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) \partial_a X^\mu (\sigma) \partial_b X^\nu(\sigma)

where T is the string tension, gμν is the metric of the target manifold and hab is the auxiliary worldsheet metric. h is the determinant of hab. 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.

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[edit] Global symmetries

The action is invariant under translations and infinitesimal Lorentz transformations

(i) X^\alpha \rightarrow X^\alpha + b^\alpha
(ii) X^\alpha \rightarrow X^\alpha + \omega^\alpha_{\ \beta} X^\beta
where ωμν = − ωνμ, which forms the Poincaré symmetry of the target manifold.

bα is constant, the action depends on the first derivative of Xα and in consequence \mathcal{S} is invariant under translations. The proof of the second relation:

\mathcal{S}' \, = {T \over 2}\int \mathrm{d}^2 \sigma \sqrt{-h} h^{ab} g_{\mu \nu} \partial_a \left( X^\mu + \omega^\mu_{\ \delta} X^\delta \right) \partial_b \left( X^\nu + \omega^\nu_{\ \delta} X^\delta \right) \,
= \mathcal{S} + {T \over 2}\int \mathrm{d}^2 \sigma \sqrt{-h} h^{ab} \left( \omega_{\mu \delta} \partial_a X^\mu \partial_b X^\delta + \omega_{\nu \delta} \partial_a X^\delta \partial_b X^\nu \right) + O(\omega^2) \,
= \mathcal{S} + {T \over 2}\int \mathrm{d}^2 \sigma \sqrt{-h} h^{ab} \left( \omega_{\mu \delta} + \omega_{\delta \mu } \right) \partial_a X^\mu \partial_b X^\delta + O(\omega^2) = \mathcal{S} + O(\omega^2)

[edit] Local symmetries

The action is invariant under diffeomorphisms or coordinates transformations and Weyl transformations.

[edit] Diffeomorphisms

Assume the following transformation:

\sigma^\alpha \rightarrow \tilde{\sigma}^\alpha\left(\sigma,\tau \right)

It transforms the Metric tensor in the following way:

h^{ab} \rightarrow \tilde{h}^{ab} = h^{cd} \frac{\partial \tilde{\sigma}^a}{\partial \sigma^c} \frac{\partial \tilde{\sigma}^b}{\partial \sigma^d}

One can see that:

\tilde{h}^{ab} \frac{\partial}{\partial \tilde{\sigma}^a} X^\mu \frac{\partial}{\partial \tilde{\sigma}^b} X^\nu = h^{cd} \frac{\partial \tilde{\sigma}^a}{\partial \sigma^c} \frac{\partial \tilde{\sigma}^b}{\partial \sigma^d} \frac{\partial}{\partial \tilde{\sigma}^a} X^\mu \frac{\partial}{\partial \tilde{\sigma}^b} X^\nu = h^{ab} \partial_a X^\mu \partial_b X^\nu

One knows that the Jacobian of this transformation is given by:

\mathrm{J} = \mathrm{det} \left( \frac{\partial \tilde{\sigma}^\alpha}{\partial \sigma^\beta} \right)

which leads to:

\mathrm{d}^2 \sigma \rightarrow \mathrm{d}^2 \tilde{\sigma} = \mathrm{J} \mathrm{d}^2 \sigma \,
h = \mathrm{det} \left( h_{ab} \right) \rightarrow \tilde{h} = \mathrm{J}^{-2} h \,

and one sees that:

\sqrt{-\tilde{h}} \mathrm{d}^2 \tilde{\sigma} = \sqrt{-h} \mathrm{d}^2 \sigma

summing up this transformation leaves the action invariant.

[edit] Weyl transformation

Assume the Weyl transformation:

h_{ab} \rightarrow \tilde{h}_{ab} = \Lambda(\sigma) h_{ab}

then:

\tilde{h}^{ab} = \Lambda^{-1}(\sigma) h^{ab}
\mathrm{det} ( \tilde{h}_{ab} ) = \Lambda^2(\sigma) h_{ab}

And finally:

\mathcal{S}' \, = {T \over 2}\int \mathrm{d}^2 \sigma \sqrt{-\tilde{h}} \tilde{h}^{ab} g_{\mu \nu} (X) \partial_a X^\mu (\sigma) \partial_b X^\nu(\sigma) \,
= {T \over 2}\int \mathrm{d}^2 \sigma \sqrt{-h} \left( \Lambda \Lambda^{-1} \right) h^{ab} g_{\mu \nu} (X) \partial_a X^\mu (\sigma) \partial_b X^\nu(\sigma) = \mathcal{S}

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} = \exp\left(\phi(\sigma)\right) \hat{h}_{ab}

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

\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 \rightarrow T_a^{\ a} = \mathrm{tr} \left( T_{ab} \right) = 0

[edit] Relation with Nambu-Goto action

Writing the Euler-Lagrange equation for the metric tensor hab 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} \right)

where G_{ab} = g_{\mu \nu} \partial_a X^\mu \partial_b X^\nu which leads to:

T_{ab} = T \left( G_{ab} - \frac12 h_{ab} h^{cd} G_{cd} \right) = 0
G_{ab} = \frac12 h_{ab} h^{cd} G_{cd}
G = \mathrm{det} \left( G_{ab} \right) = \frac14 h \left( h^{cd} G_{cd} \right)^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.

[edit] 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 \nu} (X) \partial_a X^\mu (\sigma) \partial_b X^\nu(\sigma) = {T \over 2}\int \mathrm{d}^2 \sigma \left( \dot{X}^2 - X'^2 \right)

where \eta_{ab} = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right)

Keeping in mind that Tab = 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 \right) = 0 .

Substituting X^\mu \rightarrow 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 \tau X' \delta X \right)_{\sigma=\pi} - \left( T \int d \tau X' \delta X \right)_{\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(\tau, \sigma + 2 \pi) = X^\mu(\tau, \sigma)\
  • Open strings
(i) Neumann boundary conditions: \partial_\sigma X^\mu (\tau, 0) = 0, \partial_\sigma X^\mu (\tau, \pi) = 0
(ii) Dirichlet boundary conditions: X^\mu(\tau, 0) = b^\mu, X^\mu(\tau, \pi) = b'^\mu \

[edit] See also

[edit] References

  • Polchinski (Nov, 1994). What is String Theory, NSF-ITP-94-97, 153pp, arXiv:hep-th/9411028v1
  • Ooguri, Yin (Feb, 1997). TASI Lectures on Perturbative String Theories, UCB-PTH-96/64, LBNL-39774, 80pp, arXiv:hep-th/9612254v3
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