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.