1 Introduction↩︎

One of the long-standing mysteries of theoretical physics is the origin of the three-family structure of fermion replication in the Standard Model (SM): quarks and leptons appear in exactly three families with identical gauge quantum numbers but differing masses and mixings, observed in particle physics since the 1970s [2]. While the SM itself places no restriction on the number of fermion families or generations, experimental observations indicate the family number \(N_f=3\), a fact often referred to as the Family Puzzle or Generation Problem. Understanding whether this number \(N_f=3\) is accidental or enforced by deeper consistency conditions remains an open theoretical puzzle in high-energy particle physics.

Experimentally, several independent lines of evidence establish that the number of fermion families in the SM is \(N_f=3\) [3]. The most direct constraint comes from precision measurements of the invisible decay width of the Z boson at LEP, which determine the number of light neutrino species to be 3 [4]–[6]. Since each SM family contains one left-handed neutrino, this implies three fermion families. Additional families are further strongly constrained by electroweak precision data, including the absence of deviations in Z-boson decays that would signal additional light quarks [5], as well as by flavor physics and the observed unitarity of the CKM matrix. Cosmological observations, such as Big Bang Nucleosynthesis [7], [8] and Cosmic Microwave Background measurements [9], independently support this conclusion by constraining the effective number of relativistic fermion species (primarily the 3 active light left-handed neutrinos) to be consistent with 3.

Theoretically, renewed attention to this problem has been prompted by proposals based on topological and nonperturbative global anomaly constraints, suggesting that the \(N_f=3\) family structure may follow from fundamental mathematical consistency requirements rather than from model-dependent dynamics. For example, Ref. [10] provides some interesting hints of the global anomalies from homotopy group or cobordism group constraints. There are two recent proposals based on topological constraints that have drawn our attention [11], [12]. Ref. [] approach this problem in particle physics using tools from topology and topological quantum field theory (TQFT, which has no local point operators but only extended operators), which have been rapidly developed in recent years to describe topological quantum matter [13], thereby going beyond conventional model-building frameworks in particle physics — in other words, thinking outside the box of conventional particle approaches.

1. Ref. [] proposes that when the family number is a multiple of 3, \[\begin{align} N_f = 0 \mod 3, \text{ namely,} \quad N_f \in 3 {\mathbb{Z}}, \end{align}\] the multiple of 3 families of 16 Weyl fermions per family/generation in the SM, with total \((N_f=3) \times 16 =48\) Weyl fermions in the 3+1d spacetime dimensions, are topologically constrained. This is due to

   1. Modular Invariance [14]: The dimensional-reduced 1+1d theory has a chiral central charge \[\begin{align} c_- = c_L - c_R = \frac{ N_f \times 16}{2}= \frac{N_f}{3} \times 48 = 0 \mod 24. \end{align}\]

   2. Hirzebruch signature [15], [16]: for a spacetime 4-manifold with a special orthogonal (SO) structure and purely bosonic gauge-invariant matter content, the signature \(\sigma(M)\) of the manifold \(M\) and its first Pontryagin class \(p_1(TM)\) [15], [17], [18] of the tangent bundle \(TM\) follow an integer-quantized relation \(\sigma(M)=\frac{\langle p_1(TM),[M]\rangle}{3} \in {\mathbb{Z}}.\). See more elaboration in Sec. 1.3.

   3. Rokhlin’s theorem [19]: for a spacetime 4-manifold with a Spin structure (the fermion parity \({\mathbb{Z}}_2^{\rm F}\) graded lift of the special orthogonal group SO, so \({\rm Spin}/{\mathbb{Z}}_2^{\rm F}=\mathop{\mathrm{SO}}\)), and fermionic gauge-invariant matter content, the signature \(\sigma(M)\) of the manifold \(M\) becomes \(\sigma(M)=\frac{\langle p_1(TM),[M]\rangle}{3} \in 16{\mathbb{Z}}.\)

   4. Cobordism mapping: The 48 Weyl fermions of the Standard Model, organized according to the bosonic SO and fermionic Spin structures, can be mapped to a trivial class in String cobordism (related to the framing anomaly-free [20]), or more generally to a trivial class in the \(w_1\)-\(p_1\) cobordism (related to the 2-framing anomaly-free [21]). This mapping argument holds independently of any internal global symmetry or gauge structure of the SM. Thus, this argument [11] may hold robustly, even if we destroy all the internal or gauge structure of the SM.

   Namely, the observation in Ref. [] is primarily a gravitational anomaly argument (in the dimensionally reduced 1+1d theory) or corresponds to a trivial cobordism class in a consistent quantum gravity theory [22]. It concerns a nonperturbative global gravitational anomaly argument instead of a perturbative local gravitational anomaly argument.

2. Ref. [] proposes a unique interplay between the family and color numbers, with \(N_f=N_c=3\). This approach introduces an additional internal \({\mathbb{Z}}_3\) symmetry, naturally arising from the discrete Baryon plus Lepton \({\boldsymbol{B} +L}\) symmetry in the SM [23], [24], \[\begin{align} {\mathbb{Z}}_{6,{\boldsymbol{B} +L}}^{\rm F}= {\mathbb{Z}}_2^{\rm F}\times {\mathbb{Z}}_{3,{\boldsymbol{B} +L}}. \end{align}\]

   In the absence of the three right-handed sterile neutrinos \(\nu_R\), the Standard Model exhibits a mixed \({\boldsymbol{(}B +L)}\)-gauge–gravitational nonperturbative global anomaly. The corresponding anomaly index for this SM (up to a \(\pm\) sign), \[N_f = 3 \in {\mathbb{Z}}_9 =\Omega_5^{\rm Spin\times\mathbb{Z}_3},\] can be canceled via an appropriate color center symmetry extension via \[\begin{align} \label{eq:Z3-extension} 1 \to {\mathbb{Z}}_{N_c=3}\to {\mathbb{Z}}_{N_cN_f=9}\to {\mathbb{Z}}_{N_f=3}\to 1. \end{align}\tag{1}\] Here the \({\mathbb{Z}}_{N_c=3}=Z(\mathop{\mathrm{SU}}(N_c=3))\) corresponds to the center of the color gauge group SU(3) of quantum chromodynamics (QCD). Here the symmetry extension refers to a particular Ref. []’s symmetry extension (e.g., group extension) construction, by trivializing the nontrivial anomaly index in \(G = {\rm Spin}\times {\mathbb{Z}}_{N_f=3}\) by pulling it back to \(G_{\rm Tot} = {\rm Spin}\times {\mathbb{Z}}_{N_cN_f=9}\) as a trivial anomaly class in \(G_{\rm Tot}\).

   This means that the Standard Model without the \(3\nu_R\) can still preserve the full \({\mathbb{Z}}_{6,{\boldsymbol{B} +L}}^{\rm F}\) symmetry by replacing \(3\nu_R\) by a finite gauge \({\mathbb{Z}}_{N_c=3}\) TQFT at low-energy. This demonstrates the uniqueness of \[\begin{align} N_f = N_c = 3, \end{align}\] corresponding to \(N_f=3\) families with \(N_c=3\) colors, which represent the number of quarks in a baryon B. Note that this \({\mathbb{Z}}_{6,{\boldsymbol{B} +L}}^{\rm F}= {\mathbb{Z}}_2^{\rm F}\times {\mathbb{Z}}_{3,{\boldsymbol{B} +L}}\) symmetry [23], [24] is independent of the choice of the SM gauge group (see [25] for an explanation of \(G_{{\rm SM}_{\rm q}}\)), \[G_{{\rm SM}_{\rm q}} \equiv \frac{\mathop{\mathrm{SU}}(3) \times \mathop{\mathrm{SU}}(2) \times {\rm U}(1)_{\tilde{Y}}}{{\mathbb{Z}}_{\rm q}}, \quad \quad {\rm q}=1,2,3,6.\] Consequently, the argument in [12] that \(N_f = N_c = 3\) is also independent of the choice of the SM gauge group \(G_{{\rm SM}_{\rm q}}\), which holds for any \({\rm q}=1,2,3,6\).

In this work, on one hand, we follow the setup in Ref. [], to prove some of its observations differently and more mathematically; on the other hand, we obtain some generalized theorems and we derive some general topological constraints for the hidden topologically ordered sector of the generic \(N_f\)-family \(N_c\)-color generalized SM.

We note that some previous works have also invoked potential nonperturbative global anomalies to constrain the \(N_f=3\) families of the SM, but their methodology differs fundamentally different from ours:

• Ref. [] uses the 6d homotopy group analysis, \(\pi_6(\mathop{\mathrm{SU}}(2)) = {\mathbb{Z}}_{12}\), \(\pi_6(\mathop{\mathrm{SU}}(3)) = {\mathbb{Z}}_6\) and \(\pi_6(G_2) = {\mathbb{Z}}_3\), to argue the nonperturbative global anomaly constraints from 6d to the 4d SM. However, we find that the cobordism classification of nonperturbative global anomaly constraints shows \(\Omega_7^{{\rm Spin}\times G_{{\rm SM}_{\rm q}}} = 0\) vanishes [26], [27], thus this means no 6d nonperturbative global anomalies to constrain the 4d SM.

• Ref. [] introduces an additional \({\mathbb{Z}}_3\) symmetry with a \({\mathbb{Z}}_9\) nonperturbative global anomaly in 4d, motivated by baryon triality or proton hexality. However, this extra \({\mathbb{Z}}_3\) symmetry relies on the structure of the more sophisticated supersymmetric Standard Model. Instead, Ref. [] and our present work consider the simpler discrete \({\mathbb{Z}}_{3,{\boldsymbol{B} +L}}\) arising from the universal discrete \({\boldsymbol{B} +L}\) symmetry of the non-supersymmetric SM [23], [24].

