Conformal algebra and multipoint correlation functions in 2D statistical models

Based on the conformal algebra approach, a general technique is given for the calculation of multipoint correlation functions in 2D statistical models at the critical point. Particular conformal operator algebras are found for operators of the 2D q-component Potts model (1 < q < 4), and the O(N) model (0 < N < 2) at the critical point. A number of four-point correlation functions are calculated for these models.     

Conformal Galilei algebras,symmetric polynomials and singular vectors

We classify and explicitly describe homomorphisms of Verma modules for conformal Galilei algebras \({\mathfrak {cga}}_\ell (d,\mathbb {C})\) with \(d=1\) for any integer value \(\ell \in {\mathbb {N}}\). The homomorphisms are uniquely determined by singular vectors as solutions of certain differential operators of flag type and identified with specific polynomials arising as coefficients in the expansion of a parametric family of symmetric polynomials into power sum symmetric polynomials.     

Frobenius theory for positive maps of von Neumann algebras

Frobenius theory about the cyclic structure of eigenvalues of irreducible non negative matrices is extended to the case of positive linear maps of von Neumann algebras. Semigroups of such maps and ergodic properties are also considered.     

Conformal blocks and generalized theta functions

LetSU _X r be the moduli space of rankr vector bundles with trivial determinant on a Riemann surfaceX. This space carries a natural line bundle, the determinant line bundleL. We describe a canonical isomorphism of the space of global sections ofL ^k with the space of conformal blocks defined in terms of representations of the Lie algebrasl _r (C((z))). It follows in particular that the dimension ofH ⁰(SU _X r,L ^k ) is given by the Verlinde formula.Both authors were partially supported by the European Science Project Geometry of Algebraic Varieties, Contract no. SCI-0398-C(A)     

Representations of affine Lie algebras,ellipticr-matrix systems,and special functions

The author considers an elliptic analogue of the Knizhnik-Zamolodchikov equations—the consistent system of linear differential equations arising from the elliptic solution of the classical Yang-Baxter equation for the Lie algebra . The solutions of this system are interpreted as traces of products of intertwining operators between certain representations of the affine Lie algebra. A new differential equation for such traces characterizing their behavior under the variation of the modulus of the underlying elliptic curve is deduced. This equation is consistent with the original system.It is shown that the system extended by the new equation is modular invariant, and the corresponding monodromy representations of the modular group are defined. Some elementary examples in which the system can be solved explicitly (in terms of elliptic and modular functions) are considered. The monodromy of the system is explicitly computed with the help of the trace interpretation of solutions. Projective representations of the braid group of the torus and representations of the double affine Hecke algebra are obtained.     

Spectra,eigenvectors and overlap functions for representation operators ofq-deformed algebras

Operators of representations corresponding to symmetric elements of theq-deformed algebrasU _q (su_1,1),U _q (so_2,1),U _q (so_3,1),U _q (so _n ) and representable by Jacobi matrices are studied. Closures of unbounded symmetric operators of representations of the algebrasU _q (su_1,1) andU _q (so_2,1) are not selfadjoint operators. For representations of the discrete series their deficiency indices are (1,1). Bounded symmetric operators of these representations are trace class operators or have continuous simple spectra. Eigenvectors of some operators of representations are evaluated explicitly. Coefficients of transition to eigenvectors (overlap coefficients) are given in terms ofq-orthogonal polynomials. It is shown how results on eigenvectors and overlap coefficients can be used for obtaining new results in representation theory ofq-deformed algebras.     

Conformal invariance and pion wave functions of nonleading twist

The restrictions are studied for the general structure of pion wave functions of twist 3 and twist 4 imposed by the conformal symmetry and the equations of motion. A systematic expansion of wave functions in the conformal spin is built and the first order corrections to asymptotic formulae are calculated by the QCD sum rule method. In particular, we have found a multiplicatively renormalizable contribution into the two-particle wave function of twist 4 which cannot be expanded in a finite set of Gegenbauer polynonials.     

