Local Axioms Research Articles

Trianguloids and triangulations of root polytopes

Triangulations of a product of two simplices and, more generally, of root polytopes are closely related to Gelfand-Kapranov-Zelevinsky's theory of discriminants, to tropical geometry, tropical oriented matroids, and to generalized permutohedra. We introduce a new approach to these objects, identifying a triangulation of a root polytope with a certain bijection between lattice points of two generalized permutohedra. In order to study such bijections, we define trianguloids as edge-colored graphs satisfying simple local axioms. We prove that trianguloids are in bijection with triangulations of root polytopes.

Characterization of queer supercrystals

We provide a characterization of the crystal bases for the quantum queer superalgebra recently introduced by Grantcharov et al. This characterization is a combination of local queer axioms generalizing Stembridge's local axioms for crystal bases for simply-laced root systems, which were recently introduced by Assaf and Oguz, with further axioms and a new graph G characterizing the relations of the type A components of the queer supercrystal. We provide a counterexample to Assaf's and Oguz' conjecture that the local queer axioms uniquely characterize the queer supercrystal. We obtain a combinatorial description of the graph G on the type A components by providing explicit combinatorial rules for the odd queer operators on certain highest weight elements. This also yields a new combinatorial description of the Schur expansion of the Schur P-polynomials.

Toward a Local Characterization of Crystals for the Quantum Queer Superalgebra

We define operators on semistandard shifted tableaux and use Stembridge’s local characterization for regular graphs to prove they define a crystal structure. This gives a new proof that Schur P-polynomials are Schur positive. We define queer crystal operators (also called odd Kashiwara operators) to construct a connected queer crystal on semistandard shifted tableaux of a given shape. Using the tensor rule for queer crystals, this provides a new proof that products of Schur P-polynomials are Schur P-positive. Finally, to facilitate applications of queer crystals in the context of Schur P-positivity, we give local axioms for queer regular graphs, generalizing Stembridge’s axioms, that partially characterize queer crystals.

Axioms for Shifted Tableau Crystals

We give local axioms that uniquely characterize the crystal-like structure on shifted tableaux developed by the authors and Purbhoo. These axioms closely resemble those developed by Stembridge for type A tableau crystals. This axiomatic characterization gives rise to a new method for proving and understanding Schur $Q$-positive expansions in symmetric function theory, just as the Stembridge axiomatic structure provides for ordinary Schur positivity.

Full Axiomatisation of Timed Processes of Interval-Timed Petri Nets

In this paper we use partial order semantics to express the truly concurrent behaviour of Interval-Timed Petri nets (ITPNs) in their most general setting, i.e. with autoconcurrency and zero duration, as studied with its standard maximal step semantics in [1]. First we introduce the notion of timed processes for ITPNs inductively. Then we investigate if the equivalence of inductive and axiomatic process semantics - true for classical Petri nets - could hold for ITPNs too. We will see that the notions of independence and immediate firing obligation seem to be antagonistic ones, and that local axioms, adequate to define processes of classical Petri nets, are not sufficient to caracterize timed processes of ITPNs. We propose several original “global” axioms which reveal to be an effective solution. Thus we yield finally a full axiomatic definition of timed processes for ITPNs.

Differential cohomology and locally covariant quantum field theory

We study differential cohomology on categories of globally hyperbolic Lorentzian manifolds. The Lorentzian metric allows us to define a natural transformation whose kernel generalizes Maxwell's equations and fits into a restriction of the fundamental exact sequences of differential cohomology. We consider smooth Pontryagin duals of differential cohomology groups, which are subgroups of the character groups. We prove that these groups fit into smooth duals of the fundamental exact sequences of differential cohomology and equip them with a natural presymplectic structure derived from a generalized Maxwell Lagrangian. The resulting presymplectic Abelian groups are quantized using the [Formula: see text]-functor, which yields a covariant functor from our categories of globally hyperbolic Lorentzian manifolds to the category of [Formula: see text]-algebras. We prove that this functor satisfies the causality and time-slice axioms of locally covariant quantum field theory, but that it violates the locality axiom. We show that this violation is precisely due to the fact that our functor has topological subfunctors describing the Pontryagin duals of certain singular cohomology groups. As a byproduct, we develop a Fréchet–Lie group structure on differential cohomology groups.

A C*-Algebra for Quantized Principal U(1)-Connections on Globally Hyperbolic Lorentzian Manifolds

The aim of this work is to complete our program on the quantization of connections on arbitrary principal U(1)-bundles over globally hyperbolic Lorentzian manifolds. In particular, we show that one can assign via a covariant functor to any such bundle an algebra of observables which separates gauge equivalence classes of connections. The C*-algebra we construct generalizes the usual CCR-algebras, since, contrary to the standard field-theoretic models, it is based on a presymplectic Abelian group instead of a symplectic vector space. We prove a no-go theorem according to which neither this functor, nor any of its quotients, satisfies the strict axioms of general local covariance. As a byproduct, we prove that a morphism violates the locality axiom if and only if a certain induced morphism of cohomology groups is non-injective. We show then that, fixing any principal U(1)-bundle, there exists a suitable category of subbundles for which a quotient of our functor yields a quantum field theory in the sense of Haag and Kastler. We shall provide a physical interpretation of this feature and we obtain some new insights concerning electric charges in locally covariant quantum field theory.

