# example Worked Example A: Black Hole Counting as a Field-Theory Computation

## 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:** Medium

## RU instantiation hooks

- example host lane; RU connection becomes concrete when the microstate counting is expressed through a finite positive propagation or certified truncation of a protected sector.
- Use this doc to justify the reduction target; use the RU bridge examples to execute it.

## Purpose

This document is a worked demonstration of the “universal host suitability for RU instantiation” method-stack in its cleanest form:

- **a gravitational observable** (black hole entropy)
- is computed by **counting microscopic states** in a dual description
- using **protected structure** to justify transporting the result from weak coupling to strong coupling

The goal here is not to re-derive every formula from scratch. The goal is to make the logic auditable:

- What is the system?
- What is computed?
- Why is it legitimate?
- What matches, in what regime, and with what limitations?

## The claim being showcased

In certain supersymmetric settings, string theory allows black hole entropy to be computed by counting microstates in a brane worldvolume theory, with agreement with the Bekenstein–Hawking result.

Wray highlights precisely this style of host suitability for RU instantiation and explicitly flags that the cleanest arguments rely on supersymmetry.

## The system: a standard BPS black hole construction

A canonical example is the five-dimensional extremal black hole built from branes and momentum modes in Type IIB string theory compactified so that there is an effective five-dimensional gravitational description.

A common representative is the D1–D5–P system:

- **D1-branes** wrapped on a circle \(S^{1}\)
- **D5-branes** wrapped on \(S^{1}\times T^{4}\) (or \(S^{1}\times K3\) in a closely related version)
- **Momentum (P)** along the same \(S^{1}\)

The charges \((n_{1}, n_{5}, n_{p})\) are integers measuring the number of each ingredient.

The resulting object is:

- supersymmetric (BPS) in an appropriate limit
- an extremal black hole in the strong-coupling supergravity regime for large charges

## Two regimes and one protected bridge

The logic depends on comparing two regimes of the same underlying system.

### Weak coupling regime: branes as a field theory

At sufficiently weak string coupling:

- the gravitational backreaction is small
- the physics is well described by open-string degrees of freedom on the branes
- the low-energy dynamics reduces to a lower-dimensional quantum field theory (in this example, a 1+1 dimensional effective theory)

In this regime, counting states is a QFT problem.

### Strong coupling regime: a black hole geometry

At sufficiently strong coupling and for large charges:

- the system’s gravitational backreaction is large
- the appropriate description is a classical supergravity black hole solution (with controlled corrections when charges are large)

In this regime, entropy is computed from the horizon area.

### The bridge: BPS protection and indices

The key idea is that the object being counted is protected by supersymmetry.

The protected sector can be computed at weak coupling and compared to the gravitational entropy at strong coupling because the relevant count does not change under continuous variation of the coupling, as long as the BPS conditions and the appropriate stability conditions remain satisfied.

This is where the “protected quantities playbook” becomes operational.

## What is computed on the microscopic side

The microstate count reduces to counting BPS excitations in an effective 2D theory.

A standard summary structure is:

- The low-energy theory has a large number of degrees of freedom proportional to \(n_{1}n_{5}\).
- The relevant excitations contributing to the BPS sector are effectively chiral (one-sided) in a suitable limit.
- The growth of the number of states at large excitation number is governed by universal 2D CFT asymptotics.

### The central charge and the Cardy growth law

In the D1–D5 system, the effective 2D theory has central charge

- \(c \sim 6 n_{1} n_{5}\)

The momentum charge \(n_{p}\) controls the excitation level (often denoted \(N\) in the CFT language).

For large charges, the asymptotic density of states is governed by the Cardy formula, yielding an entropy scaling of the form

- \(S_{micro} = 2\pi \sqrt{n_{1} n_{5} n_{p}}\)

This is the key operational deliverable: an explicit entropy formula from a field-theory count.

