WAYREP · Bridging W-algebras and affine Yangians- from new perspectives

Recent advances in theoretical physics have significantly deepened our understanding of quantum field theories and uncovered surprising connections between seemingly unrelated areas of mathematics. A key development is the Alday–Gaiotto–Tachikawa correspondence, which links four-dimensional supersymmetric gauge theories with two-dimensional conformal field theories, revealing profound ties between enumerative geometry and the representation theory of vertex algebras. This framework has since expanded into the broader BPS/CFT correspondence, predicting a deep yet still mysterious relationship between cohomological Hall algebras and vertex algebras, incarnating rich combinatorics, tensor categories, and representation theory from different perspectives. Supported by the Marie Skłodowska-Curie Actions programme, the WAYREP project aims to construct a rigorous bridge between these algebraic worlds. It focuses on establishing a long-sought correspondence between affine Yangians and W-algebras, leveraging recent advances in the structure and classification of the latter. Through this approach, WAYREP will contribute to a deeper understanding of moduli spaces, vertex algebras, and quantum symmetries, fostering interdisciplinary dialogue across geometry, algebra, and physics.