In comparison, we believe that the \({\mathbb{Z}}_{3,{\boldsymbol{B} +L}}\) symmetry with a \({\mathbb{Z}}_9\) nonperturbative global anomaly [12] offers a more robust trustworthy argument than the 3-family arguments in the older literature Ref. [].

In practice, our present work studies fermionic anomalies associated with discrete \({\mathbb{Z}}_n\) symmetry in 3+1 dimensions [28], [29]. Our focus is on distinguishing which anomaly classes can be canceled by anomalous \({\mathbb{Z}}_{n'}\)-symmetric gauge TQFTs through symmetry extension constructions, and which cannot; for certain choices of \(n\) and its corresponding \(n'\). Similar questions about 3+1d symmetric anomalous fermionic TQFTs are explored in [28], [30]–[36].⁴

Readers may also notice that on the physical mathematics side, some recent papers explore the relation between the \(p_1\) structure (also the \(w_1\)-\(p_1\) structure), gravitational Chern-Simons theory, and physical topological field theories in [37], [38], where Ref. [37] further points out a relation to dualizable tensor categories [39].

1.1 Main results↩︎

The main results of our present article are as follows:

1. We show that the group-cohomology subclass \(\operatorname{H}^5({\mathbb{Z}}_n,{\rm U}(1))\cong{\mathbb{Z}}_n\) anomaly can be canceled by 3+1d \({\mathbb{Z}}_n\)-gauge TQFTs, while beyond-group-cohomology contributions involving the Pontryagin class \(p_1\) generally cannot. Namely, \(A_{{\mathbb{Z}}_n}(\beta_{(n,n)}A_{{\mathbb{Z}}_n})(\beta_{(n,n)}A_{{\mathbb{Z}}_n})\) can be trivialized via the symmetry extension \[\begin{align} \label{eq:Zn942} 1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\to{\mathbb{Z}}_n\to1, \end{align}\tag{2}\] \(A_{{\mathbb{Z}}_n}p_1\) cannot be trivialized by any finite group extension except \(n =2\) or \(n=3\). Here, \(A_{{\mathbb{Z}}_n}\) is the generator of \(\operatorname{H}^1({\mathbb{Z}}_n,{\rm U}(1))\cong \operatorname{H}^1({\mathbb{Z}}_n,{\mathbb{Z}}_n)\) and \(\beta_{(n,n)}:\operatorname{H}^*(-,{\mathbb{Z}}_n)\to\operatorname{H}^{*+1}(-,{\mathbb{Z}}_n)\) is the Bockstein homomorphism. See Appendices 6 and 7.

2. More generally, for odd spacetime dimensions \(d\geqslant 3\), we prove that any cocycle \(\alpha_d\in\operatorname{H}^d({\mathbb{Z}}_n,{\rm U}(1))\) can be trivialized via the symmetry extension 2 , and we explicitly construct the corresponding anomalous boundary symmetry-extended \((d-1)\)d TQFTs via the explicit \(d-1\)-cochain \(\tilde{\beta}_{d-1}\) solution found in Appendix 8. Following Ref. []’s symmetry-extension approach, we derive the explicit \((d-1)\)-cochain \(\tilde{\beta}_{d-1}\) that splits the \(d\)-cocycle \(\tilde{\alpha}_d=\delta\tilde{\beta}_{d-1}\) as a coboundary in \(\operatorname{H}^d({\mathbb{Z}}_{n^2},{\rm U}(1))\) by the above symmetry extension eq. (2 ) for any odd \(d\geqslant 3\) and any \(n\geqslant 2\).

3. As a physical application, for \(d=5\) and \(n=3\), we construct a \({\rm Spin}\times{\mathbb{Z}}_3\)-symmetric 4d \({\mathbb{Z}}_3\)-gauge TQFT that cancels the mixed discrete \((\mathbf{B}+\mathbf{L})\)-gauge-gravitational anomaly of the Standard Model in the absence of 3 right-handed neutrinos, \(\nu_{R,e}, \nu_{R,\mu}\), and \(\nu_{R,\tau}\).

4. We consider a generic \(N_c\)-color and \(N_f\)-family SM. We aim to trivialize the nonperturbative global anomaly associated with discrete \({\boldsymbol{B} +L}\) symmetry \({\mathbb{Z}}_{2 N_f,{\boldsymbol{B} +L}}^{\rm F}\) in the absence of \(N_f\) sterile right-handed neutrinos via the appropriate \({\mathbb{Z}}_{N_c}\)-color center symmetry extension \[\begin{align} \label{eq:NcNf-extension} 1 \to {\mathbb{Z}}_{N_c}\to{\mathbb{Z}}_{N_cN_f}\to{\mathbb{Z}}_{N_f}\to 1, \end{align}\tag{3}\] or more precisely involving the spacetime-internal symmetry together:⁵ \[\begin{align} \label{eq:Spin-NcNf-extension} 1 \to {\mathbb{Z}}_{N_c}\to {\rm Spin}\times {\mathbb{Z}}_{N_cN_f}\to {\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2 N_f,{\boldsymbol{B} +L}}^{\rm F}\to 1. \end{align}\tag{4}\] Here especially when \(N_c \in \mathbb{N}_{\text{odd}}\) is an odd positive integer, the \[\begin{align} \label{eq:ZNc} {\mathbb{Z}}_{N_c}=Z(\mathop{\mathrm{SU}}(N_c)) \end{align}\tag{5}\] corresponds to the center of the color gauge group \(\mathop{\mathrm{SU}}(N_c)\) of the generalized QCD. Here \(\times_{N}\) means the direct product mod out the common normal subgroup \(N\) while \(N={\mathbb{Z}}_2^{\rm F}\) is the fermion parity. In the physical interpretation of the symmetry-extension construction [1], the normal subgroup \({\mathbb{Z}}_{N_c}\) in eq. (4 ) needs to be anomaly-free and is consistently dynamically gauged. The symmetry-extension construction of anomalous \({\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2 N_f,{\boldsymbol{B} +L}}\)-symmetry \({\mathbb{Z}}_{N_c}\)-gauge TQFT in 3+1d follows the general approach outlined in Sec. 1.2.

   We can compare eq. (4 ) with the \({\boldsymbol{B} +L}\) symmetry under the color-extension to the \({\boldsymbol{Q}} + N_c {\boldsymbol{L}}\) symmetry as \[\begin{align} 1 \to {\mathbb{Z}}_{N_c}\to {\mathbb{Z}}_{2 N_c N_f,{\boldsymbol{Q}} + N_c {\boldsymbol{L}}} \to {\mathbb{Z}}_{2 N_f,{\boldsymbol{B} +L}}^{\rm F}\to 1. \end{align}\]

   It turns out that for eq. (4 ) to work for the minimal extension, as we will explain in Theorem 1, \(N_f\) and \(N_c\) are either both odd positive integers \(\mathbb{N}_{\text{odd}}\) or both even positive integers \(\mathbb{N}_{\text{even}}\).
   \(\bullet\) When \(N_f\) and \(N_c\) are both \(\mathbb{N}_{\text{odd}}\), eq. (4 )’s extended total group coincides with the free quarks full faithful \({\boldsymbol{Q}} + N_c {\boldsymbol{L}}\) symmetry \({\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2 N_c N_f,{\boldsymbol{Q}} + N_c {\boldsymbol{L}}}^{\rm F} = {\rm Spin}\times {\mathbb{Z}}_{N_cN_f, {\boldsymbol{Q}} + N_c {\boldsymbol{L}}}\), so eq. (4 ) is also equivalent to extending \[\begin{align} \label{eq:Spin-NcNf-extension-quark} 1 \to {\mathbb{Z}}_{N_c}\to {\rm Spin}\times {\mathbb{Z}}_{N_c N_f,{\boldsymbol{Q}} + N_c {\boldsymbol{L}}} \to {\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2 N_f,{\boldsymbol{B} +L}}^{\rm F}\to 1. \end{align}\tag{6}\] The \({\mathbb{Z}}_{N_c}\) coincides with the center \(Z(\mathop{\mathrm{SU}}(N_c))\) of the color gauge group \(\mathop{\mathrm{SU}}(N_c)\).
   \(\bullet\) When \(N_f\) and \(N_c\) are both \(\mathbb{N}_{\text{even}}\), eq. (4 )’s extended total group \({\rm Spin}\times {\mathbb{Z}}_{N_cN_f}\) does not coincide with the free quarks full faithful \({\boldsymbol{Q}} + N_c {\boldsymbol{L}}\) symmetry \({\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2 N_c N_f,{\boldsymbol{Q}} + N_c {\boldsymbol{L}}}^{\rm F}\). So the \({\mathbb{Z}}_{N_c}\) may not coincide with the center \(Z(\mathop{\mathrm{SU}}(N_c))\) of the color gauge group \(\mathop{\mathrm{SU}}(N_c)\).

5. We prove that \(A_{{\mathbb{Z}}_3}p_1=0\mod3\), see Appendix 13.

6. We prove a main theorem in Sec. 4 that we will outline here,

   Theorem 1. If the following conditions hold,

   1. The color number \(N_c\) and the family number \(N_f\) are minimal nonzero positive integers,

   2. The anomaly of \(N_f\) copies of a 4d charge-\(1\) Weyl fermion (namely the unit charge “sterile” right-handed neutrinos \(\nu_R\)) with symmetry \({\rm Spin}\times_{{\mathbb{Z}}_2^{{\rm F}}}{\mathbb{Z}}_{2N_f}\) is trivialized by a minimal \({\mathbb{Z}}_{N_c}\)-extension,⁶

   3. \(N_c\) is odd, so that an SM baryon is a fermion (as it is in our SM),

   then we prove that 3 families and 3 colors, \(N_c=N_f=3\), is the unique case that stands out.

