ADHM instantons and the Gauss-Bonnet integral on non-compact hyper-Kahler manifolds

• Main Author: Burgess, Andrew E. F.
• Published: Swansea University 2004
• Subjects: 519
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.636177

The main intention of this thesis is to calculate the Gauss-Bonnet integral on the moduli space of Yang-Mills instantons and in particular to test a conjecture of Dorey, Hollowood and Khoze which relates the D-instanton partition function (a quantity arising from string-theoretic considerations), and the Gauss-Bonnet integral on the resolved moduli space of instantons. We shall present two main results. Firstly, we use the ADHM construction to determine the metric on the moduli space of a single <i>SU</i>(3) instanton. The result obtained agrees with the previous result of [20]. From this metric we calculate the spin connection and the curvature. Ultimately, we were able to evaluate the Gauss-Bonnet integral over this resolved moduli space. This involved a non-trivial integral over an eight dimensional hyper-Kahler space. The result obtained confirms the prediction of [17]. Secondly, I have also been able to verify explicitly that the D-instanton partition function derived from string theory reduces to the Gauss-Bonnet integral on the resolved instanton moduli space for the case of a single instanton in an arbitrary gauge group. In the introductory chapter, we discuss in general term the motivation for the calculations presented in this thesis. In chapter two, we discuss zero modes and collective co-ordinates and introduce the notion of a moduli space. We also verify that the instanton moduli space in hyper-Kahler. Chapter three discusses the ADHM construction and we pursue some of its consequences. Chapter four is devoted to obtaining the supersymmetric quantum mechanical sigma model on the moduli space of instantons and the elucidation of its geometrical significance. Chapter five is where we illustrate the explicit implementation of the ADHM construction and calculate the Gauss-Bonnet integral in the single instanton <i>SU</i>(3) case. The results of this calculation are compared with those obtained by [17]. Their method is reviewed in chapter six. The results of both are in agreement.