arXiv subjects

Sylvie Paycha

Publications and source records attributed to Sylvie Paycha.

At least 19 recordsLinked to original sources

arXiv2026

Metric Geometry of the Signature Group for $p$-Variation Rough Paths

The signatures of $p$-rough paths form a subgroup of sufficiently high-level truncated tensor algebras, whose inverse limit is a subgroup of the full tensor algebra. For $p \geq 1$, we provide a top-down description of the signature group as the inverse limit of finite-dimensional Carnot--Carathéodory geometries in the $p$-variation setting. We show that every compatible choice of metrics induces a topological tree structure on the inverse-limit group, under which the signature group is not a topological group. This extends the results of Enrico Le Donne and Roland Züst from bounded variation to rough paths. We also characterise the dependence of the inverse-limit groups and their metric completions on the choice of metric, identifying them with the tree-reduced path group of Horatio Boedihardjo, Xiang Geng, Terry Lyons, and Danyu Yang.

math.MG↗

arXiv2024

Generalized Euler-Maclaurin formula and Signatures

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 variation. For this purpose, new variants of the signature are introduced, such as the flip and the sawtooth signature. The counterparts of the Bernoulli numbers that arise in the classical Euler-Maclaurin formula are shown to be the integration constants in the repeated integration by parts which ``recursively minimise the error'' at every truncation level.

math.PR↗

arXiv2024

A Pseudodifferential Analytic Perspective on Getzler's Rescaling

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 differential operators acting on sections of a spinor bundle over an even-dimensional manifold. This formula is expressed in terms of another local density built from the symbol of the logarithm of a limit of rescaled differential operators acting on differential forms. When applied to complex powers of the square of a Dirac operator, it amounts to expressing the index of a Dirac operator in terms of a local density involving the logarithm of the Getzler rescaled limit of its square.

math.DG↗

arXiv2023

A topological splitting of the space of meromorphic germs in several variables and continuous evaluators

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 gives rise to a continuous evaluator (at zero) on the space of meromorphic germs in several variables. Our constructions are carried out in the framework of Silva spaces and use an inner product on the underlying space of variables. They generalise to several variables, the topological direct decomposition of meromorphic germs at zero as sums of holomorphic and polar germs previously derived by the first and third author and provide a topological refinement of a known algebraic decomposition of such spaces previously derived by the second author and collaborators.

math.CV↗

arXiv2023

Principal bundle groupoids, their gauge group and their nerve

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 discuss a functorial correspondence between PB groupoids and bundle gerbes over groupoids. From a PB groupoid over a fibre product groupoid, we build a bundle gerbe over another fibre product groupoid. Conversely, from a bundle gerbe over a Lie groupoid, we build a PB groupoid. It has a trivial base and from any PB groupoid with trivial base, we build a bundle gerbe over a Lie groupoid. In that case, the resulting bundle gerbe is isomorphic as a groupoid to a partial quotient of the PB groupoid. We describe the nerves of PB groupoids and their partial quotients, which are simplicial objects in the category of principal bundles. Applying this construction enables us to define the inner transformation group of the nerve of a partial quotient groupoid and to describe the transformations of the corresponding bundle gerbe.

math.DG↗

arXiv2023

Locality Galois groups of meromorphic germs in several variables

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 and locality characters. The key technical tool is the dependence subspace for a meromorphic germ with which we define a locality orthogonal relation between two meromorphic germs. We describe the structure of locality subalgebras generated by classes of meromorphic germs with certain types of poles. We also define and determine their group of locality transformations which fix the holomorphic germs and preserve multivariable residues, a group we call the locality Galois group. We then specialise to two classes of meromorphic germs with prescribed types of nested poles, arising from multiple zeta functions in number theory and Feynman integrals in perturbative quantum field theory respectively. We show that they are locality polynomial subalgebras with locality polynomial bases given by the locality counterpart of Lyndon words. This enables us to explicitly describe their locality Galois group. As an application, we propose a mathematical interpretation of Speer's analytic renormalisation for Feynman amplitudes. We study a class of locality characters, called generalised evaluators after Speer. We show that the locality Galois group acts transitively on generalised evaluators by composition, thus providing a candidate for a renormalisation group in this multivariable approach.

math-ph↗

arXiv2022

Tensor products and the Milnor-Moore theorem in the locality setup

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 locality vector spaces raise challenging questions, such as whether the locality tensor product of two locality vector spaces is a locality vector space. A related question is whether the quotient of locality vector spaces is a locality vector space, which we first reinterpret in a group theoretic language and then in terms of short exact sequences. We prove a universal property for the locality tensor algebra and for the locality enveloping algebra, the analogs in the locality framework of the tensor algebra and of the enveloping algebra. These universal properties hold under compatibility assumptions between the locality and the multilinearity underlying the construction of tensor products which we formulate in the form of conjectural statements. Assuming they hold true, we generalise the Milnor-Moore theorem to the locality setup and discuss some of its consequences.

