Virasoro Representations Research Articles

Degenerate operators in JT and Liouville (super)gravity

We derive explicit expressions for a specific subclass of Jackiw-Teitelboim (JT) gravity bilocal correlators, corresponding to degenerate Virasoro representations. On the disk, these degenerate correlators are structurally simple, and they allow us to shed light on the 1/C Schwarzian bilocal perturbation series. In particular, we prove that the series is asymptotic for generic weight h ∉ −ℕ/2. Inspired by its minimal string ancestor, we propose an expression for higher genus corrections to the degenerate correlators. We discuss the extension to the mathcal{N} = 1 super JT model. On the disk, we similarly derive properties of the 1/C super-Schwarzian perturbation series, which we independently develop as well. As a byproduct, it is shown that JT supergravity saturates the chaos bound λL = 2π/β at first order in 1/C. We develop the fixed-length amplitudes of Liouville supergravity at the level of the disk partition function, the bulk one-point function and the boundary two-point functions. In particular we compute the minimal superstring fixed length boundary two-point functions, which limit to the super JT degenerate correlators. We give some comments on higher topology at the end.

Gravitational edge modes: from Kac–Moody charges to Poincaré networks

We revisit the canonical framework for general relativity in its connection-vierbein formulation, recasting the Gauss law, the Bianchi identity and the space diffeomorphism bulk constraints as conservation laws for boundary surface charges, respectively electric, magnetic and momentum charges. Partitioning the space manifold into 3D regions glued together through their interfaces, we focus on a single domain and its punctured 2D boundary. The punctures carry a ladder of Kac–Moody edge modes, whose 0-modes represent the electric and momentum charges while the higher modes describe the stringy vibration modes of the 1D-boundary around each puncture. In particular, this allows to identify missing observables in the discretization scheme used in loop quantum gravity (LQG) and leads to an enhanced theory upgrading spin networks to tube networks carrying Virasoro representations. In the limit where the tubes are contracted to 1D links and the string modes neglected, we do not just recover LQG but obtain a more general structure: Poincaré charge networks, which carry a representation of the 3D diffeomorphism boundary charges on top of the fluxes and gauge transformations.

Berry phases on Virasoro orbits

We point out that unitary representations of the Virasoro algebra contain Berry phases obtained by acting on a primary state with conformal transformations that trace a closed path on a Virasoro coadjoint orbit. These phases can be computed exactly thanks to the Maurer-Cartan form on the Virasoro group, and they persist after combining left- and right-moving sectors. Thinking of Virasoro representations as particles in AdS3 dressed with boundary gravitons, the Berry phases associated with Brown-Henneaux diffeomorphisms provide a gravitational extension of Thomas precession.

BMS modules in three dimensions

We build unitary representations of the BMS algebra and its higher-spin extensions in three dimensions, using induced representations as a guide. Our prescription naturally emerges from an ultrarelativistic limit of highest-weight representations of Virasoro and [Formula: see text] algebras, which is to be contrasted with nonrelativistic limits that typically give nonunitary representations. To support this dichotomy, we also point out that the ultrarelativistic and nonrelativistic limits of generic [Formula: see text] algebras differ in the structure of their nonlinear terms.

Quantization of conical spaces in 3D gravity

We discuss the quantization and holographic aspects of a class of conical spaces in 2+1 dimensional pure AdS gravity. These appear as topological solitons in the Chern-Simons formulation of the theory and are closely related to the recently studied conical solutions in higher spin gravity. We discuss the classical fluctuations around these solutions, which form exceptional coadjoint orbits of the asymptotic Virasoro group. We argue that the quantization of these solutions leads to nonunitary representations of the Virasoro algebra, on account of their having boundary graviton fluctuations which lower the energy. We propose a framework to quantize them in a semiclassical expansion in the inverse central charge, which we use to compute their one-loop corrected energies. Interestingly, the resulting Virasoro representations contain a null vector, thus providing an appearance of Kac's degenerate representations, which are nonunitary at large central charge, in the context of gravity. We match the computed quantum corrections in the bulk with the properties of a class of primaries in Kac's classification.

SLE and Virasoro Representations: Fusion

We continue the study of null-vector equations in relation with partition functions of (systems of) Schramm-Loewner Evolutions (SLEs) by considering the question of fusion. Starting from $n$ commuting SLEs seeded at distinct points, the partition function satisfies $n$ null-vector equations (at level 2). We show how to obtain higher level null-vector equations by coalescing the seeds one by one. As an example, we extend Schramm's formula (for the position of a marked bulk point relatively to a chordal SLE trace) to an arbitrary number of SLE strands. The argument combines input from representation theory - the study of Verma modules for the Virasoro algebra - with regularity estimates, themselves based on hypoellipticity and stochastic flow arguments.

