Geometry of the $G_2$ moduli space beyond the Calabi–Yau analogy: geometric meaning of its curvature, the $CY_3\times S^1$ product case and global structure

Grigorian and Yau (Commun. Math. Phys. 287 (2009) 459–488) computed the full curvature of the Kähler metric on the complexified moduli space of torsion-free $G_2$-structures on a compact 7-manifold of holonomy $G_2$, the moduli space governing M-theory compactifications, with Kähler potential $K=-3\log V$ where $V$ is the volume. Beyond terms analogous to Calabi–Yau special geometry (a universal piece and a Yukawa-coupling-squared term), the curvature contains new terms with no Calabi–Yau analogue — a quartic integral in the deformation endomorphisms, trace terms, and a term quadratic in a coupling associated with singlet directions — whose geometric meaning they left open. They further posed: for product-type examples $CY_3\times S^1$, relate the $G_2$ moduli-space curvature to the separate Calabi–Yau complex- and Kähler-moduli curvatures (with the $b^3_7=1$ pseudo-Kähler signature and C-field-ansatz obstacles), and find how mirror symmetry manifests from the $G_2$ perspective; understand the global structure of the complexified moduli space (including the missing Calabi–Yau-type line bundle); and extend the analysis to degenerate $S^1$-bundle (D6-brane lift) geometries. All four directions remain open, with substantial partial progress on the real (Hessian) moduli space.

Background

The exceptional Lie group $G_2$ can be realized as the stabilizer, inside $GL(7,\mathbb{R})$, of a particular positive 3-form on $\mathbb{R}^7$ (equivalently, as the automorphism group of the octonions). A $G_2$-structure on an oriented 7-manifold $X$ is a 3-form $\varphi$ pointwise equivalent to this model form; $\varphi$ determines a Riemannian metric $g_\varphi$ and an orientation, but nonlinearly in $\varphi$. The structure is torsion-free exactly when $\varphi$ is parallel for the Levi-Civita connection of $g_\varphi$, in which case the holonomy group of $g_\varphi$ (the group of parallel transports, which measures how much the metric geometry is constrained) is contained in $G_2$ and the metric is Ricci-flat. Compact manifolds with holonomy exactly $G_2$ exist by Joyce's desingularization constructions, but they carry no complex or algebraic structure, and no analogue of Yau's theorem is known; these features make $G_2$ geometry much harder to organize than Calabi–Yau geometry. Joyce also proved that the moduli space $\mathcal{M}$ of torsion-free $G_2$-structures on a compact $X$ (the space of such structures modulo diffeomorphisms isotopic to the identity) is a smooth manifold of dimension $b^3(X)$ (the third Betti number), locally parametrized by the cohomology class $[\varphi]\in H^3(X,\mathbb{R})$; infinitesimal deformations are harmonic 3-forms (3-forms in the kernel of the Hodge Laplacian, one per de Rham cohomology class), which decompose into irreducible $G_2$-components $\Lambda^3=\Lambda^3_1\oplus\Lambda^3_7\oplus\Lambda^3_{27}$, and when the holonomy is exactly $G_2$ only the $\Lambda^3_1\oplus\Lambda^3_{27}$ pieces occur.

M-theory (eleven-dimensional supergravity and its conjectured UV completion) compactified on $M_4\times X$ with $X$ of $G_2$ holonomy yields a four-dimensional $\mathcal{N}=1$ supersymmetric effective theory whose scalar fields are the $G_2$ moduli together with the periods of the 11-dimensional 3-form gauge potential (the C-field). In the Kaluza–Klein reduction (the standard procedure of expanding the higher-dimensional fields in harmonics along the compact manifold to obtain the lower-dimensional effective theory), the metric moduli $s^M$ and the C-field periods $c^M$ combine into complex coordinates $z^M=s^M+ic^M$ on a complexified moduli space $\mathcal{M}_\mathbb{C}$, and the kinetic terms define a Kähler metric on it — a metric locally of the form $G_{M\bar N}=\partial_M\partial_{\bar N}K$ — with Kähler potential $K=-3\log V$, where $V$ is the total volume of $(X,g_\varphi)$; this is the $G_2$ analogue of the complexified Kähler cone (the cone of Kähler classes) of Calabi–Yau manifolds, and was used in the M-theory effective-action literature such as Beasley and Witten's flux analysis. Grigorian and Yau note that this complexification is natural while the holomorphic structure is to some extent artificial, since the $G_2$ and C-field moduli do not really mix.

This parallels Calabi–Yau special geometry, where the complex-structure moduli space of a Calabi–Yau threefold (a compact Kähler six-manifold with trivial canonical bundle, equivalently holonomy contained in $SU(3)$) is Kähler with the Weil–Petersson metric (the natural $L^2$-type Kähler metric on that moduli space), its curvature is governed by the cubic Yukawa coupling (the pointwise cubic self-coupling of three deformation fields), and both the complex- and Kähler-moduli spaces carry a line bundle whose first Chern class coincides with the Kähler class (Candelas and de la Ossa; Strominger). Under mirror symmetry — the conjectured equivalence between the complex geometry of a Calabi–Yau manifold and the symplectic geometry of a mirror Calabi–Yau manifold, which interchanges complex-structure and Kähler moduli — these two moduli geometries are exchanged; whether an analogous duality acts on $G_2$ moduli spaces is one of the source's open questions. Hitchin had earlier observed that the Hessian of the volume functional on the space of positive 3-forms is nondegenerate; on the real (uncomplexified) $G_2$ moduli space with $b^1=0$ it defines a metric of Lorentzian signature $(1,b^3-1)$, whereas the potential $-3\log V$ gives a positive-definite, volume-normalized $L^2$ metric when the holonomy is full $G_2$.

