Local regluings of families of curves and sheaves on them and the Deligne–Riemann–Roch theorem

Consider the formal punctured disc and the group which is the semidirect product of the group of invertible functions on this disk and the group of automorphisms of this disk. More precisely, this group has to be considered as a group ind-scheme ${\mathcal G}$ that assigns to every commutative ring $A$ the group ${\mathcal G}(A)$ which is the semidirect product of the group of invertible element $A((t))^*$ of the $A$-algebra of Laurent series $A((t))$ and the group of continuous $A$-automorphisms of the algebra $A((t))$. The group ind-scheme $\mathcal G$ remarkably acts on the moduli space that parameterizes quintets: a projective curve, an invertible sheaf on the curve, a smooth point on the curve, a formal local parameter at the point, a formal trivialization of the sheaf at the point. Besides, there is the Deligne–Riemann–Roch theorem for invertible sheaves on families of smooth projective curves. I will describe the local Deligne–Riemann–Roch theorem as the equivalence of two central extensions of $\mathcal G$ by the multiplicative group ${\mathbb G}_m$. Note that one of the key tools to construct one of the two central extensions is the Contou-Carrère symbol that is the bimultiplicative pairing on the group $A((t))^*$ with values in the group ${\mathbb G}_m(A)=A^*$.