   Below we sketch the proof of Theorem 1, the detailed proof can be found in the proof of Theorem 2. Assume that the anomaly of \(N_f\) copies of a 4d charge-\(1\) Weyl fermion (namely unit charge \(\nu_R\)) with symmetry \({\rm Spin}\times_{{\mathbb{Z}}_2^{{\rm F}}}{\mathbb{Z}}_{2N_f}\) is trivialized by a minimal \({\mathbb{Z}}_{N_c}\)-extension. Writing a generic family number \[N_f=2^p 3^r s,\qquad p,r\geqslant 0,\quad 2\nmid s,\quad 3\nmid s,\] the anomaly index decomposes as the subgroup of the cobordism group as \[2^p\!\cdot\! \mathbb{Z}_{2^{p+3}}\;\oplus\;3^r\!\cdot\!\mathbb{Z}_{3^{r+1}}.\] By [35], the \(2\)-power factor is trivialized by a \(\mathbb{Z}_4\)-extension, and (see Appendix 11), the \(3\)-power factor is trivialized by a \(\mathbb{Z}_3\)-extension, where the fact that \(A_{{\mathbb{Z}}_3} p_1 = 0 \mod 3\) is used. Hence, the minimal extension order is \[\left\{ \begin{array}{lll} N_c=3,&\text{if }p=0,r\geqslant 1,&\text{so } N_f \geqslant 3,\\ N_c=4, &\text{if }p\geqslant 1,r=0,&\text{so } N_f \geqslant 2,\\ N_c=12,&\text{if }p\geqslant 1,r\geqslant 1,&\text{so } N_f \geqslant 6. \end{array}\right.\] If the baryon is a fermion, then the color number \(N_c\) must be odd. So out of the above three choices, only \(N_c=3\) holds, whence the minimal family number is \(N_f=3\). So under the stated conditions, we prove that \[N_c = N_f =3\] is the only physically meaningful minimal solution.

This framework provides a systematic topological perspective on anomaly cancellation and offers a viewpoint on the distinguished role of the \(N_f=3\)-family structure of the Standard Model with the color number \(N_c=3\).

1.2 Symmetry Extension↩︎

Let us explain the symmetry extension method in [1]. For ’t Hooft anomalies of some global symmetry \(G\) to be nonperturbative global anomalies, we can potentially apply the appropriate symmetry-extension trivialization method [1], making a nonperturbative global anomaly in \(G\) becomes anomaly-free in an appropriate \(G_{\rm Tot}\) via an appropriate group extension \[\begin{align} \label{eq:extension} 1 \to K \to G_{\rm Tot} \overset{r}{\longrightarrow} G \to 1. \end{align}\tag{7}\] Namely, the precise mathematical check is, given a \(G\), we search for what an appropriate finite \(K\) and an appropriate extended \(G_{\rm Tot}\) are, such that a nonperturbative global anomaly index \[\begin{align} \nu_G \in \mathrm{TP}_d^G \end{align}\] in the Freed-Hopkins version [40] of cobordism group TP becomes the trivial anomaly class \[\begin{align} (r^*\nu)_{G_{\rm Tot}} =0 \in \mathrm{TP}_d^{G_{\rm Tot}} \end{align}\] for the cobordism group TP of the pulled back \(G_{\rm Tot}\). Here \(r\) is the reduction map from \(G_{\rm Tot} \overset{r}{\longrightarrow} G\), then the \(r^*\) with a \(*\) denote the pullback. According to [1], this provides a (3+1)d anomalous \(G\)-symmetric \(K\)-gauge topological order construction whose low-energy theory is a (3+1)d finite \(K\)-gauge TQFT, which is designed to carry the original nontrivial ’t Hooft anomaly index in \(G\), namely \(\nu_G \in \mathrm{TP}_d(G)\). By \(K\)-gauge, we mean that \(K\) is dynamically gauged with corresponding finite \(K\) gauge fields, such as in eq. (4 ) and eq. (5 ), the \(K={\mathbb{Z}}_{N_c}=Z(\mathop{\mathrm{SU}}(N_c))\) is dynamically gauged.

Note that TP cobordism group is the direct sum of a torsion bordism group in \(d\) dimensions and a free bordism group in \(d+1\) dimensions \[\begin{align} \mathrm{TP}_d(G) = (\Omega_d^G)_{\text{torsion}} \oplus (\Omega_{d+1}^G)_{\text{free}}. \end{align}\]

When we focus on the nonperturbative global anomaly of 3+1d quantum field theory (QFT) with \(G\) symmetry, we could use the classification of anomalies at the \(4+1 = 5\)d cobordism group as \[\mathrm{TP}_5(G) = (\Omega_5^G)_{\text{torsion}}\] in the case that \((\Omega_{6}^G)_{\text{free}}=0\). Then we need to check the anomaly index a nonperturbative global anomaly index \[\begin{align} \nu_G \in \mathrm{TP}_5(G) = (\Omega_5^G)_{\text{torsion}}, \end{align}\] which becomes the trivial anomaly class \[\begin{align} (r^*\nu)_{G_{\rm Tot}} =0 \in (\Omega_5^{G_{\rm Tot}})_{\text{torsion}} \end{align}\] for the bordism group of the pulled back \(G_{\rm Tot}\). In summary, when we refer to the symmetry extension, or the symmetry extension trivialization of the ’t Hooft anomaly, what we really mean is exactly the check done in this subsection, Sec. 1.2.

1.3 First Pontryagin class↩︎

We briefly introduce Pontryagin classes [15], [17], [18], which are fundamental topological invariants of real vector bundles, analogous to the Chern classes for complex vector bundles. For a real vector bundle \(E\to B\) where the total space \(E\) maps surjectively to the base space \(B\), the total Pontryagin class is defined via the complexification \(E\otimes_{\mathbb{R}}\mathbb{C}\) by \[\begin{align} p(E)=1+p_1(E)+p_2(E)+\dots ,\qquad p_k(E)=(-1)^k\,c_{2k}(E\otimes_{\mathbb{R}}\mathbb{C})\;\in \operatorname{H}^{4k}(B,\mathbb{Z}). \end{align}\] Here, \(c_{2k}(E\otimes_{\mathbb{R}}\mathbb{C})\) are the Chern classes of the complex vector bundle \(E\otimes_{\mathbb{R}}\mathbb{C}\) and \(\operatorname{H}^{4k}(B,{\mathbb{Z}})\) is the singular cohomology of \(B\), defined as the quotient group of the group of singular cocycles quotient by the group of singular coboundaries.

In the study of index theory, gravitational instantons, and gravitational anomalies [41]–[43], the first Pontryagin class \(p_1\) plays a central role. Geometrically, by the Chern-Weil theory, the first Pontryagin class \(p_1\) is the integer-valued topological invariant whose image in real cohomology is represented by (see appendices of [44] for detailed expressions written in differential forms and the relation to the gravitational Chern-Simons 3-form) \[\begin{align} p_1 (TM) =-\frac{1}{8\pi^2} \mathrm{Tr}(R \wedge R) \in \operatorname{H}^4_{\mathrm{dR}}(M,\mathbb{R}) \cong \operatorname{H}^4(M,\mathbb{R})\cong \operatorname{H}^4(M,{\mathbb{Z}})\otimes_{\mathbb{Z}}\mathbb{R}, \end{align}\] where \(R\) is the curvature 2-form of the tangent bundle \(TM\), the trace Tr is over the real Lie algebra valued 4-form, and \(\operatorname{H}^4_{\mathrm{dR}}(M,\mathbb{R})\) is the de Rham cohomology of the spacetime manifold \(M\). The de Rham cohomology \(\operatorname{H}^*_{\mathrm{dR}}(M,\mathbb{R})\) is defined as the quotient group of the group of closed differential forms quotient by the group of exact differential forms. By the Chern-Weil theory, if \(M_1\) and \(M_2\) are the same smooth manifold with different metrics and \(\alpha_1\) and \(\alpha_2\) are the closed differential forms representing the first Pontryagin class of \(TM\) with respect to the two different metrics, then \(\alpha_1\) and \(\alpha_2\) differ by an exact differential form. Therefore, \(\alpha_1\) and \(\alpha_2\) represent the same de Rham cohomology class and the first Pontryagin class is independent of the choice of metric. In particular, the first Pontryagin class is invariant under Wick rotation, regardless whether we choose the metric to be Euclidean or Lorentz/Minkoswki signature, and independent of the choice of the real Lie algebra, Euclidean \(so(4,\mathbb{R})\) or Lorentz \(so(3,1,\mathbb{R})\).⁷

For a closed oriented 4-manifold \(M\), the \(p_1\) integral over a closed oriented 4-manifold gives the Pontryagin number \[\begin{align} \label{eq:Pontryagin-number} \langle p_1(TM),[M]\rangle =-\int_{M} \frac{1}{8\pi^2} \mathrm{Tr}(R \wedge R) \in \mathbb{Z}, \end{align}\tag{8}\] which is intimately related to the signature \(\sigma(M)\) via the Hirzebruch signature theorem [16]: \[\begin{align} \sigma(M)=\frac{1}{3}\langle p_1(TM),[M]\rangle. \end{align}\] Here, \(TM\) is the tangent bundle of \(M\), \([M]\) is the fundamental class of \(M\), and \(\langle p_1(TM),[M]\rangle\) is the pairing which evaluates the cohomology class \(p_1(TM)\) on the homology class \([M]\).

Although Lorentzian signature does not admit real self-dual or anti-self-dual (Euclidean instanton) solutions due to the properties of the Hodge star operator, the topological charge (defined by the Chern number \(\langle c_2,[M]\rangle\) or the Pontryagin number \(\langle p_1,[M]\rangle\)) remains well-defined and quantized. This topological charge quantity depends only on the topology of the bundle and represents an integral characteristic class, and is therefore quantized independently of the metric or signature.