SLE and Virasoro Representations: Localization

We consider some probabilistic and analytic realizations of Virasoro highest-weight representations. Specifically, we consider measures on paths connecting points marked on the boundary of a (bordered) Riemann surface. These Schramm–Loewner evolution-type measures are constructed by the method of localization in path space. Their partition function (total mass) is the highest-weight vector of a Virasoro representation, and the action is given by Virasoro uniformization. We review the formalism of Virasoro uniformization, which allows to define a canonical action of Virasoro generators on functions (or sections) on a—suitably extended—Teichmüller space. Then we describe the construction of families of measures on paths indexed by marked bordered Riemann surfaces. Finally we relate these two notions by showing that the partition functions of the latter generate a highest-weight representation—the quotient of a reducible Verma module—for the former.

Virasoro representations with central charges 1/2 and 1 on the real neutral fermion Fock space F⊗1/2

We study a family of fermionic oscillator representations of the Virasoro algebra via 2-point-local Virasoro fields on the Fock space F⊗1/2 of a neutral (real) fermion. We obtain the decomposition of F⊗1/2 as a direct sum of irreducible highest weight Virasoro modules with central charge c = 1. As a corollary we obtain the decomposition of the irreducible highest weight Virasoro modules with central charge c = 1/2 into irreducible highest weight Virasoro modules with central charge c = 1. As an application we show how positive sum (fermionic) character formulas for the Virasoro modules of charge c = 1/2 follow from these decompositions.

Stochastic processes with ZN symmetry and complex Virasoro representations. The partition functions

In a previous letter (Alcaraz F C et al 2014 J. Phys. A: Math. Theor. 47 212003) we have presented numerical evidence that a Hamiltonian expressed in terms of the generators of the periodic Temperley–Lieb algebra has, in the finite-size scaling limit, a spectrum given by representations of the Virasoro algebra with complex highest weights. This Hamiltonian defines a stochastic process with a ZN symmetry. We give here analytical expressions for the partition functions for this system which confirm the numerics. For N even, the Hamiltonian has a symmetry which makes the spectrum doubly degenerate leading to two independent stochastic processes. The existence of a complex spectrum leads to an oscillating approach to the stationary state. This phenomenon is illustrated by an example.

Solving the 3d Ising Model with the Conformal Bootstrap II. $$c$$ c -Minimization and Precise Critical Exponents

We use the conformal bootstrap to perform a precision study of the operator spectrum of the critical 3d Ising model. We conjecture that the 3d Ising spectrum minimizes the central charge $$c$$ in the space of unitary solutions to crossing symmetry. Because extremal solutions to crossing symmetry are uniquely determined, we are able to precisely reconstruct the first several $$\mathbb {Z}_2$$ -even operator dimensions and their OPE coefficients. We observe that a sharp transition in the operator spectrum occurs at the 3d Ising dimension $$\Delta _\sigma = 0.518154(15)$$ , and find strong numerical evidence that operators decouple from the spectrum as one approaches the 3d Ising point. We compare this behavior to the analogous situation in 2d, where the disappearance of operators can be understood in terms of degenerate Virasoro representations.

Cyclic representations of the periodic Temperley–Lieb algebra, complex Virasoro representations and stochastic processes

An -dimensional representation of the periodic Temperley–Lieb algebra TLL(x) is presented. It is also a representation of the cyclic group ZN. We choose x = 1 and define a Hamiltonian as a sum of the generators of the algebra acting in this representation. This Hamiltonian gives the time evolution operator of a stochastic process. In the finite-size scaling limit, the spectrum of the Hamiltonian contains representations of the Virasoro algebra with complex highest weights. The N = 3 case is discussed in detail. We discuss briefly the consequences of the existence of complex Virasoro representations for the physical properties of the systems.

