[Paper Review] Almost contact manifolds and type II string equations

Thomas Friedrich, Stefan Ivanov|arXiv (Cornell University)|Nov 12, 2001

Geometric Analysis and Curvature Flows8 references5 citations

TL;DR

This paper classifies five-dimensional locally homogeneous quasi-Sasakian manifolds admitting a parallel spinor under a unique connection with totally skew-symmetric torsion, identifies conditions for conformal invariance of such spinors, and establishes topological obstructions to their existence in the compact case. It links these geometric structures to type II string theory via a conformal transformation framework and the dilation function.

ABSTRACT

Abstract. We classify locally homogeneous quasi-Sasakian manifolds in dimension five that admit a parallel spinor ψ of algebraic type F · ψ = 0 with respect to the unique connection ∇ preserving the quasi-Sasakian structure and with totally skew-symmetric torsion. We introduce a certain conformal transformation of almost contact metric manifolds and discuss a link between them and the dilation function in 5-dimensional string theory. We find natural conditions implying conformal invariances of parallel spinors. We present topological obstructions to the existence of parallel spinors in the compact case. Contents

Motivation & Objective

• To classify five-dimensional locally homogeneous quasi-Sasakian manifolds that admit a parallel spinor under a connection with totally skew-symmetric torsion.
• To introduce a conformal transformation for almost contact metric manifolds and relate it to the dilation function in 5D type II string theory.
• To identify natural geometric conditions under which parallel spinors remain invariant under conformal rescaling.
• To derive topological obstructions to the existence of parallel spinors in the compact case.

Proposed method

• Utilizes the unique connection ∇ preserving the quasi-Sasakian structure and equipped with totally skew-symmetric torsion.
• Analyzes parallel spinors ψ satisfying the algebraic condition F · ψ = 0 with respect to this connection.
• Applies a conformal transformation to almost contact metric structures to explore invariance properties of parallel spinors.
• Employs differential geometric techniques in contact and spin geometry to classify the underlying manifolds.
• Relies on the interplay between algebraic spinor conditions and curvature constraints in the classification.
• Connects the geometric framework to 5D string theory by relating the conformal factor to the dilation function.

Experimental results

Research questions

• RQ1 Which five-dimensional locally homogeneous quasi-Sasakian manifolds admit a parallel spinor under a connection with totally skew-symmetric torsion?
• RQ2 How does a conformal transformation of almost contact metric structures relate to the dilation function in 5D type II string theory?
• RQ3 Under what geometric conditions is a parallel spinor invariant under conformal rescaling of the metric?
• RQ4 What topological obstructions prevent the existence of parallel spinors on compact quasi-Sasakian manifolds?
• RQ5 How do the algebraic condition F · ψ = 0 and the torsion structure constrain the possible geometric realizations?

Key findings

• The classification identifies specific five-dimensional locally homogeneous quasi-Sasakian manifolds that support a parallel spinor under the specified connection with totally skew-symmetric torsion.
• A conformal transformation framework is established that links the geometry of almost contact metric manifolds to the dilation function in 5D type II string theory.
• Natural conditions are found under which the parallel spinor remains invariant under conformal rescaling, preserving its geometric and physical significance.
• Topological obstructions to the existence of parallel spinors are derived in the compact case, constraining the possible underlying manifolds.
• The algebraic condition F · ψ = 0, combined with the torsion structure, imposes strong restrictions on the curvature and holonomy of the manifold.