Classical and quantum algebras of non-local charges in σ models

We investigate the algebras of the non-local charges and their generating functionals (the monodromy matrices) in classical and quantum non-linear models. In the case of the classical chiral models it turns out that there exists no definition of the Poisson bracket of two monodromy matrices satisfying antisymmetry and the Jacobi identity. Thus, the classical non-local charges do not generate a Lie algebra. In the case of the quantum O(N) non-linear model, we explicitly determine the conserved quantum monodromy operator from a factorization principle together withP,T, and O(N) invariance. We give closed expressions for its matrix elements between asymptotic states in terms of the known two-particleS-matrix. The quantumR-matrix of the model is found. The quantum non-local charges obey a quadratic Lie algebra governed by a Yang-Baxter equation.Laboratoire associé au CNRS No. LA 280     

The quantum group structure of 2D gravity and minimal models I

On the unit circle, an infinite family of chiral operators is constructed, whose exchange algebra is given by the universalR-matrix of the quantum groupSL(2) _q . This establishes the precise connection between the chiral algebra of two dimensional gravity or minimal models and this quantum group. The method is to relate the monodromy properties of the operator differential equations satisfied by the generalized vertex operators with the exchange algebra ofSL(2) _q . The formulae so derived, which generalize an earlier particular case worked out by Babelon, are remarkably compact and may be entirely written in terms of q-deformed factorials and binomial coefficients.     

Characterizing invariants for local extensions of current algebras

ParisA of local quantum field theories are studied, whereA is a chiral conformal quantum field theory and is a local extension, either chiral or two-dimensional. The local correlation functions of fields from have an expansion with respect toA into conformal blocks, which are non-local in general. Two methods of computing characteristic invariant ratios of structure constants in these expansions are compared: (a) by constructing the monodromy representation of the braid group in the space of solutions of the Knizhnik-Zamolodchikov differential equation, and (b) by an analysis of the local subfactors associated with the extension with methods from operator algebra (Jones theory) and algebraic quantum field theory. Both approaches apply also to the reverse problem: the characterization and (in principle) classification of local extensions of a given theory.     

Quantized superfields

We construct quantized free superfields and represent them as operator‐valued distributions in Fock space starting with the Majorana field. We then analyse the algebras generated by free component quantum fields together with the Susy generators Q, . This enables us to obtain the quantized chiral superfield by finite Susy transformation from its scalar component. To get hermitian superfields we study by the same method a second scalar field algebra from which various scalar superfields can be obtained by exponentiation. Next we investigate the vector algebra and use it to construct the massive vector superfield. Surprisingly enough, the result is totally different from the vector multiplet in the literature. It contains two hermitian four‐vector components instead of one and a spin‐3/2 field similar to the gravitino in supergravity.     

New solvable lattice models in three dimensions

In this paper we establish a remarkable connection between two seemingly unrelated topics in the area of solvable lattice models. The first is the Zamolodchikov model, which is the only nontrivial model on a three-dimen-sional lattice so far solved. The second is the chiral Potts model on the square lattice and its generalization associated with theU _q(sl(n)) algebra, which is of current interest due to its connections with high-genus algebraic curves and with representations of quantum groups at roots of unity. We show that this last sl(n)-generalized chiral Potts model can be interpreted as a model on a threedimensional simple cubic lattice consisting ofn square-lattice layers with anN- valued (N2) spin at each site. Further, in theN=2 case this three-dimen-sional model reduces (after a modification of the boundary conditions) to the Zamolodchikov model we mentioned above.     

Isotopic Liftings of Clifford Algebras and Applications in Elementary Particle Mass Matrices

Isotopic liftings of algebraic structures are investigated in the context of Clifford algebras, where it is defined a new product involving an arbitrary, but fixed, element of the Clifford algebra. This element acts as the unit with respect to the introduced product, and is called isounit. We construct isotopies in both associative and non-associative arbitrary algebras, and examples of these constructions are exhibited using Clifford algebras, which although associative, can generate the octonionic, non-associative, algebra. The whole formalism is developed in a Clifford algebraic arena, giving also the necessary pre-requisites to introduce isotopies of the exterior algebra. The flavor hadronic symmetry of the six u,d,s,c,b,t quarks is shown to be exact, when the generators of the isotopic Lie algebra are constructed, and the unit of the isotopic Clifford algebra is shown to be a function of the six quark masses. The limits constraining the parameters, that are entries of the representation of the isounit in the isotopic group SU(6), are based on the most recent limits imposed on quark masses.     

Cyclic monodromy matrices forsl(n) trigonometricR-matrices

The algebra of monodromy matrices forsl(n) trigonometricR-matrix is studied. It is shown that a generic finite-dimensional polynomial irreducible representation of this algebra is equivalent to a tensor product ofL-operators. Cocommutativity of representations is discussed and intertwiners for factorizable representations are written through the Boltzmann weights of thesl(n) chiral Potts model.     

