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Some Remarks on the Finitude of Quiver Theories

Yang-Hui He

TL;DR

This work links finiteness in four-dimensional gauge theories to quiver representations arising from string-theory constructions (D-brane probes, Hanany–Witten setups, and geometrical engineering). By translating physical beta-function conditions into a discriminant function $f_{ij} = d\delta_{ij} - a_{ij}^d$ on quivers and leveraging foundational results from representation theory (Gabriel’s and Nazarova’s theorems, the Happel–Preiser–Ringel framework), it classifies allowed quiver types for finite and (tame) affine theories, showing ${\cal N}=2$ finiteness corresponds to affine $\widehat{ADE}$ or infinite Dynkin/quivers. It further argues that orbifold and related constructions cannot yield completely IR-free theories (Steinberg’s semi-definiteness constraint) and that, in general, $\mathcal{N}<2$ theories are unclassifiable within a finite schema. The paper thus provides a rigorous, cross-disciplinary bridge between string-theoretic gauge construction and the axiomatic theory of quivers, offering precise graph-theoretic constraints to guide model-building and highlighting intrinsic limitations in fully classifying four-dimensional gauge theories.

Abstract

D-brane probes, Hanany-Witten setups and geometrical engineering stand as a trichotomy of the currently fashionable techniques of constructing gauge theories from string theory. Meanwhile, asymptotic freedom, finitude and IR freedom pose as a trichotomy of the beta-function behaviour in quantum field theories. Parallel thereto is a trichotomy in set theory of finite, tame and wild representation types. At the intersection of the above lies the theory of quivers. We briefly review some of the terminology standard to the physics and to the mathematics. Then we utilise certain results from graph theory and axiomatic representation theory of path algebras to address physical issues such as the implication of graph additivity to finiteness of gauge theories, the impossibility of constructing completely IR free string orbifold theories and the unclassifiability of N<2 Yang-Mills theories in four dimensions.

Some Remarks on the Finitude of Quiver Theories

TL;DR

This work links finiteness in four-dimensional gauge theories to quiver representations arising from string-theory constructions (D-brane probes, Hanany–Witten setups, and geometrical engineering). By translating physical beta-function conditions into a discriminant function on quivers and leveraging foundational results from representation theory (Gabriel’s and Nazarova’s theorems, the Happel–Preiser–Ringel framework), it classifies allowed quiver types for finite and (tame) affine theories, showing finiteness corresponds to affine or infinite Dynkin/quivers. It further argues that orbifold and related constructions cannot yield completely IR-free theories (Steinberg’s semi-definiteness constraint) and that, in general, theories are unclassifiable within a finite schema. The paper thus provides a rigorous, cross-disciplinary bridge between string-theoretic gauge construction and the axiomatic theory of quivers, offering precise graph-theoretic constraints to guide model-building and highlighting intrinsic limitations in fully classifying four-dimensional gauge theories.

Abstract

D-brane probes, Hanany-Witten setups and geometrical engineering stand as a trichotomy of the currently fashionable techniques of constructing gauge theories from string theory. Meanwhile, asymptotic freedom, finitude and IR freedom pose as a trichotomy of the beta-function behaviour in quantum field theories. Parallel thereto is a trichotomy in set theory of finite, tame and wild representation types. At the intersection of the above lies the theory of quivers. We briefly review some of the terminology standard to the physics and to the mathematics. Then we utilise certain results from graph theory and axiomatic representation theory of path algebras to address physical issues such as the implication of graph additivity to finiteness of gauge theories, the impossibility of constructing completely IR free string orbifold theories and the unclassifiability of N<2 Yang-Mills theories in four dimensions.

Paper Structure

This paper contains 11 sections, 13 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries from the Physics
  3. D-brane Probes on Orbifolds
  4. Hanany-Witten
  5. Geometrical Engineering
  6. Preliminaries from the Mathematics
  7. Quivers and Path Algebras
  8. Representation Type of Algebras
  9. Restrictions on the Shapes of Quivers
  10. Quivers in String Theory and Yang-Mills in Graph Theory
  11. Concluding Remarks and Prospects

Figures (3)

  • Figure 1: Two examples of quivers with nodes and edges labeled.
  • Figure 2: The Finite and Infinite Dynkin Diagrams as labeled quivers. The finite cases are the well-known Dynkin-Coxeter graphs in Lie Algebras (from Chapter 4 of Rep).
  • Figure 3: The Euclidean Diagrams as labeled quivers; we recognise that this list contains the so-called Affine Dynkin Diagrams (from Chapter 4 of Rep).