## Prequantization and Lie brackets.(English)Zbl 1095.53060

The present paper begins by reviewing the relationship between the classical (Weil-Kostant) integrality condition for the cohomology class of a closed 2-form $$\omega$$ (not necessarily symplectic, i.e., non-degenerate) stemming from prequantization theory and the integrability of a certain Lie algebroid associated to $$\omega$$, and presents a path space construction of the prequantizing bundle. Subsequently, everything is subsumed by a theory concerning Lie groupoids endowed with multiplicative 2-forms. The problem of their prequantization is solved (Theorem 5.1). Especially intriguing and worth further pursuit is the relationship with gerbes [cf. J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Birkhäuser, Boston (1993; Zbl 0823.55002)], pointed out by the author. The paper is quite readable, fairly self-contained and carefully related to the existing literature.

### MSC:

 53D50 Geometric quantization 58H05 Pseudogroups and differentiable groupoids 22A22 Topological groupoids (including differentiable and Lie groupoids)

### Keywords:

prequantization; integrability for Lie algebroids; gerbes

Zbl 0823.55002
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