Exact Half-BPS Flux Solutions in M-theory II: Global solutions asymptotic to AdS_7 x S^4

TL;DR

This work constructs globally regular half-BPS flux solutions in 11D supergravity that are asymptotic to $AdS_{7}\times S^{4}$ and preserve $SO(2,2)\times SO(4)\times SO(4)$, focusing on the Case II class. The local solution is governed by a real harmonic function $h$ on a 2D base $\Sigma$ and a complex function $G$ satisfying $\partial_w G = \tfrac{1}{2}(G+\bar{G})\partial_w \ln h$, with bulk geometry encoded in $W^{2}= -|G|^{4}-(G-\bar{G})^{2}$; the paper derives explicit global regular solutions by prescribing boundary data on $\partial\Sigma$ via intervals on the $s$-axis, yielding a genus-$g$ family parameterized by $2g+1$ moduli. It proves bulk regularity by demonstrating $W^{2}>0$ under ordered boundary data (all $\eta_n$ equal) and showcases a nontrivial $g=1$ example with extra four-cycles and fluxes, illustrating richer topology than the AdS$_7\times S^{4}$ vacuum. Interpreted within AdS/CFT, these solutions describe gravity duals of 1+1D half-BPS defects in the 6D $(2,0)$ CFT, analogous to bubbling Wilson loops, and point toward deeper insights into M5-brane dynamics and possible matrix-model descriptions, with future avenues including multi-AdS asymptotics via multi-pole harmonic data.

Abstract

General local half-BPS solutions in M-theory, which have $SO(2,2)\times SO(4)\times SO(4)$ symmetry and are asymptotic to $AdS_{7}\times S^{4}$, were constructed in exact form by the authors in [arXiv:0806.0605]. In the present paper, suitable regularity conditions are imposed on these local solutions, and corresponding globally well-defined solutions are explicitly constructed. The physical properties of these solutions are analyzed, and interpreted in terms of the gravity duals to extended 1+1-dimensional half-BPS defects in the 6-dimensional CFT with maximal supersymmetry.

Figures (6)

• Figure 1: $\Sigma$ and boundary conditions for $AdS_{7}\times S^{4}$ solution
• Figure 2: The surface $\Sigma$ and boundary conditions for a general regular solution.
• Figure 3: Metric factors for a $g=1$ solution.
• Figure 4: Nontrivial four cycles for the $g= 1$ solution
• Figure 5: The fluxes for a $g= 0$ solution.