3-Manifold Invariants
In Chapter 9 we discussed the geometry and topology of moduli spaces of gauge fields on a manifold. In recent years these moduli spaces have been extensively studied for manifolds of dimensions 2, 3, and 4 (collectively referred to as low-dimensional mani
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3-Manifold Invariants
10.1 Introduction In Chapter 9 we discussed the geometry and topology of moduli spaces of gauge fields on a manifold. In recent years these moduli spaces have been extensively studied for manifolds of dimensions 2, 3, and 4 (collectively referred to as low-dimensional manifolds). This study was initiated for the 2-dimensional case in [17]. Even in this classical case, the gauge theory perspective provided fresh insights as well as new results and links with physical theories. We make only a passing reference to this case in the context of Chern–Simons theory. In this chapter, we mainly study various instanton invariants of 3-manifolds. The material of this chapter is based in part on [263]. The basic ideas come from Witten’s work on supersymmetric Morse theory. We discuss this work in Section 10.2. In Section 10.3 we consider gauge fields on a 3-dimensional manifold. The field equations are obtained from the Chern–Simons action functional and correspond to flat connections. Casson invariant is discussed in Section 10.4. In Section 10.5 we discuss the Z8 -graded instanton homology theory due to Floer and its relation to the Casson invariant. Floer’s theory was extended to arbitrary closed oriented 3-manifolds by Fukaya. When the first homology of such a manifold is torsion-free, but not necessarily zero, Fukaya also defines a class of invariants indexed by the integer s, 0 ≤ s < 3, where s is the rank of the first integral homology group of the manifold. These invariants include, in particular, the Floer homology groups in the case s = 0. The construction of these invariants is closely related to that of Donaldson polynomials of 4-manifolds, which we considered in Chapter 9. As with the definition of Donaldson polynomials a careful analysis of the singular locus (the set of reducible connections) is required in defining the Fukaya invariants. Section 10.6 is a brief introduction to an extension of Floer homology to a Z-graded homology theory, due to Fintushel and Stern, for homology 3-spheres. Floer also defined a homology theory for symplectic manifolds using Lagrangian submanifolds and used it in his proof of the K. Marathe, Topics in Physical Mathematics, DOI 10.1007/978-1-84882-939-8 10, c Springer-Verlag London Limited 2010
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10 3-Manifold Invariants
Arnold conjecture. We do not discuss this theory. For general information on various Morse homologies, see, for example, [29]. The WRT invariants, which arise as a byproduct of Witten’s TQFT interpretation of the Jones polynomial are discussed in Section 10.7. Section 10.8 is devoted to a special case of the question of relating gauge theory and string theory where exact results are available. Geometric transition that is used to interpolate between these theories is also considered here. Some of the material of this chapter is taken from [263].
10.2 Witten Complex and Morse Theory Classical Morse theory on a finite-dimensional, compact, differentiable manifold M relates the behavior of critical points of a suitable function on M with topol
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