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Sylvie Paycha

Publications and source records attributed to Sylvie Paycha.

At least 37 records · Page 2Linked to original sources
arXiv2015

Algebraic Birkhoff Factorization and the Euler-Maclaurin Formula on Cones

We equip the space of lattice cones with a coproduct which makes it a connected cograded colagebra. The exponential sum and exponential integral on lattice cones can be viewed as linear maps on this space with values in the space of meromorphic germs with linear poles at zero. We investigate the subdivision properties-- reminiscent of the inclusion-exclusion principle for the cardinal on finite sets-- of such linear maps and establish a compatibility of these properties with respect to the convolution quotient of the coalgebra. Implementing the Algebraic Birkhoff Factorization procedure on the linear maps under consideration, we factorize the exponential sum as a convolution quotient of two maps, with each of the maps in the factorization satisfying a subdivision property. Consequently, the Algebraic Birkhoff Factorization specializes to the Euler-Maclaurin formula on lattice cones and provides a simple formula for the interpolating factor by means of a projection map.

math-ph↗
arXiv2015

Traces of holomorphic families of operators on the noncommutative torus and on Hilbert modules

We revisit traces of holomorphic families of pseudodifferential operators on a closed manifold in view of geometric applications. We then transpose the corresponding analytic constructions to two different geometric frameworks; the noncommutative torus and Hilbert modules. These traces are meromorphic functions whose residues at the poles as well as the constant term of the Laurent expansion at zero (the latter when the family at zero is a differential operator) can be expressed in terms of Wodzicki residues and extended Wodzicki residues involving logarithmic operators. They are therefore local and contain geometric information. For holomorphic families leading to zeta regularised traces, they relate to the heat-kernel asymptotic coefficients via an inverse Mellin mapping theorem. We revisit Atiyah's L^2-index theorem by means of the (extended) Wodzicki residue and interpret the scalar curvature on the noncommutative two torus as an (extended) Wodzicki residue.

math.OA↗
arXiv2015

Counting an infinite number of points: a testing ground for renormalization methods

This is a leisurely introductory account addressed to non-experts and based on previous work by the authors, on how methods borrowed from physics can be used to "count" an infinite number of points. We begin with the classical case of counting integer points on the non-negative real axis and the classical Euler-Maclaurin formula. As an intermediate stage, we count integer points on product cones where the roles played by the coalgebra and the algebraic Birkhoff factorization can be appreciated in a relatively simple setting. We then consider the general case of (lattice) cones for which we introduce a conilpotent coalgebra of cones, with applications to renormalization of conical zeta values. When evaluated at zero arguments conical zeta functions indeed "count" integer points on cones.

math-ph↗
arXiv2014

The canonical trace and the noncommutative residue on the noncommutative torus

Using a global symbol calculus for pseudodifferential operators on tori, we build a canonical trace on classical pseudodifferential operators on noncommutative tori in terms of a canonical discrete sum on the underlying toroidal symbols. We characterise the canonical trace on operators on the noncommutative torus as well as its underlying canonical discrete sum on symbols of fixed (resp.\ any) non--integer order. On the grounds of this uniqueness result, we prove that in the commutative setup, this canonical trace on the noncommutative torus reduces to Kontsevich and Vishik's canonical trace which is thereby identified with a discrete sum. A similar characterisation for the noncommutative residue on noncommutative tori as the unique trace which vanishes on trace--class operators generalises Fathizadeh and Wong's characterisation in so far as it includes the case of operators of fixed integer order. By means of the canonical trace, we derive defect formulae for regularized traces. The conformal invariance of the $ζ$--function at zero of the Laplacian on the noncommutative torus is then a straightforward consequence.

math.AP↗
arXiv2013

Conical zeta values and their double subdivision relations

We introduce the concept of a conical zeta value as a geometric generalization of a multiple zeta value in the context of convex cones. The quasi-shuffle and shuffle relations of multiple zeta values are generalized to open cone subdivision and closed cone subdivision relations respectively for conical zeta values. In order to achieve the closed cone subdivision relation, we also interpret linear relations among fractions as subdivisions of decorated closed cones. As a generalization of the double shuffle relation of multiple zeta values, we give the double subdivision relation of conical zeta values and formulate the extended double subdivision relation conjecture for conical zeta values.

math.NT↗
arXiv2010

A Canonical Trace Associated with Certain Spectral Triples

In the abstract pseudodifferential setup of Connes and Moscovici, we prove a general formula for the discrepancies of zeta-regularised traces associated with certain spectral triples, and we introduce a canonical trace on operators, whose order lies outside (minus) the dimension spectrum of the spectral triple.

math.OA↗
arXiv2010

The logarithmic residue density of a generalised Laplacian

We show that the residue density of the logarithm of a generalised Laplacian on a closed manifold defines an invariant polynomial valued differential form. We express it in terms of a finite sum of residues of classical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulae provide a pedestrian proof of the Atiyah-Singer formula for a pure Dirac operator in dimension $4$ and for a twisted Dirac operator on a flat space of any dimension. These correspond to special cases of a more general formula by S. Scott and D. Zagier announced in \cite{Sc2} and to appear in \cite{Sc3}. In our approach, which is of perturbative nature, we use either a Campbell-Hausdorff formula derived by Okikiolu or a non commutative Taylor type formula.

