On supertwistor geometry and integrability in super gauge theory

TL;DR

This thesis explores the role of twistor geometry in understanding integrability in supersymmetric gauge theories, revealing hidden symmetries, hierarchies, and conservation laws in self-dual SYM models.

Contribution

It provides a comprehensive twistor-based framework for analyzing integrability and symmetries in supersymmetric gauge theories, including new hierarchies and algebraic structures.

Findings

Derived affine extensions of internal and space-time symmetries.

Constructed self-dual SYM hierarchies with infinite flows.

Identified infinitely many nonlocal conservation laws.

Abstract

In this thesis, we report on different aspects of integrability in supersymmetric gauge theories. The main tool of investigation is twistor geometry. In trying to be self-contained, we first present a brief review about the basics of twistor geometry. We then focus on the twistor description of various gauge theories in four and three space-time dimensions. These include self-dual supersymmetric Yang-Mills (SYM) theories and relatives, non-self-dual SYM theories and supersymmetric Bogomolny models. Furthermore, we present a detailed investigation of integrability of self-dual SYM theories. In particular, the twistor construction of infinite-dimensional algebras of hidden symmetries is given and exemplified by deriving affine extensions of internal and space-time symmetries. In addition, we derive self-dual SYM hierarchies within the twistor framework. These hierarchies describe an infinite number of flows on the respective solution space, where the lowest level flows are space-time translations. We also derive infinitely many nonlocal conservation laws.