Extended Symmetry Algebra Research Articles

Carroll strings with an extended symmetry algebra

Starting from the Polyakov action we consider two distinct Carroll limits in target space, keeping the string worldsheet relativistic. The resulting magnetic and chiral Carroll string models exhibit different symmetries and dynamics. Both models have an infinite dimensional symmetry algebra with Carroll symmetry included in a finite dimensional subalgebra. For the magnetic model, this is the so-called string Carroll algebra. The chiral model realises an extended version of the string Carroll algebra. The magnetic model does not have any transverse string excitations. The chiral model is less restrictive and includes arbitrary left-moving modes that carry transverse momentum but do not contribute to the energy in target space.

Extensions of the asymptotic symmetry algebra of general relativity

We consider a recently proposed extension of the Bondi-Metzner-Sachs algebra to include arbitrary infinitesimal diffeomorphisms on a 2-sphere. To realize this extended algebra as asymptotic symmetries, we work with an extended class of spacetimes in which the unphysical metric at null infinity is not universal. We show that the symplectic current evaluated on these extended symmetries is divergent in the limit to null infinity. We also show that this divergence cannot be removed by a local and covariant redefinition of the symplectic current. This suggests that such an extended symmetry algebra cannot be realized as symmetries on the phase space of vacuum general relativity at null infinity, and that the corresponding asymptotic charges are ill-defined. However, a possible loophole in the argument is the possibility that symplectic current may not need to be covariant in order to have a covariant symplectic form. We also show that the extended algebra does not have a preferred subalgebra of translations and therefore does not admit a universal definition of Bondi 4-momentum.

Zn clock models and chains of so(n)2 non-Abelian anyons: symmetries, integrable points and low energy properties

We study two families of quantum models which have been used previously to investigate the effect of topological symmetries in one-dimensional correlated matter. Various striking similarities are observed between certain quantum clock models, spin chains generalizing the Ising model, and chains of non-Abelian anyons constructed from the so(n)2 fusion category for odd n, both subject to periodic boundary conditions. In spite of the differences between these two types of quantum chains, e.g. their Hilbert spaces being spanned by tensor products of local spin states or fusion paths of anyons, the symmetries of the lattice models are shown to be closely related. Furthermore, under a suitable mapping between the parameters describing the interaction between spins and anyons the respective Hamiltonians share part of their energy spectrum (although their degeneracies may differ). This spin-anyon correspondence can be extended by fine-tuning of the coupling constants leading to exactly solvable models. We show that the algebraic structures underlying the integrability of the clock models and the anyon chain are the same. For n = 3,5,7 we perform an extensive finite size study—both numerical and based on the exact solution—of these models to map out their ground state phase diagram and to identify the effective field theories describing their low energy behaviour. We observe that the continuum limit at the integrable points can be described by rational conformal field theories with extended symmetry algebras which can be related to the discrete ones of the lattice models.

AdS2 holographic dictionary

We construct the holographic dictionary for both running and constant dilaton solutions of the two dimensional Einstein-Maxwell-Dilaton theory that is obtained by a circle reduction from Einstein-Hilbert gravity with negative cosmological constant in three dimensions. This specific model ensures that the dual theory has a well defined ultraviolet completion in terms of a two dimensional conformal field theory, but our results apply qualitatively to a wider class of two dimensional dilaton gravity theories. For each type of solutions we perform holographic renormalization, compute the exact renormalized one-point functions in the presence of arbitrary sources, and derive the asymptotic symmetries and the corresponding conserved charges. In both cases we find that the scalar operator dual to the dilaton plays a crucial role in the description of the dynamics. Its source gives rise to a matter conformal anomaly for the running dilaton solutions, while its expectation value is the only non trivial observable for constant dilaton solutions. The role of this operator has been largely overlooked in the literature. We further show that the only non trivial conserved charges for running dilaton solutions are the mass and the electric charge, while for constant dilaton solutions only the electric charge is non zero. However, by uplifting the solutions to three dimensions we show that constant dilaton solutions can support non trivial extended symmetry algebras, including the one found by Compère, Song and Strominger [1], in agreement with the results of Castro and Song [2]. Finally, we demonstrate that any solution of this specific dilaton gravity model can be uplifted to a family of asymptotically AdS2 × S 2 or conformally AdS2 × S 2 solutions of the STU model in four dimensions, including non extremal black holes. The four dimensional solutions obtained by uplifting the running dilaton solutions coincide with the so called ‘subtracted geometries’, while those obtained from the uplift of the constant dilaton ones are new.

