The operator algebra of orbifold models (original) (raw)

An operator formulation of orbifold conformal field theory

Communications in Mathematical Physics, 1990

In many applications of conformal field theory one encounters twisted conformal fields, i.e. fields which have branch cut singularities on the relevant Riemann surfaces. We present a geometrical framework describing twisted conformal fields on Riemann surfaces of arbitrary genus which is alternative to the standard method of coverings. We further illustrate the theory of twisted Grassmannians and its relation with the representation theory of the twisted oscillator algebras. As an application of the above, we expound an operator formalism for orbifold strings.

Affine Orbifolds and Rational Conformal Field Theory Extensions of W 1+&#x0221E

Communications in Mathematical Physics, 1997

Chiral orbifold models are defined as gauge field theories with a finite gauge group Γ. We start with a conformal current algebra A associated with a connected compact Lie group G and a negative definite integral invariant bilinear form on its Lie algebra. Any finite group Γ of inner automorphisms or A (in particular, any finite subgroup of G) gives rise to a gauge theory with a chiral subalgebra A Γ ⊂ A of local observables invariant under Γ. A set of positive energy A Γ modules is constructed whose characters span, under some assumptions on Γ, a finite dimensional unitary representation of SL(2, Z). We compute their asymptotic dimensions (thus singling out the nontrivial orbifold modules) and find explicit formulae for the modular transformations and hence, for the fusion rules.

Affine Orbifolds and Rational Conformal Field Theory Extensions of W1+8

Communications in Mathematical Physics, 1997

Chiral orbifold models are defined as gauge field theories with a finite gauge group Gamma\GammaGamma. We start with a conformal current algebra A associated with a connected compact Lie group G and a negative definite integral invariant bilinear form on its Lie algebra. Any finite group Gamma\GammaGamma of inner automorphisms or A (in particular, any finite subgroup of G) gives rise to a gauge theory with a chiral subalgebra AGammasubsetAA^{\Gamma}\subset AAGammasubsetA of local observables invariant under Gamma\GammaGamma. A set of positive energy AGammaA^{\Gamma}AGamma modules is constructed whose characters span, under some assumptions on Gamma\GammaGamma, a finite dimensional unitary representation of SL(2,Z)SL(2,Z)SL(2,Z). We compute their asymptotic dimensions (thus singling out the nontrivial orbifold modules) and find explicit formulae for the modular transformations and hence, for the fusion rules. As an application we construct a family of rational conformal field theory (RCFT) extensions of W1+inftyW_{1+\infty}W1+infty that appear to provide a bridge between two approaches to the quantum Hall effect.

Modular invariance of trace functions in orbifold theory

The goal of the present paper is to provide a mathematically rigorous foundation to certain aspects of rational orbifold conformal field theory, in other words the theory of rational vertex operator algebras and their automorphisms. Under a certain finiteness condition on a rational vertex operator algebra V which holds in all known examples, we determine the precise numbers of g-twisted sectors for any automorphism g of V of finite order. We prove that the trace functions and correlations functions associated with such twisted sectors are holomorphic functions in the upper half-plane and, under suitable conditions, afford a representations of the modular group of the type prescribed in string theory. We establish the rationality of conformal weights and central charge. In addition to conformal field theory itself, where our conclusions are required on physical grounds, there are applications to the generalized Moonshine conjectures of Conway-Norton-Queen and to equivariant elliptic...

The chiral ring of a symmetric orbifold and its large N limit

Journal of High Energy Physics

We analyze the chiral operator ring of the symmetric orbifold conformal field theory on the complex two-plane ℂ2. We compute the large N limit of the ring and exhibit its factorized leading order behaviour. We moreover calculate all structure constants at the subleading and sub-subleading order. These features are coded as properties of the symmetric group and we review the relevant mathematical theorems on the product of conjugacy classes in the center of the group algebra. We illustrate the efficiency of the formalism by iteratively computing broad classes of higher point extremal correlators. We point out generalizations of our simplest of models and argue that our combinatorial analysis is relevant to the organization of the large N perturbation theory of generic symmetric orbifolds.

