Some Remarks on the Finitude of Quiver Theories

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. 1 Research supported in part by the CTP and the LNS of MIT and the U.S. Department of Energy under cooperative research agreement #DE-FC02-94ER40818, the ITP of UCSB under NSF grant PHY94-07194, as well as the NSF Graduate Fellowship. 2 Address until Nov. 21st, 1999.