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

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 $Ω_1({\rm B}G)$ 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.

• The paper develops a refined Cobordism Conjecture linking the duality group's commutator width to the necessity of duality defects in 9d IIB supergravity.
• It employs advanced homological techniques and spectral sequences to classify topology-changing bordisms and quantify monodromy complexities.
• It concludes that for groups with infinite commutator width, such as SL(2,Z), gravitational solitons alone are insufficient, demanding a full spectrum of duality defects to break approximate global symmetries.

Bordisms, 9d Type IIB Supergravities, and Commutator Widths of Duality Groups

Overview and Motivation

This paper develops a sharp analysis of cobordisms between nine-dimensional (9d) type IIB gauged supergravities that arise from compactifications of ten-dimensional (10d) type IIB string theory on $\mathbb{S}^1$ with nontrivial $\mathsf{SL}(2,\mathbb{Z})$ duality bundles. The study addresses how such topology changing bordisms implement the required duality monodromies—either via the presence of $[p,q]$ 7-brane defects or through smooth gravitational soliton solutions of nontrivial topology—in the context of expectations from the Swampland program and the Cobordism Conjecture.

A central focus is the formulation and application of a refinement of the Swampland Cobordism Conjecture: the commutator width of the duality group $G$ (for IIB, $G = \mathsf{SL}(2,\mathbb{Z})$) can obstruct the realization of required bordisms by smooth topology-changing configurations, ultimately requiring additional duality defects beyond those associated to $G^{\mathrm{ab}}$. The consequences for global symmetry violation and decay rates of string backgrounds are analyzed with technical precision.

9d IIB Supergravities and Bordisms: Structure and Duality

After reviewing the derivations and classification of 9d maximal and gauged IIB supergravities via Kaluza-Klein and generalized Scherk-Schwarz reductions, the paper clarifies the connection to F-theory via compactification on twisted 3-tori with various $\mathsf{SL}(2,\mathbb{Z})$ monodromies. In particular, the discrete conjugacy classes of $\mathsf{SL}(2,\mathbb{Z})$ (parabolic, elliptic, hyperbolic) yield a rich variety of 9d SUGRAs, each associated with a specific geometry and monodromy in the compactification.

Bordisms connecting pairs of these 9d SUGRAs (or connecting to "nothing") are described in the F-theory language as codimension-one domain walls or topology-changing interpolating geometries. These are elliptic 4-manifolds $\mathcal{B}_4$ fibered over 2d bases with boundaries labeled by twisted 3-torus geometries. The monodromy content of the associated duality group bundles provides the data necessary for a precise homological and topological analysis of such bordisms.

Homological Analysis and Bordism Topology

The paper derives in detail the (co)homological constraints on the interpolating manifolds, using the Leray-Serre spectral sequence for fibrations and long exact sequences for relative homology. The key result is that the complexity of the bordism (in particular, the genus $g$ of the base) grows with the commutator length required to realize the boundary monodromies.

Boundary conditions arising from a given pair of 9d SUGRAs with fixed $\mathsf{SL}(2,\mathbb{Z})$0 conjugacy classes enforce, through the closing relation on $\mathsf{SL}(2,\mathbb{Z})$1, a required product of monodromies in the bordism interior. This requirement can be split between codimension-2 defect (7-brane) insertions, which implement abelianized monodromies, and the topology of the base, which can realize elements of the commutator subgroup through nontrivial cycles.

The explicit homology computations show that, for large monodromy (e.g., large parabolic exponent $\mathsf{SL}(2,\mathbb{Z})$2), the genus $\mathsf{SL}(2,\mathbb{Z})$3 or the number of topological features in the bordism grows linearly with the spectral norm of the monodromy matrix. In effect, gravitational solitons alone cannot produce all required monodromies in a topologically efficient way if the duality group’s commutator width is infinite.

