The Volume of a Differentiable Stack

  • Alan Weinstein
Open Access


We extend the notion of the cardinality of a discrete groupoid (equal to the Euler characteristic of the corresponding discrete orbifold) to the setting of Lie groupoids. Since this quantity is an invariant under equivalence of groupoids, we call it the volume of the associated stack rather than of the groupoid itself. Since there is no natural measure in the smooth case like the counting measure in the discrete case, we need extra data to define the volume. This data has the form of an invariant section of a natural line bundle over the base of the groupoid. Invariant sections of a square root of this line bundle constitute an “intrinsic Hilbert space” of the stack.

Mathematics Subject Classification (2000)

Primary 58H05 Secondary 53D17 


Lie groupoid Lie algebroid modular class differentiable stack 



I would like to think John Baez, Kai Behrend, Rui Loja Fernandes, Minhyong Kim, Yvette Kosmann-Schwarzbach, Eckhard Meinrenken, Martin Olsson for helpful discussion and comments. I would also like to thank the group Analyse Algébrique at the Institut Mathématique de Jussieu for their hospitality.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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