<!--HTML-->In this talk, I will discuss the TTbar deformation of fermionic theories. First, I briefly review the TTbar deformation of two-dimensional QFTs. And, I demonstrate the Hamiltonian analysis of the TTbar deformation of free fermion, and I show how this reproduces the deformed spectrum. Then, I explain how negative norm states could appear in the TTbar deformed fermionic theories depending on the energy-momentum tensor. Finally, I shortly summarize our results on the TTbar deformation of N=(1,1) SUSY theory.
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.
Smirnov and Zamolodchikov recently introduced a new class of two-dimensional quantum field theories, defined through a differential change of any existing theory by the determinant of the energy-momentum tensor. From this
TT flow equation one can find a simple expression for both the energy spectrum and the
S-matrix of the
TT deformed theories. Our goal is to find the renormalized Lagrangian of the
TT deformed theories. In the context of the
TT deformation of an integrable theory, the deformed theory is also integrable and, correspondingly, the
S-matrix factorizes into two-to-two
S-matrices. One may thus hope to be able to extract the renormalized Lagrangian from the
S-matrix. We do this explicitly for the
TT deformation of a free massive scalar, to second order in the deformation parameter. Once one has the renormalized Lagrangian one can, in principle, compute all other observables, such as correlation functions. We briefly discuss this, as well as the relation between the renormalized Lagrangian, the
TT flow equation, and the
S-matrix. We also mention a more general class of integrability-preserving deformations of a free scalar field theory.
The irrelevant composite operator
TTˉ, constructed from components of the stress-energy tensor, exhibits unique properties in two-dimensional quantum field theories and represents a distinctive form of integrable deformation. Significant progress has been made in understanding the bulk aspects of the theory, including its interpretation in terms of coordinate transformations and its connection to topological gravity models. However, the behavior of
TTˉ-deformed theories in the presence of boundaries and defects remains largely unexplored. In this note, we review analytical results obtained through various techniques. Specifically, we study the
TTˉ-deformed exact g-function within the framework of the Thermodynamic Bethe Ansatz and show that the results coincide with those obtained by solving the corresponding Burgers-type flow equation. Finally, we highlight some potentially significant open problems.
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 study
TT T T ¯ deformations of chiral bosons using the formalism due to Sen. For arbitrary numbers of left- and right-chiral bosons, we find that the
TT T T ¯ -deformed Lagrangian can be computed in closed form, giving rise to a novel non-local action in Sen’s formalism. We establish that at the limit of infinite
TT T T ¯ coupling, the equations of motion of deformed theory exhibits chiral decoupling. We then turn to a discussion of
TT T T ¯ -deformed chiral fermions, and point out that the stress tensor of the
TT T T ¯ -deformed free fermion coincides with the undeformed seed theory. We explain this behaviour of the stress tensor by noting that the deformation term in the action is purely topological in nature and closely resembles the fermionic Wess-Zumino term in the Green-Schwarz formalism. In turn, this observation also explains a puzzle in the literature, viz. why the
TT T T ¯ deformation of multiple free fermions truncate at linear order. We conclude by discussing the possibility of an interplay between
TT T T ¯ deformations and bosonisation.
Abstract The irrelevant composite operator TT, constructed from components of the stress-energy tensor, exhibits unique properties in two-dimensional quantum field theories and represents a distinctive form of integrable deformation. Significant progress has been made in understanding the bulk aspects of the theory, including its interpretation in terms of coordinate transformations and its connection to topo- logical gravity models. However, the behavior of TT-deformed theories in the pres- ence of boundaries and defects remains largely unexplored. In this note, we review analytical results obtained through various techniques. Specifically, we study the TT-deformed exact g-function within the framework of the Thermodynamic Bethe Ansatz and show that the results coincide with those obtained by solving the corre- sponding Burgers-type flow equation. Finally, we highlight some potentially signif- icant open problems.