The volume of a differentiable stack
The volume of a differentiable stack

Alan Weinstein, University of California, Berkeley
Fine Hall 314
We extend to the setting of Lie groupoids the notion of the cardinality of a finite groupoid (a rational number, equal to the Euler characteristic of the correspondingdiscrete orbifold). Since this quantity is an invariant under equivalence of groupoids, we call it the volume of the associatedstack 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 naturalline bundle over the stack. Sections of a square root of this line bundle constitute an "intrinsic Hilbert space'' of the stack. The talk will not require prior knowledge of groupoids or stacks.