$$ \mathcal{W} $$ -algebra modules, free fields, and Gukov-Witten defects

Abstract

We study the structure of modules of corner vertex operator algebras arrising at junctions of interfaces in N = 4 SYM. In most of the paper, we concentrate on truncations of W 1+∞ associated to the simplest trivalent junction. First, we generalize the Miura transformation for W N 1 to a general truncation Y N 1 ,N 2 ,N 3. Secondly, we propose a simple parametrization of their generic modules, generalizing the Yangian generating function of highest weight charges. Parameters of the generating function can be identified with exponents of vertex operators in the free field realization and parameters associated to Gukov-Witten defects in the gauge theory picture. Finally, we discuss some aspect of degenerate modules. In the last section, we sketch how to glue generic modules to produce modules of more complicated algebras. Many properties of vertex operator algebras and their modules have a simple gauge theoretical interpretation.

Gluing two affine Yangians of 𝔤𝔩1

“…Moreover note from this example that in the glued algebra, the S 3 symmetry (embodied by the ϕ 3 (u) function) is broken into a Z 2 symmetry, satisfied by ϕ 2 (u). 21 On the other hand, the hatted charge for this configuration iŝ…”

“…This is true at generic value of central charge c and 't Hooft coupling λ. At special values where null vectors arise, the W1+∞ character counts fewer states.4 See[21] for an construction of (the truncated version of) some of these VOAs using free field realization, which in principle also allows one to derive explicit algebraic relations.…”

On extensions of $$ \mathfrak{gl}\widehat{\left(\left.m\right|n\right)} $$ Kac-Moody algebras and Calabi-Yau singularities

“…Such an embedding can be conveniently carved out by a generalized version of the Miura transformation [2,3] for W N × gl (1). To define the Miura transformation, [40] first had to introduce N 1 copies of a pseudo-differential operator L (1) , N 2 copies of a pseudo-differential operator L (2) and N 3 copies of a pseudo-differential operator L (3) . By Miura transformation, we mean the process of multiplying these pseudo-differential operators and rewriting them in terms of a pseudo-differential operator in the standard form by commuting derivatives to the right.…”

“…Let us now move to the discussion of vertex operator algebras and review the generalized Miura transformations from [40]. W N × gl(1) algebras admit a well-known realization [2,3,25,51] in terms of a subalgebra of N copies of gl (1).…”

“…We will see a natural generalization of the formula for CY 3 m,n bellow. It was found in [40] that there actually exist two more Miura operators whose products give rise to Y N 1 ,N 2 ,N 3 truncations. Let us normalize the currents appearing in these two Miura operators as 8…”