In this note we quantize the usual symplectic (Kähler) form on the vortex moduli space by modifying the Quillen metric of the Quillen determinant line bundle.
Hitchin has shown that the moduli space M of the dimensionally reduced self-dual Yang-Mills equations has a hyperKähler structure. In this paper we first explicitly show the hyperKähler structure, the details of which is missing in Hitchin's paper. We show here that M admits three pre-quantum line bundles, corresponding to the three symplectic forms. We use Quillen's determinant line bundle construction and show that the Quillen curvatures of these prequantum line bundles are proportional to each of the symplectic forms mentioned above. The prequantum line bundles are holomorphic with respect to their respective complex structures. We show how these prequantum line bundles can be derived from cocycle line bundles of Chern-Simons gauge theory with complex gauge group in the case when the moduli space is smooth.
In this paper we obtain the general solution to the minimal surface equation, namely its local Weierstrass-Enneper representation, using a system of hodographic coordinates. This is done by using the method of solving the Born-Infeld equations by Whitham. We directly compute conformal coordinates on the minimal surface which give the Weierstrass-Enneper representation. From this we derive the hodographic coordinate ρ ∈ D ⊂ C and σ its complex conjugate which enables us to write the Weierstrass-Enneper representation in a new way.
Abstract. In this paper we show that the dimensionally reduced SeibergWitten equations lead to a Higgs field and study the resulting moduli spaces. The moduli space arising out of a subset of the equations, shown to be nonempty for a compact Riemann surface of genus g ≥ 1, gives rise to a family of moduli spaces carrying a hyperkähler structure. For the full set of equations the corresponding moduli space does not have the aforementioned hyperkähler structure but has a natural symplectic structure. For the case of the torus, g = 1, we show that the full set of equations has a solution, different from the "vortex solutions".
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