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The LaplaceBeltrami operator of a smooth Riemannian manifold is determined by the Riemannian metric. Conversely, the heat kernel constructed from the eigenvalues and eigenfunctions of the LaplaceBeltrami operator determines the Riemannian metric. This work proves the analogy on Euclidean polyhedral surfaces (triangle meshes), that the discrete heat kernel and the discrete Riemannian metric (unique up to a scaling) are mutually determined by each other. Given a Euclidean polyhedral surface, its Riemannian metric is represented as edge lengths, satisfying triangle inequalities on all faces. The LaplaceBeltrami operator is formulated using the cotangent formula, where the edge weight is defined as the sum of the cotangent of angles against the edge. We prove that the edge lengths can be determined by the edge weights unique up to a scaling using the variational approach.

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Distinct quantum vacua of topologically ordered states can be tunneled into each other, not by local operators, but via extended operators. The possible applications include condensed matter and quantum cosmology. We present a straightforward approach to calculate the partition function on various manifolds and ground state degeneracy (GSD), mainly based on continuum/cochain Topological Quantum Field Theories (TQFT), in any dimension. This information can be related to the counting of extended operators of bosonic/fermionic TQFT. On the lattice scale, anyonic particles/strings live at the ends of line/surface operators. Certain systems in different dimensions are related to each other through dimensional reduction schemes, analogous to decategorification. We consider situations where a TQFT lives on (1) a closed spacetime or (2) a spacetime with boundary, such that both the bulk and boundary are fully-gapped and long-range entangled (LRE). Anyonic excitations can be deconfined on the boundary. We introduce new exotic topological interfaces on which neither particle nor string excitations alone condensed, but only fuzzy-composite objects of extended operators can end (e.g. a string-like fuzzy-composite object formed by a set of particles can end on a special 2+1D boundary of 3+1D bulk). We explore the relations between group extension constructions and partially breaking constructions (e.g. 0-form/higher-form/“composite” breaking) of topological boundaries, after gauging. We comment on the implications of entanglement entropy for some of such LRE systems.

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