Bordisms between 9d type IIB supergravities and commutator widths of duality groups

This paper investigates the topological properties of bordisms between 9d type IIB supergravities, revealing that the increasing complexity of gravitational solitons for large monodromies motivates a refinement of the Swampland Cobordism Conjecture which links the necessity of infinite duality defects to the divergence of the commutator width in duality groups.

The Gist

The Big Picture: The "Landscape" of Universes

Imagine the universe as a vast, complex landscape. In the world of string theory, there isn't just one version of physics; there are millions of different "vacuum states" or versions of reality, each with its own rules. These are called Effective Field Theories (EFTs).

The authors of this paper are studying a specific neighborhood in this landscape: 9-dimensional universes created by taking our familiar 10-dimensional string theory and curling up one dimension into a tiny circle (like a garden hose).

The Problem: Connecting the Islands

In this landscape, different universes can have different "twists" in their geometry. Imagine two islands. One island has a road that loops around a mountain once; another has a road that loops around twice. In physics, these are called monodromies.

A major rule in Quantum Gravity, called the Swampland Cobordism Conjecture, says that no two valid universes should be permanently disconnected. If you have two different universes (or even a universe and "nothing"), there must be a physical process—a bordism—that allows you to travel from one to the other. Think of it as a bridge or a tunnel connecting two islands.

The paper asks: What do these bridges look like?

The Twist: The "Commutator" Game

The bridges in this theory are built using two main tools:

1. Defects (Brane Stacks): Imagine these as specific, heavy construction materials (like [p, q] 7-branes) that you can place on the map to change the rules of the road.
2. Gravitational Solitons (Topology Changes): Imagine these as the shape of the land itself. You can twist the ground, create a handle (like a donut hole), or change the shape of the bridge to accommodate the twist.

The authors discovered a mathematical game called the Commutator Game.

• In this game, you try to build a complex twist (a monodromy) by combining simple moves.
• A "commutator" is like a specific move: Do A, then B, then undo A, then undo B.
• Some twists can be built with just one or two of these moves.
• Others require a huge number of them.

The paper focuses on a group of rules called SL(2, Z). They found that for this group, the number of moves needed to build a complex twist can be arbitrarily large. This is called having an infinite commutator width.

The Discovery: The Bridge Gets Too Heavy

Here is the core conflict the paper identifies:

1. The "Lazy" Bridge (Gravitational Solitons): If you try to build a bridge between two universes with a very complex twist using only the shape of the land (topology), you have to use a massive number of commutators.

   • The Analogy: Imagine trying to build a bridge by folding a piece of paper. To make a complex knot, you have to fold the paper over and over again. If the knot is huge, you need a piece of paper so large and crumpled that it becomes a mountain.
   • The Result: The "bridge" (the gravitational soliton) becomes so topologically complex (it has a huge number of "handles" or genus) that it becomes incredibly heavy. In physics terms, the energy required to build this bridge is so high that the probability of it happening is zero. It is "arbitrarily suppressed."

2. The "Smart" Bridge (Defects/Brane Stacks): Alternatively, you can use the specific construction materials (the [p, q] 7-branes) to fix the twist.

   • The Analogy: Instead of folding the paper a million times, you just glue a specific, heavy metal plate (a brane) onto the road. It's a direct, efficient fix.
   • The Result: These bridges are much lighter and much more likely to exist.

The Main Conclusion: A New Rule for Nature

The authors propose a refinement to the Swampland Cobordism Conjecture.

The Old Idea: If a twist can be mathematically described as a product of commutators, then a gravitational bridge (a soliton) should exist to connect the universes.

The New Proposal: If the number of commutators needed to describe a twist is unbounded (infinite), then nature must provide a full spectrum of specific defects (branes) to connect these universes. You cannot rely on the "lazy" gravitational bridges because they become too heavy and impossible to form.

In simple terms: If a rule requires an infinite number of complex steps to fix, nature won't try to do it by twisting space itself. Instead, it will provide a specific "tool" (a brane) for every possible variation of that rule.

