Abstract
We study the partition function of three-dimensional ${\mathcal N}=4$ superconformal Chern-Simons theories of the circular quiver type, which are natural generalizations of the ABJM theory, the worldvolume theory of M2-branes. In the ABJM case, it was known that the perturbative part of the partition function sums up to the Airy function as $Z(N)=e^{A}C^{-1/3}\mathrm{Ai}[C^{-1/3}(N-B)]$ with coefficients $C$, $B$ and $A$ and that for the non-perturbative part the divergences coming from the coefficients of worldsheet instantons and membrane instantons cancel among themselves. We find that many of the interesting properties in the ABJM theory are extended to the general superconformal Chern-Simons theories. Especially, we find an explicit expression of $B$ for general ${\mathcal N}=4$ theories, a conjectural form of $A$ for a special class of theories, and cancellation in the non-perturbative coefficients for the simplest theory next to the ABJM theory.
Highlights
Theory, the infinite-dimensional path integral in defining these quantities is reduced to a finite-dimensional matrix integration
We study the partition function of three-dimensional N = 4 superconformal Chern-Simons theories of the circular quiver type, which are natural generalizations of the ABJM theory, the worldvolume theory of M2-branes
In the ABJM case, it was known that the perturbative part of the partition function sums up to the Airy function as Z(N ) = eAC−1/3Ai[C−1/3(N − B)] with coefficients C, B and A and that for the non-perturbative part the divergences coming from the coefficients of worldsheet instantons and membrane instantons cancel among themselves
Summary
We shall restrict ourselves to these cases It was argued in [15] that, for a large class of general N = 3 superconformal circular quiver Chern-Simons theories associated to a hermitian Hamiltonian H in the above sense, the number n(E) of states whose eigenvalue of H is smaller than E is universally given as n(E) = CE2 + n(0) + non-pert,. With C and n(0) being constants depending on k and “non-pert” standing for nonperturbative corrections From this form the authors showed that the perturbative part of the grand potential is given by a cubic polynomial. We shall calculate n(0) and C explicitly for the class of N = 4 superconformal circular quiver Chern-Simons theories, from the study of the Fermi surface as in [15]. Using (3.4) we can read off the expression of B (1.7) directly from this result
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