Extremal higher spin black holes

The gauge sector of three-dimensional higher spin gravities can be formulated as a Chern-Simons theory. In this context, a higher spin black hole corresponds to a flat connection with suitable holonomy (smoothness) conditions which are consistent with the properties of a generalized thermal ensemble. Building on these ideas, we discuss a definition of black hole extremality which is appropriate to the topological character of 3d higher spin theories. Our definition can be phrased in terms of the Jordan class of the holonomy around a non-contractible (angular) cycle, and we show that it is compatible with the zero-temperature limit of smooth black hole solutions. While this notion of extremality does not require supersymmetry, we exemplify its consequences in the context of sl(3|2) + sl(3|2) Chern-Simons theory and show that, as usual, not all extremal solutions preserve supersymmetries. Remarkably, we find in addition that the higher spin setup allows for non-extremal supersymmetric black hole solutions. Furthermore, we discuss our results from the perspective of the holographic duality between sl(3|2) + sl(3|2) Chern-Simons theory and two-dimensional CFTs with W_{(3|2)} symmetry, the simplest higher spin extension of the N=2 super-Virasoro algebra. In particular, we compute W_{(3|2)} BPS bounds at the full quantum level, and relate their semiclassical limit to extremal black hole or conical defect solutions in the 3d bulk. Along the way, we discuss the role of the spectral flow automorphism and provide a conjecture for the form of the semiclassical BPS bounds in general N=2 two-dimensional CFTs with extended symmetry algebras.

Entanglement scrambling in 2d conformal field theory

We investigate how entanglement spreads in time-dependent states of a 1+1 dimensional conformal field theory (CFT). The results depend qualitatively on the value of the central charge. In rational CFTs, which have central charge below a critical value, entanglement entropy behaves as if correlations were carried by free quasiparticles. This leads to long-term memory effects, such as spikes in the mutual information of widely separated regions at late times. When the central charge is above the critical value, the quasiparticle picture fails. Assuming no extended symmetry algebra, any theory with $c>1$ has diminished memory effects compared to the rational models. In holographic CFTs, with $c \gg 1$, these memory effects are eliminated altogether at strong coupling, but reappear after the scrambling time $t \gtrsim \beta \log c$ at weak coupling.

Infinite-dimensional symmetries of two-dimensional coset models

It has long been appreciated that the toroidal reduction of any gravity or supergravity to two dimensions gives rise to a scalar coset theory exhibiting an infinite-dimensional global symmetry. This symmetry is an extension of the finite-dimensional symmetry G in three dimensions, after performing a further circle reduction. There has not been universal agreement as to exactly what the extended symmetry algebra is, with different arguments seemingly concluding either that it is G ˆ , the affine Kac–Moody extension of G , or else a subalgebra thereof. We take the very explicit approach of Schwarz as our starting point for studying the simpler situation of two-dimensional flat-space sigma models, which nonetheless capture all the essential details. We arrive at the conclusion that the full symmetry is described by the Kac–Moody algebra G ˆ , whilst the subalgebra obtained by Schwarz arises as a gauge-fixed truncation. We then consider the explicit example of the SL ( 2 , R ) / O ( 2 ) coset, and relate Schwarz's approach to an earlier discussion that goes back to the work of Geroch.

Dualization of the principal sigma model

The first-order formulation of the principal sigma model with a Lie group target space is performed. By using the dualization of the algebra and the field content of the theory the field equations which are solely written in terms of the field strengths are realized through an extended symmetry algebra parametrization. The structure of this symmetry algebra is derived so that it generates the realization of the field equations in a Bianchi identity of the current derived from the extended parametrization.

