Reviews in Mathematical Physics

Motivated by the computation of loop space quantum mechanics as indicated in [14], here we seek a better understanding of the tubular geometry of loop space ${ℒ}{ℳ}$ corresponding to a Riemannian manifold ${ℳ}$ around the submanifold of vanishing loops. Our approach is to first compute the tubular metric of ${({{ℳ}}^{2N+1})}_C$ around the diagonal submanifold, where ${({{ℳ}}^N)}_C$ is the Cartesian product of $N$ copies of ${ℳ}$ with a cyclic ordering. This gives an infinite sequence of tubular metrics such that the one relevant to ${ℒ}{ℳ}$ can be obtained by taking the limit $N\to \infty$. Such metrics are computed by adopting an indirect method where the general tubular expansion theorem of [21] is crucially used. We discuss how the complete reparametrization isometry of loop space arises in the large-$N$ limit and verify that the corresponding Killing equation is satisfied to all orders in tubular expansion. These tubular metrics can alternatively be interpreted as some natural Riemannian metrics on certain bundles of tangent spaces of ${ℳ}$ which, for ${ℳ}\times {ℳ}$, is the tangent bundle $T{ℳ}$.

In this paper, we study the properties of weakly conformally symmetric pseudo- Riemannian manifolds focusing particularly on the $4$-dimensional Lorentzian case. First, we provide a new proof of an important result found in literature; then several new others are stated. We provide a decomposition for the conformal curvature tensor in $n\ge 5$. Moreover, some important identities involving two particular covectors are stated; for example, it is proven that under certain conditions the Ricci tensor and other tensors are Weyl compatible. Topological properties involving the vanishing of the first Pontryagin form are then stated. Further, we study weakly conformally symmetric $4$-dimensional Lorentzian manifolds (space-times): it is proven that one of the previously defined co-vectors is null and unique up to a scaling. Moreover, it is shown that under certain conditions, the same vector is an eigenvector of the Ricci tensor and its integral curves are geodesics. Finally, it is stated that such space-time is of Petrov type N with respect to the same vector.

We study quantum isometry groups, denoted by $\mathbb{Q}(\mathrm{\Gamma },S)$, of spectral triples on $C_r^{\ast }(\mathrm{\Gamma })$ for a finitely generated discrete group $\mathrm{\Gamma }$ coming from the word-length metric with respect to a symmetric generating set $S$. We first prove a few general results about $\mathbb{Q}(\mathrm{\Gamma },S)$ including:

• For a group $\mathrm{\Gamma }$ with polynomial growth property, the dual of $\mathbb{Q}(\mathrm{\Gamma },S)$ has polynomial growth property provided the action of $\mathbb{Q}(\mathrm{\Gamma },S)$ on $C_r^{\ast }(\mathrm{\Gamma })$ has full spectrum.

•$\mathbb{Q}(\mathrm{\Gamma },S)\cong QISO(\widehat{\mathrm{\Gamma }},d)$ for any discrete abelian group $\mathrm{\Gamma }$, where $d$ is a suitable metric on the dual compact abelian group $\widehat{\mathrm{\Gamma }}$.

We then carry out explicit computations of $\mathbb{Q}(\mathrm{\Gamma },S)$ for several classes of examples including free and direct product of cyclic groups, Baumslag–Solitar group, Coxeter groups etc. In particular, we have computed quantum isometry groups of all finitely generated abelian groups which do not have factors of the form ${\mathbb{Z}}_2^k$ or ${\mathbb{Z}}_4^l$ for some $k,l$ in the direct product decomposition into cyclic subgroups.

We discuss the gauge theory approach to consideration of the Nambu–Goldstone bosons as gauge and vector fields represented by the Cartan forms of spontaneously broken symmetries. The approach is generalized to describe the fundamental branes in terms of $(p+1)$-dimensional worldvolume gauge and massless tensor fields consisting of the Nambu–Goldstone bosons associated with the spontaneously broken Poincaré symmetry of the $D$-dimensional Minkowski space.