A curved lagrangian in terms of a flat one
Let us consider the following gravitational background:
and the Polyakov lagrangian in conformal gauge with Euclidean Lorentzian signature:
The Euler-Lagrange equations can be found to be:
We look at this and think how can we make these equations simpler. The first equation can be solved with:
This is a “T-duality” transformation. For the second equation we see that if then we get
with the flat-space Polyakov lagrangian in terms of the new coordinates :
We can integrate this equation to find an expression for in terms of the solutions of the equations of motion and the flat-space lagrangian:
Then, pluging this back into the classical action we get
Finally we can write
so we can write the expression in the exponential as a sum of integrals over the worldsheet coordinates.
On the other hand, if instead we have then we can write:
The lagrangian can be writen as
We have found an expression for the classical lagragian in some (not-so) arbitrary gravitational background in terms of the coordinates and the flat-space lagrangian. The problem is we do not get any information about the solution for the equations of motion.