We study some geometrical and topological aspects of the generalised dimensional reduction of supergravities in D = 11 and D = 10 dimensions, which give rise to massive theories in lower dimensions. In these reductions, a global symmetry is used in order to allow some of the fields to have a non-trivial dependence on the compactifying coordinates.Global consistency in the internal space imposes topological restrictions on the parameters of the compactification as well as the structure of the space itself. Examples that we consider include the generalised reduction of the type IIA and type IIB theories on a circle, and also the massive ten-dimensional theory obtained by the generalised reduction of D = 11 supergravity.

We study the Euclidean-signature supergravities that arise by compactifying D = 11 supergravity or type IIB supergravity on a torus that includes the time direction. We show that the usual T-duality relation between type IIA and type IIB supergravities compactified on a spatial circle no longer holds if the reduction is performed on the time direction. Thus there are two inequivalent Euclidean-signature nine-dimensional maximal supergravities.They become equivalent upon further spatial compactification to D = 8. We also show that duality symmetries of Euclidean-signature supergravities allow the harmonic functions of any single-charge or multi-charge instanton to be rescaled and shifted by constant factors.Combined with the usual diagonal dimensional reduction and oxidation procedures, this allows us to use the duality symmetries to map any single-charge or multi-charge p-brane soliton, or any intersection, into its near-horizon regime. Similar transformations can also be made on non-extremal p-branes. We also study the structures of duality multiplets of instanton and (D − 3)-brane solutions.theories are distinct, and cannot be related to one another by any valid field redefinition.It is only after a further reduction of the two Euclidean-signature theories to D = 8 that an equivalence emerges.One of the motivations for investigating Euclidean-signature supergravities is to study the instanton states, which necessarily live in Euclidean-signature space. Unlike p-branes with p ≥ 0, which are supported by higher-degree field strengths, and which form linear representations under the U-duality group, the instantons are supported by axionic scalars, which transform non-linearly under U-duality. The orbits of the higher p-branes in M-theory are much better understood, and were obtained in [3,4]. In this paper, we shall study the U-duality transformations of instanton solutions, and also the orbits of their charges, which are the Noether charges of the global symmetry group.Another of our results is concerned with the properties of the instanton solutions that are the natural end-points of a sequence of diagonal reductions of p-branes, when the reduction has encompassed the entire world-volume including the time direction. We show that all instanton solutions, including multi-charge ones and even non-extremal ones, have the property that they can be transformed, using SL(2, IR) global duality symmetries of the lower-dimensional theories, into solutions where the harmonic functions characterising the solutions are shifted and scaled by constants. In particular, the shifts can be chosen so as to remove the constant terms in the harmonic functions altogether, with the result that for extremal p-branes the entire solution is of the form that was previously approached only asymptotically in the near-horizon limit. The solutions can then be oxidised back to higher dimensions, by retracing the sequence of reduction steps. They then describe p-branes again, but now with similarly shifted harmonic functions. Thus the asymptoti...

In the usual procedure for toroidal Kaluza-Klein reduction, all the higher-dimensional fields are taken to be independent of the coordinates on the internal space. It has recently been observed that a generalisation of this procedure is possible, which gives rise to lower-dimensional massive supergravities. The generalised reduction involves allowing gauge potentials in the higher dimension to have an additional linear dependence on the toroidal coordinates. In this paper, we show that a much wider class of generalised reductions is possible, in which higher-dimensional potentials have additional terms involving differential forms on the internal manifold whose exterior derivatives yield representatives of certain of its cohomology classes. We consider various examples, including the generalised reduction of M-theory and type II strings on K3, Calabi-Yau and 7-dimensional Joyce manifolds. The resulting massive supergravities support domain-wall solutions that arise by the vertical dimensional reduction of higher-dimensional solitonic p-branes and intersecting p-branes. † Research supported in part by DOE Grant DE-FG05-91-ER40633

The gauge transformations of p-form fields in supergravity theories acquire a noncommuting character when one introduces potentials both for the theory's original field strengths and for their duals. This has previously been shown in the "doubled" formalism for maximal supergravities, where a generalised duality relation between original and dual field strengths replaces the equations of motion. In the doubled formalism, the gauge transformations generate a superalgebra, and the corresponding symmetries have accordingly been called "superdualities." The corresponding Noether charges form a representation of the cohomology ring on the spacetime manifold. In this paper, we show that the gauge symmetry superalgebra implies certain non-trivial relations among the various p-brane tensions, which can straightforwardly be read off from the superalgebra commutation relations.This provides an elegant derivation of the brane-tension relations purely within a given theory, without the need to make use of duality relations between different theories, such as the type IIA/IIB T-duality, although the results are consistent with such dualities. We present the complete set of brane-tension relations in M-theory, in the type IIA and type IIB theories, and in all the lower-dimensional maximal supergravities. We also construct a doubled formalism for massive type IIA supergravity, and this enables us to obtain the brane-tension relations involving the D8-brane, purely within the framework of the massive IIA theory. We also obtain explicit transformations for the nine-dimensional T-duality between the massive type IIA theory and the Scherk-Schwarz reduced type IIB theory.

In the usual procedure for toroidal Kaluza-Klein reduction, all the higher-dimensional fields are taken to be independent of the coordinates on the internal space. It has recently been observed that a generalisation of this procedure is possible, which gives rise to lower-dimensional massive supergravities. The generalised reduction involves allowing gauge potentials in the higher dimension to have an additional linear dependence on the toroidal coordinates. In this paper, we show that a much wider class of generalised reductions is possible, in which higher-dimensional potentials have additional terms involving differential forms on the internal manifold whose exterior derivatives yield representatives of certain of its cohomology classes. We consider various examples, including the generalised reduction of M-theory and type II strings on K3, Calabi-Yau and 7-dimensional Joyce manifolds. The resulting massive supergravities support domain-wall solutions that arise by the vertical dimensional reduction of higher-dimensional solitonic p-branes and intersecting p-branes. † Research supported in part by DOE Grant DE-FG05-91-ER40633

We show that the full global symmetry groups of all the D-dimensional maximal supergravities can be described in terms of the closure of the internal general coordinate transformations of the toroidal compactifications of D = 11 supergravity and of type IIB supergravity, with type IIA/IIB T-duality providing an intertwining between the two pictures. At the quantum level, the part of the U-duality group that corresponds to the surviving discretised internal general coordinate transformations in a given picture leaves the internal torus invariant, while the part that is not described by internal general coordinate transformations can have the effect of altering the size or shape of the internal torus.For example, M-theory compactified on a large torus T n can be related by duality to a compactification on a small torus, if and only if n ≥ 3. We also discuss related issues in the toroidal compactification of the self-dual string to D = 4. An appendix includes the complete results for the toroidal reduction of the bosonic sector of type IIB supergravity to arbitrary dimensions D ≥ 3.

The plot of allowed p and D values for p-brane solitons in D-dimensional supergravity is the same whether the solitons are extremal or nonextremal. One of the useful tools for relating different points on the plot is vertical dimensional reduction, which is possible if periodic arrays of p-brane solitons can be constructed. This is straightforward for extremal p-branes, since the no-force condition allows arbitrary multicenter solutions to be constructed in terms of a general harmonic function on the transverse space. This has also been shown to be possible in the special case of nonextremal black holes in D = 4 arrayed along an axis. In this paper, we extend previous results to include multiscalar black holes, and dyonic black holes. We also consider their oxidation to higher dimensions, discuss general procedures for constructing the solutions and study their symmetries.