The Structure of the D0-D4 Bound State

TL;DR

This paper derives and analyzes equations describing the bound state of a D0-brane and a D4-brane, revealing a unique normalizable solution and uncovering unusual features in the asymptotic behavior influenced by symmetry.

Contribution

It formulates the wavefunction equations for the D0-D4 bound state and reduces them to a single, simple equation, providing new insights into their structure and solutions.

Findings

Infinite solutions exist, but only one is normalizable.

Leading asymptotic behavior is influenced by massive modes due to symmetry.

The vacuum equations simplify to a single, elegant equation.

Abstract

We derive a set of equations for the wavefunction describing the marginal bound state of a single D0-brane with a single D4-brane. These are equations determining the vacuum of an N=8 abelian gauge theory with a charged hypermultiplet. We then solve these equations for the most general possible zero-energy solution using a Taylor series. We find that there are an infinite number of such solutions of which only one must be normalizable. We explore the structure of a normalizable solution under the assumption of an asymptotic expansion. Even the leading terms in the asymptotic series, which should reflect the supergravity solution, are unusual. Through the $Spin(5)$ flavor symmetry, the modes which are massive at long distance actually influence the leading behavior. Lastly, we show that the vacuum equations can quite remarkably be reduced to a single equation involving one unknown function. The resulting equation has a surprisingly simple and suggestive form.