Orthogonally additive functions on operator algebras

For a real or complex Hilbert spaceH, a general form of an even, orthogonally additive functional on B(H)is given.     

Frobenius manifolds and Frobenius algebra-valued integrable systems

The notion of integrability will often extend from systems with scalar-valued fields to systems with algebra-valued fields. In such extensions the properties of, and structures on, the algebra play a central role in ensuring integrability is preserved. In this paper, a new theory of Frobenius algebra-valued integrable systems is developed. This is achieved for systems derived from Frobenius manifolds by utilizing the theory of tensor products for such manifolds, as developed by Kaufmann (Int Math Res Not 19:929–952, 1996), Kontsevich and Manin (Inv Math 124: 313–339, 1996). By specializing this construction, using a fixed Frobenius algebra \({\mathcal {A}},\) one can arrive at such a theory. More generally, one can apply the same idea to construct an \({\mathcal {A}}\)-valued topological quantum field theory. The Hamiltonian properties of two classes of integrable evolution equations are then studied: dispersionless and dispersive evolution equations. Application of these ideas are discussed, and as an example, an \({\mathcal {A}}\)-valued modified Camassa–Holm equation is constructed.     

Dipole correlation functions

Glarum's relation between permanent dipole correlation functions for an isolated sphere and a spherical region in a continuous medium is derived by molecular response theory and relation time development operators for the two cases.     

Triple correlation functions

New values for double mode-mode and triple flux molecular equilibrium correlations are presented. The relevance of these results for hydrodynamics is then discussed. It is found that the divergence of Burnett coefficients has only a formal meaning.     

Finite-temperature correlation functions

Lattice measurements of the Pisa group are analyzed numerically, and the parameters of correlation functions are extracted fromthe data—both below and above the deconfinement temperature T _c. The gluon condensate is found for six temperatures in the interval (0.956–1.131)T _c, and field distributions in the deconfined phase are obtained.     

Billiards correlation functions

We discuss various experiments on the time decay of velocity autocorrelation functions in billiards. We perform new experiments and find results which are compatible with an exponential mixing hypothesis first put forward by Friedman and Martin (FM): they do not seem compatible with the stretched exponentials believed, in spite of FM and more recently of Chernov, to describe the mixing. The analysis leads to several byproducts: we obtain information about the normal diffusive nature of the motion and we consider the probability distribution of the number of collisions in timet _m (ast _m ), finding a strong dependence on some geometric characteristics of the locus of the billiard obstacles.This paper is dedicated to Philippe Choquard on his 65th birthday     

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.     

Mutual information functions versus correlation functions

This paper studies one application of mutual information to symbolic sequences: the mutual information functionM(d). This function is compared with the more frequently used correlation function(d). An exact relation betweenM(d) and(d) is derived for binary sequences. For sequences with more than two symbols, no such general relation exists; in particular,(d)=0 may or may not lead toM(d)=0. This linear, but not general, independence between symbols separated by a distance is studied for ternary sequences. Also included is the estimation of the finite-size effect on calculating mutual information. Finally, the concept of symbolic noise is discussed.     

Rank of quantized universal enveloping algebras and modular functions

We compute an intrinsic rank invariant for quasitriangular Hopf algebras in the case of general quantum groupsU _q (g). As a function ofq the rank has remarkable number theoretic properties connected with modular covariance and Galois theory. A number of examples are treated in detail, including rank (U _q (su(3))) and rank (U _q (e ₈)). We briefly indicate a physical interpretation as relating Chern-Simons theory with the theory of a quantum particle confined to an alcove ofg.     

Conformal invariant two and three-point functions for fields with arbitrary spin

The conformal invariant two and three-point functions for any “fundamental” fields with an arbitrary spin and scale dimensions are found in the Minkowsky x-space. The two-point functions for Dirac, symmetric and antisymmetric tensor fields are given. The three-point functions for two Dirac fields and one symmetrical tensor field, as well as any other field for which this function is nonvanishing, are given. In the case of conserved currents the Ward identities are considered.