Six-dimensional Origin of $\mathcal{N}=4$ SYM with Duality Defects

TL;DR

The paper shows how the 6d $(2,0)$ M5-brane theory, when compactified on an elliptically fibered Kahler three-fold with a topological twist, yields 4d ${\\cal N}=4$ SYM with a spatially varying complex coupling $\\tau$ and a rich network of duality defects. Singular fibers induce 3d duality walls and 2d surface defects, across which $SL(2,\\mathbb{Z})$ dualities act; the 2d defects support chiral modes arising from the 6d tensor multiplet, with flavor symmetries determined by fiber degenerations. The authors provide a non-abelian generalization of the 4d bulk theory, discuss the fate of walls and defects in the non-abelian context, and connect 4d defect dynamics to M5/M/F-theory descriptions, including M5- and D3-brane interpretations. They also develop a detailed picture of point defects formed at defect intersections, where flavor symmetries enhance in ways dictated by the Kodaira fiber structure, encapsulated in a Matroshka-like hierarchy of defects. This framework offers a geometric and higher-dimensional origin for duality-twisted 4d theories with position-dependent couplings and defect sectors.

Abstract

We study the topologically twisted compactification of the 6d $(2,0)$ M5-brane theory on an elliptically fibered Kähler three-fold preserving two supercharges. We show that upon reducing on the elliptic fiber, the 4d theory is $\mathcal{N}=4$ Super-Yang Mills, with varying complexified coupling $τ$, in the presence of defects. For abelian gauge group this agrees with the so-called duality twisted theory, and we determine a non-abelian generalization to $U(N)$. When the elliptic fibration is singular, the 4d theory contains 3d walls (along the branch-cuts of $τ$) and 2d surface defects, around which the 4d theory undergoes $SL(2,\mathbb{Z})$ duality transformations. Such duality defects carry chiral fields, which from the 6d point of view arise as modes of the two-form $B$ in the tensor multiplet. Each duality defect has a flavor symmetry associated to it, which is encoded in the structure of the singular elliptic fiber above the defect. Generically 2d surface defects will intersect in points in 4d, where there is an enhanced flavor symmetry. The 6d point of view provides a complete characterization of this 4d-3d-2d-0d `Matroshka'-defect configuration.

Figures (4)

• Figure 1: M-theory setup: $Y_3$ is the world-volume of the M5, and is an elliptic fibration over a base $B$. This elliptic three-fold is embedded into an elliptically fibered Calabi-Yau four-fold $X_4$, with base $M$, which contains $B_2$. The elliptic fibration of $X_4$ restricts to that of $Y_3$. We assume that the fibration has a section, i.e. there is a map from the base to a marked point in the fiber, which is origin of the fiber elliptic curve.
• Figure 2: Setup of 4d theory on $B_2$ with local coordinates $z_1$ and $z_2$. The curve $\mathcal{C}$ is given by $z_1=0$, and the wall $W^{\gamma}$ by $\theta_1 =0$, where $\theta_1$ is the angular coordinate of $z_1$. As one crosses the branch-cut $W^\gamma$, the coupling undergoes an $SL(2,\mathbb{Z})$ monodromy $\gamma$.
• Figure 3: Schematics of the setup: The total space is the elliptically fibered Calabi-Yau four-fold $X_4$, with base three-fold $M$. The restriction of the fibration to the subspace $B \subset M$ results in the elliptic three-fold $Y_3$, which will be wrapped by the M5-brane. The fiber becomes singular above the locus $\Delta\subset M$, which intersects $B$ in (a collection of) curve(s) $\mathcal{C}$.
• Figure 4: Example box graph for the codimension two fiber of the collision $I_5$ with $I_3$. The simple roots are $\alpha$ and $\tilde{\alpha}$, and the extremal curves, i.e. generators of the relative Mori cone, are written out explicitly, with $C_{\tilde{j} i}^\pm = \pm(L_{i} + \tilde{L}_j)$.