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) $$X^alpha ightarrow X^alpha + b^alpha :(ii) $$X^alpha ightarrow X^alpha + omega^alpha_{ eta} X^eta 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:::$$ilde{h}^{ab} frac{partial}{partial ilde{sigma}^a} X^mu frac{partial}{partial ilde{sigma}^b} X^ u = h^{cd} frac{partial ilde{sigma}^a}{partial sigma^c} frac{partial ilde{sigma}^b}{partial sigma^d} frac{partial}{partial ilde{sigma}^a} X^mu frac{partial}{partial ilde{sigma}^b} X^ u = h^{ab} partial_a X^mu partial_b X^ u One knows that the Jacobian of this transformation is given by:::$$mathrm{J} = mathrm{det} left( frac{partial ilde{sigma}^alpha}{partial sigma^eta} ight) 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:::$$mathcal{S} = {T over 2}int mathrm{d}^2 sigma sqrt{-eta} eta^{ab} g_{mu u} (X) partial_a X^mu (sigma) partial_b X^ u(sigma) = {T over 2}int mathrm{d}^2 sigma left( dot{X}^2 - X'^2 ight) where $$eta_{ab} = left( egin{array}{cc} 1 & 0 \ 0 & -1 end{array} ight)

Keeping in mind that $$T_{ab} = 0 one can derive the constraints:::$$T_{01} = T_{10} = dot{X} X' = 0 ::$$T_{00} = T_{11} = frac12 left( dot{X}^2 + X'^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' delta X ight)_{sigma=pi} - left( T int d au X' 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'^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,