**RESEARCH IN PAIRS**

**Associated Varieties**

**of Affine Vertex Algebras and Class**

*S*

**Theory**

**August 1-12, 2016**

In the thirty years that have passed since they were introduced by Borcherds, vertex algebras have turned out to be extremely useful in many areas of mathematics, such as algebraic geometry, the theory of finite groups, modular functions, topology, integrable system, and combinatorics.
The theory of vertex algebras also serves as the rigorous mathematical foun- dation for two-dimensional conformal field theory and string theory, extensively studied by physicists. The notion of vertex algebras was extended by Beilinson and Drinfeld to chiral algebras, which served as a foundation of the celebrated geometric Langlands program.We plan to explore yet another perspective of vertex algebras. More precisely, this project follows our previous works [Arakawa-Moreau, Joseph ideals and lisse minimal W-algebras, to appear in J. of Inst. Math. Jussieu] and [Arakawa-Moreau, Sheets and associated varieties of affine vertex algebras, preprint].The first article has surprinsgly raised the interest of some physicists (like Nishinaka, Tachikawa, Rastelli, etc.), thus opening new prospects for applications of vertex algebras, in particular in connection with class S theory, which is a rich class of four-dimensional field theories with N = 2 supersymmetry. The class S theory provides numerous interesting questions. In particular it gives conjectural relations between vertex algebras and symplectic varieties. We wish to understand these interesting relations during our stay. We also plan to study some conjectures on associated varieties to affine vertex algebras, started in our previous works. |