Quantum Riemann surfaces in Chern-Simons theory. (English) Zbl 1304.81143

The paper deals, in part, with quantizing - in the context of Chern-Simons theory – algebraic Riemannian surfaces, defined as zero-loci of polynomial functions on the semi-classical phase space. The author constructs from first principles the operators \(A^M\) that annihilate the partition functions (or wavefunctions) of three-dimensional Chern-Simons theory with gauge groups \(\mathrm{SU}(2)\), \(\mathrm{SL}(2;\mathbb{R})\), or \(\mathrm{SL}(2;\mathbb{C})\) on knot complements \(M\). The operator \({\hat A}_M\) is a quantization of a knot complement’s classical \(A\)-polynomial \(A_M(\ell;m)\). The construction proceeds by decomposing three-manifolds into ideal tetrahedra, and invoking a new, more global understanding of gluing in \(TQFT\) to put them back together. It is advocated in particular that, properly interpreted, “gluing = symplectic reduction”. It is also obtained a new finite-dimensional state integral model for computing the analytically continued “holomorphic blocks” that compose any physical Chern-Simons partition function.


81T70 Quantization in field theory; cohomological methods
81T13 Yang-Mills and other gauge theories in quantum field theory
58J28 Eta-invariants, Chern-Simons invariants
81T45 Topological field theories in quantum mechanics
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