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Consistent ${\cal N}=8$ truncation of massive IIA on $S^6$

Adolfo Guarino, Oscar Varela

TL;DR

The paper proves a consistent, fully nonlinear truncation of massive IIA supergravity on S^6 to four-dimensional ${\cal N}=8$ ISO(7) gauged supergravity with a dyonic gauging. It achieves this by recasting IIA in an ${\rm SO}(1,3)\times{\rm SL}(7)$ covariant form and employing an ${\cal N}=8$ SL(7) covariant restriction of the D=4 tensor hierarchy, then deriving explicit KK ansatze and the complete non-linear embedding of all $D=4$ fields (including the RR 3-form) into the ten-dimensional metric, form potentials, and field strengths. Consistency is established at the level of supersymmetry transformations and Bianchi identities within the restricted hierarchy, and a closed expression for the Freund–Rubin term is obtained via duality techniques. The work also analyzes a G$_2$-invariant sector, showing its uplift matches the universal nearly-Kähler truncation and detailing overlaps with NK reductions, thereby connecting maximal and half-maximal truncations in a coherent framework. These results provide a robust basis for holographic explorations of D2-brane dynamics and offer patterns applicable to other spherical truncations and higher-dimensional theories.

Abstract

Massive type IIA supergravity is shown to admit a consistent truncation on the six-sphere to maximal supergravity in four dimensions with a dyonic ISO(7) gauging. We obtain the complete, non-linear embedding of all the $D=4$ fields into the IIA metric and form potentials, and show its consistency. We first rewrite the IIA theory in an $\textrm{SO}(1,3) \times \textrm{SL}(7)$--covariant way. Then, we employ an ${\cal N}=8$ SL(7)--covariant restriction of the $D=4$ tensor hierarchy in order to find the full embedding. The redundant $D=4$ degrees of freedom introduced by the tensor hierarchy can be eliminated by writing the embedding in terms of the field strengths and exploiting the restricted duality hierarchy. In particular, closed expressions for the Freund-Rubin term are found using this technique which reveal a pattern valid for other truncations. Finally, we show that the present ${\cal N}=8$ truncation of massive IIA on $S^6$ and the ${\cal N}=2$ truncation obtained when $S^6$ is equipped with its nearly-Kähler structure, overlap in the ${\cal N}=1$, G$_2$--invariant sector of the former.

Consistent ${\cal N}=8$ truncation of massive IIA on $S^6$

TL;DR

The paper proves a consistent, fully nonlinear truncation of massive IIA supergravity on S^6 to four-dimensional ISO(7) gauged supergravity with a dyonic gauging. It achieves this by recasting IIA in an covariant form and employing an SL(7) covariant restriction of the D=4 tensor hierarchy, then deriving explicit KK ansatze and the complete non-linear embedding of all fields (including the RR 3-form) into the ten-dimensional metric, form potentials, and field strengths. Consistency is established at the level of supersymmetry transformations and Bianchi identities within the restricted hierarchy, and a closed expression for the Freund–Rubin term is obtained via duality techniques. The work also analyzes a G-invariant sector, showing its uplift matches the universal nearly-Kähler truncation and detailing overlaps with NK reductions, thereby connecting maximal and half-maximal truncations in a coherent framework. These results provide a robust basis for holographic explorations of D2-brane dynamics and offer patterns applicable to other spherical truncations and higher-dimensional theories.

Abstract

Massive type IIA supergravity is shown to admit a consistent truncation on the six-sphere to maximal supergravity in four dimensions with a dyonic ISO(7) gauging. We obtain the complete, non-linear embedding of all the fields into the IIA metric and form potentials, and show its consistency. We first rewrite the IIA theory in an --covariant way. Then, we employ an SL(7)--covariant restriction of the tensor hierarchy in order to find the full embedding. The redundant degrees of freedom introduced by the tensor hierarchy can be eliminated by writing the embedding in terms of the field strengths and exploiting the restricted duality hierarchy. In particular, closed expressions for the Freund-Rubin term are found using this technique which reveal a pattern valid for other truncations. Finally, we show that the present truncation of massive IIA on and the truncation obtained when is equipped with its nearly-Kähler structure, overlap in the , G--invariant sector of the former.

Paper Structure

This paper contains 23 sections, 118 equations, 1 table.

Table of Contents

  1. Introduction
  2. Type IIA with only $\textrm{SO}(1,3)$ symmetry manifest
  3. Redefinitions that leave $D=4$ spacetime symmetry manifest
  4. Redefinitions that make the $D=4$ tensor hierarchy manifest
  5. Truncation on $S^6$ to $D=4$ ${\cal N}=8$ ISO(7) supergravity
  6. Kaluza-Klein ansatze
  7. The full non-linear embedding
  8. Consistency: supersymmetry transformations
  9. Consistency: field strengths and duality hierarchy
  10. Truncation to the G$_2$ sector
  11. Explicit embedding formulae
  12. Overlap with the universal nearly-Kähler truncation
  13. Discussion
  14. Massive type IIA supergravity
  15. Spinor conventions
  16. ...and 8 more sections