# Topological recursion with hard edges

## TL;DR

This paper establishes a Givental-type decomposition for partition functions from topological recursion, linking spectral curve types to specific tau functions, with applications to matrix models with hard edges.

## Contribution

It introduces a novel decomposition framework connecting spectral curves to tau functions, including irregular cases like hard edges.

## Key findings

- Regular spectral curves yield Konstevich-Witten tau functions.
- Irregular spectral curves produce Brezin-Gross-Witten tau functions.
- Decomposition demonstrated for matrix models with two hard edges.

## Abstract

We prove a Givental type decomposition for partition functions that arise out of topological recursion applied to spectral curves. Copies of the Konstevich-Witten KdV tau function arise out of regular spectral curves and copies of the Brezin-Gross-Witten KdV tau function arise out of irregular spectral curves. We present the example of this decomposition for the matrix model with two hard edges and spectral curve $(x^2-4)y^2=1$