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arXiv:1904.13376 · Communications in Theoretical Physics

A pedagogical review on solvable irrelevant deformations of 2D quantum field theory

Yunfeng Jiang

Published Feb 10, 2021• 116 citations
arXivHTMLPDF

Abstract

Abstract This is a pedagogical review on T T ¯ 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 T T ¯ 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 T T ¯ / J T ¯ deformed conformal field theories and modular invariance/covariance. In the third part, different perspectives of T T ¯ 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.

DOI
10.1088/1572-9494/abe4c9

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Any two dimensional quantum field theory that can be consistently defined on a torus is invariant under modular transformations. In this paper we study families of quantum field theories labeled by a dimensionful parameter t, that have the additional property that the energy of a state at finite t is a function only of t and of the energy and momentum of the corresponding state at t = 0, where the theory becomes conformal. We show that under this requirement, the partition sum of the theory at t = 0 uniquely determines the partition sum (and thus the spectrum) of the perturbed theory, to all orders in t, to be that of a TTˉT\bar{T}TTˉ deformed CFT. Non-perturbatively, we find that for one sign of t (for which the energies are real) the partition sum is uniquely determined, while for the other sign we find non-perturbative ambiguities. We characterize these ambiguities and comment on their possible relations to holography.
Modular covariance and uniqueness of JT‾ J\overline{T} JT deformed CFTs
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TTˉT\bar{T}TTˉ Deformations of Quantum Field theory
Recent work by Zamolodchikov and others has uncovered a kind of "solvable" irrelevant deformation of two dimensional QFT generated by the composite operator T T .The irrelevant composite operator T T is built with the components of the energy-momentum tensor.The T T deformation can be regarded as a peculiar kind of integrable perturbation of integrable quantum field theories (IQFT).In the holographic dual, this deformation represents a geometric cutoff of the asymptotic region of AdS.The QFT is placed on a Dirichlet boundary at finite radial distance in the bulk.In this thesis we consider the problem of exact integration of the T T deformation of two dimensional quantum field theories, as well as some higher dimensional extensions in the form of T T deformations.When the action can be shown to only depend algebraically on the background metric the solution of the deformation equation on the Lagrangian can be given in closed form in terms of solutions of the (extended) Burgers' equation.We present such examples in two and higher dimensions.
Abstract We initiate the study of T T ¯ -like irrelevant solvable deformations in quantum field theory with boundaries and defects. For this purpose, we employ a general formalism developed in the context of spin chains, which allows us to derive both, the deformed bulk and boundary/defect scattering matrices of integrable models. Using the deformed scattering matrices, we derive the flow equation for the deformed finite volume spectrum, as well as the cylinder partition function and the exact g-function.
We consider KdV currents in a quantum field theory obtained by deforming a two dimensional conformal field theory via the irrelevant operator TT‾. In this paper we determine their torus one–point functions modular properties. We find that the one–point functions decompose into a direct sum of two non–holomorphic modular forms. We also obtain a general differential equation that the KdV generalized torus partition sum satisfies. The differential equation provides a non–perturbative description of TT‾ deformed theories.