Intersection Rules for p-Branes

TL;DR

The paper addresses how extremal $p$-branes can intersect with zero binding energy in a general $D$-dimensional setting. It develops a model-independent approach by solving the bosonic equations of motion for gravity, a dilaton, and multiple $n$-form fields, using a metric ansatz that leads to ${\cal N}$ harmonic functions and a harmonic superposition rule. The key contribution is a universal intersection condition $\bar{q}+1 = \frac{(q_A+1)(q_B+1)}{D-2} - \frac{1}{2}\varepsilon_A a_A \varepsilon_B a_B$, together with the algebraic constraints $M_{AB}=0$ that fix brane intersections; this is then specialized to $D=11$ and $D=10$ to yield explicit M-brane, D-brane, and NSNS-brane intersection patterns. The results are consistent with supersymmetry and support interpretations of open branes ending on higher-dimensional branes (e.g., membranes on M5 and NS5 acting as a D-brane for D-branes), with implications for brane bound states and potential microstate counting in black hole entropy.

Abstract

We present a general rule determining how extremal branes can interesect in a configuration with zero binding energy. The rule is derived in a model independent way and in arbitrary spacetime dimensions $D$ by solving the equations of motion of gravity coupled to a dilaton and several different $n$-form field strengths. The intersection rules are all compatible with supersymmetry, although derived without using it. We then specialize to the branes occurring in type II string theories and in M-theory. We show that the intersection rules are consistent with the picture that open branes can have boundaries on some other branes. In particular, all the D-branes of dimension $q$, with $1\leq q \leq6$, can have boundaries on the solitonic 5-brane.