Archimedean Atomic Lattice Effect Algebras with Complete Lattice of Sharp Elements

We study Archimedean atomic lattice effect algebras whose set of sharp elements is a complete lattice. We show properties of centers, compatibility centers and central atoms of such lattice effect algebras. Moreover, we prove that if such effect algebra $E$ is separable and modular then there exists a faithful state on $E$. Further, if an atomic lattice effect algebra is densely embeddable into a complete lattice effect algebra $\widehat{E}$ and the compatiblity center of $E$ is not a Boolean algebra then there exists an $(o)$-continuous subadditive state on $E$

Modularity, Atomicity and States in Archimedean Lattice Effect Algebras

Effect algebras are a generalization of many structures which arise in quantum physics and in mathematical economics. We show that, in every modular Archimedean atomic lattice effect algebra $E$ that is not an orthomodular lattice there exists an $(o)$-continuous state $\omega$ on $E$, which is subadditive. Moreover, we show properties of finite and compact elements of such lattice effect algebras

Smearing of Observables and Spectral Measures on Quantum Structures

An observable on a quantum structure is any $\sigma$-homomorphism of quantum structures from the Borel $\sigma$-algebra of the real line into the quantum structure which is in our case a monotone $\sigma$-complete effect algebras with the Riesz Decomposition Property. We show that every observable is a smearing of a sharp observable which takes values from a Boolean $\sigma$-subalgebra of the effect algebra, and we prove that for every element of the effect algebra there is its spectral measure

Holomorphic reduction of N=2 gauge theories, Wilson-'t Hooft operators, and S-duality

We study twisted N=2 superconformal gauge theory on a product of two Riemann surfaces Sigma and C. The twisted theory is topological along C and holomorphic along Sigma and does not depend on the gauge coupling or theta-angle. Upon Kaluza-Klein reduction along Sigma, it becomes equivalent to a topological B-model on C whose target is the moduli space MV of nonabelian vortex equations on Sigma. The N=2 S-duality conjecture implies that the duality group acts by autoequivalences on the derived category of MV. This statement can be regarded as an N=2 counterpart of the geometric Langlands duality. We show that the twisted theory admits Wilson-'t Hooft loop operators labelled by both electric and magnetic weights. Correlators of these loop operators depend holomorphically on coordinates and are independent of the gauge coupling. Thus the twisted theory provides a convenient framework for studying the Operator Product Expansion of general Wilson-'t Hooft loop operators.Comment: 50 pages, latex. v2: an erroneous statement about an analog of the Hitchin fibration has been fixe

Archimedean Atomic Lattice Effect Algebras with Complete Lattice of Sharp Elements

We study Archimedean atomic lattice effect algebras whose set of sharp elements is a complete lattice. We show properties of centers, compatibility centers and central atoms of such lattice effect algebras. Moreover, we prove that if such effect algebra E is separable and modular then there exists a faithful state on E. Further, if an atomic lattice effect algebra is densely embeddable into a complete lattice effect algebra ^E and the compatiblity center of E is not a Boolean algebra then there exists an (o)-continuous subadditive state on E

Archimedean atomic lattice effect algebras with complete lattice of sharp elements

We study Archimedean atomic lattice effect algebras whose set of sharp elements is a complete lattice. We show properties of centers, compatibility centers and central atoms of such lattice ef fect algebras. Moreover, we prove that if such effect algebra E is separable and modular then there exists a faithful state on E. Further, if an atomic lattice effect algebra is densely embeddable into a complete lattice effect algebra Eb and the compatibility center of E is not a Boolean algebra then there exists an (o)-continuous subadditive state on E

Algebraic conformal quantum field theory in perspective

Conformal quantum field theory is reviewed in the perspective of Axiomatic, notably Algebraic QFT. This theory is particularly developped in two spacetime dimensions, where many rigorous constructions are possible, as well as some complete classifications. The structural insights, analytical methods and constructive tools are expected to be useful also for four-dimensional QFT.Comment: Review paper, 40 pages. v2: minor changes and references added, so as to match published versio

Modularity, Atomicity and States in Archimedean Lattice Effect Algebras

Effect algebras are a generalization of many structures which arise in quantum physics and in mathematical economics. We show that, in every modular Archimedean atomic lattice effect algebra E that is not an orthomodular lattice there exists an (o)-continuous state ω on E, which is subadditive. Moreover, we show properties of finite and compact elements of such lattice effect algebras

The low-temperature phase of Kac-Ising models

We analyse the low temperature phase of ferromagnetic Kac-Ising models in dimensions $d\geq 2$. We show that if the range of interactions is \g^{-1}, then two disjoint translation invariant Gibbs states exist, if the inverse temperature \b satisfies \b -1\geq \g^\k where \k=\frac {d(1-\e)}{(2d+1)(d+1)}, for any \e>0. The prove involves the blocking procedure usual for Kac models and also a contour representation for the resulting long-range (almost) continuous spin system which is suitable for the use of a variant of the Peierls argument.Comment: 19pp, Plain Te