The nonperturbative analysis of background duality in orbifold conformal field theory

Nuclear Physics B, 1991

We study duality in simple two-dimensional orbifold models with metric and axionic backgrounds at arbitrary orders in string perturbation theory. It is shown that duality involves certain linear relations among twist correlators at dual backgrounds which are independent of the perturbative order. Relying on methods of harmonic analysis, a nonperturbative representation (in the sense of string field theory) of a class of S-matrix elements is obtained in terms of a universal background-independent spectral distribution and non_holomomhic mndiOlar vertar functions of the background parameters. At the same time, we find new realizations of the Verlinde algebra and of hypergroups characterizing the S-matrix nonperturbatively. Various examples and calculations are worked out.

Systematic Approach to Cyclic Orbifolds

International Journal of Modern Physics A, 1998

We introduce an orbifold induction procedure which provides a systematic construction of cyclic orbifolds, including their twisted sectors. The procedure gives counterparts in the orbifold theory of all the current-algebraic constructions of conformal field theory and enables us to find the orbifold characters and their modular transformation properties.

Chiral family classification of fermionic heterotic orbifold models

Physics Letters B, 2007

Free fermionic construction of four dimensional string vacua, are related to the Z 2 × Z 2 orbifolds at special points in the moduli space, and yielded the most realistic three family string models to date. Using free fermionic construction techniques we are able to classify more than 10 10 string vacua by the net family and anti-family number. Using a montecarlo technique we find a bell shaped distribution that peaks at vanishing net number of chiral families. We also observe that ∼ 15% of the models have three net chiral families. In addition to mirror symmetry we find that the distribution exhibits a symmetry under the exchange of (spinor plus anti-spinor) representations with vectorial representations.

Calculation of threepoint functions of chiral primaries in a symmetric orbifold

2005

The AdS/CFT conjecture in string theory suggests that quantum gravity in Anti de Sitter space, as a low energy limit of some string theory configuration, is equivalent to a conformal field theory on the boundary. One of the simplest examples is the D1-D5 system that has a super-gravity limit in AdS3. It is argued that the boundary theory has a point in its moduli space that is a 2 dimensional CFT with as target space the symmetric orbifold of a 4 dimensional hyperkähler space. To compare both theories one has to study objects that do not depend on the coupling constants. On the CFT side these objects can be described from a conformal field theory point of view or by dimensional reduction to a certain cohomology of the target space. In this thesis the link between these two methods is studied. An explicit description of the cohomology ring is given and the benefits of each description is discussed.

Operator algebras and conformal field theory

Communications in Mathematical Physics, 1993

We define and study two-dimensional, chiral conformal field theory by the methods of algebraic field theory. We start by characterizing the vacuum sectors of such theories and show that, under very general hypotheses, their algebras of local observables are isomorphic to the unique hyperfinite type III 1 factor. The conformal net determined by the algebras of local observables is proven to satisfy Haag duality. The representation of the Moebius group (and presumably of the entire Virasoro algebra) on the vacuum sector of a conformal field theory is uniquely determined by the Tomita-Takesaki modular operators associated with its vacuum state and its conformal net. We then develop the theory of Moebius covariant representations of a conformal net, using methods of Doplicher, Haag and Roberts. We apply our results to the representation theory of loop groups. Our analysis is motivated by the desire to find a "background-independent" formulation of conformal field theories. Contents

The topological symmetric orbifold

Journal of High Energy Physics

We analyze topological orbifold conformal field theories on the symmetric product of a complex surface M. By exploiting the mathematics literature we show that a canonical quotient of the operator ring has structure constants given by Hurwitz numbers. This proves a conjecture in the physics literature on extremal correlators. Moreover, it allows to leverage results on the combinatorics of the symmetric group to compute more structure constants explicitly. We recall that the full orbifold chiral ring is given by a symmetric orbifold Frobenius algebra. This construction enables the computation of topological genus zero and genus one correlators, and to prove the vanishing of higher genus contributions. The efficient description of all topological correlators sets the stage for a proof of a topological AdS/CFT correspondence. Indeed, we propose a concrete mathematical incarnation of the proof, relating Gromow-Witten theory in the bulk to the cohomology of the Hilbert scheme on the boun...

Asymmetric non-Abelian orbifolds and model building

1996

The rules for the free fermionic string model construction are extended to include general non-abelian orbifold constructions that go beyond the real fermionic approach. This generalization is also applied to the asymmetric orbifold rules recently introduced. These non-abelian orbifold rules are quite easy to use. Examples are given to illustrate their applications.

