Developments in non-relativistic field theory and complexity
Abstract
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 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 AdS3 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!