Some Remarks on the Finitude of Quiver Theories

TL;DR

This paper explores the mathematical structure of quiver theories in string physics, linking graph theory and representation theory to physical properties like finiteness and IR behavior of gauge theories.

Contribution

It applies graph theory and axiomatic representation theory to analyze the finiteness and classification issues of quiver gauge theories in string theory.

Findings

Graph additivity implies finiteness constraints on gauge theories.

Complete IR freedom in string orbifold theories is impossible.

N<2 Yang-Mills theories in four dimensions are unclassifiable.

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.