Chiral Algebra Research Articles

Parafermionic theory with the symmetry Z5

A parafermionic conformal theory with the symmetry Z 5 is constructed, based on the second solution of Fateev–Zamolodchikov for the corresponding parafermionic chiral algebra. The primary operators of the theory, which are the singlet, doublet 1, doublet 2, and disorder operators, are found to be accommodated by the weight lattice of the classical Lie algebra B 2. The finite Kac tables for unitary theories are defined and the formula for the conformal dimensions of primary operators is given.

D-branes in an asymmetric orbifold

We consider the asymmetric orbifold that is obtained by acting with T-duality on a 4-torus, together with a shift along an extra circle. The chiral algebra of the resulting theory has non-trivial outer automorphisms that act as permutations on its simple factors. These automorphisms play a crucial role for constructing D-branes that couple to the twisted sector of the orbifold.

Proving the PP-Wave/CFT2Duality

We study the duality between IIB string theory on a pp-wave background, arising as a Penrose limit of the AdS_3 times S^3 times M, where M is T^4 (or K3), and the 2D CFT which is given by the N=(4,4) orbifold (M)^N/S_N, resolved by a blowing-up mode. After analizying the action of the supercharges on both sides, we establish a correspondence between the states of the two theories. In particular and for the T^4 case, we identify both massive and massless oscillators on the pp-wave, with certain classes of excited states in the resolved CFT carrying large R-charge n. For the former, the excited states involve fractional modes of the generators of the N=4 chiral algebra acting on the Z_n ground states. For the latter, they involve, fractional modes of the U(1)^4_L times U(1)^4_R super-current algebra acting on the Z_n ground states. By using conformal perturbation theory we compute the leading order correction to the conformal dimensions of the first class of states, due to the presence of the blowing up mode. We find agreement, to this order, with the corresponding spectrum of massive oscillators on the pp-wave. We also discuss the issue of higher order corrections.

Axial-vector current in nuclear many-body physics

Weak-interaction currents are studied in a recently proposed effective field theory of the nuclear many-body problem. The Lorentz-invariant effective field theory contains nucleons, pions, isoscalar scalar ($\sigma$) and vector ($\omega$) fields, and isovector vector ($\rho$) fields. The theory exhibits a nonlinear realization of $SU(2)_L \times SU(2)_R$ chiral symmetry and has three desirable features: it uses the same degrees of freedom to describe the axial-vector current and the strong-interaction dynamics, it satisfies the symmetries of the underlying theory of quantum chromodynamics, and its parameters can be calibrated using strong-interaction phenomena, like hadron scattering or the empirical properties of finite nuclei. Moreover, it has recently been verified that for normal nuclear systems, it is possible to systematically expand the effective lagrangian in powers of the meson fields (and their derivatives) and to reliably truncate the expansion after the first few orders. Here it is shown that the expressions for the axial-vector current, evaluated through the first few orders in the field expansion, satisfy both PCAC and the Goldberger--Treiman relation, and it is verified that the corresponding vector and axial-vector charges satisfy the familiar chiral charge algebra. Explicit results are derived for the Lorentz-covariant, axial-vector, two-nucleon amplitudes, from which axial-vector meson-exchange currents can be deduced.

Boundary states and symplectic fermions

We investigate the set of boundary states in the symplectic fermion description of the logarithmic conformal field theory with central charge c=−2. We show that the thus constructed states correspond exactly to those derived under the restrictions of the maximal chiral symmetry algebra for this model, W(2,3,3,3). This connects our previous work to the coherent state approach of Kawai and Wheater.

On a q-Analogue of the McKay Correspondence and the ADE Classification of [formula omitted]2 Conformal Field Theories

The goal of this paper is to give a category theory based definition and classification of “finite subgroups in U q ( s l 2)” where q= e πi/ l is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of U q ( s l 2); we show that this definition is a natural generalization of the notion of a subgroup in a reductive group, and that it is also related with extensions of the chiral (vertex operator) algebra corresponding to s l ̂ 2 at level k= l−2. We show that “finite subgroups in U q ( s l 2)” are classified by Dynkin diagrams of types A n , D 2 n , E 6, E 8 with Coxeter number equal to l, give a description of this correspondence similar to the classical McKay correspondence, and discuss relation with modular invariants in ( s l ̂ 2) k conformal field theory. The results we get are parallel to those known in the theory of von Neumann subfactors, but our proofs are independent of this theory.

On the rational solutions of the $ \sf\widehat{su}(2)_k$ Knizhnik-Zamolodchikov equation

We present some new results on the rational solutions of the Knizhnik-Zamolodchikov equation for the four-point conformal blocks of isospin I primary fields in the SU(2)_k Wess-Zumino-Novikov-Witten model. The rational solutions corresponding to integrable representations of the affine algebra su(2)_k have been classified by Michel, Stanev and Todorov; provided that the conformal dimension is an integer, they are in one-to-one correspondence with the local extensions of the chiral algebra. Here we give another description of these solutions as specific braid-invariant combinations of the so called regular basis and display a new series of rational solutions for isospins I = k+1 corresponding to non-integrable representations of the affine algebra.

