nLab S-duality

duality

In QFT and String theory

The term S-duality can mean two different things:

Contents

Idea

In (super) Yang-Mills theory

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,

$S_{YM} : \nabla \mapsto \int_X \frac{1}{e^2} \langle F_\nabla \wedge \star F_\nabla\rangle + \int_{X} i \theta \langle F_\nabla \wedge F_\nabla \rangle \,,$

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

$\tau = \frac{\theta}{2 \pi} + \frac{4 \pi i}{e^2} \,.$

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 $SL(2,\mathbb{Z})$ 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 $SL(2,\mathbb{Z})$-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.

In string theory

In string theory, S-duality is supposed to apply to whole string theories and make type II string theory be S-dual to itself and make heterotic string theory be S-dual to type I string theory.

Heterotic/type I duality

Something substantial should go here, for the moment the following is copied from a discussion forum comment by some Olof here:

For the Het/I relation, the first observation is that the massless spectra of the two models agree. Moreover, if we make the identification

$\tag{1} G^I_{\mu\nu} = e^{-\Phi_h} G^h_{\mu\nu} , \qquad \Phi^I = - \Phi^h , \qquad \tilde{F}^I_3 = \tilde{H}^h_3 , \qquad A^I_1 = A^h_1$

the low energy effective supergravity actions of the two models match. Since the string coupling constants $g_s^I$ and $g_s^h$ are given as the expectation values of the exponentials of the dilatons $\exp(\Phi^I)$ and $\exp(\Phi^h)$, respectively, the above equations relates the type-I theory at strong coupling to the heterotic theory at weak coupling:

$\tag{2} g^I_s = \frac{1}{g^h_s} .$

From the relative scaling of the metric in (1) we also see that the string length in the two theories are related by

$\tag{3} l^I_s = l^h_s \sqrt{g^h_s}.$

As a non-perturbative check we can consider the tension of the type-I D1 brane. The brane is a BPS object, so for all values of the coupling $g_s^I$ the tension is given by the same formula

$T^I_{D1} = \frac{1}{g_s^I} \frac{1}{2\pi\left(l^I_s\right)^2} = \frac{g^h_s}{2\pi\left(l^h_s\sqrt{g^h_s}\right)^2} = \frac{1}{2\pi\left(l^h_s\right)^2}$

where I’ve used relations (2) and (3). But this is equal to the tension of the fundamental heterotic string

$T^h_{F1} = \frac{1}{2\pi\left(l^h_s\right)^2}.$

This indicates that it is sensible to identify the strong coupling limit of the type-I D1 brane with the heterotic string.

Type IIB self-duality

In type IIB string theory S-duality mixes the fundamental string with the D1-brane.

At least part of the S-duality in type II string theory can be seen as a system of autoequivalences of the super L-infinity algebras which defines the extended super spacetime constituted by the type II superstring (FSS 13, section 4.3).

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

S-duality acting on (p,q)-strings is discussed for instance in

• Igor Bandos, Superembedding Approach and S-Duality. A unified description of superstring and super-D1-brane, Nucl.Phys.B599:197-227,2001 (arXiv:hep-th/0008249)

S-duality in type II string theory as an operation on the extended super spacetime super L-infinity algebra is in section 4.3 of