Commutator Width, Duality Defects, and the Cobordism Conjecture

A principal result is the identification of the commutator width $\mathsf{SL}(2,\mathbb{Z})$4 of the duality group $\mathsf{SL}(2,\mathbb{Z})$5 as a controlling parameter: for $\mathsf{SL}(2,\mathbb{Z})$6, $\mathsf{SL}(2,\mathbb{Z})$7. This means there is no finite $\mathsf{SL}(2,\mathbb{Z})$8 bounding the number of commutators required to express all group elements; consequently, sectors of the moduli space requiring large monodromies would force interpolating bordisms of arbitrarily high genus (and hence, complexity).

The Swampland Cobordism Conjecture originally asserts that all cobordism classes must be dynamically trivialized in quantum gravity, to eliminate global bordism charges. The paper argues that, for infinite commutator width $\mathsf{SL}(2,\mathbb{Z})$9, gravitational instantons alone cannot efficiently mediate all necessary topology changes. The Euclidean action for such solitons becomes arbitrarily large in the large genus limit, rendering corresponding decay rates arbitrarily suppressed—contradicting the expectation that all approximate global symmetries are broken at least at the quantum gravity suppression scale.

To resolve this, the paper proposes a refined version of the Cobordism Conjecture: for $[p,q]$0 with $[p,q]$1, a complete spectrum of duality defects (e.g., all $[p,q]$2 7-branes for $[p,q]$3) is required in the theory, so that all monodromies can be trivialized by a bounded number of topological and defect insertions, keeping decay rates unsuppressed beyond quantum gravity scales.

Technical and Quantitative Results

• The bordism homology calculations exhibit that only abelianized monodromies (elements of $[p,q]$4) require duality defects; commutator monodromies can be realized topologically but at the price of arbitrarily increasing genus for large monodromy parameters.
• Explicit construction shows that for $[p,q]$5, $[p,q]$6, so only 12 D7-brane defects suffice to kill the abelianized part. However, this is insufficient in practice since infinite commutator length can render some processes non-perturbatively suppressed.
• Extensions to more general duality groups ($[p,q]$7, metaplectic or Pin$[p,q]$8 lifts, and U-duality groups in maximal supergravities) demonstrate that finite commutator width for higher-rank duality groups ($[p,q]$9, $G$0; simply connected Chevalley groups) enables avoidance of the pathologies present for $G$1.
• In the context of 4d $G$2 SUGRA from CY compactification, explicit analysis of monodromy groups (including infinite-index subgroups of $G$3) reveals models where large duality groups again force the necessity of a rich defect spectrum.

Broader Implications and Future Directions

The results have far-reaching implications for the Swampland program and the structure of quantum gravity vacua. The need for an infinite defect spectrum in cases of infinite commutator width suggests a profound interplay between algebraic properties of duality groups and the consistency conditions for quantum gravity. The analysis points to the danger that approximate global symmetries might persist in effective field theory unless the spectrum of defects is sufficiently large, thus challenging the universality of certain Swampland expectations.

From the perspective of string cosmology and non-perturbative transitions, the suppression of bubble nucleation rates for high-genus transitions further constrains the realistic possibilities for topology-changing processes in string-derived theories.

For future developments, the algebraic quantification of defect necessity via commutator widths provides a calculable criterion for assessing quantum gravity consistency in new classes of compactifications. The extension to higher-dimensional bordisms, further details on the tension and interactions of duality defects, and the exploration of these structures in phenomenologically relevant lower-dimensional models remain open avenues.

Conclusion

This paper presents a rigorous exploration of the interplay between the topology of cobordisms in 9d type IIB supergravity, the algebraic structure of duality groups, and the requirements of consistent quantum gravity. The central technical result is that the commutator width of the duality group dictates whether gravitational solitons can efficiently trivialize all bordism classes or whether an infinite set of duality defects is required. For duality groups with infinite commutator width—such as $G$4 in IIB string theory—preventing arbitrarily suppressed decay rates demands a complete spectrum of duality defects, refining the Swampland Cobordism Conjecture. This constraint must be taken into account in the analysis of both the high-dimensional landscape and its more general compactifications (2605.15276).