Polyakov action

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{{ordered list

| list-style-type=lower-roman

| $X^\backslash alpha\; \backslash to\; X^\backslash alpha\; +\; b^\backslash alpha,$

| $X^\backslash alpha\; \backslash to\; X^\backslash alpha\; +\; \backslash omega^\backslash alpha\_\{\backslash \; \backslash beta\}\; X^\backslash beta,$

}}

where $\backslash omega\_\{\backslash mu\; \backslash nu\}\; =\; -\backslash omega\_\{\backslash nu\; \backslash mu\}$, and $b^\backslash alpha$ is a constant. This forms the Poincaré symmetry of the target manifold.

The invariance under (i) follows since the action $\backslash mathcal\{S\}$ depends only on the first derivative of $X^\backslash alpha$. The proof of the invariance under (ii) is as follows:

: $\backslash begin\{align\}$

\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) + \operatorname{O}\left(\omega^2\right) \\

&= \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 + \operatorname{O}\left(\omega^2\right) \\

&= \mathcal{S} + \operatorname{O}\left(\omega^2\right).

\end{align}

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

Assume the following transformation:

: $\backslash sigma^\backslash alpha\; \backslash rightarrow\; \backslash tilde\{\backslash sigma\}^\backslash alpha\backslash left(\backslash sigma,\backslash tau\; \backslash right).$

It transforms the metric tensor in the following way:

: $h^\{ab\}(\backslash sigma)\; \backslash rightarrow\; \backslash tilde\{h\}^\{ab\}\; =\; h^\{cd\}\; (\backslash tilde\{\backslash sigma\})\backslash frac\{\backslash partial\; \{\backslash sigma\}^a\}\{\backslash partial\; \backslash tilde\{\backslash sigma\}^c\}\; \backslash frac\{\backslash partial\; \{\backslash sigma\}^b\}\{\backslash partial\; \backslash tilde\{\backslash sigma\}^d\}.$

One can see that:

: $$

\tilde{h}^{ab} \frac{\partial}{\partial {\sigma}^a} X^\mu(\tilde{\sigma}) \frac{\partial}{\partial \sigma^b} X^\nu(\tilde{\sigma}) =

h^{cd} \left(\tilde{\sigma}\right)\frac{\partial \sigma^a}{\partial \tilde{\sigma}^c} \frac{\partial \sigma^b}{\partial \tilde{\sigma}^d} \frac{\partial}{\partial \sigma^a} X^\mu(\tilde{\sigma})\frac{\partial}{\partial {\sigma}^b} X^\nu(\tilde{\sigma}) =

h^{ab}\left(\tilde{\sigma}\right)\frac{\partial}{\partial \tilde{\sigma}^a}X^\mu(\tilde{\sigma}) \frac{\partial}{\partial \tilde{\sigma}^b} X^\nu(\tilde{\sigma}).

One knows that the Jacobian of this transformation is given by

: $\backslash mathrm\{J\}\; =\; \backslash operatorname\{det\}\; \backslash left(\; \backslash frac\{\backslash partial\; \backslash tilde\{\backslash sigma\}^\backslash alpha\}\{\backslash partial\; \backslash sigma^\backslash beta\}\; \backslash right),$

which leads to

: $\backslash begin\{align\}$

\mathrm{d}^2 \tilde{\sigma} &= \mathrm{J} \mathrm{d}^2 \sigma \\

h &= \operatorname{det} \left( h_{ab} \right) \\

\Rightarrow \tilde{h} &= \mathrm{J}^2 h,

\end{align}

and one sees that

: $\backslash sqrt\{-\backslash tilde\{h\}\}\; \backslash mathrm\{d\}^2\; \{\backslash sigma\}\; =\; \backslash sqrt\{-h\; \backslash left(\backslash tilde\{\backslash sigma\}\backslash right)\}\; \backslash mathrm\{d\}^2\; \backslash tilde\{\backslash sigma\}.$

Summing up this transformation and relabeling $\backslash tilde\{\backslash sigma\}\; =\; \backslash sigma$, we see that the action is invariant.

