Class Of Conformal Field Theories Research Articles

Anyonic Chains – alpha -Induction – CFT – Defects – Subfactors

Given a unitary fusion category, one can define the Hilbert space of a so-called “anyonic spin-chain” and nearest neighbor Hamiltonians providing a real-time evolution. There is considerable evidence that suitable scaling limits of such systems can lead to 1+1-dimensional conformal field theories (CFTs), and in fact, can be used potentially to construct novel classes of CFTs. Besides the Hamiltonians and their densities, the spin chain is known to carry an algebra of symmetry operators commuting with the Hamiltonian, and these operators have an interesting representation as matrix-product-operators (MPOs). On the other hand, fusion categories are well-known to arise from a von Neumann algebra-subfactor pair. In this work, we investigate some interesting consequences of such structures for the corresponding anyonic spin-chain model. One of our main results is the construction of a novel algebra of MPOs acting on a bipartite anyonic chain. We show that this algebra is precisely isomorphic to the defect algebra of 1+1 CFTs as constructed by Fröhlich et al. and Bischoff et al., even though the model is defined on a finite lattice. We thus conjecture that its central projections are associated with the irreducible vertical (transparent) defects in the scaling limit of the model. Our results partly rely on the observation that MPOs are closely related to the so-called “double triangle algebra” arising in subfactor theory. In our subsequent constructions, we use insights into the structure of the double triangle algebra by Böckenhauer et al. based on the braided structure of the categories and on alpha -induction. The introductory section of this paper to subfactors and fusion categories has the character of a review.

Notes on tensor models and tensor field theories

Tensor models and tensor field theories admit a $1/N$ expansion and a melonic large $N$ limit which is simpler than the planar limit of random matrices and richer than the large $N$ limit of vector models. They provide examples of analytically tractable but non-trivial strongly coupled quantum field theories and lead to a new class of conformal field theories. We present a compact introduction to the topic, covering both some of the classical results in the field, like the details of the $1/N$ expansion, as well as recent developments. These notes are loosely bases on four lectures given at the Journées de physique mathématique Lyon 2019: Random tensors and SYK models.

BPS indices, modularity and perturbations in quantum K-theory

We study a perturbation family of mathcal{N} = 2 3d gauge theories and its relation to quantum K-theory. A 3d version of the Intriligator-Vafa formula is given for the quantum K-theory ring of Grassmannians. The 3d BPS half-index of the gauge theory is connected to the theory of bilateral hypergeometric q-series, and to modular q-characters of a class of conformal field theories in a certain massless limit. Turning on 3d Wilson lines at torsion points leads to mock modular behavior. Perturbed correlators in the IR regime are computed by determining the UV-IR map in the presence of deformations.

CFT in conformally flat spacetimes

A new class of conformal field theories is presented, where the background gravitational field is conformally flat. Conformally flat (CF) spacetimes enjoy conformal properties quite similar to the ones of flat spacetime. The conformal isometry group is of maximal rank and the conformal Killing vectors in conformally flat coordinates are {\em exactly} the same as the ones of flat spacetime. In this work, a new concept of distance is introduced, the {\em conformal distance}, which transforms covariantly under all conformal isometries of the CF space. It is shown that precisely for CF spacetimes, an adequate power of the said conformal distance is a solution of the non-minimal d'Alembert equation.

Correlation measures and the entanglement wedge cross-section after quantum quenches in two-dimensional conformal field theories

We consider the time evolution of mixed state correlation measures in two-dimensional conformal field theories, such as logarithmic negativity, odd entropy, and reflected entropy, after quantum quenches of various kinds. These correlation measures, in the holographic context, are all associated to the entanglement wedge cross section. We contrast various classes of conformal field theories, both rational and irrational (pure) conformal field theories. First, for rational conformal field theories, whose dynamics can be well described by the quasi-particle picture, we find all four quantities for disjoint intervals to be proportional, regardless of the specific quench protocol. Second, using the light cone boot- strap, we generalize our results to irrational conformal field theories where we find sharp distinctions from the quasi-particle results and striking differences between mutual information and the other measures. The large surplus of logarithmic negativity relative to mutual information forces us to reconsider what mutual information and logarithmic negativity really measure. We interpret these results as a signature of information scrambling and chaos in irrational theories. These CFT results perfectly agree with our gravitational (holographic) calculations. Furthermore, using holography, we are able to generalize the results to outside of the light cone limit. Finally, due to the breakdown of the quasi-particle picture for irrational theories, we appeal to the “line-tension picture,” motivated by random unitary circuits, as a phenomenological description. We observe that random unitary circuits, with local Hilbert space dimension determined by the Cardy formula, have precisely the same entanglement dynamics as irrational (including holographic) conformal field theories.

