Developments in non-relativistic field theory and complexity

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!