We investigate higher-order corrections to correlators in a general CFT (conformal field theory) with the double-trace T T ¯ deformation. Standard perturbation theory proves inadequate for this problem due to the intricate stress-tensor flow induced by the deformation. To tackle this challenge, we introduce a novel technique termed the conservation equation method. This method leverages the stress tensor’s trace relation and conservation property to establish relationships between higher and lower-order corrections, and subsequently determine the correlators by enforcing symmetry properties. As an illustration, we compute first- and higher-order corrections to various types of correlators. Our results align with existing calculations in the literature and demonstrate the viability of our method for general CFTs.
We investigate higher-order corrections to correlators in a general CFT (conformal field theory) with the double-trace T T ¯ deformation. Standard perturbation theory proves inadequate for this problem due to the intricate stress-tensor flow induced by the deformation. To tackle this challenge, we introduce a novel technique termed the conservation equation method. This method leverages the stress tensor’s trace relation and conservation property to establish relationships between higher and lower-order corrections, and subsequently determine the correlators by enforcing symmetry properties. As an illustration, we compute first- and higher-order corrections to various types of correlators. Our results align with existing calculations in the literature and demonstrate the viability of our method for general CFTs.
In this paper, we investigate the correlation functions of the conformal field theory (CFT) with the TT deformation on a torus in terms of the perturbative CFT approach, which is the extension of the previous investigations on correlation functions defined on a plane. We systematically obtain the first-order correction to the correlation functions of the CFTs with a TT deformation in both operator formalism and path integral language. As a consistency check, we compute the deformed partition function, namely, the zero-point correlation function, up to the first order, which is consistent with results in the literature. Moreover, we obtain a new recursion relation for correlation functions with multiple T's and T's inserted in generic CFTs on a torus. Based on the recursion relations, we study some correlation functions of stress tensors up to the first order under TT deformation.
Locality and conserved charges in TT-deformed CFTs
We investigate the locality properties of T \overline T T T ¯ -deformed CFTs within perturbation theory. Up to third order in the deformation parameter, we find a Hamiltonian operator which solves the flow equation, reproduces the T\overline T T T ¯ energy spectrum, and is consistent with quasi-locality of the theory. This Hamiltonian includes terms proportional to the central charge which have not appeared before and which are necessary to reproduce the correct spectrum. We show that the Hamiltonian is not uniquely defined since it contains free parameters, starting at second order, which do not spoil the above properties. We then use it to determine the full conserved stress tensor. In our approach, the KdV charges are automatically conserved to all orders but are not a priori local. Nevertheless, we show that they can be made local to first order. Our techniques allow us to further comment on the space of Hamiltonians constructed from products of KdV charges which also flow to local charges in the deformed theory in the IR.
Abstract The deformation, built from the components of the stress tensor and of a current, is a universal irrelevant deformation of two-dimensional CFTs that preserves the left-moving conformal symmetry, while breaking locality on the right-moving side. Operators in the -deformed CFT are naturally labeled by the left-moving position and right-moving momentum and transform in representations of the one-dimensional extended conformal group. We derive an all-orders formula for the spectrum of conformal dimensions and charges of the deformed CFT, which we cross-check at leading order using conformal perturbation theory. We also compute the linear corrections to the one-dimensional OPE coefficients and comment on the extent to which the correlation functions in -deformed CFTs can be obtained from field-dependent coordinate transformations.
Symmetries and operators in TT deformed CFTs
A bstract TT T T ¯ -deformed CFTs are known to possess nonlocal Virasoro×Virasoro symmetries, whose currents depend on integration of the stress tensor. Unfortunately, these symmetries do not act in a tractable manner on the undeformed operators, which are defined by evolving local operators in the undeformed CFT on a constant-time slice using the deformed Hamiltonian. In this paper, we explicitly construct two distinct classes of operators: (i) dressed operators, which are primary with respect to the Virasoro×Virasoro symmetries, and (ii) physical operators, a new type of operator that depends on fundamental fields at specific points and is therefore local at the classical level. While dressed operators preserve the conformal symmetry structure, they are themselves nonlocal in the sense that they depend on integrated quantities. Physical operators, by contrast, depend only on local fields. Moreover, physical operators can be expressed in terms of dressed operators, and this relation can be promoted to a definition of such operators at the quantum level. This enables the use of Virasoro×Virasoro symmetries to systematically compute physical quantities such as correlation functions. We calculate the two-point correlation functions of these operators in momentum space and find that our results align with both string theory predictions and field theory calculations. Additionally, we explore the relationship between physical operators and alternative operator definitions proposed in the literature.
On the flow of states under TT
We study the TT deformation of two dimensional quantum field theories from a Hamiltonian point of view, focusing on aspects of the theory in Lorentzian signature. Our starting point is a simple rewriting of the spatial integral of the TT operator, which directly implies the deformed energy spectrum of the theory. Using this rewriting, we then derive flow equations for various quantities in the deformed theory, such as energy eigenstates, operators, and correlation functions. On the plane, we find that the deformation merely has the effect of implementing successive canonical/Bogoliubov transformations along the flow. This leads us to define a class of non-local, “dressed” operators (including a dressed stress tensor) which satisfy the same commutation relations as in the undeformed theory. This further implies that on the plane, the deformed theory retains its symmetry algebra, including conformal symmetry, if the original theory is a CFT. On the cylinder the TT deformation is much more non-trivial, but even so, correlation functions of certain dressed operators are integral transforms of the original ones. Finally, we propose a tensor network interpretation of our results in the context of AdS/CFT.
We compute three-point correlators between the stress-energy tensor and the conserved currents of conformal field theories (CFTs) in 2+1 dimensions. We first compute the correlators in the large-flavor-number expansion of conformal gauge theories and then perform the computation using holography. In the holographic approach, the correlators are computed from an effective action on (3+1)-dimensional anti-de Sitter space (AdS4) and depend upon the coefficient $\ensuremath{\gamma}$ of a four-derivative term in the action. We find a precise match between the CFT and the holographic results, thus, fixing the values of $\ensuremath{\gamma}$. The CFTs of free fermions and bosons take the values $\ensuremath{\gamma}=1/12,\ensuremath{-}1/12$, respectively, and so saturate the bound $|\ensuremath{\gamma}|\ensuremath{\le}1/12$ obtained earlier from the holographic theory; the correlator of the conserved gauge flux of U(1) gauge theories takes intermediate values of $\ensuremath{\gamma}$. The value of $\ensuremath{\gamma}$ also controls the frequency dependence of the conductivity and other properties of quantum critical transport at nonzero temperatures. Our results for the values of $\ensuremath{\gamma}$ lead to an appealing physical interpretation of particlelike or vortexlike transport near quantum phase transitions of interest in condensed-matter physics. This paper includes Appendices reviewing key features of the AdS-CFT correspondence for condensed-matter physicists.