Consistent subsectors of maximal supergravity and wrapped M5-branes

This paper introduces a new family of $D=4$ $\mathcal{N}=8$ gauged supergravities featuring supersymmetric AdS vacua related to wrapped M5-branes, while formalizing sufficient conditions for consistently truncating maximal supergravity to subsectors invariant under groups not necessarily contained within the original gauge group.

The Gist

Imagine the universe as a giant, intricate machine. Physicists try to understand how this machine works by building smaller, simpler models of it. One of the most powerful tools they have is Supergravity, a theory that tries to unify gravity with the other forces of nature.

This paper is about building a new, massive model of this machine and then figuring out how to safely take it apart to study specific, interesting parts without breaking the whole thing.

Here is the breakdown using simple analogies:

1. The "Maximal" Machine (The Big Picture)

Think of Maximal Supergravity as the ultimate, fully-loaded version of a video game. It has every possible character, every weapon, and every level. It's the most complete description of the universe we can imagine (in 4 dimensions).

• The Problem: This "Maximal" model is so huge and complex that it's impossible to play with directly. It's like trying to solve a puzzle with a million pieces when you only care about the picture of the sky.
• The Usual Trick: Physicists usually "truncate" (cut down) this big model into a smaller, manageable version. They say, "Okay, let's just look at the characters that look like this specific shape." Usually, they only keep parts that fit inside the rules of the original game.

2. The New Discovery: Breaking the Rules (The "Trombone" Twist)

The authors of this paper did something bold. They built a new family of these massive models (called $N=8$ gauged supergravities).

• The Twist: They mixed two different types of rules together. One part is standard, but the other part involves something called the "Trombone Symmetry."
• The Analogy: Imagine a trombone player. When they slide the tube, the pitch changes, but the instrument itself stretches. In physics, this "trombone" symmetry means the universe can stretch or shrink its scale. Usually, this is a global rule (applies everywhere). These authors "gauged" it, meaning they made it a local rule that changes from place to place.
• The Result: This creates a new, exotic type of universe model that had never been fully explored before. It's like building a car engine that runs on both gasoline and a new, unstable fuel.

3. The "Wrapped" M5-Branes (The Real-World Connection)

Why build this weird engine? Because it might explain M5-branes.

• The Analogy: Imagine the universe is made of tiny, vibrating strings (String Theory). Sometimes, these strings can clump together to form higher-dimensional sheets called Branes. An M5-brane is a 5-dimensional sheet.
• The Setup: The paper looks at what happens when these M5-branes are "wrapped" around a curved, donut-shaped (or more accurately, negatively curved) space.
• The Connection: The authors showed that their new, weird "Trombone" model is actually the perfect description of these wrapped M5-branes. It's like finding that the strange engine they built is exactly what powers a specific type of spaceship they've been trying to understand for years.

4. The "Consistent Subsectors" (The Safe Disassembly)

This is the most technical but also the most important part of the paper.

• The Old Way: To study a specific part of the big model, you usually had to cut it in a way that respected the original rules. If you tried to cut it a different way, the math would break, and the pieces wouldn't fit back together.
• The New Rule: The authors discovered a new set of instructions (mathematical conditions) that allow you to cut the big model in any way you want, even if the way you cut it doesn't seem to fit the original rules.
• The Metaphor: Imagine you have a giant, complex Lego castle.
  • Old Rule: You can only remove blocks that are the same color as the base. If you try to remove a red block from a blue section, the castle collapses.
  • New Rule: The authors found a secret glue. Now, you can remove any block you want, even if it's a different color, as long as you follow their specific "glue instructions." The castle stays standing, and you can study that specific red block without the whole thing falling apart.

5. The "Exotic" Results

When they applied these new rules to their Trombone model, they found some strange things:

• Massive Confusion: In normal physics, if you shake a system, the vibrations (masses) are predictable. But because of the "Trombone" stretching, some of these vibrations became complex numbers (mathematical weirdness involving imaginary numbers).
• The Interpretation: It's like hearing a sound that is both a note and a silence at the same time. The authors suggest this might be a sign that the model is telling us something deep about the geometry of the universe, or perhaps that some of these "vibrations" are just mathematical ghosts that disappear when you look at the full, higher-dimensional picture.

