⚛️ high-energy theory

BMS $_3$ modules from path integrals on the coadjoint orbits

This paper presents a systematic path integral construction of quantum modules for Virasoro and BMS $_3$ coadjoint orbits, demonstrating how appropriate integration contours yield convergent Euclidean half-line integrals that define reference states and their descendants, with computed characters confirming consistency between state counting and geometric action methods.

Original authors: Pujian Mao, Xin-Cheng Mao

Published 2026-09-21

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CC BY 4.0

Original authors: Pujian Mao, Xin-Cheng Mao

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the study of gravity, the universe is often imagined as a vast, smooth fabric stretching through space and time. But when scientists look at the very edges of this fabric, where the universe meets the void, they find something surprising: the boundary itself seems to hold a life of its own. In three-dimensional universes, the bulk of space contains no moving waves of gravity; instead, all the physical action is compressed onto this boundary. This boundary is not empty; it vibrates with patterns of symmetry that act like a hidden language describing the state of the entire universe. Physicists call these patterns "asymptotic symmetries," and they are crucial for understanding how gravity works in the simplest possible settings, much like how studying the surface of a sphere helps us understand the sphere itself.

For decades, researchers have tried to translate these boundary patterns into a quantum mechanical description, essentially asking what the "notes" of this cosmic music sound like when played at the smallest scales. A major challenge has been dealing with certain patterns that seem unstable, where the energy of the system could theoretically drop infinitely low. Standard methods often discard these unstable patterns as unphysical, assuming they cannot exist in a real universe. However, a new study by Pujian Mao and Xin-Cheng Mao suggests that this dismissal might be premature. By using a sophisticated mathematical tool called a path integral—a method that sums up every possible way a system could evolve—they have shown that even these seemingly unstable patterns can be described consistently and clearly.

The researchers focused on two specific groups of symmetries that govern the boundaries of three-dimensional universes: one related to curved space and another related to flat space. They developed a systematic way to build a "module," which is a mathematical collection of all possible quantum states that can arise from a specific boundary pattern. Think of a module as a complete family tree of states, starting with a single reference state and branching out into all its possible variations or "descendants." The team constructed these families for every constant pattern found in these symmetry groups. Their most significant finding is that they could define these families even for the patterns that were previously considered too unstable to exist. By carefully choosing the mathematical path used to calculate the system's behavior, they found a way to make the calculations converge, proving that these unstable-looking patterns actually correspond to well-defined quantum states.

This work clarifies the relationship between the geometry of space and the quantum states that inhabit it. The authors found that for the flat universe case, the quantum states they constructed fall into two distinct categories. One category matches what was already known: states that behave like massive particles or empty space. The other category, however, is entirely new. It describes states with zero mass but non-zero angular momentum, arising from a specific type of symmetry that had not been fully explored before. This new category of states is induced from a non-trivial representation of a smaller symmetry group, meaning it has a more complex internal structure than the standard empty space or massive particle states.

The researchers also verified their results by counting the number of states in each family and comparing this count to a different calculation method involving the geometry of the path integral. Both methods produced identical results, confirming the accuracy of their construction. This agreement holds true even for the patterns with unbounded energy, where the calculations are most difficult. The study demonstrates that the apparent instability of these gravitational patterns is not a fundamental flaw but rather a feature that depends on how one chooses to look at the mathematical path. By showing that these patterns yield finite, consistent results, the paper suggests that the "unstable" sectors of gravity might be just as real and important as the stable ones, potentially offering new insights into the nature of flat space and the holographic principle, which posits that the information of a volume of space can be encoded on its boundary.

In the broader context of theoretical physics, this work provides a rigorous bridge between classical geometry and quantum mechanics for three-dimensional gravity. It confirms that the mathematical structures describing the boundary of the universe are robust enough to handle even the most extreme cases. The authors also noted that their approach could be extended to four dimensions, where similar symmetry groups describe the edges of our own universe. While the current study is confined to three dimensions, the methods developed here offer a clear roadmap for exploring the quantum nature of boundaries in more complex, realistic universes. The findings do not overturn existing theories but rather expand the landscape of what is possible, ensuring that no potential state of the universe is left behind simply because it looks unstable at first glance.

Technical Summary: BMS3 Modules from Path Integrals on Coadjoint Orbits

Problem Statement
In gravitational theories with boundaries, physical degrees of freedom are often encoded in the asymptotic symmetry group (ASG). For three-dimensional gravity, the ASG is either the Virasoro group (AdS $_3$ ) or the BMS $_3$ group (flat space). A central challenge in the quantization of these systems is constructing the quantum modules (representations) associated with the coadjoint orbits of these groups. While Virasoro coadjoint modules for constant orbits are well-understood as highest-weight representations (HWRs), the construction of BMS $_3$ modules is more subtle due to the semi-direct product structure of the group, which intertwines supermomentum and super-angular momentum.

Existing approaches, such as the Wigner construction of induced representations, successfully describe massive and vacuum sectors but face difficulties with orbits where the Hamiltonian is unbounded from below. Furthermore, induced representations do not always uniquely correspond to coadjoint orbits, and characters alone cannot distinguish between different infinite-dimensional representations (e.g., HWRs vs. induced representations) that may yield identical partition functions. There is a need for a systematic, direct procedure to construct modules starting from a specific coadjoint orbit, particularly for sectors with unbounded Hamiltonians where standard ground-state projections fail.

