  This thesis focuses on two research areas: non-relativistic field theories
and complexity. In the first part we review the general classification of the
trace anomaly for 2+1 dimensional field theories coupled to a Newton-Cartan
background and we apply the heat kernel method to compute the trace anomaly for
specific theories. We find a relation with the conformal anomaly of the 3+1
dimensional relativistic counterpart which suggests the existence of a
non-relativistic version of the a-theorem. We consider a model realizing a
$\mathcal{N}=2$ supersymmetric extension of the Bargmann group in 2+1
dimensions with non-vanishing superpotential, obtained by null reduction of a
relativistic Wess-Zumino model. We check that the superpotential is protected
against quantum corrections as in the relativistic parent theory, thus finding
a non-relativistic version of the non-renormalization theorem. We find evidence
that the theory is one-loop exact, due to the causal structure of the
non-relativistic propagator together with mass conservation. In the second part
of the thesis we review the holographic conjectures proposed by Susskind to
describe the time-evolution of the Einstein-Rosen bridge in gravity: the
complexity=volume and complexity=action. We investigate both the volume and the
action for black holes living in warped $\mathrm{AdS}_3$ spacetime. There exist
extensions of the proposals when the dual state from the field theory side is
mixed; we then analytically compute the subregion action complexity for a
general segment on the boundary in the BTZ black hole background, finding that
it is equal to the sum of a linearly divergent term proportional to the size of
the subregion and of a term proportional to the entanglement entropy. We also
find that mutual holographic complexity carries a different content compared to
mutual information. This means that entropy is not enough!
