Duality and Hidden Symmetries in Gravitational Theories

Abstract

Duality is presently considered the key to the Holy Grail of String Theory— it is supposed to provide links between the five known different superstring theories in ten dimensions, hoped to be just different limits of one unique eleven dimensional theory [1]. The main role of duality is to relate two different regimes— e.g. one of weak and one of strong coupling— of these theories. In most cases duality is not a very precise concept, just because the strong coupling regime is a matter of speculation. This is rather different from the duality transformations in the gravitational theories considered in this work, which have a very precise meaning— not the least because we are dealing with classical field theories (compare, however, [2] and for a modest attempt to use duality symmetry in Quantum Gravity, see [3]). The historical example of all these dualities is the duality between electric and magnetic fields in electrodynamics, which, when expressed in terms of the field strength, is just an example of the mathematical notion of Hodge duality for differential forms. Actually, the source-free Maxwell equations are not only invariant under a discrete duality, but under a continous one-parameter group of duality rotations. It is this kind of transformations, which is the subject of this article. Whereas, in general, the electromagnetic duality rotations are an ‘on-shell’ symmetry, i.e. a symmetry of the equations of motion and not of the action, the situation changes,if one considers time-independent solutions. In this case also the magnetic field can be derived from a (pseudo) scalar potential and the duality rotations expressed in terms of scalar potentials become a bona fide "off-shell' symmetry of the "dimensionally reduced' three dimensional theory. This replacement of the vector potential by a scalar one has analogues in higher dimensions playing an important role in the construction of supergravity theories through the process of dimensional reduction. A typical example is the (pseudo) scalar "axion',obtained as the dual of a gauge field 2-form in 4 dimensions. This scalar axion combines nicely with another scalar,the dilaton to a doublet giving rise to an SL(2) group of non-linear duality transformations [4]. A particular element of this group,replacing the dilaton by its inverse,lies at the heart of string duality ("S- duality')[5],where the expectation value of the dilaton plays the role of a string coupling constant.