# (q,t)-KZ equation for Ding-Iohara-Miki algebra

## TL;DR

This paper generalizes the Knizhnik-Zamolodchikov equation for the Ding-Iohara-Miki algebra, linking it to topological string amplitudes, R-matrix braiding, and Nekrasov partition functions, with extensions to elliptic cases.

## Contribution

It introduces a (q,t)-generalization of the KZE for the DIM algebra and connects it to refined topological strings and gauge theory partition functions.

## Key findings

- Refined topological string amplitudes satisfy the (q,t)-KZE.
- The braiding is governed by the R-matrix of the DIM algebra.
- Solutions relate to Nekrasov partition functions for 5d quiver gauge theories.

## Abstract

We derive the generalization of the Knizhnik-Zamolodchikov equation (KZE) associated with the Ding-Iohara-Miki (DIM) algebra U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1). We demonstrate that certain refined topological string amplitudes satisfy these equations and find that the braiding transformations are performed by the R-matrix of U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1). The resulting syste is the uplifting of the \widehat{\mathfrak{u}}_1 Wess-Zumino-Witten model. The solutions to the (q,t)-KZE are identified with the (spectral dual of) building blocks of the Nekrasov partition function for 5d linear quiver gauge theories. We also construct an elliptic version of the KZE and discuss its modular and monodromy properties, the latter being related to a dual version of KZE.