Testing the Theory

The authors tested this idea on other types of duality groups (other sets of rules for different dimensions and types of string theory):

• Groups with Finite Width: For some groups, the number of steps is limited. In these cases, gravitational bridges work fine, and you don't need a huge variety of defects.
• Groups with Infinite Width: For groups like SL(2, Z) (Type IIB string theory) and Mp(2, Z) (which includes fermions), the steps are infinite. The paper confirms that in these cases, the full spectrum of defects (all the different types of 7-branes) is indeed required to keep the theory consistent.

Summary

The paper argues that in the quantum gravity landscape, you can't always rely on "weird shapes of space" to connect different universes. If the mathematical complexity of the connection is too high (infinite commutator width), the universe forces itself to use specific physical objects (branes) to make the connection, otherwise, the connection would be so heavy it would never happen. This ensures that global symmetries are always broken and that the theory remains consistent.

Technical Summary

Technical Summary: Bordisms between 9d type IIB supergravities and commutator widths of duality groups

Problem Statement
The paper investigates the topological properties of bordisms interpolating between different 9-dimensional gauged supergravities derived from Type IIB string theory compactified on a circle ($S^1$) with a non-trivial $SL(2, \mathbb{Z})$ bundle. The central motivation stems from the Swampland Cobordism Conjecture, which posits that all cobordism classes in a consistent theory of Quantum Gravity (QG) must vanish. This implies that any two d-dimensional theories (or a theory and "nothing") must be connected by a dynamical process, often realized as a domain wall or a "bubble of nothing."

While the conjecture guarantees the existence of such interpolating configurations, it does not specify their topological complexity. The authors focus on the specific case where the boundary conditions are defined by monodromies in the duality group $G = SL(2, \mathbb{Z})$. They identify a tension: if the duality group has an infinite commutator width (meaning elements require an arbitrarily large number of commutators to be expressed), realizing these monodromies purely through gravitational solitons (topological changes of the bulk geometry) would require bordisms of arbitrarily large genus. The paper asks whether such arbitrarily complex geometries are physically viable or if they lead to a breakdown of the effective field theory (EFT) description, thereby challenging the standard interpretation of the Cobordism Conjecture.

Methodology
The authors employ a combination of algebraic topology, string theory compactification techniques, and Euclidean gravity calculations:

1. F-theory and Geometric Setup: They model the 9d gauged supergravities as arising from F-theory on a twisted 3-torus $T^3_M$ with monodromy $M \in SL(2, \mathbb{Z})$. The bordisms connecting these theories (or to nothing) are described as elliptically fibered 4-manifolds $B_4$ with base $\tilde{B}_2$ (a Riemann surface with punctures).
2. Homological Analysis: Using the Leray-Serre spectral sequence, they compute the integral homology of the bordism $B_4$ in terms of the genus $g$ of the base $\tilde{B}_2$ and the monodromies acting on the fiber. They relate the homology of the bordism to the coinvariants of the $SL(2, \mathbb{Z})$ action.
3. Algebraic Group Theory: The authors analyze the commutator subgroup $G' = [G, G]$ and the abelianization $G^{ab} = G/G'$. They utilize the concept of commutator width ($cl(G)$), defined as the supremum of the number of commutators needed to express any element in the commutator subgroup. They specifically examine $SL(2, \mathbb{Z})$, showing it has infinite commutator width.
4. Decay Rate Estimation: To assess the physical viability of these bordisms, they estimate the Euclidean action ($S_E$) for "Bubbles of Nothing" (BoN) that decay to nothing. They compare two channels:
   • Gravitational Solitons: Realizing monodromies via the topology of the base $\tilde{B}_2$ (genus $g > 0$).
   • Defects: Realizing monodromies via codimension-2 defects (stacks of $[p, q]$ 7-branes) on a trivial base.
     They calculate the scaling of the action with the genus $g$ and the spectral norm of the monodromy matrix, considering higher-curvature corrections and the breakdown of the EFT when the curvature exceeds the Quantum Gravity scale ($\Lambda_{QG}$).

