# Seminar: Why is Ding-Iohara-Miki algebra relevant?

- Date:
- Location: A73121
- Lecturer: Yegor Zenkevich (Milano B.)
- Contact person: Fabrizio Nieri
- Seminarium

In the first part of the lecture we will introduce several subjects in which similar structures seem to appear “mysteriously” and tie them together:

1. The study of gauge theories using localization. Here Nekrasov partition function (which is the Omega-deformation of the Seiberg-Witten prepotential) is given by the instanton series, where each term is labelled by a tuple of Young diagrams. The series satisfies some interesting recurrence relations, which are known as qq-characters and form a W-algebra. Also, Nekrasov functions of certain theories, which are naively quite different, turn out to actually coincide (this is known as the spectral duality).

2. 2d CFT, where conformal blocks, which are holomorphic parts of the correlators, turn out to be equal to Nekrasov partitions functions (the fact known as the AGT duality).

3. Refined topological strings are natural deformations of ordinary topological strings on toric Calabi-Yau manifolds and reproduce 5d Nekrasov partition functions. The partition functions are computed by a certain glueing algorithm, which depends on the choice of a preferred direction on the toric diagram. The final answer, however, turns out to be independent of this choice.

Having gathered the motivation from different fields, in the second part of the lecture we proceed to define the DIM algebra and describe its representations. We show how all the mysterious correspondences between various theories are explained by the properties of this remarkable mathematical object.