Abstract
We study the topologically twisted compactification of the 6d $(2,0)$ M5-brane theory on an elliptically fibered K\"ahler 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 $\tau$, 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 $\tau$) 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.
Highlights
S-duality is one of the profound characteristics of N = 4 Super-Yang Mills (SYM) theory in 4d with gauge group G
The relation between 6d (2, 0) theory or M5-branes on an elliptic curve and 4d N = 4 SYM for fixed coupling constant τ, parametrized by the complex structure of the elliptic curve, makes the SL(2, Z) duality of the 4d theory manifest in the modular action on τ
In this paper we studied what happens as we allow the elliptic curve to be fibered non-trivially over a 4d base manifold — much like the generalization of IIB to F-theory by allowing the axio-dilaton to vary
Summary
Taking a varying holomorphic coupling τ over a base manifold B2, can result in singularities in the fiber, above 2 real-dimensional surfaces/complex curves C ⊂ B, along which τ degenerates Around each such curve C, the function τ has a monodromy τ → τ given by an SL(2, Z) modular transformation γ of the elliptic curve aτ + b τ = γ.τ =. We begin with some background for studying the 6d (2, 0) super-conformal theory on an elliptic fibration, by reviewing some basic facts about such geometries, the relation of the 6d theory to N = 4 SYM in 4d, as well as the bonus symmetry of the abelian 4d N = 4 theory
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