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.[3] The action reads:

where is the string tension, is the metric of the target manifold, is the worldsheet metric, its inverse, and is the determinant of . The metric signature is chosen such that timelike directions are + and the spacelike directions are −. 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.

Global symmetries

N.B.: 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 transformations

where , and is a constant. This forms the Poincaré symmetry of the target manifold.

The invariance under (i) follows since the action depends only on the first derivative of . 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.


Assume the following transformation:

It transforms the metric tensor in the following way:

One can see that:

One knows that the Jacobian of this transformation is given by

which leads to

and one sees that

Summing up this transformation and relabeling , we see that the action is invariant.

Weyl transformation

Assume the Weyl transformation:


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:

Let's define:

Because of Weyl symmetry, the action does not depend on :

where we've used the functional derivative chain rule.

Relation with Nambu–Goto action

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.

Equations of motion

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 .

Keeping in mind that 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.

See also


  1. ^ Deser, S.; Zumino, B. (1976). "A Complete Action for the Spinning String". Phys. Lett. B. 65 (4): 369–373. doi:10.1016/0370-2693(76)90245-8.
  2. ^ Brink, L.; Di Vecchia, P.; Howe, P. (1976). "A locally supersymmetric and reparametrization invariant action for the spinning string". Physics Letters B. 65 (5): 471–474. doi:10.1016/0370-2693(76)90445-7.
  3. ^ Polyakov, A. M. (1981). "Quantum geometry of bosonic strings". Physics Letters B. 103 (3): 207–210. doi:10.1016/0370-2693(81)90743-7.
  4. ^ Friedan, D. (1980). "Nonlinear Models in 2+ε Dimensions" (PDF). Physical Review Letters. 45 (13): 1057–1060. Bibcode:1980PhRvL..45.1057F. doi:10.1103/PhysRevLett.45.1057.

Further reading