Chiral Algebra Research Articles

Notes on the Verlinde formula in nonrational conformal field theories

We review and extend evidence for the validity of a generalized Verlinde formula, in particular, nonrational conformal field theories. We identify a subset of representations of the chiral algebra in nonrational conformal field theories that give rise to an analogue of the relation between modular $S$-matrices and fusion coefficients in rational conformal field theories. To that end we review and extend the Cardy-type brane calculations in bosonic and supersymmetric Liouville theory (and its duals) as well as in ${H}_{3}^{+}$. We analyze the three-point functions of Liouville theory and of ${H}_{3}^{+}$ in detail to directly identify the fusion coefficients from the operator product expansion. Moreover, we check the validity of a proposed generic formula for localized brane one-point functions in nonrational conformal field theories.

Bosonization in Higher Dimensions via Noncommutative Field Theory

We propose the bosonization of a many-body fermion theory in D spatial dimensions through a noncommutative field theory on a (2D-1)-dimensional space. This theory leads to a chiral current algebra over the noncommutative space and reproduces the correct perturbative Hilbert space and excitation energies for the fermions. The validity of the method is demonstrated by bosonizing a two-dimensional gas of fermions in a harmonic trap.

Twisted boundary states and representation of generalized fusion algebra

The mutual consistency of boundary conditions twisted by an automorphism group G of the chiral algebra is studied for general modular invariants of rational conformal field theories. We show that a consistent set of twisted boundary states associated with any modular invariant realizes a non-negative integer matrix representation (NIM-rep) of the generalized fusion algebra, an extension of the fusion algebra by representations of the twisted chiral algebra associated with the automorphism group G. We check this result for several concrete cases. In particular, we find that two NIM-reps of the fusion algebra for su ( 3 ) k ( k = 3 , 5 ) are organized into a NIM-rep of the generalized fusion algebra for the charge-conjugation automorphism of su ( 3 ) k . We point out that the generalized fusion algebra is non-commutative if G is non-Abelian and provide some examples for G ≅ S 3 . Finally, we give an argument that the graph fusion algebra associated with simple current extensions coincides with the generalized fusion algebra for the extended chiral algebra, and thereby explain that the graph fusion algebra contains the fusion algebra of the extended theory as a subalgebra.

Logarithmic torus amplitudes

AbstractFor the example of the logarithmic triplet theory at c = −2 the chiral vacuumtorus amplitudes are analysed. It is found that the space of these torus amplitudes isspanned by the characters of the irreducible representations, as well as a function thatcan be associated to the logarithmic extension of the vacuum representation. A fewimplications and generalisations of this result are discussed. 1. Introduction. During the last twenty years much has been understood about the struc-ture of rational conformal field theories. Rational conformal field theories are characterisedby the property that they have only finitely many irreducible highest weight representationsof the chiral algebra (or vertex operator algebra), and that every highest weight represen-tation is completely decomposable into irreducible representations. The structure of thesetheories is well understood: in particular, the characters of the irreducible representationstransform into one another under modular transformations [1] (see also [2]), and the modularS-matrix determines the fusion rules via the Verlinde formula [3]. (A general proof for thishas only recently been given in [4].)On the other hand, it is clear that rational conformal field theories are rather special, andit is therefore important to understand the structure of more general classes of conformalfield theories. One such class are the (rational) logarithmic theories that possess only finitelymany indecomposable representations, but for which not all highest weight representationsare completely decomposable. The name ‘logarithmic’ comes from the fact that their chiralcorrelation functions typically have logarithmic branch cuts. The first example of a (non-rational) logarithmic conformal field theory was found in [5] (see also [6]), and the firstrational example (that shall also concern us in this paper) was constructed in [7]; for somerecent reviews see [8, 9, 10]. From a physics point of view, logarithmic conformal fieldtheories appear naturally in various models of statistical physics, for example in the theory

Conformal field theories associated to regular chiral vertex operator algebras, I: Theories over the projective line

Given a chiral vertex operator algebra satisfying a suitable finiteness condition with semisimplicity of the zero-mode algebra as well as a regularity condition for induced modules, we construct conformal field theories over the projective line and prove the factorization theorem. We appropriately generalize the arguments in [TUY] so that we are able to define sheaves of conformal blocks and study them in detail.