Multi-instanton and string loop corrections in toroidal orbifold models

Journal of High Energy Physics, 2008

We analyze N = 2 (perturbative and non-perturbative) corrections to the effective theory in type I orbifold models where a dual heterotic description is available. These corrections may play an important role in phenomenological scenarios. More precisely, we consider two particular compactifications: the Bianchi-Sagnotti-Gimon-Polchinski orbifold and a freely-acting Z 2 × Z 2 orbifold with N = 1 supersymmetry and gauge group SO(q) × SO(32 − q). By exploiting perturbative calculations of the physical gauge couplings on the heterotic side, we obtain multi-instanton and one-loop string corrections to the Kähler potential and the gauge kinetic function for these models. The non-perturbative corrections appear as sums over relevant Hecke operators, whereas the one-loop correction to the Kähler potential matches the expression proposed in [1, 2]. We argue that these corrections are universal in a given class of models where target-space modular invariance (or a subgroup of it) holds.

Correlation Functions for M N / S N Orbifolds

Communications in Mathematical Physics, 2001

We develop a method for computing correlation functions of twist operators in the bosonic 2-d CFT arising from orbifolds M N /S N , where M is an arbitrary manifold. The path integral with twist operators is replaced by a path integral on a covering space with no operator insertions. Thus, even though the CFT is defined on the sphere, the correlators are expressed in terms of partition functions on Riemann surfaces with a finite range of genus g. For large N, this genus expansion coincides with a 1/N expansion. The contribution from the covering space of genus zero is 'universal' in the sense that it depends only on the central charge of the CFT. For 3-point functions we give an explicit form for the contribution from the sphere, and for the 4-point function we do an example which has genus zero and genus one contributions. The condition for the genus zero contribution to the 3-point functions to be non-vanishing is similar to the fusion rules for an SU(2) WZW model. We observe that the 3-point coupling becomes small compared to its large N limit when the orders of the twist operators become comparable to the square root of Nthis is a manifestation of the stringy exclusion principle.

Z/iVZ Conformal Field Theories

1990

We compute the modular properties of the possible genus-one characters of some Rational Conformal Field Theories starting from their fusion rules. We show that the possible choices of S matrices are indexed by some automorphisms of the fusion algebra. We also classify the modular invariant partition functions of these theories. This gives the complete list of modular invariant partition functions of Rational Conformal Field Theories with respect to the Aff level one algebra.

Threshold Corrections for General Orbifold Models

2016

We calculate the moduli dependent part of string one-loop threshold corrections to gauge couplings for the heterotic string theory compactified on abelian toroidal orbifolds, allowing for arbitrary discrete Wilson lines. We show that the knowledge of threshold corrections for any such compactification is equivalent to solving a class of integrals. We solve a sub-class of these integrals and show how any model can be mapped onto this class by fractional linear transformations of its fixed plane moduli. Modular symmetries of the final expression are discussed.

Quantum Matrix Models for Simple Current Orbifolds

Nucl Phys B, 2003

An algebraic formulation of the stringy geometry on simple current orbifolds of the WZW models of type A_N is developed within the framework of Reflection Equation Algebras, REA_q(A_N). It is demonstrated that REA_q(A_N) has the same set of outer automorphisms as the corresponding current algebra A^{(1)}_N which is crucial for the orbifold construction. The CFT monodromy charge is naturally identified within the algebraic framework. The ensuing orbifold matrix models are shown to yield results on brane tensions and the algebra of functions in agreement with the exact BCFT data.

Higher twisted sector couplings of ZN orbifolds

Nuclear Physics B, 1993

We derive the basic correlation functions of twist fields coming from arbitrary twisted sectors in symmetric ZNZ_NZN orbifold conformal field theories, keeping all the admissible marginal perturbations, in particular those corresponding to the antisymmetric tensor background field. This allows a thorough investigation of modular symmetries in this type of string compactification. Such a study is explicitly carried out for the group generated by duality transformations. Thus, apart from being of phenomenological use, our couplings are also interesting from the mathematical point of view as they represent automorphic functions for a large class of discrete groups.

