equation for the metric tensor one obtains that Knowing also that: One can write the variational derivative of the action: where , which leads to If the auxiliary worldsheet metric tensor is calculated from the equations of motion: and substituted back to the action, it becomes the Nambu–Goto action: However, the Polyakov action is more easily quantized because it is linear. Equations of motion Using diffeomorphisms and Weyl transformation, with a Minkowskian target space, one can make the physically insignificant transformation , thus writing the action in the conformal gauge: where . Keeping in mind that one can derive the constraints: Substituting , one obtains And consequently The boundary conditions to satisfy the second part of the variation of the action are as follows. Closed strings: Periodic boundary conditions: Open strings: Working in light-cone coordinates , we can rewrite the equations of motion as Thus, the solution can be written as , 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. See also D-brane Einstein–Hilbert action References Further reading 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. Conformal field theory String theory