A crucial feature of the first Pontryagin class is that it is an oriented cobordism invariant. Two closed oriented \(n\)-manifolds \(M_0\) and \(M_1\) are oriented cobordant if there exists a compact oriented \((n+1)\)-manifold \(W\) whose boundary is the disjoint union \(M_0\sqcup \overline{M_1}\) (where the overbar denotes reversed orientation). A characteristic number is an oriented cobordism invariant if it depends only on the cobordism class; for \(n=4\), the Pontryagin number \(\langle p_1(TM),[M]\rangle\) is precisely such an invariant.

In Appendix 13.2, we prove another property of the first Pontryagin class [45], [46]: \[\begin{align} A_{{\mathbb{Z}}_3}p_1=0\mod3 \end{align}\] where \(A_{{\mathbb{Z}}_n}\) is the generator of \(\operatorname{H}^1({\mathbb{Z}}_n,{\rm U}(1))\cong\operatorname{H}^1({\mathbb{Z}}_n,{\mathbb{Z}}_n)\). The cup product \(A_{{\mathbb{Z}}_n}p_1\) is a mod \(n\) cohomology class on a manifold \(M\) when \(A_{{\mathbb{Z}}_n}\) is pulled back to \(M\). Throughout this article, \(p_1\) always means \(p_1(TM)\) for a manifold \(M\). The proof is based on the following facts. Let \(P_3^1:\operatorname{H}^*(-,{\mathbb{Z}}_3)\to\operatorname{H}^{*+4}(-,{\mathbb{Z}}_3)\) be the mod 3 Steenrod reduced power.

1. Based on the defining property of the mod 3 Steenrod reduced power, \[\begin{align} P_3^1(x)=0\text{ if }\deg(x)<2. \end{align}\]

2. This fact is nontrivial and a more general fact is proven in Appendix 13.1. \[\begin{align} P_3^1(x)=p_1\smile x=x\smile p_1 \end{align}\] for any \({\mathbb{Z}}_3\)-valued cohomology class \(x\), where \(\smile\) is the cup product.

Since \(\deg(A_{{\mathbb{Z}}_3})=1\), the above two facts imply \(A_{{\mathbb{Z}}_3}p_1=0\mod3\).

1.4 The Plan↩︎

The plan of this article goes as follows:

In Sec. 2.1, to warm up, we derive the 3+1d nonperturbative global anomaly of a Weyl fermion in \({\rm Spin}\times {\mathbb{Z}}_3\) symmetry (with \({\mathbb{Z}}_{3, {\boldsymbol{B} + L}}\) in mind) from the reduction of the perturbative local anomaly in \({\rm Spin}\times {\rm U}(1)\) symmetry.

In Sec. 2.2, we derive the 3+1d nonperturbative global anomaly of a Weyl fermion in \({\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{6}^{\rm F}}\) symmetry (with \({{\mathbb{Z}}_{6, {\boldsymbol{B} + L}}^{\rm F}}\) in mind) from the reduction of the perturbative local anomaly in \({\rm Spin}^c \equiv {\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\rm U}(1)\).

In Sec. 3, we show the global anomaly trivialization via the symmetry extension \(1 \to \mathbb{Z}_{N_c=3}\to \mathbb{Z}_{N_cN_f=9}\to \mathbb{Z}_{N_f=3}\to 1\), and we explicitly construct 3+1d anomalous \({\mathbb{Z}}_3\)-gauge TQFT as the low-energy theory of topological order. Thus this topological order may be a hypothetical quantum dark matter candidate — “quantum” in the sense that topological order is well-defined at the 0K temperature quantum limit.

In Sec. 4, we generalize to a generic \(N_c\)-color and \(N_f\)-family SM. We aim to trivialize the nonperturbative global anomaly associated with discrete \({\boldsymbol{B} +L}\) symmetry \({\mathbb{Z}}_{2 N_f,{\boldsymbol{B} +L}}^{\rm F}\) in the absence of \(N_f\) sterile right-handed neutrinos via the appropriate \({\mathbb{Z}}_{N_c}\)-color center symmetry extension \[1 \to {\mathbb{Z}}_{N_c}\to{\mathbb{Z}}_{N_cN_f}\to{\mathbb{Z}}_{N_f}\to 1,\] or more precisely involving the spacetime-internal symmetry together: \[1 \to {\mathbb{Z}}_{N_c}\to {\rm Spin}\times {\mathbb{Z}}_{N_cN_f}\to {\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2 N_f,{\boldsymbol{B} +L}}^{\rm F}\to 1.\] The symmetry-extension construction of anomalous \({\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2 N_f,{\boldsymbol{B} +L}}\)-symmetry \({\mathbb{Z}}_{N_c}\)-gauge TQFT in 3+1d follows the general approach outlined in Sec. 1.2 and [1].

In addition, we will prove various mathematical theorems in the Appendices that will be implemented in the main text.

In Appendix 6, we prove that any group cocycle \(\alpha_d \in \operatorname{H}^d({\mathbb{Z}}_n,{\rm U}(1))\) is trivialized by the symmetry extension \(1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\to{\mathbb{Z}}_n\to 1\) for odd \(d\geqslant 3\) and any \(n\geqslant 2\).

In Appendix 7, we prove that \(A_{{\mathbb{Z}}_n}p_1\) cannot be trivialized by any finite group extension except for \(n=2\) and \(n=3\) where \(A_{{\mathbb{Z}}_n}\) is defined as the generator of \(\operatorname{H}^1({\mathbb{Z}}_n,{\rm U}(1))\).

In Appendix 8, following Ref. []’s symmetry-extension approach, we derive the explicit \((d-1)\)-cochain \(\tilde{\beta}_{d-1}\) that splits the \(d\)-cocycle \(\tilde{\alpha}_d=\delta\tilde{\beta}_{d-1}\) as a coboundary in \(\operatorname{H}^d({\mathbb{Z}}_{n^2},{\rm U}(1))\) by the symmetry extension \(1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\to{\mathbb{Z}}_n\to1\) for any odd \(d\geqslant 3\) and any \(n\geqslant 2\).

In Appendix 9, we construct the path integral of \(d\)d-bulk/\((d-1)\)d-boundary coupled invertible topological field theory/symmetric anomalous gapped TQFT by the symmetry extension \(1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\to{\mathbb{Z}}_n\to 1\) for any odd \(d\geqslant 3\) and any \(n\geqslant 2\).

In Appendix 10, we derive a 3+1d nonperturbative global anomaly formula of Weyl fermion in \({\rm Spin}\times {{\mathbb{Z}}_{n}}\) symmetry for integer \(n\) with \(2 \nmid n\) and \(3 \nmid n\).

In Appendix 11, we derive a 3+1d nonperturbative global anomaly formula of Weyl fermion \({\rm Spin}\times {\mathbb{Z}}_{3^r} = {\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{2 \cdot 3^r}^{\rm F}}\) symmetry.

In Appendix 12, we derive a 3+1d nonperturbative global anomaly formula of Weyl fermion \({\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{2 \cdot 2^p}^{\rm F}}\) symmetry.

In Appendix 13, we provide a proof of \(A_{{\mathbb{Z}}_3}p_1=0\mod3\) for the mod 3 cohomology class in an appropriate context.

2 Perturbative Local Anomaly to Nonperturbative Global Anomaly↩︎

In the 3+1d Standard Model (SM), each family contains 15 Weyl fermions in the absence of the 16th Weyl fermion sterile right-handed neutrino \(\nu_R\). This SM suffers from the perturbative local mixed-gauge-gravitational anomalies [41]–[43] between the lepton number \({\boldsymbol{L}}\) symmetry and gravitational background fields, in 3+1d (or simply 4d) spacetime. Namely these anomalies are computable via perturbative triangle Feynman diagrams \({\rm U}(1)_{\boldsymbol{L}}^3\) and \({\rm U}(1)_{\boldsymbol{L}}\)-gravity-gravity, \[\includegraphics[height=.15\textwidth]{anomaly-LLL-2024.pdf} \quad \quad \quad \includegraphics[height=.15\textwidth]{anomaly-Lgravgrav-2024.pdf}\] with the anomaly index coefficient \[-N_f + n_{\nu_R},\] counting the difference between the family or generation number \(N_f\) (typically \(N_f=3\)) and the total right-hand neutrino number \(n_{\nu_R}\). See recent related expositions about this anomaly index \(-N_f + n_{\nu_R}\) for examples in [24], [44], [47]–[51]. However, because of the analogous Adler-Bell-Jackiw anomalies [52], [53] via the SM electroweak gauge instanton [54]–[57], instead of thinking of the classical lepton number \({\boldsymbol{L}}\) symmetry, only the baryon number plus or minus lepton number \({\boldsymbol{B}} \pm {\boldsymbol{L}}\) symmetries are physically meaningful quantum mechanical symmetries of the SM [23], [24]:

• For the gauge-invariant baryons, a full faithful combined symmetry of \({\boldsymbol{B}} - {\boldsymbol{L}}\) and \({\boldsymbol{B}} + {\boldsymbol{L}}\) with the Lorentz spacetime Spin group symmetry is \[\begin{align} {\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\rm U}(1)_{{\boldsymbol{B}} - {\boldsymbol{L}}} \times_{{\mathbb{Z}}_2^{\rm F}}{\mathbb{Z}}_{2 N_f, {\boldsymbol{B}} + {\boldsymbol{L}}}. \end{align}\]

• For the free quarks, a full faithful combined symmetry of \({\boldsymbol{Q}} - N_c {\boldsymbol{L}}\) and \({\boldsymbol{Q}} + N_c {\boldsymbol{L}}\) with the Lorentz spacetime Spin group symmetry is \[\begin{align} {\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\rm U}(1)_{{\boldsymbol{Q}} - N_c {\boldsymbol{L}}} \times_{{\mathbb{Z}}_2^{\rm F}}{\mathbb{Z}}_{2 N_c N_f, {\boldsymbol{Q}} + N_c {\boldsymbol{L}}}, \end{align}\] but it is unfaithful for the gauge-invariant baryons. Here in the conventional SM, the family number is \(N_f=3\), and the color number is \(N_c=3\).

