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Tri-Sasakian consistent reduction

Davide Cassani, Paul Koerber

TL;DR

This work constructs a universal consistent Kaluza–Klein truncation of M-theory on seven-dimensional tri-Sasakian manifolds, yielding a four-dimensional $ mathcal{N}=4$ gauged supergravity with three vector multiplets and an $SO(3)$ gauge factor. Consistency is established by mapping the reduction to a left-invariant coset reduction on $Sp(2)/Sp(1)$ and embedding the result into the $ ext{E}_{7(7)}$ framework via Exceptional Generalized Geometry; it also connects to the coset realizations of other universal truncations. The resulting theory admits AdS$_4$ vacua with spontaneous $ math{4} o math{3,1,0}$ supersymmetry breaking and a spectrum containing massive modes, including subtruncations to minimal $ math{N}=3$ and $ math{N}=1$ sectors, as well as an enhanced truncation on $N^{010}$ with a Betti vector multiplet. The construction has implications for AdS$_4$/CFT$_3$ holography and holographic condensed matter, illustrating a unified, universal framework for consistent truncations across multiple compactification manifolds.

Abstract

We establish a universal consistent Kaluza-Klein truncation of M-theory based on seven-dimensional tri-Sasakian structure. The four-dimensional truncated theory is an N=4 gauged supergravity with three vector multiplets and a non-abelian gauge group, containing the compact factor SO(3). Consistency follows from the fact that our truncation takes exactly the same form as a left-invariant reduction on a specific coset manifold, and we show that the same holds for the various universal consistent truncations recently put forward in the literature. We describe how the global symmetry group SL(2,R) x SO(6,3) is embedded in the symmetry group E7(7) of maximally supersymmetric reductions, and make the connection with the approach of Exceptional Generalized Geometry. Vacuum AdS4 solutions spontaneously break the amount of supersymmetry from N=4 to N=3,1 or 0, and the spectrum contains massive modes. We find a subtruncation to minimal N=3 gauged supergravity as well as an N=1 subtruncation to the SO(3)-invariant sector. We also show that a reduction on the homogeneous space N^{010} enhances the universal tri-Sasakian truncation with a Betti vector multiplet.

Tri-Sasakian consistent reduction

TL;DR

This work constructs a universal consistent Kaluza–Klein truncation of M-theory on seven-dimensional tri-Sasakian manifolds, yielding a four-dimensional gauged supergravity with three vector multiplets and an gauge factor. Consistency is established by mapping the reduction to a left-invariant coset reduction on and embedding the result into the framework via Exceptional Generalized Geometry; it also connects to the coset realizations of other universal truncations. The resulting theory admits AdS vacua with spontaneous supersymmetry breaking and a spectrum containing massive modes, including subtruncations to minimal and sectors, as well as an enhanced truncation on with a Betti vector multiplet. The construction has implications for AdS/CFT holography and holographic condensed matter, illustrating a unified, universal framework for consistent truncations across multiple compactification manifolds.

Abstract

We establish a universal consistent Kaluza-Klein truncation of M-theory based on seven-dimensional tri-Sasakian structure. The four-dimensional truncated theory is an N=4 gauged supergravity with three vector multiplets and a non-abelian gauge group, containing the compact factor SO(3). Consistency follows from the fact that our truncation takes exactly the same form as a left-invariant reduction on a specific coset manifold, and we show that the same holds for the various universal consistent truncations recently put forward in the literature. We describe how the global symmetry group SL(2,R) x SO(6,3) is embedded in the symmetry group E7(7) of maximally supersymmetric reductions, and make the connection with the approach of Exceptional Generalized Geometry. Vacuum AdS4 solutions spontaneously break the amount of supersymmetry from N=4 to N=3,1 or 0, and the spectrum contains massive modes. We find a subtruncation to minimal N=3 gauged supergravity as well as an N=1 subtruncation to the SO(3)-invariant sector. We also show that a reduction on the homogeneous space N^{010} enhances the universal tri-Sasakian truncation with a Betti vector multiplet.

Paper Structure

This paper contains 29 sections, 147 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction and discussion of the results
  2. Tri-Sasakian geometry
  3. The dimensional reduction
  4. Reduction of the geometric sector
  5. Reduction of the form sector
  6. The complete 4D action of the truncated theory
  7. Match with $\mathcal{N}=4$ gauged supergravity
  8. The general action for $\mathcal{N}=4$ gauged supergravity
  9. Identification of the coset manifold
  10. Identification of electric and magnetic vectors
  11. The gauging
  12. AdS solutions and their spectra
  13. AdS$_4$ vacua
  14. Spectrum
  15. Embedding in the spectrum on $S^7$
  16. ...and 14 more sections

Figures (1)

  • Figure 1: Web of interrelated consistent truncations of 11D, type IIA and type IIB supergravity. In each box we display the geometric structure (if any) and the coset manifold on which the truncation is based, as well as its amount of supersymmetry. Each arrow denotes a consistent subtruncation. The colored boxes highlight the truncations on which we focused in this paper.