Abstract
In supergravity compactifications, there is in general no clear prescription on how to select a finite-dimensional family of metrics on the internal space, and a family of forms on which to expand the various potentials, such that the lower-dimensional effective theory is supersymmetric. We propose a finite-dimensional family of deformations for regular Sasaki-Einstein seven-manifolds M7, relevant for M-theory compactifications down to four dimensions. It consists of integrable Cauchy-Riemann structures, corresponding to complex deformations of the Calabi-Yau cone M8 over M7. The non-harmonic forms we propose are the ones contained in one of the Kohn-Rossi cohomology groups, which is finite-dimensional and naturally controls the deformations of Cauchy-Riemann structures. The same family of deformations can be also described in terms of twisted cohomology of the base M6, or in terms of Milnor cycles arising in deformations of M8. Using existing results on SU(3) structure compactifications, we briefly discuss the reduction of M-theory on our class of deformed Sasaki-Einstein manifolds to four-dimensional gauged supergravity.
Highlights
Compactifications of superstring/M-theory to lower dimensions are often treated in terms of a reduction to lower-dimensional effective theories
The non-harmonic forms we propose are the ones contained in one of the Kohn-Rossi cohomology groups, which is finitedimensional and naturally controls the deformations of Cauchy-Riemann structures
Notice that we have considered only d < 5 because in this paper we are interested in the case of Kahler-Einstein manifolds with positive curvature; the computations apply to the case d = 5, and in particular, recalling (3.19), lead to a 45-dimensional twisted cohomology
Summary
Compactifications of superstring/M-theory to lower dimensions are often treated in terms of a reduction to lower-dimensional effective theories. The reduction can proceed in a very similar way as for the Calabi-Yau case, where the parameters describing the non-closure of the forms are viewed as charges, or gauging parameters, for the multiplets in the lower-dimensional theory This approach was taken for example in [1, 10, 11] for IIA compactifications and in [7, 12] for M-theory. Appendix A discusses some examples of Sasaki-Einstein manifolds arising from complete intersections, while appendix B applies some of the concepts introduced in section 4 to the Gibbons-Hawking metrics, that provide a useful nontrivial example in four dimensions
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