Dualities of gauge theories

Lead Research Organisation: University of Oxford
Department Name: Mathematical Institute


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 $\tau$:
\mathbb{E}_\tau = \mathbb{C} /(\mathbb{Z} \oplus \tau \mathbb{Z})\,.
This provides the relation that
($\star$)\qquad $N=4$ SYM in $\mathbb{R}^{1,3} $ = 6d $N=2$ theory on $\mathbb{E}_\tau \times \mathbb{R}^{1,3}$
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.



Max Hubner (Student)


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Braun A (2019) Higgs bundles for M-theory on G2-manifolds in Journal of High Energy Physics

Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509711/1 01/10/2016 30/09/2021
1941660 Studentship EP/N509711/1 01/10/2017 30/09/2020 Max Hubner
Description We derived the dictionary between G2 manifolds with an ALE-fibrations and their, via M-theory, associated 4d N=1 theory. This dictionary is purely geometric if the Higgs field is associated to a Higgs bundle, in other cases background fluxes and non-commutative geometry play crucial roles. More specifically we analysed singular Higgs fields is associated to a singular Higgs bundle and derived the 4d spectrum and interactions from these.
Exploitation Route We laid the foundations for 4d N=1 local models with commutative geometry, a similar analysis for non-commutative geometries is an open and interesting problem.
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