Circle actions in geometric quantisation

This chapter focuses on circle actions. Specifically, it specializes the Weil algebra and the Weil model to a circle action. In this case, all the formulas simplify. The chapter derives a simpler complex, called the Cartan model, which is isomorphic to the Weil model as differential graded algebras. It considers the theorem that for a circle action, there is a graded-algebra isomorphism. Under the isomorphism F, the Weil differential δ‎ corresponds to a differential called the Cartan differential. An element of the Cartan model is called an equivariant differential form or equivariant form for a circle action on the manifold M.

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Rationale for a Localization Formula

Introductory Lectures on Equivariant Cohomology ◽

10.23943/princeton/9780691191751.003.0029 ◽

2020 ◽

pp. 232-238

Author(s):

Loring W. Tu

Keyword(s):

Fixed Point ◽

Closed Form ◽

Blow Up ◽

Torus Action ◽

Exact Form ◽

Fixed Point Set ◽

Circle Actions ◽

Localization Formula ◽

Point Set ◽

Finite Sum

This chapter offers a rationale for a localization formula. It looks at the equivariant localization formula of Atiyah–Bott and Berline–Vergne. The equivariant localization formula of Atiyah–Bott and Berline–Vergne expresses, for a torus action, the integral of an equivariantly closed form over a compact oriented manifold as a finite sum over the fixed point set. The central idea is to express a closed form as an exact form away from finitely many points. Throughout his career, Raoul Bott exploited this idea to prove many different localization formulas. The chapter then considers circle actions with finitely many fixed points. It also studies the spherical blow-up.

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An outline of geometric quantisation (d'après Kostant)

Lecture Notes in Mathematics - Differential Geometrical Methods in Mathematical Physics ◽

10.1007/bfb0087779 ◽

1977 ◽

pp. 1-10 ◽

Cited By ~ 3

Author(s):

D. J. Simms

Keyword(s):

Geometric Quantisation

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Circle actions on homotopy spheres bounding generalized plumbing manifolds

Mathematische Annalen ◽

10.1007/bf01349230 ◽

1973 ◽

Vol 205 (3) ◽

pp. 201-210 ◽

Cited By ~ 2

Author(s):

Reinhard Schultz

Keyword(s):

Circle Actions

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Fundamental groups of 4-manifolds with circle actions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410007448x ◽

1996 ◽

Vol 119 (4) ◽

pp. 645-655 ◽

Cited By ~ 1

Author(s):

Sławomir Kwasik ◽

Reinhard Schultz

Keyword(s):

Fundamental Groups ◽

Circle Actions

AbstractTopological circle actions on 4-manifolds are studied using modifications of known techniques for smooth actions. This yields topological versions of some previously known restrictions on the fundamental groups of 4-manifolds admitting smooth circle actions.

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