On the Lagrangian Holographic Relation at $D\rightarrow2$ and $4$ Limits of Gravity

TL;DR

This paper investigates the holographic relations in gravity theories at the critical dimensions of two and four, revealing how the bulk and boundary terms relate in these limits through dimensional reduction.

Contribution

It derives holographic relations for Einstein and Gauss-Bonnet gravities at D=2 and D=4 limits, extending the understanding of bulk-boundary relations in these special cases.

Findings

Holographic relations at D=2 and D=4 are established.

Dimensional reduction yields scalar-tensor theories with similar holographic structures.

Relations match those in pure gravity in foliation-independent formalism.

Abstract

The gravitational Lagrangian can be written as a summation of a bulk and a total derivative term. For some theories of gravity such as Einstein gravity, or more general Lovelock gravities, there are Lagrangian holographic relations between the bulk and the total derivative term such that the latter is fully determined by the former. However at the $D\rightarrow 2\&4$ limit, the bulks of Einstein or Gauss-Bonnet theories become themselves total derivatives. Performing the Kaluza-Klein reduction on Einstein and Gauss-Bonnet gravities gives rise to some two-dimensional or four-dimensional scalar-tensor theories respectively. We obtain the holographic relations for the $D = 2$ and $D = 4$ cases, which have the same form as the holographic relations in pure gravity in the foliation independent formalism.