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Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions

Pavel Putrov, Juven Wang, Shing-Tung Yau

TL;DR

This work builds a unified framework to study intrinsic topological order in 2+1D and 3+1D by developing and analyzing dynamical TQFTs that include bosonic Dijkgraaf–Witten-like theories, higher-form gauge theories, and spin TQFTs arising from gauged fermionic SPTs. It provides explicit path-integral calculations of braiding statistics for both particle and string excitations, unveiling a rich set of link invariants such as Milnor’s triple linking, Milnor μ̄, Arf–Brown–Kervaire, Rokhlin, and Sato–Levine invariants, and connects them to observable data like ground-state degeneracy and modular matrices. The paper demonstrates how these invariants distinguish Z8 fermionic SPT classes in 2+1D, and maps odd ν theories to non-Abelian spin TQFTs via gluings with Ising or SU(2)2 sectors, while even ν theories yield bosonic or fermionic Abelian orders, thereby tying continuum field theories to lattice models and potential experimental signatures. Across 3+1D, higher-order linking numbers (triple, quadruple) and intersection counts classify non-Abelian string excitations and their braiding, expanding the toolkit for diagnosing topological phases in solid-state and quantum information settings. Overall, the results provide a concrete bridge between field-theoretic constructions, geometric link invariants, and condensed-matter realizations of both bosonic and fermionic topological phases, offering new diagnostics for identifying and distinguishing topological superconductors and related TQFTs.

Abstract

Topological Quantum Field Theories (TQFTs) pertinent to some emergent low energy phenomena of condensed matter lattice models in 2+1 and 3+1D are explored. Many of our field theories are highly-interacting without free quadratic analogs. Some of our bosonic TQFTs can be regarded as the continuum field theory formulation of Dijkgraaf-Witten twisted discrete gauge theories. Other bosonic TQFTs beyond the Dijkgraaf-Witten description and all fermionic spin TQFTs are either higher-form gauge theories where particles must have strings attached, or fermionic discrete gauge theories obtained by gauging the fermionic Symmetry-Protected Topological states (SPTs). We calculate both Abelian and non-Abelian braiding statistics data of anyon particle and string excitations, where the statistics data can one-to-one characterize the underlying topological orders of TQFTs. We derive path integral expectation values of links formed by line and surface operators in the TQFTs. The acquired link invariants include not only the Aharonov-Bohm linking number, but also Milnor triple linking number in 2+1D, triple and quadruple linking numbers of surfaces, and intersection number of surfaces in 3+1D. We also construct new spin TQFTs with the corresponding knot/link invariants of Arf(-Brown-Kervaire), Sato-Levine and others. We propose a new relation between the fermionic SPT partition function and Rokhlin invariant. We can use these invariants and other observables, including ground state degeneracy, reduced modular $\mathcal{S}^{xy}$ and $\mathcal{T}^{xy}$ matrices, and the partition function on $\mathbb{RP}^3$ manifold, to identify all $\mathbb{Z}_8$ classes of 2+1D gauged $\mathbb{Z}_2$-Ising-symmetric $\mathbb{Z}_2^f$-fermionic Topological Superconductors (TSC, realized by stacking layers of a pair of $p+ip$ and $p-ip$ SC, where boundary supports non-chiral Majorana-Weyl modes) with continuum spin-TQFTs.

Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions

TL;DR

This work builds a unified framework to study intrinsic topological order in 2+1D and 3+1D by developing and analyzing dynamical TQFTs that include bosonic Dijkgraaf–Witten-like theories, higher-form gauge theories, and spin TQFTs arising from gauged fermionic SPTs. It provides explicit path-integral calculations of braiding statistics for both particle and string excitations, unveiling a rich set of link invariants such as Milnor’s triple linking, Milnor μ̄, Arf–Brown–Kervaire, Rokhlin, and Sato–Levine invariants, and connects them to observable data like ground-state degeneracy and modular matrices. The paper demonstrates how these invariants distinguish Z8 fermionic SPT classes in 2+1D, and maps odd ν theories to non-Abelian spin TQFTs via gluings with Ising or SU(2)2 sectors, while even ν theories yield bosonic or fermionic Abelian orders, thereby tying continuum field theories to lattice models and potential experimental signatures. Across 3+1D, higher-order linking numbers (triple, quadruple) and intersection counts classify non-Abelian string excitations and their braiding, expanding the toolkit for diagnosing topological phases in solid-state and quantum information settings. Overall, the results provide a concrete bridge between field-theoretic constructions, geometric link invariants, and condensed-matter realizations of both bosonic and fermionic topological phases, offering new diagnostics for identifying and distinguishing topological superconductors and related TQFTs.