Grigorian and Yau computed the full curvature of the Kähler metric $G_{M\bar N}$ on $\mathcal{M}_\mathbb{C}$, which required expanding the Hodge dual $*\varphi$ to fourth order in deformations of $\varphi$ (the curvature involves fourth derivatives of $K$). They found that the covariant third derivative $A_{MNP}$ plays precisely the role of the Yukawa coupling, $\frac{1}{V}\langle D_\rho D_\mu D_\nu\varphi,*\varphi\rangle$ reproducing it just as on Calabi–Yau moduli spaces; but the Riemann curvature contains, beyond a universal Kähler-type piece and the Yukawa-squared term familiar from Calabi–Yau special geometry, genuinely new terms: a term quadratic in the coupling $C_{MN}=\partial_M\partial_N K$, which measures the singlet ($\Lambda^3_1$) content of the 3-form deformation (it is the coefficient of $\varphi$ in the second covariant derivative $D_M D_N\varphi$), a quartic integral $\int\psi^{abcd}h^a_\kappa\wedge h^b_{\bar\mu}\wedge h^c_\nu\wedge h^d_{\bar\rho}\wedge\varphi$ built from the symmetric traceless endomorphisms $h_M$ representing $\Lambda^3_{27}$-deformations (with $\psi=*\varphi$), and trace terms such as $\operatorname{Tr}(h_\kappa h_{\bar\mu}h_\nu h_{\bar\rho})$ and $\operatorname{Tr}(h_{(\kappa}h_{\bar\mu}\operatorname{Tr}h_\nu h_{\bar\rho)})$. These quartic and trace terms have no Calabi–Yau analogue, and their geometric meaning was left open. The paper also observed that the $G_2$ moduli space lacks the key Calabi–Yau feature of a natural line bundle over the moduli space (proposing only a provisional real line bundle $L$ defined by $\varphi$ with a covariant derivative), and that everything computed is local/infinitesimal.

Problem Statement

Give the geometry of the $G_2$ moduli space an understanding comparable to Calabi–Yau special geometry, along the following interlocking directions posed at the end of Grigorian and Yau's computation of the local curvature. (1) Interpretation of the curvature: identify the geometric meaning of the terms in the curvature of the Kähler metric $G_{M\bar N}=\partial_M\partial_{\bar N}K$ ($K=-3\log V$) on the complexified $G_2$ moduli space that have no Calabi–Yau analogue — the quartic integral in the deformation endomorphisms, the trace terms, and the term quadratic in the singlet coupling $C_{MN}$ — with the Calabi–Yau special geometry that expresses the Yukawa coupling and the Weil–Petersson curvature through the period geometry of Calabi–Yau threefolds (their periods — the integrals of the holomorphic 3-form over 3-cycles — locally determine the complex structure). (2) Product-type examples: for $X=CY_3\times S^1$ ($CY_3$ a Calabi–Yau threefold) with the induced torsion-free $G_2$-structure, determine the relation between the $G_2$ moduli-space metric and curvature and the Calabi–Yau complex-structure and Kähler-moduli geometries — how the two Calabi–Yau sectors intermix inside the single $G_2$ structure; whether and how the relation can be established given the obstacles identified in the source (the harmonic $\Lambda^3_7$ count $b^3_7=1$ makes $\partial\bar\partial K$ pseudo-Kähler of signature $(-+\cdots+)$, i.e. an indefinite-signature Kähler-type metric, and the C-field ansatz would have to be modified); and how mirror symmetry of the Calabi–Yau factor manifests on the $G_2$ moduli-space geometry, including how existing proposals for mirror symmetry on $G_2$ manifolds act on these moduli-space structures. (3) Global structure: extend the local theory to global statements about the complexified moduli space — whether the missing Calabi–Yau-type line-bundle/special-geometry structure exists globally, the global behavior of the Kähler potential $K=-3\log V$ and of the alternative Hessian potential $V$ (Lorentzian signature $(1,b^3_{27})$), and the metric geometry for families with nontrivial $b^3_7$. (4) Degenerate limits: analyze the moduli-space geometry of the physically relevant singular or non-compact $G_2$ spaces realized as $S^1$-bundles over a Calabi–Yau threefold degenerating over a special-Lagrangian locus — the D6-brane lifts; here a special Lagrangian submanifold is a real-three-dimensional calibrated submanifold of the Calabi–Yau threefold on which the holomorphic 3-form restricts to a fixed-phase multiple of the induced volume form, and a D6-brane is a solitonic charged membrane of type IIA string theory that fills spacetime and wraps such a submanifold, whose M-theory description is precisely this degenerating $S^1$-bundle — whose moduli are determined jointly by the Calabi–Yau and special-Lagrangian moduli, and what these degenerations imply for mirror symmetry. (1) and (2) concern the curvature and its Calabi–Yau relation, while (3) and (4) extend the same objective from the local, smooth, compact setting toward the global and singular settings.

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