[Paper Review] Gravity solutions for the D1-D5 system with angular momentum

Oleg Lunin, Juan Maldacena+1 more|ArXiv.org|Dec 17, 2002

Black Holes and Theoretical Physics31 references188 citations

TL;DR

This paper constructs exact, non-singular supergravity solutions for the D1-D5 system with angular momentum by leveraging U-duality to map to Kaluza-Klein monopole 'supertubes'. The solutions reveal that increasing angular momentum reduces the energy gap to non-BPS excitations, approaching zero near black hole formation, and demonstrate that conical singularities are inadequate long-distance approximations for generic chiral primary states.

ABSTRACT

We construct a large family of supergravity solutions that describe BPS excitations on AdS_3 x S^3 with angular momentum on S^3. These solutions take into account the full backreaction on the metric. We find that as we increase the energy of the excitation, the energy gap to the next non-BPS excitation decreases. These solutions can be viewed as Kaluza-Klein monopole ``supertubes'' which are completely non-singular geometries. We also make some remarks on supertubes in general.

Motivation & Objective

• To construct exact supergravity solutions for BPS D1-D5 states with angular momentum on S³, accounting for full metric backreaction.
• To understand the gravitational implications of large angular momentum in AdS₃×S³ compactifications.
• To clarify the relationship between non-singular supertube geometries and the long-distance behavior of chiral primary states.
• To challenge the validity of conical singularity approximations for generic chiral primaries in AdS₃×S³.
• To explore the geometric and physical properties of supertubes in various dimensions and U-duality frames.

Proposed method

• Utilize U-duality to map the D1-D5 system with angular momentum to a Kaluza-Klein monopole configuration wrapped on T⁴ and a circle.
• Construct solutions using the profile of a D2-brane with electric and magnetic fields, corresponding to a tubular geometry with arbitrary cross-section.
• Apply the known gravity solution for a circular KK monopole and generalize it to non-circular shapes via a parametric ansatz.
• Use the correspondence between left-moving string excitations and gravity solutions to model arbitrary supertube profiles.
• Derive asymptotic and near-ring metrics in various dimensions (d=3 to d=8) by expanding the solution in the limit of small ρ (distance from the ring).
• Analyze the scaling behavior of metric components, particularly g_tt and g_φφ, to assess non-degeneracy and singularity structure in different dimensions.

Experimental results

Research questions

• RQ1 How do BPS D1-D5 states with angular momentum on S³ affect the background geometry when backreaction is fully included?
• RQ2 What is the relationship between the energy gap to non-BPS excitations and increasing angular momentum in these configurations?
• RQ3 Why are conical singularity solutions inadequate for describing the long-distance physics of generic chiral primary states?
• RQ4 How do the geometric properties of supertubes differ across dimensions, particularly in terms of metric scaling near the brane?
• RQ5 In which U-duality frames do non-singular supertube solutions remain non-singular, and what does this imply for the physical interpretation of curvature invariants?

Key findings

• The energy gap to the next non-BPS excitation decreases as angular momentum increases, approaching zero for states on the verge of forming black holes.
• The solutions are non-singular because they arise from Kaluza-Klein monopoles, which are inherently regular geometries.
• For maximal angular momentum, the near-horizon geometry is AdS₃×S³ in global coordinates, matching the vacuum state of the dual CFT.
• Conical singularities with opening angles 2π/N are only valid approximations for very special chiral primaries, not generic ones.
• In dimensions d>4, the g_φφ component of the metric scales more slowly than other components near the ring, indicating non-trivial curvature effects that prevent a simple brane interpretation.
• The near-ring metrics in all dimensions show that g_tt scales to zero as (ρ/a)^(3(d-3)/2), but g_tφ remains finite, ensuring a non-degenerate geometry even in the limit.