@article{ClavierGuoPaychaetal.2019, author = {Clavier, Pierre J. and Guo, Li and Paycha, Sylvie and Zhang, Bin}, title = {An algebraic formulation of the locality principle in renormalisation}, series = {European Journal of Mathematics}, volume = {5}, journal = {European Journal of Mathematics}, number = {2}, publisher = {Springer}, address = {Cham}, issn = {2199-675X}, doi = {10.1007/s40879-018-0255-8}, pages = {356 -- 394}, year = {2019}, abstract = {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 lattice cones, we renormalise the exponential generating function which sums over the lattice points in a lattice cone. As a consequence, for a suitable multivariate regularisation, renormalisation from the algebraic Birkhoff factorisation amounts to composition by a projection onto holomorphic multivariate germs.}, language = {en} } @incollection{ClavierGuoPaychaetal.2020, author = {Clavier, Pierre J. and Guo, Li and Paycha, Sylvie and Zhang, Bin}, title = {Renormalisation and locality}, series = {Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA) Volume 2}, booktitle = {Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA) Volume 2}, publisher = {European Mathematical Society Publishing House}, address = {Z{\"u}rich}, isbn = {978-3-03719-205-4 print}, doi = {10.4171/205}, pages = {85 -- 132}, year = {2020}, language = {en} } @article{Clavier2020, author = {Clavier, Pierre J.}, title = {Double shuffle relations for arborified zeta values}, series = {Journal of algebra}, volume = {543}, journal = {Journal of algebra}, publisher = {Elsevier}, address = {San Diego}, issn = {0021-8693}, doi = {10.1016/j.jalgebra.2019.10.015}, pages = {111 -- 155}, year = {2020}, abstract = {Arborified zeta values are defined as iterated series and integrals using the universal properties of rooted trees. This approach allows to study their convergence domain and to relate them to multiple zeta values. Generalisations to rooted trees of the stuffle and shuffle products are defined and studied. It is further shown that arborified zeta values are algebra morphisms for these new products on trees.}, language = {en} } @article{Clavier2021, author = {Clavier, Pierre J.}, title = {Borel-{\´E}calle resummation of a two-point function}, series = {Annales Henri Poincar{\´e} : a journal of theoretical and mathematical physics / ed. jointly by the Institut Henri Poincar{\´e} and by the Swiss Physical Society}, volume = {22}, journal = {Annales Henri Poincar{\´e} : a journal of theoretical and mathematical physics / ed. jointly by the Institut Henri Poincar{\´e} and by the Swiss Physical Society}, number = {6}, publisher = {Springer}, address = {Cham}, issn = {1424-0637}, doi = {10.1007/s00023-021-01057-w}, pages = {2103 -- 2136}, year = {2021}, abstract = {We provide an overview of the tools and techniques of resurgence theory used in the Borel-ecalle resummation method, which we then apply to the massless Wess-Zumino model. Starting from already known results on the anomalous dimension of the Wess-Zumino model, we solve its renormalisation group equation for the two-point function in a space of formal series. We show that this solution is 1-Gevrey and that its Borel transform is resurgent. The Schwinger-Dyson equation of the model is then used to prove an asymptotic exponential bound for the Borel transformed two-point function on a star-shaped domain of a suitable ramified complex plane. This proves that the two-point function of the Wess-Zumino model is Borel-ecalle summable.}, language = {en} } @article{ClavierGuoPaychaetal.2020, author = {Clavier, Pierre and Guo, Li and Paycha, Sylvie and Zhang, Bin}, title = {Locality and renormalization: universal properties and integrals on trees}, series = {Journal of mathematical physics}, volume = {61}, journal = {Journal of mathematical physics}, number = {2}, publisher = {American Institute of Physics}, address = {College Park, Md.}, issn = {0022-2488}, doi = {10.1063/1.5116381}, pages = {19}, year = {2020}, abstract = {The purpose of this paper is to build an algebraic framework suited to regularize 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 regularization of integrals indexed by rooted forests. We study their renormalization, along the lines of Kreimer's toy model for Feynman integrals.}, language = {en} } @article{ClavierFoissyPaycha2022, author = {Clavier, Pierre J. and Foissy, Loic and Paycha, Sylvie}, title = {From non-unitary wheeled PROPs to smooth amplitudes and generalised convolutions}, series = {European journal of mathematics}, volume = {8}, journal = {European journal of mathematics}, number = {Supplement 2}, publisher = {Springer}, address = {Berlin}, issn = {2199-675X}, doi = {10.1007/s40879-022-00557-1}, pages = {411 -- 480}, year = {2022}, abstract = {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 (respectively, 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 (respectively, smoothing operators) on a closed manifold form a TRAP. From their universal properties we build smooth amplitudes associated with the graph.}, language = {en} }