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arXiv:2509.01513 · International Journal of Modern Physics A

Bosonic Bogoliubov transformations as Lorentz boosts in (c,c¯)=(1,1) conformal field theories with marginal JJ¯ deformations

Federico L. Bottesi, Guillermo R. Zemba

Published Dec 6, 2025• 0 citations
arXivHTMLPDF

Abstract

In this paper, we consider conformal field theories with central charges [Formula: see text] that are invariant under the exchange of the holomorphic and antiholomorphic sectors, in both bosonic and fermionic realizations that are meaningful for condensed matter systems. The effect of marginal current–current [Formula: see text] perturbations is to induce a deformation of the Hilbert space given by a Lorentz boost in the 2D space of currents, which is identified with a Bogoliubov transformation. The rapidity of the boost is determined by the coupling constant of the marginal perturbation. When the perturbation is diagonal in the original currents of the theory, there is a linear relation between the two, and nonlinear otherwise. In the fermionic cases, both free and with Calogero–Sutherland interactions, the marginal perturbation corresponds to backward scattering processes.

DOI
10.1142/s0217751x26500314

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In this paper, we consider conformal field theories with central charges [Formula: see text] that are invariant under the exchange of the holomorphic and antiholomorphic sectors, in both bosonic and fermionic realizations that are meaningful for condensed matter systems. The effect of marginal current–current [Formula: see text] perturbations is to induce a deformation of the Hilbert space given by a Lorentz boost in the 2D space of currents, which is identified with a Bogoliubov transformation. The rapidity of the boost is determined by the coupling constant of the marginal perturbation. When the perturbation is diagonal in the original currents of the theory, there is a linear relation between the two, and nonlinear otherwise. In the fermionic cases, both free and with Calogero–Sutherland interactions, the marginal perturbation corresponds to backward scattering processes.
It has been recently shown that the deformation of an arbitrary two-dimensional conformal field theory by the composite irrelevant operator T \bar T T T ‾ , built from the components of the stress tensor, is solvable; in particular, the finite-size spectrum of the deformed theory can be obtained from that of the original CFT through a universal formula. We study a similarly universal, Lorentz-breaking deformation of two-dimensional CFTs that possess a conserved U(1) U ( 1 ) current, J J . The deformation takes the schematic form J \bar T J T ‾ and is interesting because it preserves an SL(2,\mathbb{R}) \times U(1) S L ( 2 , ℝ ) × U ( 1 ) subgroup of the original global conformal symmetries. For the case of a purely (anti)chiral current, we find the finite-size spectrum of the deformed theory and study its thermodynamic properties. We test our predictions in a simple example involving deformed free fermions.
Abstract We study solvable deformations of two-dimensional quantum field theories driven by a bilinear operator constructed from a pair of conserved U(1) currents Ja. We propose a quantum formulation of these deformations, based on the gauging of the corresponding symmetries in a path integral. This formalism leads to an exact dressing of the S-matrix of the system, similarly as what happens in the case of a TT‾ \textrm{T}\overline{\textrm{T}} TT T T ¯ deformation. For conformal theories the deformations under study are expected to be exactly marginal. Still, a peculiar situation might arise when the conserved currents Ja are not well-defined local operators in the original theory. A simple example of this kind of system is provided by rotation currents in a theory of multiple free, massless, non-compact bosons. We verify that, somewhat unexpectedly, such a theory is indeed still conformal after deformation and that it coincides with a TsT transformation of the original system. We then extend our formalism to the case in which the conserved currents are non-Abelian and point out its connection with Deformed T-dual Models and homogeneous Yang-Baxter deformations. In this case as well the deformation is based on a gauging of the symmetries involved and it turns out to be non-trivial only if the symmetry group admits a non-trivial central extension. Finally we apply what we learned by relating the TT‾ \textrm{T}\overline{\textrm{T}} TT T T ¯ deformation to the central extension of the two-dimensional Poincaré algebra.
Abstract In this chapter we discuss two explicit examples of conformal field theories. We start our analysis with the free massless bosonic theory that we have already seen in the previous chapter. After, we discuss the conformal field theory of a complex fermion operator (a Dirac fermion) using its decomposition in the real Majorana components. The central charge of both bosonic and fermion theories is c = 1 and this suggests the existence of an equivalence between them. The transformation that maps a bosonic into a fermion theory and vice versa is known as bosonization: it provides a useful tool both for the comprehension of the conformal theories and for a wide range of applications, in particular in low-dimensional condensed matter systems.
