9,729 research outputs found
A conical approach to Laurent expansions for multivariate meromorphic germs with linear poles
We use convex polyhedral cones to study a large class of multivariate
meromorphic germs, namely those with linear poles, which naturally arise in
various contexts in mathematics and physics. We express such a germ as a sum of
a holomorphic germ and a linear combination of special non-holomorphic germs
called polar germs. In analyzing the supporting cones -- cones that reflect the
pole structure of the polar germs -- we obtain a geometric criterion for the
non-holomorphicity of linear combinations of polar germs. This yields the
uniqueness of the above sum when required to be supported on a suitable family
of cones and assigns a Laurent expansion to the germ. Laurent expansions
provide various decompositions of such germs and thereby a uniformized proof of
known results on decompositions of rational fractions. These Laurent expansions
also yield new concepts on the space of such germs, all of which are
independent of the choice of the specific Laurent expansion. These include a
generalization of Jeffrey-Kirwan's residue, a filtered residue and a coproduct
in the space of such germs. When applied to exponential sums on rational convex
polyhedral cones, the filtered residue yields back exponential integrals.Comment: 30 page
From the discrete to the continuous - towards a cylindrically consistent dynamics
Discrete models usually represent approximations to continuum physics.
Cylindrical consistency provides a framework in which discretizations mirror
exactly the continuum limit. Being a standard tool for the kinematics of loop
quantum gravity we propose a coarse graining procedure that aims at
constructing a cylindrically consistent dynamics in the form of transition
amplitudes and Hamilton's principal functions. The coarse graining procedure,
which is motivated by tensor network renormalization methods, provides a
systematic approximation scheme towards this end. A crucial role in this coarse
graining scheme is played by embedding maps that allow the interpretation of
discrete boundary data as continuum configurations. These embedding maps should
be selected according to the dynamics of the system, as a choice of embedding
maps will determine a truncation of the renormalization flow.Comment: 22 page
Honeycomb tessellations and canonical bases for permutohedral blades
This paper studies two families of piecewise constant functions which are
determined by the -skeleta of collections of honeycomb tessellations of
with standard permutohedra. The union of the codimension
cones obtained by extending the facets which are incident to a vertex of such a
tessellation is called a blade. We prove ring-theoretically that such a
honeycomb, with 1-skeleton built from a cyclic sequence of segments in the root
directions , decomposes locally as a Minkowski sum of
isometrically embedded components of hexagonal honeycombs: tripods and
one-dimensional subspaces. For each triangulation of a cyclically oriented
polygon there exists such a factorization. This consequently gives resolution
to an issue proposed and developed by A. Ocneanu, to find a structure theory
for an object he discovered during his investigations into higher Lie theories:
permutohedral blades. We introduce a certain canonical basis for a vector space
spanned by piecewise constant functions of blades which is compatible with
various quotient spaces appearing in algebra, topology and scattering
amplitudes. Various connections to scattering amplitudes are discussed, giving
new geometric interpretations for certain combinatorial identities for one-loop
Parke-Taylor factors. We give a closed formula for the graded dimension of the
canonical blade basis. We conjecture that the coefficients of the generating
function numerators for the diagonals are symmetric and unimodal.Comment: Added references; new section on configuration space
Curvature function and coarse graining
A classic theorem in the theory of connections on principal fiber bundles
states that the evaluation of all holonomy functions gives enough information
to characterize the bundle structure (among those sharing the same structure
group and base manifold) and the connection up to a bundle equivalence map.
This result and other important properties of holonomy functions has encouraged
their use as the primary ingredient for the construction of families of quantum
gauge theories. However, in these applications often the set of holonomy
functions used is a discrete proper subset of the set of holonomy functions
needed for the characterization theorem to hold. We show that the evaluation of
a discrete set of holonomy functions does not characterize the bundle and does
not constrain the connection modulo gauge appropriately.
We exhibit a discrete set of functions of the connection and prove that in
the abelian case their evaluation characterizes the bundle structure (up to
equivalence), and constrains the connection modulo gauge up to "local details"
ignored when working at a given scale. The main ingredient is the Lie algebra
valued curvature function defined below. It covers the holonomy
function in the sense that .Comment: 34 page
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