Magnetic zero-modes, vortices and Cartan geometry.
LETTERS IN MATHEMATICAL PHYSICS 2017;
108:949-983. [PMID:
29606790 PMCID:
PMC5869901 DOI:
10.1007/s11005-017-1023-2]
[Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 07/28/2017] [Revised: 10/21/2017] [Accepted: 10/23/2017] [Indexed: 06/08/2023]
Abstract
We exhibit a close relation between vortex configurations on the 2-sphere and magnetic zero-modes of the Dirac operator on [Formula: see text] which obey an additional nonlinear equation. We show that both are best understood in terms of the geometry induced on the 3-sphere via pull-back of the round geometry with bundle maps of the Hopf fibration. We use this viewpoint to deduce a manifestly smooth formula for square-integrable magnetic zero-modes in terms of two homogeneous polynomials in two complex variables.
Collapse