Exact relations between M2-brane theories with and without Orientifolds

We study partition functions of low-energy effective theories of M2-branes, whose type IIB brane constructions include orientifolds. We mainly focus on circular quiver superconformal Chern-Simons theory on $S^3$, whose gauge group is $O(2N+1)\times USp(2N)\times \cdots \times O(2N+1)\times USp(2N)$. This theory is the natural generalization of the $\mathcal{N}=5$ ABJM theory with the gauge group $O(2N+1{)}_{2k}\times USp(2N{)}_{-k}$. We find that the partition function of this type of theory has a simple relation to the one of the M2-brane theory without the orientifolds, whose gauge group is $U(N)\times \cdots \times U(N)$. By using this relation, we determine an exact form of the grand partition function of the $O(2N+1{)}_2\times USp(2N{)}_{-1}$ ABJM theory, where its supersymmetry is expected to be enhanced to $\mathcal{N}=6$. As another interesting application, we discuss that our result gives a natural physical interpretation of a relation between the grand partition functions of the $U(N+1{)}_4\times U(N{)}_{-4}$ ABJ theory and $U(N{)}_2\times U(N{)}_{-2}$ ABJM theory, recently conjectured by Grassi-Hatsuda-Mari~no. We also argue that partition functions of ${\widehat{A}}_3$ quiver theories have representations in terms of an ideal Fermi gas systems associated with $\widehat{D}$-type quiver theories and this leads an interesting relation between certain $U(N)$ and $USp(2N)$ supersymmetric gauge theories.