[Paper Review] AdS 3 solutions with from S 3 × S 3 fibrations

Andrea Legramandi, Niall T. Macpherson|arXiv (Cornell University)|Mar 1, 2020

Black Holes and Theoretical Physics41 references14 citations

TL;DR

This paper constructs new warped AdS₃ solutions in massive IIA supergravity with N=(3,0) and N=(1,0) supersymmetry by compactifying on S³×S³ fibrations over an interval. It derives necessary and sufficient conditions via systems of ordinary differential equations, yielding analytic and numerical examples that interpolate between D-brane and O-plane behaviors in the internal geometry.

ABSTRACT

We study warped AdS$_3$ solutions in massive IIA supergravity preserving $\mathcal{N}=(3,0)$ and $\mathcal{N}=(1,0)$ supersymmetry. We consider solutions whose internal spaces decompose as an S$^3 imes$S$^3$ fibration and a interval over which the rest of the solution is foliated. We present necessary and sufficient conditions for these solutions to exist, in terms of systems of ordinary differential equations and find several new analytic and numerical examples with internal spaces bounded between D-brane and O-plane behaviors.

Motivation & Objective

• To construct new supersymmetric warped AdS₃ solutions in massive IIA supergravity with N=(3,0) and N=(1,0) supersymmetry.
• To analyze solutions whose internal geometry decomposes as an S³×S³ fibration over an interval.
• To derive necessary and sufficient conditions for the existence of such solutions using differential equations.
• To identify and classify new analytic and numerical examples with bounded internal spaces exhibiting D-brane and O-plane-like behaviors.

Proposed method

• Formulate the background ansatz with an S³×S³ fibration over a one-dimensional interval, preserving the specified supersymmetries.
• Derive the system of ordinary differential equations (ODEs) that govern the warp factors and gauge fluxes in the internal space.
• Impose supersymmetry conditions to reduce the system to a set of first-order ODEs with boundary conditions at the ends of the interval.
• Solve the ODE system analytically and numerically to construct explicit solutions.
• Analyze the behavior of the internal geometry near the boundaries to identify D-brane and O-plane limits.
• Use the solutions to probe the structure of flux and curvature in the compactification manifold.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for the existence of warped AdS₃ solutions with N=(3,0) and N=(1,0) supersymmetry in massive IIA supergravity?
• RQ2 How do S³×S³ fibrations over an interval contribute to the construction of such supersymmetric solutions?
• RQ3 What types of boundary behaviors—specifically D-brane and O-plane limits—can emerge in the internal space of these solutions?
• RQ4 Can analytic and numerical solutions be constructed that interpolate between these boundary behaviors?
• RQ5 What role do the ODEs derived from the supersymmetry conditions play in classifying the solution space?

Key findings

• The paper derives a complete system of ordinary differential equations that fully characterize the existence of N=(3,0) and N=(1,0) warped AdS₃ solutions with S³×S³ fibrations.
• Several new analytic solutions are found that smoothly interpolate between D-brane and O-plane behaviors in the internal space.
• Numerical solutions are constructed that confirm the existence of bounded internal geometries with the required supersymmetry and curvature structure.
• The solutions exhibit a non-trivial fibration structure where the S³×S³ geometry evolves along the interval, preserving the desired supersymmetry.
• The boundary conditions at the ends of the interval are shown to correspond to physical sources such as D-branes and O-planes.
• The results demonstrate that the internal space is constrained by the supersymmetry conditions, leading to a finite class of possible compactifications.