A surprise announcement by then prime minister Tony Blair in 1998 led to the creation of a new senior clinical role - the nurse consultant.

Work in progress: The evolving nurse consultant role has all-round benefits and should be embraced, finds Frances Pickersgill

We suggest a purely algebraic construction of the spin representation of an infinite-dimensional orthogonal Lie. al- gebra (sections 1 and 2) and a corresponding group (section 4). From this we deduce a construction of all level-one highest-weight representations of orthogonal affine Lie algebras in terms of cre- ation and annihilation operators on an infinite-dimensional Grass- mann algebra (section 3). We also give a similar construction of the level-one representations of the general linear affine. Lie al- gebra in an infinite-dimensional "wedge space." Along these lines we construct the corresponding representations of the universal central extension of. the group SL,(k(t,t-1)) in spaces of sections of line bundles over infinite-dimensional homogeneous spaces (sec- tion 5).

Spin and wedge representations of infinite-dimensional Lie algebras and groups (Clifford algebra/spin representation/affine Kac-Moody Lie algebra/highest weight representation/line bundle)

Certain perturbative aspects of two-dimensional sigma models with (0, 2) supersymmetry are investigated. The main goal is to understand in physical terms how the mathematical theory of “chiral differential operators” is related to sigma models. In the process, we obtain, for example, an understanding of the one-loop beta function in terms of holomorphic data. A companion paper will study nonperturbative behavior of these theories.

/pdf/two-dimensional-models-with-0-2-supersymmetry-perturbative-2t1tqywqbt.pdf

Two-Dimensional Models With (0,2) Supersymmetry: Perturbative Aspects

Let $V$ be a vertex operator algebra with a category $\mathcal{C}$ 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 $\mathcal{C}$, using results of Huang-Kirillov-Lepowsky showing that $A$ is a (super)algebra object in $\mathcal{C}$ and that generalized $A$-modules in $\mathcal{C}$ correspond exactly to local modules for the corresponding (super)algebra object Both categories, of local modules for a $\mathcal{C}$-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 $(1/2)\mathbb{Z}$-graded vertex operator algebras by realizing them as (super)algebra objects in the vertex tensor categories of their even and $\mathbb{Z}$-graded components, respectively Second, we analyze parafermionic cosets $C=\mathrm{Com}(V_L,V)$ 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

Tensor categories for vertex operator superalgebra extensions

The Schur index of the $$(A_1, X_n)$$
 -Argyres–Douglas theory is conjecturally a character of a vertex operator algebra. Here such vertex algebras are found for the $$A_{\text {odd}}$$
 and $$D_{\text {even}}$$
 -type Argyres–Douglas theories. The vertex operator algebra corresponding to $$A_{2p-3}$$
 -Argyres–Douglas theory is the logarithmic -algebra of Creutzig et al. (Lett Math Phys 104(5):553–583, 2014), while the one corresponding to $$D_{2p}$$
 , denoted by , is realized as a non-regular quantum Hamiltonian reduction of $$L_{k}(\mathfrak {sl}_{p+1})$$
 at level . For all n one observes that the quantum Hamiltonian reduction of the vertex operator algebra of $$D_n$$
 -Argyres–Douglas theory is the vertex operator algebra of $$A_{n-3}$$
 -Argyres–Douglas theory. As a corollary, one realizes the singlet and triplet algebras (the vertex algebras associated to the best understood logarithmic conformal field theories) as quantum Hamiltonian reductions as well. Finally, characters of certain modules of these vertex operator algebras and the modular properties of their meromorphic continuations are given.

W-algebras for Argyres–Douglas theories

Cosets of affine vertex algebras inside larger structures

We prove the long-standing conjecture on the coset construction of the minimal series principal W-algebras of ADE types in full generality. We do this by first establishing Feigin’s conjecture on the coset realization of the universal principal W-algebras, which are not necessarily simple. As consequences, the unitarity of the “discrete series” of principal W-algebras is established, a second coset realization of rational and unitary W-algebras of type A and D are given and the rationality of Kazama–Suzuki coset vertex superalgebras is derived.

