In Kurt Strebel's book "Quadratic Differentials", in Chapter 2, $$\S4$$, he begins by saying:

"Every analytic function $$\varphi$$ is a domain $$G$$ of the $$z$$-plane defines, in a natural way, a field of line elements $$dz$$, namely by the requirement that $$\varphi(z)dz^2$$ is real and positive. This means of course that $$\text{arg }dz = -\frac{1}{2}\text{arg }\varphi(z)\mod \pi$$, and thus $$dz$$ is determined, up to sign, for every $$z$$, where $$\varphi(z)\ne 0,\infty$$. One may then ask for the integral curves of this field of line elements."

I am having some trouble with the language used here.

Note that I am not a differential geometer by training. My background in differential geometry mostly comes from Voisin's first book on Hodge theory, Bott-Tu's "Differential forms in Algebraic Topology", and a bit of Kobayashi-Nomizu and a few snippets from elsewhere.

This book began its first chapter on background material on Riemann surfaces, and the point of the book seems to be to study the differential geometry of Riemann surfaces. Thus, I'm sure $$G$$ must be a domain in $$\mathbb{C}$$, $$z$$ is a holomorphic coordinate, and "analytic" probably means "complex analytic".

Now, normally, for me, a line element should be a differential 1-form. Though, for him, since he says "field of line elements", I'm assuming he uses "line element" to refer to a cotangent vector at a point, and thus his "field of line elements" should be taken to be a differential 1-form.

Okay, fine, but what would it mean for $$\varphi(z)dz^2$$ to be real and positive? In all analogous texts, $$dz^2$$ is really short for $$dz\otimes dz$$, ie a holomorphic section of the tensor square of the complex-valued cotangent bundle, but presumably for him he really means $$\varphi(z_0)(dz|_{z_0}\otimes dz|_{z_0})$$ as an element of the tensor square of the complex-valued cotangent space at $$z_0\in G$$? In this case, if we view $$dz = dx + idy$$ as a complex-valued differential 1-form, one might interpret his requirement as saying that $$\varphi(z_0)(dz|_{z_0}\otimes dz|_{z_0})(X\otimes X) := \varphi(z_0)\cdot dz(X)\cdot dz(X)\in\mathbb{R}_{>0}\qquad\text{for all X\in T_{G,z_0}}$$ where $$T_{G,z_0}$$ is the (real) tangent space at $$z_0$$, but then this condition will never be satisfied since if it holds for $$X$$, then it will fail for $$iX$$, where $$i$$ is the complex involution on $$T_{G,z_0}$$.

But maybe there is hope, as he explains that this means $$\text{arg }dz = -\frac{1}{2}\text{arg }\varphi(z)\mod \pi$$. However, this only confuses me more. For a complex number $$z_0 = r\cdot e^{i\theta}$$ with $$r\in\mathbb{R}_{>0},\theta\in\mathbb{R}$$, $$\text{arg }(z_0) := \theta\mod 2\pi$$. But what do they mean by the argument of a differential/cotangent vector? The best I can think of is: Identify $$T_{G,z_0}$$ with $$\mathbb{C}$$ via $$\frac{\partial }{\partial x}\mapsto 1$$ and $$\frac{\partial}{\partial y}\mapsto i$$, and then $$\text{arg }dz$$ is how much the $$\mathbb{C}$$-linear map $$dz|_{z_0} : T_{G,z_0} = \mathbb{C}\rightarrow\mathbb{C}$$ "rotates" the tangent vector (viewed as an element in $$\mathbb{C}$$). However, this still does not resolve the previous issue with $$\varphi(z)dz^2$$ being "real and positive".

Lastly, what are "the integral curves of this field of line elements"? Usually one takes integral curves of a vector field. Are the $$dz$$'s really tangent vectors?

First of all, a quadratic differential on a complex manifold $$M$$ (holomorphic or not) is a smooth section of the symmetric square $$S^2(T^{*(1,0)}M)$$ of the holomorphic cotangent bundle; if $$M$$ is a complex curve (which I will assume from now on), we can identify such a section with a section of the tensor square of $$T^{*(1,0)}M$$. Every quadratic differential $$\omega$$ defines the notion of positive tangent vectors $$v\in T_pM$$, namely, vectors satisfying $$\omega(v,v)\ge 0$$. If $$\omega$$ does not vanish at $$p$$ then the set of positive tangent vectors in $$T_pM$$ is a real line $$L_p\subset T_pM$$. Therefore, one obtains a rank 1 smooth distribution (a line field) $$L$$ on the complement $$M'$$ to the set of zeroes of $$\omega$$ in $$M$$. (In other words, this is a real line subbundle in $$TM'$$.) This distribution $$L$$ is the field of "line elements" that Strebel is defining. The integrability condition of the Frobenius (integrability) theorem is trivially satisfied in this situation and, hence, $$L$$ is tangent to a foliation $${\mathcal H}$$ on $$M'$$, called the horizontal foliation of $$\omega$$. Leaves of this foliation are the "integral curves" in Strebel's book.
Similarly, given an angle $$\theta\in S^1$$ one defines the distribution $$L^{\theta}$$ on $$M'$$ by the condition that a nonzero vector $$v\in T_pM$$ is in $$L^{\theta}_p$$ iff $$arg(\omega(v,v))=\theta.$$ The line field $$L^{\theta}$$ is again tangent to a foliation on $$M'$$. Besides the horizontal foliation ($$\theta=0$$), one frequently uses the vertical foliation ($$\theta=-1$$). The horizontal and vertical foliations are orthogonal to each other with respect to the flat metric on $$M'$$ defined by $$\omega$$.