Abstract
D-brane probes, Hanany-Witten setups and geometrical engineering stand as a trichotomy of standard techniques of constructing gauge theories from string theory. Meanwhile, asymptotic freedom, conformality 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.
Highlights
In a quantum field theory (QFT), it has been known since the 1970s (e.g., [1]) that the behaviour of physical quantities such as mass and coupling constant are sensitive to the renormalization and evolve according to momentum scale as dictated by the so-called renormalisation flows
We have briefly reviewed the three contemporary techniques of obtaining four dimensional gauge theories from string theory, namely Hanany-Witten, D-brane probes and geometrical engineering
We focus on what finitude signifies for these theories and how interests in quiver diagrams arises
Summary
In a quantum field theory (QFT), it has been known since the 1970s (e.g., [1]) that the behaviour of physical quantities such as mass and coupling constant are sensitive to the renormalization and evolve according to momentum scale as dictated by the so-called renormalisation flows. Mathematics 2018, 6, 291 and, vice versa, facilitate the incorporation of the Standard Model into string unifications These finite (super-)conformal theories in four dimensions still remain a topic of fervent pursuit. Though the main results are given, we begin with some preliminaries from contemporary techniques in string theory on constructing four dimensional super-Yang-Mills, focusing on what each interprets finitude to mean: Section 2.1 on D-brane probes on orbifold singularities, Section 2.2 on Hanany-Witten setups and Section 2.3 on geometrical engineering. Since our quivers allow more edges between vertices, one should strictly use “directed pseudographs”, but, for simplicity, we will use “quiver” and “directed graph” inter-changeably
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