Analytic diffeomorphisms of the circle and topological Riemann-Roch theorem for circle fibrations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914271503319040 |
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| author | Osipov, Denis V. |
| author_facet | Osipov, Denis V. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_10517 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analytic diffeomorphisms of the circle and topological Riemann-Roch theorem for circle fibrations Osipov, Denis V. Differential Geometry Algebraic Geometry Functional Analysis Representation Theory 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. |
| title | Analytic diffeomorphisms of the circle and topological Riemann-Roch theorem for circle fibrations |
| topic | Differential Geometry Algebraic Geometry Functional Analysis Representation Theory |
| url | https://arxiv.org/abs/2503.10517 |