Summary

In short, this paper does three main things:

1. Invented a new, complex type of universe model that includes a "stretching" (Trombone) feature.
2. Proved that this model perfectly describes the physics of wrapped M5-branes (a key object in String Theory).
3. Created a new rulebook that allows physicists to safely study small, specific parts of these giant models, even when those parts don't look like they belong to the original set.

It's a bit like discovering a new type of engine, realizing it powers a specific spaceship, and then inventing a new wrench that lets you take the engine apart in ways nobody thought was possible, revealing hidden gears that were previously invisible.

Technical Summary

1. Problem Statement

The paper addresses two interconnected challenges in theoretical high-energy physics:

1. The Holographic Status of M5-branes: While the holographic duals for D3-branes (AdS$_5$/CFT$_4$) and M2-branes (AdS$_4$/CFT$_3$) are well-established, the description of M5-branes remains incomplete. Specifically, the six-dimensional $(2,0)$ superconformal field theory is not fully understood. However, wrapping M5-branes on special holonomy manifolds leads to lower-dimensional superconformal phases (e.g., $N=2$ Class S in 4D, $N=1$ phases). Existing models describing these phases are often "submaximal" (having less than maximal supersymmetry), making it difficult to compute complete spectra of operators.
2. Consistent Truncations in Maximal Supergravity: A standard method to study these systems is to embed submaximal models into a maximal ($D=4, N=8$) gauged supergravity. Traditionally, consistent truncations to invariant subsectors were assumed to require the invariance group ($G_S$) to be a subgroup of the parent theory's gauge group ($G$). The authors identify that this assumption is too restrictive; recent examples show truncations can be consistent even if $G_S \not\subset G$, but a general formalism for such cases was lacking.

2. Methodology

The authors employ a four-dimensional approach based on the embedding tensor formalism of $D=4, N=8$ gauged supergravity. Their methodology consists of three main pillars:

• Formalism for General Consistent Subsectors: They derive sufficient kinematic and dynamical conditions for truncating a maximal supergravity to a $G_S$-invariant subsector, where $G_S$ is a subgroup of the global duality group $SU(8)$ but not necessarily a subgroup of the gauge group $G$.

  • They establish that consistency requires specific linear and quadratic constraints on the embedding tensor ($\Theta, \vartheta$) when projected onto the $G_S$-invariant tensors.
  • They demonstrate that these constraints are frame-dependent: a truncation consistent in one duality frame (related by $E_{7(7)}$ transformations) may be inconsistent in another, even if the underlying physics is equivalent.

• Construction of a New Family of Supergravities ($TCSO$): They introduce a new class of $D=4, N=8$ gauged supergravities denoted as $TCSO(p, q, r; N)$.

  • These theories are mixtures of dyonic CSO gaugings (electric/magnetic) and a Scherk-Schwarz gauging involving a 3D Bianchi type $N$ group ($G_3$).
  • Crucially, when $G_3$ is non-unimodular, the theory involves the gauging of the trombone scaling symmetry ($\mathbb{R}^+$). This results in a theory that does not admit a Lagrangian description and must be formulated at the level of field equations.

• Application to M5-brane Configurations: They focus on a specific member of this family, $TCSO(5, 0, 0; V)$, which corresponds to M5-branes wrapped on negatively curved submanifolds (SLAG 3-cycles and associative 3-cycles). They explicitly construct the embedding tensor for this theory and identify two specific $N=2$ and $N=1$ subsectors that match previously known models derived from $D=11$ supergravity.

3. Key Contributions

A. Generalized Consistency Conditions

The paper formalizes the conditions under which a maximal supergravity can be consistently truncated to a $G_S$-invariant sector even when $G_S \not\subset G$.

