Anomalies and Holomorphy in Super-Chern-Simons Matter Theories

Using techniques from complex analysis and q-hypergeometric function theory, we explore a generalization of the Witten-Reshetikhin-Turaev invariant of the three sphere. Specifically, we study the partition function of N = 2 Super-Chern-Simons matter theories on an ellipsoidal three sphere.;To properly quantize Chern-Simons Matter theories, a sensitive cancellation of anomalies requires the half integrality of the Chern-Simons level k to match the parity mod 2 of the quadratic Casimir of the fermion representation. For abelian N=2 theories, Pasquetti observed that the Coulomb branch partition function of a family of ellipsoidal three-spheres admits a factorization precisely when k is properly quantized.;In this thesis we use known formulas for the Coulomb branch partition function to prove Pasquetti's conjecture for all simple Lie groups, with matter in a direct sum of multiplicity free representations. Along the way, we recover a known relationship between the Chern-Simons level and the eigenvalues of the Dirac operator in terms of the asymptotic behavior of the double sine function. Finally, we compute the analytic continuation of torus knot observables in SU(2) Chern-Simons theory in the presence of matter.