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In this paper we first show that on projective manifolds (M, {\omega}), thereare holomorphic determinant bundles (in the sense of Knusden-Mumford used byBismut, Gillet, Soule) which play the role of the geometric quantum bundle,namely one for each input data of a Hermitian holomorphic line bundle L ofnon-trivial Chern class on a compact Kahler manifold Z (with Todd genusnon-zero) and a choice of a geometric quantization of (M, {\omega}). Next wefurther study the generalization of the vortex equations on Kahler 4-manifoldwhich has been studied earlier by Bradlow. We show that when the Kahler4-manifold avoids some obstructions then the regular part of the moduli spaceis a Kahler manifold and admit a pull back of a Quillen determinant bundle asthe quantum line bundle, i.e. the curvature is proportional to the Kahler form.Thus they can be quantized geometrically. In fact we show that the moduli spaceof the usual vortex equations on a projective Kahler 4-manifold is projectivewhen the moduli space is smooth. Since in Kahler 4-manifold the vortex moduliand the Seiberg Witten moduli coincide our effort gives a quantization ofSeiberg Witten moduli by determinant bundles

Determinat Bundles and Geometric Quantization Of Vortex Moduli Spaces ON Compact Kahler Surfaces