How do I deal with a quantum field in the denominator? I am wondering how to deal with an expression like
$$ \int d^4\theta \frac{1}{T + T^\dagger} \big( \dots \big) $$
If the denominator was of the form $1 + T + T^\dagger$, I could assume that $T \ll 1$ and expand the denominator in a Taylor series.
If more context helps, this expression pops up in $5D$ SuGra abelian gauge theory (http://arxiv.org/abs/hep-th/0106256, eq. (5) on page 2).
The authors of the above mentioned paper assume the modulus to be stable ($<T> \equiv R$) before performing the superspace integration. I do not want to do this and keep the dependence on the modulus.
 A: I would like to give you some general idea about the integration procedure of class of operators, just to have an idea on how to handle these mathematical objects (what is permitted and what does not make sense). The key relations, the ones completely determining your operator, are summarized in (1) below.
In general, if you have a class of operators $\{A(s)\}_{s\in S}$, with $A(s) : D \to H$ for some common domain $D \subset H$ ($H$ being the Hilbert space of the theory), and with $S$ given by some set equipped with a positive measure $\mu$, it is possible to define an integrated operator:
$$\int_S A(s) d\mu(s)$$
with the following steps.
Assuming that, for every $\phi\in  D$, the map
$$S \ni s \mapsto ||A(s) \phi||  $$
is $\mu$ integrable
and that for every $\phi\in  D$ and $\psi \in H$, the map $$S \ni s \mapsto \langle \psi | A(s) \phi \rangle$$
is measurable, then
$$H \times D \ni (\psi,\phi) \mapsto Q(\psi, \phi):= \int_S \langle \psi | A(s) \phi \rangle d\mu(s)\:.$$ is well defined, since 
$$|\langle \psi | A(s) \phi \rangle| \leq ||\psi||  ||A(s) \phi||\: $$
It is simply proved that $Q(\cdot, \cdot)$ is linear in the right-hand slot and it is anti-linear in the left-hand one, moreover:
$$|Q(\psi, \phi)|\leq C||\psi|| \:.$$
Riesz' theorem easily implies that, for every $\phi\in D$ there is a unique  vector, indicated by
$$\int_S  A(s) \phi  d\mu(s)$$
such that, if $\psi\in H$:
$$Q(\psi, \phi) = \left\langle \psi |\int_S  A(s) \phi  d\mu(s) \right\rangle \:.$$
By construction, since $Q$ is right-linear, one finds that
$$D \ni \phi \mapsto \int_S  A(s) \phi  d\mu(s)$$
is linear, too.
Summing up, under quite mild hypotheses, there exists a unique linear operator, indicated by $\int_S A(s) d\mu(s)$ and defined on $D$, such that:
$$\left\langle \psi |\int_S  A(s)  d\mu(s)  \phi \right\rangle
= \left\langle \psi |\int_S  A(s)\phi  d\mu(s)   \right\rangle 
=  \int_S \langle \psi | A(s) \phi \rangle d\mu(s)\tag{1}$$
for every $\phi \in D$ and $\psi \in H$. From these identities you can induce some properties from $A(s)$ to $\int_S  A(s)  d\mu(s)$. For instance, if the $A(s)$s are Hermitian, $\int_S  A(s)  d\mu(s)$ is. If $||A(s)|| <K <+\infty$ for some $K$ and all $s\in S$, then $\int_S  A(s)  d\mu(s)$ is bounded, and so on.
In your case $s=\theta$, I expect that all involved operators depend on $\theta$ in some way (the paper is quite obscure to me on these details),  and  you should use (1) to define the wanted operator: There is only one operator satisfying it. Obviously $1/(T+T^\dagger)$ has to be understood as $(T+T^\dagger)^{-1}$ (inverse operator).
