Moduli spaces of vector bundles with fixed determinant over a real curve

Abstract

Let $$(\Sigma ,\tau )$$ denote a Riemann surface of genus $$g \ge 2$$ equipped with an anti-holomorphic involution $$\tau $$. In this paper we study the topology of the moduli space $$M(r,\xi )^\tau $$ of stable Real vector bundles over $$(\Sigma ,\tau )$$ of rank r and fixed determinant $$\xi $$ of degree coprime to r. We prove that $$M(r,\xi )^{\tau }$$ is an orientable and monotone Lagrangian submanifold of the complex moduli space $$M(r,\xi )$$ so it determines an object in the appropriate Fukaya category. We derive recursive formulas for the $${\mathbb {Z}}_2$$-Betti numbers of $$M(r,\xi )^\tau $$ and compute $${\mathbb {Z}}_p$$-Betti numbers for odd p through a range of degrees. We deduce that if r is even and $$ g>>0$$, then $$M(r,\xi )^{\tau }$$ and $$M(r,\xi ')^{\tau }$$ have non-isomorphic cohomology groups unless $$\xi $$ and $$\xi '$$ have equivalent Stieffel–Whitney classes modulo automorphisms of $$(\Sigma ,\tau )$$. If r is even, and $$g>>0$$ is even, we prove that the Betti numbers of $$M(r,\xi )^{\tau }$$ distinguish topological types of $$(\Sigma , \tau ; \xi )$$. If $$r=2$$ and g is odd, we compute all $${\mathbb {Z}}_p$$-Betti numbers of $$M(2,\xi )^\tau $$.