General Actions of Extended Objects and Volume-Preserving Diffeomorphism

This paper establishes that general actions for extended objects depending on both worldvolume and induced metrics are classically equivalent to the Nambu-Goto action, demonstrating that volume-preserving diffeomorphism symmetry is as physically constraining as full diffeomorphism symmetry across Riemannian, areal, and volume-metric spacetimes.

The Gist

Imagine the universe as a giant, complex stage. In physics, we often study how tiny "actors" move across this stage. The most famous actors are strings (one-dimensional lines) and branes (multi-dimensional sheets). To describe how these actors move, physicists write down "scripts" called Actions. These scripts tell the actors how to behave, what energy they have, and how they interact with the stage.

For decades, physicists have had three different scripts for strings that seemed different but produced the exact same movie. This paper asks a big question: Are there any other scripts we missed? And do they all tell the same story?

Here is a breakdown of the paper's findings using simple analogies.

1. The Three Famous Scripts (The "Big Three")

The paper starts by looking at three well-known ways to describe a string:

• The Nambu-Goto Script: This is like measuring the actual area of a soap bubble. It's direct but mathematically messy to calculate.
• The Polyakov Script: This is like using a rubber sheet underneath the bubble. You have the bubble (the string) and a separate, invisible rubber sheet (an auxiliary metric) that helps you do the math. It's easier to work with.
• The Schild Script: This is a clever shortcut. Instead of measuring the area, it measures the square of the area. It's simpler but only works if you don't stretch the rubber sheet in certain ways.

The Discovery: The authors proved that if you write down the most general possible version of these scripts, they all collapse back into the same story. Whether you use the "rubber sheet" method or the "area" method, the physics is identical.

2. The "Volume-Preserving" Rule (The VPD)

One of the paper's biggest insights is about a specific rule called Volume-Preserving Diffeomorphism (VPD).

• The Analogy: Imagine a balloon filled with water. You can squish it, stretch it, and twist it into any shape you want, as long as the total amount of water inside stays exactly the same.
• The Finding: The authors found that this "keep the water volume constant" rule is actually just as powerful as the rule that says "you can do anything to the shape." Even though VPD sounds like a weaker, more restrictive rule, it turns out to be strong enough to force the string to behave exactly like the standard Nambu-Goto string. It's like discovering that if you only allow yourself to rearrange furniture without changing the room's total square footage, you still end up with the exact same living room layout as if you could move walls.

3. The "Areal Metric" Twist (New Geometry)

The paper then asks: What if the stage itself isn't made of normal geometry?

• Normal Geometry: We usually define distance (length) and then calculate area from that.
• Areal Geometry: Imagine a universe where you can define area directly, without needing to define length first. It's like having a ruler that measures "square inches" but has no "inch" markings.

The authors tried to write string scripts for this weird, area-only universe.

• The Result: Just like in the normal universe, the different scripts (Nambu-Goto vs. Schild) turned out to be equivalent.
• The Catch: However, when they tried to add a "quantum" version (the Polyakov script) to this area-only universe, it broke. It couldn't describe a "critical" string (a stable, vibrating string that fits our universe). It's like trying to build a house using only blueprints for the roof; the walls just don't hold up. This suggests that while area-only geometry is mathematically interesting, it might not be the right foundation for our actual universe's strings.

4. The "Universal Theorem" (The Magic Trick)

In the final section, the authors prove a general theorem that applies to strings, membranes, and even higher-dimensional objects.

• The Analogy: Imagine a chef cooking a soup. The recipe says, "Add ingredients until the flavor is perfect."
• The Theorem: The authors proved that if your recipe (the Action) respects the "volume-preserving" rule, the "flavor" (the tension or energy of the string) automatically adjusts itself to be a constant number. You don't need to force it; the math forces it to happen.
• Why it matters: This explains why all these different-looking scripts are actually the same. They all force the string to settle into a state where its "density" is constant, making them indistinguishable in the real world.

Summary: What's the Takeaway?

