High Energy Physics

For any vertex operator algebra V, Y. Zhu constructed an associative algebra Zhu(V) that captures its representation theory (more generally, given a finite order automorphism g of V, there exists an algebra Zhu_g(V) that captures g-twisted representation theory of V). 

To a 4d N=2 superconformal theory T, one assigns a vertex algebra V[T] by the construction of Beem et al. We explain one role of Zhu algebra in this context. Namely, we show that a certain quotient of the Zhu algebra describes what happens to the Schur sector of the theory T under the dimensional reduction on S^1. This connects the VOA construction in 4d N=2 SCFT to the topological quantum mechanics construction in 3d N=4 SCFT, with the latter being given by the aforementioned quotient of the Zhu algebra. In the process, we will discuss how to reformulate the VOA construction on an S^3 x S^1 geometry.

We study the defect operator product expansion (OPE) of displacement operators in free and interacting conformal field theories using replica methods. We show that as n approaches 1 a contact term can emerge when the OPE contains defect operators of twist d−2. For interacting theories and general states we give evidence that the only possibility is from the defect operator that becomes the stress tensor in the n→1 limit. This implies that the quantum null energy condition (QNEC) is always saturated for CFTs with a twist gap. As a check, we show independently that in a large class of near vacuum states, the second variation of the entanglement entropy is given by a simple correlation function of averaged null energy operators as studied by Hofman and Maldacena. This suggests that sub-leading terms in the the defect OPE are controlled by a defect version of the spin-3 non-local light ray operator and we speculate about the possible origin of such a defect operator. For free theories this contribution condenses to a contact term that leads to violations of QNEC saturation.