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 $\mathrm{SO}(d+1,1)$, 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 $[L_m, L_n] = (m-n)L_{m+n} + \frac{c}{12}(m^3-m)\delta_{m+n,0}$ from infinitesimal generators $\ell_n = -z^{n+1}\partial_z$, computing the central charge explicitly for free theories (boson $c=1$, Majorana fermion $c=1/2$, $bc$-ghosts $c = 1-3(2\lambda-1)^2$) 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 $|h,\bar{h}\rangle$, deriving the transformation law $T'(w) = (dw/dz)^{-2}[T(z) - (c/12)\{w;z\}]$ that reveals the geometric origin of the conformal anomaly.
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), 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+12c(m3−m)δm+n,0 from infinitesimal generators ℓn=−zn+1∂z, computing the central charge explicitly for free theories (boson c=1, Majorana fermion c=1/2, bc-ghosts c=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⟩, deriving the transformation law T′(w)=(dw/dz)−2[T(z)−(c/12){w;z}] that reveals the geometric origin of the conformal anomaly.
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 n 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}.
In this dissertation we describe progress towards carving out the space of non-integrable conformal field theories (CFTs) in 1+1 dimensions using conformal bootstrap techniques. We develop and apply numerical and analytical techniques to explore the non-perturbative consequences of conformal symmetry, unitarity and locality on the spectrum and dynamics of 2D CFTs. Throughout we focus on compact irrational CFTs, which have a discrete spectrum consisting of an infinite number of primary operators and exhibit chaotic dynamics. In the first part of the dissertation we report on numerical advances in charting the space of 2D CFTs. We use semidefinite programming to compute bounds on the gap in the spectrum of primary operators as a function of the central charge and on spectral functions that capture the spectral density of primary operators. We then develop recursion relations for arbitrary Virasoro conformal blocks, and explore the constraints of modular invariance of the genus-two partition function, leading to ``critical surfaces'' that bound the heavy structure constants of a CFT. In the second part of the dissertation we develop the analytic bootstrap program for two dimensional conformal field theories, focusing on universal aspects of irrational unitary CFTs and holography. We show that the large-spin sector of irrational CFTs is universally governed by a ``Virasoro mean field theory,'' incorporating exact stress tensor dynamics into the analytic bootstrap and leading to non-perturbative insights about the bound-state dynamics of three-dimensional quantum gravity. We then leverage a generalized notion of modular invariance to derive a universal asymptotic formula for the average value of the structure constants whenever one or more of the operators has large conformal dimension. The large central charge limit makes contact with quantum gravity in three dimensions, where the averaging over heavy states corresponds to coarse-graining over black hole microstates. Finally, we study the Euclidean gravitational path integral of pure AdS3 quantum gravity, and use a modular bootstrap analysis to suggest new saddle point configurations that can be consistently included in the gravitational path integral and whose inclusion renders the dual torus partition function unitary.
We review conformal field theory on the plane in the conformal bootstrap approach. We introduce the main ideas of the bootstrap approach to quantum field theory, and how they apply to two-dimensional theories with local conformal symmetry. We describe the mathematical structures that appear in such theories, from the Virasoro algebra and its representations, to BPZ equations and conformal blocks. Examples include Liouville theory, (generalized) minimal models, free bosonic theories, the H3+ model, and the SU2 and SL2(R) WZW models. We also discuss relations between some of these models, and limits of these models when the central charge and/or conformal dimensions tend to particular values.
Abstract This paper is designed to be a practical tool for constructing and investigating two-point correlation functions in defect conformal field theory, directly in physical space, between any two bulk primaries or between a bulk primary and a defect primary, with arbitrary spin. Although geometrically elegant and ultimately a more powerful approach, the embedding space formalism gets rather cumbersome when dealing with mixed symmetry tensors, especially in the projection to physical space. The results in this paper provide an alternative method for studying two-point correlation functions for a generic d-dimensional conformal field theory with a flat p-dimensional defect and d − p = q co-dimensions. We tabulate some examples of correlation functions involving a conserved current, an energy momentum tensor and a Maxwell field strength, while analysing the constraints arising from conservation and the equations of motion. A method for obtaining bulk-to-defect correlators is also explained. Some explicit examples are considered: free scalar theory on ℝp× (ℝq/ℤ2) and a free four dimensional Maxwell theory on a wedge.
