Webs of W-algebras

We associate vertex operator algebras to $(p,q)$-webs of interfaces in the topologically twisted $\mathcal{N}=4$ super Yang-Mills theory. Y-algebras associated to trivalent junctions are identified with truncations of $\mathcal{W}_{1+\infty}$ 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 $\mathcal{W}_{1+\infty}$ 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, $\mathcal{N}=2$ superconformal algebra, $\mathcal{N}=2$ super-$\mathcal{W}_\infty$, Bershadsky-Polyakov $\mathcal{W}_3^{(2)}$, cosets and Drinfeld-Sokolov reductions of unitary groups can be obtained as special cases of this construction.


Gluing with examples
Zamolodchikov W 3 algebra As in illustration, the W 3 algebra constructed by Zamolodchikov (1984) has a stress-energy tensor (Virasoro algebra) with OPE together with spin 3 primary field W (z) We need to find the OPE of W with itself such that the resulting chiral algebra is associative.
The result: Λ is a quasiprimary 'composite' (spin 4) field, The algebra is non-linear, not a Lie algebra in the usual sense (in fact linearity should not be expected for spins ≥ 3).

Construction of W-algebras
Many different ways of constructing W-algebras: solving the associativity conditions for a given spin content; e.g. Zamolodchikov W 3 , Gaberdiel-Gopakumar W ∞ affine Lie algebra Casimir subalgebra sl(N) 1 /sl(N) or more generally GKO coset Hamiltonian reduction via BRST: Drinfeld-Sokolov reduction; e.g. W assuming the spin content 2, 3, 4, . . . a two-parametric family of algebras is found: in this sense W ∞ interpolates between all W N triality symmetry: for each value of c three solutions λ j giving the same algebra the vacuum character is the MacMahon function (counting three-dimensional partitions) satisfies the Yang-Baxter equation algebraic Bethe ansatz Construction #3 (Yangian): (Schiffmann-Vasserot, Tsymbaliuk) an associative algebra with generators ψ j , e j , f j , j = 0, 1, . . . and relations of the form . . .

Intro W 3 W∞ Gaiotto-Rapčák Gluing Conclusion
Brane construction (Gaiotto-Rapčák) trivalent junction of codim 1 interfaces in the Kapustin-Witten twist of N = 4 SYM vertex operator algebra at the corner

Gluing
We can now consider more complicated interfaces and their associated VOAs: vertices: degrees of freedom at junction described by Y LMN algebras internal edges: degrees of freedom associated to line operators along the interfaces -bi-modules the resulting VOA is obtained by conformally extending Y LMN algebras (vertices) by bi-modules associated to internal edges the total central charge is a sum of individual central charges (no contribution from the line ops) and can be read off directly from the diagram Intro W 3 W∞ Gaiotto-Rapčák Gluing Conclusion Vertex and edge five-brane charge conservation at vertex j (p j , q j ) = 0 parameters of the vertex algebra are given by (Ψ ≡ − 2 / 1 ) the conformal dimension of is (half-)integral (independent of Ψ) and the statistics of the gluing matter (boson/fermion) depends on the relative orientation of the two vertices Gaiotto-Rapčák Gluing Conclusion Example -N = 2 superconformal algebra An extension of Virasoro algebra by spin 1 current J and two charge ±1 fermionic spin 3 2 supercurrents G ± The vacuum character (generic c) is up to an U(1) factor exactly what one gets from the gluing 1 2 Intro W 3 W∞ Gaiotto-Rapčák Gluing Conclusion Example -Bershadsky-Polyakov W (2) 3 an extension of Virasoro algebra by spin 1 current J and two charge ±1 bosonic spin 3 2 currents G ± the same spin content as N = 2 SCA but different statistics of the gluing fields can be obtained from U(3) Kac-Moody by DS reduction using the non-principal embedding 3 = 2 + 1 3 1 Intro W 3 W∞ Gaiotto-Rapčák Gluing Conclusion N = 2 W ∞ N = 2 W ∞ symmetry algebra of Kazama-Suzuki coset models is glued from two copies of bosonic W ∞ the central charge can be decomposed as corresponding to central charges of the two W ∞ vertices the symmetry of the diagram is Z 2 × Z 2 which is also the duality symmetry of the algebra the four basic minimal representations correspond to four external legs Other examples and questions