# nLab S-duality

duality

## In QFT and String theory

The term S-duality can mean two different things:

# Contents

## Idea

In its original form, S-duality refers to Montonen-Olive duality , which is about the following phenomenon:

The Lagrangian of Yang-Mills theory has two summands,

${S}_{\mathrm{YM}}:\nabla ↦{\int }_{X}\frac{1}{{e}^{2}}⟨{F}_{\nabla }\wedge \star {F}_{\nabla }⟩+{\int }_{X}i\theta ⟨{F}_{\nabla }\wedge {F}_{\nabla }⟩\phantom{\rule{thinmathspace}{0ex}},$S_{YM} : \nabla \mapsto \int_X \frac{1}{e^2} \langle F_\nabla \wedge \star F_\nabla\rangle + \int_{X} i \theta \langle F_\nabla \wedge F_\nabla \rangle \,,

each pairing the curvature 2-form with itself in an invariant polynomial, but the first involving the Hodge star operator dual, and the second not. One can combine the coefficients $\frac{1}{{e}^{2}}$ and $i\theta$ into a single complex number

$\tau =\frac{\theta }{2\pi }+\frac{4\pi i}{{e}^{2}}\phantom{\rule{thinmathspace}{0ex}}.$\tau = \frac{\theta}{2 \pi} + \frac{4 \pi i}{e^2} \,.

Montonen-Olive duality asserts that the quantum field theories induced from one such parameter value and another one obtained from it by an action of $\mathrm{SL}\left(2,ℤ\right)$ on the upper half plane are equivalent.

This is actually not quite true for ordinary Yang-Mills theory, but seems to be true for super Yang-Mills theory.

Edward Witten has suggested that this is to be understood geometrically by understand Yang-Mills theory as a compactification of a conformal quantum field theory in 6-dimensions – that instead of a gauge field given by a principal bundle with connection involves a principal 2-bundle with 2-connection – on a torus. The $\mathrm{SL}\left(2,ℤ\right)$-invariance of the resulting 4-dimensional theory is then the remnant of the invariance of the 6-dimensional theory under conformal transformations of that torus.

Moreover, Witten has suggested that this S-duality secretly drives a host of other subtle phenomena, notably that the geometric Langlands duality is just an aspect of a special case of this.

## References

The understanding of Montonen-Olive duality 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