An Sl(2,Z) Multiplet of Nine-Dimensional Type II Supergravity Theories

TL;DR

The paper addresses how to construct a complete set of nine-dimensional massive Type II supergravities by employing generalized dimensional reduction that captures all brane degrees of freedom, including D-7 branes and their S-duals. It develops an SL(2,R)-covariant reduction for Type IIB, yielding a triplet of massive 9D theories whose mass parameters form an SL(2,R) adjoint and transform under SL(2,Z) quantum monodromies, with brane backgrounds as their origin. It then links this IIB construction to an 11D massive supergravity with two KK-9M branes, showing that T-duality and reductions reproduce Romans-like massive IIA backgrounds and generalized Buscher rules, while mapping the brane spectrum (D-7, Q-7, KK-8A, KK-9M, etc.) onto the 9D mass parameters. The work outlines a rich network of dualities among branes and higher-dimensional theories and suggests avenues for extending to more mass parameters and deeper symmetry structures such as U-duality and higher-dimensional algebras.

Abstract

We show that only by performing generalized dimensional reductions all possible brane configurations are taken into account and one gets the complete lower-dimensional theory. We apply this idea to the reduction of type IIB supergravity in an SL(2,R)-covariant way and establish T duality for the type II superstring effective action in the context of generalized dimensional reduction giving the corresponding generalized Buscher's T duality rules. The full (generalized) dimensional reduction involves all the S duals of D-7-branes: Q-7-branes and a sort of composite 7-branes. The three species constitute an SL(2,Z) triplet. Their presence induces the appearance of the triplet of masses of the 9-dimensional theory. The T duals, including a ``KK-8A-brane'', which must have a compact transverse dimension have to be considered in the type IIA side. Compactification of 11-dimensional KK-9M-branes (a.k.a. M-9-branes) on the compact transverse dimension give D-8-branes while compactification on a worldvolume dimension gives KK-8A-branes. The presence of these KK-monopole-type objects breaks translation invariance and two of them given rise to an SL(2,R)-covariant ``massive 11-dimensional supergravity'' whose reduction gives the massive 9-dimensional type II theories.

Figures (5)

• Figure 1: This diagram represents two different ways of obtaining the same result: Generalized dimensional reduction and "dual" standard dimensional reduction.
• Figure 2: Scheme of the different dimensional reductions with the names of the respective compact coordinates $z,x,y$.
• Figure 3: If we place a 7-brane on a cylinder, one has to take into account that automatically 7-branes are created at the boundaries. This can be easily seen by conformally transforming the cylinder into a punctured sphere. Consistency of the monodromy implies that the total sum of the charges in the sphere is nil.
• Figure 4: This figure is a magnified and more detailed piece of Figure \ref{['fig:dualbran']} in which a general picture of all the known extended objects of M/string theory and their duality relations is given. Only well-established relations are shown, and so no duality connections between the conjectured KK-8B-brane and other objects are drawn. In the triplets $(m,n,p)$$m$ stands for the number of standard spacelike dimensions of the object, $n$ for the number of special isometric directions ($z$) and $p$ for the number of standard transverse dimensions. The double arrows indicate on which directions T duality acts.
• Figure 5: Duality relations between classical solutions of 10- and 11-dimensional supergravity theories describing string/M-theory solitons: p-branes, M-branes, D-branes, gravitational waves, Kaluza-Klein monopoles and other KK-type solutions. Lines with two arrows denote T duality relations. Dashed lines denote S duality relations. Lines with a single arrow denote relations of dimensional reduction, either vertical (direct dimensional reduction) or diagonal (double dimensional reduction).