Mathematical Aspects of Field Theory: Nahm's Equations and Jacobi Forms

Abstract

This thesis consists of two projects wherein we explore some mathematical aspects of field theory. In the first project, we address Nahm's equations, which is an integrable system with a Lax pair. We consider boundary conditions on Nahm's equations that correspond to the Dirac multimonopole in Yang-Mills theory. The algebro-geometric integration method is to construct solutions via a linear flow in the Jacobian of the spectral curve associated to the Lax pair. We construct a frame of sections of this linear flow, which allows us to obtain exact solutions to Nahm's equations for arbitrary rank n. Nahm's equations with our boundary conditions correspond to the Dirac multimonopole via the ADHMN construction. The ADHMN construction requires us to find normalizable zero modes of Dirac operators. We again use the frame of sections of the linear flow on the Jacobian of the spectral curve to construct these normalizable zero modes. In the second project, we consider weak Jacobi forms of weight 0. The polar coefficients of such weak Jacobi form are known to uniquely determine the weak Jacobi form, and we improve on the number of polar coefficients that determine the weight 0 form. Weak Jacobi forms of weight 0 may be exponentially lifted to Siegel modular forms, which appear in the string-theory of black holes. In connection to this, the growth of a certain sum about a term q^a y^b in the Fourier-Jacobi expansion of the underlying weak Jacobi form is of interest to us. We discover that the weak Jacobi forms which are quotients of theta functions give us a large class of forms that are slow growing about their most polar term. Additionally, the characteristics of growth behavior for a weak Jacobi form about a term y^b are known, here we investigate growth behavior about an arbitrary q^a y^b term and find several analogues.