Noether identities, β-functions and symmetries in DFT

Given the beta functions of the closed string sigma model up to one loop in alpha', the effective action implements 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 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 (x) over tilde space-time coordinates from DFT.