# nLab electric-magnetic duality

Contents

## Surveys, textbooks and lecture notes

#### Differential cohomology

differential cohomology

## Application to gauge theory

#### Duality in string theory

duality in string theory

general mechanisms

string-fivebrane duality

string-string dualities

M-theory

F-theory

string-QFT duality

QFT-QFT duality:

# Contents

## Idea

Electric-magnetic duality is a lift of Hodge duality from de Rham cohomology to ordinary differential cohomology.

## Description

Consider a circle n-bundle with connection $\nabla$ on a space $X$. Its higher parallel transport is the action functional for the sigma-model of $(n-1)$-dimensional objects ($(n-1)$-branes) propagating in $X$.

For $n = 1$ this is the coupling of the electromagnetic field to particles. For $n = 2$ this is the coupling of the Kalb-Ramond field to strings.

The curvature $F_\nabla \in \Omega^{n+1}(X)$ is a closed $(n+1)$-form. The condition that its image $\star F_\nabla$ under the Hodge star operator is itself closed

$d_{dR} \star F_\nabla = 0$

is the Euler-Lagrange equation for the standard (abelian Yang-Mills theory-action functional on the space of circle n-bundle with connection.

If this is the case, it makes sense to ask if $\star F_\nabla$ itself is the curvature $(d-(n+1))$-form of a circle $(d-(n+1)-1)$-bundle with connection $\tilde \nabla$, where $d = dim X$ is the dimension of $X$.

If such $\tilde \nabla$ exists, its higher parallel transport is the gauge interaction action functional for $(d-n-3)$-dimensional objects propagating on $X$.

In the special case of ordinary electromagnetism with $n=1$ and $d = 4$ we have that electrically charged 0-dimensional particles couple to $\nabla$ and magnetically charged $(4-(1+1)-2) = 0$-dimensional particles couple to $\tilde \nabla$.

In analogy to this case one calls generally the $d-n-3$-dimensional objects coupling to $\tilde \nabla$ the magnetic duals of the $(n-1)$-dimensional objects coupling to $\nabla$.

## Generalizations

For $d= 4$ EM-duality is the special abelian case of S-duality for Yang-Mills theory. Witten and Kapustin argued that this is governed by the geometric Langlands correspondence.

## Examples

• M2-brane and M5-brane: the 4-form $G_4$ which the M2-brane couples to is the Hodge-dual of the 7-form $G_7$ which the M5-brane couples to
$G_7 = \star G_4$

as imposed by the equations of motion for 11d supergravity (see here). The associated C3-field and C6-field are electric-magnetic duals.

Detailed review is in

It was originally noticed in

• P. Goddard, J. Nuyts, and David Olive, Gauge Theories And Magnetic Charge, Nucl. Phys. B125 (1977) 1-28.

that where electric charge in Yang-Mills theory takes values in the weight lattice of the gauge group, then magnetic charge takes values in the lattice of what is now called the Langlands dual group.

This led to the electric/magnetic duality conjecture formulation in

According to (Kapustin-Witten 06, pages 3-4) the observation that the Montonen-Olive dual charge group coincides with the Langlands dual group is due to

The insight that the Montonen-Olive duality works more naturally in super Yang-Mills theory is due to

and that it works particularly for N=4 D=4 super Yang-Mills theory is due to

• H. Osborn, Topological Charges For $N = 4$ Supersymmetric Gauge Theories And Monopoles Of Spin 1, Phys. Lett. B83 (1979) 321-326.

The observation that the $\mathbb{Z}_2$ electric/magnetic duality extends to an $SL(2,\mathbb{Z})$-action in this case is due to

• John Cardy, E. Rabinovici, Phase Structure Of Zp Models In The Presence Of A Theta Parameter, Nucl. Phys. B205 (1982) 1-16;

• John Cardy, Duality And The Theta Parameter In Abelian Lattice Models, Nucl. Phys. B205 (1982) 17-26.

• A. Shapere and Frank Wilczek, Selfdual Models With Theta Terms, Nucl. Phys. B320 (1989) 669-695.

and specifically the embedding of this into string theory S-duality originates in

• Ashoke Sen, Dyon - Monopole Bound States, Self-Dual Harmonic Forms on the Multi-Monopole Moduli Space, and $SL(2,\mathbb{Z})$ Invariance in String Theory (arXiv:hep-th/9402032)

The understanding of this $SL(2,\mathbb{Z})$-symmetry as a remnant conformal transformation on a 6-dimensional principal 2-bundle-theory – the 6d (2,0)-superconformal QFT – compactified on a torus is described in

The relation of S-duality to geometric Langlands duality was understood in

Exposition of this is in

• Edward Frenkel, What Do Fermat’s Last Theorem and Electro-magnetic Duality Have in Common? KITP talk 2011 (web)

A perspective unifying EM-duality of higher gauge theories with non-abelian T-duality: