# Gauge theory - where particle physics and mathematics meet

Oxford Mathematician Yuuji Tanaka describes his part in the advances in our understanding of gauge theory.

"Gauge theory originated in physics, emerging as a unified theory of weak interaction such as appears in beta decay and electro-magnetism via the framework of Yang-Mills gauge theory together with the "Higgs mechanism" which wonderfully attaches mass to matters and forces. It became a mainstream of particle physics after the great discovery of the renormalisable property by Veltman and 't Hooft, and it gives precise descriptions of the experiments. Nowadays all fundamental interactions (electro-magnetism, weak force, strong force, gravity) can be described by gauge theory.

These development certainly stimulated the mathematical studies of gauge fields, particularly in the field of principal or vector bundles. In this context, the curvature of a connection corresponds to the field strength of a gauge field. In the early 80s Donaldson looked into the moduli space of solutions to a certain type of Yang-Mills gauge field (called self-dual or anti-self-dual connections), and produced surprising ways to distinguish differential structures of 4-dimensional curved spaces with the same topology type by using the moduli space or by defining invariants of smooth structures through the moduli space.

After Donaldson's work, Witten skilfully reinterpreted it in terms of a certain quantum field theory. Subsequently Atiyah and Jeffrey mathematically reformulated Witten's work by using the Mathai-Quillen formalism. These exchanges of ideas were one of the stepping stones which led to the discovery of the Seiberg-Witten equations and their invariants around 1994 through a generalisation of the electro-magnetic duality, a hidden symmetry in the theory of electro-magnetism. Seiberg and Witten presented a striking application of this on a super Yang-Mills theory in the quantum level, called strong-weak duality, which enables one to calculate things in the strong coupling region of the theory in terms of ones in the weak coupling region.

Vafa and Witten further analysed Seiberg and Witten's work in a more symmetric model, and conjectured that the partition function of the invariants in this case would have a modular property, which is a mathematical enhancement of the strong-weak duality mentioned above and originally discovered in the theory of elliptic curves in the 19th century. They examined this property in examples by using results from mathematics under the assumption that the Higgs fields automatically vanish.

However, even a mathematically rigorous definition of this, especially including Higgs fields, was not produced for over 20 years. Richard Thomas and myself recently defined deformation invariants of projective surfaces, from the moduli space of solutions to the gauge-theoretic equations of the Vafa and Witten theory, by using modern techniques in algebraic geometry. We then computed partition functions of the invariants coming from non-vanishing Higgs fields as well. Surprisingly, our calculations match the conjecture by Vafa and Witten more than two decades ago despite ours also including sheaves on the surface."