2012/10/31 by J. M. Baptista, Baptista, J. M.
Mathematics · Physics and Astronomy · #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #hep-th #math.DG
paper · pdf · doi:10.48550/arxiv.1211.0012
v2: 48 pages; significant changes; description of the vortex moduli spaces of the GLSM extended to allow general values of the parameters, beyond the generic values of v1
arxiv created 2013/08/20 · arxiv updated 2013/08/21
We consider the self-dual vortex equations on a positive line bundle L --> M over a compact Kaehler manifold of arbitrary dimension. When M is simply connected, the moduli space of vortex solutions is a projective space. When M is an abelian variety, the moduli space is the projectivization of the Fourier-Mukai transform of L. We extend this description of the moduli space to the abelian GLSM, i.e. to vortex equations with a torus gauge group acting linearly on a complex vector space. After establishing the Hitchin-Kobayashi correspondence appropriate for the general abelian GLSM, we give explicit descriptions of the vortex moduli space in the case where the manifold M is simply connected or is an abelian variety. In these examples we compute the Kaehler class of the natural L2-metric on the moduli space. In the simplest cases we compute the volume and total scalar curvature of the muduli space. Finally, we note that for abelian GLSM the vortex moduli space is a compactification of the space of holomorphic maps from M to toric targets, just as in the usual case of M being a Riemann surface. This leads to various natural conjectures, for instance explicit formulae for the volume of the space of maps CPm --> CPn.