UC Berkeley

This dissertation has two parts. The first part is devoted to the study of a line(ray) operator in 5d SCFTs with exceptional group global symmetry. 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 $n=2,\dots,7$) of operators of low spin, thus verifying that while the expression for the index is only SO$(2n-2)\times$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 $n\leq 7$, in contrast to local operators, they carry nontrivial charge under the $\mathbb{Z}_{9-n}\subset E_n$ center of the global symmetry. The representations that appear in the ray operator index are therefore different, for $n\le 7$, from those appearing in the previously computed superconformal index. For $3\le n\le 7$, 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.

The second part discusses line defects in 5d non-commutative Chern-Simons theory. We studied aspects of a topologically twisted supergravity under Omega background and its interpretation as the bulk side of topological subsector of AdS/CFT correspondence. The field theory side is a protected sub-sector of a specific 3d $\mathcal{N}=4$ gauge theory(from M2-branes), especially its Higgs branch, whose chiral ring deformation-quantizes into an algebra via the $\Omega-$background. The line defect comes from this 3d system. The bulk side is interestingly captured by a field theory again, a 5d Chern-Simons theory, which is topological in 1 dimension and holomorphic in 4 dimensions. The statement of topological holography is an isomorphism between the operator algebras. It is possible to introduce M5-brane to decorate the relation. It acts like a module of the algebra(of M2-brane) in the field theory side and a chiral algebra that interacts with 5d Chern-Simons in the gravity side.