We present a version of ten-dimensional IIA supergravity containing a 9-form potential for which the field equations are equivalent to those of the standard, massless, IIA theory for vanishing 10-form field strength, F 10 , and to those of the 'massive' IIA theory for non-vanishing F 10 . We present a multi 8-brane solution of these equations that generalizes the 8-brane of Polchinski and Witten. We show that this solution is T-dual to a new multi 7-brane solution of S 1 compactified IIB supergravity, and that the latter is T-dual to the IIA 6-brane. When combined with the Sl(2; Z) U-duality of the type IIB superstring, the T-duality between type II 7-branes and 8-branes implies a quantization of the cosmological constant of type IIA superstring theory. These results are made possible by the construction of a new massive N=2 D=9 supergravity theory. We also discuss the 11-dimensional interpretation of these type II p-branes.2

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We present the complete structure of extended conformal supergravlty for N ~< 4. The relation with the graded algebra SU(2, 2iN) and with the multlplet of currents is &scussed. The N = 4 superconformal theory has a formulation with local SU(4) x U(1) and rigid SU(1, 1) mvarlance. We give the linearized mvariant actions for all N. For N = 2 we exhibit the complete non-hnear lagrangian.

We show how the Newton-Cartan formulation of Newtonian gravity can be obtained from gauging the Bargmann algebra, i.e., the centrally extended Galilean algebra. In this gauging procedure several curvature constraints are imposed. These convert the spatial (time) translational symmetries of the algebra into spatial (time) general coordinate transformations, and make the spin connection gauge fields dependent. In addition we require two independent Vielbein postulates for the temporal and spatial directions. In the final step we impose an additional curvature constraint to establish the connection with (on-shell) Newton-Cartan theory. We discuss a few extensions of our work that are relevant in the context of the AdS-CFT correspondence.

An arbitrary number of abelian vector multiplets is coupled to N = 4 supergravity. The resulting action is invariant under global SO(n, 6), where n is the number of vector multiplets, and under local SU(4) × U(1) transformations. The scalar fields of the theory parametrize the manifold [SO(n,6)/SO(n) × SO(6)] x [SU(1,1)/U(1)]. The role of the matter fields of the N = 4 Weyl muhiplet in the Poincare supergravity theory is clarified.