Near-horizon conformal symmetry and black hole entropy.

Near an event horizon, the action of general relativity acquires a new asymptotic conformal symmetry. For two-dimensional dilaton gravity, this symmetry results in a chiral Virasoro algebra, and Cardy's formula for the density of states reproduces the Bekenstein-Hawking entropy. This lends support to the notion that black hole entropy is controlled universally by conformal symmetry near the horizon.

Extended chiral algebras in the SU(2)0WZNW model

We investigate the W-algebras generated by the integer dimension chiral primary operators of the SU(2)_0 WZNW model. These have a form almost identical to that found in the c=-2 model but have, in addition, an extended Kac-Moody structure. Moreover on Hamiltonian reduction these SU(2)_0 W-algebras exactly reduce to those found in c=-2. We explicitly find the free field representations for the chiral j=2 and j=3 operators which have respectively a fermionic doublet and bosonic triplet nature. The correlation functions of these operators accounts for the rational solutions of the Knizhnik-Zamolodchikov equation that we find. We explicitly compute the full algebra of the j=2 operators and find that the associativity of the algebra is only guaranteed if certain null vectors decouple from the theory. We conjecture that these algebras may produce a quasi-rational conformal field theory.

Logarithmic conformal field theories via logarithmic deformations

We construct logarithmic conformal field theories starting from an ordinary conformal field theory—with a chiral algebra C and the corresponding space of states V—via a two-step construction: (i) deforming the chiral algebra representation on V⊗End K[[ z, z −1]], where K is an auxiliary finite-dimensional vector space, and (ii) extending C by operators corresponding to the endomorphisms End K. For K= C 2 , with End K being the two-dimensional Clifford algebra, our construction results in extending C by an operator that can be thought of as ∂ −1 E, where ∮ E is a fermionic screening. This covers the (2, p) Virasoro minimal models as well as the sl (2) WZW theory.

Branching rules of semi-simple Lie algebras using affine extensions

We present a closed formula for the branching coefficients of an embedding p in g of two finite-dimensional semi-simple Lie algebras. The formula is based on the untwisted affine extension of p. It leads to an alternative proof of a simple algorithm for the computation of branching rules which is an analog of the Racah-Speiser algorithm for tensor products. We present some simple applications and describe how integral representations for branching coefficients can be obtained. In the last part we comment on the relation of our approach to the theory of NIM-reps of the fusion rings of WZW models with chiral algebra g_k. In fact, it turns out that for these models each embedding p in g induces a NIM-rep at level k to infinity. In cases where these NIM-reps can be be extended to finite level, we obtain a Verlinde-like formula for branching coefficients.

Stability and activation gaps of the parafermionic Hall states in the second Landau level

Analyzing the effective conformal field theory for the parafermionic Hall states, corresponding to filling fractions ν k =2+ k/( kM+2), k=2,3,… , M odd, we show that the even k plateaux are expected to be more stable than their odd k neighbors. The reason is that the parafermion chiral algebra can be locally extended for k even. This reconciles the theoretical implication, that the bigger the k the less stable the fluid, with the experimental fact that, for M=1, the k=2 and k=4 plateaux are already observed at electron temperature T e ≃8 mK, while the Hall resistance for k=3 is not precisely quantized at that temperature in the sample of Pan et al. Using a heuristic gap ansatz we estimate the activation energy gap for ν 3=13/5 to be approximately 0.015 K, which implies that the quantization of the Hall conductance could be observed for temperature below 1 mK in the same sample. We also find an appealing exact relation between the fractional electric charge and fractional statistics of the quasiholes. Finally, we argue that besides the Moore–Read phase for the ν 2=5/2 state there is another relevant phase, in which the fundamental quasiholes obey abelian statistics and carry half-integer electric charge.

Differential operators on the loop group via chiral algebras

Differential operators on the loop group via chiral algebras

Vertex Algebras, Mirror Symmetry, and D-Branes: The Case of Complex Tori

A vertex algebra is an algebraic counterpart of a two-dimensional conformal field theory. We give a new definition of a vertex algebra which includes chiral algebras as a special case, but allows for fields which are neither meromorphic nor anti-meromorphic. To any complex torus equipped with a flat Kahler metric and a closed 2-form we associate an N=2 superconformal vertex algebra (N=2 SCVA) in the sense of our definition. We find a criterion for two different tori to produce isomorphic N=2 SCVA's. We show that for algebraic tori isomorphism of N=2 SCVA's implies the equivalence of the derived categories of coherent sheaves corresponding to the tori or their noncommutative generalizations (Azumaya algebras over tori). We also find a criterion for two different tori to produce N=2 SCVA's related by a mirror morphism. If the 2-form is of type (1,1), this condition is identical to the one proposed by Golyshev, Lunts, and Orlov, who used an entirely different approach inspired by the Homological Mirror Symmetry Conjecture of Kontsevich. Our results suggest that Kontsevich's conjecture must be modified: coherent sheaves must be replaced with modules over Azumaya algebras, and the Fukaya category must be ``twisted'' by a closed 2-form. We also describe the implications of our results for BPS D-branes on Calabi-Yau manifolds.

Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions

We study two-dimensional nonlinear sigma models in which the target spaces are the coset supermanifolds U(n+m|n)/[ U(1)× U(n+m−1|n)]≅ CP n+m−1|n (projective superspaces) and OSp(2 n+ m|2 n)/OSp(2 n+ m−1|2 n)≅ S 2 n+ m−1|2 n (superspheres), n, m integers, −2⩽ m⩽2; these quantum field theories live in Hilbert spaces with indefinite inner products. These theories possess non-trivial conformally-invariant renormalization-group fixed points, or in some cases, lines of fixed points. Some of the conformal fixed-point theories can also be obtained within Landau–Ginzburg theories. We obtain the complete spectra (with multiplicities) of exact conformal weights of states (or corresponding local operators) in the isolated fixed-point conformal field theories, and at one special point on each of the lines of fixed points. Although the conformal weights are rational, the conformal field theories are not, and (with one exception) do not contain the affine versions of their superalgebras in their chiral algebras. The method involves lattice models that represent the strong-coupling region, which can be mapped to loop models, and then to a Coulomb gas with modified boundary conditions. The results apply to percolation, dilute and dense polymers, and other statistical mechanics models, and also to the spin quantum Hall transition in non-interacting fermions with quenched disorder.

Causal Poisson brackets of the WZNW model and its coset theories

From the basic chiral and anti-chiral Poisson bracket algebra of the SL(2,R) WZNW model, non-equal time Poisson brackets are derived. Through Hamiltonian reduction we deduce the corresponding brackets for its coset theories.

On chiral differential operators over homogeneous spaces

We give a classification and construction of chiral algebras of differential operators over semisimple algebraic groups G and over homogeneous spaces G/N and G/P where N is a nilpotent and P a parabolic subgroup.

The Role of Mathematics in the Experimental/Theoretical/Computational Trichotomy of Chemistry

The drastically increasing availability ofmodern computers coupled with the equally drasticallylower cost of a given amount of computer power inrecent years has resulted in the evolution of thetraditional experimental/theoretical dichotomy inchemistry into anexperimental/theoretical/computational trichotomy. This trichotomy can be schematically represented by atriangle (the ETC triangle) with experimental,theoretical, and computational chemistry at the threevertices. The ET and EC edges of the ETC triangledepict the uses of theoretical and computationalchemistry, respectively, to predict and interpretexperimental results. The TC edge depicts therelationship between theoretical and computationalchemistry. Mathematics plays an increasing role in allaspects of chemistry, particularly theoreticalchemistry, and has led to the evolution of thediscipline of mathematical chemistry. Research inmathematical chemistry can be considered to lie on achemistry-mathematics continuum depending on therelative depths of the underlying chemistry andmathematics. Examples of the author's own researchlying near each end of the chemistry-mathematicscontinuum include his work on applications of graphtheory and topology in inorganic coordination andcluster chemistry lying near the chemistry end and hiswork on chirality algebra lying near the mathematicsend. The general points in this essay are illustratedby an analysis of the roles of computational andtheoretical chemistry in developing an understandingof structure and bonding in deltahedral boranes andrelated carboranes. This work has allowed extensionof the concept of aromaticity from two dimensions asin benzene and other planar hydrocarbons to the thirddimension in deltahedral boranes.

Solution of the Dirac equation on the homogeneous manifoldSL(2,c)/GL(1,c) in the presence of a magnetic monopole field

Equations of shape invariance have been derived on the homogeneous manifold SL (2,c )/GL (1,c ), by means of which the Dirac equation is solved for a charged spin-½ particle in the presence of a magnetic monopole. The Dirac spinors on this manifold are written in terms of the master function. It is shown that these spinors represent an N = 1 chiral supersymmetry algebra and a unitary parasuperalgebra of arbitrary order p .

PSL( n| n) sigma model as a conformal field theory

We discuss the sigma model on the PSL( n| n) supergroup manifold. We demonstrate that this theory is exactly conformal. The chiral algebra of this model is given by some extension of the Virasoro algebra, similar to the W algebra of Zamolodchikov. We also show that all group invariant correlation functions are coupling constant independent and can be computed in the free theory. The non-invariant correlation functions are highly non-trivial and coupling dependent. At the end we compare two- and three-point correlation functions of the PSL(1, 1|2) sigma model with the correlation functions in the boundary theory of AdS 3 × S 3 and find a qualitative agreement.