SU(3)-structures on quotients of 3-Sasakian manifolds

Technical Summary: SU(3)-Structures on Quotients of 3-Sasakian Orbifolds

Problem Statement
The paper addresses the construction of $SU(3)$-structures on six-dimensional manifolds and orbifolds, specifically focusing on those with non-vanishing intrinsic torsion. In the context of string theory and supergravity, particularly flux compactifications, $SU(3)$-structures are crucial as their intrinsic torsion encodes background fluxes. While torsion-free structures correspond to Calabi-Yau manifolds, the authors target a specific class known as LT-structures (defined by non-zero torsion classes $W_1$ and $W_2$, with $W_3=W_4=W_5=0$), which are relevant for $N=1$ compactifications of IIA supergravity to $AdS_4$. The paper seeks to systematically generate such structures on quotients of 7-dimensional 3-Sasakian manifolds and orbifolds, extending previous results from toric settings to the broader context of 3-Sasakian geometry.

Methodology
The authors employ a two-pronged approach combining the geometry of 3-Sasakian cones with a specific "twist construction" for $SU(n)$-structures:

1. 3-Sasakian Geometry: They utilize the definition of a 3-Sasakian manifold $(S, g, \xi_i, \eta_i, \Phi_i)$ of dimension 7 as a manifold whose Riemannian cone $C(S)$ is hyperkähler. The cone admits three complex structures $J_i$ satisfying quaternionic relations. The authors fix one Reeb vector field, $\xi_3$, to define a quotient space $Z = S / \xi_3$, which serves as the twistor space of the underlying quaternionic Kähler base.
2. Twist Construction: Building on the work of Larfors, Lüst, and Tsimpis [LLT10] and generalized by Peres [Per26], the authors apply a method to reverse a direction of the almost complex structure using a "twist form."
   • They identify a holomorphic volume form $\Omega^{(4)}$ on the hyperkähler cone $C(S)$ (viewed as a Calabi-Yau 4-fold).
   • They construct a specific 1-form $\eta$ on the cone, derived from the contact forms $\eta_1$ and $\eta_2$ of the 3-Sasakian structure, which acts as a twist form with specific charge properties under the $S^1$ action generated by $\xi_3$.
   • By applying the twist theorem (Theorem 2.2.4), they descend the structure from the cone to the quotient $Z$.

Key Contributions and Results

• Theorem A (Specific Construction): The authors prove that for a quasi-regular or regular 7-dimensional 3-Sasakian manifold $S$, the quotient $Z$ by the foliation generated by $\xi_3$ inherits an $SU(3)$-structure. This structure is explicitly defined by a 2-form $\omega$ and a complex 3-form $\Omega$ constructed from the cone's Kähler forms and the twist form. The resulting structure is an LT-structure with specific non-vanishing torsion classes:

  • $W_1 = 8/3$
  • $W_2 = \frac{W_1}{2}(\omega + \frac{3i}{2}\eta \wedge \bar{\eta})$
  • $W_3 = W_4 = W_5 = 0$.

• Theorem B (Parametric Family): The authors generalize Theorem A to a parametric family of $SU(3)$-structures on $Z$. By introducing real parameters $a, b$ and a complex parameter $c$ (related by $b^2 = |c|^2/a^2$), they define a family of forms $(\omega_{a,b}, \Omega_c)$.

  • They derive the general expressions for the torsion classes $W_1$ and $W_2$ in terms of these parameters.
  • They establish the conditions under which the induced metric is positive definite ($a > 0$) and identify specific subcases:
    • Nearly Kähler: Occurs when $a = 2b^2$, yielding $W_2 = 0$ and $W_1 \neq 0$.
    • Symplectic $SU(1,2)$: Occurs when $a = -b^2$, yielding $W_1 = 0$ and $W_2 \neq 0$ (resulting in a metric of signature $(2,4)$).

• Orbifold Generalization: The methodology is explicitly extended to the quasi-regular and orbifold cases. The authors demonstrate that basic forms on the 3-Sasakian orbifold descend to well-defined forms on the quotient orbifold $Z$, allowing the construction of LT-structures on singular spaces.

• Examples:

  • Regular Case: The construction recovers known LT-structures on $\mathbb{CP}^3$ and the flag manifold $SU(3)/T^2$, relating them to the Eells-Salamon construction of almost complex structures on twistor spaces.
  • Quasi-regular/Orbifold Case: The authors generate infinite families of examples using 3-Sasakian moment map reductions (weighted actions on spheres) and by lifting metrics from weighted projective spaces (which are quaternionic Kähler orbifolds). These examples yield topologically distinct LT-structures, some with singularities.

Significance and Claims
The paper claims to provide a systematic geometric mechanism for generating $SU(3)$-structures with specific torsion properties (LT-structures) from the rich class of 3-Sasakian manifolds and orbifolds.

• Unification: It unifies the construction of nearly Kähler structures and other LT-structures under a single parametric framework derived from 3-Sasakian geometry.
• Extension to Orbifolds: A significant contribution is the successful application of these constructions to orbifolds, thereby producing new examples of spaces (potentially with singularities) that admit these structures, which are of interest in string theory compactifications involving orientifolds.
• Physical Relevance: The authors note that the derived torsion classes satisfy specific constraints related to $AdS_4$ solutions in supergravity. They observe that violating certain inequalities involving the torsion moduli ($3|W_1|^2 - |W_2|^2 \geq 0$) signals the presence of orientifold sources, linking the geometric parameters directly to physical source configurations.

The authors conclude modestly, noting that while their results are specific to dimension 7 (due to charge constraints of the twist form), they open the door to investigating whether similar LT-structures exist in higher dimensions, particularly in the context of spaces with singularities.