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Bratteli networks and the Spectral Action on quivers

Carlos I. Perez-Sanchez

TL;DR

This work develops a novel framework linking quiver representations with finite-dimensional prespectral triples via Bratteli networks, producing a combinatorial description of Rep$(Q)$, the gauge group, and their quotient. By constructing spectral triples directly from representation data, it expresses the spectral action as a sum over closed paths on the quiver, recovering lattice Yang–Mills theory and Weisz–Wohlert improvements in the lattice setting, with a Higgs field arising from self-loops. The formalism enables a path integral over Dirac operators by integrating over ${ m Rep} olimits(Q)$, aligning with random noncommutative geometry ideas and yielding a continuum YM–Higgs limit on flat space. The framework carefully distinguishes genuine representations from spurious vertex-labels via Bratteli networks, and offers a principled route to quantisation of quiver-based geometries through finite-dimensional truncations and Dirac-operator ensembles.

Abstract

In the context of noncommutative geometry, we consider quiver representations -- not on vector spaces, as traditional, but on finite-dimensional prespectral triples (`discrete topological noncommutative spaces'). A similar idea appeared in the original work of Marcolli-van Suijlekom on quiver representations in spectral triples (`discrete noncommutative geometries'), which paved the way for some of our results in independent directions. We introduce Bratteli networks, a structure that yields a neat combinatorial characterisation of the space $\mathrm{Rep}~Q$ of prespectral-triple-representations of a quiver $\mathrm{Rep}~Q$, as well as of the gauge group and of their quotient. Not only these claims that make it possible to `integrate over $\mathrm{Rep}~Q$' are, as we now argue, in line with the spirit of random noncommutative geometry -- formulating path integrals over Dirac operators -- but they also contain a physically relevant case. Namely, the equivalence between quiver representations and path algebra modules, established here for the new category, inspired the following construction: Only from representation theory data, we build a spectral triple for the quiver and evaluate the spectral action functional from a general formula over closed paths. When we apply this construction to lattice-quivers, we obtain not only Wilsonian Yang-Mills lattice gauge theory, but also the Weisz-Wohlert-cells in the context of Symanzik's improved gauge theory. We show that a hermitian (`Higgs') matrix field emerges from the self-loops of the quiver and derive the Yang-Mills--Higgs theory on flat space as a smooth limit.

Bratteli networks and the Spectral Action on quivers

TL;DR

This work develops a novel framework linking quiver representations with finite-dimensional prespectral triples via Bratteli networks, producing a combinatorial description of Rep, the gauge group, and their quotient. By constructing spectral triples directly from representation data, it expresses the spectral action as a sum over closed paths on the quiver, recovering lattice Yang–Mills theory and Weisz–Wohlert improvements in the lattice setting, with a Higgs field arising from self-loops. The formalism enables a path integral over Dirac operators by integrating over , aligning with random noncommutative geometry ideas and yielding a continuum YM–Higgs limit on flat space. The framework carefully distinguishes genuine representations from spurious vertex-labels via Bratteli networks, and offers a principled route to quantisation of quiver-based geometries through finite-dimensional truncations and Dirac-operator ensembles.

Abstract

In the context of noncommutative geometry, we consider quiver representations -- not on vector spaces, as traditional, but on finite-dimensional prespectral triples (`discrete topological noncommutative spaces'). A similar idea appeared in the original work of Marcolli-van Suijlekom on quiver representations in spectral triples (`discrete noncommutative geometries'), which paved the way for some of our results in independent directions. We introduce Bratteli networks, a structure that yields a neat combinatorial characterisation of the space of prespectral-triple-representations of a quiver , as well as of the gauge group and of their quotient. Not only these claims that make it possible to `integrate over ' are, as we now argue, in line with the spirit of random noncommutative geometry -- formulating path integrals over Dirac operators -- but they also contain a physically relevant case. Namely, the equivalence between quiver representations and path algebra modules, established here for the new category, inspired the following construction: Only from representation theory data, we build a spectral triple for the quiver and evaluate the spectral action functional from a general formula over closed paths. When we apply this construction to lattice-quivers, we obtain not only Wilsonian Yang-Mills lattice gauge theory, but also the Weisz-Wohlert-cells in the context of Symanzik's improved gauge theory. We show that a hermitian (`Higgs') matrix field emerges from the self-loops of the quiver and derive the Yang-Mills--Higgs theory on flat space as a smooth limit.
Paper Structure (29 sections, 20 theorems, 144 equations, 12 figures)

This paper contains 29 sections, 20 theorems, 144 equations, 12 figures.

Table of Contents

  1. Motivation and Introduction
  2. Preliminary work and two observations
  3. A remark about the Higgs field
  4. A remark about the gauge group of the quiver
  5. Strategy and Results
  6. Prespectral triples
  7. The category of prespectral triples
  8. Characterisation of morphisms
  9. Involutive algebra morphisms
  10. Hilbert space compatible morphisms
  11. Quiver representations on prespectral triples
  12. Quivers weighted by operators
  13. Path algebras
  14. Quiver ${\mathscr{pS}}$-representations and path algebra modules
  15. Combinatorial description of $\mathop{\mathrm{\mathrm{Rep}}}\nolimits (Q)$
  16. ...and 14 more sections

Key Result

Lemma 2.6

If $*$-$\mathsf{alg}$ denotes the category of unital, involutive algebras, for $\mathbf{m}\in \mathbb{Z}_{> 0 }^{l_s}$ and $\mathbf n\in \mathbb{Z}_{> 0 }^{l_t}$,

Figures (12)

  • Figure 1: Naive examples with graphs and quivers
  • Figure 2: The $3$-cycle quiver $C_3$, its augmented quiver $C_3^\star$ and their weights
  • Figure 3: On the proof of Proposition \ref{['thm:Wk']} and Corollary \ref{['thm:Wksym']}.
  • Figure 4: Bratteli networks from $\mathop{\mathrm{\mathrm{Rep}_{{\mathscr{pS}}}}}\nolimits(Q)$ with light gray vertices over a quiver $Q$ in blue vertices. Any path in $Q$ should lift to a sequence of Bratteli diagrams. This constrains e.g. the edges between $v$ and $z$ to have a symmetric Bratteli diagram (thus the identity).
  • Figure 5: Left: The commutative diagram defining invertible natural transformation $R=(X,\Phi)\to R'=(X',\Phi')$. Middle: A path on a quiver $Q$ is shown, and two lifts $\Phi'_p = \Phi_{e_k}' \circ \cdots \circ \Phi_{e_2}' \circ \Phi_{e_1}'$ (uppermost) and $\Phi_p=\Phi_{e_k} \circ \cdots \circ \Phi_{e_2} \circ \Phi_{e_1}$ (lower). As vertical morphisms are invertible, the diagram on the left commutes for any path $p$ if and only if it does so for any edge $e$, for intermediate curved arrows cancel out after concatenation of single-edge diagrams. Right: The (as explained below w.l.o.g.) reduced case of the two representations $R, R'$ coinciding in objects, $X_v=X_v'$ for all $v\in Q_0$. Black arrows represent the parameters $\Upsilon_v\in \mathop{\mathrm{Aut}}\nolimits_{{\mathscr{pS}}}(X_v)$ of the gauge group $\mathcal{G}(Q)$ (given a Bratteli network).
  • ...and 7 more figures

Theorems & Definitions (75)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.8: Consequence of Artin-Wedderburn Theorem
  • proof
  • Example 2.9
  • ...and 65 more