Non-Lorentzian Supergravity from Matrix Theory

This paper demonstrates that the decoupling limits of matrix theory on D-particles give rise to non-Lorentzian supergravity, whose dynamics are linked to worldsheet current algebra anomalies and which, through D-particle backreaction, holographically connects to both Lorentzian IIA supergravity at large N and the null reduction of eleven-dimensional supergravity at moderately large N.

The Gist

Imagine the universe as a giant, complex machine. For decades, physicists have tried to understand how this machine works by looking at its smallest gears (quantum mechanics) and its biggest gears (gravity). Usually, these two worlds don't get along; they speak different languages.

This paper is about finding a secret translation manual between these two worlds, specifically for a special type of machine called "Matrix Theory."

Here is the story of the paper, broken down into simple concepts and analogies.

1. The Two Worlds: The Quantum Computer vs. The Gravity Universe

• The Matrix Theory (The Quantum Computer): Imagine a super-complex video game running on a computer. It's made of "D-particles" (tiny, point-like bits of energy) interacting with each other. This is a quantum mechanical system. It doesn't have gravity in the traditional sense; it's just numbers and matrices dancing around.
• The Supergravity (The Gravity Universe): This is the "real world" version of the game. It has space, time, and gravity. Usually, when you have a lot of these D-particles, they create a heavy gravitational field, warping space and time like a bowling ball on a trampoline. This is the standard "Lorentzian" universe we know (where time and space are mixed together).

2. The Problem: The "No-Geometry" Zone

The authors noticed something weird. If you zoom in on the quantum computer (Matrix Theory) and look at it under a specific set of rules (a "decoupling limit"), the usual rules of space and time break down.

In this zoomed-in view:

• Time is absolute: It flows like a river, one way, everywhere.
• Space is separate: It doesn't mix with time.
• Gravity is instant: There are no gravitational waves traveling at the speed of light. Instead, gravity acts like a telepathic connection; if you move a particle here, another particle instantly feels it over there.

This is called Non-Lorentzian Geometry. It's like a world where the "speed of light" is infinite. In this world, the usual map (the metric) doesn't work. You can't draw a standard grid. It's a "flat" world where time is a rigid ruler, and space is just a collection of points.

3. The Discovery: A New Kind of Gravity

The paper asks: If we strip away the "heavy" gravity that comes from having too many particles, what is left?

They found that even without the heavy gravity, there is still a "skeleton" of gravity left behind. They call this Non-Lorentzian Supergravity.

• The Analogy: Imagine a heavy, complex sculpture made of clay (the full universe with gravity). If you carefully chip away the outer layers until only the wireframe remains, you still have a shape, but it's very different. That wireframe is this new Non-Lorentzian gravity.
• The "Moderately Large" N: The authors realized this happens when you have a "moderately large" number of particles. Not so few that nothing happens, but not so many that they crush the universe into a black hole. In this "Goldilocks" zone, the universe behaves like this instant-gravity, non-relativistic system.

4. The Secret Code: The String Worldsheet

How do we describe the rules of this weird, instant-gravity world? The authors used a tool called Ambitwistor String Theory.

• The Analogy: Think of the universe as a movie. Usually, we study the movie by looking at the screen (the 3D world). But this paper suggests we should look at the film reel (the string worldsheet) to understand the rules.
• They found that the "glitches" or "anomalies" in the film reel (mathematical inconsistencies in the string theory) actually tell us exactly how this new gravity works. It's like finding that the errors in a code reveal the secret instructions for the program.

5. The Twist: Who Can Move the Furniture?

In this new Non-Lorentzian world, not everything can create gravity.

• The D-Particles (The Points): If you try to put a single point particle (a D-particle) in this world, it doesn't create a gravity field. It's like trying to make a dent in a steel plate with a needle; the plate is too stiff. The D-particles are "decoupled" from the gravity in this specific regime.
• The Extended Objects (The Sheets and Strings): However, if you put in a String (a 1D line) or a D4-brane (a 4D sheet), then the gravity reacts! These extended objects can bend this weird, instant-gravity space.

The Metaphor: Imagine a trampoline.

