Exact relations between M2-brane theories with and without Orientifolds

TL;DR

This paper establishes exact relations between M2-brane theories with and without orientifolds, revealing simplified partition functions and connections to Fermi gas systems, advancing understanding of supersymmetric gauge theories and their dualities.

Contribution

It derives exact relations between partition functions of orientifolded and non-orientifolded M2-brane theories, including explicit forms and physical interpretations.

Findings

Partition function of orientifolded theories relates simply to non-orientifolded ones.

Exact grand partition function for $O(2N+1)_{2} imes USp(2N)_{-1}$ ABJM theory determined.

Relation between grand partition functions of ABJ and ABJM theories explained.

Abstract

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 $\hat{A}_3$ quiver theories have representations in terms of an ideal Fermi gas systems associated with $\hat{D}$-type quiver theories and this leads an interesting relation between certain $U(N)$ and $USp(2N)$ supersymmetric gauge theories.