Metric approach to a T(T)over-bar-like deformation in arbitrary dimensions

We consider a one-parameter family of composite fields - bi-linear in the components of the stress-energy tensor - which generalise the T (T) over bar operator to arbitrary space-time dimension d >= 2. We show that they induce a deformation of the classical action which is equivalent - at the level of the dynamics - to a field-dependent modification of the background metric tensor according to a specific flow equation. Even though the starting point is the flat space, the deformed metric is generally curved for any d > 2, thus implying that the corresponding deformation can not be interpreted as a coordinate transformation. The central part of the paper is devoted to the development of a recursive algorithm to compute the coefficients of the power series expansion of the solution to the metric flow equation. We show that, under some quite restrictive assumptions on the stress-energy tensor, the power series yields an exact solution. Finally, we consider a class of theories in d = 4 whose stress-energy tensor fulfils the assumptions above mentioned, namely the family of abelian gauge theories in d = 4. For such theories, we obtain the exact expression of the deformed metric and the vierbein. In particular, the latter result implies that ModMax theory in a specific curved space is dynamically equivalent to its Born-Infeld-like extension in flat space. We also discuss a dimensional reduction of the latter theories from d = 4 to d = 2 in which an interesting marginal deformation of d = 2 field theories emerges.

Automated summary

This paper considers a one-parameter family of composite fields, which generalize the T(T) operator to arbitrary space-time dimension. It shows that these fields induce a deformation of the classical action equivalent to a field-dependent modification of the background metric tensor. A recursive algorithm is developed to compute the coefficients of the power series expansion of the solution to the metric flow equation, and exact solutions are obtained under certain assumptions. The paper also discusses a specific class of theories in d=4 and their dimensional reduction.