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.
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 In this paper we explicitly carry out the perturbative renormalization of the
TT T T ¯ -deformed free massive Dirac fermion in two dimensions up to second order in the coupling constant. This is done by computing the two-to-two S-matrix using the LSZ reduction formula and canceling out the divergences by introducing counterterms. We demonstrate that the renormalized Lagrangian is unambiguously determined by demanding that it gives the correct S-matrix of a
TT T T ¯ -deformed integrable field theory. Remarkably, the renormalized Lagrangian is qualitatively very different from its classical counterpart.
This thesis focuses on renormalization of quantum field theories. Its first part considers three tensor models in three dimensions, a Fermionic quartic with tensors of rank-3 and two Bosonic sextic, of ranks 3 and 5. We rely upon the large-
N melonic expansion of tensor models. For the first model, invariant under
U(N)3, we obtain the RG flow of the two melonic couplings and the vacuum phase diagram, from a reformulation with a diagonalizable matrix intermediate field. The discrete chiral symmetry breaks spontaneously and we compare with the three-dimensional Gross-Neveu model. Beyond the massless
U(N)3 symmetric phase, we also observe a massive phase of same symmetry and another where the symmetry breaks into
U(N2)×U(N/2)×U(N/2). A matrix model invariant under
U(N)×U(N2), with close properties, is also studied. For the other models, with symmetry groups
U(N)3 and
O(N)5, a non-melonic coupling (the "wheel") with an optimal scaling in
N drives us to a generalized melonic expansion. The kinetic terms are taken of short- and long-range, and we analyze perturbatively, at large-
N, the RG flows of the sextic couplings up to four loops. Only the rank-3 model displays non-trivial fixed points (two real Wilson-Fisher-like in the short-range case and a line of fixed points in the other). We finally obtain the real conformal dimensions of the primary bilinear operators. In the second part, we establish the first results of perturbative multi-scale renormalization for a quartic scalar field on critical Galton-Watson trees, with a long-range kinetic term. At criticality, an emergent infinite spine provides a space of effective dimension
4/3 on which to compute averaged correlation fonctions. This approach formalizes the notion of a QFT on a random geometry. We use known probabilistic bounds on the heat-kernel on a random graph reviewed in detail.
A bstract We consider a perturbative expansion of the Lanczos coefficients and the Krylov complexity for two-dimensional conformal field theories under integrable deformations. Specifically, we explore the consequences of
TT T T ¯ ,
JT J T ¯ , and
JJ J J ¯ deformations, focusing on first-order corrections in the deformation parameter. Under
TT T T ¯ deformation, we demonstrate that the Lanczos coefficients b n exhibit unexpected behavior, deviating from linear growth within the valid perturbative regime. Notably, the Krylov exponent characterizing the rate of exponential growth of complexity surpasses that of the undeformed theory for positive value of deformation parameter, suggesting a potential violation of the conjectured operator growth bound within the realm of perturbative analysis. One may attribute this to the existence of logarithmic branch points along with higher order poles in the autocorrelation function compared to the undeformed case. In contrast to this, both
JJ J J ¯ and
JT J T ¯ deformations induce no first order correction to either the linear growth of Lanczos coefficients at large- n or the Krylov exponent and hence the results for these two deformations align with those of the undeformed theory.
It was noticed many years ago, in the framework of massless RG flows, that the irrelevant composite operator
TT , built with the components of the energy-momentum tensor, enjoys very special properties in 2D quantum field theories, and can be regarded as a peculiar kind of integrable perturbation. Novel interesting features of this operator have recently emerged from the study of effective string theory models. In this paper we study further properties of this distinguished perturbation. We discuss how it affects the energy levels and one-point functions of a general 2D QFT in finite volume through a surprising relation with a simple hydrodynamic equation. In the case of the perturbation of CFTs, adapting a result by Lüscher and Weisz we give a compact expression for the partition function on a finite-length cylinder and make a connection with the exact g-function method. We argue that, at the classical level, the deformation naturally maps the action of N massless free bosons into the Nambu-Goto action in static gauge, in N + 2 target space dimensions, and we briefly discuss a possible interpretation of this result in the context of effective string models.