Würzburg Seminar on Quantum Field Theory and Gravity

We will consider the algebraic properties of the von Neumann algebra of low-energy observables in the operator spectrum of 2d conformal field theory with large central charge. Through means of a direct first-principles CFT computation we will derive thermal properties of expectation values of these low-energy operators on ultra-high energy pure states. We will use this as input for the application of Tomita-Takesaki theory to corroborate an earlier claim in the literature, that there exists a class of non-trivial excited states that are indistinguishable from the ultra-high energy state from the perspective of low-energy observables. In the process we find a mathematical intuition for this claim in terms of a discrete transformation that maps the low-energy observables to a non-trivial commutant algebra. From an AdS/CFT perspective, it will be argued that these `invisible' excited states correspond to bulk behind-the-event horizon excitations.