on $phi$:::$frac\{delta\; S\}\{delta\; phi\}\; =\; frac\{delta\; S\}\{delta\; h^\{ab\; frac\{delta\; h^\{ab\{delta\; phi\}\; =\; frac12\; sqrt\{-h\}\; T\_\{ab\}\; h^\{ab\}\; =\; frac12\; sqrt\{-h\}\; T\_a^\{\; a\}\; =\; 0\; ightarrow\; T\_a^\{\; a\}\; =\; mathrm\{tr\}\; left(\; T\_\{ab\}\; ight)\; =\; 0$

Relation with Nambu-Goto action

Writing the Euler-Lagrange equation for the metric tensor $h^\{ab\}$ one obtains that:::$frac\{delta\; S\}\{delta\; h^\{ab\; =\; T^\{ab\}\; =\; 0$Knowing also that:::$delta\; sqrt\{-h\}\; =\; -frac12\; sqrt\{-h\}\; h\_\{ab\}\; delta\; h^\{ab\}$One can write the variational derivative of the action:::$frac\{delta\; S\}\{delta\; h^\{ab\; =\; frac\{T\}\{2\}\; sqrt\{-h\}\; left(\; G\_\{ab\}\; -\; frac12\; h\_\{ab\}\; h^\{cd\}\; G\_\{cd\}\; ight)$where $G\_\{ab\}\; =\; g\_\{mu\; u\}\; partial\_a\; X^mu\; partial\_b\; X^\; u$ which leads to:::$T\_\{ab\}\; =\; T\; left(\; G\_\{ab\}\; -\; frac12\; h\_\{ab\}\; h^\{cd\}\; G\_\{cd\}\; ight)\; =\; 0$::$G\_\{ab\}\; =\; frac12\; h\_\{ab\}\; h^\{cd\}\; G\_\{cd\}$::$G\; =\; mathrm\{det\}\; left(\; G\_\{ab\}\; ight)\; =\; frac14\; h\; left(\; h^\{cd\}\; G\_\{cd\}\; ight)^2$

If the auxiliary worldsheet metric tensor $sqrt\{-h\}$ is calculated from the equations of motion:::$sqrt\{-h\}\; =\; frac\{2\; sqrt\{-G\{h^\{cd\}\; G\_\{cd$and substituted back to the action, it becomes the Nambu-Goto action:::$S\; =\; \{T\; over\; 2\}int\; mathrm\{d\}^2\; sigma\; sqrt\{-h\}\; h^\{ab\}\; G\_\{ab\}\; =\; \{T\; over\; 2\}int\; mathrm\{d\}^2\; sigma\; frac\{2\; sqrt\{-G\{h^\{cd\}\; G\_\{cd\; h^\{ab\}\; G\_\{ab\}\; =\; T\; int\; mathrm\{d\}^2\; sigma\; sqrt\{-G\}$

However, the Polyakov action is more easily quantized because it is linear.

Equations of motion

Using diffeomorphisms and Weyl transformation one can transform the action into the following form:::$mathcal\{S\}\; =\; \{T\; over\; 2\}int\; mathrm\{d\}^2\; sigma\; sqrt\{-eta\}\; eta^\{ab\}\; g\_\{mu\; u\}\; (X)\; partial\_a\; X^mu\; (sigma)\; partial\_b\; X^\; u(sigma)\; =\; \{T\; over\; 2\}int\; mathrm\{d\}^2\; sigma\; left(\; dot\{X\}^2\; -\; X\text{'}^2\; ight)$where

Keeping in mind that $T\_\{ab\}\; =\; 0$ one can derive the constraints:::$T\_\{01\}\; =\; T\_\{10\}\; =\; dot\{X\}\; X\text{'}\; =\; 0$::$T\_\{00\}\; =\; T\_\{11\}\; =\; frac12\; left(\; dot\{X\}^2\; +\; X\text{'}^2\; ight)\; =\; 0$.

Substituting $X^mu\; ightarrow\; X^mu\; +\; delta\; X^mu$ one obtains:::$delta\; mathcal\{S\}\; =\; T\; int\; mathrm\{d\}^2\; sigma\; eta^\{ab\}\; partial\_a\; X^mu\; partial\_b\; delta\; X\_mu\; =$

:::$=\; -T\; int\; mathrm\{d\}^2\; sigma\; eta^\{ab\}\; partial\_a\; partial\_b\; X^mu\; delta\; X\_mu\; +\; left(\; T\; int\; d\; au\; X\text{'}\; delta\; X\; ight)\_\{sigma=pi\}\; -\; left(\; T\; int\; d\; au\; X\text{'}\; delta\; X\; ight)\_\{sigma=0\}\; =\; 0$

And consequently:::$square\; X^mu\; =\; eta^\{ab\}\; partial\_a\; partial\_b\; X^mu\; =\; 0$

With the boundary conditions in order to satisfy the second part of the variation of the action.
* Closed strings: Periodic boundary conditions: $X^mu(\; au,\; sigma\; +\; 2\; pi)\; =\; X^mu(\; au,\; sigma)$
* Open strings:(i) Neumann boundary conditions: $partial\_sigma\; X^mu\; (\; au,\; 0)\; =\; 0,\; partial\_sigma\; X^mu\; (\; au,\; pi)\; =\; 0$:(ii) Dirichlet boundary conditions: $X^mu(\; au,\; 0)\; =\; b^mu,\; X^mu(\; au,\; pi)\; =\; b\text{'}^mu$

See also

* D-brane
* Einstein-Hilbert action

References

* Polchinski (Nov, 1994). "What is String Theory", NSF-ITP-94-97, 153pp,
* Ooguri, Yin (Feb, 1997). "TASI Lectures on Perturbative String Theories", UCB-PTH-96/64, LBNL-39774, 80pp,