Advances in Theoretical and Mathematical Physics

Quantum Riemann surfaces in Chern-Simons theory

Tudor Dimofte

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We construct from first principles the operators $\hat A_M$ that annihilate the partition functions (or wavefunctions) of three-dimensional Chern-Simons theory with gauge groups $SU(2)$, $SL(2,\mathbb{R})$, or $SL(2,\mathbb{C})$ 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 topological quantum field theory to put them back together. We advocate in particular that, properly interpreted, "gluing $=$ symplectic reduction." We also arrive at a new finite-dimensional state integral model for computing the analytically continued "holomorphic blocks" that compose any physical Chern-Simons partition function.

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Adv. Theor. Math. Phys., Volume 17, Number 3 (2013), 479-599.

First available in Project Euclid: 20 August 2014

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Dimofte, Tudor. Quantum Riemann surfaces in Chern-Simons theory. Adv. Theor. Math. Phys. 17 (2013), no. 3, 479--599.

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