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View article: Generalized Euler-Maclaurin formula and Signatures
Generalized Euler-Maclaurin formula and Signatures Open
The Euler-Maclaurin formula which relates a discrete sum with an integral, is generalised to the setting of Riemann-Stieltjes sums and integrals on stochastic processes whose paths are a.s. rectifiable, namely, continuous and with bounded …
View article: A Pseudodifferential Analytic Perspective on Getzler's Rescaling
A Pseudodifferential Analytic Perspective on Getzler's Rescaling Open
Inspired by Gilkey's invariance theory, Getzler's rescaling method and Scott's approach to the index via Wodzicki residues, we give a localisation formula for the $\mathbb Z_2$-graded Wodzicki residue of the logarithm of a class of differe…
View article: Mathematical Reflections on Locality
Mathematical Reflections on Locality Open
Starting from the principle of locality in quantum field theory, which states that an object is influenced directly only by its immediate surroundings, we review some features of the notion of locality arising in physics and mathematics. W…
View article: Locality Galois groups of meromorphic germs in several variables
Locality Galois groups of meromorphic germs in several variables Open
Meromorphic germs in several variables with linear poles naturally arise in mathematics in various disguises. We investigate their rich structures under the prism of locality, including locality subalgebras, locality transformation groups …
View article: A topological splitting of the space of meromorphic germs in several variables and continuous evaluators
A topological splitting of the space of meromorphic germs in several variables and continuous evaluators Open
We prove a topological decomposition of the space of meromorphic germs at zero in several variables with prescribed linear poles as a sum of spaces of holomorphic and polar germs. Evaluating the resulting holomorphic projection at zero giv…
View article: Tensor products and the Milnor-Moore theorem in the locality setup
Tensor products and the Milnor-Moore theorem in the locality setup Open
The present exploratory paper deals with tensor products in the locality framework {developed in previous work}, a natural setting for an algebraic formulation of the locality principle in quantum field theory. Locality tensor products of …
View article: Smooth rough paths, their geometry and algebraic renormalization
Smooth rough paths, their geometry and algebraic renormalization Open
We introduce the class of "smooth rough paths" and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer-Cartan perspective is th…
View article: Principal bundle groupoids, their gauge group and their nerve
Principal bundle groupoids, their gauge group and their nerve Open
We consider groupoids in the category of principal bundles, which we call principal bundles (PB) groupoids. Inspired by work by Th. Nikolaus and K. Waldorf, we generalise bundle gerbes over manifolds to bundle gerbes over groupoids and dis…
View article: From Orthocomplementations to Locality
From Orthocomplementations to Locality Open
After some background on lattices, the locality framework introduced in earlier work by the authors is extended to cover posets and lattices. We then extend the correspondence between Euclidean structures on vector spaces and orthogonal co…
View article: From non-unitary wheeled PROPs to smooth amplitudes and generalised convolutions
From non-unitary wheeled PROPs to smooth amplitudes and generalised convolutions Open
We introduce the concept of TRAP (Traces and Permutations), which can roughly be viewed as a wheeled PROP (Products and Permutations) without unit. TRAPs are equipped with a horizontal concatenation and partial trace maps. Continuous morph…
View article: A conical approach to Laurent expansions for multivariate meromorphic germs with linear poles
A conical approach to Laurent expansions for multivariate meromorphic germs with linear poles Open
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…
View article: From orthocomplementations to locality lattices
From orthocomplementations to locality lattices Open
Using the \loc framework of \cite{CGPZ}, we extend the correspondence between Euclidean structures on vector spaces and orthogonal
complementation to a one to one correspondence between a class of \loc structures and orthocomplementation…
View article: ProPs of graphs and generalised traces
ProPs of graphs and generalised traces Open
We assign generalised convolutions (resp. traces) to graphs whose edges are decorated by smooth kernels (resp. smoothing operators) on a closed manifold. To do so, we introduce the concept of TraPs (Traces and Permutations), which roughly …
View article: Spectral 𝜁-invariants lifted to coverings
Spectral 𝜁-invariants lifted to coverings Open
The canonical trace and the Wodzicki residue on classical pseudodifferential operators on a closed manifold are characterised by their locality and shown to be preserved under lifting to the universal covering as a result of their local fe…
View article: Renormalisation and locality: branched zeta values
Renormalisation and locality: branched zeta values Open
Multivariate renormalisation techniques are implemented in order to build,\nstudy and then renormalise at the poles, branched zeta functions associated\nwith trees. For this purpose, we first prove algebraic results and develop\nanalytic t…
View article: Renormalisation via locality morphisms
Renormalisation via locality morphisms Open
This is a survey on renormalisation in algebraic locality setup highlighting the role that locality morphisms can play for renormalisation purposes. After describing the general framework to build locality regularisation maps, we illustrat…
View article: Locality and renormalisation: universal properties and integrals on trees
Locality and renormalisation: universal properties and integrals on trees Open
The purpose of this paper is to build an algebraic framework suited to regularise branched structures emanating from rooted forests and which encodes the locality principle. This is achieved by means of the universal properties in the loca…
View article: Renormalisation via locality morphisms
Renormalisation via locality morphisms Open
This is a survey on renormalisation in the locality setup highlighting the role that locality morphisms can play for renormalisation purposes. Having set up a general framework to build regularisation maps, we illustrate renormalisation by…
View article: Renormalisation and locality: branched zeta values
Renormalisation and locality: branched zeta values Open
Multivariate renormalisation techniques are implemented in order to build, study and then renormalise at the poles, branched zeta functions associated with trees. For this purpose, we first prove algebraic results and develop analytic tool…
View article: An algebraic formulation of the locality principle in renormalisation
An algebraic formulation of the locality principle in renormalisation Open
We study the mathematical structure underlying the concept of locality which lies at the heart of classical and quantum field theory, and develop a machinery used to preserve locality during the renormalisation procedure. Viewing renormali…
View article: Renormalised conical zeta values
Renormalised conical zeta values Open
Conical zeta values associated with rational convex polyhedral cones generalise multiple zeta values. We renormalise conical zeta values at poles by means of a generalisation of Connes and Kreimer's Algebraic Birkhoff Factorisation. This p…
View article: Lifted trace defect formulae on coverings
Lifted trace defect formulae on coverings Open
The canonical trace and the Wodzicki residue on classical pseudodifferential operators on a closed manifold are shown to be preserved under lifting to the universal covering as a result of their local feature. In contrast, regularised trac…
View article: Renormalised conical zeta values
Renormalised conical zeta values Open
Conical zeta values associated with rational convex polyhedral cones generalise multiple zeta values. We renormalise conical zeta values at poles by means of a generalisation of Connes and Kreimer's Algebraic Birkhoff Factorisation. This p…