Advances in Theoretical and Mathematical Physics

Volume 17 (2013)

Number 3

Quantum Riemann surfaces in Chern-Simons theory

Pages: 479 – 599



Tudor Dimofte (Institute for Advanced Study, Princeton, New Jersey, U.S.A.; Trinity College, University of Cambridge, United Kingdom; Max-Planck-Institut für Mathematik, Bonn, Germany)


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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