Abstract

Topological Quantum Field Theories (TQFTs) pertinent to some emergent low energy phenomena of condensed matter lattice models in 2+1 and 3+1D are explored. Many of our field theories are highly-interacting without free quadratic analogs. Some of our bosonic TQFTs can be regarded as the continuum field theory formulation of Dijkgraaf-Witten twisted discrete gauge theories. Other bosonic TQFTs beyond the Dijkgraaf-Witten description and all fermionic spin TQFTs are either higher-form gauge theories where particles must have strings attached, or fermionic discrete gauge theories obtained by gauging the fermionic Symmetry-Protected Topological states (SPTs). We calculate both Abelian and non-Abelian braiding statistics data of anyon particle and string excitations, where the statistics data can one-to-one characterize the underlying topological orders of TQFTs. We derive path integral expectation values of links formed by line and surface operators in the TQFTs. The acquired link invariants include not only the Aharonov-Bohm linking number, but also Milnor triple linking number in 2+1D, triple and quadruple linking numbers of surfaces, and intersection number of surfaces in 3+1D. We also construct new spin TQFTs with the corresponding knot/link invariants of Arf(-Brown-Kervaire), Sato-Levine and others. We propose a new relation between the fermionic SPT partition function and Rokhlin invariant. We can use these invariants and other observables, including ground state degeneracy, reduced modular and matrices, and the partition function on manifold, to identify all classes of 2+1D gauged -Ising-symmetric -fermionic Topological Superconductors (TSC, realized by stacking layers of a pair of and SC, where boundary supports non-chiral Majorana-Weyl modes) with continuum spin-TQFTs.

Paper Structure

This paper contains 17 sections, 93 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction and Summary
  2. The plan of the article and the convention of notation
  3. $\int BdA$ in any dimension and Aharonov-Bohm's linking number
  4. $\int K_{IJ} A_I dA_J$ and $\int BdA+AdA$ in 2+1D and the linking number
  5. $\int BdA+A^3$ in 2+1D, non-Abelian anyons and Milnor's triple linking number
  6. $\int BdA+A^2dA$ in 3+1D and the triple linking number of 2-surfaces
  7. $\int BdA+A^4$ in 3+1D, non-Abelian strings and the quadruple linking number of 2-surfaces
  8. $\int BdA+BB$ in 3+1D and the intersection number of open surfaces
  9. Fermionic TQFT/ spin TQFT in 2+1D and 3+1D
  10. 2+1D ${\mathbb{Z}}_2^f \times {\mathbb{Z}}_2$ symmetric fermionic SPTs
  11. Ground state degeneracy (GSD): Distinguish the odd-$\nu$ and even-$\nu$ classes
  12. $Z[{\mathbb{RP}}^3]$: Distinguish $\nu\in\mathbbm{Z}_8$ classes
  13. ${\cal S} ^{xy}$ and ${\cal T} ^{xy}$: The mutual- and self-exchange braiding statistics for $\nu\in\mathbbm{Z}_8$ classes
  14. Fermionic Topological Superconductor and Rokhlin invariant
  15. Other examples of 2+1D/3+1D spin-TQFTs and ${\mathbb{Z}}_2^f\times ({\mathbb{Z}}_2)^n$ fermionic SPTs: Sato-Levine invariant and more
  16. ...and 2 more sections

Figures (9)