Z/NZ Conformal Field Theories

Communications in Mathematical Physics, 1990

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Bethe ansatz equations for general orbifolds of Script N = 4 SYM

Journal of High Energy Physics, 2008

We consider the Bethe Ansatz Equations for orbifolds of Script N = 4 SYM w.r.t. an arbitrary discrete group. Techniques used for the Abelian orbifolds can be extended to the generic non-Abelian case with minor modifications. We also illustrate the interplay with the quiver gauge theory notation.

Edge excitations of paired fractional quantum Hall states

Physical Review B, 1996

The Hilbert spaces of the edge excitations of several ``paired'' fractional quantum Hall states, namely the Pfaffian, Haldane-Rezayi and 331 states, are constructed and the states at each angular momentum level are enumerated. The method is based on finding all the zero energy states for those Hamiltonians for which each of these known ground states is the exact, unique, zero-energy eigenstate of lowest angular momentum in the disk geometry. For each state, we find that, in addition to the usual bosonic charge-fluctuation excitations, there are fermionic edge excitations. The edge states can be built out of quantum fields that describe the fermions, in addition to the usual scalar bosons (or Luttinger liquids) that describe the charge fluctuations. The fermionic fields in the Pfaffian and 331 cases are a non-interacting Majorana (i.e., real Dirac) and Dirac field, respectively. For the Haldane-Rezayi state, the field is an anticommuting scalar. For this system we exhibit a chiral Lagrangian that has manifest SU(2) symmetry but breaks Lorentz invariance because of the breakdown of the spin statistics connection implied by the scalar nature of the field and the positive definite norm on the Hilbert space. Finally we consider systems on a cylinder where the fluid has two edges and construct the sectors of zero energy states, discuss the projection rules for combining states at the two edges, and calculate the partition function for each edge excitation system at finite temperature in the thermodynamic limit. It is pointed out that the conformal field theories for the edge states are examples of orbifold constructions.

Fourier transform and the Verlinde formula for the quantum double of a finite group

Journal of Physics A-mathematical and General, 1999

A Fourier transform S is defined for the quantum double D(G) of a finite group G. Acting on characters of D(G), S and the central ribbon element of D(G) generate a unitary matrix representation of the group SL(2,Z). The characters form a ring over the integers under both the algebra multiplication and its dual, with the latter encoding the fusion rules of D(G). The Fourier transform relates the two ring structures. We use this to give a particularly short proof of the Verlinde formula for the fusion coefficients.

Mirror symmetry and the moduli space for generic hypersurfaces in toric varieties

Nuclear Physics B, 1995

The moduli dependence of (2; 2) superstring compactications based on Calabi{ Yau hypersurfaces in weighted projective space has so far only been investigated for Fermat-type polynomial constraints. These correspond to Landau-Ginzburg orbifolds with c = 9 whose potential is a sum of A-type singularities. Here we consider the generalization to arbitrary quasi-homogeneous singularities at c = 9 . W e use mirror symmetry to derive the dependence of the models on the complexied K ahler moduli and check the expansions of some topological correlation functions against explicit genus zero and genus one instanton calculations. As an important application we give examples of how non-algebraic (\twisted") deformations can be mapped to algebraic ones, hence allowing us to study the full moduli space. We also study how moduli spaces can be nested in each other, thus enabling a (singular) transition from one theory to another. Following the recent w ork of Greene, Morrison and Strominger we show that this corresponds to black hole condensation in type II string theories compactied on Calabi-Yau manifolds.

U(1)×U(1)⋊Z_{2} Chern-Simons theory and Z_{4} parafermion fractional quantum Hall states

