Developments in non-relativistic field theory and complexity

This thesis focuses on the investigation of two broad research areas: non-relativistic field theories and holographic 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 also review the heat kernel method, which is used to study one-loop effective actions and then allows to compute anomalies for a given theory.We apply this technique to extract the exact coefficients of the curvature terms of the trace anomaly for both a non-relativistic free scalar and a fermion, finding 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 on which we comment. We continue the analysis of non-relativistic free scalar and fermion with the heat kernel method by turning on a source for the particle mass: on this background, we find that there is no gravitational anomaly, but the trace anomaly is not gauge invariant.We then consider a specific model realizing a 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.Moreover, we find strong 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 gravitational theories: the complexity=volume and complexity=action.These quantities may be used as a tool to investigate dualities, and we investigate both the volume and the action for black holes living in warped AdS_3 spacetime, which is a non-trivial modification of usual AdS_3 with non-relativistic boundary isometries.In particular, we analytically compute the time dependence of complexity finding an asymptotic growth rate proportional to the product of Hawking temperature and Bekenstein-Hawking entropy.In this context, there exist extensions of the holographic proposals when the dual state from the field theory side is mixed, i.e. we consider only a subregion on the boundary.We study the structure of UV divergences, the sub/super-additivity behaviour of complexity and its temperature dependence for warped black holes in 2+1 dimensions when the subregion is taken to be one of the two disconnected boundaries.Finally, we 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 termproportional to the entanglement entropy.While this result suggests a strong relation of complexity with entanglement entropy, we find after investigating the case of two disjoint segments in the BTZ background that there are additional finite contributions: then the previous elegant structure holds only for the divergent parts.