For the conventional SM with \(N_f=3\), below we determine the \({\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{6,{{\boldsymbol{B}} + {\boldsymbol{L}}}}^{\rm F}= {\rm Spin}\times {\mathbb{Z}}_{3,{{\boldsymbol{B}} + {\boldsymbol{L}}}}\) mixed gauge-gravitational anomaly for the right-handed “sterile” neutrino \(\nu_R\) (sterile to the SM gauge force but not sterile to \({\boldsymbol{B}} \pm {\boldsymbol{L}}\) gauge field). In fact, we shall treat the anti-particle of right-handed neutrino \(\bar{\nu}_R\) as the left-handed particle, whose quantum numbers (here discrete charges of abelian global symmetries) are given by: \[\begin{align} \begin{tabular}{| c | c | c | c | c | c | c | c | c | c | } \hline & {\rm U}(1)_{{\boldsymbol{B}}-{\boldsymbol{L}}} & {\mathbb{Z}}_{6,{{\boldsymbol{B}} + {\boldsymbol{L}}}}^{\rm F} & {\mathbb{Z}}_{3,{{\boldsymbol{B}} + {\boldsymbol{L}}}} & {\mathbb{Z}}_2^{\rm F} & {\mathbb{Z}}_{18, {\boldsymbol{Q}} + 3 {\boldsymbol{L}}}^{\rm F} & {\mathbb{Z}}_{9, {\boldsymbol{Q}} + 3 {\boldsymbol{L}}} \\ \hline \hline \bar{\nu}_R & 1 & -1 & -1 & 1 & -3 & -3\\ \hline \end{tabular}. \end{align}\] So the charge \(Q_{{\mathbb{Z}}_{6,{{\boldsymbol{B}} + {\boldsymbol{L}}}}^{\rm F}} = Q_{{\mathbb{Z}}_{3,{{\boldsymbol{B}} + {\boldsymbol{L}}}}} \mod 3\), and \(Q_{{\mathbb{Z}}_{18,{{\boldsymbol{B}} + {\boldsymbol{L}}}}^{\rm F}} = Q_{{\mathbb{Z}}_{9,{{\boldsymbol{B}} + {\boldsymbol{L}}}}} \mod 9\).⁸

In Subsection 2.1, we start with a perturbative local anomaly of U(1) charge \(q=1\) left-handed Weyl fermion in \({\rm Spin}\times {\rm U}(1)\) to derive the the nonperturbative global anomaly of \({\mathbb{Z}}_{3, {\boldsymbol{B} + L}}\) charge \(q=1\) Weyl fermion in \({\rm Spin}\times {\mathbb{Z}}_{3, {\boldsymbol{B} + L}}\).

In Subsection 2.2, we will make a comparison to a perturbative local anomaly in \({\rm Spin}^c \equiv {\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\rm U}(1)\) and a nonperturbative global anomaly in \({\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{6,{\boldsymbol{B} + L}}^{\rm F}}\).

2.1 \({\rm Spin}\times {\rm U}(1)\) to \({\rm Spin}\times {\mathbb{Z}}_3\)↩︎

The perturbative local anomaly of U(1) charge \(q=1\) left-handed Weyl fermion of \({\rm Spin}\times {\rm U}(1)\) symmetry in 3+1d or 4d is captured by a 5d invertible field theory (iTFT) with the anomaly index \(k=1\) [42], [44], as an invertible U(1)-valued partition function: \[\exp( \mathrm{i}k \int_{M^5} A \frac{c_1^2}{6}-A \frac{p_1}{24}),\] with the first Chern class \(c_1\) and the first Pontryagin class \(p_1\). Now we redefine the U(1) gauge field \(A\) as a \({\mathbb{Z}}_3\) cohomology class gauge field \(A_{{\mathbb{Z}}_3} \in\operatorname{H}^1(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3)= {\mathbb{Z}}_3\) with the following replacement: \[\begin{align} A &\mapsto& \frac{2 \pi}{3} A_{{\mathbb{Z}}_3}.\cr c_1 = \frac{\mathrm{d}A}{2 \pi} &\mapsto& \frac{\mathrm{d}A_{{\mathbb{Z}}_3}}{3} \equiv \beta_{(3,3) } A_{{\mathbb{Z}}_3}. \end{align}\] The \(\beta_{(n,m)}: \operatorname{H}^*(-,{\mathbb{Z}}_m) \mapsto \operatorname{H}^{*+1}(-,{\mathbb{Z}}_n)\) is the Bockstein homomorphism associated with the extension \({\mathbb{Z}}_n \stackrel{\cdot m}{\to} {\mathbb{Z}}_{nm} \to {\mathbb{Z}}_m\). Thus we get the 5d topological invariant of the \({\rm Spin}\times {\mathbb{Z}}_3\) that captures the 4d anomaly as: \[\begin{align} \label{eq:Spin-Z3} &&\exp \big( \mathrm{i}2 \pi k \int_{M^5} ( \frac{1}{18} {A_{{\mathbb{Z}}_3}} (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) -\frac{1}{3 \cdot 24} {A_{{\mathbb{Z}}_3}} p_1 ) \big) \cr &=&\exp \big( \mathrm{i}\frac{2 \pi}{9} k \int_{M^5} ( \frac{1}{2} {A_{{\mathbb{Z}}_3}} (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) -\frac{1}{8} {A_{{\mathbb{Z}}_3}} p_1 ) \big)\cr &=&\exp \big( \mathrm{i}\frac{2 \pi}{9} k \int_{M^5} ( -4 {A_{{\mathbb{Z}}_3}} (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) +3\cdot {} \frac{A_{{\mathbb{Z}}_3}p_1}{3} ) \big)\cr &=&\exp \big( \mathrm{i}\frac{2 \pi}{9} (-4k) \int_{M^5} ( {A_{{\mathbb{Z}}_3}} (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) -3\cdot {} \frac{A_{{\mathbb{Z}}_3}p_1}{3} ) \big). \end{align}\tag{9}\] Here \(\beta_{(3,3)}:\operatorname{H}^1(-,{\mathbb{Z}}_3)\to \operatorname{H}^2(-,{\mathbb{Z}}_3)\) is the Bockstein homomorphism. Here \({} \frac{A_{{\mathbb{Z}}_3}p_1}{3}\) is a mod 3 class that involves Pontryagin class because \(A_{{\mathbb{Z}}_3}p_1=0\mod3\) [45], [46] (see Appendix 13 for the proof), while \({A_{{\mathbb{Z}}_3}} (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}})\) is a mod 3 class.

In 9 , the first quality rewrites the coefficients \(\frac{1}{18}=\frac{1}{9}\cdot\frac{1}{2}\) and \(-\frac{1}{3\cdot 24}=\frac{1}{9}\cdot(-\frac{1}{8})\), since the anomaly of 4d Weyl fermion with symmetry \({\rm Spin}\times{\mathbb{Z}}_3\) contains only 3-torsion [10], [28]–[30], [58], [59], we can regard \(2\) and \(8\) as invertible in \({\mathbb{Z}}_9\).
The second equality uses the fact that \(1=-8\mod9\) to obtain \(\frac{1}{2}=-4\mod9\) and \(-\frac{1}{8}=1\mod9\) and uses the fact that \(A_{{\mathbb{Z}}_3}p_1=0\mod3\) [45], [46] to rewrite \(A_{{\mathbb{Z}}_3}p_1=3\cdot \frac{A_{{\mathbb{Z}}_3}p_1}{3}\).
The third equality uses the fact that \(3 = 4 \cdot 3 \mod 9\) to rewrite \(3\cdot \frac{A_{{\mathbb{Z}}_3}p_1}{3}=4\cdot 3\cdot \frac{A_{{\mathbb{Z}}_3}p_1}{3}\mod9\) and factors out the common factor \(-4k\) of the two terms.

Thus the 4d fermionic anomaly has the anomaly index \(k \in {\mathbb{Z}}_{9}\), agreeing with the bordism group classification by \(\Omega_5^{{\rm Spin} \times {\mathbb{Z}}_{3}}={\mathbb{Z}}_9\) [10], [28]–[30], [58], [59].

When we have three right-handed neutrinos (3\(\nu_R\)), we need to consider \(k=3\) instead of \(k=1\), so eq. (9 ), with \(4k=-12 = -3 \mod 9\), becomes \[\begin{align} \label{eq:alpha5} && \exp \big( \mathrm{i}\frac{2 \pi}{3} (-1) \int_{M^5} ( {A_{{\mathbb{Z}}_3}} (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) ) \big), \end{align}\tag{10}\] which is the generator of the bosonic group cohomology 5d iTFT from \({\rm H}^5(\mathbb{Z}_n,{\rm U}(1))\cong\mathbb{Z}_n\) [60] with \(n=3\). Later Sec. 3 shows that for this specific case with \(N_f=3\), the symmetry-extension eq. (1 ) can be used to construct the 3+1d \({\mathbb{Z}}_{N_c=3}\)-gauge topological order with a low-energy 3+1d fermionic \({\mathbb{Z}}_{N_c=3}\)-gauge TQFT (as a hypothetical sector of 3+1d dark matter).

2.2 \({\rm Spin}^c \equiv {\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\rm U}(1)\) to \({\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{6}^{\rm F}}\)↩︎

In this section, we start with a perturbative local anomaly of U(1) charge \(q=1\) left-handed Weyl fermion in \({\rm Spin}^c \equiv {\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\rm U}(1)\) to derive the the nonperturbative global anomaly of \({{\mathbb{Z}}_{6,{\boldsymbol{B} + L}}^{\rm F}}\) charge \(q=1\) Weyl fermion in \({\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{6,{\boldsymbol{B} + L}}^{\rm F}}\).

