Let V be a vertex operator algebra with a category C of (generalized) modules that has vertex tensor category structure, and thus braided tensor category structure, and let A be a vertex operator (super)algebra extension of V. We employ tensor categories to study untwisted (also called local) A-modules in C, using results of Huang-Kirillov-Lepowsky showing that A is a (super)algebra object in C and that generalized A-modules in C correspond exactly to local modules for the corresponding (super)algebra object. Both categories, of local modules for a C-algebra and (under suitable conditions) of generalized A-modules, have natural braided monoidal category structure, given in the first case by Pareigis and Kirillov-Ostrik and in the second case by Huang-Lepowsky-Zhang. Our main result is that the Huang-Kirillov-Lepowsky isomorphism of categories between local (super)algebra modules and extended vertex operator (super)algebra modules is also an isomorphism of braided monoidal (super)categories. Using this result, we show that induction from a suitable subcategory of V-modules to A-modules is a vertex tensor functor. We give two applications. First, we derive Verlinde formulae for regular vertex operator superalgebras and regular (1/2)ℤ-graded vertex operator algebras by realizing them as (super)algebra objects in the vertex tensor categories of their even and ℤ-graded components, respectively. Second, we analyze parafermionic cosets C=Com(V_L,V) where L is a positive definite even lattice and V is regular. If the category of either V-modules or C-modules is understood, then our results classify all inequivalent simple modules for the other algebra and determine their fusion rules and modular character transformations. We illustrate both directions with several examples.

Given a gauged linear sigma model (GLSM) $\mathcal{T}_{X}$ realizing a projective variety $X$ in one of its phases, i.e. its quantum K\"ahler moduli has a geometric point, we propose an \emph{extended} GLSM $\mathcal{T}_{\mathcal{X}}$ realizing the homological projective dual category $\mathcal{C}$ to $D^{b}Coh(X)$ as the category of B-branes of the Higgs branch of one of its phases. In most of the cases, the models $\mathcal{T}_{X}$ and $\mathcal{T}_{\mathcal{X}}$ are anomalous and the analysis of their Coulomb and mixed Coulomb-Higgs branches gives information on the semiorthogonal/Lefschetz decompositions of $\mathcal{C}$ and $D^{b}Coh(X)$. We also study the models $\mathcal{T}_{X_{L}}$ and $\mathcal{T}_{\mathcal{X}_{L}}$ that correspond to homological projective duality of linear sections $X_{L}$ of $X$. This explains why, in many cases, two phases of a GLSM are related by homological projective duality. We study mostly abelian examples: linear and Veronese embeddings of $\mathbb{P}^{n}$ and Fano complete intersections in $\mathbb{P}^{n}$. In such cases, we are able to reproduce known results as well as produce some new conjectures. In addition, we comment on the construction of the HPD to a nonabelian GLSM for the Pl\"ucker embedding of the Grassmannian $G(k,N)$.

We introduce a web of strongly correlated interacting 3+1D topological superconductors/ insulators of 10 particular global symmetry groups of Cartan classes, realizable in electronic condensed matter systems, and their generalizations. The symmetries include SU(N), SU(2), U(1), fermion parity, time reversal and relate to each other through symmetry embeddings. We overview the lattice Hamiltonian formalism. We complete the list of bulk symmetry-protected topological invariants (SPT invariants/partition functions that exhibit boundary 't Hooft anomalies) via cobordism calculations, matching their full classification. We also present explicit 4-manifolds that detect these SPTs. On the other hand, once we dynamically gauge part of their global symmetries, we arrive in various new phases of $SU(N)$ Yang-Mills (YM) theories, realizable as quantum spin liquids with emergent gauge fields. We discuss how coupling YM theories to time reversal-SPTs affects the strongly coupled theories at low energy. For example, we point out a possibility of having two deconfined gapless time-reversal symmetric $SU(2)$ YM theories at $\theta=\pi$ as two distinct conformal field theories, which although are secretly indistinguishable by gapped SPT states nor by correlators of local operators on oriented spacetimes, can be distinguished on non-orientable spacetimes or potentially by correlators of extended operators.

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We show that the category of finite-length generalized modules for the singlet vertex algebra M(p), p∈Z_+, is equal to the category O_M(p) of C_1-cofinite M(p)-modules, and that this category admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. Since O_M(p) includes the uncountably many typical M(p)-modules, which are simple M(p)-module structures on Heisenberg Fock modules, our results substantially extend our previous work on tensor categories of atypical M(p)-modules. We also introduce a tensor subcategory O_M(p)^T, graded by an algebraic torus T, which has enough projectives and is conjecturally tensor equivalent to the category of finite-dimensional weight modules for the unrolled restricted quantum group of sl_2 at a 2pth root of unity. We compute all tensor products involving simple and projective M(p)-modules, and we prove that both tensor categories O_M(p) and O_M(p)^T are rigid and thus also ribbon. As an application, we use vertex operator algebra extension theory to show that the representation categories of all finite cyclic orbifolds of the triplet vertex algebras W(p) are non-semisimple modular tensor categories, and we confirm a conjecture of Adamović-Lin-Milas on the classification of simple modules for these finite cyclic orbifolds.