Spinning States and Unitarity in 3D Gravity

This paper proposes that negative densities of states in the 3D gravitational path integral can be cured by incorporating spinning states interpreted as bulk defects and overspinning BTZ geometries, which arise from mixed elliptic-hyperbolic identifications and exhibit causal pathologies despite being smooth pure gravity quotients of AdS$_3$.

The Gist

Imagine you are trying to bake the perfect cake (a theory of the universe) using a specific recipe (the laws of gravity in a 3D world). You follow the instructions carefully, but when you weigh the ingredients, something is wrong: the scale shows negative weight.

In physics, a "negative weight" (or negative density of states) is a disaster. It means the theory is broken, predicting impossible things like probabilities less than zero. This is exactly the problem physicists faced with 3D Gravity: a simplified model of our universe that, while mathematically elegant, keeps spitting out these "negative weights" in its calculations.

This paper by Ziyi Li is like a master baker trying to fix the recipe. The author proposes adding a new, strange ingredient to the mix: Spinning States.

Here is the breakdown of the paper using everyday analogies:

1. The Problem: The "Ghost" Ingredients

In the standard recipe for 3D gravity (called the MWK partition function), the math works great for most things. But when you look at two specific scenarios—very heavy objects (black holes) and objects spinning extremely fast—the math breaks down. It starts counting "ghosts" that have negative mass.

• The Analogy: Imagine a music playlist. Most songs are great, but suddenly, the playlist starts playing "negative silence." It's a glitch that ruins the whole experience.

2. The Solution: Adding "Spinning" Ingredients

To fix the glitch, the author suggests adding new "ingredients" to the universe. These aren't normal particles; they are objects that spin, and their spin is tied to the fundamental "size" of the universe (the central charge).

The author explores three types of these spinning ingredients:

A. The "Heavy Spinning Defects" (Sub-extremal & Extremal)

• What they are: Imagine a heavy point in space that is spinning so fast it creates a tiny, twisted knot in the fabric of space-time.
• The Catch: These knots are surrounded by a "time-loop zone" (Closed Timelike Curves). If you flew into this zone, you could theoretically travel back in time and meet your past self.
• Why use them? Even though time travel sounds scary and weird, the math shows that adding these spinning knots perfectly cancels out the "negative weights" in the recipe. They act like a counter-weight that balances the scale.
• The Verdict: They work, but they are a bit "messy" because they require a physical singularity (a point of infinite density) to hold them together.

B. The "Overspinning Black Holes" (The Smooth Fix)

• What they are: This is the paper's most exciting discovery. Usually, if a black hole spins too fast, it's supposed to be impossible or break apart. But the author found a way to interpret these "overspinning" objects as smooth, perfect geometries.
• The Analogy: Think of a spinning top. If it spins too fast, it usually flies apart. But imagine a top made of pure light that spins so fast it creates a new, stable shape that doesn't break. These are "Overspinning BTZ geometries."
• The Magic: Unlike the "defects" above, these don't have a messy knot in the middle. They are smooth everywhere. They are pure gravity, no extra matter needed.
• The Catch: Like the defects, they also have that "time-loop zone" where causality breaks down. But the author argues: So what? If the math works and the universe is consistent, maybe we can accept a little time-travel weirdness in the background.
• Bonus Feature: These objects have a "temperature" on one side (like a warm breeze) but not the other, behaving like a one-way thermal engine.

3. The "Time-Loop" Dilemma

The biggest hurdle is the Closed Timelike Curves (CTCs). In our real world, time travel is generally considered impossible.

• The Author's View: The paper argues that we should judge these geometries by their Euclidean form (a mathematical snapshot of the universe) rather than their Lorentzian form (the movie of time flowing).
• The Analogy: Think of a movie reel. The "Euclidean" version is just a single, perfect photograph of the scene. The "Lorentzian" version is the movie playing, where time moves forward. The author says, "Even if the movie has a glitch where a character walks backward in time, if the photograph is perfect and the math balances, we should still use it in our recipe."