The fundamentals of the conformal symmetry transformations in two dimensions are introduced. From the adapted \textit{Ward identities under conformal constraints}, the \textit{operator product expansion} (OPE) is defined for free fields and the energy momentum tensor (EM). After demonstrating how the energy momentum tensor transforms under conformal invariance, the notion of the \textit{central charge c} is defined. Conformal fields in two dimensions are \textit{radially} quantized on the cylinder within the framework of the second quantization of quantum field theory and are then mapped onto complex plane via a conformal mapping. The \textit{radial ordering} of conformal fields is adopted. The \textit{Virasoro algebra} is derived from the identification of the \textit{conformal generators}, which follow from relations between equal-time commutators and contour integrals of conformal fields. The radial quantization is applied to the free fermionic and bosonic fields, where for the latter the \textit{Vertex operator} is defined and specified. The Fock space of the free boson is constructed through the \textit{operator state correspondence}. Various boundary conditions are applied to free fields, and their effects on the ground-state energy density are detailed, leading to the notion of the \textit{Casimir energy}. The OPEs are regularized for the \textit{interacting} conformal fields, namely the E-M tensor, and the related definition of \textit{normal ordering} is revisited. Utilization of the Virasoro algebra led first to the \textit{descendant states} and subsequently to the \textit{conformal families}. The OPEs are generalized for the conformal families, and it is demonstrated how the three-point function coefficients may be obtained in terms of the two-point functions. The same procedure is employed generically through the definition of the \textit{conformal blocks}. The \textit{conformal Bootstrap method} is introduced to calculate the coefficients of nnn point functions. The Hilbert space of the conformal fields in two dimensions is formally constructed through the \textit{representation theory of Virasoro algebra}. Accordingly, the reducible and irreducible \textit{Verma modules} are specified. Due to the unitarity requirement of the Hilbert space, the famous \textit{Kac determinant} is introduced to obtain constraints on the inner products of the Verma modules, whence on the \textit{conformal weights} and the central charge. The decoupling of the \textit{null states} from rest of the their conformal families is used to obtain \textit{fusion rules} of the families. The \textit{Kac table} for the constrained conformal weights is constructed, and its analysis under the requirement of unitarity leads to the identification of the \textit{minimal models}. The correspondence between the discrete statistical systems at criticality and the minimal models is studied, where special attention is given on the \textit{Ising model}.
We have gathered quite a large amount of technology to construct conformal boundary conditions and study their properties. So far, most of the tools relied on detailed knowledge of the underlying algebraic structure of conformal field theories (CFTs), such as representations of symmetry algebras and modular transformations. In almost all cases, this route will lead to a restricted class of boundary conditions only, favouring those that preserve a lot of symmetry. In “real life applications” of boundary CFT, arising in string theory, in condensed matter physics, or in the study of integrable models, we are often faced with situations where we start from a simple given boundary condition and then “deform” by “turning on perturbations”. In this chapter, we will discuss methods aimed at making this procedure well defined and at determining the perturbed boundary theory. Most of the time, we will restrict ourselves to pure boundary perturbations, which leave the bulk CFT untouched. We will sometimes be led to new boundary conditions that would have been difficult to find directly. The total space of boundary CFTs takes the form of a fibration over the moduli space of bulk theories, with the fibre over each point made up from all boundary conditions (branes) that exist for that bulk theory. Marginal boundary deformations, if they exist, will generate, starting from a single boundary condition, whole continuous families within this boundary moduli space.
This article presents a systematic development of conformal field theory from first principles in quantum field theory, emphasizing complete derivations of the conformal group in arbitrary dimensions from the conformal Killing equation, proving its isomorphism to SO(d+1,1)\mathrm{SO}(d+1,1)SO(d+1,1), and developing the embedding space formalism as a tool for constructing conformally covariant correlation functions constrained by Ward identities. In two dimensions, conformal symmetry enhances to an infinite-dimensional algebra: we derive the Virasoro algebra [Lm,Ln]=(m−n)Lm+n+c12(m3−m)δm+n,0[L_m, L_n] = (m-n)L_{m+n} + \frac{c}{12}(m^3-m)\delta_{m+n,0}[Lm​,Ln​]=(m−n)Lm+n​+12c​(m3−m)δm+n,0​ from infinitesimal generators ℓn=−zn+1∂z\ell_n = -z^{n+1}\partial_zℓn​=−zn+1∂z​, computing the central charge explicitly for free theories (boson c=1c=1c=1, Majorana fermion c=1/2c=1/2c=1/2, bcbcbc-ghosts c=1−3(2λ−1)2c = 1-3(2\lambda-1)^2c=1−3(2λ−1)2) and demonstrating its physical interpretation as vacuum energy. Through radial quantization, we establish the operator-state correspondence and construct the Hilbert space via Virasoro descendants of primary states ∣h,hˉ⟩|h,\bar{h}\rangle∣h,hˉ⟩, deriving the transformation law T′(w)=(dw/dz)−2[T(z)−(c/12){w;z}]T'(w) = (dw/dz)^{-2}[T(z) - (c/12)\{w;z\}]T′(w)=(dw/dz)−2[T(z)−(c/12){w;z}] that reveals the geometric origin of the conformal anomaly.