Conjectured Z2-Orbifold Constructions of Self-Dual Conformal Field Theories at Central Charge 24 – the Neighborhood Graph

By considering constraints on the dimensions of the Lie algebra corresponding to the weight 1-states of Z₂ and Z₃ orbifold models arising from imposing the appropriate modular properties on the graded characters of the automorphisms on the underlying conformal field theory, we propose a set of constructions of all but one of the 71 self-dual meromorphic bosonic conformal field theories at central charge 24. In the Z₂ case, this leads to an extension of the neighborhood graph of the even self-dual lattices in 24 dimensions to conformal field theories, and we demonstrate that the graph becomes disconnected.     

Monstrous Branes

 We study D-branes in the bosonic closed string theory whose automorphism group is the Bimonster group (the wreath product of the Monster simple group with ℤ₂). We give a complete classification of D-branes preserving the chiral subalgebra of Monster invariants and show that they transform in a representation of the Bimonster. Our results apply more generally to self-dual conformal field theories which admit the action of a compact Lie group on both the left- and right-moving sectors. Received: 20 February 2002 / Accepted: 17 August 2002 Published online: 19 December 2002 Communicated by R.H. Dijkgraaf     

Heisenberg groups and noncommutative fluxes

We develop a group-theoretical approach to the formulation of generalized abelian gauge theories, such as those appearing in string theory and M-theory. We explore several applications of this approach. First, we show that there is an uncertainty relation which obstructs simultaneous measurement of electric and magnetic flux when torsion fluxes are included. Next, we show how to define the Hilbert space of a self-dual field. The Hilbert space is Z₂-graded and we show that, in general, self-dual theories (including the RR fields of string theory) have fermionic sectors. We indicate how rational conformal field theories associated to the two-dimensional Gaussian model generalize to (4k + 2)-dimensional conformal field theories. When our ideas are applied to the RR fields of string theory we learn that it is impossible to measure the K-theory class of a RR field. Only the reduction modulo torsion can be measured.     

Four-dimensional heterotic strings and conformal field theory

The techniques of (super) conformal field theory are applied to 4-dimensional heterotic string theories. We discuss certain aspects of 4-dimensional strings in the framework of the bosonic lattice approach such as the realization of superconformal symmetry, character valued partition functions, construction of vertex operators and ghost picture changing. As an application we compute all possible 3- and 4-point tree amplitudes of the massless fields and derive from them the low energy effective action of the massless modes. Some effects for the massless spectrum due to one-loop string effects are also mentioned.     

Hamiltonian Anomalies from Extended Field Theories

We develop a proposal by Freed to see anomalous field theories as relative field theories, namely field theories taking value in a field theory in one dimension higher, the anomaly field theory. We show that when the anomaly field theory is extended down to codimension 2, familiar facts about Hamiltonian anomalies can be naturally recovered, such as the fact that the anomalous symmetry group admits only a projective representation on the Hilbert space, or that the latter is really an abelian bundle gerbe over the moduli space. We include in the discussion the case of non-invertible anomaly field theories, which is relevant to six-dimensional (2, 0) superconformal theories. In this case, we show that the Hamiltonian anomaly is characterized by a degree 2 non-abelian group cohomology class, associated to the non-abelian gerbe playing the role of the state space of the anomalous theory. We construct Dai-Freed theories, governing the anomalies of chiral fermionic theories, and Wess-Zumino theories, governing the anomalies of Wess-Zumino terms and self-dual field theories, as extended field theories down to codimension 2.     

Integrable quantum field theories and conformal field theories from lattice models in the light-cone approach

The light-cone lattice approach to two-dimensional quantum field theories is generalized to a large class of vertex models with any number of possible states per link and any simple Lie group of symmetry. Starting from a given lattice model, different scaling limits are defined leading to conformal field theories or to massive integrable quantum field theories, for which the lattice hamiltonian, momentum and currents are constructed. For a large set of models, the complete mass spectrum is also exhibited. Our approach applies equally well to chiral fermionic theories (like the chiral Gross-Neveu) and to bosonic models like the principal chiral model.     

