**Modular forms and quantum invariants of 3-manifolds**

**Ruth Lawrence and Don Zagier**

**Abstract: **Most of the paper concentrates on the case of the
Poincar\'e homology sphere, where it is shown that there the
Witten-Reshetikhin-Turaev invariant arrives as the limiting value of a
holomorphic function defined inside the unit disc. This function is found
to be an Eichler integral of a modular form of half-integral weight (in
fact it is the first component of a 2-dimensional modular form), so that
although the function itself does not behave in a modular way, the
descrepancy from modularity is defined by a continuous function SL(2,Z) -->
C[S^1]^2. In this particular case, we report on a close connection with
Ramanujan's mock theta functions observed by Sander Zwegers. The
possibility of generalising such results to other Seifert fibred manifolds
is also discussed.

**Keywords: **knot theory, manifold invariants, perturbative
expansions, modular forms, TQFT

**AMS subject classification: **57M25 11F37 81Q30

**Length: **15 pages

**Reference: *** Asian
Journal of Mathematics*** 3** (1999) 93-107 dedicated to Sir Michael Atiyah on the occasion
of his 70th birthday.
MR1701924 (2000j:11057) (review by * Justin Roberts *.)

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*Last updated May 23rd, 1999.*

ruthel@ma.huji.ac.il