First, we compare the \({\rm Spin}^c\) gauge field and the U(1) gauge field.

For \({\rm Spin}^c\), the \({\rm U}(1) \supset {{\mathbb{Z}}_2^{\rm F}}\) contains the fermion parity as a normal subgroup.

For the original \({\rm U}(1)\) with \(c_1({\rm U}(1))\), the gauge bundle constraint is \(w_2(TM)= 2 c_1 \mod 2\). In the original \({\rm U}(1)\), fermions have odd charges under \({\rm U}(1)\), while bosons have even charges under \({\rm U}(1)\). Call the original U(1) gauge field \(A\), then \(c_1=\frac{\mathrm{d}A}{2 \pi} \in \frac{1}{2}{\mathbb{Z}}\).
For the new \({\rm U}(1)'=\frac{{\rm U}(1)}{{\mathbb{Z}}_2^{\rm F}}\) with \(c_1({\rm U}(1)')\), the gauge bundle constraint is \(w_2(TM)= c_1' = 2 c_1 \mod 2\). Call the new \({\rm U}(1)'\) gauge field \(A'\), then \(c_1'=\frac{\mathrm{d}A'}{2 \pi}=\frac{\mathrm{d}(2A)}{2 \pi} = 2 c_1\in 2\frac{1}{2}{\mathbb{Z}}= {\mathbb{Z}}\).
To explain why \(A' = 2 A\) or \(c_1' = 2 c_1\), we look at the Wilson line operator \(\text{\exp(\mathrm{i}q' \oint A') and \exp(\mathrm{i}q \oint A).}\) The original \({\rm U}(1)\) has charge transformation \(\exp(\mathrm{i}q \theta)\) with \(\theta \in [0, 2 \pi)\), while the new \({\rm U}(1)'\) has charge transformation \(\exp(\mathrm{i}q' \theta')\) with \(\theta' \in [0, 2 \pi)\). But the \({\rm U}(1)'=\frac{{\rm U}(1)}{{\mathbb{Z}}_2^{\rm F}}\), so the \(\theta=\pi\) in the old \({\rm U}(1)\) is identified as \(\theta'=2\pi\) as a trivial zero in the new \({\rm U}(1)'\). In the original \({\rm U}(1)\), the \(q \in {\mathbb{Z}}\) to be compatible with \(\theta \in [0, 2 \pi)\). In the new \({\rm U}(1)'\), the original \(q\) is still allowed to have \(2{\mathbb{Z}}\) to be compatible with \(\theta \in [0, \pi)\); but the new \(q'=\frac{1}{2} q \in {\mathbb{Z}}\) and the new \(\theta'= 2 \theta \in [0, 2 \pi)\) are scaled accordingly. Since the new \(q'=\frac{1}{2} q \in {\mathbb{Z}}\), we show the new \(A'=2 A\).

The perturbative local anomaly of charge \(q=1\) left-handed Weyl fermion of \({\rm Spin}^c\) symmetry in 3+1d or 4d is captured by a 5d invertible field theory (iTFT) with the anomaly index \(k'=1\) [44]: \[\exp( \mathrm{i}k' \int_{M^5} A' \frac{(2c_1)^2}{48}-A' \frac{p_1}{48}).\] Now we redefine the U(1) gauge field \(A'\) as a \({\mathbb{Z}}_3\) gauge field \(A'_{{\mathbb{Z}}_3} \in\operatorname{H}^1(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3)= {\mathbb{Z}}_3\) with the following replacement: \[\begin{align} A' &\mapsto& \frac{2 \pi}{3} A'_{{\mathbb{Z}}_3}.\cr 2c_1=c_1' = \frac{\mathrm{d}A'}{2 \pi} &\mapsto& \frac{\mathrm{d}A'_{{\mathbb{Z}}_3}}{3} \equiv \beta_{(3,3) } A'_{{\mathbb{Z}}_3}. \end{align}\] The \(\beta_{(n,m)}: \operatorname{H}^*(-,{\mathbb{Z}}_m) \mapsto \operatorname{H}^{*+1}(-,{\mathbb{Z}}_n)\) is the Bockstein homomorphism associated with the extension \({\mathbb{Z}}_n \stackrel{\cdot m}{\to} {\mathbb{Z}}_{nm} \to {\mathbb{Z}}_m\). Thus we get the 5d topological invariant of the \({\rm Spin}\times {\mathbb{Z}}_3\) as: \[\begin{align} \label{eq:Spin-Z6-Z2} &&\exp \big( \mathrm{i}2 \pi k' \int_{M^5} ( \frac{1}{144} {A'_{{\mathbb{Z}}_3}} (\beta_{(3,3)} {A'_{{\mathbb{Z}}_3}}) (\beta_{(3,3)} {A'_{{\mathbb{Z}}_3}}) -\frac{1}{3 \cdot 48} {A'_{{\mathbb{Z}}_3}} p_1 ) \big) \cr &=&\exp \big( \mathrm{i}\frac{2 \pi}{9} k' \int_{M^5} ( \frac{1}{16} {A'_{{\mathbb{Z}}_3}} (\beta_{(3,3)} {A'_{{\mathbb{Z}}_3}}) (\beta_{(3,3)} {A'_{{\mathbb{Z}}_3}}) -\frac{1}{16} {A'_{{\mathbb{Z}}_3}} p_1 ) \big)\cr &=&\exp \big( \mathrm{i}\frac{2 \pi}{9} k' \int_{M^5} ( 4{A'_{{\mathbb{Z}}_3}} (\beta_{(3,3)} {A'_{{\mathbb{Z}}_3}}) (\beta_{(3,3)} {A'_{{\mathbb{Z}}_3}}) -4\cdot 3\cdot {} \frac{A'_{{\mathbb{Z}}_3}p_1}{3} ) \big)\cr &=&\exp \big( \mathrm{i}\frac{2 \pi}{9} (4k') \int_{M^5} ( {A'_{{\mathbb{Z}}_3}} (\beta_{(3,3)} {A'_{{\mathbb{Z}}_3}}) (\beta_{(3,3)} {A'_{{\mathbb{Z}}_3}}) - 3\cdot {} \frac{A'_{{\mathbb{Z}}_3}p_1}{3} ) \big). \end{align}\tag{11}\] Here \(\beta_{(3,3)}:\operatorname{H}^1(-,{\mathbb{Z}}_3)\to \operatorname{H}^2(-,{\mathbb{Z}}_3)\) is the Bockstein homomorphism. Here \({} \frac{A'_{{\mathbb{Z}}_3}p_1}{3}\) is a mod 3 class that involves Pontryagin class because \(A'_{{\mathbb{Z}}_3}p_1=0\mod3\) [45], [46] (see Appendix 13 for the proof), while \({A'_{{\mathbb{Z}}_3}} (\beta_{(3,3)} {A'_{{\mathbb{Z}}_3}}) (\beta_{(3,3)} {A'_{{\mathbb{Z}}_3}})\) is a mod 3 class.

In 11 , the first equality rewrites the coefficients \(\frac{1}{144}=\frac{1}{9}\cdot\frac{1}{16}\) and \(-\frac{1}{3\cdot48}=\frac{1}{9}\cdot(-\frac{1}{16})\) since the anomaly of 4d Weyl fermion with symmetry \({\rm Spin}\times_{{\mathbb{Z}}_2^{{\rm F}}}{\mathbb{Z}}_6\) contains only 3-torsion [10], [28]–[30], [58], [59], we can regard \(16\) as invertible in \({\mathbb{Z}}_9\).
The second equality uses the fact that \(1=64\mod9\) to obtain \(\frac{1}{16}=4\mod9\) and uses the fact that \(A'_{{\mathbb{Z}}_3}p_1=0\mod3\) [45], [46] to rewrite \(A'_{{\mathbb{Z}}_3}p_1=3\cdot \frac{A'_{{\mathbb{Z}}_3}p_1}{3}\).
The third equality factors out the common factor \(4k'\) of the two terms.

Eq.@eq:eq:Spin-Z3 and 11 give the same 5d topological invariant since \[\begin{align} A'_{{\mathbb{Z}}_3}=-A_{{\mathbb{Z}}_3} \mod 3, \text{ and } k' = k. \end{align}\] Here, \(A'_{{\mathbb{Z}}_3}=2A_{{\mathbb{Z}}_3}=-A_{{\mathbb{Z}}_3}\mod 3\), because \(c_1'=2c_1\) and \(A'=2A\).
Here, \(k' = k\), because we derive from the same perturbative anomaly from the same charge \(q=1\) Weyl fermion for both \({\rm Spin}\times {\rm U}(1)\) and \({\rm Spin}^c \equiv {\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\rm U}(1)\) symmetries.

Thus, the conclusion here in Sec. 2.2 follows the same as Sec. 2.1.

