A calculation on moduli stabilization

In section 21.6 "Moduli stabilization and the landscape" in Zwiebach's string theory book, I read the sentence "Deriving the potential $V(R)$ associated with $R$ is a straightforward but technical calculation in general relativity". At this point I did not understand what the calculation was. I vaguely remembered a paper by Witten about instabilities in Kaluza-Klein spacetimes related to instantons. A calculation with instantons is indeed technical, but perhaps straightforward for experts. I found more information in a paper by Denef [1]. The calculation has nothing to do with instantons, but is indeed a straightforward calculation in differential geometry. In the rest of this blog post I set out the calculation in the form of a new exercise for Zwiebach's book.

Lorentz invariance of string theory in the light-cone gauge

On page 261 in his book [1] Zwiebach writes ''There is much at stake in this calculation. It is in fact, one of the most important calculations in string theory. [...] The calculation is long and uses many of our previously derived results''. Then Zwiebach states the result \begin{align} \left[ M^{- I}, M^{- J}\right] = &- \frac{1}{\alpha'\ { p^+ }^2} \sum_{m=1}^{\infty} \left(\alpha^I_{-m} \alpha^J_m - \alpha^J_{-m} \alpha^I_m \right)\nonumber\\ &\times \left\{ m \left[1 - \dfrac{1}{24} (D-2) \right] + \dfrac{1}{m} \left[ \dfrac{1}{24} (D-2) + a \right] \right\}\label{eq:20170103} \end{align} This is the commutator of two Lorentz transformations in the light-cone gauge. The commutator should be zero for string theory to be Lorentz invariant. The calculation of the commutator is indeed very tedious and after scribling too many pages I gave up. I googled for a quick way to obtain \eqref{eq:20170103}. Here are the more interesting results that I found.