  Given the $\beta$ functions of the closed string sigma model up to one loop
in $\alpha'$, the effective action implement the condition $\beta=0$ to
preserve conformal symmetry at quantum level. One of the more powerful and
striking results of string theory is that this effective action contains
Einstein gravity as an emergent dynamics in space-time. We show from the
$\beta$ functions and its relation with the equations of motion of the
effective action, that the differential identities [1] are the Noether
identities associated with the effective action and its gauge symmetries. From
here, we reconstruct the gauge and space time symmetries of the effective
action. In turn, we can show that the differential identities are the
contracted Bianchi identities of the the field strength $H$ and Riemann tensor
$R$. Next, we apply the same ideas to DFT. Taking as starting point that the
generalized $\beta$ functions in DFT are proportional to the equations of
motion, we construct the generalized differential identities in DFT. Relating
the Noether identities with the contracted Bianchi identities of DFT, we were
able to reconstruct the generalized gauge and space time symmetries. Finally,
we recover the original $\beta$ functions, effective action, differential
identities, and symmetries when we turn off the $\tilde x$ space time
coordinates from DFT.