Fusion algebras of logarithmic minimal models

We present explicit conjectures for the chiral fusion algebras of the logarithmic minimal models considering Virasoro representations with no enlarged or extended symmetry algebra. The generators of fusion are countably infinite in number but the ensuing fusion rules are quasi-rational in the sense that the fusion of a finite number of representations decomposes into a finite direct sum of representations. The fusion rules are commutative, associative and exhibit an sℓ(2) structure but require so-called Kac representations which are typically reducible yet indecomposable representations of rank 1. In particular, the identity of the fundamental fusion algebra p ≠ 1 is a reducible yet indecomposable Kac representation of rank 1. We make detailed comparisons of our fusion rules with the results of Gaberdiel and Kausch for p = 1 and with Eberle and Flohr for (p, p′) = (2, 5) corresponding to the logarithmic Yang–Lee model. In the latter case, we confirm the appearance of indecomposable representations of rank 3. We also find that closure of a fundamental fusion algebra is achieved without the introduction of indecomposable representations of rank higher than 3. The conjectured fusion rules are supported, within our lattice approach, by extensive numerical studies of the associated integrable lattice models. Details of our lattice findings and numerical results will be presented elsewhere. The agreement of our fusion rules with the previous fusion rules lends considerable support for the identification of the logarithmic minimal models with the augmented (minimal) models defined algebraically.

Fusion algebra of critical percolation

We present an explicit conjecture for the chiral fusion algebra of critical percolationconsidering Virasoro representations with no enlarged or extended symmetry algebra. Therepresentations that we take to generate fusion are countably infinite in number. Theensuing fusion rules are quasi-rational in the sense that the fusion of a finitenumber of these representations decomposes into a finite direct sum of theserepresentations. The fusion rules are commutative, associative and exhibit an structure. They involve representations which we call Kac representations of which someare reducible yet indecomposable representations of rank 1. In particular, the identity ofthe fusion algebra is a reducible yet indecomposable Kac representation of rank1. We make detailed comparisons of our fusion rules with the recent results ofEberle–Flohr and Read–Saleur. Notably, in agreement with Eberle–Flohr, we find theappearance of indecomposable representations of rank 3. Our fusion rules aresupported by extensive numerical studies of an integrable lattice model of criticalpercolation. Details of our lattice findings and numerical results will be presentedelsewhere.

Centrally extended symmetry algebra of asymptotically Gödel spacetimes

We define an asymptotic symmetry algebra for three-dimensional Goedel spacetimes supported by a gauge field which turns out to be the semi-direct sum of the diffeomorphisms on the circle with two loop algebras. A class of fields admitting this asymptotic symmetry algebra and leading to well-defined conserved charges is found. The covariant Poisson bracket of the conserved charges is then shown to be centrally extended to the semi-direct sum of a Virasoro algebra and two affine algebras.

Boundary states, extended symmetry algebra, and module structure for certain rational torus models

The massless bosonic field compactified on the circle of rational R2 is reexamined in the presense of boundaries. A particular class of models corresponding to R2=1/2k is distinguished by demanding the existence of a consistent set of Neumann boundary states. The boundary states are constructed explicitly for these models and the fusion rules are derived from them. These are the ones prescribed by the Verlinde formula from the S-matrix of the theory. In addition, the extended symmetry algebra of these theories is constructed, which is responsible for the rationality of these theories. Finally, the chiral space of these models is shown to split into a direct sum of irreducible modules of the extended symmetry algebra.

Boundary Structure and Module Decomposition of the Bosonic Z2 Orbifold Models with R2=1/2 k

The Z 2 bosonic orbifold models with compactification radius R 2=1/2 k are examined in the presence of boundaries. Demanding the extended algebra characters to have definite conformal dimension and to consist of an integer sum of Virasoro characters, one arrives at the right splitting of the partition function. This is used to derive a free field representation of a complete, consistent set of boundary states, compatible with the modular transformations of the characters. Finally the modules of the extended symmetry algebra that correspond to the finitely many characters are identified inside the direct sum of Fock modules that constitute the space of states of the theory.

