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Arrangements in Boston: A Conference in Hyperplane Arrangements
June 12-15, 1999
Northeastern University
Boston, MA, USA |
|
Organizers
Dan Cohen, David Massey, Alex Suciu
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Sums and integrals over polytopes and quantum invariants
by
Ruth Lawrence
University of Michigan (Ann Arbor) and Hebrew University (Jerusalem)
We discuss recent results on the structure of quantum 3-manifold
invariants in the context of properties of sums over integer
points contained in a polytope.
The Witten-Reshetikhin-Turaev quantum invariants ZK(M) of
3-manifolds, M, are complex number invariants dependent on the
choice of a Lie algebra, and of a root of unity, q, of order K.
For rational homology spheres, it is known that the collection of
invariants of a fixed manifold as K varies has additional structure:
-
*
-
It is a family of algebraic integers. (Murakami)
-
*
- It is connected to the Ohtsuki power series invariant via
K-adic convergence for prime K. (Rozansky)
-
*
- For some simple manifolds, it has a simple holomorphic
extension. (Jeffrey; L.; L.-Rozansky; L.-Zagier)
We will see that these properties are related to the general structure
of a sum of the form
|
|
å
x in KR \cap Zn
|
f(q, x) |
|
where R is a rational polytope and f is periodic in x of
period K (for qK=1).
In particular we discuss when such a sum can be represented by an
integral of f over a region (or finite collection of regions) whose
dependence on K is limited to its congruence class mod P, some
fixed P.
http://www.math.lsa.umich.edu/~ruthjl
Date received: June 3, 1999
Copyright © 1999 by the author(s).
The author(s) of this document and the organizers of the conference
have granted their consent to include this abstract in
Atlas Mathematical Conference Abstracts.
Document # cadi-24.