$$T\bar{T}$$ deformation of chiral bosons and Chern–Simons $$\hbox {AdS}_3$$ gravity
Hao Ouyang, Hongfei Shu
Abstract
Abstract We study the $$T\bar{T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mover> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> </mml:math> deformation of the chiral bosons and show the equivalence between the chiral bosons of opposite chiralities and the scalar fields at the Hamiltonian level under the deformation. We also derive the deformed Lagrangian of more generic theories which contain an arbitrary number of chiral bosons to all orders. By using these results, we derive the $$T\bar{T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mover> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> </mml:math> deformed boundary action of the $$\hbox {AdS}_3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>AdS</mml:mtext> <mml:mn>3</mml:mn> </mml:msub> </mml:math> gravity theory in the Chern–Simons formulation. We compute the deformed one-loop torus partition function, which satisfies the $$T\bar{T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mover> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> </mml:math> flow equation up to the one-loop order. Finally, we calculate the deformed stress–energy tensor of a solution describing a BTZ black hole in the boundary theory, which coincides with the boundary stress–energy tensor derived from the BTZ black hole with a finite cutoff.