Holomorphicity of Functions of unitary matrices I am studying lattice QCD and there I encounter functions of unitary matrices. For ex. The action, $S = \sum$Tr( plaquettes), where each plaquette, $P$ is written as,
$$ P = U_{\mu}(x)U_{\nu}(x+\mu){U}_{\mu}^\dagger(x+\nu){U}_{\nu}^\dagger(x) $$
here $\mu$, $\nu$ represent directions and $x$ represents lattice sites.
While going over the literature [1], i encounter terms like holomorphicity of these functions of unitary matrices which I am unable to make any sense of.
If anyone can help me understand how to check if a given function $f(U)$ (where $U$ is a unitary matrix) is holomorphic or not, or can direct me to the topic i need to study for this. It would be a great help.\
References:

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*So for example https://arxiv.org/abs/1808.04400 here in equation (14) they propose a function, $f(U)$ and in paragraph below mention this function is not holomorphic.

 A: The set of matrices is a complex manifold, i.e. locally described by $\mathbb{C}^n$. Therefore one needs to check what holomorphicity means on $\mathbb{C}^n$. Indeed, consider a function $f:\mathbb{C}^n \rightarrow \mathbb{C}^m$ and pick a point $z_0 \in \mathbb{C}^n$. Then $f$ is complex differentiable at $z_0$ if there is a a linear map $L_z : \mathbb{C}^n \rightarrow \mathbb{C}^m$ continuous at $z_0$ such that
$$ f(z) = f(z_0) + L_z(z-z_0)  \ . $$
Then $L_{z_0}$ is the derivative of $f$ at $L_{z_0}$.
The important part is of course the absence of $\overline{z} - \overline{z}_0$.
In the paper you gave, they enlarge their gauge group from the unitary group to the special linear group. The function they define depends on expressions of the form $UU^\dagger$. This is not holomorphic on the special linear group, but it is holomorphic on the unitary group. Indeed, in the unitary group we can replace daggers, that is, complex conjugations, by taking the inverse, which is holomorphic. That is the reason why the action, which is of the form $U_1U_2U_3^\dagger U_4^\dagger = U_1U_2U_3^{-1}U_4^{-1}$ is holomorphic on the unitary matrices, but not the special linear transformations.
