A degeneracy bound for homogeneous topological order

We introduce a notion of homogeneous topological order, which is obeyed by most, if not all, known examples of topological order including fracton phases on quantum spins (qudits). The notion is a condition on the ground state subspace, rather than on the Hamiltonian, and demands that given a collection of ball-like regions, any linear transformation on the ground space be realized by an operator that avoids the ball-like regions. We derive a bound on the ground state degeneracy \mathcal D𝒟 for systems with homogeneous topological order on an arbitrary closed Riemannian manifold of dimension dd, which reads [ D c (L/a)^{d-2}.] Here, LL is the diameter of the system, aa is the lattice spacing, and cc is a constant that only depends on the isometry class of the manifold, and \muμ is a constant that only depends on the density of degrees of freedom. If d=2d=2, the constant cc is the (demi)genus of the space manifold. This bound is saturated up to constants by known examples.examples.

The Trace Form Over Cyclic Number Fields

In the mid 80’s Conner and Perlis showed that for cyclic number fields of prime degree p the isometry class of integral trace is completely determined by the discriminant. Here we generalize their result to tame cyclic number fields of arbitrary degree. Furthermore, for such fields, we give an explicit description of a Gram matrix of the integral trace in terms of the discriminant of the field.

APPLICATIONS OF SYSTEMS OF QUADRATIC FORMS TO GENERALISED QUADRATIC FORMS

A system of quadratic forms is associated to every generalised quadratic form over a division algebra with involution of the first kind in characteristic two. It is shown that this system determines the isotropy behaviour and the isometry class of generalised quadratic forms. An application of this construction to the Witt index of generalised quadratic forms is also given.

On the large-scale geometry of diffeomorphism groups of 1-manifolds

We apply the framework of Rosendal to study the large-scale geometry of the topological groups {\operatorname{Diff}_{+}^{k}(M^{1})} , consisting of orientation-preserving {C^{k}} -diffeomorphisms (for {1\leq k\leq\infty} ) of a compact 1-manifold {M^{1}} ( {=I} or {\mathbb{S}^{1}} ). We characterize the relative property (OB) in such groups: {A\subseteq\operatorname{Diff}_{+}^{k}(M^{1})} has property (OB) relative to {\operatorname{Diff}_{+}^{k}(M^{1})} if and only if {\sup_{f\in A}\sup_{x\in M^{1}}\lvert\log f^{\prime}(x)|<\infty} and {\sup_{f\in A}\sup_{x\in M^{1}}|f^{(j)}(x)|<\infty} for every integer j with {2\leq j\leq k} . We deduce that {\operatorname{Diff}_{+}^{k}(M^{1})} has the local property (OB), and consequently a well-defined non-trivial quasi-isometry class, if and only if {k<\infty} . We show that the groups {\operatorname{Diff}_{+}^{1}(I)} and {\operatorname{Diff}_{+}^{1}(\mathbb{S}^{1})} are quasi-isometric to the infinite-dimensional Banach space {C[0,1]} .

Single-class genera of positive integral lattices

We give an enumeration of all positive definite primitive $ \mathbb{Z} $-lattices in dimension $n\geq 3$ whose genus consists of a single isometry class. This is achieved by using bounds obtained from the Smith–Minkowski–Siegel mass formula to computationally construct the square-free determinant lattices with this property, and then repeatedly calculating pre-images under a mapping first introduced by G. L. Watson.We hereby complete the classification of single-class genera in dimensions 4 and 5 and correct some mistakes in Watson’s classifications in other dimensions. A list of all single-class primitive $ \mathbb{Z} $-lattices has been compiled and incorporated into the Catalogue of Lattices.

ANNIHILATING IDEALS OF QUADRATIC FORMS OVER LOCAL AND GLOBAL FIELDS

We study annihilating polynomials and annihilating ideals for elements of Witt rings for groups of exponent 2. With the help of these results and certain calculations involving the Clifford invariant, we are able to give full sets of generators for the annihilating ideal of both the isometry class and the equivalence class of an arbitrary quadratic form over a local field. By applying the Hasse–Minkowski theorem, we can then achieve the same for an arbitrary quadratic form over a global field.

On compactness of isospectral conformal, metrics of 4-manifolds

In this paper, we are interested in the compactness of isospectral conformal metrics in dimension 4.Let us recall the definition of the isospectral metrics. Two Riemannian metrics g, g′ on a compact manifold are said to be isospectral if their associated Laplace operators on functions have identical spectrum. There are now numeruos examples of compact Riemannian manifolds which admit more than two metrics such that they are isospectral but not isometric. That is to say that the eigenvalues of the Laplace operator Δ on the functions do not necessarily determine the isometry class of (M, g). If we further require the metrics stay in the same conformal class, the spectrum of Laplace operator still does not determine the metric uniquely ([BG], [BPY]).