I have the full Polyakov sigma model action:

\begin{equation} \begin{split} &S=S_P + S_B + S_\Phi = \\ &- {1 \over 4 \pi \alpha'} \Big[ \int_\Sigma d^2\sigma \sqrt{-g} g^{ab} \partial_a X^\mu \partial_b X^\nu G_{\mu\nu}(X)\, + \\ &+\epsilon^{ab} B_{\mu\nu}(X) \partial_a X^\mu \partial_b X^\nu \, +\alpha'\Phi(X) R^{(2)}(\sigma) \Big] \,. \end{split} \end{equation}

and I want to derive the classical equations of motion by varying $X \mapsto X + \delta X$. I am confused as to what to do with the last term. It is of a higher power of $\alpha '$, so I am thinking it can just be ignored, as it's variation will be of a higher order. Is this thinking correct?

Does this question even make sense, as I'm trying to derive classical equations from a sigma-model, which as far as I have seen, is used when quantizing the string?

  • $\begingroup$ Under $X \to X + \delta X$, we have $\Phi(X) \to \Phi(X+\delta X) = \Phi(X) + \delta X^\mu \partial_\mu \Phi(X)$. $\endgroup$ – Prahar Jun 6 '19 at 15:10

Strings in strong gravitational fields - Gary T. Horowitz and Alan R. Steif, Phys. Rev. D 42, 1950 confirms the argument in the question:

"For the remainder of this section we consider a purely classical string. Since the dilaton term is multiplied by $\alpha'$, it is a quantum correction and does not directly affect the motion of a classical string."

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