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Brane webs and 1/4-BPS geometries

Oleg Lunin

TL;DR

This work provides a comprehensive gravitational treatment of brane webs preserving eight supercharges, spanning IIB string theory and M-theory. It shows that probe analyses and supergravity impose consistent, testable restrictions on brane sources: planar straight-line string webs with charge-correlated orientations for (p,q) strings, holomorphic membranes and D3/D5 webs, and holomorphic brane embeddings in 1/4-BPS backgrounds. The authors construct explicit geometries, prove the existence and uniqueness of solutions via perturbative (multipole) expansions, and establish boundary conditions for regular bubbling geometries with AdS asymptotics. They also formulate unified descriptions that connect 1/2- and 1/4-BPS regimes across IIB and M-theory, and explore the topology and flux quantization of bubbling solutions. The results reinforce the open/closed duality between probe brane dynamics and closed-string geometries, and provide a framework to investigate lower-supersymmetry generalizations and their field theory duals.

Abstract

We discuss brane webs preserving eight supercharges and derive geometries produced by them. Consistency conditions of supergravity are shown to impose certain requirements on the locations of the sources, and these restrictions are found to be in a perfect agreement with results of the probe analysis. In particular, solutions of IIB SUGRA describing (p,q) stings are inconsistent, unless the web consists of straight line segments whose orientation is correlated with charges of the string. The geometries produced by membranes and D3 branes are only consistent if brane profiles are holomorphic. Using perturbation theory, we show that a unique gravity solution exists for any allowed distribution of sources. We also revisit 1/4-BPS geometries with AdS_p x S^q asymptotics and derive the boundary conditions leading to regular geometries. All degenerate limits of regular solutions are shown to agree with expectations from the brane probe analysis.

Brane webs and 1/4-BPS geometries

TL;DR

This work provides a comprehensive gravitational treatment of brane webs preserving eight supercharges, spanning IIB string theory and M-theory. It shows that probe analyses and supergravity impose consistent, testable restrictions on brane sources: planar straight-line string webs with charge-correlated orientations for (p,q) strings, holomorphic membranes and D3/D5 webs, and holomorphic brane embeddings in 1/4-BPS backgrounds. The authors construct explicit geometries, prove the existence and uniqueness of solutions via perturbative (multipole) expansions, and establish boundary conditions for regular bubbling geometries with AdS asymptotics. They also formulate unified descriptions that connect 1/2- and 1/4-BPS regimes across IIB and M-theory, and explore the topology and flux quantization of bubbling solutions. The results reinforce the open/closed duality between probe brane dynamics and closed-string geometries, and provide a framework to investigate lower-supersymmetry generalizations and their field theory duals.

Abstract

We discuss brane webs preserving eight supercharges and derive geometries produced by them. Consistency conditions of supergravity are shown to impose certain requirements on the locations of the sources, and these restrictions are found to be in a perfect agreement with results of the probe analysis. In particular, solutions of IIB SUGRA describing (p,q) stings are inconsistent, unless the web consists of straight line segments whose orientation is correlated with charges of the string. The geometries produced by membranes and D3 branes are only consistent if brane profiles are holomorphic. Using perturbation theory, we show that a unique gravity solution exists for any allowed distribution of sources. We also revisit 1/4-BPS geometries with AdS_p x S^q asymptotics and derive the boundary conditions leading to regular geometries. All degenerate limits of regular solutions are shown to agree with expectations from the brane probe analysis.

Paper Structure

This paper contains 39 sections, 432 equations, 7 figures.

Table of Contents

  1. Introduction and summary.
  2. Webs in the probe approximation
  3. Geometries produced by string webs
  4. Webs of membranes
  5. Structure of the solution
  6. Solution in perturbation theory
  7. Webs of five-- and three--branes
  8. Summary of the solutions
  9. Alternative description
  10. 1/4--BPS bubbling solutions of IIB SUGRA
  11. Local description
  12. Examples
  13. AdS$_5\times$S$^5$ as a 1/4--BPS state in a theory on $R\times S^3$.
  14. AdS$_5\times$S$^5$ as a 1/4--BPS geometry with $SO(4)$ R--symmetry.
  15. 1/2--BPS bubbling solutions.
  16. ...and 24 more sections

Figures (7)

  • Figure 1: String webs: (a) an elementary junction, (b) a connected web, (c) a generic supersymmetric web.
  • Figure 2: Boundary conditions corresponding to two embeddings of $AdS_5\times S^5$: (a) spacial sphere $S^3$ is preserved, (b) $SO(4)$--part of the R--symmetry group is unbroken.
  • Figure 3: Boundary conditions for the 1/2--BPS geometries of LLM: (a) giant graviton and a dual giant, (b) generic distribution of droplets.
  • Figure 4: Correspondence between boundary conditions in 1/2--BPS (a) and 1/4--BPS (b) cases.
  • Figure 5: Giant gravitons and holomorphic surfaces. To construct a worldvolume of a D3 brane, one should take an intersection of the holomorphic surface with $\{y=0,Z=\frac{1}{2}\}$ region (this intersection is shown in red), and fiber $\psi$ and $t$ over it. In the case of $AdS_5\times S^5$, this picture gives a geometric interpretation of the radial coordinate introduced in mikhail
  • ...and 2 more figures