The locality axiom in quantum field theory and tensor products of C*-algebras

The prototypes of mutually independent systems are systems which are localized in spacelike separated regions. In the framework of locally covariant quantum field theory, we show that the commutativity of observables in spacelike separated regions can be encoded in the tensorial structure of the functor which associates unital C*-algebras (the local observable algebras) to globally hyperbolic spacetimes. This holds under the assumption that the local algebras satisfy the split property and involves the minimal tensor product of C*-algebras.

Erratum to: Local axioms in disguise: Hilbert on Minkowski diagrams

During the production of the article, some errors and misprints have been overlooked. The author only noticed them after the paper was already online. On page 7 of the online version, lines 14–17, the last sentence of the quote from Hilbert should read: “In this same way, electromagnetic inertia and Einsteinian gravitation, [as conceived] with the corresponding concepts of the classical theories, are compatible [vertraglich], inasmuch as these classical concepts can be conceived as limiting cases of the more general concepts in the new theories.” Page 9, line 14 should read: “an axiom system having all the known facts of a given scientific domain represented as deducible from the axioms ...”. Page 16, line 9 should read: “Furthermore, one may ask ...”. Page 17, line 6 should read: “he took primitive evidence to be akin to proto-arithmetical-algebraical evidence”. Page 18, line 4 should read: “certain questions are formulated, to which geometrical intuition contributes”. Pages 27, 28 and 30, one should read: “discriminant” instead of “determinant” with regard to binary quadratic forms. Although terminology is not unanimously settled on this score and varies from one author to another, consistent choices should be made in the paper when talking about Lagrange’s reduction algorithm on the one hand and Hermite-Minkowski’s setting on the other hand. Therefore on page 27, lines 12, 17 and

Local axioms in disguise: Hilbert on Minkowski diagrams

While claiming that diagrams can only be admitted as a method of strict proof if the underlying axioms are precisely known and explicitly spelled out, Hilbert praised Minkowski’s Geometry of Numbers and his diagram-based reasoning as a specimen of an arithmetical theory operating “rigorously” with geometrical concepts and signs. In this connection, in the first phase of his foundational views on the axiomatic method, Hilbert also held that diagrams are to be thought of as “drawn formulas”, and formulas as “written diagrams”, thus suggesting that the former encapsulate propositional information which can be extracted and translated into formulas. In the case of Minkowski diagrams, local geometrical axioms were actually being produced, starting with the diagrams, by a process that was both constrained and fostered by the requirement, brought about by the axiomatic method itself, that geometry ought to be made independent of analysis. This paper aims at making a twofold point. On the one hand, it shows that Minkowski’s diagrammatic methods in number theory prompted Hilbert’s axiomatic investigations into the notion of a straight line as the shortest distance between two points, which start from his earlier work focused on the role of the triangle inequality property in the foundations of geometry, and lead up to his formulation of the 1900 Fourth Problem. On the other hand, it purports to make clear how Hilbert’s assessment of Minkowski’s diagram-based reasoning in number theory both raises and illuminates conceptual compatibility concerns that were crucial to his philosophy of mathematics.

Finite axiomatizability of local set theory

The need for modifying axiomatic set theories was caused, in particular, by the development of category theory. The ZF and NBG axiomatic theories turned out to be unsuitable for defining the notion of a model of category theory. The point is that there are constructions such as the category of categories in naive category theory, while constructions like the set of sets are strongly restricted in the ZF and NBG axiomatic theories. Thus, it was required, on the one hand, to restrict constructions similar to the category of categories and, on the other hand, adapt axiomatic set theory in order to give a definition of a category which survives restricted construction similar to the category of categories. This task was accomplished by promptly inventing the axiom of universality (AU) asserting that each set is an element of a universal set closed under all NBG constructions. Unfortunately, in the theories ZF + AU and NBG + AU, there are toomany universal sets (as many as the number of all ordinals), whereas to solve the problem stated above, a countable collection of universal sets would suffice. For this reason, in 2005, the first-named author introduced local-minimal set theory, which preserves the axiom AU of universality and has an at most countable collection of universal sets. This was achieved at the expense of rejecting the global replacement axiom and using the local replacement axiom for each universal class instead. Local-minimal set theory has 14 axioms and one axiom scheme (of comprehension). It is shown that this axiom scheme can be replaced by finitely many axioms that are special cases of the comprehension scheme. The proof follows Bernays’ scheme with significant modifications required by the presence of the restricted predicativity condition on the formula in the comprehension axiom scheme.

