The Volume of a Differentiable Stack

Abstract

Abstract. 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.

Measures on differentiable stacks

“…Finally, we point out that Proposition 6.3 immediately implies Conjecture 5.2 from [33] (even without the simplifying assumptions from loc.cit).…”

“…We will show that a rather straightforward extension of Haefliger's approach to transverse measures for foliations [17] allows one to talk about measures and geometric measures ( = densities) for differentiable stacks. For geometric measures (densities), when we compute the resulting volumes, we will recover the formulas that are taken as definition by Weinstein [33]; also, using our viewpoint, we prove the conjecture left open in loc.cit.…”

“…For instance, if the symplectic groupoid has 1-connected s-fibers, then the variation of the leafwise symplectic areas gives rise (because of properness) to a lattice in ν (a transverse integral affine structure), see e.g. [33]; then β Haar is just the corresponding density. If also the leaves are 1-connected -condition that ensures that B = M/G is smooth, then this induces an integral affine structure on B and β Haar is the canonical density associated to it.…”

“…In general we prove Morita invariance, a Stokes formula which provides reinterpretations in terms of (Ruelle-Sullivan type) algebroid currents, and a Van Est isomorphism. In the proper case we reduce the theory to classical (Radon) measures on the underlying space, we provide explicit (Weyl-type) formulas that shed light on Weinstein's notion of volumes of differentiable stacks; in particular, in the symplectic case, we prove the conjecture left open in [33]. We also revisit the notion of modular class and of Haar systems (and the existence of cut-off functions).…”

Modular Classes of Lie Groupoid Representations up to Homotopy

“…The modular class of a Lie groupoid was introduced by Evens, Lu, and Weinstein [5,Appendix B], simultaneously generalizing the modular character of a Lie group, which represents the failure of the Haar measure to be bi-invariant, and the modular class of a foliation [14]. In light of Weinstein's notion of volume of a differentiable stack [12], the modular class of a Lie groupoid can be interpreted as the obstruction to the existence of a volume form.…”