Moduli of generic $\mathrm{SU}(7)$ flux backgrounds in exceptional generalised geometry: localised sources, stability, and smoothness of the physical moduli space

Generic four-dimensional $\mathcal{N}=1$ flux compactifications of M-theory and type II string theory are characterised, in $E_{7(7)}\times\mathbb{R}^+$ exceptional generalised geometry, by an integrable $\mathrm{SU}(7)$ structure: an involutive subbundle of the generalised tangent bundle together with a vanishing moment map for the action of generalised diffeomorphisms and gauge symmetries. The source paper (Ashmore, Strickland-Constable, Tennyson, Waldram, JHEP 2021) built this framework, computed the moduli of $G_2$ and GMPT backgrounds — GMPT being the flux backgrounds of Graña–Minasian–Petrini–Tomasiello, defined by a pair of compatible pure spinors obeying first-order supersymmetry equations — and left a precise open core: (i) how localised sources (branes, orientifolds, boundaries) modify the involutivity and moment-map conditions and the moduli spaces; (ii) whether the proposed GIT-like stability condition on the pseudo-Kähler space of $\mathrm{SU}(7)$ structures is necessary and sufficient for existence and uniqueness of solutions, given that the Kähler metric is indefinite; (iii) whether critical points of the Kähler potential are non-degenerate transverse to generalised-diffeomorphism orbits, so that the physical moduli space is a smooth finite-dimensional quotient; and (iv) an unconditional cohomological description of the deformations of an arbitrary integrable $\mathrm{SU}(7)$ structure, removing the assumed generalised $\partial'\bar\partial'$-lemma. Later work by the same group resolved the heterotic analogue, the conditional infinitesimal type-3 moduli computation, and the type II all-orders moduli, but the four core questions remain open.

Background

String compactifications preserving minimal four-dimensional supersymmetry ($\mathcal{N}=1$ flux backgrounds) are the default setting for string model building. In M-theory compactified on a seven-dimensional internal manifold, the fluxless supersymmetric backgrounds are manifolds of $G_2$ holonomy — seven-manifolds whose metric has special holonomy group $G_2\subset\mathrm{SO}(7)$ and a closed, coclosed defining three-form $\varphi$. Turning on four-form flux $F$ while keeping a four-dimensional Minkowski factor leads to warped backgrounds that are the natural extension of $G_2$ holonomy; in type II theories the analogous backgrounds extend Calabi–Yau geometry and include the flux backgrounds of Graña–Minasian–Petrini–Tomasiello (GMPT), defined by a pair of compatible pure spinors, i.e. differential forms annihilated by a Clifford action, obeying first-order supersymmetry equations.

Generalised geometry reformulates such structures: the generalised tangent bundle $E$ combines vectors and differential forms, and its section algebra carries the Dorfman derivative $L_V$, a natural bilinear, non-skew derivative encoding diffeomorphisms and gauge transformations together. In $E_{7(7)}\times\mathbb{R}^+$ exceptional generalised geometry on a seven-manifold, $E$ carries the 56-dimensional exceptional representation and the symmetry group is the group of generalised diffeomorphisms $\mathrm{GDiff}$ (diffeomorphisms plus the M-theory three-form gauge symmetry). The source paper (Ashmore–Strickland-Constable–Tennyson–Waldram, JHEP 01 (2021) 158) proved that generic Minkowski $\mathcal{N}=1$ flux backgrounds of M-theory and type II are exactly integrable $\mathrm{SU}(7)\subset E_{7(7)}$ structures: supersymmetry is equivalent to (a) involutivity of a distinguished complex rank-seven subbundle $L\subset E_{\mathbb{C}}$ — the analogue of the integrability of the $(1,0)$ tangent bundle of a complex manifold, equivalently the vanishing of an F-term superpotential, the holomorphic part of the four-dimensional vacuum equations — and (b) the vanishing of a moment map $\mu$ for the $\mathrm{GDiff}$ action — the analogue of a D-term, the second, non-holomorphic class of vacuum equations — where a moment map is an equivariant map whose zero locus realises a symplectic quotient of the space of structures. The space of structures carries a natural pseudo-Kähler structure (a Kähler structure whose metric is indefinite rather than positive definite) with Kähler potential $K$, an extension of Hitchin's $G_2$ functional (a topological volume functional whose critical points are torsion-free $G_2$ structures). The physical moduli space is the quotient $\mathcal{M}_{\mathrm{phys}}=\mathcal{M}_\psi/\mathrm{GDiff}^{\mathbb{C}}$ of torsion-free $\mathrm{SU}(7)$ structures by the complexified symmetry group, realised as critical points of $K$.

This suggests an infinite-dimensional analogue of Geometric Invariant Theory (GIT), the finite-dimensional framework in which a symplectic quotient by a compact group equals an algebraic quotient by the complexified group and existence of zero modes is controlled by a stability condition (as in the Hitchin–Kobayashi correspondence for Hermitian Yang–Mills connections). However, as the authors stress, the Kähler metric on the space of $\mathrm{SU}(7)$ structures is not positive definite — only in the ordinary $G_2$ case does the partially solved moment map appear to remove the negative directions — so the stability picture may be only sufficient, not necessary. Moreover the moduli-space identification assumes the critical points of $K$ are non-degenerate in directions transverse to the $\mathrm{GDiff}$ orbits.

