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1/2-BPS states in M theory and defects in the dual CFTs

Oleg Lunin

TL;DR

This work advances the holographic understanding of 1/2-BPS states in M-theory by classifying brane configurations in $AdS_7\times S^4$ and $AdS_4\times S^7$, showing that probe membranes can polarize into M5-branes with fluxes and that such states backreact to yield geometries governed by a harmonic function (in the $SO(2,2)\times SO(4)^2$ sector) or a Toda equation (in the $SO(2,1)\times SO(6)$ sector). It provides a unified framework linking probe branes to fully backreacted supergravity solutions, including a complete boundary-data–to–geometry map, perturbative constructions around both AdS backgrounds, and a Schwarz–Christoffel map description for multi-membrane seeds. The results describe defects in 6D (2,0) theories and 3D CFTs, revealing how giant- and dual-giant-type branes encode different regimes of defect representations and how topology and fluxes manifest in the bulk via boundary conditions on a two-dimensional harmonic function. The paper also explores various decompactification limits, reproducing Russo–Tseytlin-type backgrounds and mass-deformed M2 geometries, thereby connecting disparate M-theory solutions within a single lightlike–to–defect dictionary and enriching the AdS/CFT correspondence with explicit backreacted brane geometries.

Abstract

We study supersymmetric branes in AdS_7 x S^4 and AdS_4 x S^7. We show that in the former case the membranes should be viewed as M5 branes with fluxes and we identify two types of such fivebranes (they are analogous to giant gravitons and to dual giants). In AdS_4 x S^7 we find both M5 branes with fluxes and freestanding stacks of membranes. We also go beyond probe approximation and construct regular supergravity solutions describing geometries produced by the branes. The metrics are completely specified by one function which satisfies either Laplace or Toda equation and we give a complete classification of boundary conditions leading to smooth geometries. The brane configurations discussed in this paper are dual to various defects in three- and six-dimensional conformal field theories.

1/2-BPS states in M theory and defects in the dual CFTs

TL;DR

This work advances the holographic understanding of 1/2-BPS states in M-theory by classifying brane configurations in and , showing that probe membranes can polarize into M5-branes with fluxes and that such states backreact to yield geometries governed by a harmonic function (in the sector) or a Toda equation (in the sector). It provides a unified framework linking probe branes to fully backreacted supergravity solutions, including a complete boundary-data–to–geometry map, perturbative constructions around both AdS backgrounds, and a Schwarz–Christoffel map description for multi-membrane seeds. The results describe defects in 6D (2,0) theories and 3D CFTs, revealing how giant- and dual-giant-type branes encode different regimes of defect representations and how topology and fluxes manifest in the bulk via boundary conditions on a two-dimensional harmonic function. The paper also explores various decompactification limits, reproducing Russo–Tseytlin-type backgrounds and mass-deformed M2 geometries, thereby connecting disparate M-theory solutions within a single lightlike–to–defect dictionary and enriching the AdS/CFT correspondence with explicit backreacted brane geometries.

Abstract

We study supersymmetric branes in AdS_7 x S^4 and AdS_4 x S^7. We show that in the former case the membranes should be viewed as M5 branes with fluxes and we identify two types of such fivebranes (they are analogous to giant gravitons and to dual giants). In AdS_4 x S^7 we find both M5 branes with fluxes and freestanding stacks of membranes. We also go beyond probe approximation and construct regular supergravity solutions describing geometries produced by the branes. The metrics are completely specified by one function which satisfies either Laplace or Toda equation and we give a complete classification of boundary conditions leading to smooth geometries. The brane configurations discussed in this paper are dual to various defects in three- and six-dimensional conformal field theories.

Paper Structure

This paper contains 40 sections, 344 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Branes in $AdS_7\times S^4$
  3. Probe M2 brane
  4. M5 brane wrapping ${S}^3$.
  5. M5 brane wrapping ${\tilde{S}}^3$.
  6. Branes in $AdS_4\times S^7$
  7. One--dimensional intersection.
  8. Two--dimensional intersections.
  9. Geometries with $SO(2,2)\times SO(4)\times SO(4)$ symmetry
  10. $AdS_4\times S^7$ branch.
  11. Recovering $AdS_4\times S^7$.
  12. Perturbation theory.
  13. Topology, fluxes and brane probes.
  14. Generalization: multiple membrane seeds and Schwarz--Christoffel map
  15. Structure of solutions with one cut
  16. ...and 25 more sections

Figures (12)

  • Figure 1: Boundary condition for the harmonic function describing a typical regular geometry. The dark regions correspond to $\partial_y\Phi|_{y=0}=2\pi~\hbox{sign}(c_1c_2)$ and light regions correspond to $\partial_y\Phi|_{y=0}=0$ (see equation (\ref{['y0Regul']})). This is the most general boundary condition unless $c_1=c_2=-1/2$.
  • Figure 2: Boundary condition for the harmonic function on $c_1=c_2=-1/2$ branch. The normal derivatives of $\Phi$ are fixed on the $x$ axis and along the branch cuts (see equations (\ref{['y0Regul']}), (\ref{['xCutRegul']})).
  • Figure 3: (a) Boundary conditions for the harmonic function (\ref{['HarmA4S7']}) describing $AdS_4\times S^7$. (b) Boundary conditions corresponding to a typical excitation (\ref{['A4S7GenHarm']}) of $AdS_4\times S^7$.
  • Figure 4: Two types of non--contractible four--spheres.
  • Figure 5: Construction of a non--contractible seven--manifold: one has to fiber both three--spheres over the one--dimensional contour.
  • ...and 7 more figures