Weyl covariance, and proposals for superconformal prepotentials in 10D superspaces

Proposals are made to describe the Weyl scaling transformation laws of supercovariant derivatives ∇A, the torsion supertensors TABC, and curvature supertensors RABcd in 10D superspaces. Starting from the proposal that an unconstrained supergravity prepotential for the 11D, $$ \mathcal{N} $$ N = 1 theory is described by a scalar superfield, considerations for supergravity prepotentials in the 10D theories are enumerated. We derive infinitesimal 10D superspace Weyl transformation laws and discover ten possible 10D, $$ \mathcal{N} $$ N = 1 superfield supergravity prepotentials. The first identification of all off-shell ten dimensional supergeometrical Weyl field strength tensors, constructed from respective torsions, is presented.

Geometrizing $$ T\overline{T} $$

The $$ T\overline{T} $$ T T ¯ deformation can be formulated as a dynamical change of coordinates. We establish and generalize this relation to curved spaces by coupling the undeformed theory to 2d gravity. For curved space the dynamical change of coordinates is supplemented by a dynamical Weyl transformation. We also sharpen the holographic correspondence to cutoff AdS3 in multiple ways. First, we show that the action of the annular region between the cutoff surface and the boundary of AdS3 is given precisely by the $$ T\overline{T} $$ T T ¯ operator integrated over either the cutoff surface or the asymptotic boundary. Then we derive dynamical coordinate and Weyl transformations directly from the bulk. Finally, we reproduce the flow equation for the deformed stress tensor from the cutoff geometry.

Renormalizing spacetime

We propose that the consistent field renormalization of gravity requires a specific Weyl transformation of the metric tensor. As a consequence, proper length and time, as well as energy and momentum, become functions of scale. We estimate the functional form of the field renormalization factor by imposing a minimum resolvable distance scale under an infinitesimal Weyl transformation. The derived transformation is applied to two key problems in quantum gravity: its nonconformal scaling and nonrenormalizability.