G / H CONFORMAL FIELD THEORY

Following the 1984 seminal work of Belavin, Polyakov and Zamolodchikov on two-dimensional conformal field theories, Toda conformal field theories were introduced in the physics literature as a family of two-dimensional conformal field theories that enjoy, in addition to conformal symmetry, an extended level of symmetry usually referred to as W-symmetry or higher-spin symmetry. More precisely Toda conformal field theories provide a natural way to associate to a finite-dimensional simple and complex Lie algebra a conformal field theory for which the algebra of symmetry contains the Virasoro algebra. In this document we use the path integral formulation of these models to provide a rigorous mathematical construction of Toda conformal field theories based on probability theory. By doing so we recover expected properties of the theory such as the Weyl anomaly formula with respect to the change of background metric by a conformal factor and the existence of Seiberg bounds for the correlation functions.

Affine.m—Mathematica package for computations in representation theory of finite-dimensional and affine Lie algebras

This thesis is devoted to the study of a new construction of two-dimensional topological conformal field theories by gauge fixing two-dimensional topological gauge theories. We study in detail the Lorenz-gauged abelian and non-abelian BF theory, which are topological conformal classically and on the quantum level. We find that the abelian model corresponds to Witten’s B-model with a parity shifted flat target space. It is therefore obtained by twisting a N = (2,2) supersymmetric model, while no such twist exists in the non-abelian case. Furthermore, we study an analogue of Gromov-Witten periods in the abelian model. Finally, we show that the non-abelian model allows non-trivial Jordan blocks of the Hamiltonian and thus defines a logarithmic conformal field theory. The existence of infinite-dimensional Jordan blocks allows an explicit construction of primary fields, whose conformal weights are subject to quantum corrections.

We compute genus two partition functions in two-dimensional conformal field theories at large central charge, focusing on surfaces that give the third Rényi entropy of two intervals. We compute this for generalized free theories and for symmetric orbifolds, and compare it to the result in pure gravity. We find a new phase transition if the theory contains a light operator of dimension . This means in particular that unlike the second Rényi entropy, the third one is no longer universal.

We present a survey of two-dimensional conformal field theory and show how the mathematical structures underlying conformal field theory can be used to construct invariants of links imbedded in a general class of three-dimensional manifolds. After a general introduction, we discuss chiral algebras and their representation theory. Chiral vertices are introduced as analogues of Clebsch-Gordan operators in group theory. Braiding and fusing of chiral vertices is analyzed, and it is sketched how to define conformal field theory on arbitrary Riemann surfaces by a sewing procedure. We then show how to construct link invariants from the data provided by a conformal field theory and sketch connections with quantum group theory.

We study the symmetry resolution of the entanglement entropy of an interval in two-dimensional conformal field theories (CFTs), by relating the bipartition to the geometry of an annulus with conformal boundary conditions. In the presence of extended symmetries such as Kac-Moody type current algebrae, symmetry resolution is possible only if the boundary conditions on the annulus preserve part of the symmetry group, i.e. if the factorization map associated with the spatial bipartition is compatible with the symmetry in question. The partition function of the boundary CFT (BCFT) is then decomposed in terms of the characters of the irreducible representations of the symmetry group preserved by the boundary conditions. We demonstrate that this decomposition already provides the symmetry resolution of the entanglement spectrum of the corresponding bipartition. Considering the various terms of the partition function associated with the same representation, or charge sector, the symmetry-resolved Rényi entropies can be derived to all orders in the UV cutoff expansion without the need to compute the charged moments. We apply this idea to the theory of a free massless boson with U(1) U ( 1 ) , \mathbb{R} ℝ and \mathbb{Z}_2 ℤ 2 symmetry.

We study quantum chaos in two dimensional conformal field theories, building on the work analyzing the out-of-time order thermal correlation functions using large-c Virasoro blocks. Our work investigates the contribution of light intermediate channels and smearing length scales to the four-point function and scrambling. Precise relations for how light intermediate channels increase the scrambling time and how smearing length scales smaller than the thermal length scale decrease the scrambling time are derived.

