[Paper Review] Instantons, five-branes and fractional strings

Christoph Nölle|arXiv (Cornell University)|Jul 31, 2012

Black Holes and Theoretical Physics43 references3 citations

TL;DR

This paper proposes a method to resolve the ambiguity in identifying the worldvolume of instanton-based NS5-branes in heterotic supergravity by constructing a smooth, regular supergravity solution through a conformal deformation and extension to a higher-dimensional Lorentzian manifold. It shows that smeared 5-branes on G₂ and Spin(7) cones can be obtained via conformal deformation, while Calabi-Yau and hyperkähler manifolds require case-by-case treatment, and demonstrates that the singular support of the instanton in the small instanton limit can be reinterpreted as a 6-dimensional calibrated submanifold, consistent with 5-brane physics.

ABSTRACT

In this note I review the construction of higher-dimensional instantons and heterotic NS5-branes on Ricci-flat cones from arXiv:1109.3552, as well as fractional strings from arXiv:1202.5046. The focus is on methods and interpretation. I show, furthermore, that smeared 5-brane supergravity solutions on manifolds with holonomy G2 or Spin(7) can be obtained quite generally by a conformal deformation of the Ricci-flat metric, whereas Calabi-Yau and hyperkähler manifolds require more sophisticated deformations and a case by case treatment.

Motivation & Objective

• To resolve the interpretational ambiguity in identifying the worldvolume of instanton-based NS5-branes in heterotic supergravity, where the singular support of the gauge field does not uniquely determine the brane geometry due to metric singularities.
• To generalize the construction of smeared 5-brane solutions on Ricci-flat cones with reduced holonomy (G₂, Spin(7), SU(2), etc.) beyond flat or symmetric spaces.
• To develop a systematic method to extend finite-scale instantons to a regular supergravity solution by introducing background strings and analyzing the small instanton limit on a modified spacetime manifold.
• To demonstrate that the singular support of the gauge field in the small instanton limit can be reinterpreted as a calibrated 6-dimensional submanifold, consistent with 5-brane worldvolume geometry.
• To explore the holographic duality of the resulting solutions, particularly the appearance of AdS₃ × S² × T⁵-like near-horizon geometries and their associated superisometry algebras.

Proposed method

• Constructing smeared 5-brane solutions on Ricci-flat cones with G₂ or Spin(7) holonomy via conformal deformation of the Ricci-flat metric, which preserves the instanton equation due to conformal invariance.
• Using the heterotic Bianchi identity dH = α′/4 Tr(R∧R − F∧F) to relate Yang-Mills instantons to 5-brane sources, where the instanton scale parameter ρ corresponds to brane thickness.
• Extending finite-scale instanton solutions to a higher-dimensional Lorentzian manifold by adding background strings, thereby removing singularities at r=0 and enabling a smooth supergravity solution.
• Analyzing the small instanton limit (ρ→0) on the extended spacetime to determine the singular support of the gauge field, which now lies at r=0 and can be identified with a 6-dimensional calibrated submanifold.
• Applying the method to specific cases such as X=S³, where the singular support becomes a smeared 3-cycle, and showing that the resulting geometry matches the expected 5-brane worldvolume.
• Deriving the superisometry algebra of the near-horizon AdS₃ solution, which takes the form 𝔤₀⊕𝔤, suggesting a holographic dual CFT with heterotic sigma model structure.

Experimental results

Research questions

• RQ1 Can smeared 5-brane solutions on G₂ and Spin(7) cones be systematically constructed via conformal deformation of the Ricci-flat metric?
• RQ2 Why do standard instanton-based supergravity solutions fail to clearly identify the brane worldvolume, and how can this ambiguity be resolved?
• RQ3 What is the role of background strings in removing singularities and enabling a consistent interpretation of the small instanton limit?
• RQ4 How does the singular support of the gauge field in the small instanton limit relate to the calibrated submanifold structure of the brane worldvolume?
• RQ5 What is the holographic dual of the resulting supergravity solution, and how does its isometry algebra support a CFT interpretation?

Key findings

• Smeared 5-brane solutions on G₂ and Spin(7) cones can be obtained via conformal deformation of the Ricci-flat metric, which preserves the instanton equation due to conformal invariance.
• In contrast, Calabi-Yau and hyperkähler manifolds require more sophisticated, case-by-case deformations due to their distinct geometric constraints.
• The singular support of the gauge field in the small instanton limit is located at r=0, which is not part of physical spacetime in the original solution, leading to ambiguity in brane worldvolume identification.
• By extending the solution to include background strings and analyzing the small instanton limit on the new spacetime, the singular support can be reinterpreted as a 6-dimensional calibrated submanifold, consistent with a 5-brane worldvolume.
• The near-horizon geometry of the solution exhibits AdS₃×S²×T⁵-like structure, with an asymptotic isometry algebra of the form 𝔤₀⊕𝔤, suggesting a holographic dual CFT with heterotic sigma model structure.
• For the case X=S³, the singular support corresponds to a smeared 3-cycle, and the 5-brane is evenly distributed over S³, consistent with a localized 5-brane interpretation in the small instanton limit.