[Paper Review] Exact string solutions and duality

A.A. Tseytlin|ArXiv.org|Jul 17, 1994

Algorithms and Data Compression4 references18 citations

TL;DR

This paper reviews exact classical solutions in bosonic string theory, focusing on three classes: coset models (gauged WZW), plane wave backgrounds with covariantly constant null Killing vectors, and F-models with two null Killing vectors. It establishes connections between these solutions via duality symmetries and presents explicit D=4 solutions, demonstrating that certain backgrounds—particularly those with null isometries—admit exact conformal field theory interpretations despite the complexity of higher-order α′ corrections.

ABSTRACT

We review known exact classical solutions in (bosonic) string theory. The main classes of solutions are `cosets' (gauged WZW models), `plane wave'-type backgrounds (admitting a covariantly constant null Killing vector) and `$F$-models' (backgrounds with two null Killing vectors generalising the `fundamental string' solution). The recently constructed $D=4$ solutions with Minkowski signature are given explicitly. We consider various relations between these solutions and, in particular, discuss some aspects of the duality symmetry. [To appear in the Proceedings of the 2nd Journe'e Cosmologie, Observatoire de Paris, June 2-4, 1994.]

Motivation & Objective

• To systematize and review known exact classical solutions in bosonic string theory that are valid to all orders in α′.
• To clarify the role of null Killing vectors in organizing exact solutions and their duality relations.
• To present explicit D=4 solutions with Minkowski signature and analyze their CFT interpretation.
• To investigate the interplay between different solution classes—cosets, plane waves, and F-models—via duality and field redefinition symmetries.
• To address the open problem of determining whether all exact solutions correspond to regular conformal field theories, especially in light of α′-corrections and scheme dependence.

Proposed method

• Classify exact string solutions into three main types: gauged WZW models (cosets), plane wave-type backgrounds with covariantly constant null Killing vectors, and F-models with two null Killing vectors.
• Use conformal field theory (CFT) as the fundamental criterion: exact string solutions must correspond to conformally invariant 2D sigma models.
• Apply field redefinition techniques to show that certain α′-dependent corrections can be absorbed into a redefinition of background fields, preserving physical content.
• Construct explicit D=4 solutions by generalizing the fundamental string solution, including cases with Euclidean D=2 black hole transverse geometry.
• Demonstrate duality between K-models (with null Killing vectors) and F-models, showing that dual solutions describe the same CFT under T-duality.
• Analyze the structure of the string effective action, emphasizing scheme dependence and the role of non-locality in α′-corrections.

Experimental results

Research questions

• RQ1 Which classes of string backgrounds are exactly solvable to all orders in α′, and what geometric or algebraic structures underlie them?
• RQ2 How do duality symmetries—particularly T-duality—relate different solution classes such as F-models, K-models, and gauged WZW models?
• RQ3 To what extent can α′-corrections be removed via local field redefinitions, and what does this imply for the physical equivalence of solutions?
• RQ4 Can all exact string solutions be associated with a regular conformal field theory, or are there obstructions in the CFT interpretation of α′-deformed backgrounds?
• RQ5 What is the role of null Killing vectors in organizing the space of exact solutions, and why do they appear to be central to duality and conformal invariance?

Key findings

• The paper presents explicit D=4 exact solutions with Minkowski signature, including a generalization of the fundamental string solution with a D=2 Euclidean black hole in the transverse space.
• Solutions with a covariantly constant null Killing vector (K-models) are shown to be dual to F-models, establishing a duality chain between different solution classes.
• A subclass of F-models can be realized as nilpotently gauged WZW models, providing a direct CFT interpretation for these backgrounds.
• The existence of null Killing vectors—especially two in F-models—appears to be a key organizing principle for exact solutions, suggesting a deeper geometric or algebraic structure.
• Field redefinitions can absorb α′-corrections in certain backgrounds, implying that the leading-order solution may be exact in a suitable scheme, though this does not hold universally.
• The fundamental string solution may not correspond to a regular CFT in the standard scheme, but can be interpreted as a non-perturbative solitonic solution arising from a mixed action involving source terms and effective field theory.