3 Symmetry Extension \(1 \to \mathbb{Z}_{N_c=3}\to \mathbb{Z}_{N_cN_f=9}\to \mathbb{Z}_{N_f=3}\to 1\), Anomaly Trivialization, and 3+1d Anomalous \({\mathbb{Z}}_3\)-Gauge Topologically Ordered Dark Matter↩︎

The fermionic anomaly in 3+1d or 4d spacetime with \({\mathbb{Z}}_n\) symmetry is classified by fermionic Spin bordism group \(\Omega_5^{{\rm Spin} \times {\mathbb{Z}}_{n}}\) [10], [28]–[30], [58], [59] which is isomorphic to bosonic SO bordism \(\Omega_5^{\rm SO}(\mathrm{B}{\mathbb{Z}}_{n})\) up to 2-torsion term when \(2 \nmid n\), namely \[\begin{align} &&\Omega_5^{\rm Spin}(\mathrm{B}{\mathbb{Z}}_n) \cong \tilde{\Omega}_5^{\rm SO}(\mathrm{B}{\mathbb{Z}}_n), \quad 2 \nmid n \\ && \Omega_5^{\rm Spin}(\mathrm{B}{\mathbb{Z}}_{3^r \cdot s} ) \cong \tilde{\Omega}_5^{\rm SO}(\mathrm{B}{\mathbb{Z}}_{3^r \cdot s} ) \cong {\mathbb{Z}}_{3^{r+1}}\oplus {\mathbb{Z}}_{3^{r-1}}\oplus {\mathbb{Z}}_s \oplus {\mathbb{Z}}_s, \quad 2\nmid s, \quad 3\nmid s. \end{align}\] Here \(\tilde{\Omega}_5^{\rm SO}(\mathrm{B}G):=\Omega_5^{\rm SO}(\mathrm{B}G)/\Omega_5^{\rm SO}\) is the reduced bordism group, modding out the \(\Omega_5^{\rm SO}=\Omega_5^{\rm SO}(pt)\).

In Appendix 6, we prove that only the group cohomology subclass (\(\operatorname{H}^5({\mathbb{Z}}_n,{\rm U}(1))\cong {\mathbb{Z}}_n\)) anomaly⁹ can be canceled by anomalous \(G\)-symmetric \(K={\mathbb{Z}}_n\)-gauge 4d TQFTs, via the appropriate symmetry-extension construction [1] of \[\begin{align} 1\to K \to G_{\rm Tot}\to G \to 1. \end{align}\] as \[\begin{align} \label{eq:Zn-extension} 1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\to{\mathbb{Z}}_n\to 1. \end{align}\tag{12}\] On the other hand, the beyond-group-cohomology subclass anomaly that involves \(A p_1\) (the first cohomology class \(A\) and the first Pontryagin class \(p_1\)) allows no such symmetric TQFTs.

More generally and mathematically, in this work, for odd \(d\geqslant 3\) and any \(n\geqslant 2\), we prove that any group cohomology cocycle \[\begin{align} \alpha_d \in \operatorname{H}^d({\mathbb{Z}}_n,{\rm U}(1)) \cong {\mathbb{Z}}_n \end{align}\] is trivialized by the group extension as eq. (12 )’s \(1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\to{\mathbb{Z}}_n\to 1\) [1].

In Appendix 8, we find an explicit \((d-1)\)-cochain \(\beta_{d-1}\) that splits the \(d\)-cocycle \(\alpha_d\) by that extension for odd \(d\geqslant 3\) and any \(n\geqslant 2\). Namely, \(\alpha_d=\delta \beta_{d-1}\) holds when pulling back the quotient \({\mathbb{Z}}_n\) to the total \({\mathbb{Z}}_{n^2}\) group, from the cocycle \(\alpha_d\) in \(\operatorname{H}^d({\mathbb{Z}}_n,{\rm U}(1))\) to the coboundary \[\begin{align} \tilde{\alpha}_d=\delta \tilde{\beta}_{d-1}\in \operatorname{H}^d({\mathbb{Z}}_{n^2},{\rm U}(1)). \end{align}\]

3.1 Explicit construction of 3+1d anomalous TQFT by the symmetry extension \(1\to{\mathbb{Z}}_3\to{\mathbb{Z}}_{9}\to{\mathbb{Z}}_3\to 1\)↩︎

As an application, for \(d=5\) and \(n=3\), we prove Ref. []’s statement that the symmetry-extension via eq. (1 )’s \[1 \to {\mathbb{Z}}_{N_c=3}\to{\mathbb{Z}}_{N_cN_f=9}\to{\mathbb{Z}}_{N_f=3}\to 1\] can construct a \(G={\rm Spin}\times {\mathbb{Z}}_{N_f=3}\)-symmetric \(K={\mathbb{Z}}_{N_c=3}\)-gauge 4d low-energy TQFT of gapped anomalous topologically ordered dark matter via canceling the missing \(N_f=3\) right-handed neutrinos \(\nu_R\)’s \({\mathbb{Z}}_{6,\boldsymbol{B} + L}^{\rm F}\)- or \({\mathbb{Z}}_{3,\boldsymbol{B} + L}\)-gauge-gravitational anomaly in the 4d SM. This proves a claim in Ref. [].¹⁰

More explicitly, for the \(\alpha_5\) given in eq. (10 ), \[\begin{align} \alpha_5= \exp \big( \mathrm{i}\frac{2 \pi}{3} \int_{M^5} ( {A_{{\mathbb{Z}}_3}} (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) ) \big), \end{align}\] we have \(\beta_4\) with \(\alpha_5= \delta \beta_4\), obtained in Appendix 8, which suggests a construction of the 5d iTFT on the bulk 5-manifold \(M^5\) and the 4d noninvertible TQFT on the 4d boundary \(M^4= \partial M^5\) with dynamical 1-cochain gauge field \(a_{{\mathbb{Z}}_3} \in C^1(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3)\) and 2-cochain (dual) gauge field \(b_{{\mathbb{Z}}_3} \in C^2(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3)\), such that the full 5d/4d coupled path integral is given by \[\begin{align} \label{eq:5d-4d} &&\exp \big( \mathrm{i}\frac{2 \pi}{3} \int_{M^5} ( {A_{{\mathbb{Z}}_3}} (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) ) \big) \cdot \cr && \cdot \sum_{ \substack{ a_{{\mathbb{Z}}_3} \in C^1(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3) \\ b_{{\mathbb{Z}}_3} \in C^2(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3) }}\exp \big( \mathrm{i}\frac{2 \pi}{3} \int_{M^4=\partial M^5} ( b_{{\mathbb{Z}}_3} \mathrm{d}a_{{\mathbb{Z}}_3} - b_{{\mathbb{Z}}_3} \beta_{(3,3)} {A_{{\mathbb{Z}}_3}} - a_{{\mathbb{Z}}_3} {A_{{\mathbb{Z}}_3}} \beta_{(3,3)} {A_{{\mathbb{Z}}_3}} ) \big) \cr &=&\exp \big( \mathrm{i}\frac{2 \pi}{3} \int_{M^5} ( {A_{{\mathbb{Z}}_3}} (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) ) \big) \cdot \cr && \cdot\sum_{ \substack{ a_{{\mathbb{Z}}_3} \in C^1(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3) \\ b_{{\mathbb{Z}}_3} \in C^2(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3) }}\exp \big( \mathrm{i}\frac{2 \pi}{3} \int_{M^4=\partial M^5}( a_{{\mathbb{Z}}_3}(\mathrm{d}b_{{\mathbb{Z}}_3}- {A_{{\mathbb{Z}}_3}} \beta_{(3,3)} {A_{{\mathbb{Z}}_3}})-b_{{\mathbb{Z}}_3}\beta_{(3,3)} {A_{{\mathbb{Z}}_3}} ) \big). \end{align}\tag{13}\] This 5d/4d coupled path integral analogously matches the discrete cocycle forms or cochain forms (e.g., [60]) derived in Appendix 8, as the 5-cocycle \[\alpha_5(g_1,g_2,g_3,g_4,g_5)=\zeta_n^{g_1 [\frac{g_2+g_3}{n}] [\frac{g_4+g_5}{n}]}\] and the 4-cochain \[\tilde{\beta}_4(h_1,h_2,h_3,h_4)=\zeta_n^{g_1k_2[\frac{g_3+g_4}{n}]}\] at \(n=3\), where \(\zeta_n\) is an \(n\)-th root of unity such as \(\zeta_n = \exp(\frac{2 \pi \mathrm{i}}{n})\), with variables \(g \in {\mathbb{Z}}_n\) and \(k \in {\mathbb{Z}}_n\).

The 5d bulk partition function on a 5d manifold with a 4d boundary is not gauge-invariant, but the full 5d/4d coupled path integral eq. (13 ) is gauge-invariant under the following gauge transformation: \[\begin{align} \label{eq:gauge-transformation} && A_{{\mathbb{Z}}_3} \mapsto A_{{\mathbb{Z}}_3} + \mathrm{d}\lambda_{0,{\mathbb{Z}}_3}, \cr && a_{{\mathbb{Z}}_3} \mapsto a_{{\mathbb{Z}}_3} + \mathrm{d}\mu_{0,{\mathbb{Z}}_3}, \cr && b_{{\mathbb{Z}}_3} \mapsto b_{{\mathbb{Z}}_3} +\lambda_{0,{\mathbb{Z}}_3}\beta_{(3,3)}A_{{\mathbb{Z}}_3} + \mathrm{d}\mu_{1,{\mathbb{Z}}_3}, \cr && A_{{\mathbb{Z}}_3} \in\operatorname{H}^1(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3)= {\mathbb{Z}}_3,\cr && a_{{\mathbb{Z}}_3} \in C^1(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3),\cr && b_{{\mathbb{Z}}_3} \in C^2(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3),\cr &&\lambda_{0,{\mathbb{Z}}_3} \in C^0(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3),\cr &&\mu_{0,{\mathbb{Z}}_3} \in C^0(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3),\cr &&\mu_{1,{\mathbb{Z}}_3} \in C^1(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3). \end{align}\tag{14}\] Here, \(C^k(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3)\) is the group of \({\mathbb{Z}}_3\)-valued \(k\)-cochains of the classifying space \(\mathrm{B}{\mathbb{Z}}_3\). The cohomology group \(\operatorname{H}^k(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3)\) is defined as the quotient group \(Z^k(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3)/B^k(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3)\) where \(Z^k(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3)\) is the group of \({\mathbb{Z}}_3\)-valued \(k\)-cocycles of the classifying space \(\mathrm{B}{\mathbb{Z}}_3\) and \(B^k(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3)\) is the group of \({\mathbb{Z}}_3\)-valued \(k\)-coboundaries of the classifying space \(\mathrm{B}{\mathbb{Z}}_3\).