math.RA↗

arXiv2021

Smooth rough paths, their geometry and algebraic renormalization

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 the key to a purely algebraic form of Lyons extension theorem, the renormalization of rough paths in the spirit of [Bruned, Chevyrev, Friz, Preiß, A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019] as well as a related notion of "sum of rough paths". We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting.

math.PR↗

arXiv2021

From orthocomplementations to locality

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 complementations to a one-one correspondence between a class of locality structures and orthocomplementations on bounded lattices. This recasts in the context of renormalisation classical results in lattice theory.

math.RA↗

arXiv2021

From non-unitary wheeled PROPs to smooth amplitudes and generalised convolutions

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 morphisms on an infinite dimensional topological space and smooth kernels (resp. smoothing operators) on a closed manifold form a TRAP but not a wheeled PROP. We build the free objects in the category of TRAPs as TRAPs of graphs and show that a TRAP can be completed to a unitary TRAP (or wheeled PROP). We further show that it can be equipped with a vertical concatenation, which on the TRAP of linear homomorphisms of a vector space, amounts to the usual composition. The vertical concatenation in the TRAP of smooth kernels gives rise to generalised convolutions. Graphs whose vertices are decorated by smooth kernels (resp. smoothing operators) on a closed manifold form a TRAP. From their universal properties we build smooth amplitudes associated with the graph.

math.CO↗

arXiv2020

ProPs of graphs and generalised traces

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 correspond to ProPs (Products and Permutations) without vertical concatenation and equipped with families of generalised partial traces. They can be equipped with a ProP structure in deriving vertical concatenation from the partial traces and we relate TraPs to wheeled ProPs first introduced by Merkulov. We further build their free object and give precise proofs of universal properties of ProPs and TraPs.

math.CO↗

arXiv2019

Locality and renormalisation: universal properties and integrals on trees

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 locality framework of properly decorated rooted forests. These universal properties are then applied to derive the multivariate regularisation of integrals indexed by rooted forests. We study their renormalisation, along the lines of Kreimer's toy model for Feynman integrals.

math-ph↗

arXiv2018

Renormalisation and locality: branched zeta values

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 tools, which we then combine to study branched zeta functions. The algebraic aspects concern universal properties for locality algebraic structures, some of which had been discussed in previous work; we "branch/ lift" to trees operators acting on the decoration set of trees, and factorise branched maps through words by means of universal properties for words which we prove in the locality setup. The analytic tools are multivariate meromorphic germs of pseudodifferential symbols with linear poles which generalise the meromorphic germs of functions with linear poles studied in previous work. Multivariate meromorphic germs of pseudodifferential symbols form a locality algebra on which we build various locality maps in the framework of locality structures. We first show that the finite part at infinity defines a locality character from the latter symbol valued meromorphic germs to the scalar valued ones. We further equip the locality algebra of germs of pseudodifferential symbols with locality Rota-Baxter operators given by regularised sums and integrals. By means of the universal properties in the framework of locality structures we can lift Rota-Baxter operators to trees, and use the lifted discrete sums in order to build and study renormalised branched zeta values associated with trees. By construction these renormalised branched zeta values factorise on mutually independent (for the locality relation) trees.

math-ph↗

arXiv2018

Renormalisation via locality morphisms

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 locality algebra homomorphisms on three examples, the renormalisation at poles of conical zeta functions, branched zeta functions and iterated integrals stemming from Kreimer's toy model.

math-ph↗

arXiv2017

An algebraic formulation of the locality principle in renormalisation

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 renormalisation in the framework of Connes and Kreimer as the algebraic Birkhoff factorisation of characters on a Hopf algebra with values in a Rota-Baxter algebra, we build locality variants of these algebraic structures, leading to a locality variant of the algebraic Birkhoff factorisation. This provides an algebraic formulation of the conservation of locality while renormalising. As an application in the context of the Euler-Maclaurin formula on cones, we renormalise the exponential generating function which sums over the lattice points in convex cones. For a suitable multivariate regularisation, renormalisation from the algebraic Birkhoff factorisation amounts to composition by a projection onto holomorphic multivariate functions.

math-ph↗

arXiv2017

Spectral $ζ$-invariants lifted to coverings

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 feature. As a consequence, we lift a class of spectral $ζ$-invariants using lifted defect formulae which express discrepancies of $ζ$-regularised traces in terms of Wodzicki residues. We derive Atiyah's $L^2$-index theorem as an instance of the $\mathbb Z_2$-graded generalisation of the canonical lift of spectral $ζ$-invariants and we show that certain lifted spectral $ζ$-invariants for geometric operators are integrals of Pontryagin and Chern forms.

math.DG↗

arXiv2017

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.

math.CV↗

arXiv2016

Renormalised conical zeta values

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 paper serves as a motivation for and an application of this generalised renormalisation scheme. The latter also yields an Euler-Maclaurin formula on rational convex polyhedral lattice cones which relates exponential sums to exponential integrals. When restricted to Chen cones, it reduces to Connes and Kreimer's Algebraic Birkhoff Decomposition for maps with values in the algebra of ordinary meromorphic functions in one variable.

math-ph↗