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 $G(k,N)$.

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.

Let V be an N-graded, simple, self-contragredient, C_2-cofinite vertex operator algebra. We show that if the S-transformation of the character of V is a linear combination of characters of V-modules, then the category C of grading-restricted generalized V-modules is a rigid tensor category. We further show, without any assumption on the character of V but assuming that C is rigid, that C is a factorizable finite ribbon category, that is, a not-necessarily-semisimple modular tensor category. As a consequence, we show that if the Zhu algebra of V is semisimple, then C is semisimple and thus V is rational. The proofs of these theorems use techniques and results from tensor categories together with the method of Moore-Seiberg and Huang for deriving identities of two-point genus-one correlation functions associated to V. We give two main applications. First, we prove the conjecture of Kac-Wakimoto and Arakawa that C_2-cofinite affine W-algebras obtained via quantum Drinfeld-Sokolov reduction of admissible-level affine vertex algebras are strongly rational. The proof uses the recent result of Arakawa and van Ekeren that such W-algebras have semisimple (Ramond twisted) Zhu algebras. Second, we use our rigidity results to reduce the "coset rationality problem" to the problem of C2-cofiniteness for the coset. That is, given a vertex operator algebra inclusion U⊗V↪A with A, U strongly rational and U, V a pair of mutual commutant subalgebras in A, we show that V is also strongly rational provided it is C_2-cofinite.

Let O_25 be the vertex algebraic braided tensor category of finite-length modules for the Virasoro Lie algebra at central charge 25 whose composition factors are the irreducible quotients of reducible Verma modules. We show that O_25 is rigid and that its simple objects generate a semisimple tensor subcategory that is braided tensor equivalent to an abelian 3-cocycle twist of the category of finite-dimensional sl_2-modules. We also show that this sl_2-type subcategory is braid-reversed tensor equivalent to a similar category for the Virasoro algebra at central charge 1. As an application, we construct a simple conformal vertex algebra which contains the Virasoro vertex operator algebra of central charge 25 as a PSL_2(C)-orbifold. We also use our results to study Arakawa's chiral universal centralizer algebra of SL_2 at level -1, showing that it has a symmetric tensor category of representations equivalent to Rep PSL_2(C). This algebra is an extension of the tensor product of Virasoro vertex operator algebras of central charges 1 and 25, analogous to the modified regular representations of the Virasoro algebra constructed earlier for generic central charges by I. Frenkel-Styrkas and I. Frenkel-M. Zhu.

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.

Let O_c be the category of finite-length central-charge-c modules for the Virasoro Lie algebra whose composition factors are irreducible quotients of reducible Verma modules. Recently, it has been shown that O_c admits vertex algebraic tensor category structure for any c∈C. Here, we determine the structure of this tensor category when c=13−6p−6/p for an integer p>1. For such c, we prove that O_c is rigid, and we construct projective covers of irreducible modules in a natural tensor subcategory O_c^0. We then compute all tensor products involving irreducible modules and their projective covers. Using these tensor product formulas, we show that O_c has a semisimplification which, as an abelian category, is the Deligne product of two tensor subcategories that are tensor equivalent to the Kazhdan-Lusztig categories for affine sl_2 at levels −2+p^±1. Next, as a straightforward consequence of the braided tensor category structure on O_c together with the theory of vertex operator algebra extensions, we rederive known results for triplet vertex operator algebras W(p), including rigidity, fusion rules, and construction of projective covers. Finally, we prove a recent conjecture of Negron that O_c^0 is braided tensor equivalent to the PSL(2,C)-equivariantization of the category of W(p)-modules.

We introduce fusion bialgebras and their duals and systematically study their Fourier analysis. As an application, we discover new efficient analytic obstructions on the unitary categorification of fusion rings. We prove the Hausdorff-Young inequality, uncertainty principles for fusion bialgebras and their duals. We show that the Schur product property, Young's inequality and the sum-set estimate hold for fusion bialgebras, but not always on their duals. If the fusion ring is the Grothendieck ring of a unitary fusion category, then these inequalities hold on the duals. Therefore, these inequalities are analytic obstructions of categorification. We classify simple integral fusion rings of Frobenius type up to rank 8 and of Frobenius-Perron dimension less than 4080. We find 34 ones, 4 of which are group-like and 28 of which can be eliminated by applying the Schur product property on the dual. In general, these inequalities are obstructions to subfactorize fusion bialgebras.

We construct a new subfactor planar algebra, and as a corollary a new subfactor, with the ‘extended Haagerup’ principal graph pair. This completes the classification of irreducible amenable subfactors with index in the range ( $$ {4},{3} + \sqrt {{3}} $$ ), which was initiated by Haagerup in 1993. We prove that the subfactor planar algebra with these principal graphs is unique. We give a skein-theoretic description, and a description as a subalgebra generated by a certain element in the graph planar algebra of its principal graph. In the skein-theoretic description there is an explicit algorithm for evaluating closed diagrams. This evaluation algorithm is unusual because intermediate steps may increase the number of generators in a diagram. This is the published version ofarXiv:0909.4099 [math.OA].

We show that direct limit completions of vertex tensor categories inherit vertex and braided tensor category structures, under conditions that hold for example for all known Virasoro and affine Lie algebra tensor categories. A consequence is that the theory of vertex operator (super)algebra extensions also applies to infinite-order extensions. As an application, we relate rigid and non-degenerate vertex tensor categories of certain modules for both the affine vertex superalgebra of osp(1|2) and the N=1 super Virasoro algebra to categories of Virasoro algebra modules via certain cosets.

We construct two non-semisimple braided ribbon tensor categories of modules for each singlet vertex operator algebra M(p), p≥2. The first category consists of all finite-length M(p)-modules with atypical composition factors, while the second is the subcategory of modules that induce to local modules for the triplet vertex operator algebra W(p). We show that every irreducible module has a projective cover in the second of these categories, although not in the first, and we compute all fusion products involving atypical irreducible modules and their projective covers.