Issues on Orientifolds: On the brane construction of gauge theories with SO(2n) global symmetry

TL;DR

This work develops a unified brane-based framework for gauge theories with orthogonal and symplectic groups by focusing on the $ON^{0}$ plane and its role in strong coupling and T-duality for $D_n$ singularities. It derives and tests 3d $\mathcal{N}=4$ mirror pairs, constructs Brane Box Models, and extends the analysis to four and six dimensions, establishing systematic rules for spectra, dualities, and anomaly cancellation. The results connect orientifold dynamics, Dynkin-diagram structures, small instanton theories, and multiple dual descriptions (Type IIB/I IIA, M-theory) across dimensions, while clarifying how global $SO(2n)$ symmetry emerges and is realized in brane setups. The work provides explicit quiver realizations for a broad class of models and identifies open problems in the SO/Sp mirror sector, offering a pathway toward a more complete dictionary between branes, singularities, and gauge theories.

Abstract

We discuss issues related to orientifolds and the brane realization for gauge theories with orthogonal and symplectic groups. We specifically discuss the case of theories with (hidden) global SO(2n) symmetry, from three to six dimensions. We analyze mirror symmetry for three dimensional N=4 gauge theories, Brane Box Models and six-dimensional gauge theories. We also discuss the issue of T-duality for D_n space-time singularities. Stuck D branes on ON^0 planes play an interesting role.

Figures (22)

• Figure 2: Two useful ways of thinking of $ON^{0}$ planes. In the S-dual description (figure B) the different signs for the charges are explained by considering branes ending on the D5 brane from the right or from the left. We may also put extra D5-branes (and NS-branes in the S-dual configuration) for future reference. In figure A, we put two kinds of D5-branes (represented as dots). One of them is bounded to the $ON^{0}$ plane; in the S-dual picture (B) it has the interpretation of a NS-branes which lives in between $O5^{-}$ plane and the D5-brane.
• Figure 3: Two sets of D-branes ($n_{1}$ with positive charge and $n_{2}$ with negative charge) ending on $ON^{0}$ plane. The resulting gauge theory and matter fields are indicated below the figure. In figure A there are two hypermultiplets in the bi-fundamental representation of the two gauge groups. These hypermultiplets parametrize fluctuations transverse to the orbifold plane. In figure B, the two hypermultiplet is projected out by the presence of the NS brane.
• Figure 4: The gauge theory can be read from the quiver diagram on the right. Nodes represent the gauge group factors, links represent bi-fundamental matter fields and external lines represent fields in the fundamental representation of the corresponding gauge group.
• Figure 5: In the case 'without vector structure' (figure A), the states living on $ON^{0}$ and responsible for absorbing the charge of Dp-branes are projected out by $\Omega$. Therefore there is only one type of Dp-brane, living at the intersection of $Oq^{-}$ and $Op^{-}$ planes. This is the theory discussed in gp. In the case 'with vector structure' (figure B), the surviving states living on $ON^{0}$ allow and require the existence of two kinds of Dp-branes. Each set of branes supports a $USp(k)$ group due to the existence of an $Op^{+}$ plane.
• Figure 6: $Z_{2}$ orbifold/orientifold models, without (A) or with vector structure (B). Details are exhaustively discussed in hzsix. Lines are Dq-branes and points are NS-branes.