Webs of W-algebras

Abstract

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.

Rectangular W-algebras, extended higher spin gravity and dual coset CFTs

“…In this appendix, we reproduce the decomposition in (4.26) from the W 2 -algebra with su(2) symmetry obtained in subsection 3.3. Similar analysis for other type of W-algebras has been done in [24], see also [76]. First, we look for two commuting energy-momentum tensors T η (η = 0, 1), whose OPEs are…”

“…in stead of (1.4), and this implies a duality relation of the coset model (1.3) as in the cases with M = 1 [19][20][21]. We give an explanation of the duality by decomposing the W-algebra in terms of the coset description as in [22][23][24][25][26].…”

Instanton R-matrix and $$ \mathcal{W} $$-symmetry

“…There is a nice combinatorial description of the truncation in terms of plane partitions (box-counting) which is discussed in [72,73]. The free field representation of Y N 1 ,N 2 ,N 3 was constructed in [67].…”

Cohomological Hall Algebras, Vertex Algebras and Instantons

“…If X = C 3 and D = 1≤i≤3 r i C 2 i , then W C 3 ,r1,r2,r3 := W r1,r2,r3 coincides with the vertex algebra Y r1,r2,r3 defined in [42]. It was later used in [88,89] as a building block for construction of more complicated vertex algebras associated with toric CY 3-folds via the gluing procedure proposed in the loc. cit.…”

“…There are at least two ways to approach the construction of W X,r1,...,r k for general toric X. We have already mentioned the approach of [88] which is analogous to the topological vertex formula in the sense that one constructs vertex algebras associated to the colored toric diagrams starting with the basic one for C 3 . The idea of gluing complicated vertex algebras from the basic ones was discussed in the case of toric surfaces for example in [21] and [30].…”