1. Unity: Whether you look at strings through the lens of area, volume, or auxiliary rubber sheets, they all tell the same physical story. The different mathematical tools are just different ways of looking at the same object.
2. Power of Constraints: A rule that seems restrictive (keeping volume constant) is actually powerful enough to define the entire physics of the string.
3. Limits of New Geometries: While we can imagine universes based purely on "area" or "volume" without standard length, our current attempts to put strings into those universes hit a wall. They don't seem to work as stable quantum theories without adding extra, unknown ingredients.

In short, the paper is a massive "unification" project. It says: "Don't worry about which mathematical script you use; they all lead to the same destination. But be careful if you try to change the geometry of the stage itself, because the actors might not know how to perform there."

Technical Summary

Here is a detailed technical summary of the paper "General Actions of Extended Objects and Volume-Preserving Diffeomorphism" by Ho, Kawai, and Liao.

1. Problem Statement

The paper addresses the classification and classical equivalence of actions for extended objects (strings and higher-dimensional branes) embedded in spacetime. Specifically, it investigates:

• Generalized Actions: What are the most general forms of worldsheet/worldvolume actions that depend on the induced metric and an auxiliary worldvolume metric, respecting specific symmetry groups: Volume-Preserving Diffeomorphisms (VPD), general Diffeomorphisms (Diff), and Diffeomorphisms with Weyl symmetry (Diff-Weyl)?
• Equivalence: Are these generalized actions classically equivalent to the standard Nambu-Goto (NG), Schild, and Polyakov actions?
• Generalized Geometries: How do these results extend when the background spacetime is not a standard Riemannian manifold but is equipped with an areal metric (for strings) or a volume metric (for higher-dimensional objects)?
• Quantum Consistency: Can a Polyakov-like action defined on an areal-metric background describe critical strings?

2. Methodology

The authors employ a systematic classical field theory approach involving the following steps:

• Construction of General Lagrangians: They construct the most general Lagrangian densities $L(\gamma_{ab}, h_{ab})$ (where $\gamma_{ab}$ is the induced metric and $h_{ab}$ is the auxiliary metric) that are invariant under the three specified symmetry groups. They utilize invariants formed from the matrix product $\Gamma H^{-1}$ (where $\Gamma$ and $H$ are the matrix forms of the metrics).
• Equations of Motion (EOM) Analysis: They derive the EOM for the auxiliary metric $h_{ab}$. By solving these equations, they express $h_{ab}$ in terms of $\gamma_{ab}$ (usually finding $h_{ab} = \phi \gamma_{ab}$ for some scalar factor $\phi$) and substitute this back into the action to eliminate the auxiliary field.
• Gauge Fixing and Equivalence Proofs:
  • For Diff and Diff-Weyl symmetries, they show the resulting actions reduce to the Nambu-Goto form.
  • For VPD symmetry, they derive a "Generalized Schild" action. They then prove that the EOM of the Generalized Schild action implies that the area/volume density is constant on the worldvolume, thereby establishing its equivalence to the standard Schild and Nambu-Goto actions.
• Generalization to Areal/Volume Metrics: They replace the standard Riemannian metric $g_{\mu\nu}$ with an areal metric $G_{\mu\nu\rho\lambda}$ (for 2D) or volume metric $V_{[\mu_1...\mu_d][\nu_1...\nu_d]}$ (for $d$-dimensions). They construct the corresponding NG and Schild actions in these backgrounds and prove their equivalence using similar EOM analysis.
• Perturbative Quantum Analysis: They analyze the Polyakov action perturbed by an areal-metric deformation. Using Conformal Field Theory (CFT) techniques, they check if the deformation operator is a primary field of weight $(1,1)$, which is a necessary condition for preserving conformal symmetry at the quantum level.
• Theorem on VPD: They formulate and prove a general theorem regarding actions invariant under simultaneous diffeomorphisms of dynamical fields and a background scalar density.

