Line bundles and divisors on a super Riemann surface
The de Witt supermanifold is a topological manifold M such that: 1) there exists a (maximal) atlas \(\{(U_ i,\psi_ i)\}\) on M whose charts \(U_ i\) are homeomorphic, by \(\psi_ i\), to open sets of the Cartesian product \[ C^{m,n}=C^{1,0}\times...\times C^{1,0} \times C^{0,1}\times...\times C^{0,\quad 1} \] where \(C^{1,0}\) are taken m times and \(C^{0,1}\) are taken n times where \(C^{1,0}\) \((C^{0,1})\) is the even (odd) sector of a complex Grassmann algebra with L generators \(\Lambda (C_ L).\) 2) the topology on \(C^{m,n}\) is the weakest topology such that the projection to the zero-elements of the Grassmann algebra \(\epsilon\) : \(C^{m,n}\to C\) m is continuous. 3) the transition functions of M are superanalytic. An \((m,n)=(1,1)\) supermanifold is called a super Riemann surface (SRS) if the transition functions satisfy \(\tilde z=F_ 0(z)+\theta G\sqrt{F_ 0'(z)}\), \({\tilde \theta}=G_ 0(z)+\theta \sqrt{F_ 0'+G_ 0G_ 0'}\) where each of the \(F_ 0,G_ 0\) is uniquely defined by an analytic function f \[ F_ 0(z)=f(\epsilon z)+f'(\epsilon z)(z-\epsilon z)+f''(\epsilon z)(z-\epsilon z)^ 2+...\quad. \] Let \((C^{1,0})\) * be the Abelian group of invertible elements of \(C^{1,0}\). Given a supermanifold M, a line bundle over M is a supermanifold L such that: 1) there exists a superanalytic projection \(p: L\to M\) with \(p^{-1}(x)\) isomorphic to \(C^{1,0}\), for any x on M, 2) there exists a locally trivializing covering of M that is a set \(\{(U_{\alpha},\phi_{\alpha})\}\) where \(\phi_{\alpha}: p^{- 1}(U_{\alpha})\to U_{\alpha}\times C^{1,0}\) are superanalytic diffeomorphisms such that the maps \(g_{\alpha \beta}: U_{\alpha}\cap U_{\beta}\to (C^{1,0})\quad *\) defined by restriction of \(\phi_{\alpha}\circ \phi^{\beta}_{-1}\) to the fibres, \(g_{\alpha \beta}(x)=\phi_{\alpha}\circ \phi^{-1}_{\beta}| p^{-1}(x)\), are superanalytic. The \(g_{\alpha \beta}'s\) are called transition functions of the line bundle L. Let \(O_ s\) * be the sheaf of germs of superanalytic invertible functions on M. Now the author proves the following Theorem: There exists an isomorphism between the cohomology group H \(1(M,O\) \(*_ s)\) and the class of isomorphic line bundles on M. After the generalization of the concept of divisor on an SRS the author proves the following Theorem: To any divisor on an SRS corresponds a line bundle on M and vice-versa. - An interesting example of a line bundle over an SRS is considered.
germs of superanalytic invertible functions
scientific article; zbMATH DE number 4058461