On the Quantization Constraints for a D3 Brane in the Geometry of NS5 Branes

TL;DR

This paper investigates the semi-classical behavior of a D3 brane in the presence of NS5 branes, deriving conditions for supersymmetry preservation and revealing quantization constraints on the brane's configuration.

Contribution

It introduces four methods to derive quantization conditions for D3 branes near NS5 branes, linking discrete approach angles to brane positions.

Findings

Discrete approach angles for D3 branes are derived.

Quantization conditions relate brane positions to intersection angles.

Multiple methods confirm the quantization constraints.

Abstract

A D3 brane in the background of NS5 branes is studied semi-classically. The conditions for preserved supersymmetry are derived, leading to a differential equation for the shape of the D3 brane. The solutions of this equation are analyzed. For a D3 brane intersecting the NS5 branes, the angle of approach is known to be restricted to discrete values. Four different ways to obtain this quantization are described. In particular, it is shown that, assuming the D3 brane avoids intersecting a {\em single} NS5 brane, the above discrete values correspond to the different possible positions of the D3 branes among the NS5 branes.