Stringy corrections of Einstein's vacuum field equations From string theory, the vacuum field equations obtain correction of the order $O[\alpha'R]^n$ such that they can be written as
$$
R_{\alpha\beta} -\frac{1}{2}g_{\alpha\beta}R + O[\alpha'R] = 0
$$
where $\alpha' = \frac{1}{2\pi T_s}$, when including just the first order term for example.
What is the physical interpretation of these corrections, what do they look like more explicitely, and how can they be obtained? 
 A: As Witten explains in his NOTICES OF THE AMS article (please see also his more 
recent lecture), the fully quantum string theory is characterized by two coupling constants (or in the language of deformation quantization: two deformation parameters). The string coupling $g_s$ and the string tension $\alpha^{'}$. In perturbation theory, one gets dependence of the string amplitudes on powers of $g_s$ (or equivalently in $\hbar$) through the genus expansion. 
The dependence of the amplitudes $\alpha^{'}$ is obtained once one takes into account that in the presence of background fields, the string Lagrangian is not free, it is described by a sigma model. If we compute the quantum correction to this sigma model we get terms with more and more derivatives multiplied by more powers of the  of the string tension (as in chiral perturbation theory). When the  quantum corrections to the trace of the energy momentum tensors are calculated then here also terms depending on powers of $\alpha^{'}$ will  appear and the condition of vanishing of the beta function will results  Einstein's equations with correction terms proportional to $\alpha^{'}$.
Please see equations 3.7.14. in Polchinsky's first volume, where the  beta functions are given to the first power of $\alpha^{'}$.
Witten explains that for a while, the work on string theory concentrated on finding candidates of $\alpha^{'}$ deformed theories (as conformal  field theories), then $\hbar$-quantize them as in ordinary quantum theory. 
But, as Witten explains, after the discovery of the full set of string dualities and the role of membranes, it was realized that in order to fully quantize string theory, the two quantizations or two deformations($\hbar$, $\alpha^{'}$)  must be perfomed together. 
This route has profound consequences, for example, it leads to the  conclusion that the string full quantum theory should be in the realm of noncommutative geometry, because in the presence of a $B$-field and  brane boundary conditions, the position-position commutation relations will obtain $\alpha^{'}$ deformation and become noncommutative. As a consequence, the ordinary uncertainty relation will get $\alpha^{'}$ deformation and turns into a generalized (minimal scale) uncertainty, in which the  position uncertainty has a nonvanishing minimum:
$\Delta x \geqslant \frac{\hbar}{\Delta p}+ \alpha^{'}\frac{\Delta p}{\hbar}$
