The Chronicles of IIBordia: Dualities, Bordisms, and the Swampland

TL;DR

This work applies the Swampland Cobordism Conjecture to type IIB string theory with nontrivial duality bundles, computing twisted Spin bordism groups for the duality groups Mp(2,Z) and GL^+(2,Z) and verifying that many generators correspond to known F-theory backgrounds (e.g., [p,q]-7-branes, non-Higgsable clusters, S-folds). The authors develop and deploy a detailed Adams spectral sequence program to derive Ω_k^{Spin}(B SL(2,Z)), Ω_k^{Spin-Mp(2,Z)}(pt), and Ω_k^{Spin-GL^+(2,Z)}(pt) for k ≤ 11, including explicit generators and their geometric realizations (Arcana). A major prediction is the necessity of a new non-supersymmetric reflection 7-brane (R7) to bound the GL^+-duality sector, along with discrete θ-angles and potential anomaly-cancellation mechanisms, providing a rich testing ground for quantum gravity constraints in non-supersymmetric backgrounds. The results extend beyond string theory, offering new topological data for twisted bordism theories and charting a path to applying these methods to broader duality groups and higher k, with potential connections to Levine–Morel bordism and elliptic cohomology.

Abstract

In this work we investigate the Swampland Cobordism Conjecture in the context of type IIB string theory geometries with non-trivial duality bundle. Quite remarkably, we find that many non-trivial bordism classes with duality bundles in Mp$(2,\mathbb{Z})$, a double cover of SL$(2,\mathbb{Z})$ related to fermions, correspond to asymptotic boundaries of well-known supersymmetric F-theory backgrounds. These include $[p,q]$-7-branes, non-Higgsable clusters, S-folds, as well as various lower-dimensional generalizations. These string theoretic objects break the global symmetries associated to the non-trivial bordism groups, providing a strong test of the Cobordism Conjecture. Further including worldsheet orientation reversal promotes the duality group to the Pin$^+$ cover of GL$(2,\mathbb{Z})$. The corresponding bordism groups require a new non-supersymmetric "reflection 7-brane" and its compactifications to ensure the absence of global symmetries, thus providing an interesting prediction of the Cobordism Conjecture for non-supersymmetric type IIB backgrounds. A major component of the present work is the explicit derivation of the involved bordism groups as well as their generators, which correspond to asymptotic boundaries of explicit string theory backgrounds. The main tool is the Adams spectral sequence, to which we provide a detailed introduction. We anticipate that the same techniques can be applied in a wide variety of settings.

Figures (36)

• Figure 1: Various extensions of the duality group of type IIB string theory.
• Figure 2: Oriented bordism between a 2-sphere and a 2-torus (can be understood as a $3$-ball with an interior torus cut out).
• Figure 3: Left: Depiction of a bordism group factor $\mathbb Z/3\mathbb Z$, where three copies of a generator bound a manifold. Right: Null-bordism of the 2-sphere, i.e., a 3-ball.
• Figure 4: The odd-dimensional Spin-$\mathrm{Mp}(2,\mathbb Z)$ manifolds often appear as the asymptotic boundary of the base manifold of a genus-one fibered, non-compact, singular Calabi-Yau manifold.
• Figure 5: Bulk / Boundary characterization of bordism generators and the corresponding defects predicted by the CC. Viewing the bordism generator for manifolds equipped with a structure defined by $\mathcal{G}$ as the boundary of some bulk space $\mathcal{B}$ equipped with a $\mathcal{G}$-bundle, the CC predicts the existence of a $\mathcal{G}$-defect which allows the class to be trivialized in $\Omega^{\mathrm{QG}}_{k}$.

Theorems & Definitions (152)

• Definition 10.2
• Remark 10.4: Multiplicative structure
• Example 10.12: Eilenberg-Mac Lane spectra
• Example 10.13: Suspension spectra
• Example 10.14: Thom spectra
• Definition 10.17
• Lemma 10.18: Shearing
• proof
• Corollary 10.19
• Lemma 10.21