| This thesis covers two distinct parts: Holomorphic Anomaly in Gauge Theory on ALE Space and Freudenthal Gauge Theory.;In part I, I presented a concise review of the Seiberg-Witten curve, Nekrasov's background, geometric engineering and the holomorphic anomaly equation followed by my published work: Holomorphic Anomaly in Gauge Theory on ALE Space, where an deformed N = 2 SU(2) gauge theory on A1 space and its five dimension lift is studied. We find that the partition functions can be reproduced via special geometry and the holomorphic anomaly equation. Schwinger type integral expressions for the boundary conditions at the monopole/dyon point in moduli space are inferred. The interpretation of the five dimensional partition function as the partition function of a refined topological string on A1x(local P1 x P1) is suggested.;In part II, I give a comprehensive review of the Freudenthal Triple System, including Freudenthal's orginal construction from Jordan Triple Systems and its relation to Lie algebra, Yang-Baxter equation, and 4d N = 2 BPS black holes, where the novel Freudenthal-dual was discovered. I also present my published work on the Freudenthal Gauge Theory, where we construct the most generic gauge theory admitting F-dual, and prove a no-go theorem that forbids coupling of a F-dual invariant gauge theory to supersymmetry. |
