Types of quantum field thories
abstract duality: opposite category,
The term S-duality can mean two different things:
In its original form, S-duality refers to Montonen-Olive duality , which is about the following phenomenon:
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 and into a single complex number
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 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 -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.
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
the low energy effective supergravity actions of the two models match. Since the string coupling constants and are given as the expectation values of the exponentials of the dilatons and , respectively, the above equations relates the type-I theory at strong coupling to the heterotic theory at weak coupling:
From the relative scaling of the metric in (1) we also see that the string length in the two theories are related by
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 the tension is given by the same formula
where I’ve used relations (2) and (3). But this is equal to the tension of the fundamental heterotic string
This indicates that it is sensible to identify the strong coupling limit of the type-I D1 brane with the heterotic string.
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).
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
On S-Duality in Abelian Gauge Theory (arXiv:hep-th/9505186)
Conformal Field Theory In Four And Six Dimensions (arXiv:0712.0157)
See also electro-magnetic duality.