Assume the Weyl transformation:

: $h\_\{ab\}\; \backslash to\; \backslash tilde\{h\}\_\{ab\}\; =\; \backslash Lambda(\backslash sigma)\; h\_\{ab\},$

then

: $\backslash begin\{align\}$

\tilde{h}^{ab} &= \Lambda^{-1}(\sigma) h^{ab}, \\

\operatorname{det} \left( \tilde{h}_{ab} \right) &= \Lambda^2(\sigma) \operatorname{det} (h_{ab}).

\end{align}

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:

: $T^\{ab\}\; =\; \backslash frac\{-2\}\{\backslash sqrt\{-h\}\}\; \backslash frac\{\backslash delta\; S\}\{\backslash delta\; h\_\{ab\}\}.$

Let's define:

: $\backslash hat\{h\}\_\{ab\}\; =\; \backslash exp\backslash left(\backslash phi(\backslash sigma)\backslash right)\; h\_\{ab\}.$

Because of Weyl symmetry, the action does not depend on $\backslash phi$:

: $\backslash frac\{\backslash delta\; S\}\{\backslash delta\; \backslash phi\}\; =\; \backslash frac\{\backslash delta\; S\}\{\backslash delta\; \backslash hat\{h\}\_\{ab\}\}\; \backslash frac\{\backslash delta\; \backslash hat\{h\}\_\{ab\}\}\{\backslash delta\; \backslash phi\}\; =\; -\backslash frac12\; \backslash sqrt\{-h\}\; \backslash ,T\_\{ab\}\backslash ,\; e^\{\backslash phi\}\backslash ,\; h^\{ab\}\; =\; -\backslash frac12\; \backslash sqrt\{-h\}\; \backslash ,T^a\_\{\backslash \; a\}\; \backslash ,e^\{\backslash phi\}\; =\; 0\; \backslash Rightarrow\; T^\{a\}\_\{\backslash \; a\}\; =\; 0,$

where we've used the functional derivative chain rule.

Writing the Euler–Lagrange equation for the metric tensor $h^\{ab\}$ one obtains that

: $\backslash frac\{\backslash delta\; S\}\{\backslash delta\; h^\{ab\}\}\; =\; T\_\{ab\}\; =\; 0.$

Knowing also that:

: $\backslash delta\; \backslash sqrt\{-h\}\; =\; -\backslash frac12\; \backslash sqrt\{-h\}\; h\_\{ab\}\; \backslash delta\; h^\{ab\}.$

One can write the variational derivative of the action:

: $\backslash frac\{\backslash delta\; S\}\{\backslash delta\; h^\{ab\}\}\; =\; \backslash frac\{T\}\{2\}\; \backslash sqrt\{-h\}\; \backslash left(\; G\_\{ab\}\; -\; \backslash frac12\; h\_\{ab\}\; h^\{cd\}\; G\_\{cd\}\; \backslash right),$

where $G\_\{ab\}\; =\; g\_\{\backslash mu\; \backslash nu\}\; \backslash partial\_a\; X^\backslash mu\; \backslash partial\_b\; X^\backslash nu$, which leads to

: $\backslash begin\{align\}$

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 &= \operatorname{det} \left( G_{ab} \right) = \frac14 h \left( h^{cd} G_{cd} \right)^2.