G/u(1)dparafermions from constrained WZNW theories

The conformal field theory based on the $g/u(1)^d$ coset construction is treated as the WZNW theory for the affine Lie algebra $\hat g$ with the constrained $\hat u(1)^d$ subalgebra.Using a modification of the generalized canonical quantization method generators and primary fields of an extended symmetry algebra are found for arbitrary d.

Conformal field theory approach to gapless 1D fermion systems and application to the edge excitations of ν=1/(2p+1) quantum Hall sequences

We present a comprehensive study of the effective Conformal Field Theory (CFT) describing the low energy excitations of a gas of spinless interacting fermions on a circle in the gapless regime (Luttinger liquid). Functional techniques and modular transformation properties are used to compute all correlation functions in a finite size and at finite temperature. Forward scattering disorder is treated exactly. Laughlin experiments on charge transport in a Quantum Hall Fluid on a cylinder are reviewed within this CFT framework. Edge excitations above a given bulk excitation are described by a twisted version of the Luttinger effective theory. Luttinger CFTs corresponding to the nu =1/(2p+1) filling fractions appear to be rational CFTs (RCFT). Generators of the extended symmetry algebra are identified as edge fermions creators and annihilators, thus giving a physical meaning to the RCFT point of view on edge excitations of these sequences.

Modular invariant partition functions in the quantum Hall effect

We study the partition function for the low-energy edge excitations of the incompressible electron fluid. On an annular geometry, these excitations have opposite chiralities on the two edges; thus, the partition function takes the standard form of rational conformal field theories. In particular, it is invariant under modular transformations of the toroidal geometry made by the angular variable and the compact Euclidean time. The Jain series of plateaus have been described by two types of edge theories: the minimal models of the W 1+∞ algebra of quantum area-preserving diffeomorphisms, and their non-minimal version, the theories with ▪ affine algebra. We find modular invariant partition functions for the latter models. Moreover, we relate the Wen topological order to the modular transformations and the Verlinde fusion algebra. We find new, non-diagonal modular invariants which describe edge theories with extended symmetry algebra; their Hall conductivities match the experimental values beyond the Jain series.

A rational logarithmic conformal field theory

We analyse the fusion of representations of the triplet algebra, the maximally extended symmetry algebra of the Virasoro algebra at c = −2. It is shown that there exists a finite number of representations which are closed under fusion. These include all irreducible representations, but also some reducible representations which appear as indecomposable components in fusion products.

Twist symmetry and open-string Wilson lines

The Möbius amplitude plays an important role in open-string theories, since it determines which sectors of a given model consist of unoriented open strings. It also fixes the Chan-Paton representations of all their states, according to the behavior under the interchange of the ends of open strings (“twist”). In this paper we discuss the role played by conventional Wilson lines in Chan-Paton symmetry breaking, and we show that the presence of an extended symmetry algebra allows, in general, a number of choices for the behavior of massive states under twist. This freedom may be ascribed to additional discrete Wilson lines, and yields consistent modifications of the group assignments, that are illustrated in a number of examples.

Unitary construction of extended conformal algebras

Particular examples and the general structure of extended conformal symmetries in coset conformal field theories are discussed. Discrete series of unitary representations, whose existence had been previously conjectured, are constructed for a class of extended conformal algebras introduced by Fateev and Zamolodchikov (FZ). The construction is a generalisation of the coset construction of the discrete series for the superconformal algebra using the coset spaces so(N) ⊕ su(N)/so(N), for fixed N. The N = 3 series is the FZ S3 algebra and the N = 4 series consists of two commuting copies of the superconformal algebra. A general method for analysing the extended conformal symmetries present in a particular coset theory and of constructing discrete series of representations of extended symmetry algebras is outlined.

Infinite number of conserved quantities and extended conformal algebra in the Thirring model

It is shown that the Thirring model has an infinite number of local conserved quantities, explicit forms of which are presented. These quantities are shown to be expressed in terms of scattering parameters. It will be shown that in this model there exists an extended symmetry algebra that includes the Virasoro algebra as its subalgebra.