Holographic Fidelity Susceptibility

TL;DR

The paper addresses holographic fidelity susceptibility for deformations by a relevant operator in a theory with a gravitational dual. By computing the backreaction of a massive scalar on the bulk metric in Einstein gravity and identifying fidelity susceptibility with the volume difference of extremal time slices, it extends the complexity/volume proposal to relevant deformations. The main result is the leading UV-divergent contribution $\delta C_\Delta$ for a deformation of dimension $\\Delta$, which scales as $\\delta C_\\Delta\\sim-\\frac{L^{2\\Delta-d-2}}{\\epsilon^{2\\Delta-d-2}}\\int d^d y\\sqrt{h^{(0)}}\\phi_0^2$ with a coefficient $-\\frac{1}{64(2\\Delta-d-2)G_N}$, and exhibits a logarithmic divergence at $\\Delta=(d+2)/2$; the authors also generalize to Lifshitz geometries with dynamical exponent $z$, yielding $G_\\Delta\\sim\\epsilon^{-2\\Delta+d+2z}$. They further discuss the thermodynamic (heat capacity) interpretation for thermal states, noting agreement up to an order-one factor and special behavior in $d=1$, and acknowledge that finite, scheme-dependent terms require the full backreacted geometry.

Abstract

For a field theory with a gravitational dual, we study holographic fidelity susceptibility for two states related by a deformation of a relevant operator. To do so, we study back reaction of a massive scalar field on the asymptotic behavior of the metric in an Einstein gravity coupled to a massive scalar field. Identifying the two states, holographically, with the original and back reacted geometries, the corresponding holographic fidelity susceptibility is given by the difference of the volume of an extremal time slice evaluated on the original and back reacted geometries.