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arxiv: 2605.15276 · v1 · pith:JKKZ724Qnew · submitted 2026-05-14 · ✦ hep-th

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

Camilo las Heras , Ignacio Ruiz This is my paper

Pith reviewed 2026-05-19 15:32 UTC · model grok-4.3

classification ✦ hep-th
keywords bordismswampland cobordism conjectureduality groupscommutator widthtype IIB supergravity9d gauged supergravitiesmonodromiesgravitational solitons
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The pith

When the commutator width of a duality group diverges, infinitely many duality defects must be included.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines bordisms that interpolate between different 9d gauged supergravities obtained from type IIB string theory compactified on a circle with a nontrivial SL(2,Z) bundle. These bordisms realize the required monodromies either through stacks of [p,q] 7-branes or through gravitational solitons whose topology grows more complex as the monodromy becomes larger, resulting in arbitrary suppression of the bordism. This suppression conflicts with the quantum gravity expectation that global symmetries must be broken. The authors therefore propose a refinement of the Swampland Cobordism Conjecture for the first bordism group with a duality bundle G, arguing that an infinite number of duality defects is needed precisely when the commutator width of G diverges.

Core claim

We propose a refinement of the Swampland Cobordism Conjecture for the first bordism group Ω1(BG) with a G duality bundle. We argue that even if gravitational solitons can realize the monodromies associated with elements of the commutator subgroup of G, if the number of needed commutators is unbounded (in other words, the commutator width of G diverges) then an infinite number of duality defects realizing elements in G need to be included. We test this proposal for different duality groups G, and see that our expectations are realized, often in non-trivial ways.

What carries the argument

The refined Swampland Cobordism Conjecture for Ω1(BG) that uses the commutator width of the duality group G to decide whether a finite or infinite number of duality defects must be added to realize all monodromies.

If this is right

  • For duality groups whose commutator width remains finite, only a finite collection of duality defects is required to generate all monodromies.
  • Bordisms realized solely by gravitational solitons are suppressed when the required number of commutators grows without bound.
  • The refinement applies across the different duality groups that arise in 9d supergravities descending from type IIB.
  • Elements of the commutator subgroup can be realized by solitons, while elements outside this subgroup generally require explicit duality defects.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same logic may extend to higher bordism groups or to other global symmetries in quantum gravity.
  • Consistent string compactifications could be restricted to those duality groups whose commutator width is finite unless supplemented by infinite defect towers.
  • Explicit model-building in lower-dimensional supergravities with known duality groups offers a route to further tests of the refinement.

Load-bearing premise

The topology of bordisms that realize large monodromies through gravitational solitons grows arbitrarily complicated and therefore suffers arbitrary suppression that must be offset by adding infinitely many duality defects exactly when the commutator width diverges.

What would settle it

An explicit construction of a gravitational-soliton bordism that realizes an element outside the commutator subgroup or requiring arbitrarily many commutators while keeping the suppression factor finite and without adding extra duality defects.

read the original abstract

We study the topological properties of bordisms interpolating between different 9d gauged supergravities obtained from compactification of type IIB string theory on $\mathbb{S}^1$ with a non-trivial $\mathsf{SL}(2,\mathbb{Z})$ bundle. We describe how such bordisms implement the needed monodromies through stacks of $[p,q]$ 7-branes or gravitational solitons of non-trivial topology. For the later mechanism, we see that the topology of the bordism becomes increasingly complicated for large monodromies, which results in the associated bordisms being arbitrarily suppressed, against expectations on the breaking of global symmetries in Quantum Gravity. Motivated by this, we propose a refinement of the Swampland Cobordism Conjecture for the first bordism group $\Omega_1({\rm B}G)$ with a $G$ duality bundle. We argue that even if gravitational solitons can realize the monodromies associated with elements of the commutator subgroup of $G$, if the number of needed commutators is unbounded (in other words, the commutator width of $G$ diverges) then an infinite number of duality defects realizing elements in $G$ need to be included. We test this proposal for different duality groups $G$, and see that our expectations are realized, often in non-trivial ways.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies bordisms interpolating between 9d gauged supergravities from type IIB on S¹ with non-trivial SL(2,ℤ) bundles. Monodromies are realized either by stacks of [p,q] 7-branes or by gravitational solitons whose bordism topology grows more complex (higher genus, additional handles) for large monodromies, leading to arbitrary suppression. This observation motivates a refinement of the Swampland Cobordism Conjecture for Ω₁(BG) with a G-duality bundle: even if solitons realize commutator-subgroup elements, a diverging commutator width of G requires infinitely many duality defects realizing elements of G. The proposal is tested for several duality groups G, with expectations realized in non-trivial ways.

