Localization of quantum topology in the presence of matter and gauge fields

Abstract

In this paper a toy model of quantum topology is reviewed to study effects of matter and gauge fields on the topology fluctuations. In the model a collection of N one-dimensional manifolds is considered where a set of boundary conditions on states of Hilbert space specifies a set of all topologies perceived by quantum particle and probability of having a specific topology is determined by a partition function over all the topologies in the context of noncommutative spectral geometry. In general the topologies will be fuzzy with the exception of a particular case which is localized by imposing a specific boundary condition. Here fermions and bosons are added to the model. It is shown that in the presence of matter, the fuzziness of topology will be dependent on N, however for large N the dependence is removed similar to the case without matter. Also turning on a particular background gauge field can overcome the fuzziness of topology to reach a localized topology with classical interpretation. It can be seen that for large N more opportunities can be provided for choosing the background gauge field to localize the fuzzy topology.