\end{align}

If the auxiliary worldsheet metric tensor $\backslash sqrt\{-h\}$ is calculated from the equations of motion:

: $\backslash sqrt\{-h\}\; =\; \backslash frac\{2\; \backslash sqrt\{-G\}\}\{h^\{cd\}\; G\_\{cd\}\}$

and substituted back to the action, it becomes the Nambu–Goto action:

: $S\; =\; \{T\; \backslash over\; 2\}\backslash int\; \backslash mathrm\{d\}^2\; \backslash sigma\; \backslash sqrt\{-h\}\; h^\{ab\}\; G\_\{ab\}\; =\; \{T\; \backslash over\; 2\}\backslash int\; \backslash mathrm\{d\}^2\; \backslash sigma\; \backslash frac\{2\; \backslash sqrt\{-G\}\}\{h^\{cd\}\; G\_\{cd\}\}\; h^\{ab\}\; G\_\{ab\}\; =\; T\; \backslash int\; \backslash mathrm\{d\}^2\; \backslash sigma\; \backslash sqrt\{-G\}.$

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 $\backslash sqrt\{-h\}\; h^\{ab\}\; \backslash rightarrow\; \backslash eta^\{ab\}$, thus writing the action in the conformal gauge:

: $\backslash mathcal\{S\}\; =\; \{T\; \backslash over\; 2\}\backslash int\; \backslash mathrm\{d\}^2\; \backslash sigma\; \backslash sqrt\{-\backslash eta\}\; \backslash eta^\{ab\}\; g\_\{\backslash mu\; \backslash nu\}\; (X)\; \backslash partial\_a\; X^\backslash mu\; (\backslash sigma)\; \backslash partial\_b\; X^\backslash nu(\backslash sigma)\; =\; \{T\; \backslash over\; 2\}\backslash int\; \backslash mathrm\{d\}^2\; \backslash sigma\; \backslash left(\; \backslash dot\{X\}^2\; -\; X\text{'}^2\; \backslash right),$

where .

Keeping in mind that $T\_\{ab\}\; =\; 0$ one can derive the constraints:

: $\backslash begin\{align\}$

T_{01} &= T_{10} = \dot{X} X' = 0, \\

T_{00} &= T_{11} = \frac12 \left( \dot{X}^2 + X'^2 \right) = 0.

\end{align}

Substituting $X^\backslash mu\; \backslash to\; X^\backslash mu\; +\; \backslash delta\; X^\backslash mu$, one obtains

: $\backslash begin\{align\}$

\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.

\end{align}

And consequently

: $\backslash square\; X^\backslash mu\; =\; \backslash eta^\{ab\}\; \backslash partial\_a\; \backslash partial\_b\; X^\backslash mu\; =\; 0.$

The boundary conditions to satisfy the second part of the variation of the action are as follows.

• Closed strings:
• : Periodic boundary conditions: $X^\backslash mu(\backslash tau,\; \backslash sigma\; +\; \backslash pi)\; =\; X^\backslash mu(\backslash tau,\; \backslash sigma).$
• Open strings:{{ordered list

| list-style-type=lower-roman

| Neumann boundary conditions: $\backslash partial\_\backslash sigma\; X^\backslash mu\; (\backslash tau,\; 0)\; =\; 0,\; \backslash partial\_\backslash sigma\; X^\backslash mu\; (\backslash tau,\; \backslash pi)\; =\; 0.$

| Dirichlet boundary conditions: $X^\backslash mu(\backslash tau,\; 0)\; =\; b^\backslash mu,\; X^\backslash mu(\backslash tau,\; \backslash pi)\; =\; b\text{'}^\backslash mu.$

}}

Working in light-cone coordinates $\backslash xi^\backslash pm\; =\; \backslash tau\; \backslash pm\; \backslash sigma$, we can rewrite the equations of motion as

: $\backslash begin\{align\}$

\partial_+ \partial_- X^\mu &= 0, \\

(\partial_+ X)^2 = (\partial_- X)^2 &= 0.

\end{align}

Thus, the solution can be written as $X^\backslash mu\; =\; X^\backslash mu\_+\; (\backslash xi^+)\; +\; X^\backslash mu\_-\; (\backslash xi^-)$, 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.

• D-brane
• Einstein–Hilbert action

{{Reflist}}

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

{{Quantum field theories}}

{{String theory topics}}

{{DEFAULTSORT:Polyakov Action}}

Category:Conformal field theory

Category:String theory