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How to Build Anomalous (3+1)d Topological Quantum Field Theories

Arun Debray, Weicheng Ye, Matthew Yu

TL;DR

The paper develops a systematic framework to realize prescribed anomalies of finite symmetries in (3+1)d fermionic theories by extending the Wang–Wen–Witten symmetry-extension paradigm to fermionic settings and embedding it in a higher-categorical cobordism framework. Central to the approach is twisted supercohomology SH^5(BG,s,ω), which classifies fermionic G-anomalies, together with fusion 2-categories (specifically 2sVect-enriched theories) that encode (3+1)d fermionic G-SETs; the authors also introduce a hastened Adams spectral sequence to compute these groups and navigate extension problems. The main results provide explicit constructions for anomalous TQFTs saturating given SH^5 obstructions in several abelian symmetry cases (e.g., G=Z/n, Z/(2n)^F, and Z/(2^{k+1})^F×Z/2^T), demonstrating how to lift anomalies via symmetry extensions H and K, and then gauge subgroups to obtain relative TQFTs realizing the anomalies as boundary theories. This establishes a concrete bridge between anomaly realization, cobordism, and higher-categorical data, yielding a principled route to constructing (3+1)d fermionic TQFTs with controlled finite-group anomalies and anomaly-matching interpretations for fermionic UV theories. The work lays groundwork for systematic anomaly saturation in fermionic topological orders and highlights both achieved results and avenues for extending the framework to time-reversal and nonabelian symmetries.

Abstract

We develop a systematic framework for constructing (3+1)-dimensional topological quantum field theories (TQFTs) that realize specified anomalies of finite symmetries, as encountered in gauge theories with fermions or fermionic lattice systems. Our approach generalizes the Wang-Wen-Witten symmetry-extension construction to the fermionic setting, building on two recent advances in the study of fermionic TQFTs and related homotopy theory. The first is the categorical classification of anomalous TQFTs in (3+1)d. The second, which we develop further in a planned sequel to this paper, is a hastened Adams spectral sequence for computing supercohomology groups, closely paralleling techniques from cobordism theory. By integrating supercohomology and cobordism methods within the recently developed categorical framework of fusion 2-categories, we provide a concrete and systematic route to constructing fermionic TQFTs with specified anomalies, thereby establishing a conceptual bridge between anomaly realization, cobordism, and higher-categorical structures.

How to Build Anomalous (3+1)d Topological Quantum Field Theories

TL;DR

The paper develops a systematic framework to realize prescribed anomalies of finite symmetries in (3+1)d fermionic theories by extending the Wang–Wen–Witten symmetry-extension paradigm to fermionic settings and embedding it in a higher-categorical cobordism framework. Central to the approach is twisted supercohomology SH^5(BG,s,ω), which classifies fermionic G-anomalies, together with fusion 2-categories (specifically 2sVect-enriched theories) that encode (3+1)d fermionic G-SETs; the authors also introduce a hastened Adams spectral sequence to compute these groups and navigate extension problems. The main results provide explicit constructions for anomalous TQFTs saturating given SH^5 obstructions in several abelian symmetry cases (e.g., G=Z/n, Z/(2n)^F, and Z/(2^{k+1})^F×Z/2^T), demonstrating how to lift anomalies via symmetry extensions H and K, and then gauge subgroups to obtain relative TQFTs realizing the anomalies as boundary theories. This establishes a concrete bridge between anomaly realization, cobordism, and higher-categorical data, yielding a principled route to constructing (3+1)d fermionic TQFTs with controlled finite-group anomalies and anomaly-matching interpretations for fermionic UV theories. The work lays groundwork for systematic anomaly saturation in fermionic topological orders and highlights both achieved results and avenues for extending the framework to time-reversal and nonabelian symmetries.

