Equivariant Prequantization

Abstract

If (S, ω) is a symplectic manifold, a prequantization of S is a principal circle bundle over S together with a connection form whose curvature is −ω. Such a circle bundle exists iff the period group of ω is contained in ℤ; i.e., the class [ω] ∈ H ²(S, ℝ) comes from an integral class. If S is simply connected it follows from the universal coefficient theorem that the integral class is unique. Also note that for simply connected S, the period group of ω is in ℤ iff the spherical period group is in ℤ; i.e., π ₂(S) ≅ H ₂(S).¹ If S is not simply connected it may have inequivalent prequantizations. Inequivalent prequantizations of S also induce inequivalent prequantizations of S × $\bar S$, $\bar S$ denoting (S, −ω). But one can show such prequantizations become equivalent when pulled back to the fundamental groupoid ($\pi \left( S \right) = \tilde S \times \tilde S/{\pi _1}\left( S \right),\lambda :\tilde S \to S$ the universal cover, with the form induced from S × $\bar S$ by λ × λ). Further if we only assume ω is integral on spherical classes, no prequantization may exist. In his preprint [10], Alan Weinstein gives a method for prequantizing the fundamental groupoid of a symplectic manifold (S, ω) when ω is integral on spherical classes, using connection theory. His result is equivalent to the statement (Corollary 1.3): For any symplectic manifold (S, ω) the period group of the fundamental groupoid π(S) is contained in ℤ iff the spherical period group of S is contained in ℤ. Since this is a statement about cohomology, Weinstein raises the question of giving a purely algebraic topology proof of this result.