2D superconductivity: Classification of universality classes by infinite symmetry

I consider superconducting condensates which become incompressible in the infinite gap limit. Classical 2D incompressible fluids possess the dynamical symmetry of area-preserving diffeomorphisms. I show that the corresponding infinite dynamical symmetry of 2D superconducting fluids is the coset W 1 + ∞ ⊗ W ¯ 1 + ∞ U ( 1 ) diagonal , with W 1 + ∞ the chiral algebra of quantum area-preserving diffeomorphisms and I derive its minimal models. These define a discrete set of 2D superconductivity universality classes which fall into two main categories: conventional superconductors with their vortex excitations and unconventional superconductors. These are characterized by a broken U ( 1 ) vector ⊗ U ( 1 ) axial symmetry and are labeled by an integer level m. They possess neutral spinon excitations of fractional spin and statistics S = θ 2 π = m − 1 2 m , which carry also an SU ( m ) isospin quantum number; this hidden SU ( m ) symmetry implies that these anyon excitations are non-Abelian. The simplest unconventional superconductor is realized for m = 2 : in this case the spinon excitations are semions (half-fermions). My results show that spin–charge separation in 2D superconductivity is a universal consequence of the infinite symmetry of the ground state. This infinite symmetry and its superselection rules realize a quantum protectorate in which the neutral spinons can survive even as soft modes on a rigid, spinless charge condensate.

The third parafermionic chiral algebra with the symmetry [formula omitted

We have constructed the parafermionic chiral algebra with the principal parafermionic fields Ψ,Ψ+ having the conformal dimension ΔΨ=8/3 and realizing the symmetry Z3.

Generalised discrete torsion and mirror symmetry for G2manifolds

A generalisation of discrete torsion is introduced in which different discrete torsion phases are considered for the different fixed points or twist fields of a twisted sector. The constraints that arise from modular invariance are analysed carefully. As an application we show how all the different resolutions of the T^7/Z_2^3 orbifold of Joyce have an interpretation in terms of such generalised discrete torsion orbifolds. Furthermore, we show that these manifolds are pairwise identified under G_2 mirror symmetry. From a conformal field theory point of view, this mirror symmetry arises from an automorphism of the extended chiral algebra of the G_2 compactification.

LOCAL FIELDS IN BOUNDARY CONFORMAL QFT

Conformal quantum field theory on the half-space x>0 of Minkowski space-time ("boundary CFT") is analyzed from an algebraic point of view, clarifying in particular the algebraic structure of local algebras and the bi-localized charge structure of local fields. The field content and the admissible boundary conditions are characterized in terms of a non-local chiral field algebra.

Symmetry breaking, permutation D-branes on group manifolds: boundary states and geometric description

We use the permutation symmetry between the product of several group manifolds in combination with orbifolds and T-duality to construct new classes of symmetry breaking branes on products of group manifolds. The resulting branes mix the submanifolds and break part of the diagonal chiral algebra of the theory. We perform a Langrangian analysis as well as a boundary CFT construction of these branes and find agreement between the two methods.

Conformal field theories with ZN and Lie algebra symmetries

We construct two-dimensional conformal field theories with a ZN symmetry, based on the second solution of Fateev–Zamolodchikov for the parafermionic chiral algebra. Primary operators are classified according to their transformation properties under the dihedral group (ZN×Z2, where Z2 stands for the ZN charge conjugation), as singlets, ⌊(N−1)/2⌋ different doublets, and a disorder operator. In an assumed Coulomb gas scenario, the corresponding vertex operators are accommodated by the Kac table based on the weight lattice of the Lie algebra B(N−1)/2 when N is odd, and DN/2 when N is even. The unitary theories are representations of the coset SOn(N)×SO2(N)/SOn+2(N), with n=1,2,…. We suggest that physically they realize the series of multicritical points in statistical systems having a ZN symmetry.

Topologizations of Chiral Representations

We analyze and compare two families of topologies that have been proposed for representation spaces of chiral algebras by Huang and Gaberdiel & Goddard respectively. We show, in particular, that for suitable pairs the topology of Gaberdiel & Goddard is coarser. We also give a new proof that the chiral two-point blocks are continuous in the topology of Huang.

Quantum mechanical considerations on the algebraic structure of central molecular chirality.

The chiral algebra of tetrahedral molecules, derived from Fischer's projections, is discussed in the framework of quantum mechanics. A "quantum chiral algebra" is obtained whose operators, acting as rotations or inversions, commute with the Hamiltonian of the system. It is shown that energy and chirality eigenstates are strictly related through the Heisenberg relations, while chirality operators "conserve" parity eigenstates.

