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Symmetry protected topological (SPT) states have boundary 't Hooft anomalies that obstruct an effective boundary theory realized in its own dimension with UV completion and with an on-site $G$-symmetry. In this work, yet we show that a certain anomalous non-on-site $G$ symmetry along the boundary becomes on-site when viewed as an extended $H$ symmetry, via a suitable group extension $1\to K\to H\to G\to1$. Namely, a non-perturbative global (gauge/gravitational) anomaly in $G$ becomes anomaly-free in $H$. This guides us to construct exactly soluble lattice path integral and Hamiltonian of symmetric gapped boundaries, \emph{always existent} for \emph{any} SPT state in any spacetime dimension $d \geq 2$ of \emph{any} finite symmetry group, including on-site unitary and anti-unitary time-reversal symmetries. The resulting symmetric gapped boundary can be described either by an $H$-symmetry extended boundary of bulk $d \geq 2$, or more naturally by a topological {emergent} $K$-gauge theory with a global symmetry $G$ on a 3+1D bulk or above. The excitations on such a symmetric topologically ordered boundary can carry fractional quantum numbers of the symmetry $G$, described by representations of $H$. (Apply our approach to a 1+1D boundary of 2+1D bulk, we find that a deconfined gauge boundary indeed has \emph{spontaneous symmetry breaking} with long-range order. The deconfined symmetry-breaking phase crosses over smoothly to a confined phase without a phase transition.) In contrast to known gapped boundaries/interfaces obtained via \emph{symmetry breaking} (either global symmetry breaking or Anderson-Higgs mechanism for gauge theory), our approach is based on \emph{symmetry extension}. More generally, applying our approach to SPT states, topologically ordered gauge theories and symmetry enriched topologically ordered (SET) states, leads to generic boundaries/interfaces constructed with a mixture of \emph{symmetry breaking}, \emph{symmetry extension}, and \emph{dynamical gauging}.

We study the moduli stack of degree 0 semistable G-bundles on an irreducible curve E of arithmetic genus 1, where G is a connected reductive group. Our main result describes a partition of this stack indexed by a certain family of connected reductive subgroups H of G (the E-pseudo-Levi subgroups), where each stratum is computed in terms of bundles on H together with the action of the relative Weyl group. We show that this result is equivalent to a Jordan–Chevalley theorem for such bundles equipped with a framingat a fixed basepoint. In the case where E has a single cusp (respectively, node), this gives a new proof of the Jordan–Chevalley theorem for the Lie algebra g (respectively, group G). We also provide a Tannakian description of these moduli stacks and use it to show that if E is an ordinary elliptic curve, the collection of framed unipotent bundles on E is equivariantly isomorphic to the unipotent cone in G. Finally, we classify the E-pseudo-Levi subgroups using the Borel–de Siebenthal algorithm, and compute some explicit examples.

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Calculation of band structures of three dimensional photonic crystals amounts to solving large-scale Maxwell eigenvalue problems, which are notoriously challenging due to high multiplicity of zero eigenvalues. In this paper, we try to address this problem in such a broad context that band structures of three dimensional isotropic photonic crystals in all $14$ Bravais lattices can be efficiently computed in a unified framework. In this work, we uncover the delicate machinery behind several key results of our framework and on the basis of this new understanding we drastically simplify the derivations, proofs and arguments. Particular effort is made on reformulating the Bloch condition for all $14$ Bravais lattices in the redefined orthogonal coordinate system, and establishing eigen-decomposition of discrete partial derivative operators by identifying the hierarchical structure of the underlying normal (block) companion matrix, and reducing the eigen-decomposition of the double-curl operator to a simple factorization of a $3$-by-$3$ complex skew-symmetric matrix. With the validity of the novel nullspace free method in the broad context, we perform some calculations on one benchmark system to demonstrate the accuracy and efficiency of our algorithm to solve Maxwell eigenvalue problems.