Need help to understand integration notation

Question

Could anyone help me understand what this notation is highlighted in yellow?

This motto is from the following article in page 4: https://arxiv.org/pdf/0804.4211.pdf

I suspect that there are three possible outcomes for the integral, but I'm not sure.

Answer 1

I think you are supposed to read

$\begin{pmatrix} -\\-\\- \end{pmatrix}$

as a vector with three components, and integration is happening in each component separately. You might prefer the notation

$\Phi(z) = \begin{pmatrix} \mathrm{Re} \int_{z_0}^{z} (1-g^2)f \; d\zeta\\\mathrm{Re} \int_{z_0}^{z}i(1+g^2)f\; d\zeta \\\mathrm{Re} \int_{z_0}^{z}2gf\; d\zeta \end{pmatrix},$

or even

$\Phi(z) = \left(\mathrm{Re} \int_{z_0}^{z} \dots, \;\;\;\mathrm{Re} \int_{z_0}^{z}\dots,\;\;\;\mathrm{Re} \int_{z_0}^{z}\dots \right)$.

However you write it, the intention is the same: the output of $\Phi$ is an ordered triple (three real numbers), i.e. $\Phi$ is a map $\widetilde{\Sigma\setminus \{p_j\}} \to \mathbb{R}^3$.