Physical Review B, 2010

We study U (1) × U (1) ⋊ Z2 Chern-Simons theory with integral coupling constants (k, l) and its relation to certain non-Abelian fractional quantum Hall (FQH) states. For the U (1) × U (1) ⋊ Z2 Chern-Simons theory, we show how to compute the dimension of its Hilbert space on genus g surfaces and how this yields the quantum dimensions of topologically distinct excitations. We find that Z2 vortices in the U (1) × U (1) ⋊ Z2 Chern-Simons theory carry non-Abelian statistics and we show how to compute the dimension of the Hilbert space in the presence of n pairs of Z2 vortices on a sphere. These results allow us to show that l = 3 U (1) × U (1) ⋊ Z2 Chern-Simons theory is the low energy effective theory for the Z4 parafermion (Read-Rezayi) fractional quantum Hall states, which occur at filling fraction ν = 2 2k−3 . The U (1) × U (1) ⋊ Z2 theory is more useful than an alternative SU (2)4 × U (1)/U (1) Chern-Simons theory because the fields are more closely related to physical degrees of freedom of the electron fluid and to an Abelian bilayer phase on the other side of a two-component to single-component quantum phase transition. We discuss the possibility of using this theory to understand further phase transitions in FQH systems, especially the ν = 2/3 phase diagram. phase transition at ν = 2/3, the phase diagram of which has attracted both theoretical and experimental attention.

Phase transitions in Z_{N} gauge theory and twisted Z_{N} topological phases

Physical Review B, 2012

We find a series of non-Abelian topological phases that are separated from the deconfined phase of ZN gauge theory by a continuous quantum phase transition. These non-Abelian states, which we refer to as the "twisted" ZN states, are described by a recently studied U (1) × U (1) ⋊ Z2 Chern-Simons (CS) field theory. The U (1) × U (1) ⋊ Z2 CS theory provides a way of gauging the global Z2 electric-magnetic symmetry of the Abelian ZN phases, yielding the twisted ZN states. We introduce a parton construction to describe the Abelian ZN phases in terms of integer quantum Hall states, which then allows us to obtain the non-Abelian states from a theory of Z2 fractionalization. The non-Abelian twisted ZN states do not have topologically protected gapless edge modes and, for N > 2, break time-reversal symmetry.

Bilayer quantum Hall phase transitions and the orbifold non-Abelian fractional quantum Hall states

Physical Review B, 2011

We study continuous quantum phase transitions that can occur in bilayer fractional quantum Hall (FQH) systems as the interlayer tunneling and interlayer repulsion are tuned. We introduce a slave-particle gauge theory description of a series of continuous transitions from the (ppq) Abelian bilayer states to a set of non-Abelian FQH states, which we dub the orbifold FQH states, of which the Z4 parafermion (Read-Rezayi) state is a special case. This provides an example in which Z2 electron fractionalization leads to non-Abelian topological phases. The naive "ideal" wave functions and ideal Hamiltonians associated with these orbifold states do not in general correspond to incompressible phases, but instead lie at a nearby critical point. We discuss this unusual situation from the perspective of the pattern of zeros/vertex algebra frameworks and discuss implications for the conceptual foundations of these approaches. Due to the proximity in the phase diagram of these non-Abelian states to the (ppq) bilayer states, they may be experimentally relevant, both as candidates for describing the plateaus in single-layer systems at filling fraction 8/3 and 12/5, and as a way to tune to non-Abelian states in double-layer or wide quantum wells.

The Outer-Automorphic WZW Orbifolds on (2n), including Five Triality Orbifolds on (8)

Journal of High Energy Physics, 2002

Following recent advances in the local theory of current-algebraic orbifolds we present the basic dynamics — including the twisted KZ equations — of each twisted sector of all outer-automorphic WZW orbifolds on (2n). Physics-friendly cartesian bases are used throughout, and we are able in particular to assemble two ℤ3 triality orbifolds and three S3 triality orbifolds on (8).

A new rational conformal field theory extension of the fully degenerate W 1+∞ (m)

Journal of High Energy Physics, 2006

A new Rational Conformal Field Theory extension of the fully degenerate W (m) 1+∞ Abstract We found new identities among the Dedekind η-function, the characters of the W m algebra and those of the level 1 affine Lie algebra su(m) 1 . They allow to characterize the Z m -orbifold of the m-component free bosons u(1) Km,p (our theory TM) as an extension of the fully degenerate representations of W (m) 1+∞ . In particular, TM is proven to be a Γ θ -RCFT extension of the chiral fully degenerate W (m) 1+∞ .