Achronal averaged null energy condition, weak cosmic censorship, and AdS/CFT duality

We examine the achronal averaged null energy condition~(ANEC) for a class of conformal field theories~(CFT) at strong coupling in curved spacetime. By applying the AdS/CFT duality, we find holographic models which violate the achronal ANEC for $3+1$ and $4+1$-dimensional boundary theories. In our model, the bulk spacetime is an asymptotically AdS vacuum bubble solution with neither causality violation nor singularities. The conformal boundary of our bubble solution is asymptotically flat and is causally proper in the sense that a "fastest null geodesics" connecting any two points on the boundary must lie entirely on the boundary. We show that conversely, if the spacetime fails to have this causally proper nature, then there must be a naked singularity in the bulk.

Notes on scrambling in conformal field theory

The onset of quantum chaos in quantum field theory may be studied using out-of-time-order correlators at finite temperature. Recent work argued that a timescale logarithmic in the central charge emerged in the context of two-dimensional conformal field theories, provided the intermediate channel was dominated by the Virasoro identity block. This suggests a wide class of conformal field theories exhibit a version of fast scrambling. In the present work we study this idea in more detail. We begin by clarifying to what extent correlators of wavepackets built out of superpositions of primary operators may be used to quantify quantum scrambling. Subject to certain caveats, these results concur with previous work. We then go on to study the contribution of intermediate states beyond the Virasoro identity block. We find that at late times, time-ordered correlators exhibit a familiar decoupling theorem, suppressing the contribution of higher dimension operators. However this is no longer true of the out-of-time-order correlators relevant for the discussion of quantum chaos. We compute the contributions of these conformal blocks to the relevant correlators, and find they are able to dominate in many interesting limits. Interpreting these results in the context of holographic models of quantum gravity, sheds new light on the black hole information problem by exhibiting a class of correlators where bulk effective field theory does not predict its own demise.

Nonlinear gravity from entanglement in conformal field theories

In this paper, we demonstrate the emergence of nonlinear gravitational equations directly from the physics of a broad class of conformal field theories. We consider CFT excited states defined by adding sources for scalar primary or stress tensor operators to the Euclidean path integral defining the vacuum state. For these states, we show that up to second order in the sources, the entanglement entropy for all ball-shaped regions can always be represented geometrically (via the Ryu-Takayanagi formula) by an asymptotically AdS geometry. We show that such a geometry necessarily satisfies Einstein’s equations perturbatively up to second order, with a stress energy tensor arising from matter fields associated with the sourced primary operators. We make no assumptions about AdS/CFT duality, so our work serves as both a consistency check for the AdS/CFT correspondence and a direct demonstration that spacetime and gravitational physics can emerge from the description of entanglement in conformal field theories.

Spinon bases in supersymmetric CFTs

We present a novel way to organise the finite size spectra of a class of conformal field theories (CFT) with or (nonlinear) superconformal symmetry. Generalising the spinon basis of the WZW theories, we introduce supersymmetric spinons , which form a representation of the supersymmetry algebra. In each case, we show how to construct a multi-spinon basis of the chiral CFT spectra. The multi-spinon states are labelled by a collection of (discrete) momenta. The state-content for given choice of is determined through a generalised exclusion principle, similar to Haldane's ‘motif’ rules for the theories. In the simplest case, which is the superconformal theory with central charge c = 1, we develop an algebraic framework similar to the Yangian symmetry of the theory. It includes an operator H2, akin to a CFT Haldane–Shastry Hamiltonian, which is diagonalised by multi-spinon states. In all cases studied, we obtain finite partition sums by capping the spinon-momenta to some finite value. For the superconformal CFTs, this finitisation precisely leads to the so-called Mk supersymmetric lattice models with characteristic order-k exclusion rules on the lattice. Finitising the c = 2 CFT with nonlinear superconformal symmetry similarly gives lattice model partition sums for spin-full Fermions with on-site and nearest neighbour exclusion.

