# Finite-Temperature Holography and Transport

## Overview

RU is a certificate engine for universality and obstructions in constrained nonnegative propagation. Host structures supply finite inputs. Outputs are audited certificates or finite obstruction witnesses.

## RU fit

- **Fit level:** Low

## RU instantiation hooks

- Transport computations are an important host story but are not, by themselves, an RU instance.
- Use this doc only as a pointer toward sectors where discrete operator data and positivity can be extracted.

## Purpose

This document explains a concrete, repeatable application: using the gravity side of AdS/CFT to compute **real-time response** and **transport coefficients** of strongly coupled quantum field theories at finite temperature.

The emphasis is not on claiming that these computations directly predict QCD or laboratory materials. The emphasis is on showing that string theory exports a **working calculation pipeline** for regimes where standard perturbation theory fails.

## What counts as an application here

A string-theory-derived method is being used when all of the following happen:

- A strongly coupled QFT observable is reformulated as a **boundary response function** (a retarded Green’s function).
- That response function is computed from a **classical gravitational problem** (linearized perturbations of a black brane geometry).
- The result is extracted by a fixed recipe that is the same across many theories, with clearly stated regime assumptions.

The canonical example is the shear viscosity to entropy density ratio, computed from horizon physics.

## The operational pipeline

The pipeline below is the key “host suitability for RU instantiation claim” in its most literal form: it is a recipe that can be executed and checked.

### Inputs

- A holographic pair (or a controlled model): a QFT state at temperature **T** that is represented by a black hole or black brane spacetime with Hawking temperature **T**.
- A boundary operator **O** whose response you want, and the corresponding bulk field **ϕ** (from the AdS/CFT dictionary).

### Steps

- Choose the bulk background describing the thermal state (often an AdS black brane).
- Linearize the bulk equations for fluctuations of the relevant field **ϕ** about that background.
- Impose **ingoing boundary conditions at the horizon** (this is what selects the retarded correlator).
- Solve the fluctuation equation and read off the boundary “source and response” coefficients near the asymptotic boundary.
- Assemble the retarded Green’s function **G_R(ω,k)** and extract transport coefficients using Kubo formulas.

This prescription for Minkowski-space correlators is standard and widely used.

## Why horizons matter

At finite temperature the bulk geometry contains a horizon, and dissipative physics on the boundary is encoded in how perturbations fall into that horizon.

Two practical consequences:

- Transport is often controlled by **near-horizon universality**. Many details of the UV completion drop out of the final transport coefficient.
- Real-time poles of correlators on the boundary correspond to **quasinormal modes** of the bulk black hole background, giving a clean physical interpretation of relaxation times.

## A example output: η/s from black hole physics

The shear viscosity **η** of the boundary theory can be extracted from the stress-tensor correlator via a Kubo formula. In a wide class of holographic theories whose gravity dual is described by Einstein gravity with minimal matter, the result is

- **η/s = 1 / (4π)**

This was shown using string-theory and black-hole methods and became a benchmark for “perfect fluid” behavior.

### Why this is a real export beyond string theory

Even when one does not believe the boundary theory is literally QCD, the computation:

- Demonstrates a concrete mechanism for very low viscosity in a controlled, strongly coupled setting.
- Provides a toolkit for computing whole families of transport coefficients that are hard to obtain otherwise.

This is a methodological application: a reproducible computational pipeline.

## What else you can compute with the same pipeline

The following are standard finite-temperature “deliverables” in holography:

- **Diffusion constants** for conserved charges (from current correlators).
- **Conductivity** and frequency-dependent response (from gauge-field perturbations).
- **Relaxation times** and spectral functions (from quasinormal modes).
- **Chern–Simons diffusion / topological rates** in certain settings.

## Use-case cards

| Boundary quantity | Bulk computation | Output type | Typical maturity | Main regime flags |
|---|---|---|---|---|
| Shear viscosity η | Metric shear perturbation at the horizon | A universal ratio η/s in Einstein gravity | Very solid in its regime | Large N, strong coupling, classical gravity |
| Charge diffusion D | Maxwell perturbations, ingoing at horizon | Diffusion constant and poles of G_R | Solid in its regime | Same |
| Conductivity σ(ω) | Gauge-field perturbations in black brane | Frequency-dependent response | Solid in models; details depend on matter content | Same + model choice |
| Relaxation time τ | Quasinormal mode spectrum | Decay rates, ringdown behavior | Solid in controlled duals | Same |

## Honest scope: what you must say out loud

These applications are real, but they are not regime-free.

The most common scope constraints are:

- **Large N and strong ’t Hooft coupling** are the clean classical-gravity limit.
- The best-controlled examples are often **conformal** or nearly conformal.
- Many “applied” models are **bottom-up** effective theories rather than fully derived from a specific string compactification.

None of these limitations destroy the host suitability for RU instantiation claim. They only clarify what kind of host suitability for RU instantiation is being asserted:

- A reliable strong-coupling computation method in a class of theories, not an automatic laboratory prediction.

## How this connects back to the supplied lecture notes

Kevin Wray’s notes emphasize AdS/CFT as a duality between a gravity theory in (d+1) dimensions and a boundary gauge theory in d dimensions, with a concrete generating-functional dictionary for correlators.

Finite-temperature transport is an application of that same dictionary, plus the additional thermal ingredient: the bulk dual includes a horizon.

## References to start with

- Son and Starinets, “Minkowski-space correlators in AdS/CFT correspondence: recipe and applications,” arXiv:hep-th/0205051.
- Policastro, Son, Starinets, “From AdS/CFT correspondence to hydrodynamics,” arXiv:hep-th/0205052.
- Kovtun, Son, Starinets, “Viscosity in Strongly Interacting Quantum Field Theories from Black Hole Physics,” arXiv:hep-th/0405231; PRL 94 (2005) 111601.