Reviews in Mathematical Physics

Nonlinear sigma models are quantum field theories describing, in the large deviation sense, random fluctuations of harmonic maps between a Riemann surface and a Riemannian manifold. Via their formal renormalization group analysis, they provide a framework for possible generalizations of the Hamilton–Perelman Ricci flow. By exploiting the heat kernel embedding introduced by Gigli and Mantegazza, we show that the Wasserstein geometry of the space of probability measures over Riemannian metric measure spaces provides a natural setting for discussing the relation between nonlinear sigma models and Ricci flow theory. In particular, we analyze the embedding of Ricci flow into a heat kernel renormalization group flow for dilatonic nonlinear sigma models, and characterize a non-trivial generalization of the Hamilton–Perelman version of the Ricci flow. We discuss in detail the monotonicity and gradient flow properties of this extended flow.

Let ${ℬ}$ be a conformal net. We give the notion of a proper action of a finite hypergroup $K$ acting by vacuum preserving unital completely positive (so-called stochastic) maps on ${ℬ}$ which generalizes the proper action of a finite group $G$. Taking the fixed point under such an action gives a finite index subnet ${{ℬ}}^K$ of ${ℬ}$, which generalizes the $G$-orbifold net. Conversely, we show that if ${𝒜}\subset {ℬ}$ is a finite inclusion of conformal nets, then ${𝒜}$ is a generalized orbifold ${𝒜}={{ℬ}}^K$ of the conformal net ${ℬ}$ by a unique finite hypergroup $K$. There is a Galois correspondence between intermediate nets ${{ℬ}}^K\subset {𝒜}\subset {ℬ}$ and subhypergroups $L\subset K$ given by ${𝒜}={{ℬ}}^L$. In this case, the fixed point of ${{ℬ}}^K\subset {𝒜}$ is the generalized orbifold by the hypergroup of double cosets $L\setminus K/L$.

If ${𝒜}\subset {ℬ}$ is a finite index inclusion of completely rational nets, we show that the inclusion ${𝒜}(I)\subset {ℬ}(I)$ is conjugate to an intermediate subfactor of a Longo–Rehren inclusion. This implies that if ${ℬ}$ is a holomorphic net, and $K$ acts properly on ${ℬ}$, then there is a unitary fusion category ${ℱ}$ which is a categorification of $K$ and $\mathrm{\text{Rep}}({{ℬ}}^K)$ is braided equivalent to the Drinfel’d center $Z({ℱ})$. More generally, if ${ℬ}$ is a completely rational conformal net and $K$ acts properly on ${ℬ}$, then there is a unitary fusion category ${ℱ}$ extending $\mathrm{\text{Rep}}({ℬ})$, such that $K$ is given by the double cosets of the fusion ring of ${ℱ}$ by the Verlinde fusion ring of ${ℬ}$ and $\mathrm{\text{Rep}}({{ℬ}}^K)$ is braided equivalent to the Müger centralizer of $\mathrm{\text{Rep}}({ℬ})$ in the Drinfel’d center $Z({ℱ})$.

We study differential cohomology on categories of globally hyperbolic Lorentzian manifolds. The Lorentzian metric allows us to define a natural transformation whose kernel generalizes Maxwell's equations and fits into a restriction of the fundamental exact sequences of differential cohomology. We consider smooth Pontryagin duals of differential cohomology groups, which are subgroups of the character groups. We prove that these groups fit into smooth duals of the fundamental exact sequences of differential cohomology and equip them with a natural presymplectic structure derived from a generalized Maxwell Lagrangian. The resulting presymplectic Abelian groups are quantized using the $\mathrm{\text{CCR}}$-functor, which yields a covariant functor from our categories of globally hyperbolic Lorentzian manifolds to the category of $C^{\ast }$-algebras. We prove that this functor satisfies the causality and time-slice axioms of locally covariant quantum field theory, but that it violates the locality axiom. We show that this violation is precisely due to the fact that our functor has topological subfunctors describing the Pontryagin duals of certain singular cohomology groups. As a byproduct, we develop a Fréchet–Lie group structure on differential cohomology groups.