Help with strange notation involving fractions of tensors

Question

I'm currently reading the paper Open strings in background gauge fields by Callan et.al. It is frequently used a notation that is not explained anywhere. If $F_{\mu\nu}$ is the electromagnetic field strength tensor, the author write two expressions:

$$\tag1 \left(\frac{1 - F}{1+F} \right)_{\mu \nu},$$

which firstly appears in eq. ($2.10$), and

$$\left(\frac{1}{1-F^2}\right)_{\lambda \nu} \tag2,$$ which appears for the first time in eq. ($2.14$).

Does anyone know the precise meaning of these expression and how to perform correct manipulations with it (such as derivatives)?

Answer 1

$f(F)^{\mu}{}_{\nu}$ refers to a component of the $(1,1)$ tensor $f(F)=\sum_{n=0}^{\infty}a_nF^n$, where components $F^{\mu}{}_{\nu}$ of the $(1,1)$ field-strength tensor is viewed as a matrix $F$ that can be multiplied and added together.

Here $f(x)=\sum_{n=0}^{\infty}a_nx^n$ is an analytic function, e.g. $f(x)=\frac{1-x}{1+x}$ or $f(x)=\frac{1}{1-x^2}$.

Finally, $f(F)_{\mu\nu}:=g_{\mu\lambda}f(F)^{\lambda}{}_{\nu}$.