In physics, the Polyakov action is an 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 L. Brink, P. Di Vecchia and P. S. Howe in 1976,[1][2] and has become associated with Alexander Polyakov after he made use of it in quantizing the string in 1981.
is the metric of the target manifold,
The spacelike worldsheet coordinate is called
, whereas the timelike worldsheet coordinate is called
This is also known as the nonlinear sigma model.
[4] The Polyakov action must be supplemented by the Liouville action to describe string fluctuations.
: Here, a symmetry is said to be local or global from the two dimensional theory (on the worldsheet) point of view.
For example, Lorentz transformations, that are local symmetries of the space-time, are global symmetries of the theory on the worldsheet.
The action is invariant under spacetime translations and infinitesimal Lorentz transformationswhere
This forms the Poincaré symmetry of the target manifold.
The proof of the invariance under (ii) is as follows: The action is invariant under worldsheet diffeomorphisms (or coordinates transformations) and Weyl transformations.
, we see that the action is invariant.
Assume the Weyl transformation: then 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.
: where we've used the functional derivative chain rule.
Writing the Euler–Lagrange equation for the metric tensor
one obtains that Knowing also that: One can write the variational derivative of the action: where
, which leads to If the auxiliary worldsheet metric tensor
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 because it is linear.
Using diffeomorphisms and Weyl transformation, with a Minkowskian target space, one can make the physically insignificant transformation
, thus writing the action in the conformal gauge: where
one can derive the constraints: Substituting
, one obtains And consequently The boundary conditions to satisfy the second part of the variation of the action are as follows.
Working in light-cone coordinates
, we can rewrite the equations of motion as Thus, the solution can be written as
, and the stress-energy tensor is now diagonal.
By Fourier-expanding the solution and imposing canonical commutation relations on the coefficients, applying the second equation of motion motivates the definition of the Virasoro operators and lead to the Virasoro constraints that vanish when acting on physical states.