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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
##### Keywords:
line bundle; regular singularities; moduli space
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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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