We introduce and study generalized Rényi entropies defined through the traces of products of TrB(| Ψi⟩⟨Ψj| ) where ∣Ψi⟩ are eigenstates of a two-dimensional conformal field theory (CFT). When ∣Ψi⟩ = ∣Ψj⟩ these objects reduce to the standard Rényi entropies of the eigenstates of the CFT. Exploiting the path integral formalism, we show that the second generalized Rényi entropies are equivalent to four point correlators. We then focus on a free bosonic theory for which the mode expansion of the fields allows us to develop an efficient strategy to compute the second generalized Rényi entropy for all eigenstates. As a byproduct, our approach also leads to new results for the standard Rényi and relative entropies involving arbitrary descendent states of the bosonic CFT.

We consider the unoriented two-dimensional Abelian sandpile model from a perspective based on two-dimensional (conformal) field theory. We compute lattice correlation functions for various cluster variables (at and off criticality), from which we infer the field-theoretic description in the scaling limit. We find perfect agreement with the predictions of a c=-2 conformal field theory and its massive perturbation, thereby providing direct evidence for conformal invariance and more generally for a description in terms of a local field theory. The question of the height 2 variable is also addressed, with, however, no definite conclusion yet.

Topological order in Quantum hall effect and two-dimensional conformal field theory

The scaling dimension of the first excited state in two-dimensional conformal field theories (CFTs) satisfies a universal upper bound. Using the modular bootstrap, we extend this result to CFTs with $W_3$ algebras which are generically dual to higher spin theories in AdS$_3$. Assuming unitarity and modular invariance, we show that the conformal weights $h$, $\bar{h}$ of the lightest charged state satisfy $h < c/12 + O(1)$ and $\bar{h} < \bar{c}/12 + O(1)$ in the limit where the central charges $c$, $\bar{c}$ are large. Furthermore, we show that in this limit any consistent CFT with $W_3$ currents must contain at least one state whose $W_3$ charge $w$ obeys $|w| > 4 |h-c/24| /\sqrt{10 \pi c} + O(1)$. We discuss hints on the existence of stronger bounds and comment on the interpretation of our results in the dual higher spin theory.

Six-dimensional supergravity on S3 × AdS 3 and 2d conformal field theory

A consistent quantization of two-dimensional (2D) massless light-front fields (scalar and fermion) is formulated. Their two-point functions exactly reproduce the massless limit of the two-point functions of the corresponding massive fields. The novel formalism incorporates bosonization in a natural way and also provides us with elements needed for an independent light-front (LF) study of the exactly solvable models (the Thirring or Thirring-Wess model, e.g.). Moreover, it displays closeness of the 2D massless LF quantum fields to conformal field theory (CFT). We calculate a few correlators including those between the components of the LF energy-momentum tensor and derive the Virasoro algebra in the LF operator form. Going over to the euclidean time, we can direcly transform all calculated quantities to the (anti)holomorphic form, in agreement with those from CFT.

We investigate the one-loop entanglement entropy of two short intervals with small cross ratio $x$ on a complex plane in two-dimensional conformal field theory (CFT) using operator product expansion of twist operators. We focus on the one-loop entanglement entropy instead of the general order $n$ R\'enyi entropy, and this makes the calculation much easier. We consider the contributions of stress tensor to order $x^{10}$, contributions of $W_3$ operator to order $x^{12}$, and contributions of $W_4$ operator to order $x^{14}$. The CFT results agree with the ones in gravity.

Since the definition of $$T\bar T$$ deformation in the curved Riemann surface is obstructive in the literature, we propose a way to do the deformation in the genus two Riemann surfaces by sewing prescription. We construct the correlation functions of conformal field theories (CFTs) on genus two Riemann surfaces with the $$T\bar T$$ deformation in terms of the perturbative CFT approach. Thanks to sewing construction to form higher genus Riemann surfaces from lower genus ones and conformal symmetry, we systematically obtain the first-order $$T\bar T$$ correction to the genus two correlation functions in the $$T\bar T$$ deformed CFTs, e.g., partition function and one/higher-point correlation functions.