math.DG↗
arXiv2009

Nested sums of symbols and renormalised multiple zeta functions

We define discrete nested sums over integer points for symbols on the real line, which obey stuffle relations whenever they converge. They relate to Chen integrals of symbols via the Euler-MacLaurin formula. Using a suitable holomorphic regularisation followed by a Birkhoff factorisation, we define renormalised nested sums of symbols which also satisfy stuffle relations. For appropriate symbols they give rise to renormalised multiple zeta functions which satisfy stuffle relations at all arguments. The Hurwitz multiple zeta functions fit into the framework as well. We show the rationality of multiple zeta values at nonpositive integer arguments, and a higher-dimensional analog is also investigated.

math.NT↗
arXiv2009

Double shuffle relations and renormalization of multiple zeta values

In this paper we present some of the recent progresses in multiple zeta values (MZVs). We review the double shuffle relations for convergent MZVs and summarize generalizations of the sum formula and the decomposition formula of Euler for MZVs. We then discuss how to apply methods borrowed from renormalization in quantum field theory and from pseudodifferential calculus to partially extend the double shuffle relations to divergent MZVs.

math.NT↗
arXiv2009

The multiplicative anomaly for determinants revisited; locality

Observing that the logarithm of a product of two elliptic operators differs from the sum of the logarithms by a finite sum of operator brackets, we infer that regularised traces of this difference are local as finite sums of noncommutative residues. From an explicit local formula for such regularised traces, we derive an explicit local formula for the multiplicative anomaly of zeta-determinants which sheds light on its locality and yields back previously known results.

math-ph↗
arXiv2008

Discrete sums of classical symbols on Z^d and zeta functions associated with Laplacians on tori

We prove the uniqueness of a translation invariant extension to non integer order classical symbols of the ordinary discrete sum on $L^1$-symbols, which we then describe using an Hadamard finite part procedure for sums over integer points of infinite unions of nested convex polytopes in $\R^d$. This canonical regularised sum is the building block to construct meromorphic extensions of the ordinary sum on holomorphic symbols. Explicit formulae for the complex residues at their poles are given in terms of noncommutative residues of classical symbols, thus extending results of Guillemin, Sternberg and Weitsman. These formulae are then applied to zeta functions associated with quadratic forms and with Laplacians on tori.

math.SP↗
arXiv2008

Renormalised iterated integrals of symbols with linear constraints

Given a holomorphic regularisation procedure (e.g. Riesz or dimensional regularisation) on classical symbols, we define renormalised multiple integrals of radial classical symbols with linear constraints. To do so, we first prove the existence of meromorphic extensions of multiple integrals of holomorphic perturbations ofradial symbols with linear constraints and then implement either generalised evaluators or a Birkhoff factorisation. Renormalised multiple integrals are covariant and factorise over independent sets of constraints.

math-ph↗
arXiv2007

The noncommutative residue and canonical trace in the light of Stokes' and continuity properties

We show that the noncommutative residue density, resp. the cut-off regularised integral are the only closed linear, resp. continuous closed linear forms on certain classes of symbols. This leads to alternative proofs of the uniqueness of the noncommutative residue, resp. the canonical trace as linear, resp. continuous linear forms on certain classes of classical pseudodifferential operators which vanish on brackets. The uniqueness of the canonical trace actually holds on classes of classical pseudodifferential with vanishing residue density which include non integer order operators in all dimensions and odd-class (resp. even-class) operators in odd (resp. even) dimensions. The description of the canonical trace for non integer order operators as an integrated global density on the manifold is extended to odd-class (resp. even-class) operators in odd (resp. even) dimensions on the grounds of defect formulae for regularised traces of classical pseudodifferential operators.

math.OA↗
arXiv2006

Renormalised Chern-Weil forms associated with families of Dirac operators

We provide local expressions for Chern-Weil type forms built from superconnections associated with families of Dirac operators previously investigated in work by S. Scott and later work by S. Scott and the second author. When the underlying fibration of manifolds is trivial, the even degree forms can be interpreted as renormalised Chern-Weil forms in as far as they coincide with regularised Chern-Weil forms up to residue correction terms. Similarly, a new formula for the curvature of the local fermionic vacuum line bundles is derived using a residue correction term added to the naive curvature formula. We interpret the odd degree Chern-Weil type forms built from superconnections as Wodzicki residues and establish a transgression formula along the lines of known transgression formulae for eta-forms.

math.DG↗
arXiv2006

Stokes' formulae on classical symbol valued forms and applications

The Wodzicki residue and the cut-off integral extend to classical symbol-valued forms. We show that they obey a Stokes' type property and that the extended Wodzicki residue can be interpreted as a complex residue like the ordinary one. In the case of cut-off integrals, Stokes' property (i.e. vanishing on exact forms) only holds for non-integer order symbol-valued forms and leads to an integration by parts formula and translation invariance for cut-off integrals on non-integer order classical symbols. The extended Wodzicki residue yields an even residue cycle on classical symbols and an odd cochain (the cosphere cochain) which measures an obstruction to Stokes' property of the cut-off integral on integer order symbol-valued forms.

math.DG↗
arXiv2006

Shuffle relations for regularised integrals of symbols

We prove shuffle relations which relate a product of regularised integrals of classical symbols to regularised nested (Chen) iterated integrals, which hold if all the symbols involved have non-vanishing residue. This is true in particular for non-integer order symbols. In general the shuffle relations hold up to finite parts of corrective terms arising from renormalisation on tensor products of classical symbols, a procedure adapted from renormalisation procedures on Feynman diagrams familiar to physicists. We relate the shuffle relations for regularised integrals of symbols with shuffle relations for multizeta functions adapting the above constructions to the case of symbols on the unit circle.

math-ph↗