## Advances in Theoretical and Mathematical Physics

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

### Quantum Riemann surfaces in Chern-Simons theory

#### Abstract

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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 20 August 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.atmp/1408562471

**Mathematical Reviews number (MathSciNet)**

MR3250765

**Zentralblatt MATH identifier**

1304.81143

#### Citation

Dimofte, Tudor. Quantum Riemann surfaces in Chern-Simons theory. Adv. Theor. Math. Phys. 17 (2013), no. 3, 479--599. https://projecteuclid.org/euclid.atmp/1408562471