Holographic operator mapping indS/CFTand cluster decomposition

The bulk-to-boundary mapping for massive scalar fields is constructed, providing a de Sitter analog of the Lehmann-Symanzik-Zimmermann reduction formula. The set of boundary correlators thus obtained defines a potentially new class of conformal field theories based on principal series representations of the global conformal group. Conversely, we show that bulk field operators in de Sitter may be reconstructed from boundary operators. While consistent at the level of the free field theory, the boundary conformal field theory does not satisfy cluster decomposition. The resulting conformal field theory does not satisfy the basic axioms of Euclidean quantum field theory due to Osterwalder and Schrader, so it is likely not well defined once interactions are included.

Constraints on Conformal Field Theories in Diverse Dimensions from the Bootstrap Mechanism

Recently an efficient numerical method has been developed to implement the constraints of crossing symmetry and unitarity on the operator dimensions and operator product expansion coefficients of conformal field theories in diverse space-time dimensions. It appears that the calculations can be done only for theories lying at the boundary of the allowed parameter space. Here it is pointed out that a similar method can be applied to a larger class of conformal field theories, whether unitary or not, and no free parameter remains, provided we know the fusion algebra of the low lying primary operators. As an example we calculate using first principles, with no phenomenological input, the lowest scaling dimensions of the local operators associated with the Yang-Lee edge singularity in three and four space dimensions. The edge exponents compare favorably with the latest numerical estimates. A consistency check of this approach on the 3D critical Ising model is also made.

Thermal valence-bond-solid transition of quantum spins in two dimensions

We study the $S=1/2$ Heisenberg ($J$) model on the two-dimensional (2D) square lattice in the presence of additional higher-order spin interactions ($Q$) which lead to a valence-bond-solid (VBS) ground state. Using quantum Monte Carlo simulations, we analyze the thermal VBS transition. We find continuously varying exponents, with the correlation-length exponent $\ensuremath{\nu}$ close to the Ising value for large $Q/J$ and diverging when $Q/J$ approaches the quantum-critical point (the critical temperature ${T}_{c}\ensuremath{\rightarrow}0$). We identify the transition with a class of conformal field theories with charge $c=1$ and critical exponents varying between those of the 2D Ising model and the Kosterlitz-Thouless (KT) fixed point. We find explicit evidence for KT physics by studying the emergence of $U(1)$ symmetry of the order parameter at $T={T}_{c}$ when ${T}_{c}\ensuremath{\rightarrow}0$.

Ashkin-Teller Criticality and Pseudo-First-Order Behavior in a Frustrated Ising Model on the Square Lattice

We study the challenging thermal phase transition to stripe order in the frustrated square-lattice Ising model with couplings J(1) < 0 (nearest-neighbor, ferromagnetic) and J(2) > 0 (second-neighbor, antiferromagnetic) for g = J(2)/|J(1| > 1/2. Using Monte Carlo simulations and known analytical results, we demonstrate Ashkin-Teller criticality for g ≥ g*; i.e., the critical exponents vary continuously between those of the 4-state Potts model at g = g* and the Ising model for g → ∞. Thus, stripe transitions offer a route to realizing a related class of conformal field theories with conformal charge c = 1 and varying exponents. The transition is first order for g < g* = 0.67 ± 0.01, much lower than previously believed, and exhibits pseudo-first-order behavior for |g* ≤ g </~1.

A-D-E Classification of Conformal Field Theories

The ADE classification scheme is encountered in many areas of mathematics, most notably in the study of Lie algebras. Here such a scheme is shown to describe families of two-dimensional conformal field theories.

Some mirror partners with complex multiplication

In theoretical physics rational conformal field theories are considered as particularly interesting class of conformal field theories. Let (X ,Y ) be a pair of families of Calabi–Yau 3-manifolds, which are mirror partners, X be a fiber of X and Y be a fiber of Y . In [7] Gukov and Vafa explain that X and Y yield a rational conformal field theory, if and only if both fibers have complex multiplication (CM). A family of Calabi–Yau manifolds over a Shimura variety has a dense set of CM fibers, if the variation of Hodge structures (VHS) is related to the Shimura datum of the base space in a natural way as in [10]. At present several of such families of Calabi–Yau 3-manifolds over Shimura varieties are known [2, 6, 10, 11, 12]. In general, one does not know a Shimura subvariety of the base space on the mirror side.1 Here we give new examples of pairs of families of Calabi–Yau 3-manifolds over Shimura varieties, which are subfamilies of mirror partners. We start with a family C3 of degree 3 covers of P1 with six different ramification points over an open Shimura subvariety M3 ⊂ (P1)3. By using the Fermat curve of degree 3 and C3, one can construct a family of K3 surfaces with a non-symplectic involution over M3 as described in [10, Section 8]. The Borcea–Voisin construction yields a family W of Calabi–Yau 3-manifolds, which has a dense set of CM fibers. Garbagnati and van Geemen [5] have given a more general method to construct K3 surfaces, which yields the same K3 surfaces for the fibers of C3. The latter method allows one to