Abstract

We develop a systematic framework for constructing (3+1)-dimensional topological quantum field theories (TQFTs) that realize specified anomalies of finite symmetries, as encountered in gauge theories with fermions or fermionic lattice systems. Our approach generalizes the Wang-Wen-Witten symmetry-extension construction to the fermionic setting, building on two recent advances in the study of fermionic TQFTs and related homotopy theory. The first is the categorical classification of anomalous TQFTs in (3+1)d. The second, which we develop further in a planned sequel to this paper, is a hastened Adams spectral sequence for computing supercohomology groups, closely paralleling techniques from cobordism theory. By integrating supercohomology and cobordism methods within the recently developed categorical framework of fusion 2-categories, we provide a concrete and systematic route to constructing fermionic TQFTs with specified anomalies, thereby establishing a conceptual bridge between anomaly realization, cobordism, and higher-categorical structures.

Paper Structure

This paper contains 24 sections, 33 theorems, 90 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Fermionic symmetries and supercohomology
  3. Notions for Anomalies of 3+1d Fermionic TQFTs
  4. Main Results
  5. Outline
  6. Preliminaries
  7. Symmetry Extension By Wang--Wen--Witten
  8. Twisted Supercohomology in Two Ways
  9. Anomalous Topological Orders
  10. Constructing the Relative TQFTs
  11. Computations for $\mathbb{Z}/n\times\mathbb{Z}/2^F$ (\ref{['ex:cyclic']})
  12. Computations for $\mathbb{Z}/(2n)^F$ (\ref{['ex:cyclicfermion']})
  13. Computations for $\mathbb{Z}/(2^{k+1})^F\times \mathbb{Z}/2^T$ (\ref{['ex:timerev']})
  14. Conclusion and Discussion
  15. Anomalies of topological orders from obstruction theory
  16. ...and 9 more sections

Key Result

Theorem 1.7

Figures (7)

  • Figure 1: Left: The $\mathcal{A}(1)$-module structure on $R_1 \cong H_{0,x^2}^*(B\mathbb{Z}/2; \mathbb{Z}/2)$Cam17. Right: $\mathcal{Q}(R_1)$, computed in DYY2.
  • Figure 2: Left: $\mathcal{Q}(C\eta)$, computed in DYY2. Center: the $E_k$-page of the HASS computing $\tau_{\le 2}\mathit{ko}(B\mathbb{Z}/2^k, 0, y)$ (here $k = 3$). Right: the $E_{k+1} = E_\infty$-page.
  • Figure 3: Left: $\mathcal{Q}(R_6)$, computed in DYY2. Center: the $E_2$-page of the HASS computing $\tau_{\le 2}\mathit{ko}_*(B\mathbb{Z}/2\times B\mathbb{Z}/2^k, x_1, y)$. We calculate the $d_2$s in range in \ref{['d265']}. Right: the $E_\infty$-page.
  • Figure 4: Left: the $\mathcal{A}(1)$-module structure on $P\coloneqq H^*((B\mathbb{Z}/2)^{\sigma-1};\mathbb{Z}/2)$. Center: $\mathrm{Ext}_{\mathcal{A}(1)}(P)$, the $E_2$-page of the Adams spectral sequence computing $\mathit{ko}_*((B\mathbb{Z}/2)^{\sigma-1})$. Right: $\mathcal{Q}(P)$, the $E_2$-page of the HASS computing $(\tau_{\le 2}\mathit{ko})_*((B\mathbb{Z}/2)^{\sigma-1})$. The classes $\kappa$, $k_1$, and $k_2$ are in the image of the map of $E_2$-pages induced by the truncation $\mathit{ko}\to\tau_{\le 2}\mathit{ko}$. We use this in the proof of \ref{['pinm_SH']}.
  • Figure 5: Left: the $E_2$-page of the HASS computing $(\tau_{\le 2}\mathit{ko})_*((B\mathbb{Z}/2)^{\sigma-1}\wedge (B\mathbb{Z}/2^k)^{V-2})$. We use this spectral sequence in the proof of \ref{['cNSH']}, where we show that the pictured $d_2$ is nonzero. Right: the $E_3 = E_\infty$-page.
  • ...and 2 more figures

Theorems & Definitions (73)

  • Example 1.6
  • Theorem 1.7
  • Corollary 1.10
  • Example 1.11
  • Theorem 1.13
  • Corollary 1.17
  • Remark 1.18
  • Example 1.19
  • Remark 1.22
  • Theorem 1.23
  • ...and 63 more