## What is computed on the gravitational side

In the strong-coupling regime, one computes:

- a classical black hole solution carrying the same charges \((n_{1}, n_{5}, n_{p})\)
- its horizon area \(A\)
- the Bekenstein–Hawking entropy \(S_{BH} = A/4\) (in appropriate units)

For large charges, the supergravity solution yields an entropy with the same scaling:

- \(S_{BH} = 2\pi \sqrt{n_{1} n_{5} n_{p}}\)

The remarkable point is not the existence of an entropy formula. It is that the coefficient matches, not just the scaling.

## Why the match is meaningful

The match is meaningful because the two computations are done in radically different descriptions:

- a weak-coupling brane QFT count
- a strong-coupling gravitational area computation

The protection logic explains why comparing the two is legitimate.

### What is actually being compared

A careful statement is:

- the microscopic computation gives an asymptotic count of BPS states (often captured robustly as an index or an asymptotically equivalent protected count)
- the gravitational computation gives the leading macroscopic entropy for the corresponding extremal black hole
- the agreement is in the regime of large charges where the gravitational description is controlled

This is an “audit-ready” bridge. It does not require believing that string theory is the unique theory of nature. It demonstrates a working method for relating gravity to QFT.

## What is universal about this example

This example is not just a one-off curiosity. It establishes a template that has been reused in many contexts.

### Template: entropy as a controlled counting problem

- Build a gravitational object from branes/fluxes.
- Identify a weakly coupled description of the same object.
- Find a protected quantity (index or protected sector) that can be computed reliably.
- Use universal CFT growth or index technology to obtain asymptotic degeneracy.
- Compare to the macroscopic gravitational entropy in the controlled regime.

The “universal host suitability for RU instantiation” is the template, not the particular brane configuration.

### Where the same template reappears

- BPS black holes in different dimensions and compactifications
- refined microstate counts and subleading corrections
- holographic entropy matches and supersymmetric black holes in AdS settings
- duality checks where an index computed in one description matches a geometric quantity in another

## Limitations and scope flags

This example is strong precisely because its limitations are clear.

### The clean argument is supersymmetric

- The protected-count logic is strongest in BPS settings.
- Extending such computations to fully non-supersymmetric astrophysical black holes remains a major challenge.

### The match is asymptotic and regime-dependent

- The microstate derivation uses large-charge asymptotics.
- The gravitational side is controlled when curvature and quantum corrections are small relative to the classical solution.

### Subleading structure requires more tools

- Higher-derivative corrections, finite-charge effects, and refined degeneracies require more delicate machinery than the leading Cardy growth.

None of these limitations weaken the core takeaway. They clarify the domain where the method is genuinely operational.

## A compact “inputs → outputs” summary

| Component | Input | Output |
|---|---|---|
| Physical system | Brane charges \((n_{1}, n_{5})\) and momentum \(n_{p}\) in a compactification with an effective 5D gravity description | A BPS extremal black hole at strong coupling; a brane QFT at weak coupling |
| Microscopic computation | Effective 2D theory with large central charge \(c\sim 6n_{1}n_{5}\); excitation level tied to \(n_{p}\) | Asymptotic microstate entropy \(S_{micro}=2\pi\sqrt{n_{1}n_{5}n_{p}}\) |
| Macroscopic computation | Supergravity solution with horizon area determined by charges | Bekenstein–Hawking entropy \(S_{BH}=2\pi\sqrt{n_{1}n_{5}n_{p}}\) |
| Bridge | Supersymmetry protection of the relevant sector | Legitimacy of comparing weak-coupling count to strong-coupling geometry |

## Why this belongs in a “universal host suitability for RU instantiation” engine

This example earns its place because it shows a complete cycle:

- specify a system
- compute a hard gravitational observable
- using a tractable field theory method
- with explicit assumptions and explicit limits

That is exactly the kind of auditable host suitability for RU instantiation the overall engine is trying to showcase.