Conformal quantum field theory and half-sided modular inclusions of von-Neumann-Algebras

LetN, be von-Neumann-Algebras on a Hilbert space , a comon cyclic and separarting vector. Assume to be cyclic and separating also forN . Denote byJ , J _N the modular conjugations to (, ), and _N the associated modular operators. If and these data define in a canonical way a conformal quantum field theory in a cricle. Conversely, the chiral part of a conformal quantum field theory in two dimensions always yields such data in a natural way.Partly supported by the DFG, SFB 288 Differentiageometrie und Quantenphysik     

Symmetries and half-sided modular inclusions of von Neumann algebras

LetN, be a von Neumann algebras on a Hilbert space , a common cyclic and separating vector. Assume to be cyclic and also separating forN . Denote by , _N , _N the modular operators to (, ), (N, ), resp (N , ). Assume now ^-it N ^it N for allt 0. (Such type of inclusions ((N U, ) , ) are called half-sided modular.) Then the modular groups ^it , _N ^ir , _N ^is ,t, r, s generate a unitary representation of the group S1(2, )/Z ₂ of positive energy.Another result is related to two half-sided modular inclusions (₁ , ) and (₂ , ). Under proper conditions the three modular groups ^it , ₁ ^ir , ₂ ^is ,t, r, s generate the three-dimensional subgroup of O(2, 1) of two commuting translations and the Lorentz transformation.Partly supported by the DFG, SFB 288 Differentialgeometrie und Quantenphysik.     

Half-Sided Translations and tye Type of von Neumann Algebras

Let M be a von Neumann algebra acting on a Hilbert space H and H a cyclic and separating vector for M. If there exists a half-sided translation for M, i.e. a continuous unitary group U(t) with U(t)=, a non-negative spectrum fulfilling Ad U(t)M M for t 0 (or 0), then we will show that either M is of type III₁ or U(t) is trivial.     

A note on strongly additive conformal field theory and half-sided modular conormal standard inclusions

Given a half-sided modular standard inclusion (N , ) of von-Neumann algebras which is conormal, we construct a chiral conformal quantum field theory on the circle. Conversely, a conformal quantum field theory on the circle which fulfills, in addition, the property of strong additivity, yields such data in a natural way.Partly supported by the DFG, SFB 288 Differentialgeometrie und Quantenphysik.     

Maximum modular graphs

Modularity has been explored as an important quantitative metric for community and cluster detection in networks. Finding the maximum modularity of a given graph has been proven to be NP-complete and therefore, several heuristic algorithms have been proposed. We investigate the problem of finding the maximum modularity of classes of graphs that have the same number of links and/or nodes and determine analytical upper bounds. Moreover, from the set of all connected graphs with a fixed number of links and/or number of nodes, we construct graphs that can attain maximum modularity, named maximum modular graphs. The maximum modularity is shown to depend on the residue obtained when the number of links is divided by the number of communities. Two applications in transportation networks and data-centers design that can benefit of maximum modular partitioning are proposed.     

Polynomial modular Frobenius manifolds

The moduli space of Frobenius manifolds carries a natural involutive symmetry, and a distinguished class–so-called modular Frobenius manifolds–lie at the fixed points of this symmetry. In this paper a classification of semi-simple modular Frobenius manifolds which are polynomial in all but one of the variables is begun, and completed for three and four dimensional manifolds. The resulting examples may also be obtained from higher dimensional manifolds by a process of folding. The relationship of these results with orbifold quantum cohomology is also discussed.     

Membrane-mediated interactions of rod-like inclusions

Inclusions embedded in lipid membranes undergo a mediated force, due to the tendency of the membrane to relax its excess of elastic energy. In this paper we determine the exact shape of a two-dimensional vesicle hosting two different inclusions, and we analyse how the inclusion conformation influences the mediated interaction. We find non-trivial equilibrium configurations for the inclusions along the hosting membrane, and we derive the complete phase diagram of the mediated interaction. In particular, we find a non-vanishing mediated force even when the distance between the inclusions is much greater than their size. Our model can be applied to describe the mediated interactions of parallel, elongated inclusions embedded in three-dimensional membranes. Received 22 October 2001 and Received in final form 8 March 2002     

Galois modular invariants of WZW models

The set of modular invariants that can be obtained from Galois transformations is investigated systematically for WZW models. It is shown that a large subset of Galois modular invariants coincides with simple current invariants. For algebras of type B and D infinite series of previously unknown exceptional automorphism invariants are found.     

Automorphism modular invariants of current algebras

We consider those two-dimensional rational conformal field theories (RCFTs) whose chiral algebras, when maximally extended, are isomorphic to the current algebra formed from some untwisted affine Lie algebra at fixed level. In this case the partition function is specified by an automorphism of the fusion ring and corresponding symmetry of the Kac-Peterson modular matrices. We classify all such partition functions when the underlying finite-dimensional Lie algebra is simple. This gives all possible spectra for this class of RCFTs. While accomplishing this, we also find the primary fields with second smallest quantum dimension.Supported in part by NSERC.     

Perturbations of the modular automorphism group

It is shown, under a necessary condition, that strong (pointwise) convergence of modular automorphism groups to a one parameter family of maps implies weak convergence of the respective states in the factor case. Moreover the limiting one parameter family of maps is the modular automorphism group for the limiting state. In the type I case weak convergence of the automorphism groups suffices. Norm convergence of the states is obtained in some cases.     

Orbifold models and modular transformation

The orbifold models of the heterotic string are constructed on the quotient spaces of generalized tori by translational and rotational discrete symmetries. In order to obtain the consistent orbifold models, the conditions of the modular invariance are derived from a one-loop vacuum amplitude. Z₃ orbifold models satisfying such conditions are searched systematically. It is shown that there are infinite possible models with N = 2 supersymmetry. Among these models, two examples having E₆ and E₇ gauge groups are discussed. The orbifold models with N = 1 supersymmetry are also discussed in detail. It is shown that there are only five consistent models in the class of these models based on E₈ ⊗ E′₈ heterotic string in which the extra six-dimensional torus and the E₈ ⊗ E′₈ maximal torus are modded out by the rotational and the translational Z₃ symmetries respectively.     

Counting curves with modular forms

We consider the type IIA string compactified on the Calabi-Yau space given by a degree 12 hypersurface in the weighted projective space P⁴_(1,1,2,2,6). We express the prepotential of the low-energy effective supergravity theory in terms of a set of functions that transform covariantly under PSL(2, Z) modular transformations. These functions are then determined by monodromy properties, by singularities at the massless monopole point of the moduli space, and by S → T exchange symmetry.     

Two-loop modular invariance and spin-statistics

Within the context of a class of fermionic closed string theories I explicitly show some relationships between two-loop modular invariance, factorization, correct spin-statistics and the existence of a massless gravity multiplet.     

Liapunov theorem for modular functions

We extend Liapunov Theorem to modular functions on complemented lattices.     

Distribution of eigenvalues for the modular group

The two-point correlation functions of energy levels for free motion on the modular domain, both with periodic and Dirichlet boundary conditions, are explicitly computed using a generalization of the Hardy-Littlewood method. It is shown that in the limit of small separations they show an uncorrelated behaviour and agree with the Poisson distribution but they have prominent number-theoretical oscillations at larger scale. The results agree well with numerical simulations.