By Maxim Jeffs, The Australian National University
In 1960, the physicist Eugene Wigner published his essay ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’, partly as a result of his surprise at the connections he had discovered between quantum mechanics and the theory of group representations – a seemingly unrelated area of mathematics developed some 50 years earlier. He could hardly have predicted what would occur 20 years later in the opposite direction: the unreasonable effectiveness of physics in mathematics. Conjectures coming from theoretical physics now pervade numerous disciplines of pure mathematics, often relating well-understood subjects in new ways and providing explanations for many surprising coincidences. The construction of the extended topological quantum eld theory of the Fukaya category in Yang-Mills theory, my project for the AMSI VRS, provides an excellent example.
In the 1980s, physicists were considering gauge theories as models for the fundamental interactions between particles, governed in the non-quantum limit by the ‘Yang-Mills equations’. The mathematicians Sir Michael Atiyah and Raoul Bott found that these equations also displayed remarkable mathematical properties and were related in a fundamental way to classical geometric problems. Atiyah’s student, Sir Simon Donaldson was able to build upon this work to prove Donaldson’s Theorem, which has as a remarkable consequence the existence of an exotic smooth structure on four-dimensional Cartesian space. Roughly speaking, this means that there is a consistent way to put coordinates on all of four-dimensional space in such a way that they cannot be converted smoothly into the usual Cartesian coordinates in a global fashion! The invariants of four-dimensional spaces that Donaldson introduced based on gauge theory revolutionised the study of geometry and topology in four dimensions.
When it comes to calculating Donaldson’s invariants, it is useful to consider so-called gluing formulas that describe what happens to the Donaldson invariant under cutting and stitching operations on the associated space. The systematic formalism for this is called an extended topological quantum eld theory. One of the beautiful observations of this theory is that the Chern-Simons function, arising in condensed matter physics, is closely related to Donaldson’s theory; its critical points and gradient flows correspond to special solutions of the Yang-Mills equations! As well as condensed matter physics, this extended topological quantum eld theory also pulls in ideas from areas of modern theoretical physics such as supersymmetry and string theory, and the remarkable coincidences that make the mathematics work ultimately derive from physics.
I would like to thank the Australian Mathematical Sciences Institute for giving me the opportunity to explore these fascinating areas of mathematics as part of my project over this summer.
Maxim Jeffs was one of the recipients of a 2015/16 AMSI Vacation Research Scholarship.