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Modular forms and quantum invariants of 3-manifolds.

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Zagier,  Don
Max Planck Institute for Mathematics, Max Planck Society;

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Lawrence, R., & Zagier, D. (1999). Modular forms and quantum invariants of 3-manifolds. The Asian Journal of Mathematics, 3(1), 93-107.


Cite as: http://hdl.handle.net/21.11116/0000-0004-3810-B
Abstract
From the introduction: The Witten-Reshetikhin-Turaev (WRT) invariant of a compact connected oriented 3-manifold M may be formally defined by Zk+\check c≥rm g(M)=\int\cal A/\cal Ge^ik\over 4π \intM \bigl\langle A,dA+[A,A]/3 \bigr\rangle dτ\cal DA, as a Feynman integral over an infinite-dimensional moduli space of G-connections on M, in terms of additional data, namely a Lie algebra ≥rm g (with dual Coxeter number \check c≥rm g) and a level k\in\bbfN. In this paper we only consider WRT invariants with ≥rm g=≥rm s≥rm l2,and it is convenient to put K=k+2. This functional integral is not well-defined in a literal sense, being an integral over an infinite-dimensional space on which there is no naturally defined measure. However, there are two meaningful statements which can be made about the object represented by the path integral, both of which derive from its form.\par 1. TQFT: Since the exponent in the integrand (the Chern-Simons action) is additive under gluing of manifolds, one would expect a certain gluing property to be satisfied by the value of ZK(M) with respect to the gluing together of 3-manifolds along a common boundary. Indeed, the invariant ZK(M) fits into a more general picture of invariants of arbitrary links in 3-manifolds (or more generally of slices of these, tangles in 3-manifolds with boundary) and as such describes a functor from the category of Riemann surfaces with punctures, whose morphisms are cobordisms, to the category of vector spaces. When this property is formalised, one obtains the mathematical structure of a topological quantum field theory.\par 2. PE: The fact that the dependence on k enters only via a scaling of the exponent in the integrand leads one to expect (as there would be for complex path integrals of a similar form) the existence of a stationary phase expansion for K large of the form ZK(M)∼ \sumAZ^AK(M), where the terms ZK^A(M) are labelled by the stationary points of the Chern-Simons action (i.e., by the equivalence classes A of flat connections on M) and each term is the product of an asymptotic series in K^-1, an exponential in K (determined by the Chern-Simons number for the connection), and a power of K (determined by homological data). There is always one contribution ZK^0(M), coming from the trivial connection, which has the form of a formal power series in K^-1.\par For most of the paper we concentrate on one single example, the Poincaré homology sphere Σ (2,3,5). The WRT-invariant of this manifold has been calculated in the literature in several different ways and is a particularly beautiful number-theoretical function. In \S 2 we collect a number of these formulae and give a table of values of W(ξ) (a normalised version of ZK(M)) for roots of unity ξ of small order. In \S 3 we obtain a formula of a different sort for W(ξ) by introducing a certain theta series of weight 3/2 and showing that the limiting values of the associated Eichler integral coincide with the numbers W(ξ). This property is used in \S 4 to determine the behaviour of the function W(e^2π iα) (α\in\bbfQ) under the action of modular transformations. In \S 5 we describe briefly a surprising connection, found by Sander Zwegers, between the WRT-invariant of Σ(2,3,5) and one of Ramanujan's ``mock theta functions of order 5''. This leads to yet another formula for W(ξ) as an infinite sum of polynomials in ξ^-1 which becomes finite (and hence makes sense) exactly when ξ is a root of unity; this is similar to the phenomenon appearing in the ``strange identity'' of [\it D. Zagier, Topology 40, 945-960 (2001; Zbl 0989.57009)]. Finally, in \S 6 we describe the generalisation to other Seifert fibrations. If these have exactly three exceptional fibres, then the picture is quite analogous to the one for the Poincaré homology sphere, while in the case of four or more exceptional fibres the same type of ideas apply but the picture becomes more complicated.