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

Dimofte, Tudor

Adv. Theor. Math. Phys. 17, No. 3, 479-599 (2013).

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.

Reviewer: Vladimir Balan (Bucureşti)

MSC:

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

Keywords:

quantum Riemann surface; Chern-Simons theory; three-manifolds; ideal tetrahedra; topological QFT; symplectic gluing; topological strings; triangulation; quantization; wavefunction; Weil representation

Software:

SnapPy; HolonomicFunctions