nLab
S-duality

The term S-duality can mean two different things:

• in mathematics it is short for Spanier-Whitehead duality : monoidal duality in the stable homotopy category;

• in physics it denotes a certain equivalence between quantum field theories, this is what we discuss below

---

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,

$$S_{\mathrm{YM}}:\nabla \mapsto {\int }_X\frac{1}{e^2}⟨F_{\nabla }\wedge \star F_{\nabla }⟩+{\int }_{X}i\theta ⟨F_{\nabla }\wedge F_{\nabla }⟩,$$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

$$\tau =\frac{\theta }{2\pi }+\frac{4\pi i}{e^2}.$$\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 $\mathrm{SL}(2,\mathbb{Z})$ 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 $\mathrm{SL}(2,\mathbb{Z})$-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.

Related concepts

• Yang-Mills theory

• S-duality

  • electro-magnetic duality

    • Montonen-Olive duality

  • Seiberg duality

  • geometric Langlands correspondence

  • quantum geometric Langlands correspondence

• duality in string theory

  • T-duality, U-duality

  • AdS/CFT correspondence

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

• Edward Witten,

  On S-Duality in Abelian Gauge Theory (arXiv:hep-th/9505186)

  Conformal Field Theory In Four And Six Dimensions (arXiv:0712.0157)

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

See also electro-magnetic duality.