A pedagogical review on solvable irrelevant deformations of 2D quantum field theory
Yunfeng Jiang
Abstract
Abstract This is a pedagogical review on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic">T</mml:mi> <mml:mover accent="true"> <mml:mrow> <mml:mi mathvariant="italic">T</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">¯</mml:mo> </mml:mrow> </mml:mover> </mml:math> deformation of two dimensional quantum field theories. It is based on three lectures which the author gave at ITP-CAS in December 2018. This review consists of four parts. The first part is a general introduction to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic">T</mml:mi> <mml:mover accent="true"> <mml:mrow> <mml:mi mathvariant="italic">T</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">¯</mml:mo> </mml:mrow> </mml:mover> </mml:math> deformation. Special emphasises are put on the deformed classical Lagrangian and the exact solvability of the spectrum. The second part focuses on the torus partition sum of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic">T</mml:mi> <mml:mover accent="true"> <mml:mrow> <mml:mi mathvariant="italic">T</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">¯</mml:mo> </mml:mrow> </mml:mover> </mml:math> / <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic">J</mml:mi> <mml:mover accent="true"> <mml:mrow> <mml:mi mathvariant="italic">T</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">¯</mml:mo> </mml:mrow> </mml:mover> </mml:math> deformed conformal field theories and modular invariance/covariance. In the third part, different perspectives of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic">T</mml:mi> <mml:mover accent="true"> <mml:mrow> <mml:mi mathvariant="italic">T</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">¯</mml:mo> </mml:mrow> </mml:mover> </mml:math> deformation are presented, including its relation to random geometry, 2D topological gravity and holography. We summarize more recent developments until January 2021 in the last part.