4. The Result: A Balanced Recipe

By adding these spinning states (either the "knots" or the "smooth overspinning shapes"), the author shows that:

1. The negative weights disappear. The scale is balanced.
2. The spectrum of states (the list of all possible particles) becomes positive and healthy.
3. We can calculate how these objects interact (correlators) and get real, sensible answers, even if the math requires some complex "time-travel" tricks to solve.

Summary

This paper is a bold attempt to save a broken theory of gravity. The author says: "Our current recipe for the universe is missing a key ingredient. If we add these weird, spinning objects—which might allow for time loops but are mathematically beautiful and smooth—we can fix the broken math and get a consistent theory of quantum gravity."

It's a proposal to embrace the weirdness of the universe (spinning, time-loops, negative weights) to find a deeper, more consistent truth.

Technical Summary

1. Problem Statement

The paper addresses the issue of non-unitarity in the partition function of pure three-dimensional gravity with a negative cosmological constant (AdS$_3$), specifically the Maloney-Witten-Kraus (MWK) partition function.

• The Issue: The MWK partition function, constructed by summing over all $SL(2, \mathbb{Z})$ images of the thermal AdS$_3$ vacuum (Euclidean saddles), yields a density of states that is not positive definite.
• Specific Regimes of Negativity:
  1. Threshold Negativity: At the BTZ black hole threshold (near $M=0$), the regularized density of states exhibits a negative delta-function contribution ($-6\delta(t)$) and negative continuous terms.
  2. Large-Spin Negativity: In the near-extremal, large-spin limit ($|j| \to \infty$), the density of states becomes negative due to the alternating signs of Kloosterman sums in the Poincaré series expansion.
• Context: Previous attempts to cure these negativities involved adding scalar states (conical defects) or compact bosons. However, a recent proposal by [23] suggested adding spinning states with spin scaling linearly with the central charge ($J \sim O(c)$) to cancel both negativities simultaneously. The authors of [23] interpreted these states as classical spinning strings (matter sources).
• Goal: Li revisits this proposal, offering alternative bulk interpretations (pure gravity geometries vs. defects), analyzing their causal structures, and generalizing the computation of scalar correlators to these new backgrounds.

2. Methodology

The paper employs a combination of conformal field theory (CFT) techniques and bulk gravitational geometry analysis:

• CFT Analysis:
  • The author constructs the total density of states by adding "seed" spinning states to the MWK partition function.
  • The seed states are chosen to have specific conformal weights $(H, \bar{H})$ such that their Poincaré series contributions (involving Kloosterman sums and Bessel functions) exactly cancel the negative terms in the MWK spectrum at both the threshold and large-spin limits.
  • The author rigorously checks for positivity of the total density of states ($\rho_{total} = \rho_{MWK} + \rho_{added}$) across different regimes (finite $t$, $t \to 0$, and large $j$).
• Bulk Geometry Analysis:
  • The author interprets the added states as specific quotients of AdS$_3$ (generated by elements of $SL(2, \mathbb{R})$).
  • Sub-extremal/Extremal States: Interpreted as spinning defects (conical singularities supported by delta-function sources).
  • Overspinning States: Interpreted as overspinning BTZ geometries (smooth quotients with no fixed points, $|M| < |J|$).
  • Scalar Correlators: The paper computes scalar field propagators on these backgrounds. For overspinning geometries, this requires analytic continuation of the radial coordinate into the complex plane to handle the complex event horizons ($r_\pm$).

3. Key Contributions and Results

A. Cancellation of Negativities

The paper demonstrates that adding specific spinning states cures the MWK negativities without introducing new ones, provided certain constraints on the central charge $\xi = \frac{c-1}{24}$ and degeneracy $d$ are met.

