D-brane Solutions in Non-Commutative Gauge Theory on Fuzzy Sphere

TL;DR

The paper studies non-commutative gauge theory on a fuzzy $S^2$ as the low-energy description of a spherical D2-brane in $S^3$ with background $b$-field, identifying a simple subset of solutions that behave like unstable flat-space solitons and correspond to D0-branes not dissolved in the D2. By matching energies between the non-commutative gauge theory and the ordinary Born-Infeld description, it confirms the D0/D2 interpretation and analyzes the stability of a single D0 near the D2, revealing tachyonic 0–2 fluctuations when proximity is close. The authors diagonalize the 0–2 fluctuation spectrum, predicting a discrete set of massive modes that inform the low-energy 0–2 CFT on $S^3$ in the Seiberg–Witten limit, and discuss how the fuzzy-sphere instability alters the usual Higgs mechanism for small brane separations. Overall, the work illustrates brane polarization effects in curved backgrounds, tests brane democracy via energy matching, and provides concrete targets for future CFT and SUSY extensions in this setting.

Abstract

Non-commutative gauge theory on fuzzy sphere was obtained by Alekseev et al. as describing the low energy dynamics of a spherical D2-brane in S^3 with the background b-field. We identify a subset of solutions of this theory which are analogs of ``unstable'' solitons on a non-commutative flat D2-brane found by Gopakumar et al. Analogously to the flat case, these solutions have the interpretation as describing D0-branes ``not yet dissolved'' by the D2-brane. We confirm this interpretation by showing the precise agreement of the binding energy computed in the non-commutative and ordinary Born-Infeld descriptions. We then study stability of the solution describing a single D0-brane off a D2-brane. Similarly to the flat case, we find an instability when the D0-brane is located close to the D2-brane. We furthermore obtain the complete mass spectrum of 0-2 fluctuations, which thus gives a prediction for the low energy spectrum of the 0-2 CFT in S^3. We also discuss in detail how the instability to a formation of the fuzzy sphere modifies the usual Higgs mechanism for small separation between the branes.