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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 Conferences Inc. Document # cadi-24.