State Type	Bulk Interpretation	Conformal Weights $(H, \bar{H})$	Constraints on $\xi$	Degeneracy Bound ($d$)
Sub-extremal	Spinning Defect ($|M| > |J|$)	$(\frac{19}{20}\xi, \frac{3}{4}\xi)$	$\xi \in 5\mathbb{Z}^+$	None (Any $d$ works)
Extremal	Extremal Defect ($|M| = |J|$)	$(\xi, \frac{3}{4}\xi)$	$\xi \in 4\mathbb{Z}^+$	None (Any $d$ works)
Overspinning	Smooth BTZ ($|M| < |J|$)	$(\frac{17}{12}\xi, \frac{3}{4}\xi)$	$\xi \in \frac{3}{2}\mathbb{Z}^+$	$d \lesssim 2.195$ (approx)

• Key Finding: Unlike the proposal in [23] which required specific integer constraints, the sub-extremal and extremal spinning defects allow for a broader range of central charges (though still quantized) and impose no upper bound on the degeneracy $d$.
• Overspinning Constraint: For overspinning states (which preserve the spectral gap), the degeneracy $d$ must be small ($d \lesssim 2.2$) to prevent the scalar density from becoming negative in the regime $\xi t \lesssim O(1)$.

B. Bulk Geometries and Causal Pathologies

• Spinning Defects (Sub/Extremal): These geometries possess conical singularities supported by matter sources. Crucially, the region surrounding the defect contains Closed Timelike Curves (CTCs). The author argues that despite these Lorentzian pathologies, they are valid Euclidean saddles.
• Overspinning Geometries: These are smooth, pure gravity solutions (no matter sources, no conical singularities) obtained by identifying AdS$_3$ via a mixed elliptic-hyperbolic generator.
  • They are free of fixed points in the real section.
  • However, they also exhibit causal pathologies (CTCs) in their Lorentzian continuation.
  • Thermality: Due to the hyperbolic component of the identification, these geometries possess a real right-moving temperature ($T_R$) and quasinormal modes, while the left-moving sector remains imaginary (similar to a defect). This makes them "partially thermal."

C. Scalar Correlators

The paper generalizes the computation of scalar correlators to these new backgrounds:

• Defects: The correlator is a sum over image geodesics, matching the heavy-light Virasoro vacuum block in the dual CFT.
• Overspinning: The computation is more complex. The radial coordinate must be analytically continued to the complex plane to define the mode sum.
  • The resulting propagator is real, despite intermediate complex steps.
  • The leading geodesic corresponds to the analytically continued Virasoro vacuum block.
  • The presence of quasinormal modes (from the hyperbolic sector) complicates the $s$-channel expansion, as OPE coefficients become complex.

4. Significance and Implications

• Pure Gravity Interpretation: The paper provides a strong argument that the "overspinning" states proposed to fix unitarity can be interpreted as pure gravity solutions (smooth quotients of AdS$_3$) rather than requiring exotic matter (like spinning strings). This supports the idea that pure 3D gravity might be consistent if one accepts these specific Euclidean saddles.
• Causal Pathologies: The work reinforces the viewpoint that causal pathologies (CTCs) in the Lorentzian continuation do not necessarily invalidate a geometry's contribution to the Euclidean path integral. This aligns with the treatment of $SL(2, \mathbb{Z})$ black holes, which also lack well-behaved Lorentzian interpretations in all sectors.
• Non-Uniqueness: The paper highlights a lack of uniqueness in the solution. Multiple types of spinning states (sub-extremal, extremal, overspinning) can cure the negativities, each imposing different quantization conditions on the central charge $c$. This suggests that the "correct" theory of 3D gravity might require additional principles (e.g., specific quantization of $\xi$) to select a unique spectrum.
• Thermodynamics of Overspinning: The discovery of real right-moving temperatures and quasinormal modes in smooth, horizonless overspinning geometries opens new avenues for understanding the thermodynamics of pure gravity and potential connections to JT gravity via dimensional reduction.

Conclusion

Ziyi Li's paper successfully demonstrates that adding spinning states with spin scaling with the central charge can render the 3D gravitational path integral unitary. By reinterpreting these states as smooth overspinning BTZ geometries and spinning defects, the author provides a "minimal matter" perspective. While these geometries introduce causal pathologies in Lorentzian signature, the paper argues they are admissible Euclidean saddles, offering a potential resolution to the long-standing unitarity problem in pure 3D quantum gravity.