Abstract
We show that superpositions of classical states in quantum gravity with fixed topology can lead to new classical states with a different topology. We study this phenomenon in a particular limit of the LLM geometries. In this limit, the UV complete minisuperspace of allowed quantum states is exactly given by the Hilbert space of a free chiral boson in two dimensions. We construct this chiral boson purely in terms of combinatorial objects associated with the permutation group. As a byproduct of this analysis, we rederive the Murnaghan-Nakayama rule for characters of the permutation group. We are able to express this rule in terms of operator relations for raising and lowering operators on the Hilbert space of states in a free fermion basis. Our construction provides a preferred notion of bulk locality by studying an appropriate notion of D-brane state generating functions. We describe how multi-droplet LLM geometries with different topologies give new classical limits of the free chiral boson, even though they can be written as superpositions of coherent states with trivial topology. As a consequence, topology cannot be accessed by a single operator measurement in this quantum system. We study other non-linear measurements in the quantum wave-function, based on uncertainty and entanglement between modes of the chiral boson, that can be used as order parameters to measure the topology of such states.
Highlights
1.1 The question: can topology be measured by an operator measurement in quantum gravity?The AdS/CFT correspondence [1] has provided a detailed model of quantum gravity in terms of a dual quantum field theory
We show that superpositions of classical states in quantum gravity with fixed topology can lead to new classical states with a different topology
We describe how multi-droplet LLM geometries with different topologies give new classical limits of the free chiral boson, even though they can be written as superpositions of coherent states with trivial topology
Summary
1.1 The question: can topology be measured by an operator measurement in quantum gravity?. The simplest multi-D-brane generating series can be shown to be exactly given by a coherent state of the chiral boson In this sense, the natural D-brane states are classical solutions of the free field theory and end up being associated with states that have the same topology as the vacuum (that is, the same topology as AdS5 ×S5, which we call the trivial topology– the one that we already know). The notion of new topology and geometry only makes sense for these states (and coherent excitations of their collective dynamics) when we study their physical properties at low and intermediate scales They can be regarded as semiclassical states (superpositions of quantum excitations around a vacuum) for high energy observables. This factorization makes it possible to compute entanglement entropies that have a physical meaning
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