  Symmetry Breaking is used as an "underlying principle", bringing different
features of QFT to the foreground. However, the understanding of Symmetry
Breaking that is used here is quite different from what is done in the
mainstream: Symmetry Breaking is understood as the solution set of a given QFT,
its vacuum manifold, or, more modernly, its Moduli Space. Distinct solutions
correspond to different sectors, phases, of the theory, which are nothing but
distinct foliations of the vacuum manifold, or points in the Moduli Space (for
all possible values of the parameters of the theory). Under this framework,
three different problems will be attacked: "Mollifying QFT", "Topological
Transitions and Geometric Langlands Duality" and "Three-dimensional Gravity and
its Phase Transitions". The first makes use of the Moduli Space of the theory
in order to construct an appropriate mollification of it, rendering it viable
to simulate a QFT in Lorentzian spaces, tackling the "sign problem" heads-on.
The connections with Lee-Yang zeros and Stokes Phenomena will be made clear.
The second will show that each different phase has its own topology which can
be used as Superselection Rule; moreover, the Euler Characteristic of each
phase gives it quantization condition. The mechanism via which several
dualities work will also be elucidated. The last one will generalize a
0-dimensional QFT, via dimensional construction through its D-Module, and
conjecture several connections between the Lie-algebra-valued extension of the
Airy function and the recent Partition Function found for three-dimensional
gravity with a negative cosmological constant. These three problems, put
together, should exhibit a solid and robust framework for treating QFT under
this new paradigm.