From buildings to point-line geometries and back again

A chamber system is a particular type of edge-labeled graph. We discu ss when such chamber systems are or are not associated with a geometry, and when they are buildings. Buildings can give rise to point-line geometries under constraints imposed by how a line should behave with respect to the point-shadows of the other geometric objects (Pasini (P)). A recent theorem o f Kasikova (K) shows that Pasini's choice is the right one. So, in a general w ay, one has a procedure for getting point-line geometries from buildings. In the other direction, we describe how a class of point-line geometries with elementary local axioms (certain parapolar spaces) successfully char- acterize many buildings and their homomorphic images. A recent result of K. Thas (KT) makes this theory free of Tits' the classification of polar spaces o f rank three (T1). One notes that parapolar spaces alone will not cover all o f the point-line geometries arising from buildings by the Pasini-Kasikova construction, so the door is wide open for further research with points and lines.

The crossing model for regular [formula omitted]-crystals

A regular A n -crystal is an edge-colored directed graph, with n colors, related to an irreducible highest weight integrable module over U q ( sl n + 1 ) . Based on Stembridge's local axioms for regular simply-laced crystals and a structural characterization of regular A 2 -crystals in [V.I. Danilov, A.V. Karzanov, G.A. Koshevoy, Combinatorics of regular A 2 -crystals, J. Algebra 310 (2007) 218–234], we present a new combinatorial construction, the so-called crossing model, and prove that this model generates precisely the set of regular A n -crystals. Using the model, we obtain a series of results on the combinatorial structure of such crystals and properties of their subcrystals.

A Riesz decomposition theorem on harmonic spaces without positive potentials

In this paper, we give a new definition of the flux of a superharmonic function defined outside a compact set in a Brelot space without positive potentials. We also give a new notion of potential in a BS space (that is, a harmonic space without positive potentials containing the constants) which leads to a Riesz decomposition theorem for the class of superharmonic functions that have a harmonic minorant outside a compact set. Furthermore, we give a characterization of the local axiom of proportionality in terms of a global condition on the space.

Combinatorics of free vertex algebras

This paper illustrates the combinatorial approach to vertex algebra - study of vertex algebras presented by generators and relations. A necessary ingredient of this method is the notion of vertex algebra. Borcherds \cite{bor} was the first to note that vertex algebras do not exist in general. The reason for this is that vertex algebras do not form a variety of algebras, because the locality axiom (see sec 2 below) is not an identity. However, a certain subcategory of vertex algebras, obtained by restricting the order of locality of generators, has a universal object, which we call the vertex algebra corresponding to the given locality bound. In [J. of Algebra, 217(2):496-527] some vertex algebras were constructed and in certain special cases their linear bases were found. In this paper we generalize this construction and find linear bases of an arbitrary vertex algebra. It turns out that vertex algebras are closely related to the vertex algebras corresponding to integer lattices. The latter algebras play a very important role in different areas of mathematics and physics. Here we explore the relation between vertex algebras and lattice vertex algebras in much detail. These results comply with the use of the word in physical literature refering to some elements of lattice vertex algebras, like in free field, free bozon or free fermion. Among other things, we find a nice presentation of lattice vertex algebras in terms of generators and relations, thus giving an alternative construction of these algebras without using vertex operators. We remark that our construction works in a very general setting; we do not assume the lattice to be positive definite, neither non-degenerate, nor of a finite rank.

Covariant SPDEs and quantum field structures

Covariant stochastic partial differential equations (SPDEs) are studied in any dimension. A special class of such equations is selected and it is proved that the solutions can be analytically continued to Minkowski spacetime yielding tempered Wightman distributions which are covariant, obey the locality axiom and a weak form of the spectral axiom.

Fuzzy neighborhood convergence spaces

We introduce a supercategory of the category NFS of fuzzy neighborhood spaces. This supercategory is defined by means of local axioms similar to those of FNS, but the objects can also nicely be characterized internally via a notion of convergence. In future work it shall be shown that this new category allows for a good treatment of compactifications.

An invariance argument for confinement

The possibility of constructing gauge invariant charge densities in classical gauge theory is investigated. For the Yang-Mills-Dirac fields in the Minkowski space, it is shown that, unless the colour label belongs to the centre of the colour algebra, gauge invariant charge densities depend non-locally on the field variables and have non-local Poisson brackets. A quantization leading to observable charge density operators with commutators related to the classical Poisson bracket would violate the locality axiom of quantum field theory. For the Yang-Mills-Dirac fields in a bag the colour algebra is trivial and the colour charges vanish identically. In this sense the Yang-Mills-Dirac theory has to be considered as confining on the basis of a purely mathematical reasoning.

Hyperfunction quantum field theory

The quantum field theory in terms of Fourier hyperfunctions is constructed. The test function space for hyperfunctions does not containC∞ functions with compact support. In spite of this defect the support concept ofH-valued Fourier hyperfunctions allows to formulate the locality axiom for hyperfunction quantum field theory.