Exact Solutions in SFT and Marginal Deformation in BCFT

In this note we will study solution of open bosonic string field theory based on action of operators from chiral algebra of boundary conformal field theory on identity element of string field theory star algebra. We will demonstrate that the string field theory action for fluctuation fields around this classical solution can be mapped to the string field theory action defined through the new boundary conformal field theory that arises from the original one through the marginal deformation inserted on the world-sheet boundary.

Parafermionic theory with the symmetry ZN for N even

Following our previous papers [Nucl. Phys. B 656 (2003) 259, Nucl. Phys. B 664 (2003) 477] we complete the construction of the parafermionic theory with the symmetry Z N based on the second solution of Fateev–Zamolodchikov for the corresponding parafermionic chiral algebra. In the present paper we construct the Z N parafermionic theory for N even. Primary operators are classified according to their transformation properties under the dihedral group ( Z N × Z 2, where Z 2 stands for the Z N charge conjugation), as two singlets, doublet 1,2,…, N/2−1, and a disorder operator. In an assumed Coulomb gas scenario, the corresponding vertex operators are accommodated by the Kac table based on the weight lattice of the Lie algebra D N/2 . The unitary theories are representations of the coset SO n ( N)× SO 2( N)/ SO n+2 ( N), with n=1,2,…. We suggest that physically they realise the series of multicritical points in statistical systems having a Z N symmetry.

Twisted modules over vertex algebras on algebraic curves

We extend the geometric approach to vertex algebras developed by the first author to twisted modules, allowing us to treat orbifold models in conformal field theory. Let V be a vertex algebra, H a finite group of automorphisms of V, and C an algebraic curve such that H⊂Aut( C). We show that a suitable collection of twisted V-modules gives rise to a section of a certain sheaf on the quotient X= C/ H. We introduce the notion of conformal blocks for twisted modules, and analyze them in the case of the Heisenberg and affine Kac–Moody vertex algebras. We also give a chiral algebra interpretation of twisted modules.

C = 2 Rational Toroidal Conformal Field Theories via the Gauss Product

We find a concise relation between the moduli $\tau, \rho$ of a rational Narain lattice $\Gamma(\tau,\rho)$ and the corresponding momentum lattices of left and right chiral algebras via the Gauss product. As a byproduct, we find an identity which counts the cardinality of a certain double coset space defined for isometries between the discriminant forms of rank two lattices.

Extended chiral algebras and the emergence of SU(2) quantum numbers in the Coulomb gas

We study a set of chiral symmetries contained in degenerate operators beyond the `minimal' sector of the c(p,q) models. For the operators h_{(2j+2)q-1,1}=h_{1,(2j+2)p-1} at conformal weight [ (j+1)p-1 ][ (j+1)q -1 ], for every 2j \in N, we find 2j+1 chiral operators which have quantum numbers of a spin j representation of SU(2). We give a free-field construction of these operators which makes this structure explicit and allows their OPEs to be calculated directly without any use of screening charges. The first non-trivial chiral field in this series, at j=1/2, is a fermionic or para-fermionic doublet. The three chiral bosonic fields, at j=1, generate a closed W-algebra and we calculate the vacuum character of these triplet models.

Conformal Covariance Subalgebras

We give a direct Lie algebraic characterisation of conformal inclusions of chiral current algebras associated with compact, reductive Lie algebras. We use straightforward quantum field theoretic arguments and prove a long standing conjecture of Schellekens and Warner on grounds of unitarity and positivity of energy. We explore the structures found to characterise ``conformal covariance subalgebras'' and ``coset current algebras''.

Parafermionic theory with the symmetry ZN, for N odd

We construct a parafermionic conformal theory with the symmetry Z N , for N odd, based on the second solution of Fateev–Zamolodchikov for the corresponding parafermionic chiral algebra. Primary operators are classified according to their transformation properties under the dihedral group D N , as singlet, doublet 1,2,…, N−1 2 , and disorder operators. In an assumed Coulomb gas scenario, the corresponding vertex operators are accommodated by the weight lattice of the Lie algebra B ( N−1)/2 . The unitary theories are representations of the coset SO n ( N)× SO 2( N)/ SO n+2 ( N), with n=1,2,… . Physically, they realise the series of multicritical points in statistical theories having a D N symmetry.