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Bubbling 1/4 BPS solutions in type IIB and supergravity reductions on S^n x S^n

James T. Liu, Diana Vaman, W. Y. Wen

TL;DR

The authors extend bubbling geometries by analyzing reductions of IIB and 6d supergravity on S^n×S^n to four dimensions. They show that a consistent S^3×S^3 reduction of IIB supports bubbling 1/2 BPS solutions, while a minimal 6d S^1×S^1 reduction is constrained by F∧F=0, blocking bubbling AdS3×S3 within that framework. By including an axion-dilaton, they construct bubbling 1/4 BPS solutions corresponding to backreacted D7 branes, with a conformally flat two-dimensional base and a deficit angle encoding fractional statistics in the dual gauged quantum mechanics. The analysis highlights a Weyl-like harmonic reduction structure and clarifies when supersymmetry guarantees full equations of motion. Overall, the work maps precise conditions under which bubbling vacua exist in lower-dimensional reductions and elucidates the role of additional fields in enriching BPS sectors.

Abstract

We extend the construction of bubbling 1/2 BPS solutions of Lin, Lunin and Maldacena (hep-th/0409174) in two directions. First we enquire whether bubbling 1/2 BPS solutions can be constructed in minimal 6d supergravity and second we construct solutions that are 1/4 BPS in type IIB. We find that the S^1 x S^1 bosonic reduction of (1,0) 6d supergravity to 4d gravity coupled to 2 scalars and a gauge field is consistent only provided that the gauge field obeys a constraint (F \wedge F=0). This is to be contrasted to the case of the S^3 x S^3 bosonic reduction of type IIB supergravity to 4d gravity, 2 scalars and a gauge field, where consistency is achieved without imposing any such constraints. Therefore, in the case of (1,0) 6d supergravity we are able to construct 1/2 BPS solutions, similar to those derived in type IIB, provided that this additional constraint is satisfied. This ultimately prohibits the construction of a family of 1/2 BPS solutions corresponding to a bubbling AdS_3 x S^3 geometry. Returning to type IIB solutions, by turning on an axion-dilaton field we construct a family of bubbling 1/4 BPS solutions. This corresponds to the inclusion of back-reacted D7 branes to the solutions of Lin, Lunin and Maldacena.

Bubbling 1/4 BPS solutions in type IIB and supergravity reductions on S^n x S^n

TL;DR

The authors extend bubbling geometries by analyzing reductions of IIB and 6d supergravity on S^n×S^n to four dimensions. They show that a consistent S^3×S^3 reduction of IIB supports bubbling 1/2 BPS solutions, while a minimal 6d S^1×S^1 reduction is constrained by F∧F=0, blocking bubbling AdS3×S3 within that framework. By including an axion-dilaton, they construct bubbling 1/4 BPS solutions corresponding to backreacted D7 branes, with a conformally flat two-dimensional base and a deficit angle encoding fractional statistics in the dual gauged quantum mechanics. The analysis highlights a Weyl-like harmonic reduction structure and clarifies when supersymmetry guarantees full equations of motion. Overall, the work maps precise conditions under which bubbling vacua exist in lower-dimensional reductions and elucidates the role of additional fields in enriching BPS sectors.

Abstract

We extend the construction of bubbling 1/2 BPS solutions of Lin, Lunin and Maldacena (hep-th/0409174) in two directions. First we enquire whether bubbling 1/2 BPS solutions can be constructed in minimal 6d supergravity and second we construct solutions that are 1/4 BPS in type IIB. We find that the S^1 x S^1 bosonic reduction of (1,0) 6d supergravity to 4d gravity coupled to 2 scalars and a gauge field is consistent only provided that the gauge field obeys a constraint (F \wedge F=0). This is to be contrasted to the case of the S^3 x S^3 bosonic reduction of type IIB supergravity to 4d gravity, 2 scalars and a gauge field, where consistency is achieved without imposing any such constraints. Therefore, in the case of (1,0) 6d supergravity we are able to construct 1/2 BPS solutions, similar to those derived in type IIB, provided that this additional constraint is satisfied. This ultimately prohibits the construction of a family of 1/2 BPS solutions corresponding to a bubbling AdS_3 x S^3 geometry. Returning to type IIB solutions, by turning on an axion-dilaton field we construct a family of bubbling 1/4 BPS solutions. This corresponds to the inclusion of back-reacted D7 branes to the solutions of Lin, Lunin and Maldacena.

Paper Structure

This paper contains 15 sections, 113 equations.

Table of Contents

  1. Introduction and Summary
  2. $S^3\times S^3$ compactification of IIB supergravity
  3. Supersymmetry variations
  4. $S^1\times S^1$ compactification of Minimal $D=6$ Supergravity
  5. Supersymmetry variations
  6. Supersymmetry analysis
  7. Specialization of $\eta$ and $\widetilde{\eta}$
  8. Both $\eta$ and $\widetilde{\eta}$ non-vanishing
  9. Only one of $\eta$ and $\widetilde{\eta}$ non-vanishing
  10. Both $\eta$ and $\widetilde{\eta}$ vanishing
  11. $S^1\times S^1$ reduction in the presence of a tensor multiplet
  12. Bubbling 1/4 BPS solutions: turning on an axion-dilaton
  13. Bosonic reductions on $S^n\times S^n$
  14. Integrability of the Killing spinor equations
  15. Differential identities for the spinor bilinears