Methodology
The authors propose a systematic construction of quantum modules via path integral quantization of the geometric action defined on the coadjoint orbits. The methodology proceeds as follows:

Geometric Action and Quantization: The classical phase space is identified with the coadjoint orbit equipped with the Kirillov–Kostant symplectic form. The geometric action serves as the kinetic term. The authors perform perturbative quantization restricted to the one-loop order.
Handling Unbounded Hamiltonians: A key technical innovation addresses orbits where the Hamiltonian is unbounded from below (leading to negative modes in the quadratic expansion). Instead of failing to define a ground state via a standard $\beta \to \infty$ limit, the authors demonstrate that an appropriate choice of integration contour in the complex plane of the dynamical variables (Darboux coordinates) can render the Gaussian path integral convergent. This contour choice depends on the sign of the coefficients in the quadratic form.
Reference State Definition: The reference state (the "vacuum" of the module) is defined by a Euclidean half-line path integral ( $t \in (-\infty, 0]$ ). The convergence of this integral imposes regularity conditions on the reference state.
Annihilation Conditions and Stabilizers:
- The path integral quantization determines which generators annihilate the reference state.
- Generators corresponding to the stabilizer of the orbit (the little group) are treated as gauge redundancies and are quotiented out; they do not generate independent excitations.
- The remaining generators act as creation operators, generating the "descendants" of the reference state.
Character Computation: The character of the resulting module is computed in two ways to ensure consistency:
- State Counting: Directly counting the number of states generated by the descendants.
- Periodic Path Integral: Evaluating the path integral of the geometric action with a periodic Euclidean direction ( $t \sim t + \beta$ ), which naturally implements the trace structure.

Key Contributions and Results

Virasoro Group: The authors apply the formalism to the Virasoro group, recovering the known results for constant orbits.
- For orbits with stabilizer $u(1)$ , the resulting module is a standard highest-weight representation (HWR).
- For orbits with stabilizer $sl(k)(2, \mathbb{R})$ , the module is spanned by descendants excluding the $L_{\pm k}$ modes, consistent with previous literature.
- The characters derived via state counting match those from the periodic path integral.
BMS $_3$ Group: The formalism is extended to the BMS $_3$ group, covering all constant orbits classified by their stabilizers.
- Generic Orbits ( $B_n \neq 0$ ): The stabilizer is $\{L_0, M_0\}$ . The resulting module is induced from a $U(1)$ little group representation, reproducing the standard massive BMS $_3$ representations.
- Exceptional Orbits ( $B_k = 0, A_k = 0$ ): The stabilizer enlarges to include $L_{\pm k}, M_{\pm k}$ . The resulting module corresponds to the vacuum representation (induced from the trivial representation of $SL(k)(2, \mathbb{R})$ ).
- New Sector ( $B_k = 0, A_k \neq 0$ ): This sector features a Hamiltonian bounded from below but with a non-trivial little group $SL(k)(2, \mathbb{R})$ . The authors construct a new module where the reference state is annihilated by $M_n$ ( $n \neq 0$ ) and $L_k$ , but not by $L_{-k}$ . This corresponds to a module induced from a non-trivial representation of the little group $SL(k)(2, \mathbb{R})$ , distinct from standard massive or vacuum representations.
- Unbounded Hamiltonians: The authors successfully construct modules for orbits with Hamiltonians unbounded from below (conical excesses) by utilizing the specific integration contours that ensure convergence.
Consistency Checks: For all cases, the characters computed via state counting agree perfectly with those derived from the periodic path integral. This agreement holds even for sectors with unbounded Hamiltonians, validating the contour choice method.
Ultra-Relativistic (UR) Limit: The BMS $_3$ modules are shown to be recoverable from two copies of the Virasoro algebra via the ultra-relativistic limit ( $\epsilon \to 0$ ), providing a bridge between the AdS $_3$ and flat-space holographic descriptions.

Significance and Claims
The paper claims to provide a systematic and direct procedure for constructing coadjoint modules for both Virasoro and BMS $_3$ groups, bypassing the ambiguities of the Wigner construction.

Resolution of Unbounded Sectors: The work demonstrates that the apparent instability of gravitational saddles with unbounded Hamiltonians (often excluded from path integrals due to negative modes) may be an artifact of the integration contour. By choosing an appropriate contour, a finite, convergent path integral can be defined, suggesting these sectors are physically relevant and possess well-defined quantum modules.
One-Loop Exactness: For orbits with Hamiltonians bounded from below, the authors note that previous studies have established the one-loop exactness of the orbit path integral. Thus, their results for these sectors are claimed to be exact quantum results, not merely semiclassical approximations. For unbounded sectors, one-loop exactness remains an open question.
Distinction of Representations: The paper highlights that while different representations (e.g., induced vs. HWR) can yield identical characters (and thus partition functions), the path integral construction on the orbit reveals their structural differences, specifically regarding the action of the little group generators on the reference state.
Future Directions: The authors suggest that this formalism could be extended to four-dimensional flat gravity (BMS $_4$ ) and that investigating the one-loop exactness of the unbounded sectors is a promising direction for future research. They also comment on the relationship between these induced modules and Highest-Weight Representations in Carrollian CFTs, noting that while they can share characters, their structural definitions differ.

Technical Summary: BMS3 Modules from Path Integrals on Coadjoint Orbits

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