Reference for Quantum field theory in $S^1 \times R$ I am looking for a reference where it is considered QFT on a space given in a circle, plus a time coordinate, namely QFT in $S^1\times R$.
 A: Quantum field theory on a cylinder? I think you are talking about conformal field theory in two dimensions. You can learn almost everything about this huge field from the book Conformal Field Theory by Philippe Francesco. Conformal field theory is a quantum field theory that has conformal invariance. 
If you are interested in quantum field theories that has the ordinary Poincare symmetry, then you may be interested in integrable models in two dimensions, such as Sine-Gordon model and Sinh-Gordon model in two dimensions. These theories are called integrable because they have an infinite number of conserved charges. You can find detailed introductions from Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics by Giuseppe Mussardo.
A: You should check out basically any introductions to string theory. QFT on $S^1 \times R$ is exactly what you study when looking at closed strings.
Ibanez and Uranga have published a Text book that I liked - you could also start by checking out these lecture notes:
https://www.thphys.uni-heidelberg.de/~weigand/Skript-strings11-12/Strings.pdf
A: For completeness I mention an alternative approach to quantum field theories on a cylinder ($S^1 \times \mathbb{R}$). They can be equivalently formulated as worldline theories.
Let us consider a generic scalar field theory
$$ S=\frac{1}{4\pi}\int d^2\sigma \sqrt{g}\, g^{ab}\left( \partial_a\phi\partial_b\phi+4\pi V(\phi) \right) $$
If we put the theory on a cylinder of radius $r$ by choosing $\sigma_1\in[0,2\pi r], \sigma_2\equiv\tau\in[0,1]$, by fixing the metric to
$$
g_{ab}=\begin{pmatrix}1 & 0\\ 0 & T^2
    \end{pmatrix},
$$
and by expanding $X$ as
$$
 X(\sigma_1,\sigma_2)=\sum_{n\in \mathbb{Z}}X_n(\sigma_2)e^{ \frac{i n\sigma_1}{r}},
$$
then we obtain, in the low energy regime, a worldline theory of particles  described by the action
$$
S(X_0,X_n)=\int_0^1 d\tau\, \left(\frac{r}{2T}\dot{X}_0+V(X_0)\right)+\sum_{n=1}^{+\infty}\int_0^1d\tau\, \left\{\frac{r}{T}|\dot{X_n}|^2+m_n^2|X_n|^2\right\}
$$
These worldline theories have some universal features, independent of the form of the potential.
More details can be found in this paper:
Dogaru, A.I., Delgado, R.C. Cylinder quantum field theories at small coupling. J. High Energ. Phys. 2022, 110 (2022)
