[Paper Review] Toric manifolds for Flux compactification

Sujan P. Dabholkar|arXiv (Cornell University)|Jan 11, 2013

Geometric and Algebraic Topology8 references3 citations

TL;DR

This paper investigates the use of smooth, compact toric manifolds for supersymmetric AdS₄ flux compactifications in Type IIA string theory by analyzing SU(3)-structures and topological constraints. It shows that non-vanishing holomorphic 3-forms and adjustable torsion classes in the SU(3)-structure equations enable the construction of new flux vacua, with topological conditions—particularly on the first and top Chern classes—acting as key constraints on viable manifolds.

ABSTRACT

We study how to use smooth, compact toric varieties for supersymmetric $AdS_4$ flux compactifications using tools of SU(3) structures, similar to $CP3$ solution. A non-vanishing globally well defined complex 3-form plays a key role in such constructions. Necessary topological conditions associated with it will be understood to put constraints on large class of these manifolds for supersymmetric flux compactification. Local analysis of SU(3)-structure is carried out, which might help to explore more flux vacua.

Motivation & Objective

• To extend the use of toric manifolds beyond Calabi-Yau hypersurfaces for flux compactifications in string theory.
• To identify topological obstructions—especially related to the first and top Chern classes—that restrict which smooth, compact toric varieties can support supersymmetric AdS₄ flux vacua.
• To analyze the local differential system of SU(3)-structures to determine the degrees of freedom in tuning torsion classes for new flux vacuum solutions.
• To generalize the $\mathbb{CP}^3$ construction, which uses a non-integrable almost complex structure, to a broader class of toric manifolds.
• To explore the potential for constructing classical dS vacua via similar methods, though the focus remains on AdS₄ solutions.

Proposed method

• Utilizes the symplectic quotient construction of smooth toric varieties via moment map constraints: $\sum_i Q^a_i |z^i|^2 = \xi^a$ modulo $U(1)^s$.
• Applies SU(3)-structure formalism with a globally defined non-vanishing spinor, encoded in a real (1,1) form $J$ and a complex (3,0) form $\Omega$, satisfying $J \wedge \Omega = 0$ and $i\Omega \wedge \bar{\Omega} = \frac{4}{3}J^3 \neq 0$.
• Analyzes the differential equations for the torsion classes: $dJ = \frac{3}{2} \text{Im}(\bar{W}_1 \Omega) + W_4 \wedge J + W_3$ and $d\Omega = W_1 J^2 + W_2 \wedge J + \bar{W}_5 \wedge \Omega$, where $W_i$ are torsion forms.
• Employs Chern class theory—particularly $c_1$ and the top Chern class $c_3$—to derive global topological constraints on the existence of nowhere-vanishing holomorphic 3-forms.
• Performs local analysis of the SU(3)-structure system to show that torsion classes $W_i$ can be tuned case-by-case, especially when $W_4$ is closed.
• Uses toric geometry via fans and lattice data to compute Betti numbers and divisor classes, linking combinatorial data to cohomological invariants.

Experimental results

Research questions

• RQ1 Which smooth, compact toric manifolds can support a globally well-defined holomorphic 3-form required for SU(3)-structure in flux compactifications?
• RQ2 What topological constraints—especially on Chern classes—arise from requiring a non-vanishing holomorphic 3-form in SU(3)-structure compactifications?
• RQ3 To what extent can the torsion classes $W_i$ in the SU(3)-structure equations be adjusted to generate new flux vacua in Type IIA supergravity?
• RQ4 Can the $\mathbb{CP}^3$ construction with a non-integrable almost complex structure be generalized to other toric manifolds?
• RQ5 Are there systematic ways to construct classical dS vacua using this framework, given the success in generating AdS₄ solutions?

Key findings

• The first Chern class $c_1$ and the top Chern class $c_3$ are critical topological invariants that constrain the existence of globally well-defined holomorphic 3-forms on smooth toric manifolds.
• For the $\mathbb{CP}^3$ case, the construction of an SU(3)-structure with a non-integrable almost complex structure is possible due to specific topological properties, particularly the vanishing of the first Chern class.
• The local differential system for SU(3)-structures allows for tunable torsion classes, especially when $W_4$ is closed, enabling the possibility of constructing new flux vacua by adjusting $W_i$ parameters.
• The analysis shows that many smooth, compact toric manifolds can potentially support SU(3)-structures with non-vanishing $W_1$ and $W_2$, indicating a broad class of possible flux vacua.
• The method provides a systematic framework to explore classical dS solutions in the future, though the current results are limited to AdS₄ vacua.
• Odd Betti numbers vanish for smooth toric manifolds, and even Betti numbers are computed via the formula $\beta_{2k} = \sum_{i=k}^n (-1)^{i-k} \binom{i}{k} d_{n-i}$, where $d_k$ counts $k$-dimensional cones in the fan.