Coset construction and character sum rules for the doubly extended N = 4 superconformal algebras

Nuclear Physics B, 1993

Character sumrules associated with the realization of the N = 4 superconformal algebraÃ γ on manifolds corresponding to the group cosets SU(3)k + /U(1) are derived and developed as an important tool in obtaining the modular properties ofÃ γ characters as well as information on certain extensions of that algebra. Their structure strongly suggests the existence of rational conformal field theories with central charges in the range 1 ≤ c ≤ 4. The corresponding characters appear in the massive sector of the sumrules and are completely specified in terms of the characters for the parafermionic theory SU(3)/(SU(2)×U(1)) and in terms of the branching functions of masslessÃ γ characters into SU(2)k + × SU(2) 1 characters.

A matrix S for all simple current extensions

Nuclear Physics B, 1996

A formula is presented for the modular transformation matrix S for any simple current extension of the chiral algebra of a conformal field theory. This provides in particular an algorithm for resolving arbitrary simple current fixed points, in such a way that the matrix S we obtain is unitary and symmetric and furnishes a modular group representation. The formalism works in principle for any conformal field theory. A crucial ingredient is a set of matrices S J ab , where J is a simple current and a and b are fixed points of J. We expect that these input matrices realize the modular group for the torus one-point functions of the simple currents. In the case of WZW-models these matrices can be identified with the S-matrices of the orbit Lie algebras that were introduced recently in [1]. As a special case of our conjecture we obtain the modular matrix S for WZW-theories based on group manifolds that are not simply connected, as well as for most coset models. § In particular we do not study 'heterotic' invariants or fusion rule automorphisms, since our interest is in defining the matrix S for the chiral half of a theory.

Systematic Approach to Cyclic Orbifolds

International Journal of Modern Physics A, 1998

Twisted quantum double model of topological phases in two dimensions

Physical Review B, 2013

We propose a new discrete model-the twisted quantum double model-of 2D topological phases based on a finite group G and a 3-cocycle α over G. The detailed properties of the ground states are studied, and we find that the ground-state subspace can be characterized in terms of the twisted quantum double D α (G) of G. When α is the trivial 3-cocycle, the model becomes Kitaev's quantum double model based on the finite group G, in which the elementary excitations are known to be classified by the quantum double D(G) of G. Our model can be viewed as a Hamiltonian extension of the Dijkgraaf-Witten topological gauge theories to the discrete graph case with gauge group being a finite group. We also demonstrate a duality between a large class of Levin-Wen string-net models and certain twisted quantum double models, by mapping the string-net 6j symbols to the corresponding 3-cocycles. The paper is presented in a way such that it is accessible to a wide range of physicists.

Asymmetric non-Abelian orbifolds and model building

Physical Review D, 1996

The Drinfel'D Double and Twisting in Stringy Orbifold Theory

International Journal of Mathematics, 2009

This paper exposes the fundamental role that the Drinfel'd double D(k[G]) of the group ring of a finite group G and its twists Dβ(k[G]), β ∈ Z3(G,k*) as defined by Dijkgraaf–Pasquier–Roche play in stringy orbifold theories and their twistings. The results pertain to three different aspects of the theory. First, we show that G-Frobenius algebras arising in global orbifold cohomology or K-theory are most naturally defined as elements in the braided category of D(k[G])-modules. Secondly, we obtain a geometric realization of the Drinfel'd double as the global orbifold K-theory of global quotient given by the inertia variety of a point with a G action on the one hand and more stunningly a geometric realization of its representation ring in the braided category sense as the full K-theory of the stack [pt/G]. Finally, we show how one can use the co-cycles β above to twist the global orbifold K-theory of the inertia of a global quotient and more importantly, the stacky K-theory of a...

State counting on fibered CY 3-folds and the non-Abelian weak gravity conjecture

Journal of High Energy Physics, 2021

We extend the dictionary between the BPS spectrum of Heterotic strings and the one of F-/M-theory compactifications on K3 fibered Calabi-Yau 3-folds to cases with higher rank non-Abelian gauge groups and in particular to dual pairs between Heterotic CHL orbifolds and compactifications on Calabi-Yau 3-folds with a compatible genus one fibration. We show how to obtain the new supersymmetric index purely from the Calabi-Yau geometry by reconstructing the Noether-Lefschetz generators, which are vector-valued modular forms. There is an isomorphism between the latter objects and vector-valued lattice Jacobi forms, which relates them to the elliptic genera and twisted-twined elliptic genera of six- and five-dimensional Heterotic strings. The meromorphic Jacobi forms generate the dimensions of the refined cohomology of the Hilbert schemes of symmetric products of the fiber and allow us to refine the BPS indices in the fiber and therefore to obtain, conjecturally, actual state counts. Using ...

