$S^1/T^2$ Compactifications of 6d $\mathcal{N}=(1,0)$ Theories and Brane Webs

TL;DR

<3-5 sentence high-level summary>We study the $S^1$ and $T^2$ compactifications of a class of 6d $\mathcal{N}=(1,0)$ SCFTs that are Higgsable to higher-rank $E$-string theories. Using Type I'–Type IIB T-duality, the authors show that the $S^1$ compactification is described by a 5d brane web that uplifts a specific class S theory, and the $T^2$ compactification yields a 4d class S theory, with the 5d and 4d descriptions written as $\widehat{\mathsf{T}}_{K}\{Y_1,Y_2,Y_3\}$ and $\mathsf{T}_{K}\{Y_1,Y_2,Y_3\}$, respectively. They verify the proposal by computing 4d conformal and flavor central charges from both the 6d anomaly polynomial and class S formulas, finding perfect agreement. The results generalize the Benini–Benvenuti– Tachikawa construction from rank-$N$ E-string compactifications to a broader family, and establish a coherent bridge between 6d tensor branches, 5d brane webs, and 4d class S theories. The paper also outlines future directions for extending these dualities to other gauge groups and brane configurations.

Abstract

We consider the circle and torus compactification of a certain subclass of 6d $\mathcal{N}=(1,0)$ SCFTs which are Higgsable to the higher rank E-string theories. Using the T-duality between Type I' and Type IIB, we found that the $S^1$ compactification of the theories can be realized by 5-brane webs describing the 5d uplifting of a specified class S theory, generalizing the result by Benini, Benvenuti and Tachikawa. We checked the above result by calculating conformal and flavor central charges of the 4d torus compactified theory both from the tensor branch structure of the 6d theory and from the predicted class S description.

Figures (12)

• Figure 1: The tensor branch quiver of the theory $\mathcal{T}^\text{6d}\{u_i\}$ which we consider in this paper. The $N-1$ positive integers $u_i$, $m_i$ for $i=2,\cdots,N$ specify the gauge and flaovr groups $\mathfrak{g}_i=\mathfrak{su}(u_i)$, $\mathfrak{f}_i=\mathfrak{su}(m_i)$ of the quiver. As usual, the solid lines represent the bifundamental hypers between connected gauge or flavor groups. The gauge group $\mathfrak{su}(u_2)$ also couples to the rank 1 E-string theory represented by the dashed line. For each gauge node, there is also a tensor multiplet, whose vev determines the coupling of the gauge group on the node.
• Figure 2: Upper: straighforward Type I' brane engineering Brunner:1997gkHanany:1997gh. The $\times$ mark represents an NS 5 brane, the horizontal line represents the stack of D6 branes, and the vertical lines represent D8 branes or the stack of O8${}^-$ plane and D8 branes. The symbols in the circles are the numbers of the branes there. Lower: Type I' configuration after the Hanany-Witten transitions. There are two D8 branes near the O8${}^-$ plane, each has $n_7$ and $n_8$ D6 branes ending on it, and $u_N+6$ D8 branes on the right side of the $N$th NS5 brane. The $K=n_8+n_7+6N$ D6 branes end on the stack of $u_N+6$ D8 branes, and the pattern of the ending is specified by the Young diagram $Y_1$\ref{['eq:Y1def']}Gaiotto:2014lca.
• Figure 3: The 5-brane web configuration introduced in Benini:2009gi. It has three legs made up of $K$ 5-branes of type $(1,0)$, $(0,1)$ and $(1,-1)$ respectively. The 5-branes in each leg terminate on 7-branes of the same type. The ending pattern of each leg at the 7-branes determines the Young diagram $Y_i$. Since the internal 5-brane web configuration is determined (up to flop transitions) by the boundary data $K$ and $Y_i \; (i=1,2,3)$, we do not write it explicitly. The 5d SCFT from this web is the 5d uplift $\widehat{\mathsf{T}}_{K}\{Y_1,Y_2,Y_3\}$ of the class S theory $\mathsf{T}_{K}\{Y_1,Y_2,Y_3\}$.
• Figure 4: The Hanany-Witten effect between a 7-brane and a 5-brane.
• Figure 5: T-dual of the Type I' brane configuration realizing $S^1$ compactified higher rank E-string theory. The O8${}^-$ plane wrapping $S^1$ becomes two O7${}^-$ planes and the eight D8s become eight D7 branes, while the NS5 branes in type I' remain to be NS5.