# Homological and monodromy representations of framed braid groups

## TL;DR

This paper introduces two new classes of representations for framed braid groups, one homological and one monodromic, and explores a conjectural equivalence between them, advancing understanding of their algebraic and geometric structures.

## Contribution

It presents novel homological and monodromy representations of framed braid groups and proposes a conjectural link between these two classes.

## Key findings

- Construction of homological representations via mapping class group actions
- Development of monodromy representations from confluent KZ equations
- Proposed conjectural equivalence between the two representation classes

## Abstract

In this paper, we introduce two new classes of representations of the framed braid groups. One is the homological representation constructed as the action of a mapping class group on a certain homology group. The other is the monodromy representation of the confluent KZ equation, which is a generalization of the KZ equation to have irregular singularities. We also give a conjectural equivalence between these two classes of representations.