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Moscow Mathematical Journal

Volume 16, Issue 1, January–March 2016  pp. 95–124.

Higher Spin Klein Surfaces

Authors:  Sergey Natanzon (1) and Anna Pratoussevitch (2)
Author institution: (1) National Research University Higher School of Economics, Vavilova Street 7, 117312 Moscow, Russia and Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
(2) Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL


A Klein surface is a generalisation of a Riemann surface to the case of non-orientable surfaces or surfaces with boundary. The category of Klein surfaces is isomorphic to the category of real algebraic curves. An m-spin structure on a Klein surface is a complex line bundle whose m-th tensor power is the cotangent bundle. We describe all mspin structures on Klein surfaces of genus greater than one and determine the conditions for their existence. In particular we compute the number of m-spin structures on a Klein surface in terms of its natural topological invariants.

2010 Math. Subj. Class. Primary: 30F50, 14H60, 30F35; Secondary: 30F60.

Keywords: Higher spin bundles, higher Theta characteristics, real forms, Riemann surfaces, Klein surfaces, Arf functions, lifts of Fuchsian groups.

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