On a smooth line bundle L over a compact Kähler Riemann surface Σ, we study the family of vortex equations with a parameter s. For each s ∈ [1,∞], we invoke techniques in Bradlow (Commun Math Phys 135:1-17, 1990) by turning the s-vortex equation into an s-dependent elliptic partial differential equation, studied in Kazdan and Warner (Ann Math 2:14-47, 1978), providing an explicit moduli space description of the space of gauge classes of solutions. We are particularly interested in the bijective correspondence between the open subset of vortices without common zeros and the space of holomorphic maps. For each s, the correspondence is uniquely determined by a smooth function us on Σ, and we confirm its convergent behaviors as s → ∞. Our results prove a conjecture posed by Baptista in Baptista (Nucl Phys B 844:308-333, 2010), stating that the s-dependent correspondence is an isometry between the open subsets when s = ∞, with L2 metrics appropriately defined.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics