Abstract
The variety of consistent ‘gauging’ deformations of supergravity theories in four dimensions depends on the choice of Lagrangian formulation. One important goal is to get the most general deformations without making hidden assumptions. Ignoring supersymmetry we consider in this paper nv abelian vector potentials in four spacetime dimensions with non-minimal kinetic coupling to ns uncharged (possibly nonlinear) scalar fields. As in the case of extended supergravities, one model may possess different formulations related by . The symplectic group mixes its electric and magnetic potentials. The model admits a global duality symmetry subgroup G which acts also on the scalars. We recall first how the general second order Lagrangian, its local deformations and those of its abelian gauge group will depend on the choice of 2nv directions (choice of ‘Darboux frame’). We start from a general frame defined by the symplectic transformation relating it to a fixed ‘reference’ one. Combinations of symplectic matrix coefficients appear then as constant parameters in the second order Lagrangians. Another gauging method uses an ‘embedding tensor’ that characterizes the realization of the gauge group via the global duality group. It involves additional 2-form gauge fields. A suitable zero charge limit of this realization has abelian gauge group and the ‘gauging’ can be viewed as a consistent deformation of that limit. We show that the two methods applied to the corresponding ungauged models have equivalent local deformations—and more generally, have isomorphic local BRST cohomology at all ghost numbers. We finally consider manifestly duality invariant first order actions with abelian gauge group. We point out that obstructions to non-abelian deformations of the Yang–Mills type exhibited in a previous work remain present when couplings to scalar fields are included.
Highlights
It has been realized since the early days of supersymmetry that electric-magnetic duality plays a central role in four dimensional extended supergravity models [1,2,3,4,5,6,7]
We show that the two methods applied to the corresponding ungauged models have equivalent local deformations—and more generally, have isomorphic local BRST cohomology at all ghost numbers
A necessary condition for a linear transformation in the internal space of the potentials to be a symmetry of the action is that it should leave the kinetic term invariant, which in turn implies that ΩMN must remain invariant under these transformations, i.e. they must belong to the symplectic group Sp(2nv, R)
Summary
It has been realized since the early days of supersymmetry that electric-magnetic duality plays a central role in four dimensional extended supergravity models [1,2,3,4,5,6,7]. Returning to 4 dimensions we show that the obstructions to local non-abelian deformations of the Yang–Mills type exhibited in [21], remain valid when couplings to scalar fields are included This result, which was already sketched in [14], forces one to go to second-order formulations in order to allow the local gaugings. We demonstrate that the latter’s space of local deformations is isomorphic to the space of local deformations of the standard second-order action with the appropriate choice of the corre sponding duality frame This is because one of the conditions fulfilled by the embedding tensor of [22,23,24,25,26], namely their ‘locality’ constraint, implies precisely that there exists a frame in which all the gauge couplings are electric. The only exception is the no-go theorem of section 2, which forbids Yang–Mills deformations in the first-order formalism with both electric and magnetic potentials
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