An integrable Lorentz-breaking deformation of two-dimensional CFTs

Abstract

It has been recently shown that the deformation of an arbitrary two-dimensional conformal field theory by the composite irrelevant operator T \bar 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> , built from the components of the stress tensor, is solvable; in particular, the finite-size spectrum of the deformed theory can be obtained from that of the original CFT through a universal formula. We study a similarly universal, Lorentz-breaking deformation of two-dimensional CFTs that possess a conserved U(1) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> current, J <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>J</mml:mi> </mml:math> . The deformation takes the schematic form J \bar T <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>J</mml:mi> <mml:mover> <mml:mi>T</mml:mi> <mml:mo accent="true">‾</mml:mo> </mml:mover> </mml:mrow> </mml:math> and is interesting because it preserves an SL(2,\mathbb{R}) \times U(1) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℝ</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo>×</mml:mo> <mml:mi>U</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> subgroup of the original global conformal symmetries. For the case of a purely (anti)chiral current, we find the finite-size spectrum of the deformed theory and study its thermodynamic properties. We test our predictions in a simple example involving deformed free fermions.