3,336 research outputs found
Causal posets, loops and the construction of nets of local algebras for QFT
We provide a model independent construction of a net of C*-algebras
satisfying the Haag-Kastler axioms over any spacetime manifold. Such a net,
called the net of causal loops, is constructed by selecting a suitable base K
encoding causal and symmetry properties of the spacetime. Considering K as a
partially ordered set (poset) with respect to the inclusion order relation, we
define groups of closed paths (loops) formed by the elements of K. These groups
come equipped with a causal disjointness relation and an action of the symmetry
group of the spacetime. In this way the local algebras of the net are the group
C*-algebras of the groups of loops, quotiented by the causal disjointness
relation. We also provide a geometric interpretation of a class of
representations of this net in terms of causal and covariant connections of the
poset K. In the case of the Minkowski spacetime, we prove the existence of
Poincar\'e covariant representations satisfying the spectrum condition. This is
obtained by virtue of a remarkable feature of our construction: any Hermitian
scalar quantum field defines causal and covariant connections of K. Similar
results hold for the chiral spacetime with conformal symmetry
The toggle group, homomesy, and the Razumov-Stroganov correspondence
The Razumov-Stroganov correspondence, an important link between statistical
physics and combinatorics proved in 2011 by L. Cantini and A. Sportiello,
relates the ground state eigenvector of the O(1) dense loop model on a
semi-infinite cylinder to a refined enumeration of fully-packed loops, which
are in bijection with alternating sign matrices. This paper reformulates a key
component of this proof in terms of posets, the toggle group, and homomesy, and
proves two new homomesy results on general posets which we hope will have
broader implications.Comment: 14 pages, 10 figures, final versio
A matroid-friendly basis for the quasisymmetric functions
A new Z-basis for the space of quasisymmetric functions (QSym, for short) is
presented. It is shown to have nonnegative structure constants, and several
interesting properties relative to the space of quasisymmetric functions
associated to matroids by the Hopf algebra morphism (F) of Billera, Jia, and
Reiner. In particular, for loopless matroids, this basis reflects the grading
by matroid rank, as well as by the size of the ground set. It is shown that the
morphism F is injective on the set of rank two matroids, and that
decomposability of the quasisymmetric function of a rank two matroid mirrors
the decomposability of its base polytope. An affirmative answer is given to the
Hilbert basis question raised by Billera, Jia, and Reiner.Comment: 25 pages; exposition tightened, typos corrected; to appear in the
Journal of Combinatorial Theory, Series
Braids, posets and orthoschemes
In this article we study the curvature properties of the order complex of a
graded poset under a metric that we call the ``orthoscheme metric''. In
addition to other results, we characterize which rank 4 posets have CAT(0)
orthoscheme complexes and by applying this theorem to standard posets and
complexes associated with four-generator Artin groups, we are able to show that
the 5-string braid group is the fundamental group of a compact nonpositively
curved space.Comment: 33 pages, 16 figure
Pseudograph associahedra
Given a simple graph G, the graph associahedron KG is a simple polytope whose
face poset is based on the connected subgraphs of G. This paper defines and
constructs graph associahedra in a general context, for pseudographs with loops
and multiple edges, which are also allowed to be disconnected. We then consider
deformations of pseudograph associahedra as their underlying graphs are altered
by edge contractions and edge deletions.Comment: 25 pages, 22 figure
Determinants Associated to Zeta Matrices of Posets
We consider the matrix , where the entries of
are the values of the zeta function of the finite poset . We give a
combinatorial interpretation of the determinant of and establish
a recursive formula for this determinant in the case in which is a boolean
algebra.Comment: 14 pages, AMS-Te
Deformations of bordered Riemann surfaces and associahedral polytopes
We consider the moduli space of bordered Riemann surfaces with boundary and
marked points. Such spaces appear in open-closed string theory, particularly
with respect to holomorphic curves with Lagrangian submanifolds. We consider a
combinatorial framework to view the compactification of this space based on the
pair-of-pants decomposition of the surface, relating it to the well-known
phenomenon of bubbling. Our main result classifies all such spaces that can be
realized as convex polytopes. A new polytope is introduced based on truncations
of cubes, and its combinatorial and algebraic structures are related to
generalizations of associahedra and multiplihedra.Comment: 25 pages, 31 figure
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