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Analytic diffeomorphisms of the circle and topological Riemann-Roch theorem for circle fibrations

Denis V. Osipov

TL;DR

This work constructs and compares determinant and related central extensions for the circle-diffeomorphism group, embedding them in a robust infinite-dimensional (Silva-space) framework. It introduces explicit 2-cocycles via block-operator determinants, bimultiplicative pairings, and cup-products, then connects these to Deligne cohomology and Gysin push-forwards. The central result is a topological Riemann-Roch theorem for circle fibrations, expressing $12$ times the determinant-gerbe class in terms of the push-forward of cup products of first Chern classes, thereby linking analytic, algebraic, and topological data. The methods unify formal and analytic analogues, using conformal welding and precise operator-theoretic constructions to derive a cohomological equality in $H^3(B,\mathbb{Z})$ with clear geometric implications for circle bundles and line bundles over base manifolds.

Abstract

We consider the group $\mathcal G$ which is the semidirect product of the group of analytic functions with values in ${\mathbb C}^*$ on the circle and the group of analytic diffeomorphisms of the circle that preserve the orientation. Then we construct the central extensions of the group $\mathcal G$ by the group ${\mathbb C}^*$. The first central extension, so-called the determinant central extension, is constructed by means of determinants of linear operators acting in infinite-dimensional locally convex topological $\mathbb C$-vector spaces. Other central extensions are constructed by $\cup$-products of group $1$-cocycles with the application to them the map related with algebraic $K$-theory. We prove in the second cohomology group, i.e. modulo of a group $2$-coboundary, the equality of the $12$th power of the $2$-cocycle constructed by the first central extension and the product of integer powers of the $2$-cocycles constructed above by means of \linebreak $\cup$-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 $π:M \to B$ 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 $6 \, π_* ( c_1(L) \cup c_1(L))$, where $[{\mathcal Det} (L)]$ is the class of the determinant gerbe on $B$ constructed by $L$ and the determinant central extension.

Analytic diffeomorphisms of the circle and topological Riemann-Roch theorem for circle fibrations

TL;DR

This work constructs and compares determinant and related central extensions for the circle-diffeomorphism group, embedding them in a robust infinite-dimensional (Silva-space) framework. It introduces explicit 2-cocycles via block-operator determinants, bimultiplicative pairings, and cup-products, then connects these to Deligne cohomology and Gysin push-forwards. The central result is a topological Riemann-Roch theorem for circle fibrations, expressing times the determinant-gerbe class in terms of the push-forward of cup products of first Chern classes, thereby linking analytic, algebraic, and topological data. The methods unify formal and analytic analogues, using conformal welding and precise operator-theoretic constructions to derive a cohomological equality in with clear geometric implications for circle bundles and line bundles over base manifolds.

Abstract

We consider the group which is the semidirect product of the group of analytic functions with values in on the circle and the group of analytic diffeomorphisms of the circle that preserve the orientation. Then we construct the central extensions of the group by the group . The first central extension, so-called the determinant central extension, is constructed by means of determinants of linear operators acting in infinite-dimensional locally convex topological -vector spaces. Other central extensions are constructed by -products of group -cocycles with the application to them the map related with algebraic -theory. We prove in the second cohomology group, i.e. modulo of a group -coboundary, the equality of the th power of the -cocycle constructed by the first central extension and the product of integer powers of the -cocycles constructed above by means of \linebreak -products (in multiplicative notation). As an application of this result we obtain a new topological Riemann-Roch theorem for a complex line bundle on a smooth manifold , where is a fibration in oriented circles. More precisely, we prove that in the group the element is equal to the element , where is the class of the determinant gerbe on constructed by and the determinant central extension.

Paper Structure

This paper contains 43 sections, 26 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. Infinite-dimensional Lie groups
  3. First central extension
  4. Formal analog and linear operators with the trace
  5. Other central extensions
  6. Comparison of central extensions and the local Deligne-Riemann-Roch isomorphism
  7. The topological Riemann-Roch theorem
  8. Organization of the paper
  9. Analytic functions on a closed curve in $\mathbb{C}$
  10. A closed curve $\Gamma$ in ${\mathbb C}$
  11. When $\Gamma$ is the unit circle
  12. Grothendieck--Köthe--Sebastião e Silva duality
  13. Functions from $S^1$ to ${\mathbb C}^*$ and diffeomorphisms of $S^1$
  14. Lie groups of functions from $S^1$ to ${\mathbb C}^*$
  15. Lie group of analytic diffeomorphisms of $S^1$
  16. ...and 28 more sections

Key Result

Proposition 2.1

There is a canonical decomposition

Theorems & Definitions (76)

  • Proposition 2.1
  • proof
  • Remark 2.1
  • Remark 2.2
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • Proposition 4.2
  • proof
  • Definition 4.1
  • ...and 66 more