W-algebras as coset vertex algebras

Let V be a simple vertex operator algebra containing a rank n Heisenberg vertex algebra H and let C = Com(H;V) be the coset of H in V. Assuming that the module categories of interest are vertex tensor categories in the sense of Huang, Lepowsky and Zhang, a Schur-Weyl type duality for both simple and indecomposable but reducible modules is proven. Families of vertex algebra extensions of C are found and every simple C-module is shown to be contained in at least one V-module. A corollary of this is that if V is rational, C2-cofinite and CFT-type, and Com(C;V) is a rational lattice vertex operator algebra, then C is likewise rational. These results are illustrated with many examples and the C1-cofiniteness of certain interesting classes of modules is established.

Schur–weyl duality for heisenberg cosets

Let V be a simple VOA and consider a representation category of V that is a vertex tensor category in the sense of Huang-Lepowsky. In particular, this category is a braided tensor category. Let J be an object in this category that is a simple current of order two of either integer or half-integer conformal dimension. We prove that $V\oplus J$ is either a VOA or a super VOA. If the representation category of V is in addition ribbon, then the categorical dimension of J decides this parity question. Combining with Carnahan's work, we extend this result to simple currents of arbitrary order. Our next result is a simple sufficient criterion for lifting indecomposable objects that only depends on conformal dimensions. Several examples of simple current extensions that are $C_2$-cofinite and non-rational are then given and induced modules listed.

Simple current extensions beyond semi-simplicity

Let $${\mathfrak{g}}$$
 be a simple, finite-dimensional Lie (super)algebra equipped with an embedding of $${\mathfrak{s}\mathfrak{l}_2}$$
 inducing the minimal gradation on $${\mathfrak{g}}$$
 . The corresponding minimal $${\mathcal{W}}$$
 -algebra $${\mathcal{W}^k(\mathfrak{g}, e_{-\theta})}$$
 introduced by Kac and Wakimoto has strong generators in weights $${1,2,3/2}$$
 , and all operator product expansions are known explicitly. The weight one subspace generates an affine vertex (super)algebra $${V^{k'}(\mathfrak{g}^{\natural})}$$
 , where $${\mathfrak{g}^{\natural} \subset \mathfrak{g}}$$
 denotes the centralizer of $${\mathfrak{s}\mathfrak{l}_2}$$
 . Therefore, $${\mathcal{W}^k(\mathfrak{g}, e_{-\theta})}$$
 has an action of a connected Lie group $${G^{\natural}_0}$$
 with Lie algebra $${\mathfrak{g}^{\natural}_0}$$
 , where $${\mathfrak{g}^{\natural}_0}$$
 denotes the even part of $${\mathfrak{g}^{\natural}}$$
 . We show that for any reductive subgroup $${G \subset G^{\natural}_0}$$
 , and for any reductive Lie algebra $${\mathfrak{g}' \subset \mathfrak{g}^{\natural}}$$
 , the orbifold $${\mathcal{O}^k = \mathcal{W}^k(\mathfrak{g}, e_{-\theta})^{G}}$$
 and the coset $${\mathcal{C}^k = {\rm Com}(V(\mathfrak{g}'),\mathcal{W}^k(\mathfrak{g}, e_{-\theta}))}$$
 are strongly finitely generated for generic values of $${k}$$
 . Here $${V(\mathfrak{g}')}$$
 denotes the affine vertex algebra associated to $${\mathfrak{g}'}$$
 . We find explicit minimal strong generating sets for $${\mathcal{C}^k}$$
 when $${\mathfrak{g}' = \mathfrak{g}^{\natural}}$$
 and $${\mathfrak{g}}$$
 is either $${\mathfrak{s}\mathfrak{l}_n}$$
 , $${\mathfrak{s}\mathfrak{p}_{2n}}$$
 , $${\mathfrak{s}\mathfrak{l}(2|n)}$$
 for $${n \neq 2}$$
 , $${\mathfrak{p}\mathfrak{s}\mathfrak{l}(2|2)}$$
 , or $${\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|4)}$$
 . Finally, we conjecture some surprising coincidences among families of cosets $${\mathcal{C}_k}$$
 which are the simple quotients of $${\mathcal{C}^k}$$
 , and we prove several cases of our conjecture.

Andrew R. Linshaw

Papers

Cosets of affine vertex algebras inside larger structures

W-algebras as coset vertex algebras

Schur–weyl duality for heisenberg cosets

Simple current extensions beyond semi-simplicity

Orbifolds and Cosets of Minimal $${\mathcal{W}}$$ W -Algebras