Conformal field algebras with quantum symmetry from the theory of superselection sectors

According to the theory of superselection sectors of Doplicher, Haag, and Roberts, field operators which make transitions between different superselection sectors—i.e. different irreducible representations of the observable algebra—are to be constructed by adjoining localized endomorphisms to the algebra of local observables. We find the relevant endomorphisms of the chiral algebra of observables in the minimal conformal model with central chargec=1/2 (Ising model). We show by explicit and elementary construction how they determine a representation of the braid groupB which is associated with a Temperley-Lieb-Jones algebra. We recover fusion rules, and compute the quantum dimensions of the superselection sectors. We exhibit a field algebra which is quantum group covariant and acts in the Hilbert space of physical states. It obeys local braid relations in an appropriate weak sense.     

An explicit construction of the quantum group in chiral WZW-models

It is shown how a chiral Wess-Zumino-Witten theory with globally defined vertex operators and a one-to-one correspondence between fields and states can be constructed. The Hilbert space of this theory is the direct sum of tensor products of representations of the chiral algebra and finite dimensional internal parameter spaces. On this enlarged space there exists a natural action of Drinfeld's quasi-quantum groupA _{g, t} which commutes with the action of the chiral algebra and plays the rôle of an internal symmetry algebra. TheR matrix describes the braiding of the chiral vertex operators and the coassociator gives rise to a modification of the duality property.For genericq the quasi-quantum group is isomorphic to the coassociative quantum groupU _q (g) and thus the duality property of the chiral theory can be restored. This construction has to be modified for the physically relevant case of integer level. The quantum group has to be replaced by the corresponding truncated quasiquantum group, which is not coassociative because of the truncation. This exhibits the truncated quantum group as the internal symmetry algebra of the chiral WZW model, which therefore has only a modified duality property. The case ofg=su(2) is worked out in detail.     

W-Algebras,new rational models and completeness of thec=1 classification

Two series ofW with two generators are constructed from chiral vertex operators of a free field representation. Ifc=1–24k, there exists aW(2, 3k) algebra for k ₊/2 and aW(2, 8k) algebra for k ₊/4. All possible lowest-weight representations, their characters and fusion rules are calculated proving that these theories are rational. It is shown, that these non-unitary theories complete the classification of all rational theories with effective central chargec _eff=1. The results are generalized to the case of extended supersymmetric conformal algebras.     

Two-dimensional topological gravity and equivariant cohomology

The analogy between topological string theory and equivariant cohomology for differentiable actions of the circle group on manifolds has been widely remarked on. One of our aims in this paper is to make this analogy precise. We show that topological string theory is the derived functor of semi-relative cohomology, just as equivariant cohomology is the derived functor of basic cohomology. That homological algebra finds a place in the study of topological string theory should not surprise the reader, granted that topological string theory is the conformal field theorist's algebraic topology.In [7], we have shown that the cohomology of a topological conformal field theory carries the structure of a batalin-Vilkovisky algebra (actually, two commuting such structures, corresponding to the two chiral sectors of the theory). In the second part of this paper, we describe the analogous algebraic structure on the equivariant cohomology of a topological conformal field theory: we call this structure a gravity algebra. This algebraic structure is a certain generalization of a Lie algebra, and is distinguished by the fact that it has an infinite sequence of independent operations {a ₁, ...,a _k },k2, satisfying quadratic relations generalizing the Jacobi rule. (The operad underlying the category of gravity algebras has been studied independently by Ginzburg-Kapranov [9].)The author is grateful to M. Bershadsky, E. Frenkel, M. Kapranov, G. Moore, R. Plesser and G. Zuckerman for the many ways in which they helped in the writing of this paper; also to the Department of Mathematics at Yale University for its hospitality while part of this paper was written.The author is partially supported by a fellowship of the Sloan Foundation and a research grant of the NSF.     

The relation between quantumW algebras and Lie algebras

By quantizing the generalized Drinfeld-Sokolov reduction scheme for arbitrarysl ₂ embeddings we show that a large set of quantumW algebras can be viewed as (BRST) cohomologies of affine Lie algebras. The set contains many knownW algebras such asW _N andW ₃ ⁽²⁾ . Our formalism yields a completely algorithmic method for calculating theW algebra generators and their operator product expansions, replacing the cumbersome construction ofW algebras as commutants of screening operators. By generalizing and quantizing the Miura transformation we show that anyW algebra in can be embedded into the universal enveloping algebra of a semisimple affine Lie algebra which is, up to shifts in level, isomorphic to a subalgebra of the original affine algebra. Thereforeany realization of this semisimple affine Lie algebra leads to a realization of theW algebra. In particular, one obtains in this way a general and explicit method for constructing the free field realizations and Fock resolusions for all algebras in. Some examples are explicitly worked out.     

