$T\bar{T}$ in JT Gravity and BF Gauge Theory

Abstract

JT gravity has a first-order formulation as a two-dimensional BF theory, which can be viewed as the dimensional reduction of the Chern-Simons description of 3d <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> gravity. We consider {T\overbar{T}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mover> <mml:mi>T</mml:mi> <mml:mo accent="true">¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> -type deformations of the (0+1) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> -dimensional dual to this 2d <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> BF theory and interpret the deformation as a modification of the BF theory boundary conditions. The fundamental observables in this deformed BF theory, and in its 3d <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> Chern-Simons lift, are Wilson lines and loops. In the 3d <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> Chern-Simons setting, we study modifications to correlators involving boundary-anchored Wilson lines which are induced by a {T\overbar{T}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mover> <mml:mi>T</mml:mi> <mml:mo accent="true">¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> deformation on the 2d <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> boundary; results are presented at both the classical level (using modified boundary conditions) and the quantum-mechanical level (using conformal perturbation theory). Finally, we calculate the analogous deformed Wilson line correlators in 2d <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> BF theory below the Hagedorn temperature where the principal series dominates over the discrete series.

Topics & Concepts