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

2017/03/02 by Baird, Thomas John
#14P25 #32L05 #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.1703.00778

Abstract

Let (Σ,τ) denote a Riemann surface of genus g ≥ 2 equipped with an anti-holomorphic involution τ. In this paper we study the topology of the moduli space M(r,ξ)^τ of stable Real vector bundles over (Σ,τ) of rank r and fixed determinant ξ of degree coprime to r. We prove that M(r,ξ)^τ is an orientable and monotone Lagrangian submanifold of the complex moduli space M(r,ξ) so it determines an object in the appropriate Fukaya category. We derive recursive formulas for the mod 2 Betti numbers of M(r,ξ)^τ and compute mod p Betti numbers for odd p through a range of degrees. We deduce that if r is even and g >>0, then M(r,ξ)^τ and M(r,ξ')^τ have non-isomorphic cohomology groups unless ξ and ξ' have equivalent Stieffel-Whitney classes modulo automorphisms of (Σ,τ). If r is even, and g>>0 is even, we prove that the Betti numbers of M(r,ξ)^τ distinguish topological types of (Σ, τ; ξ). If r=2 and g is odd, we compute all mod p Betti numbers of M(2,ξ)^τ. MR 32L05, 14P25.

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