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We study the gluing theory of Riemann surfaces using formal algebraicgeometry, and give computable relations between the associated parameters fordifferent gluing processes. As its application to the Liouville conformal fieldtheory, we construct the sheaf of tempered conformal blocks on the moduli spaceof pointed Riemann surfaces which satisfies the factorization property and hasa natural action of the Teichm\"{u}ller groupoid.

Gluing theory of Riemann surfaces and Liouville conformal field theory