We present an investigation of the massless, two-dimentional, interacting field theories. Their basic property is their invariance under an infinite-dimensional group of conformal (analytic) transformations. It is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of Virasoro algebra, and that the correlation functions are built up of the “conformal blocks” which are completely determined by the conformal invariance. Exactly solvable conformal theories associated with the degenerate representations are analyzed. In these theories the anomalous dimensions are known exactly and the correlation functions satisfy the systems of linear differential equations.
Tensor network methods are playing a central role in multiple disciplines of modern quantum physics, such as condensed matter physics, quantum information and quantum gravity. Based on renormalization group (RG) ideas, they provide a large class of variational ansatz for quantum many-body systems. There are various tensor network structures which are suited for different systems based on their entanglement structures, such as matrix product states (MPS), projected entangled pair states (PEPS), multi-scale entanglement renormalization ansatz (MERA). Numerous techniques involving coarse-graining tensor networks have been developed, such as tensor renormalization group (TRG), tensor network renormalization (TNR) and its variants. \n \nFirst, we study the generalization of MERA to continuous systems, or cMERA, which is expected to become a powerful variational ansatz for the ground state of strongly interacting quantum field theories. We investigate, in the simpler context of Gaussian cMERA for free theories, the extent to which the cMERA state ketPsicM with finite UV cut-off Lambda can capture the spacetime symmetries of the ground state ketPsi. For a free boson conformal field theory (CFT) in 1+1 dimensions as a concrete example, we build a quasi-local unitary transformation V that maps ketPsi into ketPsicM and show two main results. (i) Any spacetime symmetry of the ground state ketPsi is also mapped by V into a spacetime symmetry of the cMERA ketPsicM. However, while in the CFT the stress-energy tensor Tmunu(x) (in terms of which all the spacetime symmetry generators are expressed) is local, the corresponding cMERA stress-energy tensor TmunucM(x)=VTmunu(x)Vdagger is quasi-local. (ii) From the cMERA, we can extract quasi-local scaling operators OalphacM(x) characterized by the exact same scaling dimensions Deltaalpha, conformal spins salpha, operator product expansion coefficients Calphabetagamma, and central charge c as the original CFT. We argue that these results should also apply to interacting theories. \n \nSecond we extend TNR to field theories in the continuum. A short-distance length scale 1/Lambda is introduced in the continuum partition function by smearing the fields. The resulting object is still defined in the continuum but has no fluctuations at distances shorter than 1/Lambda. An infinitesimal coarse-graining step is then generated by the combined action of a \\textit{rescaling} operator mathfrakL and a \\textit{disentangling} operator mathfrakK that implements a quasilocal field redefinition. As demonstrated for a free boson in two dimensions, continuous TNR exactly preserves translation and rotation symmetries and can generate a proper RG flow. Moreover, from a critical fixed point of this RG flow one can then extract the conformal data of the underlying conformal field theory. \n \nLeaving behind tensor network methods, we explore the universal structure that emerges from the eigenvectors of the reduced density matrix (RDM) for 1+1 dimensional critical lattice systems. Using the Ising model as an example, we demonstrate that the large-weight eigenvectors are related to the low-lying states of the underlying CFT with suitable boundary conditions by a conformal map, and they transform accordingly under the Virasoro generators. \n \nFinally, using the quantum renormalization group (QRG), we derive the bulk geometry that emerges in the holographic dual of the fermionic U(N) vector model at a nonzero charge density. The obstruction that prohibits the metallic state from being smoothly deformable to the direct product state under the renormalization group flow gives rise to a horizon at a finite radial coordinate in the bulk. The region outside the horizon is described by the Lifshitz geometry with a higher-spin hair determined by microscopic details of the boundary theory. On the other hand, the interior of the horizon is not described by any Riemannian manifold, as it exhibits an algebraic non-locality. The non-local structure inside the horizon carries the information on the shape of the filled Fermi sea.