On bosonization of 2d conformal field theories

We show how bosonic (free field) representations for so-called degenerate conformal theories are built by singular vectors in Verma modules. Based on this construction, general expressions of conformal blocks are proposed. As an example, we describe new modules for the SL(2) Wess-Zumino-Witten model. They are, in fact, the simplest nontrivial modules in a full set of bosonized highest weight representations of the ŝl₂ algebra. The Verma and Wakimoto modules appear as boundary modules of this set. Our construction also yields a new kind of bosonization in 2d conformal field theories.     

Numerical study on schramm-loewner evolution in nonminimal conformal field theories

The Schramm-Loewner evolution (SLE) is a powerful tool to describe fractal interfaces in 2D critical statistical systems, yet the application of SLE is well established for statistical systems described by quantum field theories satisfying only conformal invariance, the so-called minimal conformal field theories (CFTs). We consider interfaces in Z(N) spin models at their self-dual critical point for N = 4 and N = 5. These lattice models are described in the continuum limit by nonminimal CFTs where the role of a ZN symmetry, in addition to the conformal one, should be taken into account. We provide numerical results on the fractal dimension of the interfaces which are SLE candidates for nonminimal CFTs. Our results are in excellent agreement with some recent theoretical predictions.     

Classification of conformal field theories based on Coulomb gases. Application to loop models

We present a method for classifying conformal field theories based on Coulomb gases (bosonic free-field construction). Given a particular geometric configuration of the screening charges, we give necessary conditions for the existence of degenerate representations and for the closure of the vertex-operator algebra. The resulting classification contains, but is more general than, the standard one based on classical Lie algebras. We then apply the method to the Coulomb gas theory for the two-flavoured loop model of Jacobsen and Kondev. The purpose of the study is to clarify the relation between Coulomb gas models and conformal field theories with extended symmetries.     

Operator algebras and conformal field theory

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 III₁ 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.     

Twisted parafermions

A new type of nonlocal currents (quasi-particles), which we call twisted parafermions, and its corresponding twisted Z-algebra are found. The system consists of one spin-1 bosonic field and six nonlocal fields of fractional spins. Jacobi-type identities for the twisted parafermions are derived, and a new conformal field theory is constructed from these currents. As an application, a parafermionic representation of the twisted affine current algebra A⁽²⁾₂ is given.     

Modular invariance and the spectrum of two dimensional conformal field theories

Modular invariance has recently emerged as a powerful tool in conformal field theory. In conjunction with the representation theory of infinite dimensional Lie algebras, the study of modular invariance gave the spectrum of several families of theories. These include the minimal conformal models (Cardy and others), WZW theories which describe string propagation on group manifolds (Gepner and Witten) and parafermionic field theories (Gepner and Qiu). The minimal conformal models models were shown to be a product of two SU(2) WZW theories (Gepner). These results represent a step towards a complete classification of conformal field theories, an important goal both for the study of critical phenomena and string theory.     

Dual equivalence between self-dual and Maxwell–Chern–Simons models coupled to dynamical U(1) charged matter

We study the equivalence between the self-dual and the Maxwell–Chern–Simons (MCS) models coupled to dynamical, U(1) charged matter, both fermionic and bosonic. This is done through an iterative procedure of gauge embedding that produces the dual mapping of the self-dual vector field theory into a Maxwell–Chern–Simons version. In both cases, to establish this equivalence a current–current interaction term is needed to render the matter sector unchanged. Moreover, the minimal coupling of the original self-dual model is replaced by a non-minimal magnetic like coupling in the MCS side. Unlike the fermionic instance however, in the bosonic example the dual mapping proposed here leads to a Maxwell–Chern–Simons theory immersed in a field dependent medium.     

New boundary conformal field theories indexed by the simply laced Lie algebras

We consider the field theory of N massless bosons which are free except for an interaction localized on the boundary of their (1+1)-dimensional world. The boundary action is the sum of two pieces: a periodic potential and a coupling to a uniform abelian gauge field. Such models arise in open-string theory and dissipative quantum mechanics, and possibly in edge state tunneling in the fractional quantized Hall effect. We explicitly show that conformal invariance is unbroken for certain special choices of the gauge field and the periodic potential. These special cases are naturally indexed by semi-simple, simply laced Lie algebras. For each such algebra, we have a discrete series of conformally invariant theories where the potential and gauge field are conveniently given in terms of the weight lattice of the algebra. We compute the exact boundary state for these theories, which explicitly shows the group structure. The partition function and correlation functions are easily computed using the boundary state result.     

