  By means of an appropriate re-scaling of the metric in a Lagrangian, we are
able to reduce it to a kinetic term only. This form enables us to examine the
extended complexified solution set (complex moduli space) of field theories by
finding all possible geodesics of this metric. This new geometrical standpoint
sheds light on some foundational issues of QFT and brings to the forefront
non-perturbative core aspects of field theory. In this process, we show that
different phases of the theory are topologically inequivalent, i.e., their
moduli space has distinct topologies. Moreover, the different phases are
related by "duality transformations", which are established by the modular
structure of the theory. In conclusion, after the topological structure is
elucidated, it is possible to use the Euler Characteristic in order to
topologically quantize the theory, in resonance with the content of the
Atiyah-Singer Index theorem.
