# Moscow Mathematical Journal

Volume 17, Issue 2, April–June 2017 pp. 327–349.

Moduli Spaces of Higher Spin Klein Surfaces

**Authors**:
Sergey Natanzon (1) and Anna Pratoussevitch (2)

**Author institution:**(1) National Research University Higher School of Economics (HSE), Myasnitskaya Ulitsa 20, Moscow 101000, Russia

Institute of Theoretical and Experimental Physics (ITEP), B. Cheremushkinskaya 25, Moscow 117218, Russia

(2) Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL

**Summary: **

We study connected components of the space of higher spin
bundles on hyperbolic Klein surfaces. 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 bundle on a Klein surface is
a complex line bundle whose *m*-th tensor power is the cotangent bundle.
Spaces of higher spin bundles on Klein surfaces are important because of
their applications in singularity theory and real algebraic geometry, in
particular for the study of real forms of Gorenstein quasi-homogeneous
surface singularities. In this paper we describe all connected components
of the space of higher spin bundles on hyperbolic Klein surfaces in terms
of their topological invariants and prove that any connected component
is homeomorphic to the quotient of ℝ^{d} by a discrete group. We also
discuss applications to real forms of Brieskorn–Pham singularities.

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

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

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