Dualities of gauge theories

Dualities of gauge theories, such as the Montonen-Olive duality of $N=4$ SYM can be lifted to dualities in supersymmetric string theories, and have a purely geometric origin in this context. This will allow a concrete study of these systems, using geometric tools.
Thus, to make progress in understanding of the S-duality of $N=4$ SYM, the connnection to string theory and M-theory will be central. There will be two particularly important implications that the relation to string theory opens up: the first is the the idenficiation of $N=4$ SYM with the dimensional reduction of a unique 6d superconformal theory. The second is the realization of both the 4d SYM as well as 6d theory in the context of string/M-theory.

$N=4$ SYM arises as a dimensional reduction on an elliptic curve $\mathbb{E}$ (i.e. a two-torus) from a 6d theory: as shown by Nahm, there is unique, up to choice of gauge group, 6d superconformal theory with $N=2$ supersymmetry, which in string theory lives on the so-called M5-brane. In the reduction, $\tau$ is the complex structure (``shape parameter") of the elliptic curve, which can be thought of as quotient of the complex plane by a 2d lattice, parametrized by $\tau$:
\begin{equation}
\mathbb{E}_\tau = \mathbb{C} /(\mathbb{Z} \oplus \tau \mathbb{Z})\,.
\end{equation}
This provides the relation that
\begin{center}
($\star$)\qquad $N=4$ SYM in $\mathbb{R}^{1,3} $ = 6d $N=2$ theory on $\mathbb{E}_\tau \times \mathbb{R}^{1,3}$
\end{center}
and from this point of view the coupling $\tau$ has a purely geometric interpretation.

The second important input will be the realization of $N=4$ SYM into string theory. This is possible through the embedding of supersymmetric gauge theories in terms of D$p$-branes, which are $(p+1)$-dimensional membranes, whose physical excitations are characterized in terms of $(p+1)$-dimensional SYM theories. Specifically, 4d $N=4$ SYM, the maximally supersymmetric Yang-Mills theory in 4d, is realized on D3-branes, which are dynamical branes in IIB string theory or more precisely, it's non-perturbative generalization, F-theory. The S-duality group in turn has a natural realization in terms of the self-duality of IIB string theory under an $SL_2\mathbb{Z}$ symmetry, which acts on the complexified string coupling in the same fashion as given in (\ref{SL2Z}).

The goal of the project is to develop a formulation of 4d $N=4$ SYM where the coupling $\tau$ is not necessarily constant, but is allowed to vary over the 4d spacetime. Some related configurations are known as Janus configurations, however here we wish to allow $\tau$ to vary compatibly with the $SL_2\mathbb{Z}$ action. Mathematically, this means that the relation $(\star)$ is relaced by studying the 6d theory on an elliptic fibration. In string theory, this would correspond to studying D3-branes in the context of F-theory, i.e. the non-perturbative version of IIB string theory, where the complexified string coupling varies over space-time.

Such configurations have been initiated with related configurations for 4d $N=2$ gauge theories in \cite{b}, as well as in lower dimensional compactifications of $N=4$ SYM \cite{c}, \cite{d}. But a complete understanding of the theories remains to be uncovered. This is the goal of the present project.

This project falls within the EPSRC Mathematical Physics research area.