Holographic Fisher Information Metric in Schr\"odinger Spacetime

In this paper we study the Fisher Information Metric on the space of the coupling constants on both sides of the duality between non-relativistic dipole field theories and string theory in Schr\"odinger spacetime. We consider the following setup. In the gauge theory side, one can deform a given conformal field theory by a proper scalar operator and compute the Quantum Information Metric via the two-point correlation function between two such operators. On the string side, the deformation corresponds to a scalar field probing the background. In the large $N$ limit of the theory the probing can be done without backreaction on the original spacetime, thus one can construct a perturbative scheme for the calculation of the dual Holographic Fisher Information Metric as shown by \cite{Trivella:2016brw}. In particular, we prove that the Holographic Information Metric reduces to its dual quantum counterpart on the boundary of the Schr\"odinger background, thus showing exact match between the two quantities on both sides of the correspondence.