The low-energy bosonic effective actions of string theory lead to Einstein-Hilbert gravity, along with scalars and $p$-form Maxwell fields. For example, the action for type IIA string theory is

$S = \int d^{10} x \sqrt{-g} \left[ e^{-2\Phi} \left(R + 4(\nabla\Phi^2) - \frac{1}{2}|H_3|^2 \right) - \frac{1}{2} |\tilde{F}_2|^2 - \frac{1}{2} |\tilde{F}_4|^2 \right] - \int \frac{1}{2} B_2 \wedge F_4 \wedge F_4 .$

These actions are in fact the leading terms in an expansion in the string length $\sqrt{\alpha'}$, and more generally the low-energy actions consist of an infinite series of higher curvature terms. For example, one would expect terms like $R^2, R_{\mu\nu}R^{\mu\nu}, R_{\mu\nu\rho\lambda}R^{\mu\nu\rho\lambda}$ to show up at next order (as well as higher derivative terms in the $p$-form field strengths).

My question is this: what is known about the convergence properties of the $\alpha'$ expansion? Is it an asymptotic series, like those typically encountered in QFT, or is there a chance that it converges to some interesting expression?

As an aside, there are some known solutions to all orders in $\alpha'$, for example plane waves. These remain solutions to all orders essentially because the curvature tensors are null, and so all contractions necessarily involve the square of a null vector--which is zero. In addition to the general question stated above, I'm curious about the possibility of finding exact non-null solutions to all orders in $\alpha'$


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