For infinitesimal moduli, the paper constructed a deformation complex: decomposing a compatible generalised connection into exceptional Dolbeault-type operators $D=D_3+D_{-1}+D_1+D_{-3}$ generalising the $\bar\partial$ operator of complex geometry, the integrable deformations are computed by a complex built from $D_{-3}$ and $D_{-1}$. They completed this computation for $G_2$ holonomy, their type-0 extension, and GMPT backgrounds (the latter a first), but for a generic integrable $\mathrm{SU}(7)$ structure the action of $D_1$ mixing two terms of the complex makes the analysis subtler; they conjectured that, under a generalised $\partial\bar\partial$-lemma (the classical $\partial\bar\partial$-lemma relates $d$-, $\partial$- and $\bar\partial$-exact forms on compact Kähler manifolds and underlies cohomological deformation theory), the moduli should be given by the cohomology of the Lie-algebroid differential $D_{-3}$, without proving it.

Finally, all backgrounds treated are warped Minkowski backgrounds without localised sources, so they evade no-go theorems only non-compactly: compact flux backgrounds require negative-tension sources (branes and orientifold planes, i.e. loci with boundary conditions and modified Bianchi identities $dF=j$ for the flux), and how such sources modify the involutivity condition, the moment map and the moduli spaces was explicitly left open, along with prospective AdS, heterotic/type I, generalised-calibration and higher-derivative extensions.

Problem Statement

The problem is to complete the geometric characterisation of generic four-dimensional $\mathcal{N}=1$ flux compactifications in $E_{7(7)}\times\mathbb{R}^+$ exceptional generalised geometry, as initiated by Ashmore–Strickland-Constable–Tennyson–Waldram. These backgrounds are integrable $\mathrm{SU}(7)$ structures on a seven-manifold: an involutive complex rank-seven subbundle of the generalised tangent bundle (equivalently a vanishing superpotential) together with a vanishing moment map for the action of the group $\mathrm{GDiff}$ of generalised diffeomorphisms; in this framework the terms "torsion-free" and "integrable" are used interchangeably to mean exactly these two conditions. The physical moduli space is described as the critical set of the Kähler potential $K$ modulo $\mathrm{GDiff}^{\mathbb{C}}$. The open core, whose components jointly serve this single objective of determining when the physical moduli space exists as a smooth finite-dimensional space and what it is, is: (1) Sources — extend the involutivity and moment-map conditions, and the corresponding moduli-space computation, to backgrounds with localised sources (branes, orientifold planes, and the boundary conditions they induce), determining how the source terms in the four-form Bianchi identity $dF=j$ deform the supersymmetry conditions and the naive moduli spaces (those computed from the unmodified, source-free conditions); (2) Stability — decide whether the GIT-like stability picture for the pseudo-Kähler space of $\mathrm{SU}(7)$ structures is valid: prove or disprove that (notwithstanding the indefinite Kähler metric) the proposed GIT-like stability condition on complexified $\mathrm{GDiff}$ orbits is necessary and sufficient for the existence and uniqueness of torsion-free $\mathrm{SU}(7)$ structures, including the conjectured Futaki-type obstructions from circle actions (a Futaki-type invariant being an analogue of the classical Futaki invariant, a Lie-algebra character of holomorphic circle actions that obstructs the existence of Kähler–Einstein metrics); (3) Smoothness — establish, or disprove, the non-degeneracy of the critical points of $K$ transverse to the $\mathrm{GDiff}$ orbits, so that the quotient $\mathcal{M}_{\mathrm{phys}}=\mathcal{M}_\psi/\mathrm{GDiff}^{\mathbb{C}}$ is a smooth finite-dimensional moduli space, and identify when it fails; (4) Cohomological moduli — prove the generalised $\partial'\bar\partial'$-lemma for the exceptional Dolbeault operators $D_3$, $D_{-1}$, $D_1$, $D_{-3}$ (or otherwise remove the lemma as an assumption), giving an unconditional cohomological computation of the infinitesimal deformations of an arbitrary integrable $\mathrm{SU}(7)$ structure. The open core is posed at the generality of the source framework (generic warped Minkowski flux backgrounds in $E_{7(7)}\times\mathbb{R}^+$ generalised geometry), not only for special classes such as $G_2$ holonomy or GMPT backgrounds.

References

1. Ashmore, Strickland-Constable, Tennyson, Waldram, "Generalising G$_2$ geometry: involutivity, moment maps and moduli", JHEP 01 (2021) 158, DOI 10.1007/JHEP01(2021)158, arXiv:1910.04795, https://arxiv.org/abs/1910.04795
2. Ashmore, Strickland-Constable, Tennyson, Waldram, "Heterotic backgrounds via generalised geometry: moment maps and moduli", JHEP 11 (2020) 071, DOI 10.1007/JHEP11(2020)071, arXiv:1912.09981, https://arxiv.org/abs/1912.09981
3. Smith, Waldram, "M-theory moduli from exceptional complex structures", JHEP 08 (2023) 022, DOI 10.1007/JHEP08(2023)022, arXiv:2211.09517, https://arxiv.org/abs/2211.09517
4. Smith, Tennyson, Waldram, "All-orders moduli for type II flux backgrounds", arXiv:2409.03847, https://arxiv.org/abs/2409.03847
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8. Ashmore, Coimbra, Strickland-Constable, Svanes, Tennyson, "Topological G$_2$ and Spin(7) strings at 1-loop from double complexes", JHEP 02 (2022) 089, DOI 10.1007/JHEP02(2022)089, arXiv:2108.09310, https://arxiv.org/abs/2108.09310