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We prove a local analog of the Deligne-Riemann-Roch isomorphism in the case of line bundles and relative dimension $1$. This local analog consists in computation of the class of $12$th power of the determinant central extension of a group ind-scheme $\mathcal G$ by the multiplicative group scheme over $\mathbb Q$ via the product of $2$-cocyles in the second cohomology group. These $2$-cocycles are the compositions of the Contou-Carr\`{e}re symbol with the $\cup$-product of $1$-cocycles. The group ind-scheme $\mathcal G$ represents the functor which assigns to every commutative ring $A$ the group that is the semidirect product of the group $A((t))^*$ of invertible elements of $A((t))$ and the group of continuous $A$-automorphisms of $A$-algebra $A((t))$. The determinant central extension naturally acts on the determinant line bundle on the moduli stack of geometric data (proper quintets). A proper quintet is a collection of a proper family of curves over $\mathop{\rm Spec} A$, a line bundle on this family, a section of this family, a relative formal parameter at the section, a formal trivialization of the bundle at the section that satisfy further conditions., Comment: 50 pages; corrected misprints; to appear in Izvestiya: Mathematics

The Bott-Thurston cocycle is a $2$-cocycle on the group of orientation-preserving diffeomorphisms of the circle. We introduce and study a formal analog of Bott-Thurston cocycle. The formal Bott-Thurston cocycle is a $2$-cocycle on the group of continuous $A$-automorphisms of the algebra $A((t))$ of Laurent series over a commutative ring $A$ with values in the group $A^*$ of invertible elements of $A$. We prove that the central extension given by the formal Bott-Thurston cocycle is equivalent to the $12$-fold Baer sum of the determinantal central extension when $A$ is a $\mathbb Q$-algebra. As a consequence of this result we prove a part of new formal Riemann-Roch theorem. This Riemann-Roch theorem is applied to a ringed space on a separated scheme $S$ over $\mathbb Q$, where the structure sheaf of the ringed space is locally on $S$ isomorphic to the sheaf ${\mathcal O}_S((t))$ and the transition automorphisms are continuous. Locally on $S$ this ringed space corresponds to the punctured formal neighbourhood of a section of a smooth morphism to $U$ of relative dimension $1$, where an open subset $U \subset S$., Comment: 43 pages; corrected misprints