A physical approach to the classification of indecomposable Virasoro representations from the blob algebra

In the context of Conformal Field Theory (CFT), many results can be obtained from the representation theory of the Virasoro algebra. While the interest in Logarithmic CFTs has been growing recently, the Virasoro representations corresponding to these quantum field theories remain dauntingly complicated, thus hindering our understanding of various critical phenomena. We extend in this paper the construction of Read and Saleur (2007) [1,2], and uncover a deep relationship between the Virasoro algebra and a finite-dimensional algebra characterizing the properties of two-dimensional statistical models, the so-called blob algebra (a proper extension of the Temperley–Lieb algebra). This allows us to explore vast classes of Virasoro representations (projective, tilting, generalized staggered modules, etc.), and to conjecture a classification of all possible indecomposable Virasoro modules (with, in particular, L0 Jordan cells of arbitrary rank) that may appear in a consistent physical Logarithmic CFT where Virasoro is the maximal local chiral algebra. As by-products, we solve and analyze algebraically quantum-group symmetric XXZ spin chains and sl(2|1) supersymmetric spin chains with extra spins at the boundary, together with the “mirror” spin chain introduced by Martin and Woodcock (2003) [3].

Combinatorial expansions of conformal blocks

In a recent paper (arXiv:0906.3219) the representation of Nekrasov partition function in terms of nontrivial two-dimensional conformal field theory has been suggested. For non-vanishing value of the deformation parameter \epsilon=\epsilon_1+\epsilon_2 the instanton partition function is identified with a conformal block of Liouville theory with the central charge c = 1+ 6\epsilon^2/\epsilon_1\epsilon_2. If reversed, this observation means that the universal part of conformal blocks, which is the same for all two-dimensional conformal theories with non-degenerate Virasoro representations, possesses a non-trivial decomposition into sum over sets of the Young diagrams, different from the natural decomposition studied in conformal field theory. We provide some details about this intriguing new development in the simplest case of the four-point correlation functions.

Polynomial fusion rings of W-extended logarithmic minimal models

The countably infinite number of Virasoro representations of the logarithmic minimal model LM(p,p′) can be reorganized into a finite number of W-representations with respect to the extended Virasoro algebra symmetry W. Using a lattice implementation of fusion, we recently determined the fusion algebra of these representations and found that it closes, albeit without an identity for p>1. Here, we provide a fusion-matrix realization of this fusion algebra and identify a fusion ring isomorphic to it. We also consider various extensions of it and quotients thereof and introduce and analyze commutative diagrams with morphisms between the involved fusion algebras and the corresponding quotient polynomial fusion rings. One particular extension is reminiscent of the fundamental fusion algebra of LM(p,p′) and offers a natural way of introducing the missing identity for p>1. Working out explicit fusion matrices is facilitated by a further enlargement based on a pair of mutual Moore–Penrose inverses intertwining between the W-fundamental and enlarged fusion algebras.

Integrable boundary conditions and -extended fusion in the logarithmic minimal models

We consider the logarithmic minimal models as ‘rational’ logarithmic conformal field theories with extended symmetry. To make contact with the extended picture starting from the lattice, we identify 4p − 2 boundary conditions as specific limits of integrable boundary conditions of the underlying Yang–Baxter integrable lattice models. Specifically, we identify 2p integrable boundary conditions to match the 2p known irreducible -representations. These 2p extended representations naturally decompose into infinite sums of the irreducible Virasoro representations (r, s). A further 2p − 2 reducible yet indecomposable -representations of rank 2 are generated by fusion and these decompose as infinite sums of indecomposable rank-2 Virasoro representations. The fusion rules in the extended picture are deduced from the known fusion rules for the Virasoro representations of and are found to be in agreement with previous works. The closure of the fusion algebra on a finite number of representations in the extended picture is remarkable confirmation of the consistency of the lattice approach.

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.

Representations of the Twisted Heisenberg–Virasoro Algebra and the Full Toroidal Lie Algebras

An explicit construction of indecomposable modules for the twisted Heisenberg–Virasoro algebra and representations for the full toroidal Lie algebras are given.

Virasoro representations and fusion for general augmented minimal models

In this paper we present explicit results for the fusion of irreducible and higher rank representations in two logarithmically conformal models, the augmented c_{2,3} = 0 model as well as the augmented Yang-Lee model at c_{2,5} = -22/5. We analyse their spectrum of representations which is consistent with the symmetry and associativity of the fusion algebra. We also describe the first few higher rank representations in detail. In particular, we present the first examples of consistent rank 3 indecomposable representations and describe their embedding structure. Knowing these two generic models we also conjecture the general representation content and fusion rules for general augmented c_{p,q} models.

Virasoro representations on fusion graphs

For any non-unitary minimal model with central charge c(2, q) the path spaces associated to a certain fusion graph are isomorphic to the irreducible Virasoro highest weight modules.