Abstract:We discuss the representation theory of the non-linear chiral algebra W 1+∞ of Gaberdiel and Gopakumar and its connection to the Yangian of u(1) whose presentation was given by Tsymbaliuk. The characters of completely degenerate representations of W 1+∞ are given by the topological vertex. The Yangian picture provides an infinite number of commuting charges which can be explicitly diagonalized in W 1+∞ highest weight representations. Many properties that are difficult to study in the W 1+∞ picture turn out to have a simple combinatorial interpretation, once translated to the Yangian picture.

We associate vertex operator algebras to (p, q)-webs of interfaces in the topologically twisted N = 4 super Yang-Mills theory. Y-algebras associated to trivalent junctions are identified with truncations of W 1+∞ algebra. Starting with Y-algebras as atomic elements, we describe gluing of Y-algebras analogous to that of the topological vertex. At the level of characters, the construction matches the one of counting D0-D2-D4 bound states in toric Calabi-Yau threefolds. For some configurations of interfaces, we propose a BRST construction of the algebras and check in examples that both constructions agree. We define generalizations of W 1+∞ algebra and identify a large class of glued algebras with their truncations. The gluing construction sheds new light on the structure of vertex operator algebras conventionally constructed by BRST reductions or coset constructions and provides us with a way to construct new algebras. Many well-known vertex operator algebras, such as U(N) k affine Lie algebra, N = 2 superconformal algebra, N = 2 super-W ∞ , Bershadsky-Polyakov W (2) 3 , cosets and Drinfeld-Sokolov reductions of unitary groups can be obtained as special cases of this construction.

Myotonic dystrophy type 1 is caused by the expansion of a CTG repeat in the 3' UTR of the DMPK gene. A length exceeding 50 CTG triplets is pathogenic. Intermediate alleles with 35-49 triplets are not disease-causing but show instability in intergenerational transmissions. We report on the identification of multiple patients with different patterns of CCG and CTC interruptions in the DMPK CTG repeat tract that display unique intergenerational instability. In patients bearing interrupted expanded alleles, the location of the interruptions changed dramatically between generations and the repeats tended to contract. The phenotype for these patients corresponded to the classical form of the disease, but in some cases without muscular dystrophy and possibly with a later onset than expected. Symptomatic patients bearing interrupted intermediate length repeat tracts were also identified, although the role of the interruptions in their phenotype remains unclear. The identification of interruptions in the DMPK repeat has important consequences for molecular genetic testing where they can lead to false negative conclusions.

We study the operator product expansions in the chiral algebra W ∞ , first using the associativity conditions in the basis of primary generating fields and then using a different basis coming from the free field representation in which the OPE takes a simpler quadratic form. The results in the quadratic basis can be compactly written using certain bilocal combinations of the generating fields and we conjecture a closed-form expression for the complete OPE in this basis. Next we show that the commutation relations as well as correlation functions can be easily computed using properties of these bilocal fields. In the last part we verify the consistency with results derived previously by studying minimal models of W ∞ and comparing them to known reductions of W ∞ to W N . The results we obtain illustrate nicely the role of triality symmetry in the representation theory of W ∞ .

We propose a refinement of the Gaberdiel-Gopakumar duality conjecture between W N conformal field theories and 2+1-dimensional higher spin gravity. We make an identification of generic representations of the W N CFT in the semiclassical limit with bulk configurations. By studying the spectrum of the semiclassical limit of the W N theories and mapping to solutions of Euclidean Vasiliev gravity at λ = −N , we propose that the 'light states' of the W N minimal models in the 't Hooft limit map not to the conical defects of the Vasiliev theory, but rather to bound states of perturbative scalar fields with these defects. Evidence for this identification comes from comparing charges and from holographic relations between CFT null states and bulk symmetries. We also make progress in understanding the coupling of scalar matter to sl(N ) gauge fields.