• Constraints: They derive linear constraints ($C_1, C_2$) and quadratic constraints ($C_3$) involving the embedding tensor and $G_S$-invariant tensors.
• Duality Frame Dependence: A novel finding is that these constraints are not satisfied in all duality frames. For a given theory, a specific duality frame (often determined by higher-dimensional origin) must be chosen for the truncation to be consistent. This challenges the notion of "frame democracy" in the context of subtruncations.

B. New Trombone-Gauged Supergravities

The authors construct the $TCSO(p, q, r; N)$ family.

• Trombone Gauging: They show that non-unimodular Bianchi groups lead to the gauging of the trombone symmetry. This introduces an embedding tensor component in the $\mathbf{56}$ representation of $E_{7(7)}$, preventing a Lagrangian formulation.
• Structure: The gauge group is $(G_3 \times CSO(p,q,r)) \ltimes \mathbb{R}^{15}$.

C. M5-brane AdS Vacua and Mass Spectra

They analyze the $TCSO(5, 0, 0; V)$ theory, which relates to M5-branes wrapped on:

1. SLAG 3-cycles: Leading to an $N=2$ AdS vacuum.
2. Associative 3-cycles: Leading to an $N=1$ AdS vacuum.

They compute the full mass spectrum of these vacua within the maximal $N=8$ theory, extending previous results obtained in submaximal models.

4. Key Results

• Recovery of Known Models: The $N=2$ and $N=1$ subsectors of $TCSO(5, 0, 0; V)$ exactly reproduce the field content and vacua of the submaximal models found in references [26, 27], confirming that these models are consistent subsectors of a maximal theory.
• Exotic Mass Spectra: The mass spectra computed within the full $N=8$ theory exhibit features absent in submaximal models due to the trombone gauging:
  • Non-Symmetric Mass Matrices: Because the trombone gauging breaks the symmetry of the mass matrices, they are not guaranteed to be diagonalizable over the reals.
  • Jordan Blocks: The gauge field and scalar mass matrices exhibit generalized eigenvalues associated with Jordan blocks (defective matrices), indicating the presence of non-diagonalizable modes.
  • Complex Masses: In the $N=1$ associative 3-cycle vacuum, some scalar states possess complex masses (conjugate pairs). While these sit at the Breitenlohner-Freedman (BF) bound, their complex nature is unusual for supersymmetric vacua. The authors suggest these may be unphysical modes projected out by global topology (compactification of the hyperboloid $H^3$) in the full M-theory uplift.
• Symmetry Breaking Patterns:
  • In the $N=2$ vacuum, the residual symmetry is $OSp(4|2) \times U(1)_S$. Notably, the $U(1)_S$ symmetry (Cartan of the invariance group) is not a symmetry of the vacuum in the sense of being a subgroup of the gauge group, yet the spectrum organizes into representations of this larger group.
  • In the $N=1$ vacuum, the gauge group breaks to $SO(3)^+ \subset SO(5)$, preserving $OSp(4|1) \times SO(3)^+ \times U(1)'_S$.

5. Significance

• Holographic Completeness: By embedding M5-brane related vacua into a maximal $N=8$ framework, the paper provides a pathway to compute the complete spectrum of operators (including those missed by submaximal truncations) for the dual CFTs. This is crucial for understanding the large-$N$ limit of these theories.
• Theoretical Framework: The derivation of consistency conditions for $G_S \not\subset G$ truncations resolves a long-standing ambiguity. It provides a rigorous toolkit for constructing consistent subsectors in maximal supergravity without relying on the restrictive assumption that the invariance group must be part of the gauge group.
• Trombone Physics: The work highlights the physical consequences of trombone gauging, specifically the loss of Lagrangian descriptions and the emergence of non-diagonalizable mass matrices. This opens new avenues for studying non-standard supergravity phases and their stability.
• M-Theory Connection: The paper bridges the gap between 4D effective field theories and 11D M-theory configurations, offering a concrete realization of wrapped M5-brane geometries within the maximal supergravity formalism.

In summary, this paper advances the understanding of maximal supergravity by generalizing the theory of consistent truncations and applying it to construct a new class of trombone-gauged theories that successfully describe the holographic duals of wrapped M5-branes, revealing exotic spectral features inherent to these maximal models.