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Towards reduction of type II theories on SU(3) structure manifolds

Amir-Kian Kashani-Poor, Ruben Minasian

TL;DR

This work develops a principled framework for reducing type II supergravity on SU(3) structure manifolds toward gauged ${\cal N}=2$ supergravity in four dimensions by expanding the higher-dimensional fields in a shared basis of invariant forms $\{\omega_i, \alpha_A, \beta^A, \tilde{\omega}^i\}$ alongside the SU(3) data $J$ and $\Omega$. It identifies the differential constraints required for a base point dependent reduction and isolates additional, more restrictive conditions needed for a base point independent reduction, with a detailed analysis of the metric sector that yields a vector multiplet special-Kähler geometry and a quaternionic hypermultiplet sector. The paper then investigates expansions in eigenforms of the Laplacian, showing a naive minimal system cannot realize the desired ${\cal N}=2$ gauged structure, and derives necessary symplectic-consistency conditions for any such expansion to be viable. Collectively, these results provide a first-principles basis for reducing type II theories on SU(3) structure manifolds and clarify the role of intrinsic torsion, moduli dependence, and Laplacian-based expansions in achieving consistent four-dimensional gauged supergravity theories.

Abstract

We revisit the reduction of type II supergravity on SU(3) structure manifolds, conjectured to lead to gauged N=2 supergravity in 4 dimensions. The reduction proceeds by expanding the invariant 2- and 3-forms of the SU(3) structure as well as the gauge potentials of the type II theory in the same set of forms, the analogues of harmonic forms in the case of Calabi-Yau reductions. By focussing on the metric sector, we arrive at a list of constraints these expansion forms should satisfy to yield a base point independent reduction. Identifying these constraints is a first step towards a first-principles reduction of type II on SU(3) structure manifolds.

Towards reduction of type II theories on SU(3) structure manifolds

TL;DR

This work develops a principled framework for reducing type II supergravity on SU(3) structure manifolds toward gauged supergravity in four dimensions by expanding the higher-dimensional fields in a shared basis of invariant forms alongside the SU(3) data and . It identifies the differential constraints required for a base point dependent reduction and isolates additional, more restrictive conditions needed for a base point independent reduction, with a detailed analysis of the metric sector that yields a vector multiplet special-Kähler geometry and a quaternionic hypermultiplet sector. The paper then investigates expansions in eigenforms of the Laplacian, showing a naive minimal system cannot realize the desired gauged structure, and derives necessary symplectic-consistency conditions for any such expansion to be viable. Collectively, these results provide a first-principles basis for reducing type II theories on SU(3) structure manifolds and clarify the role of intrinsic torsion, moduli dependence, and Laplacian-based expansions in achieving consistent four-dimensional gauged supergravity theories.

Abstract

We revisit the reduction of type II supergravity on SU(3) structure manifolds, conjectured to lead to gauged N=2 supergravity in 4 dimensions. The reduction proceeds by expanding the invariant 2- and 3-forms of the SU(3) structure as well as the gauge potentials of the type II theory in the same set of forms, the analogues of harmonic forms in the case of Calabi-Yau reductions. By focussing on the metric sector, we arrive at a list of constraints these expansion forms should satisfy to yield a base point independent reduction. Identifying these constraints is a first step towards a first-principles reduction of type II on SU(3) structure manifolds.

Paper Structure

This paper contains 12 sections, 1 theorem, 60 equations.

Table of Contents

  1. Introduction
  2. Conditions on the expansion forms
  3. Reduction of the metric sector
  4. Special geometry
  5. Quaternionic geometry
  6. The metric scalars
  7. The RR scalars
  8. Expanding in eigenforms of the Laplacian
  9. Our conditions and the Laplacian
  10. A first attempt at constructing a set of expansion forms
  11. The scope of minimality
  12. A variation on conditions 3 and 4

Key Result

Theorem 1

The completion $L^2 A^p(M)$ of $A^p(M)$ with respect to the $L^2$ norm has an orthonormal basis $\phi_{1,p}, \phi_{2,p}, \ldots$ consisting of eigenforms of $\triangle_p$. One can order the eigenforms so that the corresponding eigenvalues $\lambda_{k,p}$ satisfy in particular, the multiplicities are finite.

Theorems & Definitions (1)

  • Theorem