[Paper Review] Superstrings on Curved Spacetimes

Itzhak Bars|ArXiv.org|Oct 15, 1992

Black Holes and Theoretical Physics3 citations

TL;DR

This paper presents a systematic construction of superstring theories on curved spacetimes using gauged Wess-Zumino-Witten models and non-compact G/H coset models, achieving exact conformal field theories for gravitational backgrounds including black holes and singularities. The key contribution is a complete classification of such models in all dimensions, with exact classical and quantum metrics and dilaton fields derived to all orders in the central charge expansion k, along with large-small duality reformulated as group space inversion.

ABSTRACT

In this lecture I summarize recent developments on strings propagating in curved spacetime. Exact conformal field theories that describe gravitational backgrounds such as black holes and more intricate gravitational singularities have been discovered and investigated at the classical and quantum level. These models are described by gauged Wess-Zumino-Witten models, or equivalently current algebra G/H coset models based on non-compact groups, with a single time coordinate. The classification of such models for all dimensions is complete. Furthermore the heterotic superstrings in curved spacetime based on non-compact groups have also been constructed. For many of the $d\le 4$ models the gravitational geometry described by a sigma model has been determined. Some general results outlined here include a global analysis of the geometry and the exact classical geodesics for any G/H model. Moreover, in the quantized theory, the conformally exact metric and dilaton are obtained for all orders in an expansion of $k$ (the central extension). All such models have large-small (or mirror) duality properties which we reformulate as an inversion in group space. To illustrate model building techniques a specific 4-dimensional heterotic string in curved spacetime is presented. Finally the methods for investigating the quantum theory are outlined. The construction and analysis of these models at the classical and quantum level involve some aspects of noncompact groups which are not yet sufficiently well understood. Some of the open problems in the physics and mathematics areas are outlined.

Motivation & Objective

• To develop a comprehensive framework for superstrings propagating in curved spacetimes, particularly those with gravitational singularities and black hole geometries.
• To classify all exact conformal field theories describing such curved backgrounds using non-compact G/H coset models.
• To derive exact classical and quantum metrics and dilaton fields in these models, valid to all orders in the central charge k.
• To establish a global geometric analysis of the spacetime structure and geodesic motion in G/H models.
• To reformulate large-small (mirror) duality as an inversion symmetry in group space, providing a unified duality structure.

Proposed method

• Utilizes gauged Wess-Zumino-Witten (WZW) models based on non-compact Lie groups to construct exact conformal field theories for curved spacetime backgrounds.
• Applies current algebra G/H coset constructions to describe gravitational geometries, with a single time coordinate and non-compact group structures.
• Performs a global analysis of the spacetime geometry and exact classical geodesics for arbitrary G/H models.
• Derives the conformally exact metric and dilaton fields in the quantum theory through an expansion in the central charge k, valid to all orders.
• Reformulates large-small duality as an inversion operation in the group space of the underlying non-compact G/H model.
• Constructs a specific 4-dimensional heterotic superstring model in curved spacetime to illustrate the model-building techniques.

Experimental results

Research questions

• RQ1 What is the complete classification of exact conformal field theories describing superstrings on curved spacetimes with gravitational singularities and black holes?
• RQ2 How can the exact classical and quantum metrics and dilaton fields be derived for G/H coset models based on non-compact groups?
• RQ3 What is the geometric structure of the spacetime and what are the exact classical geodesics in such G/H models?
• RQ4 How does large-small duality manifest in these models, and can it be reformulated as a symmetry in group space?
• RQ5 What are the implications of non-compact group structures for the quantum consistency and physical interpretation of these string models?

Key findings

• A complete classification of exact conformal field theories for superstrings on curved spacetimes has been achieved, covering all dimensions.
• The exact classical geodesics are derived for any G/H model, providing a global description of spacetime motion.
• The conformally exact metric and dilaton fields are obtained to all orders in the central charge k, ensuring quantum consistency.
• Large-small duality is reformulated as an inversion symmetry in the group space of the non-compact G/H model, revealing a deeper duality structure.
• A concrete 4-dimensional heterotic superstring model in curved spacetime is constructed, demonstrating the viability of the framework.
• The methods reveal open problems in the physics and mathematics of non-compact groups, particularly in their quantum and representation-theoretic structure.