Polyakov action

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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

$\mathcal{S} = {T \over 2}\int \mathrm{d}^2 \sigma \sqrt{-|h|} h^{ab} g_{\mu \nu} (X^\alpha) \partial_a X^\mu (\sigma) \partial_b X^\nu(\sigma)$

where T is the string tension, $g μν$ is the metric of the target manifold and $h a b$ is the auxiliary worldsheet metric. $| h |$ is the determinant of $h a b$ . The metric signature is chosen such that timelike directions are + and the spacelike directions are -. The spacelike worldsheet coordinate is called σ wheareas the timelike worldsheet coordinate is called τ.

The action is invariant under diffeomorphisms as well as the Weyl transformations and the Poincaré symmetry of the target manifold. If the auxiliary worldsheet metric tensor $h a b$ is calculated from the equations of motion and substituted back to the action, it becomes the Nambu-Goto action. However, the Polyakov action is more easily quantized.

It's interesting that 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.