Significance. If the proposed refinement is substantiated, it would furnish a new algebraic-topological criterion for the inclusion of duality defects in quantum gravity, directly tying the commutator width of duality groups to the breaking of global symmetries. The explicit tests across multiple G provide concrete illustrations that could guide further swampland studies. The work usefully connects bordism theory with group-theoretic invariants, though its impact depends on making the suppression mechanism quantitative.

major comments (2)
  1. [Abstract / proposal for Ω₁(BG)] Abstract and the paragraph introducing the refined conjecture for Ω₁(BG): the inference that a diverging commutator width necessitates infinitely many duality defects (rather than finitely many but arbitrarily heavy ones) rests on the assertion of 'arbitrary suppression' from increasing bordism complexity. No explicit formula, bound, or scaling relation is supplied that converts a topological invariant (genus, number of handles, or other bordism data) into a suppression factor such as an instanton action or exponential weight that demonstrably diverges with commutator number.
  2. [Gravitational solitons and bordism topology] Discussion of gravitational solitons realizing monodromies: the statement that 'the topology of the bordism becomes increasingly complicated for large monodromies' is presented qualitatively. A concrete example or estimate showing how the required number of commutators maps to a quantifiable suppression that becomes unbounded would be needed to secure the central claim that this forces inclusion of infinitely many defects precisely when the commutator width diverges.
minor comments (2)
  1. [Introduction / notation] The definition and basic properties of commutator width for the relevant duality groups (e.g., SL(2,ℤ) and its extensions) should be recalled or referenced at the first appearance to aid readers unfamiliar with the group-theoretic notion.
  2. [Figures and bordism descriptions] Figure captions or the text describing the bordism constructions could explicitly label which features (handles, genus, etc.) correspond to additional commutators to make the complexity argument easier to follow.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comments point by point below. Revisions have been made to clarify the qualitative nature of our arguments where feasible.

read point-by-point responses

  1. Referee: [Abstract / proposal for Ω₁(BG)] Abstract and the paragraph introducing the refined conjecture for Ω₁(BG): the inference that a diverging commutator width necessitates infinitely many duality defects (rather than finitely many but arbitrarily heavy ones) rests on the assertion of 'arbitrary suppression' from increasing bordism complexity. No explicit formula, bound, or scaling relation is supplied that converts a topological invariant (genus, number of handles, or other bordism data) into a suppression factor such as an instanton action or exponential weight that demonstrably diverges with commutator number.

    Authors: We appreciate the referee's observation. The refined conjecture is motivated by the general expectation in quantum gravity that bordisms of increasing topological complexity (higher genus or additional handles) correspond to contributions that are arbitrarily suppressed in the path integral. We do not supply an explicit formula or scaling relation in this work, as a quantitative derivation of the instanton action or exponential weight from the bordism data would require a detailed effective-field-theory analysis of the 9d gravitational solitons that is beyond the present scope. The central claim remains that, for duality groups whose commutator width diverges, the suppression can be made arbitrarily strong, thereby necessitating infinitely many duality defects to realize all group elements without violating expectations on global symmetries. We have revised the abstract and the relevant introductory paragraph to state explicitly that the suppression argument is qualitative and to flag the need for future quantitative work. revision: partial

  2. Referee: [Gravitational solitons and bordism topology] Discussion of gravitational solitons realizing monodromies: the statement that 'the topology of the bordism becomes increasingly complicated for large monodromies' is presented qualitatively. A concrete example or estimate showing how the required number of commutators maps to a quantifiable suppression that becomes unbounded would be needed to secure the central claim that this forces inclusion of infinitely many defects precisely when the commutator width diverges.

    Authors: We agree that the discussion of gravitational solitons is presented qualitatively. The manuscript illustrates the trend for SL(2,ℤ) by noting that larger commutator widths require bordisms with higher genus or extra handles, which are expected to be more suppressed. No explicit numerical mapping or estimate is provided here, as constructing such a quantitative relation would demand concrete computations of soliton actions in the 9d theory. We have added a clarifying sentence and a reference to related literature on topological complexity and instanton suppression to make the qualitative reasoning more transparent, while acknowledging that a full quantitative treatment remains an open task. revision: partial

Circularity Check

0 steps flagged

No circularity; proposal is a motivated refinement with independent content

full rationale

The paper's chain consists of an observational claim about bordism topology growing complicated for large monodromies (leading to asserted arbitrary suppression) followed by a proposal to refine the Swampland Cobordism Conjecture for Ω1(BG) when commutator width diverges. This is presented as motivation rather than a closed derivation or first-principles result that reduces to its inputs by construction. No equations, fitted parameters renamed as predictions, or load-bearing self-citations appear in the provided text. The central refinement has independent content as a suggested adjustment to account for global symmetry breaking expectations, and the argument remains self-contained without reducing the conclusion to a tautology or prior author result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The proposal relies on standard bordism theory and properties of SL(2,Z) and other duality groups; no free parameters are introduced in the abstract. The refinement itself functions as an ad-hoc adjustment to the conjecture.

axioms (2)
  • domain assumption Bordisms can interpolate between 9d gauged supergravities with non-trivial SL(2,Z) bundles.
    Stated in the first sentence of the abstract as the setup for the study.
  • domain assumption Gravitational solitons realize monodromies via their topology.
    Abstract states this as one of the two mechanisms for implementing monodromies.
invented entities (1)
  • Refined Swampland Cobordism Conjecture for Ω1(BG) no independent evidence
    purpose: To account for suppression of bordisms when commutator width diverges by requiring infinite duality defects.
    Introduced in the abstract as the main proposal motivated by the soliton suppression observation.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A missing link: Brane networks and the Cobordism Conjecture

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    Defects tied to discrete symmetries via bordism groups Ω^ξ_2(BG) and homology H_2(BG;Z) are codimension-two branes that participate in networks with junctions, expanding the Cobordism Conjecture's predictions in strin...

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