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Global Structure in the Presence of a Topological Defect

Arun Debray, Weicheng Ye, Matthew Yu

TL;DR

The paper develops a unified framework to study global defect structure in quantum field theories by embedding topological defects on submanifolds via the Pontryagin–Thom construction. It introduces characteristic structures and a conjectural characteristic long exact sequence to capture how defects and bulk data interrelate, and it computes low-degree characteristic bordism groups for FK, FK$^{O}$, GM, KT$^{-}$, and KT$^{+}$. Using Smith long exact sequences as computable approximations, it derives physical implications for anomaly matching and obstructions to spontaneous breaking of finite higher-form $\\mathbb{Z}/2$ symmetries, including explicit obstructions in 1-form and 2-form cases and their dependence on manifold tangential structures (e.g., spin, pin, and KT twists). The results illuminate how global manifold structure and defect topology constrain QFTs and their anomalies, offering a pathway to connect bordism data with physical observable invariants and potential extensions to quantum gravity contexts.

Abstract

We investigate the global structure of topological defects which wrap a submanifold $F\subset M$ in a quantum field theory defined on a closed manifold $M$. The Pontryagin-Thom construction oversees the interplay between the global structure of $F$ and the global structure of $M$. We will employ this construction to two distinct mathematical frameworks with physical applications. The first framework is the concept of a characteristic structure, consisting of the data of pairs of manifolds $(M,F)$ where $F$ is Poincaré dual to some characteristic class. This concept is discussed in the mathematics literature, and shown here to have meaningful physical interpretations related to defects. In our examples we will mainly focus on the case where $M$ is 4-dimensional and $F$ has codimension 2. The second framework uses obstruction theory and the fact that spontaneously broken finite symmetries leave behind domain walls, to determine the conditions on which dimensions a higher-form finite symmetry can spontaneously break. We explicitly study the cases of higher-form $\mathbb Z/2$ symmetry, but the method can be generalized to other groups.

Global Structure in the Presence of a Topological Defect

TL;DR

The paper develops a unified framework to study global defect structure in quantum field theories by embedding topological defects on submanifolds via the Pontryagin–Thom construction. It introduces characteristic structures and a conjectural characteristic long exact sequence to capture how defects and bulk data interrelate, and it computes low-degree characteristic bordism groups for FK, FK, GM, KT, and KT. Using Smith long exact sequences as computable approximations, it derives physical implications for anomaly matching and obstructions to spontaneous breaking of finite higher-form symmetries, including explicit obstructions in 1-form and 2-form cases and their dependence on manifold tangential structures (e.g., spin, pin, and KT twists). The results illuminate how global manifold structure and defect topology constrain QFTs and their anomalies, offering a pathway to connect bordism data with physical observable invariants and potential extensions to quantum gravity contexts.

Abstract

We investigate the global structure of topological defects which wrap a submanifold in a quantum field theory defined on a closed manifold . The Pontryagin-Thom construction oversees the interplay between the global structure of and the global structure of . We will employ this construction to two distinct mathematical frameworks with physical applications. The first framework is the concept of a characteristic structure, consisting of the data of pairs of manifolds where is Poincaré dual to some characteristic class. This concept is discussed in the mathematics literature, and shown here to have meaningful physical interpretations related to defects. In our examples we will mainly focus on the case where is 4-dimensional and has codimension 2. The second framework uses obstruction theory and the fact that spontaneously broken finite symmetries leave behind domain walls, to determine the conditions on which dimensions a higher-form finite symmetry can spontaneously break. We explicitly study the cases of higher-form symmetry, but the method can be generalized to other groups.

Paper Structure

This paper contains 14 sections, 59 theorems, 60 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Summary of Main Results
  3. Topological Preliminaries
  4. Tangential structure
  5. Pontryagin-Thom construction
  6. Characteristic bordism
  7. Long exact sequences and approximate symmetries
  8. Defects and Characteristic Structures
  9. Freedman-Kirby Characteristic Bordism
  10. Guillou-Marin Characteristic Bordism
  11. Symmetry Approximations to GM
  12. Symmetry Approximations to KT
  13. Topological Obstructions to Spontaneous Symmetry breaking
  14. Discussion

Key Result

Theorem 2.13

There is an isomorphism $\pi_n(MT\xi) \cong \Omega^\xi_*$.The usual formulation of the Pontryagin-Thom theorem uses the Thom spectrum of $\xi^*(V)\to B$, not $-V$, and identifies its homotopy groups with the bordism groups of manifolds with a $\xi$-structure on the stable normal bundle $\nu\to M$, r

Figures (13)

  • Figure 1: Left: the $\mathcal{A}(1)$-module structure on $H_\mathit{ko}^*(M^\mathit{ko}(\widehat{A}\circ f_{0,U}))$ in low degrees. The pictured module contains all classes in degrees $6$ and below. As will be the case in all the figures, curved lines that join points separated by two degrees denote actions by $\mathrm{Sq}^2$, and straight lines joining points separated by one degree denote $\mathrm{Sq}^1$ actions. Right: The $E_2$-page of the Baker-Lazarev Adams spectral sequence computing $2$-completed $(M\mathrm O_2, 0, U)$-twisted spin bordism groups. We use this spectral sequence to prove \ref{['prop:bordcompGM']}.
  • Figure 2: Left: The $\mathcal{A}(1)$-module structure on $H^*((B\mathrm O_2)^{V\oplus 3\det(V)-5})$; this summand includes all classes in degree $6$ and below. Right: The $E_2$-page of the Adams spectral sequence computing $2$-completed spin-$\mathrm O_2$ bordism. We use this in \ref{['lem:bordismspinO2']}.
  • Figure 3: Left: The $\mathcal{A}(1)$-module structure on $H^*((B\mathrm O_2)^{\sigma-1};\mathbb Z/2)/H^*((B\mathrm O_1)^{\sigma-1}; \mathbb Z/2)$; this summand includes all classes in degree $5$ and below. Right: The $E_2$-page of the corresponding Adams spectral sequence, which computes a summand of the $2$-completion of $\Omega_*^{\mathrm{Spin}}((B\mathrm O_2)^{\sigma-1})$. We use this in \ref{['smith_spO2']}.
  • Figure 4: The characteristic long exact sequence for Guillou-Marin's characteristic structure, which we prove in \ref{['GM_LES_thm']}. The map $\phi$ sends $(1,0)\mapsto (0,8)$ and $(0,1)\mapsto 0$.
  • Figure 5: Left: The $\mathcal{A}(1)$-module $M_1$ (\ref{['M1defn']}). In \ref{['GMA1']} we showed that, modulo classes of degree at least $7$, $M_1$ is a summand of $H_\mathit{ko}^*(M^\mathit{ko} (\widehat{A}\circ f_{0,U}))$. Right: the $\mathcal{A}(1)$-module structure on $H^*((B\mathrm O_1)^{\sigma-1};\mathbb Z/2)$.
  • ...and 8 more figures

Theorems & Definitions (120)

  • Definition 1.2
  • Conjecture 1.3
  • Conjecture
  • Definition 2.1: Lashof Las63
  • Definition 2.2: Hason:2020yqf
  • Remark 2.3
  • Definition 2.5
  • Example 2.6
  • Definition 2.7
  • Example 2.8
  • ...and 110 more