An index for ray operators in 5d E n SCFTs

Abstract

We construct an index for BPS operators supported on a ray in five dimensional superconformal field theories with exceptional global symmetries. We compute the E n representations (for n = 2, . . . , 7) of operators of low spin, thus verifying that while the expression for the index is only SO(2n − 2)×U(1) invariant, the index itself exhibits the full E n symmetry (at least up to the order we expanded). The ray operators we studied in 5d can be viewed as generalizations of operators constructed in a Yang-Mills theory with fundamental matter by attaching an open Wilson line to a quark. For n ≤ 7, in contrast to local operators, they carry nontrivial charge under the Z 9−n ⊂ E n center of the global symmetry. The representations that appear in the ray operator index are therefore different, for n ≤ 7, from those appearing in the previously computed superconformal index. For 3 ≤ n ≤ 7, we find that the leading term in the index is a character of a minuscule representation of E n . We also discuss the case n = 8, which presents a unique technical challenge, and remains an open problem. arXiv:1608.06284v2 [hep-th]

Quantum geometry and θ-angle in five-dimensional super Yang-Mills

“…It was argued by Seiberg [1] that such a theory has a UV fixed point where the symmetry group SO(2N f ) × U(1) gets enhanced to E N f +1 . With the recent success of localization methods, this symmetry enhancement was checked explicitly through a computation of the Sp(1) superconformal index [2], and an index for 1/2-BPS ray operators [3].…”

“…Treating the SU(2) gauge group as Sp(1), an index for ray operators was defined based on a type I string theory construction in [3], building on the computation of the superconformal index in the absence of defect carried out in [2].…”

“…When θ = π, the physics of supersymmetric line defects in 5d SU(2) π has received little attention in comparison. 3 The partition function and superconformal index of the pure 5d N = 1 SU(2) π gauge theory, in the absence of defects, were computed in [2,32], and from the topological string in [33].…”

“…For our purposes, it will be enough to consider a single D3 brane 4 wrapping the X 0,7,8,9 directions and sitting at X 1,2,3,4 = 0, see table 1. The web is drawn in the X 5,6 plane, so the 3 For a recent notable exception, see [30] and the related investigation of Wilson loops in SU(N ) gauge…”

“…This is not the approach we follow in this paper; instead, relying on a D-brane quantum mechanics, we argue that we should modify the definition of the index altogether when a Wilson loop is present. See the related discussion in [3,7,22,23,31]. 4 The number of D3 branes determines which representation of SU(2) the Wilson loop transforms in [41].…”