Path integral quantization of tensionless bosonic strings with Carroll-Weyl ghosts

This paper revisits the path integral quantization of tensionless bosonic strings by demonstrating that treating Carroll-Weyl scaling as a genuine local gauge symmetry necessitates extending the standard BMS $bc$ ghost system to a $bcs$ system, thereby fundamentally altering the BRST complex and the conditions for anomaly cancellation.

The Gist

Imagine you are trying to take a perfect photograph of a very strange, invisible object: a tensionless string. In physics, a "string" is usually thought of as a tiny, vibrating piece of rubber. But a tensionless string is like a piece of rubber that has lost all its stretchiness; it's completely limp and floppy.

For decades, physicists have tried to take a "quantum photograph" of this limp string using a method called Path Integral Quantization. Think of this method as a way of summing up every possible way the string could wiggle to figure out how it behaves.

However, there's a catch. The string has a lot of "redundant" ways it can wiggle that don't actually change its physical state. It's like trying to count the number of ways a shadow can move on a wall when the object casting it hasn't moved at all. To get a clear picture, you have to "fix" these redundancies. In the old way of doing this, physicists used a specific set of mathematical tools called ghosts (not scary ghosts, but invisible mathematical variables that cancel out the redundant movements).

The Problem: A Missing Piece of the Puzzle
The authors of this paper, Sarthak Duary and Sourav Maji, realized that the old method was missing a crucial piece of the puzzle. They found that the "worldsheet" (the 2D surface the string sweeps out) has a hidden symmetry called Carroll-Weyl scaling.

To use an analogy: Imagine you are trying to measure a room.

• The Old Method: You fixed the length of the walls (diffeomorphisms) and the angle of the corners (Weyl scaling). You thought you had fixed the room completely.
• The New Discovery: The authors realized that in this specific "Carrollian" universe, you can also stretch or shrink the entire volume of the room without changing its shape, and this is a separate, independent rule. The old method ignored this rule.

Because they ignored this rule, the old "ghost" system was incomplete. It was like trying to lock a door with a key that only has two teeth when the lock actually needs three.

The Solution: The "bcs" Ghost System
The paper argues that to get the math right, you need to add a third "ghost" to the mix.

• Old System: Had two ghosts, named b and c.
• New System: Adds a third ghost named s.

The authors call this the bcs system.

• The b and c ghosts handle the usual movements of the string.
• The new s ghost (and its partner bs) handles the "Carroll-Weyl scaling"—the stretching of the volume.

Why This Matters (The "Mixing" Effect)
The most interesting part of the paper is how these ghosts talk to each other. In the old system, the ghosts were like two separate teams working in different rooms. In this new system, the new ghost s and the old ghost b are in the same room and they constantly bump into each other.

The paper shows a specific mathematical term, $-2sb_0$, which represents this interaction. It's like a gear mechanism where turning one gear (the scaling) forces the other gear (the time movement) to turn. This interaction wasn't there before because the old method didn't account for the scaling symmetry.

The Big Picture: A New Rulebook
Because of this new ghost, the "rulebook" for the string changes:

1. The BRST Charge: This is the master equation that ensures the theory makes sense. The old master equation is now incomplete; it needs a new term to account for the s ghost.
2. The Anomaly Problem: In string theory, if the math doesn't add up perfectly, the theory "breaks" (an anomaly). The old calculation said the theory works in 26 dimensions. The authors show that this calculation was only checking half the rules. Now that the full rulebook (including the s ghost) is in place, the check for 26 dimensions is just a "partial" check. We don't know the final answer yet; we have to redo the math with the new ghost included.

Summary
Think of the tensionless string as a complex machine. For years, physicists tried to fix it using a wrench (the bc ghosts). The authors of this paper found a hidden bolt (the Carroll-Weyl symmetry) that was loose. They realized that to fix the machine properly, you need a new tool, a screwdriver (the s ghost), and that this screwdriver is tightly connected to the wrench.

They haven't fixed the machine's final destination (the critical dimension) yet, but they have written the correct instruction manual for how to fix it. They proved that the old manual was missing a chapter, and without that chapter, the machine might not work at all.

Technical Summary

Technical Summary: Path Integral Quantization of Tensionless Bosonic Strings with Carroll-Weyl Ghosts

Problem Statement
The paper addresses a gap in the path integral quantization of the tensionless (null) bosonic string, specifically within the ILST (Isaacson-Lindström-Sundborg-Taylor) formalism. While previous treatments successfully identified the BMS$_3$ (Bondi-Metzner-Sachs) $bc$ ghost system as the Faddeev-Popov determinant for fixing diffeomorphism redundancies, they treated the Carrollian worldsheet geometry as having only these gauge symmetries. The authors argue that a Carrollian worldsheet possesses an additional, genuine local gauge symmetry: volume-preserving Carroll-Weyl scaling. This symmetry, generated by the constraint $C_3 = P \cdot X$, was previously omitted in the path integral measure. Consequently, the standard BMS $bc$ ghost system represents a partially gauge-fixed theory rather than the fully covariant quantum theory. The omission of this symmetry leads to an incomplete BRST complex and an incomplete formulation of the anomaly problem.