• If you drop a marble (D-particle) on a super-stiff, non-relativistic trampoline, it doesn't sink.
• But if you lay a heavy rope (String) or a sheet of fabric (D-brane) across it, the trampoline does bend.
• The paper maps out exactly how these ropes and sheets bend this strange, instant-gravity trampoline.

6. The Big Picture: Holography

The paper ties this all together with the Holographic Principle. This is the idea that a 3D world with gravity can be described by a 2D world without gravity (like a hologram on a credit card).

• The Conclusion: The authors show that this "weird, instant-gravity" world is the holographic dual of the "moderately large" Matrix Theory.
• If you have a few particles, you get the weird, instant gravity.
• If you add more particles (making N very large), the "instant" gravity warps and turns back into the normal, relativistic gravity (Einstein's gravity) that we are used to.

Summary

This paper is a guidebook for a parallel universe that exists inside our quantum theories.

1. It's a place where time is rigid and gravity is instant.
2. It's the "skeleton" left behind when you strip away the heavy gravity of a crowded universe.
3. It's described by a special kind of math found in the "film reel" of string theory.
4. In this world, point particles are invisible to gravity, but strings and sheets can still bend space.

It's a step toward understanding how the smooth, curved space-time of Einstein emerges from the chaotic, quantum jitters of the fundamental building blocks of the universe.

Technical Summary

Here is a detailed technical summary of the paper "Non-Lorentzian Supergravity from Matrix Theory" by Maskalaniec, Yan, and Zorba.

1. Problem Statement

The paper addresses the emergence of gravity from quantum mechanical systems, specifically focusing on the Banks-Fischler-Shenker-Susskind (BFSS) matrix theory, which conjectures that the large-$N$ limit of $N$ non-relativistic D-particles describes M-theory in asymptotically flat spacetime.

While the full holographic dual of the BFSS matrix theory at large $N$ is known to be Type IIA supergravity (a Lorentzian theory), the behavior of the theory in specific decoupling limits and at "moderately large" $N$ remains less understood. Specifically, the authors investigate:

• Whether a non-Lorentzian sector exists within the supergravity dual to the BFSS matrix theory.
• What the dynamics of this non-Lorentzian supergravity are.
• Whether there is a string worldsheet formalism (analogous to ambitwistor strings) that encodes these non-Lorentzian gravitational dynamics.
• How extended brane objects (other than D-particles) source this non-Lorentzian geometry.

2. Methodology

The authors employ a multi-faceted approach combining holographic duality, effective field theory expansions, and string worldsheet techniques:

• Holographic Expansion ($r_\omega$-expansion): They analyze the near-horizon geometry of D-particles in Type IIA supergravity. By introducing a dimensionless parameter $r_\omega$ (related to the speed of light or an infinite boost parameter $\omega$), they perform a large-$r_\omega$ expansion of the supergravity action. This isolates a leading-order sector that lacks a ten-dimensional metric description, instead requiring a Newton-Cartan formalism (vielbein fields).
• Decoupling Limits (M0T): They study the "Matrix 0-brane Theory" (M0T), a corner of Type IIA string theory obtained by zooming in on D-particle modes via a BPS decoupling limit. This limit effectively corresponds to the Discrete Light Cone Quantization (DLCQ) of M-theory.
• Ambitwistor String Techniques: To understand the origin of the non-Lorentzian dynamics, they utilize techniques from ambitwistor string theory. They argue that the equations of motion for non-Lorentzian supergravity arise from the cancellation of anomalies in the current algebra of a curved $\beta\gamma$-system (a chiral string worldsheet theory).
• Source Analysis: They systematically derive the coupling of various extended objects (F-strings, D4-branes, NS5-branes) to the truncated non-Lorentzian supergravity to determine which objects can source non-trivial curved solutions without breaking the non-Lorentzian structure.

3. Key Contributions and Results

A. Identification of Non-Lorentzian Supergravity

The authors demonstrate that the leading-order term in the large-$r_\omega$ expansion of Type IIA supergravity yields a non-Lorentzian supergravity action ($S_{NL}$).