3. Key Contributions and Results

A. Equivalence of Generalized Actions on Riemannian Manifolds

• VPD Symmetry: The most general VPD-invariant action reduces to a Generalized Schild Action of the form $L = F(\sqrt{\gamma})$.
• Diff and Diff-Weyl Symmetry: The most general actions respecting these symmetries reduce to the Nambu-Goto Action ($L \propto \sqrt{\gamma}$).
• Main Conclusion 1: All non-trivial self-consistent actions (Generalized Schild, Generalized NG, Generalized Polyakov) are classically equivalent.
• Main Conclusion 2: VPD symmetry is as strong as full Diff symmetry in the context of classical equivalence. While VPD is a smaller symmetry group than Diff, the dynamical constraints imposed by the VPD-invariant action force the system to behave exactly like the Diff-invariant Nambu-Goto theory (specifically, by dynamically fixing the area density to a constant).

B. Extension to Areal and Volume Metrics

• Areal Metric Backgrounds: The authors define the Nambu-Goto and Schild actions using the areal metric $G_{\mu\nu\rho\lambda}$. They prove that the Generalized Areal Schild action is classically equivalent to the Areal Nambu-Goto action.
• Volume Metric Backgrounds: They generalize the concept to $d$-dimensional objects using volume metrics. They prove that Generalized Schild actions (defined via Nambu brackets) are classically equivalent to Volume-Nambu-Goto actions.
• Universal Theorem (Theorem 7.1): They prove that for any action $S[\phi; \rho]$ invariant under simultaneous diffeomorphisms of dynamical fields $\phi$ and a background scalar density $\rho$, the quantity $\delta S / \delta \rho$ is a spacetime constant on-shell. This theorem provides a universal mechanism explaining why the area/volume density becomes constant in Schild-type theories, ensuring equivalence to NG actions in arbitrary dimensions and metric backgrounds.

C. Quantum Consistency of Areal-Metric Deformations

• The authors investigate a Polyakov-like action perturbed by an areal-metric term (proposed in previous literature).
• Result: They demonstrate that this perturbation cannot describe critical strings. By analyzing the operator corresponding to the deformation in the conformal gauge, they show it fails to be a primary field of weight $(1,1)$ unless the deformation is trivial (zero). Thus, the conformal symmetry is broken at the quantum level without additional interaction terms.

D. Higher-Dimensional Objects

• The analysis extends naturally to $d$-dimensional worldvolumes. The equivalence between Generalized Schild and Nambu-Goto actions holds for any dimension $d$, provided the background is a Riemannian or volume-metric manifold. The tension in these theories is dynamically determined as an integration constant.

4. Significance

• Unification of String Actions: The paper rigorously establishes that the distinction between Schild, Nambu-Goto, and Polyakov actions is largely a matter of gauge choice and symmetry presentation. At the classical level, they describe the same physics, even when generalized to arbitrary functions of the metrics.
• Role of VPD: It highlights the surprising power of Volume-Preserving Diffeomorphisms. Despite being a subgroup of the full diffeomorphism group, VPD is sufficient to enforce the classical equivalence to the Nambu-Goto theory, suggesting that the "area-preserving" nature is the fundamental geometric constraint for strings.
• Constraints on Generalized Geometries: The finding that areal-metric deformations of the Polyakov action do not yield critical strings places a strong constraint on attempts to formulate string theory in non-Riemannian (areal) backgrounds. It suggests that such backgrounds require non-perturbative formulations or additional fields to be consistent.
• Theoretical Tool: The general theorem on VPD (Theorem 7.1) offers a powerful tool for analyzing theories with background scalar densities, with potential applications in unimodular gravity and cosmology (where the cosmological constant arises as an integration constant).

Summary of Conclusions

1. Classical Equivalence: Generalized Schild, Nambu-Goto, and Polyakov actions are classically equivalent on both Riemannian and Areal/Volume metric manifolds.
2. VPD Strength: VPD symmetry is dynamically equivalent to full Diff symmetry for these systems.
3. Quantum Obstruction: Simple areal-metric perturbations of the Polyakov action break conformal invariance and do not define critical strings.
4. Universality: The equivalence holds for extended objects of any dimension $d$ in spacetimes defined by volume metrics.