bibkey: clavier2025localityhdr
authors: Pierre J. Clavier
year: 2025
title: "TRAPs, Generalisations of MZVs, Locality and Resurgence for Quantum Field Theories"
doi: null
url: https://arxiv.org/abs/2506.09493v1
claim: "The habilitation thesis surveys the locality framework for renormalisation and states as its Conjecture conj:loc_tensor that for any locality vector space (E,⊤) and any two locality subspaces V and W, the locality tensor product V⊗_⊤W with the quotient locality relation is a locality vector space."
strata_touched:
  - D5/S3/HomologicalAlgebra/LocalityTensorProductRefutation
license: citation-only
triage: anchor

TRAPs, Generalisations of MZVs, Locality and Resurgence for Quantum Field Theories

P. J. Clavier, habilitation thesis (Université de Haute-Alsace), arXiv:2506.09493v1 [math-ph]. Quotations are from the v1 TeX source.

The quotient locality (Definition defn:quotientlocality):

For a subspace $W$ of a locality vector space $(V,\top )$, we call {\bf quotient locality} on the quotient $V/W$, the final locality relation $\left([u]\text{ttop}[v]⟺\exists (u^{\prime },v^{\prime })\in [u]\times [v]:u^{\prime }\top v^{\prime }\right)$

The conjecture (Conjecture conj:loc_tensor), introduced as the most important open gap of the locality framework:

Let $(E,\top )$ be any locality vector space, $V$ and $W$ any two locality vector subspaces of $E$. Then $(V{\otimes }_{\top }W,\text{ttop})$ is a locality vector space.

and, among the listed evidence:

We could not find counterexamples, although we did spend time and effort to build them.

The locality cartesian power (Eq. eq:lprod):

$X^{{}_{\top }k}:=\underset{k\text{ factors}}{\underbrace{X{\times }_{\top }\cdots {\times }_{\top }X}}:=\left\{(x_1,\cdots ,x_k)\in X^k|(x_i,x_j)\in \top ,1\le i\ne j\le k\right\}.$

and the explicit description of the tensor locality:

Then for $X,Y\in E{\otimes }_{\top }E$, we have $X{\top }_{\otimes 2}Y$ if we can write $X={\sum }_i{v}_i\otimes w_i,Y={\sum }_i{v}_i^{\prime }\otimes w_i^{\prime }$ such that $(v_i,w_i,v_j^{\prime },w_j^{\prime })\in E^{{\times }_{\top }4}$ for any $i$ and $j$.

(the displays are written inline here).

Verified locator

Source: https://arxiv.org/abs/2506.09493v1

• URL: https://arxiv.org/abs/2506.09493v1 (source chapter LOC.tex, md5 41f5e772e40d5eaf8583e8903293dbbb, 1831 lines):
  • the locality cartesian power, Eq. eq:lprod, l. 219–221;
  • the quotient locality, Definition defn:quotientlocality, l. 606–612;
  • the explicit description of the tensor locality through $E^{{\times }_{\top }4}$, l. 743–747;
  • the statement that $E{\otimes }_{\top }E$ being a locality vector space is still a conjecture, l. 749;
  • the conjecture and its evidence, Conjecture conj:loc_tensor, l. 1794–1806.