Key Contributions and Results

• Topological Complexity of Bordisms: The authors demonstrate that for $SL(2, \mathbb{Z})$ monodromies, the genus $g$ of the gravitational soliton required to realize a monodromy $M$ (modulo abelian defects) grows linearly with the spectral norm $\|M\|_2$. Since $cl(SL(2, \mathbb{Z})) = \infty$, there exist monodromies requiring an arbitrarily large number of commutators, implying bordisms of arbitrarily large genus.
• Suppression of Decay Channels: They argue that for large genus $g$, the Euclidean action $S_E$ becomes arbitrarily large due to the contribution of the scalar kinetic terms (axio-dilaton) wrapping the fundamental domain of the duality group. This leads to an arbitrarily suppressed decay rate ($\Gamma \sim e^{-S_E}$). Such suppression implies an approximate global symmetry, which contradicts the "No Global Symmetries" conjecture in QG.
• Dominance of Defect Channels: In contrast, realizing the same monodromies via stacks of $[p, q]$ 7-branes on a trivial base results in a decay rate that is not arbitrarily suppressed. The tension of these defects scales linearly with the number of branes, which is much more favorable than the exponential suppression associated with large-genus solitons.
• Refined Cobordism Conjecture: Based on these findings, the authors propose a refinement of the Swampland Cobordism Conjecture for the first bordism group $\Omega_1(BG)$:

  If the commutator width of the duality group $G$ is infinite ($cl(G) = \infty$), consistency requires the existence of an infinite number of duality defects capable of killing elements in $G$, rather than just those associated with the abelianization $G^{ab}$.
  Without this full spectrum of defects, the theory would possess arbitrarily suppressed decay channels, effectively preserving a global symmetry.

• Testing on Other Groups: The proposal is tested against other duality groups:
  • $GL(2, \mathbb{Z})$ and $Mp(2, \mathbb{Z})$: These groups also have infinite commutator width. The authors show that while the full spectrum of defects is theoretically required, the specific monodromies relevant to 9d gauged supergravities can often be realized with low genus (e.g., $g \le 2$) if the full defect spectrum is available.
  • Maximal Supergravity (U-duality): For dimensions $D \le 7$, the U-duality groups (e.g., $SL(n, \mathbb{Z})$ for $n \ge 3$, $E_{n(n)}(\mathbb{Z})$) have finite commutator width. In these cases, the conjecture predicts that a finite set of defects (or purely gravitational solitons) suffices, which is consistent with the known properties of these groups.
  • 4d $\mathcal{N}=2$ Theories: The analysis extends to vector and hypermultiplet sectors, where monodromy groups involving free products and Heisenberg groups also exhibit infinite commutator width, reinforcing the need for a complete spectrum of axionic strings (duality defects).

Significance and Claims
The paper claims to provide a quantitative refinement of the Swampland Cobordism Conjecture by linking the topological complexity of bordisms to the algebraic property of commutator width. The primary significance lies in the argument that gravitational solitons alone are insufficient to satisfy the Cobordism Conjecture for groups with infinite commutator width without violating the "No Global Symmetries" principle via excessive suppression of decay rates.

The authors assert that their results explain why Type IIB string theory requires the full spectrum of $[p, q]$ 7-branes: not merely to satisfy charge quantization, but to ensure that the Cobordism Conjecture is realized via physically viable (non-suppressed) decay channels. They emphasize that this is a non-trivial constraint on the spectrum of defects in any consistent QG completion. The paper concludes modestly, noting that while the qualitative picture is robust, precise numerical factors in the decay rates and the exact bounds on commutator widths for specific U-duality groups require further investigation. They also suggest that their geometric framework could be extended to study junctions between multiple theories and higher-dimensional bordism groups.