[Paper Review] Electric/Magnetic Deformations of S^3 and AdS_3, and Geometric Cosets

Dan Israël, Costas Kounnas+2 more|arXiv (Cornell University)|May 24, 2004

Black Holes and Theoretical Physics51 references75 citations

TL;DR

This paper investigates asymmetric marginal deformations of SU(2)k and SL(2,R)k WZW models, generating exact string vacua via electric or magnetic background fields. It demonstrates that these deformations yield geometric cosets like S², AdS², and H² as consistent, exact CFTs, with explicit constructions of spectra, partition functions, and supersymmetry in the AdS₂×S² background, confirming its status as an exact string vacuum with potential holographic applications.

ABSTRACT

We analyze asymmetric marginal deformations of SU(2)_k and SL(2,R)_k WZW models. These appear in heterotic string backgrounds with non-vanishing Neveu--Schwarz three-forms plus electric or magnetic fields, depending on whether the deformation is elliptic, hyperbolic or parabolic. Asymmetric deformations create new families of exact string vacua. The geometries which are generated in this way, deformed S^3 or AdS_3, include in particular geometric cosets such as S^2, AdS_2 or H_2. Hence, the latter are consistent, exact conformal sigma models, with electric or magnetic backgrounds. We discuss various geometric and symmetry properties of the deformations at hand as well as their spectra and partition functions, with special attention to the supersymmetric AdS_2 x S^2 background. We also comment on potential holographic applications.

Motivation & Objective

• Understand the moduli space of near-horizon NS5-brane geometries (S³ and AdS³) beyond supergravity.
• Analyze asymmetric marginal deformations of SU(2)k and SL(2,R)k WZW models driven by electric or magnetic backgrounds.
• Establish that resulting geometries—such as S², AdS², and H²—form consistent, exact conformal field theories with non-trivial fluxes.
• Construct the spectra and partition functions for these deformed backgrounds, particularly for the AdS₂×S² solution.
• Explore holographic implications and the role of space-time supersymmetry in the AdS₂×S² background.

Proposed method

• Use asymmetric marginal deformations of WZW models via left-right current bilinears J·J̄, where one current is from the affine algebra and the other from a U(1) outside the chiral algebra.
• Apply geometric deformation techniques to preserve part of the original isometry, mapping SU(2) and SL(2,R) group manifolds to squashed spheres and deformed AdS₃.
• Utilize different coordinate systems (elliptic, hyperbolic, Poincaré) to describe the three-sphere, AdS₃, and electromagnetic wave backgrounds.
• Construct the deformed background fields (metric, B-field, gauge field) from the current algebra and Kaluza-Klein reduction, identifying the effective target-space geometry.
• Derive the spectrum of primary states and compute the partition function using character formulas and modular invariance conditions.
• Verify that the deformed backgrounds solve the low-energy equations of motion (β-functions) at leading order in α′, with corrections absorbed in a shift k → k+2.

Experimental results

Research questions

• RQ1 How do asymmetric marginal deformations of SU(2)k and SL(2,R)k WZW models generate new exact string vacua with electric or magnetic backgrounds?
• RQ2 What is the resulting geometry after deformation, and how do these geometries (e.g., S², AdS², H²) arise as geometric cosets?
• RQ3 Are the deformed backgrounds, particularly AdS₂×S², consistent and unitary CFTs, and do they preserve space-time supersymmetry?
• RQ4 What is the spectrum and partition function of the deformed CFTs, and how is modular invariance maintained?
• RQ5 Can the near-horizon geometry of the extremal Reissner-Nordström black hole (AdS₂×S²) be realized as an exact string vacuum via these deformations?

Key findings

• Electric and magnetic deformations of AdS₃ and S³ generate new exact string vacua, including geometric cosets such as S², AdS₂, and H².
• The AdS₂×S² background, relevant to the near-horizon geometry of the four-dimensional extremal Reissner-Nordström black hole, is shown to be an exact CFT with N=(1,0) worldsheet supersymmetry.
• Geometric cosets like AdS₂ and H² arise as limiting cases of asymmetric deformations: AdS₂ via hyperbolic (electric) deformation and H² via imaginary magnetic field in the elliptic deformation.
• The spectrum of primary states in the deformed CFTs is computed explicitly, with characters and modular invariance verified for the partition function.
• The deformed backgrounds solve the leading-order equations of motion of the heterotic string, with all α′ corrections captured by the shift k → k+2 in the level.
• Supersymmetric AdS₂×S² is a consistent, exact string vacuum, with the full spectrum and partition function constructed, supporting its potential use in holography.