Editing Polyakov action (section)

== Equations of motion ==

Using [[diffeomorphism]]s and [[Weyl transformation]], with a [[Minkowski space|Minkowskian target space]], one can make the physically insignificant transformation <math>\sqrt{-h} h^{ab} \rightarrow \eta^{ab}</math>, thus writing the action in the ''conformal gauge'':
: <math> \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), </math>
where <math> \eta_{ab} = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) </math>.

Keeping in mind that <math> T_{ab} = 0 </math> one can derive the constraints:
: <math>\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}</math>

Substituting <math> X^\mu \to X^\mu + \delta X^\mu </math>, one obtains
: <math>\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}</math>

And consequently
: <math> \square X^\mu = \eta^{ab} \partial_a \partial_b X^\mu = 0. </math>

The boundary conditions to satisfy the second part of the variation of the action are as follows.
* Closed strings:
*: [[Periodic boundary conditions]]: <math> X^\mu(\tau, \sigma + \pi) = X^\mu(\tau, \sigma). </math>
* Open strings:{{ordered list
 | list-style-type=lower-roman
 | [[Neumann boundary conditions]]: <math> \partial_\sigma X^\mu (\tau, 0) = 0, \partial_\sigma X^\mu (\tau, \pi) = 0. </math>
 | [[Dirichlet boundary conditions]]: <math> X^\mu(\tau, 0) = b^\mu, X^\mu(\tau, \pi) = b'^\mu. </math>
 }}
Working in [[light-cone coordinates]] <math>\xi^\pm = \tau \pm \sigma</math>, we can rewrite the equations of motion as

: <math>\begin{align}
          \partial_+ \partial_- X^\mu &= 0, \\
  (\partial_+ X)^2 = (\partial_- X)^2 &= 0.
\end{align}</math>

Thus, the solution can be written as <math>X^\mu = X^\mu_+ (\xi^+) + X^\mu_- (\xi^-)</math>, and the stress-energy tensor is now diagonal. By [[Fourier series|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 [[N = 2 superconformal algebra#Free field construction|Virasoro constraints]] that vanish when acting on physical states.