[Paper Review] KT and HKT Geometries in Strings and in Black Hole Moduli Spaces

G. Papadopoulos|ArXiv.org|Jan 15, 2002

Black Holes and Theoretical Physics16 references11 citations

TL;DR

This paper establishes that the moduli space of five-dimensional supersymmetric black holes preserving four supersymmetries is a weak HKT manifold, and proves that compact, conformally balanced, strong KT manifolds with SU(n)-holonomy of the KT connection are Calabi-Yau. These results connect torsionful geometries in string theory and supergravity to special holonomy and Calabi-Yau structures, with implications for supersymmetric compactifications and dualities in type II string theory and supergravity compactifications.

ABSTRACT

Some selected applications of KT and HKT geometries in string theory, supergravity, black hole moduli spaces and hermitian geometry are reviewed. It is shown that the moduli spaces of a large class of five-dimensional supersymmetric black holes are HKT spaces. In hermitian geometry, it is shown that a compact, conformally balanced, strong KT manifold whose associated KT connection has holonomy contained in SU(n) is Calabi-Yau. The implication of this result in the context of some string compactifications is explained.

Motivation & Objective

• To establish the geometric structure of moduli spaces of five-dimensional supersymmetric black holes preserving four supersymmetries.
• To determine the conditions under which compact, conformally balanced, strong KT manifolds with SU(n)-holonomy of the KT connection are Calabi-Yau.
• To clarify the implications of these geometries for supersymmetric compactifications in type II string theory and supergravity.
• To provide a non-compact example of a KT manifold with SU(3)-holonomy of the KT connection, relevant to holographic duals of Yang-Mills theories.

Proposed method

• Analysis of the moduli space geometry of five-dimensional black holes using supersymmetry constraints and the structure of hypercomplex manifolds with torsion.
• Application of the KT-connection and HKT-connection to hermitian and hyperhermitian manifolds, with focus on holonomy reduction to SU(n).
• Use of the Chern connection and its torsion to derive curvature and holonomy conditions, particularly under the assumption of closed H-form and conformal balance.
• Derivation of a vanishing theorem via the Hopf maximum principle applied to a scalar quantity constructed from the Nijenhuis tensor and torsion components.
• Construction of a non-compact, strong, conformally balanced KT manifold with SU(3)-holonomy using a specific IIA supergravity solution with left-invariant forms on S³.
• Use of the Kähler form and metric ansatz to verify the KT structure and holonomy condition in the non-compact example.

Experimental results

Research questions

• RQ1 Is the moduli space of five-dimensional supersymmetric black holes preserving four supersymmetries a weak HKT manifold?
• RQ2 Under what conditions does a compact, conformally balanced, strong KT manifold with SU(n)-holonomy of the KT connection become Calabi-Yau?
• RQ3 Can smooth supersymmetric compactifications of the common sector of type II string theory exist with non-zero NS-NS three-form and SU(n)-holonomy of the KT connection?
• RQ4 What is the geometric structure of the gravitational dual of pure N=1 supersymmetric Yang-Mills theory in four dimensions?
• RQ5 Are there non-compact examples of strong, conformally balanced KT manifolds with SU(3)-holonomy of the KT connection?

Key findings

• The moduli space of five-dimensional supersymmetric black holes preserving four supersymmetries is a weak HKT manifold, as shown through analysis of the hypercomplex structure and torsionful connection.
• Compact, conformally balanced, strong KT manifolds with holonomy of the KT connection contained in SU(n) are necessarily Calabi-Yau, proven via a vanishing theorem using the Hopf maximum principle.
• No smooth supersymmetric compactifications of the common sector of type II string theory exist with non-vanishing NS⊗NS three-form and SU(n)-holonomy of the KT connection, due to the Calabi-Yau conclusion.
• A non-compact, strong, conformally balanced KT manifold with SU(3)-holonomy of the KT connection is constructed using an IIA supergravity solution with specific left-invariant forms on S³.
• The constructed non-compact example is complete and admits a Kähler form and metric that realize the KT structure with SU(3) holonomy, relevant to holographic duals of N=1 Yang-Mills theories.
• The torsion H-form in the non-compact example is closed (dH=0), and the manifold satisfies the conformal balance condition, confirming it as a strong KT structure.