We consider topological defect lines (TDLs) in two-dimensional conformal field theories. Generalizing and encompassing both global symmetries and Verlinde lines, TDLs together with their attached defect operators provide models of fusion categories without braiding. We study the crossing relations of TDLs, discuss their relation to the ’t Hooft anomaly, and use them to constrain renormalization group flows to either conformal critical points or topological quantum field theories (TQFTs). We show that if certain non-invertible TDLs are preserved along a RG flow, then the vacuum cannot be a non-degenerate gapped state. For various massive flows, we determine the infrared TQFTs completely from the consideration of TDLs together with modular invariance.

We investigate the problem of counting $1/16$ Bogomol'nyi-Prasad-Sommerfield (BPS) operators in $\mathcal{N}=4$ super-Yang-Mills theory at weak coupling. We present the complete set of $1/16$ BPS operators in the infinite-$N$ limit, which agrees with the counting of free BPS multigraviton states in the gravity dual ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$. Further, we conjecture that all $1/16$ BPS operators in $\mathcal{N}=4$ super-Yang-Mills theory are of the multigraviton form, and give numerical evidence for this conjecture. We discuss the implication of our conjecture and the seeming failure in reproducing the entropy of large $1/16$ BPS black holes in ${\mathrm{AdS}}_{5}$.

We bootstrap \( \mathcal{N}=\left(1,0\right) \) superconformal field theories in six dimensions, by analyzing the four-point function of flavor current multiplets. Assuming E₈ flavor group, we present universal bounds on the central charge C_T and the flavor central charge C_J. Based on the numerical data, we conjecture that the rank-one E-string theory saturates the universal lower bound on C_J, and numerically determine the spectrum of long multiplets in the rank-one E-string theory. We comment on the possibility of solving the higher-rank E-string theories by bootstrap and thereby probing M-theory on \( {\mathrm{AdS}}_7\times {\mathrm{S}}^4/{\mathrm{\mathbb{Z}}}_2 \).

We revamp the constructive enumeration of 1/16-BPS states in the maximally supersymmetric Yang-Mills in four dimensions, and search for ones that are not of multi-graviton form. A handful of such states are found for gauge group SU(2) at relatively high energies, resolving a decade-old enigma. Along the way, we clarify various subtleties in the literature, and prove a non-renormalization theorem about the exactness of the cohomological enumeration in perturbation theory. We point out a giant-graviton-like feature in our results, and envision that a deep analysis of our data will elucidate the fundamental properties of black hole microstates.

We consider topological defect lines (TDLs) in two-dimensional fermionic conformal field theories (CFTs). Besides inheriting all the properties of TDLs in bosonic CFTs, TDLs in fermionic CFTs could host fermionic defect operators at their endpoints and junctions. Furthermore, there is a new type of TDLs, called q-type TDLs, that have no analog in bosonic CFTs. Their distinguishing feature is an extra one-dimensional Majorana fermion living on the worldline of the TDLs. The properties of TDLs in fermionic CFTs are captured in the mathematical language of the super fusion category. We propose a classification of the rank-2 super fusion categories generalizing the \mathbb{Z}_8 ℤ 8 classification for the anomalies of \mathbb{Z}_2 ℤ 2 symmetries. We explicitly solve the F-moves for all the nontrivial categories, and derive the corresponding spin selection rules that constrain the spectrum of the defect operators. We find the full set of TDLs in the standard fermionic minimal models and a partial set of TDLs in the two exceptional models, which give CFT realizations to the rank-2 super fusion categories. Finally, we discuss a constraint on the renormalization group flow that preserves a q-type TDL.