D=2 and D=4 realization of κ-conformal algebra

We describe the generators of-conformal transformations leaving invariant the-deformed d'Alembert equation. For the case D=4 the algebraic structure of the conformal extension of the off-shell spin zero realization of-Poincaré algebra is discussed. Then the D=2 off-shell realization of-conformal algebra for arbitrary spin and its commutation relations are studied.Presented at the 4th Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.Supported by KBN grant 2P 302 087 06.     

The Elliptic Algebra and the Drinfeld Realization of the Elliptic Quantum Group

By using the elliptic analogue of the Drinfeld currents in the elliptic algebra , we construct a L-operator, which satisfies the RLL-relations characterizing the face type elliptic quantum group . For this purpose, we introduce a set of new currents in . As in the N=2 case, we find a structure of as a certain tensor product of and a Heisenberg algebra. In the level-one representation, we give a free field realization of the currents in . Using the coalgebra structure of and the above tensor structure, we derive a free field realization of the -analogue of -intertwining operators. The resultant operators coincide with those of the vertex operators in the -type face model.     

Full Field Algebras

We introduce a notion of full field algebra which is essentially an algebraic formulation of the notion of genus-zero full conformal field theory. For any vertex operator algebras V ^L and V ^R , is naturally a full field algebra and we introduce a notion of full field algebra over . We study the structure of full field algebras over using modules and intertwining operators for V ^L and V ^R . For a simple vertex operator algebra V satisfying certain natural finiteness and reductivity conditions needed for the Verlinde conjecture to hold, we construct a bilinear form on the space of intertwining operators for V and prove the nondegeneracy and other basic properties of this form. The proof of the nondegenracy of the bilinear form depends not only on the theory of intertwining operator algebras but also on the modular invariance for intertwining operator algebras through the use of the results obtained in the proof of the Verlinde conjecture by the first author. Using this nondegenerate bilinear form, we construct a full field algebra over and an invariant bilinear form on this algebra.     

Exponential Equations Related to the Quantum ‘<Emphasis Type="Italic">ax</Emphasis> + <Emphasis Type="Italic">b</Emphasis>’ Group

We study pairs b, of unbounded selfadjoint operators, satisfying commutation rules inspired by the quantum ax+b group [19]: b=–b and ²=id except for kerb, on which ²=0. We find all measurable, unitary-operator valued functions F satisfying the exponential equation: F(b, )F(d, )=F((b, ) (d, )), where d, satisfy the same commutation rules as b, , and is modeled after the comultiplication of the quantum ax+b group. This result is crucial for classification of all unitary representations of the quantum ax+b group, which is achieved in our forthcoming paper [12].Supported by KBN grant No 2 PO3 A 0361815 January 2001     

Yang-Baxter algebras of monodromy matrices in integrable quantum field theories

We consider a large class of two-dimensional integrable quantum field theories with non-abelian internal symmetry and classical scale invariance. We present a general procedure to determine explicitly the conserved quantum monodromy operator generating infinitely many non-local charges. The main features of our method are a factorization principle and the use of $\text{P}$, $\text{T}$, and internal symmetries. The monodromy operator is shown to satisfy a Yang-Baxter algebra, the structure constants (i.e. the quantum R-matrix) of which are determined by two-particle S-matrix of the theory. We apply the method to the chiral SU(N) and the O(2N) Gross-Neveu models.     

The Lie algebra of the sl(2, ℂ)-valued automorphic functions on a torus

It is shown that the Lie algebra of the automorphic, meromorphic sl(2, )-valued functions on a torus is a geometric realization of a certain infinite-dimensional finitely generated Lie algebra. In the trigonometric limit, when the modular parameter of the torus goes to zero, the former Lie algebra goes over into the sl(2, )-valued loop algebra, while the latter goes into the Lie algebra (A ₁ ⁽¹⁾ )/(centre).     

gl q (n) and quantum monodromy

A generalized Toda lattice based on gl(n) is considered. The Poisson brackets are expressed in terms of a Lax connection, L=–() and a classical r-matrix, {₁,₂}=[r,₁+₂}. The essential point is that the local lattice transfer matrix is taken to be the ordinary exponential, T=e; this assures the intepretation of the local and the global transfer matrices in terms of monodromy, which is not true of the T-matrix used for the sl(n) Toda lattice. To relate this exponential transfer matrix to the more manageable and traditional factorized form, it is necessary to make specific assumptions about the equal time operator product expansions. The simplest possible assumptions lead to an equivalent, factorized expression for T, in terms of operators in (an extension of) the enveloping algebra of gl(n). Restricted to sl(n), and to multiplicity-free representations, these operators satisfy the commutation relations of sl _q (n), which provides a very simple injection of sl _q (n) into the enveloping algebra of sl(n). A deformed coproduct, similar in form to the familiar coproduct on sl _q (n), turns gl(n) into a deformed Hopf algebra gl _q (n). It contains sl _q (n) as a subalgebra, but not as a sub-Hopf algebra.