A note on background independence in AdS3 string theory

A bstract In this note, we comment on the path integral formulation of string theory on $$ \mathcal{M} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> × S 3 × $$ {\mathbbm{T}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> where $$ \mathcal{M} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> is any hyperbolic 3-manifold. In the special case of k = 1 units of NS-NS flux, we provide a covariant description of the worldsheet theory and argue that the path integral depends only on the details of the conformal boundary $$ \partial \mathcal{M} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∂</mml:mi> <mml:mi>M</mml:mi> </mml:math> , making the background independence of this theory manifest. We provide a simple path integral argument that the path integral localizes onto holomorphic covering maps from the worldsheet to the boundary. For closed manifolds $$ \mathcal{M} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> , the gravitational path integral is argued to be trivial. Finally, we comment on the effect of continuous deformations of the worldsheet theory which introduce non-minimal string tension.