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Non-abelian localization for Chern-Simons theory. (English) Zbl 1097.58012

The authors present a topological interpretation for the Chern-Simons partition functions, in the case of the \(4\)-dimensional Seifert manifolds (these manifolds are the total spaces of the nontrivial circle bundles over Riemann surfaces). They reinterpret the Chern-Simons partition function as a topological quantity determined entirely by a suitable equivariant cohomology ring on the moduli space of flat connections on the Seifert manifold \(M\). Thus, the Chern-Simons theory on \(M\) can be interpreted as a two-dimensional topological theory on the base Riemann surface \(\Sigma\). This interpretation of the Chern-Simons theory is obtained by applying non-abelian localization to the Chern-Simons path integral \(Z(k)=\int{\mathcal{D}}A\;\exp[i {k\over{4\pi}} \;\int_M \text{Tr}(A\wedge dA+ {2\over{3}} A\wedge A\wedge A)]\). The path integral of Yang-Mills theory on a Riemann surface takes the canonical form \(Z(\varepsilon)=\int_X \exp[\Omega - {1\over 2}\varepsilon (\mu,\mu)]\), where the role of the symplectic manifold \(X\) with the symplectic form \(\Omega\) is played by the affine space of all connections on a fixed principal bundle and \(\mu\) is the moment map. The authors discuss the localization for Yang-Mills theory considering the contribution to the path integral from flat Yang-Mills connections, corresponding to the stable minima of the Yang-Mills action. Then they extend this result to compute precisely the contributions from the higher unstable critical points as well. Then they apply localization to perform path integral computations in Chern-Simons theory on a Seifert manifold. These computations depend on the nature of the local symplectic geometry near each critical point. The authors study the localization at the trivial connection on a Seifert homology sphere and find a formula found by R. Lawrence and L. Rozansky in [Commun. Math. Phys. 205, 287–314 (1999; Zbl 0966.57017)] and generalized by M. Marino, [Commun. Math. Phys. 253, 25–49 (2004)]. Next, they consider localization on a smooth component of the moduli space of flat connections, obtaining a formula obtained by L. Rozansky in [Commun. Math. Phys. 178, 27–60 (1996; Zbl 0864.57017)].

MSC:

58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
53D20 Momentum maps; symplectic reduction
81T13 Yang-Mills and other gauge theories in quantum field theory
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