This dissertation covers topics in the intersection of high energy and condensed matter

physics. It is motivated by the question, ‘Given information about physics at the highest energy

scale, how does that constrain the theory at low energies?’ This is a difficult question as complexity

can be ‘emergent’, leading to a rich and unpredictable variety of possibilities.

In the first half of the thesis we discuss ‘symmetry protected topological phases’; states of

matter whose low-energy physics is described by an ‘invertible’ topological field theory. Such

theories encode ‘anomalies’ and imply exotic surface states when defined on a manifold with

boundary. We study a model in five dimensions whose anomalous boundary is electromagnetism,

but where the elementary electric and magnetically charged particles are fermions.

In the second half we discuss ‘non-relativistic conformal field theories’ at finite charge

density. One possibility for the low-energy physics of such systems is that of a superfluid

ground-state, realized experimentally in systems of ultra cold fermi gases. Additionally, such

theories have a ‘state-operator correspondence’ which relates their operator spectrum to states in

a harmonic trap. This enables us to use the field theory of the superfluid to calculate properties of

the operator spectrum systemically in the limit of large charge.