# Dualities of gauge theories

Lead Research Organisation:
University of Oxford

Department Name: Mathematical Institute

### Abstract

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.

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.

## People |
## ORCID iD |

Max Hubner (Student) |

### Publications

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. |

Sectors | Other |