Abstract
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.
