Multiple Intersections of D-branes and M-branes

TL;DR

The paper classifies all threshold BPS bound-state intersections of $D$-branes in ten dimensions and $M$-branes in eleven dimensions, determining their residual supersymmetry. It uses the harmonic-function rule with independent harmonic functions $H_i$ for each brane and analyzes dependence on transverse coordinates, focusing on cases with a single scalar and a single harmonic function in $D\ge 2$. The main results show a maximal eight participating branes in these intersections (nine with an $n=8$ pair) and identify three inequivalent eight-brane $D$-brane configurations, including one that cannot be lifted to non-boosted $M$-branes. The work also carries out detailed dimensional reduction to lower-dimensional dilatonic $p$-branes, deriving the general relation $\\Delta = a^2 + 2 { (p+1)(D-p-3) \\over D-2 }$ and the supersymmetry condition $\\Delta = 4/N$, thereby linking high-dimensional brane intersections to bound-state realizations in $D=10,11$ and illuminating implications for black hole microphysics. Overall, the results provide a unified framework connecting $D$- and $M$-brane intersections across dimensions and identifying which configurations originate in eleven dimensions.

Abstract

We give a classification of all multiple intersections of D-branes in ten dimensions and M-branes in eleven dimensions that corresponds to threshold BPS bound states. The residual supersymmetry of these composite branes is determined. By dimensional reduction composite p-branes in lower dimensions can be constructed. We emphasize in dimensions D greater or equal than two, those solutions which involve a single scalar and depend on a single harmonic function. For these extremal branes we obtain the strength of the coupling between the scalar and the gauge field. In particular we give a D-brane and M-brane interpretation of extreme p-branes in two, three and four dimensions.

Figures (4)

• Figure 1: $D$-brane intersections with $n=4$ in 10 dimensions:the numbers $(n_1, n_2,\ldots)$ label the number of times a building block with $(1,2,\ldots)$ worldvolume directions is used. The subscript in the Figure indicates the amount of supersymmetry preserved in each solution. The number $N$ indicates the number of independent harmonics. The lines between solutions indicate how one configuration follows from another by adding (or deleting) a harmonic function. The configuration (0,0,0,7) cannot be extended to 11 dimensions in terms of (non-boosted) 2- and 5-branes only.
• Figure 2: $M$-brane intersections with $n=4,5$ in 11 dimensions:the numbers $(n_1, \cdots ,n_{[N/2]})$ are the same labels used in $D=10$, and indicate to which $D$-brane intersection the $D=11$ solution reduces. The configurations in gray rectangles only reduce to $D=10$ intersections involving NS-NS branes. For these configurations we use the eleven-dimensional notation $\{n_1,\cdots ,n_N\}$ explained in the text. The subscripts indicate the amount of residual supersymmetry.
• Figure 3: $D$-brane intersections with $n=4,8$ in 10 dimensions:The solutions are labelled by $(n_1,\ldots n_{[N/2]})$, as explained in section 2. For $N=5$ an extra superscript is added to distinguish between the two sets of labels. Subscripts indicate the supersymmetry of the configurations. The $(1,0,0,7)$ configuration given in the grey rectangle cannot be extended to eleven dimensions in terms of (non-boosted) 2- and 5-branes.
• Figure 4: $M$-brane intersections with $n=4,5,8$ in 11 dimensions: Since all con-fi-gu-ra-tions reduce to $D$-branes in $D=10$ with $n=4,8$ we use $D=10$ labels to classify the solutions. For $N=5$ an extra superscript is added to distinguish between the two sets of labels. Subscripts indicate the unbroken supersymmetry.