Holographic duality with a view toward many-body physics

TL;DR

McGreevy provides a pedagogical tour of holographic duality with a focus on many-body physics, introducing the GKPW prescription that equates CFT generating functionals with bulk gravity actions in asymptotically AdS spaces. The notes explain how large-N limits render the bulk gravity classical and how operator dimensions Δ arise from bulk masses via m^2 L^2 = Δ(Δ−d), including BF and unitarity constraints. They then develop the bulk computation of vacuum and finite-temperature correlators, real-time holography, and the bulk-to-boundary propagator, extending to n-point functions and geodesic limits, before addressing finite temperature, finite density, and hydrodynamic transport such as η/s = 1/(4π) in Einstein gravity. The work culminates with discussions of other observables like entanglement entropy, the role of supersymmetry, and pragmatic guidance for applying AdS/CFT to strongly coupled many-body systems, highlighting both universality and UV sensitivity of certain observables. Overall, the notes present a coherent framework for using holography to study strongly correlated quantum matter, emphasizing the emergence of bulk gravity as a tool for understanding real-time dynamics, transport, and entanglement in strongly coupled field theories.

Abstract

These are notes based on a series of lectures given at the KITP workshop "Quantum Criticality and the AdS/CFT Correspondence" in July, 2009. The goal of the lectures was to introduce condensed matter physicists to the AdS/CFT correspondence. Discussion of string theory and of supersymmetry is avoided to the extent possible.

Figures (18)

• Figure 1: The extra ('radial') dimension of the bulk is the resolution scale of the field theory. The left figure indicates a series of block spin transformations labelled by a parameter $z$. The right figure is a cartoon of AdS space, which organizes the field theory information in the same way. In this sense, the bulk picture is a hologram: excitations with different wavelengths get put in different places in the bulk image. The connection between these two pictures is pursued further in Swingle:2009bg. This paper contains a useful discussion of many features of the correspondence for those familiar with the real-space RG techniques developed recently from quantum information theory.
• Figure 2: This diagram consists of $4$ three point vertices, $6$ propagators, and $4$ index loops
• Figure 3: planar graphs that contribute to the vacuum$\rightarrow$vacuum amplitude.
• Figure 4: Non-planar (but still oriented!) graph that contributes to the vacuum$\rightarrow$vacuum amplitude.
• Figure 5: Direct surfaces constructed from the vacuum diagram in (a) Fig. \ref{['fig:vac_planer']}a and (b) Fig. \ref{['fig:vac_nonplaner']}.