  We show that the construction of the higher Gaudin Hamiltonians associated to
the Lie algebra $\mathfrak{gl}_{n}$ admits an interpolation to any complex $n$.
We do this using the Deligne's category $\mathcal{D}_{t}$, which is a formal
way to define the category of finite-dimensional representations of the group
$GL_{n}$, when $n$ is not necessarily a natural number.
  We also obtain interpolations to any complex $n$ of the no-monodromy
conditions on a space of differential operators of order $n$, which are
considered to be a modern form of the Bethe ansatz equations. We prove that the
relations in the algebra of higher Gaudin Hamiltonians for complex $n$ are
generated by our interpolations of the no-monodromy conditions.
  Our constructions allow us to define what it means for a pseudo-deifferential
operator to have no monodromy. Motivated by the Bethe ansatz conjecture for the
Gaudin model associated with the Lie superalgebra $\mathfrak{gl}_{n\vert n'}$,
we show that a ratio of monodromy-free differential operators is a
pseudo-differential operator without monodromy.
