# The duality of $\mathfrak{gl}_{m|n}$ and $\mathfrak{gl}_k$ Gaudin models

## TL;DR

This paper reveals a duality between two types of Gaudin models, one associated with the superalgebra gl_{m|n} and the other with gl_k, through Capelli identities and matrix determinants.

## Contribution

It establishes a novel duality between gl_{m|n} and gl_k Gaudin models via Capelli identities and matrix expansions.

## Key findings

- Duality between gl_{m|n} and gl_k Gaudin models proven.
- Hamiltonians expressed through Berezinian and column determinants.
- Capelli identities used to compare and establish the duality.

## Abstract

We establish a duality of the non-periodic Gaudin model associated with superalgebra $\mathfrak{gl}_{m|n}$ and the non-periodic Gaudin model associated with algebra $\mathfrak{gl}_k$. The Hamiltonians of the Gaudin models are given by expansions of a Berezinian of an $(m+n)\times(m+n)$ matrix in the case of $\mathfrak{gl}_{m|n}$ and of a column determinant of a $k\times k$ matrix in the case of $\mathfrak{gl}_k$. We obtain our results by proving Capelli type identities for both cases and comparing the results.