Equivariant Prequantization

  • R. Lashof
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 20)

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.

Keywords

Spectral Sequence Symplectic Manifold Connection Form Maximal Compact Subgroup Period Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 1991

Authors and Affiliations

  • R. Lashof
    • 1
  1. 1.University of California at BerkeleyBerkeleyUSA

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