Branes And Supergroups

TL;DR

This work develops a two-sided D3-NS5 brane system where theta-terms induce a three-dimensional Chern-Simons theory for supergroups such as U(m|n) and OSp(m|2n). By applying S-duality, and in some cases a subsequent T-duality, it derives magnetic dual descriptions and strong/weak coupling relations, and, after lifting, connects to Khovanov-type homologies for supergroups. The paper also extends these constructions to orthosymplectic cases with orientifolds, analyzes line and surface operators under dualities, and discusses symmetry breaking and lifts to D4-D6 and M-theory, outlining two analogs of Khovanov homology for supergroups. Collectively, these results provide a rich, duality-driven framework linking higher-dimensional gauge theories, supergroup Chern-Simons theory, and categorified knot invariants. The framework has potential implications for knot theory via supergroup invariants and for understanding gauge/brane dualities in contexts involving boundary defects and orientifolds.

Abstract

Extending previous work that involved D3-branes ending on a fivebrane with $θ_{\mathrm{YM}}\not=0$, we consider a similar two-sided problem. This construction, in case the fivebrane is of NS type, is associated to the three-dimensional Chern-Simons theory of a supergroup U$(m|n)$ or OSp$(m|2n)$ rather than an ordinary Lie group as in the one-sided case. By $S$-duality, we deduce a dual magnetic description of the supergroup Chern-Simons theory; a slightly different duality, in the orthosymplectic case, leads to a strong-weak coupling duality between certain supergroup Chern-Simons theories on $\mathbb{R}^3$; and a further $T$-duality leads to a version of Khovanov homology for supergroups. Some cases of these statements are known in the literature. We analyze how these dualities act on line and surface operators.

Figures (14)

• Figure 1: An NS5-brane (sketched as a vertical dotted line) with $m$ D3-branes ending on it from the left and $n$ from the right -- sketched here for $m=3$, $n=4$. The D3-branes but not the NS5-brane extend in the $x_3$ direction, which is plotted horizontally, and the NS5-brane but not the D3-branes extend in the $x_{4,5,6}$ directions, which are represented symbolically by the vertical direction in this figure.
• Figure 2: A "quiver" associated to a chain of $w$ NS5-branes, with $n_0,\dots,n_w$ D3-branes in the regions bounded by the NS5-branes; $w=4$ in the example sketched here.
• Figure 3: Dynkin diagram for the $\mathfrak{su}(m|n)$ superalgebra. The subscripts are expressions for the roots in terms of the orthogonal basis $\delta_\bullet$, $\epsilon_\bullet$. The superscripts represent the Dynkin labels of a weight. The middle root denoted by a cross is fermionic.
• Figure 4: Dynkin diagram for the $\mathfrak{osp}(2m+1|2n)$ superalgebra, $m\ge1$. The subscripts are expressions for the roots in terms of the orthogonal basis $\delta_\bullet$, $\epsilon_\bullet$. The superscripts represent the Dynkin labels of a weight. The arrows point in the direction of a shorter root. The middle root denoted by a cross is fermionic. Roots of the $\mathfrak{sp}(2n)$ and $\mathfrak{so}(2m+1)$ subalgebras are on the left and on the right of the fermionic root. The site shown in grey and labeled $a_n$ is the long simple root of the $\mathfrak{sp}(2n)$ subalgebra, which does not belong to the set of simple roots of the superalgebra.
• Figure 5: Examples of dominant weights for $\mathfrak{u}(3|4)$. a. A typical weight. b. A weight of atypicality two, which is part of a block of atypical weights. The block is labeled by $\widetilde{x}_1$, $\widetilde{x}_2$, and $\widetilde{y}_1$, which correspond to a dominant weight of $\mathfrak{u}(1|2)$. The weights that make up this block are parametrized by $z_1$ and $z_2$, which can be thought of as labels of a maximally atypical weight of $\mathfrak{u}(2|2)$.