We introduce a class of Vertex Operator Algebras which arise at junctions of supersymmetric interfaces in N = 4 Super Yang Mills gauge theory. These vertex algebras satisfy non-trivial duality relations inherited from S-duality of the four-dimensional gauge theory. The gauge theory construction equips the vertex algebras with collections of modules labelled by supersymmetric interface line defects. We discuss in detail the simplest class of algebras Y L,M,N , which generalizes W N algebras. We uncover tantalizing relations between Y L,M,N , the topological vertex and the W 1+∞ algebra.

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.

We introduce a class of Vertex Operator Algebras which arise at junctions of supersymmetric interfaces in N = 4 Super Yang Mills gauge theory. These vertex algebras satisfy non-trivial duality relations inherited from S-duality of the fourdimensional gauge theory. The gauge theory construction equips the vertex algebras with collections of modules labelled by supersymmetric interface line defects. We discuss in detail the simplest class of algebras Y L,M,N , which generalizes W N algebras. We uncover tantalizing relations between Y L,M,N , the topological vertex and the W 1+∞ algebra.

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.

We discuss a class of vertex operator algebras W m|n×∞ generated by a supermatrix of fields for each integral spin 1, 2, 3,. .. . The algebras admit a large family of truncations that are in correspondence with holomorphic functions on the Calabi-Yau singularity given by solutions to xy = z m w n. We propose a free-field realization of such truncations generalizing the Miura transformation for W N algebras. Relations in the ring of holomorphic functions lead to bosonization-like relations between different free-field realizations. The discussion provides a concrete example of a non-trivial interplay between vertex operator algebras, algebraic geometry and gauge theory.

We define an action of the (double of) Cohomological Hall algebra of Kontsevich and Soibelman on the cohomology of the moduli space of spiked instantons of Nekrasov. We identify this action with the one of the affine Yangian of gl(1). Based on that we derive the vertex algebra at the corner Wr 1 ,r 2 ,r 3 of Gaiotto and Rapčák. We conjecture that our approach works for a big class of Calabi-Yau categories, including those associated with toric Calabi-Yau 3-folds.

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These lecture notes cover a brief introduction into some of the algebro-geometric techniques used in the construction of BPS algebras. The first section introduces the derived category of coherent sheaves as a useful model of branes in toric Calabi-Yau three-folds. This model allows a rather simple derivation of quiver quantum mechanics describing low-energy dynamics of various brane systems. Vacua of such quantum mechanics can be identified with the critical equivariant cohomology of the moduli space of quiver representations. These are often counted by various crystal configurations. Using correspondences in algebraic geometry, one can construct rich families of affine-Yangian representations. We conclude with an exploration of different algebraic structures naturally appearing in our story. The material was covered in a 4-lecture mini-course within the Second PIMS Summer School on Algebraic Geometry in High-Energy Physics. The text contains some new ideas, examples and remarks that are going to be covered in detail in a joint work with Dylan Butson. Contents 1. Physical motivation 2. Quivers from branes 2.1. Derived category of coherent sheaves as a brane category 2.2. Morphisms in the brane category 2.3. Framed quivers 2.4. Higgs field 2.5. Potential 2.6. Examples of framed quivers with potential 3. Supersymmetric vacua 3.1. Quiver quantum mechanics 3.2. Twisted masses and equivariance 3.3. Critical equivariant cohomology 3.4. Example of equivariant cohomology 3.5. D2 and 1d partitions 3.6. D4 and 2d partitions 3.7. D6 and 3d partitions 4. Modules from correspondences 4.1. Correspondences 4.2. D2-brane and the Weyl algebra 4.3. D4-brane and the p gl 1 Kac-Moody algebra 4.4. D6-brane and the MacMahon module 5. Cherednik algebras and W-algebras 5.1. Affine Yangian of gl 1 and its shifts 5.2. Coproduct