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The Hilbert Space of large $N$ Chern-Simons matter theories

We demonstrate that the known expressions for the thermal partition function of large $N$ Chern-Simons matter theories admit a simple Hilbert space interpretation as the partition function of an associated ungauged large $N$ matter theory with one additional condition: the Fock space of this associated theory is projected down to the subspace of its \emph{quantum} singlets i.e.~singlets under the Gauss law for Chern-Simons gauge theory. Via the Chern-Simons / WZW correspondence, the space of quantum singlets are equivalent to the space of WZW conformal blocks. One step in our demonstration involves recasting the Verlinde formula for the dimension of the space of conformal blocks in $SU(N)_k$ and $U(N)_{k,k'}$ WZW theories into a simple and physically transparent form, which we also rederive by evaluating the partition function and superconformal index of pure Chern-Simons theory in the presence of Wilson lines. A particular consequence of the projection of the Fock space of Chern-Simons matter theories to quantum (or WZW) singlets is the `Bosonic Exclusion Principle': the number of bosons occupying any single particle state is bounded above by the Chern-Simons level. The quantum singlet condition (unlike its Yang-Mills Gauss Law counterpart) has a nontrivial impact on thermodynamics even in the infinite volume limit. In this limit the projected Fock space partition function reduces to a product of partition functions, one for each single particle state. These single particle state partition functions are $q$-deformations of their free boson and free fermion counterparts and interpolate between these two special cases. We also propose a formula for the large $N$ partition function that is valid for arbitrary finite volume of the spatial $S^2$ and not only at large volume.