Brane Dynamics in Background Fluxes and Non-commutative Geometry

TL;DR

The paper demonstrates that D-branes wrapping spheres inside $S^3$ with NS-NS flux realize non-commutative world-volumes in the form of fuzzy $S^2$ algebras, leading to a low-energy action that combines non-commutative Yang–Mills and Chern–Simons terms. Through boundary CFT analyses of the SU(2) WZW model, it shows that stacks of D0-branes can dynamically condense into higher-dimensional branes, with the fixed-point structures governed by su(2) representations and a consistency check via $g$-factors. The results extend to supersymmetric cases and suggest universal features for branes in curved backgrounds, including potential connections to quantum-group structures and finite-radius effects, with broader implications for tachyon condensation and bound-state formation in string theory.

Abstract

Branes in non-trivial backgrounds are expected to exhibit interesting dynamical properties. We use the boundary conformal field theory approach to study branes in a curved background with non-vanishing Neveu-Schwarz 3-form field strength. For branes on an $S^3$, the low-energy effective action is computed to leading order in the string tension. It turns out to be a field theory on a non-commutative `fuzzy 2-sphere' which consists of a Yang-Mills and a Chern-Simons term. We find a certain set of classical solutions that have no analogue for flat branes in Euclidean space. These solutions show, in particular, how a spherical brane can arise as bound state from a stack of D0-branes.