Quantum Riemann Surfaces in Chern-Simons Theory

By Tudor Dimofte. 2011

149 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/8e2b58617a6bdbaa76b453f5ae66bdb6dfcfed5f

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Citations in the corpus

The Coulomb Branch of 3d N = 4 Theories (2015, background): “35) we still have (9)Such a quantum symplectic reduction also appeared in [43], in a rather different context.”
Gluing equations for PGL(n, ℂ)–representations of 3–manifolds (2012, methodology): “We will list three types of invariants here, and refer to the literature for more details: (a) The Quantum Riemann surfaces of [4]”