Renormalization flow fixed points for higher-dimensional abelian gauge fields

A connection modulo gauge symmetry on the trivial principal bundle \(M\times G\) is a morphism from the loop group of \(M\) into \(G\). Thus, considering only loops around the 2-cells of a distinguished family of progressively refined cellular structures on \(M\), the observable algebra \(A\) of an...

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Bibliographic Details
Published in:arXiv.org 2020-01
Main Author: Rodrigo Vargas Le-Bert
Format: Article
Language:English
Subjects:
Cellular structure
Differential equations
Interaction parameters
Mathematical analysis
Operators (mathematics)
Polynomials
Quotients
Yang-Mills fields
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Summary:A connection modulo gauge symmetry on the trivial principal bundle \(M\times G\) is a morphism from the loop group of \(M\) into \(G\). Thus, considering only loops around the 2-cells of a distinguished family of progressively refined cellular structures on \(M\), the observable algebra \(A\) of an abelian gauge field can be presented as an inductive limit of quotients of polynomial algebras. In that context, it turns out that the state \(\mu_\lambda:A\rightarrow\mathbb{C}\) of the Yang-Mills field on the sphere can be written \(\mu_\lambda = \mu_0\mathrm{e}^{\lambda L}\) with \(\lambda\) an interaction strength parameter, \(L:A\rightarrow A\) an explicit second-order partial differential operator and \(\mu_0\) the state of an almost surely flat connection. Extrapolating, we provide analogous states for the case of abelian gauge fields on \(\mathbb{R}^d\).
ISSN:2331-8422