Towards a Finite AdS$${ }_{\mathbf {3}}$$ Topological Gravity Landscape

Abstract

Abstract We investigate Swampland constraints on the three dimensional topological Anti-de Sitter gravity in the Chern-Simons formulation with connection valued in various real forms of Lie algebras. We consider the general boundary conditions proposed by Grumiller and Riegler for the spin 2 gravity based on sl $$_{\mathbf {2}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow> <mml:mstyle> <mml:mn>2</mml:mn> </mml:mstyle> </mml:mrow> <mml:mrow/> </mml:mmultiscripts> </mml:math> and extend the asymptotic symmetry from the usual two copies of the affine algebras to four copies by considering the contribution from the chemical potentials. We discuss the emerging gauge anomaly and study its cancellation using boundary 2D strings introduced by means of the Chern Simons and Wess Zumino Witten duality. We define an AdS $$_{\mathbf {3}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow> <mml:mstyle> <mml:mn>3</mml:mn> </mml:mstyle> </mml:mrow> <mml:mrow/> </mml:mmultiscripts> </mml:math> Landscape and lay the groundwork to argument the validity for the finiteness Swampland conjecture in 3D by requiring the unitarity of the boundary CFT with the additional extended algebra. We discuss the implications of this constraint on the spectrum of higher spin gravity theories.