• Geometry: The target space is described by Newton-Cartan geometry with a temporal vielbein $\tau_\mu$ and spatial vielbeins $e^i_\mu$. It possesses an absolute time direction and lacks a Lorentzian metric.
• Symmetries: The action is invariant under Galilei boosts and an emergent dilatation symmetry.
• Relation to DLCQ: This truncated theory is shown to be equivalent to the null reduction of eleven-dimensional supergravity.

B. The "Moderately Large $N$" Regime

A crucial distinction is made between the "large $N$" limit (where D-particle backreaction creates a full Lorentzian AdS-like geometry) and a "moderately large $N$" regime ($N \ll r_\omega$).

• In this regime, the D-particles are decoupled at the leading order.
• Consequently, the D-particles do not backreact to deform the non-Lorentzian geometry into a Lorentzian one.
• The holographic dual at this order is a classical non-Lorentzian supergravity on a flat background, representing the leading-order contribution of weakly coupled bulk gravity.

C. Worldsheet Origin and Anomalies

The paper provides evidence that the dynamics of this non-Lorentzian gravity are encoded in the current algebra anomalies of a fundamental string worldsheet theory.

• By analyzing the T-dual frame (Matrix (-1)-brane Theory), they connect the system to a chiral string (ambitwistor string).
• The requirement that the constraint algebra of the worldsheet theory remains closed (anomaly-free) at the quantum level yields the equations of motion for the target space non-Lorentzian supergravity. This involves a curved $\beta\gamma$-system where the vanishing of higher-order poles in the Operator Product Expansion (OPE) dictates the gravitational dynamics.

D. Brane Dynamics and Curved Solutions

The authors investigate which extended objects can source non-trivial curved solutions within this non-Lorentzian framework:

• D-particles: They cannot source the non-Lorentzian supergravity at leading order; their backreaction forces a deformation back to full Type IIA supergravity (restoring Lorentzian geometry).
• F-strings and D4-branes: These objects form 1/4-BPS states with the D-particles. Their tension terms scale as $O(r_\omega)$, allowing them to source the non-Lorentzian supergravity.
  • They derive explicit curved non-Lorentzian geometries for F-string solitons and D4-branes.
  • For example, a probe D-particle moving orthogonally to an F-string background experiences an attractive potential $\sim v^2/r^6$, matching scattering amplitudes in the full theory.
• Other Branes: They show that for general $Dp$-branes, only specific configurations (like $D(p+4)$-branes) preserve the non-Lorentzian structure. Non-BPS configurations (like D2-branes) induce torsion that destabilizes the non-Lorentzian limit, driving the system back to the relativistic theory.

E. Generalization to Other Holographies

The framework is generalized to:

• Dp-brane Holography: Extending the $r_\omega$ expansion to Matrix $p$-brane theories (MpT), identifying non-Lorentzian supergravities dual to these gauge theories.
• String Soliton Holography: Applying the analysis to matrix string theory (dual to F-strings), identifying a non-Lorentzian NS5-brane solution.

4. Significance

• Emergent Geometry: The paper provides a concrete realization of how non-Lorentzian geometry emerges as a low-energy effective description of quantum mechanical systems (matrix theories) before the full Lorentzian spacetime is reconstructed via large-$N$ backreaction.
• New Holographic Duals: It establishes a precise holographic dictionary for a "moderately large $N$" regime, linking truncated non-Lorentzian supergravity to the leading-order contributions of matrix theories.
• Worldsheet-Gravity Connection: It strengthens the link between ambitwistor string theory and non-Lorentzian gravity, suggesting that non-relativistic gravitational dynamics can be derived from chiral string worldsheet anomalies.
• Tractable Quantum Gravity: By isolating a sector where the fundamental degrees of freedom (D-particles) decouple but extended objects (F-strings, D4-branes) still interact, the authors identify a regime of quantum gravity that is simpler than full M-theory but retains rich non-trivial dynamics (curved solutions, scattering amplitudes).

In summary, the paper successfully constructs a consistent non-Lorentzian supergravity as the holographic dual to a specific sector of matrix theory, derives its equations of motion from string worldsheet anomalies, and maps out the landscape of curved solutions sourced by BPS extended objects.