Analytic Diffeomorphisms of the Circle and Topological Riemann–Roch Theorem for Circle Fibrations

We consider the group $$\mathcal G$$ which is the semidirect product of the group of analytic $$\mathbb C^*$$ -valued functions on the circle and the group of analytic orientation-preserving diffeomorphisms of the circle. We construct central extensions of the group $$\mathcal G$$ by the group $$\mathbb C^*$$ . The first central extension, referred to as the determinant central extension, is constructed using the determinants of linear operators acting on infinite-dimensional locally convex topological $${\mathbb C}$$ -vector spaces. Other central extensions are obtained via $$\cup$$ -products of group $$1$$ -cocycles to which a map related to algebraic $$K$$ -theory is applied. We show that in the second cohomology group, i.e., modulo the group $$2$$ -coboundary, the 12th power of the $$2$$ -cocycle constructed via the first central extension is equal to a product of integer powers of $$2$$ -cocycles constructed via $$\cup$$ -products (in multiplicative notation). As an application of this result, we obtain a new topological Riemann–Roch theorem for a complex line bundle $$L$$ on a smooth manifold $$M$$ , where $$\pi \colon\, M\to B$$ is a fibration in oriented circles. More precisely, we prove that in the group $$H^3(B,\mathbb Z)$$ the element $$12[\mathcal Det(L)]$$ is equal to the element $$6\,\pi_*(c_1(L)\cup c_1(L))$$ , where $$[\mathcal Det(L)]$$ is the class of the determinant gerbe on $$B$$ constructed from $$L$$ and the determinant central extension.