Integrability in large N matrix models

Understanding the large N limit of multi-matrix models in the Hamiltonian formalism is central to answering many important questions in the studies of Yang-Mills theories, with or without supersymmetry, and string theories. In the present work, we present an algebraic approach by which several physically interesting matrix model Hamiltonians can be diagonalized in the large N limit. This approach makes use of the connection between Hamiltonians of large N matrix models and those of quantum spin chains introduced by Rajeev and Lee in the recent past. A large class of quantum spin chains have been proven to exactly solvable quantum many body systems, and there also exists an understanding of their exact solvability in terms of novel Hopf algebraic symmetries. We present, in this work, a detailed account of how a particular kind of Hopf algebra, known as the Yangian algebra, can be realized as the symmetry of a reasonably large class of large N matrix models. We then utilize these symmetries to compute the spectra of the matrix model Hamiltonians that possess Yangian symmetries. The choice of matrix model Hamiltonians studied in the present work is governed by their applications to higher dimensional gauge theories. Four dimensional gauge theories when studied in the scheme of radial quantization produce matrix models as the effective radial Hamiltonians order by order in perturbation theory. In particular, we study matrix model Hamiltonians which are the effective Hamiltonians of superconformal Yang-Mills theory. As a by-product of the study of integrability of large N matrix models, we shall present the explicit spectrum of anomalous dimensions of superconformal Yang-Mills theory up to the third order in perturbation theory. Apart from the three loop solutions of gauge theories, several string theoretical implications of these results will also be discussed.