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 prequantum 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

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... h 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. e-print archive: http://lanl.arXiv.org/abs/arXiv:math-ph/0605027 820 RUKMINI DEY But P −1 = L 2 ⊗ R −2 has the A (0,1) -term as it is and the Φ (0,1) -term in the inverse bundle. Thus P −1 is I-holomorphic. Secondly, J (α (0,1) ) = −iγ (0,1) , Thus w.r.t. J , the A (0,1) − Φ (0,1) -term is holomorphic and the A (0,1) + Φ (0,1) -term is anti-holomorphic. Thus E −1 = E −1 + ⊗ E − is holomorphic since the anti-holomorphic term comes in the inverse. Thirdly, K(α (0,1) ) = γ (0,1) , Thus w.r.t. K, the A (0,1) + iΦ (0,1) -term is anti-holomorphic and theA (0,1) − iΦ (0,1) -term is holomorphic. Thus N −1 = N −1 + ⊗ N − is holomorphic. Polarization Since the symplectic forms are all Kähler, we can take square integrable I-holomorphic sections of P −1 , J -holomorphic sections of E −1 and K-holomorphic sections of N −1 as our Hilbert spaces. But we are still not guaranteed finite dimensional Hilbert spaces.