Gauge It or Break It: Lectures on Global Symmetries and the Cobordism Conjecture
Abstract
These lecture notes introduce the No Global Symmetries Conjecture in the context of the Swampland Program, with particular emphasis on the Cobordism Conjecture. We begin with a review of ordinary and higher-form symmetries, their realization by extended operators, and mechanisms for gauging or breaking them. We then discuss the no global symmetries principle, including evidence from black hole physics, string perturbation theory, and AdS/CFT, and its relation to other Swampland constraints. A central theme is the reformulation of symmetry constraints in topological terms: we introduce bordism groups, with special focus on spin bordism, and explain how they provide a natural framework for organizing would-be global symmetries in gravitational theories. With this at hand, we discuss the Cobordism Conjecture, which proposes that nontrivial bordism classes are eliminated by dynamical processes in quantum gravity. We conclude with examples and a discussion of dynamical aspects involving defects, boundaries, and related open questions. These notes are based on lectures delivered by Matilda Delgado at the 21st Modave Summer School on Mathematical Physics. A similar series of lectures was also presented at the KMPB School on D-branes in Physics and Mathematics.
Related
- Nilpotent Symmetries and Conserved Charges: BRST-Quantized Version of the 4D Abelian 2-Form Gauge Theory
- Gauge versus (hidden) physical symmetries of FLRW cosmologies
- Five-flavor udscb¯ molecular pentaquarks from heavy-quark and local hidden gauge symmetries
- Observation of robust corner states in photonic crystals without global symmetries
- Spectral Emergence from Topological Solitons: A Unified Conjecture for Gauge Representations
BibTeX
@misc{delgado2026gauge,
title = {{Gauge It or Break It: Lectures on Global Symmetries and the Cobordism Conjecture}},
author = {Matilda Delgado and Corinne Englert and Guillermo Mera Álvarez and Eduardo Velasco-Aja},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31749},
eprint = {2610.31749},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31749},
}
RIS
TY - UNPB
TI - Gauge It or Break It: Lectures on Global Symmetries and the Cobordism Conjecture
AU - Matilda Delgado
AU - Corinne Englert
AU - Guillermo Mera Álvarez
AU - Eduardo Velasco-Aja
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31749
UR - https://arxiv.org/abs/2610.31749
AB - These lecture notes introduce the No Global Symmetries Conjecture in the context of the Swampland Program, with particular emphasis on the Cobordism Conjecture. We begin with a review of ordinary and higher-form symmetries, their realization by extended operators, and mechanisms for gauging or breaking them. We then discuss the no global symmetries principle, including evidence from black hole physics, string perturbation theory, and AdS/CFT, and its relation to other Swampland constraints. A central theme is the reformulation of symmetry constraints in topological terms: we introduce bordism groups, with special focus on spin bordism, and explain how they provide a natural framework for organizing would-be global symmetries in gravitational theories. With this at hand, we discuss the Cobordism Conjecture, which proposes that nontrivial bordism classes are eliminated by dynamical processes in quantum gravity. We conclude with examples and a discussion of dynamical aspects involving defects, boundaries, and related open questions. These notes are based on lectures delivered by Matilda Delgado at the 21st Modave Summer School on Mathematical Physics. A similar series of lectures was also presented at the KMPB School on D-branes in Physics and Mathematics.
ER -