A toy black hole S-matrix in the D1-D5 CFT

Journal of High Energy Physics, 2013

To model the process of absorption and emission of quanta by an extremal D1-D5 black hole in the dual CFT, we consider transitions between different Ramond vacua via absorption and emission of chiral primaries. We compute the probabilities to reach different CFT states starting with a special Ramond vacuum, using techniques of the orbifold CFT. It is found that the processes involving the change of angular momentum by k units are suppressed as ∼ 1/N k .

Quantum symmetry and braid group statistics inG-spin models

Communications in Mathematical Physics, 1993

In two-dimensional lattice spin systems in which the spins take values in a finite group G we find a non-Abelian "parafermion" field of the form order x disorder that carries an action of the Hopf algebra ^(G), the double of G. This field leads to a "quantization" of the Cuntz algebra and allows one to define amplifying homomorphisms on the ^(G)-invariant subalgebra that create the ^(G)-charges and generalize the endomorphisms in the Doplicher-Haag-Roberts program. The so-obtained category of representations of the observable algebra is shown to be equivalent to the representation category of &(G). The representation of the braid group generated by the statistics operator and the corresponding statistics parameter are calculated in each sector.

On the quantum symmetry of the chiral Ising model

Nuclear Physics B, 1994

We introduce the notion of rational Hopf algebras that we think are able to describe the superselection symmetries of two dimensional rational quantum field theories. As an example we show that a six dimensional rational Hopf algebra H can reproduce the fusion rules, the conformal weights, the quantum dimensions and the representation of the modular group of the chiral Ising model. H plays the role of the global symmetry algebra of the chiral Ising model in the following sense: 1) a simple field algebra F and a representation π on H π of it is given, which contains the c = 1/2 unitary representations of the Virasoro algebra as subrepresentations; 2) the embedding U : H → B(H π) is such that the observable algebra π(A) − is the invariant subalgebra of B(H π) with respect to the left adjoint action of H and U (H) is the commutant of π(A); 3) there exist H-covariant primary fields in B(H π), which obey generalized Cuntz algebra properties and intertwine between the inequivalent sectors of the observables.

Non-invertible global symmetries and completeness of the spectrum

Journal of High Energy Physics, 2021

It is widely believed that consistent theories of quantum gravity satisfy two basic kinematic constraints: they are free from any global symmetry, and they contain a complete spectrum of gauge charges. For compact, abelian gauge groups, completeness follows from the absence of a 1-form global symmetry. However, this correspondence breaks down for more general gauge groups, where the breaking of the 1-form symmetry is insufficient to guarantee a complete spectrum. We show that the correspondence may be restored by broadening our notion of symmetry to include non-invertible topological operators, and prove that their absence is sufficient to guarantee a complete spectrum for any compact, possibly disconnected gauge group. In addition, we prove an analogous statement regarding the completeness of twist vortices: codimension-2 objects defined by a discrete holonomy around their worldvolume, such as cosmic strings in four dimensions. We discuss how this correspondence is modified in vari...

Lectures on branes in curved backgrounds

Classical and Quantum Gravity, 2002

These lectures provide an introduction to the microscopic description of branes in curved backgrounds. After a brief reminder of the flat space theory, the basic principles and techniques of (rational) boundary conformal field theory are presented in the second lecture. The general formalism is then illustrated through a detailed discussion of branes on compact group manifolds. In the final lecture, many more recent developments are reviewed, including some results for non-compact target spaces.

Space, Matter and Interactions in a Quantum Early Universe Part I: Kac–Moody and Borcherds Algebras

Symmetry, 2021

We introduce a quantum model for the universe at its early stages, formulating a mechanism for the expansion of space and matter from a quantum initial condition, with particle interactions and creation driven by algebraic extensions of the Kac-Moody Lie algebra e 9 . We investigate Kac-Moody and Borcherds algebras, and we propose a generalization that meets further requirements that we regard as fundamental in quantum gravity.

The operator algebra of orbifold models (original) (raw)

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