Prefactorization and vertex algebras associated to holomorphic fibrations, the toroidal algebra, and averages of unlabeled networks
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Abstract
This thesis consists of two distinct parts. The first concerns prefactorization and vertex algebras associated to holomorphic fibrations and the second describes a notion of averaging on the space of unlabeled networks. Factorization algebras provide a geometric encoding of the algebra of observables and their symmetries in perturbative quantum field theory. Vertex algebras provide a concrete algebraic realization of the symmetry algebra of two dimensional conformal field theories. A theorem of Costello-Gwilliam connects these two worlds. Given a translation invariant holomorphic (pre)factorization algebra on the complex plane, one can associate a unique vertex algebra. The aim of the first part of this thesis is to construct a prefactorization algebra associated to a holomorphic fibration and describe the corresponding vertex algebra. Specializing to the case in which the fiber is a torus, we recover a vertex algebra naturally associated to an (n+1)-toroidal algebra, a multi-loop generaliztion of Kac-Moody algebras. The second part of this dissertation concerns averages of unlabeled, undirected networks with edge weights. It is becoming increasingly common to see large collections of network data objects, and as a result there is a need to develop basic statistical tools. We introduce a space parameterizing such networks, characterize some relevant topological and geometric properties, and use these properties to establish the asymptotic behavior of a generalized notion of an empirical mean. The lack of vertex labeling necessitates working with a quotient space in which we mod out permutations of labels, resulting in a nontrivial geometry which has implications on the types of probabilistic and statistical results that may be obtained and the techniques needed to obtain them.