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Notes on prequantization of moduli of \(G\)-bundles with connection on Riemann surfaces. (English) Zbl 1078.53095
Summary: Let \({\mathcal X}\to S\) be a smooth proper family of complex curves (i.e., family of Riemann surfaces), and \({\mathcal F}\) a \(G\)-bundle over \({\mathcal X}\) with connection along the fibres \({\mathcal X}\to S\). We construct a line bundle with connection \(({\mathcal L}_{\mathcal F}, \nabla_{\mathcal F})\) on \(S\) (also in cases when the connection on \({\mathcal F}\) has regular singularities). We discuss the resulting \(({\mathcal L}_{\mathcal F},\nabla_{\mathcal F})\), mainly in the case \(G=\mathbb{C}^*\). For instance when \(S\) is the moduli space of line bundles with connection over a Riemann surface \(X\), \({\mathcal X}=X\times S\), and \({\mathcal F}\) is the Poincaré bundle over \({\mathcal X}\), we show that \(({\mathcal L}_{\mathcal F},\nabla_{\mathcal F})\) provides a prequantization of \(S\).
MSC:
53D50 Geometric quantization
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References:
[1] Deligne, P., Théorie de Hodge. III., Inst. Hautes Ètudes Sci. Publ. Math., 44, 5-77, (1974) · Zbl 0237.14003
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