Asymptotic Completeness for Infraparticles in Two-Dimensional Conformal Field Theory

We formulate a new concept of asymptotic completeness for two-dimensional massless quantum field theories in the spirit of the theory of particle weights. We show that this concept is more general than the standard particle interpretation based on Buchholz’ scattering theory of waves. In particular, it holds in any chiral conformal field theory in an irreducible product representation and in any completely rational conformal field theory. This class contains theories of infraparticles to which the scattering theory of waves does not apply.     

Entanglement and topological interfaces

In this paper we consider entanglement entropies in two‐dimensional conformal field theories in the presence of topological interfaces. Tracing over one side of the interface, the leading term of the entropy remains unchanged. The interface however adds a subleading contribution, which can be interpreted as a relative (Kullback‐Leibler) entropy with respect to the situation with no defect inserted. Reinterpreting boundaries as topological interfaces of a chiral half of the full theory, we rederive the left/right entanglement entropy in analogy with the interface case. We discuss WZW models and toroidal bosonic theories as examples.     

Non-supersymmetric gauge theories from D-branes in type 0 string theory

We construct non-supersymmetric four-dimensional gauge theories arising as effective theories of D-branes placed on various singularities in Type 0B string theory. We mostly focus on models which are conformal in the large-N limit and present both examples appearing on self-dual D3-branes on orbifold singularities and examples including orientifold planes. Moreover, we derive type 0 Hanany-Witten setups with NS 5-branes intersected by D-branes and the corresponding rules for determining the massless spectra. Finally, we discuss possible duality symmetries (Seiberg duality) for non-supersymmetric gauge theories within this framework.     

New principles for string/membrane unification

The target space theory of the N = (2,1) heterotic string may be interpreted as a theory of gravity coupled to matter in either 1 + 1 or 2 + 1 dimensions. Among the target space theories in 1 + 1 dimensions are the bosonic, type II, and heterotic string world-sheet field theories in a physical gauge. The (2 + 1)-dimensional version describes a consistent quantum theory of supermembranes in 10 + 1 dimensions. The unifying framework for all of these vacua is a theory of (2 + 2)-dimensional self-dual geometries embedded in 10 + 2 dimensions. There are also indications that the N = (2,1) string describes the strong-coupling dynamics of compactifications of critical string theories to two dimensions, and may lead to insights about the fundamental degrees of freedom of the theory.     

Critical boundary sine-Gordon revisited

We revisit the exact solution of the two space-time dimensional quantum field theory of a free massless boson with a periodic boundary interaction and self-dual period. We analyze the model by using a mapping to free fermions with a boundary mass term originally suggested in Ref. [J. Polchinski, L. Thorlacius, Phys. Rev. D 50 (1994) 622]. We find that the entire SL (2, C) family of boundary states of a single boson are boundary sine-Gordon states and we derive a simple explicit expression for the boundary state in fermion variables and as a function of sine-Gordon coupling constants. We use this expression to compute the partition function. We observe that the solution of the model has a strong–weak coupling generalization of T-duality. We then examine a class of recently discovered conformal boundary states for compact bosons with radii which are rational numbers times the self-dual radius. These have simple expression in fermion variables. We postulate sine-Gordon-like field theories with discrete gauge symmetries for which they are the appropriate boundary states.     

Realizations of pseudo bosonic theories with non-diagonal automorphisms

Pseudo conformal field theories are theories with the same fusion rules, but different modular matrix as some conventional field theory. One of the authors defined these and conjectured that, for bosonic systems, they can all be realized by some actual RCFT, which is that of free bosons. We complete the proof here by treating the non-diagonal automorphism case. It is shown that for characteristic p≠2, they are equivalent to a diagonal case, fully classified in our previous publication. For p=2ⁿ we realize the non-diagonal cases, establishing this theorem.