- Polyakov action
In
physics , the Polyakov action is the two-dimensional action of aconformal field theory describing theworldsheet of a string instring 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 withAlexander 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}. Themetric 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 spacetimetranslation s andinfinitesimal Lorentz transformation s::(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 worldsheetdiffeomorphism s (or coordinates transformations) andWeyl transformation s.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 theJacobian 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 underWeyl 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 ofWeyl 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) = 0Relation with Nambu-Goto action
Writing the
Euler-Lagrange equation for themetric 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)^2If 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 theNambu-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
diffeomorphism s andWeyl 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'^muSee 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,
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