  • Figure 1: Particular choice of surfaces $\Sigma_I$ for Borromean rings. The red lines show pairwise intersections $\Sigma_J\cap \Sigma_K$. The endpoints of the redlines are intersection points which pairs are counted in (\ref{['int-pair-count']}).
  • Figure 2: An example of configuration with a triple linking number Eq.(\ref{['triple-surface-linking']}) of three 2-surfaces being $\text{Tlk}(\Sigma_1,\Sigma_3,\Sigma_2)\equiv \# ({\cal V}_1 \cap {\cal V}_2 \cap \Sigma_3)=1$. (The same link figure is shown in Table \ref{['table:TQFTlink']}.) Take the $\Sigma_2,\Sigma_3$ to be a spun of Hopf link with Seifert hypersurfaces ${\cal V}_2,{\cal V}_3$ being spuns of Seifert surfaces. The surface $\Sigma_1$ is a torus embedded at a fixed value of the spin angle and encircling the Hopf link. Choose Seifert hypersurface ${\cal V}_1$ to be the interior the torus $\Sigma_1$. The intersection $\# ({\cal V}_1 \cap {\cal V}_2 \cap \Sigma_3)=1$ contains one point shown bold in the figure.
  • Figure 3: An example of computing $t(\Sigma_1;{\cal V}_2,{\cal V}_3,{\cal V}_4)$ by formula (\ref{['t-surface-formula']}). The red numbers $0,1$ denote the value of the weight with which the intersection points of $\gamma_2^{(1)}$ with $\gamma_3^{(1)}$ are counted in different domains separated by $\gamma_4^{(1)}$. The points $a\in \gamma^{(1)}_2\cap \gamma^{(1)}_3$ that enter into the sum with non-zero weight are shown as bold black points.
  • Figure 4: An illustration of invariance of (\ref{['quadruple-surface-linking']}) under deformation of Seifert hypersurfaces ${\cal V}_I$. A local configuration of $\Sigma_1,{\cal V}_2,{\cal V}_3,{\cal V}_4$ in shown in ${\mathbb{R}}^4\cong {\mathbb{R}}^3 \times {\mathbb{R}}_\text{time}$, where ${\mathbb{R}}_\text{time}$ is not shown in the picture. The hypersurfaces ${\cal V}_2,{\cal V}_3,{\cal V}_4$ are locally repsented by planes $\times {\mathbb{R}}_\text{time}$, while $\Sigma_1$ is locally a plane $\times$ point and ${\cal V}_1$ is locally a half ${\mathbb{R}}^3$ bounded by $\Sigma^1$ and spanned in the direction of the reader. The right side shows a local deformation of ${\cal V}_4$ which results in increasing both $\# ({\cal V}_1 \cap {\cal V}_2 \cap {\cal V}_3\cap {\cal V}_4)$ and $t(\Sigma_1;{\cal V}_2,{\cal V}_3,{\cal V}_4)$ by 1 (the contributing intersection points are shown bold and black). The total sum (\ref{['quadruple-surface-linking']}) stays intact.
  • Figure 5: An example of configuration with a quadruple linking number Eq.(\ref{['quadruple-surface-linking']}) being $\text{Qlk}(\Sigma_1,\Sigma_2,\Sigma_3,\Sigma_4)=1$. Take the triple $\Sigma_2,\Sigma_3,\Sigma_4$ to be a spun of Borromean rings with Seifert hypersurfaces ${\cal V}_2,{\cal V}_3,{\cal V}_4$ being the spun of Seifert surfaces in Fig. \ref{['fig:borromean-rings-int']}. The surface $\Sigma_1$ is a torus embedded at a fixed value of the spin angle and encircling the Borromean rings. Choose Seifert hypersurface ${\cal V}_1$ to be the interior the torus $\Sigma_1$. It is easy to see that for this choice of Seifert hypersurfaces all $t(\Sigma_I;{\cal V}_J,{\cal V}_K,{\cal V}_L)$ vanish just because for each $\Sigma_I$ one of the three curves $\gamma_J^{(I)}\,,J\neq I$ is empty. The quadruple intersection $\# ({\cal V}_1 \cap {\cal V}_2 \cap {\cal V}_3\cap {\cal V}_4)=1$ contains one point shown bold in the figure.
  • ...and 4 more figures