Methodology
The authors apply the rigorous Faddeev-Popov quantization procedure to the Carroll-Weyl gauged null string action.

1. Gauge Fixing: They define a gauge slice that fixes both the temporal gauge of the Carroll vector density ($V^a = (1, 0)$) and the Carroll-Weyl connection ($W = V^a W_a = 0$). This requires fixing three gauge conditions: $G_0 = V^0 - 1$, $G_1 = V^1$, and $G_s = W$.
2. Faddeev-Popov Operator: They derive the Faddeev-Popov operator $M$ by computing the variation of these gauge conditions under the combined infinitesimal diffeomorphisms ($\epsilon^a$) and Carroll-Weyl scalings ($\lambda$). Unlike the tensile string (2 rows) or the previous BMS treatment (2 rows), this operator is a three-row matrix acting on the gauge parameters $(\epsilon^0, \epsilon^1, \lambda)$.
3. Ghost System Construction: The determinant of this $3 \times 3$ operator is exponentiated using Grassmann variables. This introduces a new set of ghosts: the standard BMS ghosts $(c^0, c^1)$ and a new fermionic scalar ghost $s$ (and corresponding antighosts $b_0, b_1, b_s$).
4. Action Derivation: The authors explicitly derive the local ghost action $S_{bcs}$ by integrating the operator $M$ against the ghost fields, carefully handling off-diagonal terms and Grassmann signs.
5. Algebraic Analysis: They analyze the residual symmetry equations, mode expansions, and the resulting constraint algebra, demonstrating how the new ghost sector couples to the existing BMS sector.

Key Contributions and Results

• The $bcs$-Ghost System: The central result is the identification of the correct ghost system for the Carroll-Weyl covariant theory as a $bcs$ system: $(b, c) \to (b, c, s)$. The ghost $s$ corresponds to the Carroll-Weyl scaling parameter.
• Revised Faddeev-Popov Operator: The operator $M_{CW}$ is derived as:
  $$M_{CW} = \begin{pmatrix} -\frac{1}{2}\partial_0 & \frac{1}{2}\partial_1 & -1 \\ 0 & -\partial_0 & 0 \\ 0 & 0 & -\partial_0 \end{pmatrix}$$
  The off-diagonal entry $-1$ (in the $M_{0s}$ position) is crucial; it arises because the temporal component $V^0$ transforms under Carroll-Weyl scaling.
• Ghost Action: The gauge-fixed action is derived as:
  $$S_{gf} = \frac{1}{2\pi} \int d^2\sigma \left[ \dot{X}^2 + i \left( c^0 \partial_0 b_0 - c^1 \partial_1 b_0 + 2c^1 \partial_0 b_1 + s \partial_0 b_s - 2s b_0 \right) \right]$$
  The term $-2s b_0$ is highlighted as a non-removable algebraic mixing term. It reflects the non-trivial bracket between the Carroll-Weyl generator and the supertranslation constraint.
• Equations of Motion and Modes: The equations of motion for the ghosts are modified. Specifically, the evolution of the temporal ghost $c^0$ and the antighost $b_s$ are coupled to the new sector:
  $$\partial_0 c^0 - \partial_1 c^1 + 2s = 0, \quad \partial_0 b_s - 2b_0 = 0$$
  This coupling ensures that the $s, b_s$ sector is not decoupled from the BMS temporal ghosts.
• Extended BMS Algebra: The constraints close into an extended BMS algebra containing a weight-one current $S_n$ (associated with $C_3$). The algebra includes non-trivial brackets:
  $$\{L_m, S_n\} = in S_{m+n}, \quad \{M_m, S_n\} = -2 M_{m+n}$$
  This confirms that the Carroll-Weyl symmetry is not a spectator but actively mixes with the BMS generators.
• BRST Structure: The BRST charge must be extended to include the ghost $s$ and the constraint $C_3$. The authors state that the standard $D=26$ critical dimension check based solely on the BMS $bc$ ghosts is a calculation in a partially gauge-fixed theory. The full anomaly cancellation condition must be re-evaluated for the extended algebra including the $s, b_s$ sector.

Significance and Claims
The paper claims to provide the foundational step for the consistent quantization of tensionless strings by ensuring the path integral measure accounts for the complete gauge orbit of the Carrollian worldsheet.

• Correction of Previous Formalism: The authors assert that the omission of the Carroll-Weyl ghost in previous literature meant the path integral was not divided by all local gauge orbits. The $bcs$ system is the "missing measure degree of freedom."
• Impact on Physical States: The physical state conditions must now include $S_n |\Psi\rangle = 0$ (for $n \geq 0$), which enforces the constraint $P \cdot X = 0$ at the quantum level.
• Future Directions: The paper outlines that this result necessitates a re-evaluation of vertex operators, scattering amplitudes, and the critical dimension. It suggests that the $bcs$ system serves as the "minimal bosonic backbone" for any future Carroll-Weyl covariant quantization of tensionless superstrings, where the ghost sector would further extend to $(bcs) \oplus (\beta\gamma\delta)$.

The authors do not claim to have solved the critical dimension problem or computed the full spectrum; rather, they claim to have established the correct gauge-fixed action and BRST structure required to perform such calculations consistently.