The Smith Fiber Sequence and Invertible Field Theories

Abstract

Smith homomorphisms are maps between bordism groups that change both the dimension and the tangential structure. We give a completely general account of Smith homomorphisms, unifying the many examples in the literature. We provide three definitions of Smith homomorphisms, including as maps of Thom spectra, and show they are equivalent. Using this, we identify the cofiber of the spectrum-level Smith map and extend the Smith homomorphism to a long exact sequence of bordism groups, which is a powerful computation tool. We discuss several examples of this long exact sequence, relating them to known constructions such as Wood's and Wall's sequences. Furthermore, taking Anderson duals yields a long exact sequence of invertible field theories, which has a rich physical interpretation. We developed the theory in this paper with applications in mind to symmetry breaking in quantum field theory, which we study in a companion paper.

arXiv comment77 pages, 3 figures. This is a companion paper to arXiv:2309.16749.

References in corpus (64)

• Reflection positivity and invertible topological phases
• Dai-Freed anomalies in particle physics
• Setting the quantum integrand of M-theory
• Twisted differential String and Fivebrane structures
• Cobordism Conjecture, Anomalies, and the String Lamppost Principle
• A New SU(2) Anomaly
• Higher Anomalies, Higher Symmetries, and Cobordisms I: Classification of Higher-Symmetry-Protected Topological States and Their Boundary Fermionic/Bosonic Anomalies via a Generalized Cobordism Theory
• On the cobordism classification of symmetry protected topological phases
• Why are fractional charges of orientifolds compatible with Dirac quantization?
• Many-body topological invariants in fermionic symmetry protected topological phases: Cases of point group symmetries
• Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms
• Time Reversal, SU(N) Yang-Mills and Cobordisms: Interacting Topological Superconductors/Insulators and Quantum Spin Liquids in 3+1D
• Bulk Anyons as Edge Symmetries: Boundary Phase Diagrams of Topologically Ordered States
• Beyond Standard Models and Grand Unifications: Anomalies, Topological Terms, and Dynamical Constraints via Cobordisms
• Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group
• An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology
• Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory
• Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map
• Anomalies of Non-Invertible Symmetries in (3+1)d
• Anomaly Matching in the Symmetry Broken Phase: Domain Walls, CPT, and the Smith Isomorphism
• A note on the $(\infty ,n)$-category of cobordisms
• Decorated ${\mathbb{Z}}_2$ Symmetry Defects and Their Time-Reversal Anomalies
• Quaternionic Monopoles
• Gauging Noninvertible Defects: A 2-Categorical Perspective
• A Non-Perturbative Definition of the Standard Models
• Fivebrane Structures
• Revisiting Wess-Zumino-Witten terms
• An Exactly Solvable Model for a $4+1D$ Beyond-Cohomology Symmetry Protected Topological Phase
• Higher Anomalies, Higher Symmetries, and Cobordisms II: Lorentz Symmetry Extension and Enriched Bosonic/Fermionic Quantum Gauge Theory
• The algebra of anomaly interplay
• The classification of 2-connected 7-manifolds
• Topological modular forms and the absence of all heterotic global anomalies
• SO(3) Monopoles, Level-One Seiberg-Witten Moduli Spaces, and Witten's Conjecture in Low Degrees
• The character map in (twisted differential) non-abelian cohomology
• Twisted topological structures related to M-branes
• Twisted Morava K-theory and E-theory
• Real pinor bundles and real Lipschitz structures
• Gauge Enhanced Quantum Criticality Beyond the Standard Model
• When the moduli space is an orbifold: Spontaneous breaking of continuous non-invertible symmetries
• Twisted topological structures related to M-branes II: Twisted Wu and Wu^c structures
• Twisted Spin cobordism and positive scalar curvature
• On the infinite loop space structure of the cobordism category
• On counting associative submanifolds and Seiberg-Witten monopoles
• Relative Thom Spectra Via Operadic Kan Extensions
• Toric 2-group anomalies via cobordism
• Bordism for the 2-group symmetries of the heterotic and CHL strings
• The Arf-Brown TQFT of Pin${}^{-}$ Surfaces
• Spin, statistics, orientations, unitarity
• Ninebrane structures
• The 2-dimensional stable homotopy hypothesis
• Diagrammatic State Sums for 2D Pin-Minus TQFTs
• ${\mathrm{H}}^4({\mathrm{Co}}_0;\mathbf{Z})=\mathbf{Z}/24$
• Twisted iterated algebraic K-theory and topological T-duality for sphere bundles
• Variations of rational higher tangential structures
• What bordism-theoretic anomaly cancellation can do for U
• On Exotic Consistent Anomalies in (1+1)$d$: A Ghost Story
• Reflection Structures and Spin Statistics in Low Dimensions
• A Long Exact Sequence in Symmetry Breaking: order parameter constraints, defect anomaly-matching, and higher Berry phases
• Interacting SPT phases are not Morita invariant
• Anomaly constraint on chiral central charge of (2+1)d topological order
• Positive scalar curvature on simply connected spin pseudomanifolds
• Bosonization and Anomaly Indicators of (2+1)-D Fermionic Topological Orders
• Smith homomorphisms and Spin${}^h$ structures
• Monodromy and mapping class groups of 3-dimensional hypersurfaces

Cited by in corpus (1)

• Crystallography, Group Cohomology, and Lieb-Schultz-Mattis Constraints