Viscosity Bound and Causality Violation

In recent work we showed that, for a class of conformal field theories (CFT) with Gauss-Bonnet gravity dual, the shear viscosity to entropy density ratio, eta/s, could violate the conjectured Kovtun-Starinets-Son viscosity bound, eta/s > or = 1/4 pi. In this Letter we argue, in the context of the same model, that tuning eta/s below (16/25)(1/4 pi) induces microcausality violation in the CFT, rendering the theory inconsistent. This is a concrete example in which inconsistency of a theory and a lower bound on viscosity are correlated, supporting the idea of a possible universal lower bound on eta/s for all consistent theories.

Black Holes and Universality Classes of Critical Points

We argue that there exists an infinite class of conformal field theories in diverse dimensions having a universal ratio of the central charge c to the normalized entropy density c. The universality class includes all conformal theories which possess a classical gravity dual according to the AdS/CFT correspondence. From the practical point of view, the universality of c/c provides an explicit test which can be applied to determine whether a given critical point may admit a dual description in terms of classical gravity.

String theory: exact solutions, marginal deformations and hyperbolic spaces

AbstractThis thesis is almost entirely devoted to studying string theory backgrounds characterized by simple geometrical and integrability properties. The archetype of this type of system is given by Wess‐Zumino‐Witten models, describing string propagation in a group manifold or, equivalently, a class of conformal field theories with current algebras. We study the moduli space of such models by using truly marginal deformations. Particular emphasis is placed on asymmetric deformations that, together with the CFT description, enjoy a very nice spacetime interpretation in terms of the underlying Lie algebra. Then we take a slight detour so to deal with off‐shell systems. Using a renormalization‐group approach we describe the relaxation towards the symmetrical equilibrium situation. In he final chapter we consider backgrounds with Ramond‐Ramond field and in particular we analyze direct products of constant‐curvature spaces and find solutions with hyperbolic spaces.

Three Dimensional Nonlinear Sigma Models in the Wilsonian Renormalization Method

The three dimensional nonlinear sigma model is unrenormalizable in perturbative method. By using the $\beta$ function in the nonperturbative Wilsonian renormalization group method, we argue that ${\cal N}=2$ supersymmetric nonlinear $\sigma$ models are renormalizable in three dimensions. When the target space is an Einstein-K\"{a}hler manifold with positive scalar curvature, such as C$P^N$ or $Q^N$, there are nontrivial ultraviolet (UV) fixed point, which can be used to define the nontrivial continuum theory. If the target space has a negative scalar curvature, however, the theory has only the infrared Gaussian fixed point, and the sensible continuum theory cannot be defined. We also construct a model which interpolates between the C$P^N$ and $Q^N$ models with two coupling constants. This model has two non-trivial UV fixed points which can be used to define the continuum theory. Finally, we construct a class of conformal field theories with ${\bf SU}(N)$ symmetry, defined at the fixed point of the nonperturbative $\beta$ function. These conformal field theories have a free parameter corresponding to the anomalous dimension of the scalar fields. If we choose a specific value of the parameter, we recover the conformal field theory defined at the UV fixed point of C$P^N$ model and the symmetry is enhanced to ${\bf SU}(N+1)$.

A New Class of Conformal Field Theories with Anomalous Dimensions

The Wilsonian renormalization group (WRG) equation is used to derive a new class of scale invariant field theories with nonvanishing anomalous dimensions in 2-dimensional ${\cal N}=2$ supersymmetric nonlinear sigma models. When the coordinates of the target manifolds have nontrivial anomalous dimensions, vanishing of the $\beta$ function suggest the existence of novel conformal field theories whose target space is not Ricci flat. We construct such conformal field theories with ${\bf U}(N)$ symmetry. The theory has one free parameter a corresponding to the anomalous dimension of the scalar fields. The new conformal field theories are well behaved for positive a and have the central charge 3N, while they have curvature singularities at the boundary for a<0. When the target space is of complex 1-dimension, we obtain the explicit form of the Lagrangian, which reduces to two different kinds of free field theories in weak and in strong coupling limit. As a consistency test, the anomalous dimensions are reproduced in these two limits. The target space in this case looks like a semi-infinite cigar with one-dimension compactified to a circle.