Below we check that 13 is gauge-invariant under 14 . Because \(\beta_{(3,3)} \mathrm{d}\lambda_{0,{\mathbb{Z}}_3}=0\), \(\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}\) and \(\mathrm{d}b_{{\mathbb{Z}}_3}- {A_{{\mathbb{Z}}_3}} \beta_{(3,3)} {A_{{\mathbb{Z}}_3}}\) are gauge-invariant under the gauge transformation 14 , hence 13 transforms under 14 as \[\begin{align} && \exp \big( \mathrm{i}\frac{2 \pi}{3} \int_{M^5} ( {A_{{\mathbb{Z}}_3}} (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) ) \big) \cdot \cr && \cdot\sum_{ \substack{ a_{{\mathbb{Z}}_3} \in C^1(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3) \\ b_{{\mathbb{Z}}_3} \in C^2(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3) }}\exp \big( \mathrm{i}\frac{2 \pi}{3} \int_{M^4=\partial M^5}( a_{{\mathbb{Z}}_3}(\mathrm{d}b_{{\mathbb{Z}}_3}- {A_{{\mathbb{Z}}_3}} \beta_{(3,3)} {A_{{\mathbb{Z}}_3}})-b_{{\mathbb{Z}}_3}\beta_{(3,3)} {A_{{\mathbb{Z}}_3}} ) \big)\cr &\mapsto& \exp \big( \mathrm{i}\frac{2 \pi}{3} \int_{M^5} ( ({A_{{\mathbb{Z}}_3}}+\mathrm{d}\lambda_{0,{\mathbb{Z}}_3}) (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) (\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}) ) \big) \cdot \cr && \cdot\sum_{ \substack{ a_{{\mathbb{Z}}_3} \in C^1(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3) \\ b_{{\mathbb{Z}}_3} \in C^2(\mathrm{B}{\mathbb{Z}}_3,{\mathbb{Z}}_3) }}\exp \big( \mathrm{i}\frac{2 \pi}{3} \int_{M^4=\partial M^5}( (a_{{\mathbb{Z}}_3}+\mathrm{d}\mu_{0,{\mathbb{Z}}_3})(\mathrm{d}b_{{\mathbb{Z}}_3}- {A_{{\mathbb{Z}}_3}} \beta_{(3,3)} {A_{{\mathbb{Z}}_3}})\cr &&-(b_{{\mathbb{Z}}_3}+\lambda_{0,{\mathbb{Z}}_3}\beta_{(3,3)}A_{{\mathbb{Z}}_3} + \mathrm{d}\mu_{1,{\mathbb{Z}}_3})\beta_{(3,3)} {A_{{\mathbb{Z}}_3}} ) \big). \end{align}\] Since by the Stokes theorem, we have \[\begin{align} \int_{M^4=\partial M^5}(\mathrm{d}\mu_{0,{\mathbb{Z}}_3})(\mathrm{d}b_{{\mathbb{Z}}_3}- {A_{{\mathbb{Z}}_3}} \beta_{(3,3)} {A_{{\mathbb{Z}}_3}})=0, \end{align}\] \[\begin{align} \int_{M^4=\partial M^5}(\mathrm{d}\mu_{1,{\mathbb{Z}}_3})\beta_{(3,3)} {A_{{\mathbb{Z}}_3}}=0, \end{align}\] and \[\begin{align} \int_{M^4=\partial M^5}\lambda_{0,{\mathbb{Z}}_3}\beta_{(3,3)}A_{{\mathbb{Z}}_3}\beta_{(3,3)}A_{{\mathbb{Z}}_3}=\int_{M^5}(\mathrm{d}\lambda_{0,{\mathbb{Z}}_3})\beta_{(3,3)}A_{{\mathbb{Z}}_3}\beta_{(3,3)}A_{{\mathbb{Z}}_3}, \end{align}\] 13 is gauge-invariant under 14 .

In Appendix 9, we construct the \(d\)d-bulk/\((d-1)\)d-boundary coupled invertible topological field theory/symmetric anomalous gapped TQFT by the symmetry extension \(1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\to{\mathbb{Z}}_n\to 1\) for any odd \(d\geqslant 3\) and any \(n\geqslant 2\) explicitly.

Many more 3+1d anomalous fermionic TQFTs, which carry a mixed gauge-gravitational nonperturbative global anomaly of the \(G =\mathrm{Spin} \times_{\mathbb{Z}_2^{\rm F}} {\mathbb{Z}}_{2m}\) or \(G= \mathrm{Spin} \times \mathbb{Z}_n\) symmetry, can be found in: Cheng-Wang-Yang’s 3+1d anomalous fermionic \({\mathbb{Z}}_4\)-gauge theory [32] (see also the bosonic analogous discussion in [62]), the recent work of Décoppet-Yu [33] and Debray-Ye-Yu [34], and Wan-Wang [35]. General obstructions and constraints on the existence of these anomalous symmetric (3+1)d TQFTs are discussed in Cordova-Ohmori [31]. General properties of the anomalies of \(G =\mathrm{Spin} \times_{\mathbb{Z}_2^{\rm F}} \mathbb{Z}_{2m}^{\rm F}\) or \(\mathrm{Spin} \times \mathbb{Z}_n\) are discussed in Hsieh [28] and Wan [29], and other related nonperturbative global anomalies are discussed in Brennan-Intriligator [63].

These TQFTs can have BSM applications [12], [47], [49] for canceling SM’s nonperturbative global anomalies [10], [28], [64], [65]. For future directions, it will be interesting to explore how other nonperturbative global anomalies can constrain other QFT-coupling-to-TQFT systems, with other potential BSM applications in mind.

4 Conclusion and Discussions: General \(N_f\) family and General \(N_c\) color Standard Model: Topologically Ordered Dark Matter via symmetry extension \(1 \to {\mathbb{Z}}_{N_c}\to{\mathbb{Z}}_{N_cN_f}\to{\mathbb{Z}}_{N_f}\to 1\)↩︎

Consider the following \(N_f\) family and \(N_c\) color version of the Standard Model (SM), which is a 4d chiral gauge theory with Yang-Mills spin-1 gauge fields of the Lie algebra \[\begin{align} \label{eq:SMLieAlgebra} {\cal G} _{\rm SM} \equiv su(N_c) \times su(2) \times u(1)_{Y} \end{align}\tag{15}\] coupling to \(N_f\) families of 15 or 16 Weyl fermions (spin-\(\frac{1}{2}\) Weyl spinor is in the \({\boldsymbol{2}}_L^\mathbb{C}\) representation of the spacetime symmetry Spin(1,3), written as a left-handed 15- or 16-plet \(\psi_L\)) in the following \({\cal G} _{\rm SM}\) representation [66], [67] \[\begin{gather} \label{eq:SMrep} ({\psi_L})_{\rm I} = ( \bar{d}_R \oplus {l}_L \oplus q_L \oplus \bar{u}_R \oplus \bar{e}_R )_{\rm I} \oplus n_{\nu_{{\rm I},R}} {\bar{\nu}_{{\rm I},R}} \\ \sim \big((\overline{\boldsymbol{N}}_c,{\boldsymbol{1}})_{- (1-r) h} \oplus ({\boldsymbol{1}},{\boldsymbol{2}})_{-N_c h} \oplus ({\boldsymbol{N}}_c,{\boldsymbol{2}})_{h} \oplus (\overline{\boldsymbol{N}}_c,{\boldsymbol{1}})_{- (1+r) h} \oplus ({\boldsymbol{1}},{\boldsymbol{1}})_{2 N_c h} \big)_{\rm I} \oplus n_{\nu_{{\rm I},R}} {({\boldsymbol{1}},{\boldsymbol{1}})_{0}} \\ \sim \big((\overline{\boldsymbol{N}}_c,{\boldsymbol{1}})_{N_c -1} \oplus ({\boldsymbol{1}},{\boldsymbol{2}})_{-N_c } \oplus ({\boldsymbol{N}}_c,{\boldsymbol{2}})_{1} \oplus (\overline{\boldsymbol{N}}_c,{\boldsymbol{1}})_{- (N_c + 1) } \oplus ({\boldsymbol{1}},{\boldsymbol{1}})_{2 N_c } \big)_{\rm I} \oplus n_{\nu_{{\rm I},R}} {({\boldsymbol{1}},{\boldsymbol{1}})_{0}} \end{gather}\tag{16}\] for each family, while \(h\) is an overall normalization and \(r\) is the splitting parameter that can be solved by \(u(1)_Y^3\) cubic anomaly cancellation to find \(h=\pm N_c\) where \(h=N_c\) gives the correct choice of \(u\) and \(d\) quark charges. Here our generic \(u(1)_{Y}\) hypercharges are solved by the following anomaly-cancellation conditions \[\begin{align} u(1)_{Y}\text{-}su(N_c)^2 &:& 2 Y_{q_L} + Y_{\bar{u}_R} + Y_{\bar{d}_R} =0, \cr u(1)_{Y}\text{-}su(2)^2 &:& N_c Y_{q_L} + Y_{l_L} =0, \cr u(1)_{Y}\text{-}(\text{gravity})^2 &:& 2 N_c Y_{q_L} + N_c Y_{\bar{u}_R} + N_c Y_{\bar{d}_R} + 2 Y_{l_L}+ Y_{\bar{e}_R} + Y_{\bar{\nu}_R}=0,\cr u(1)_{Y}^3 &:& 2 N_c Y_{q_L}^3 + N_c Y_{\bar{u}_R}^3 + N_c Y_{\bar{d}_R}^3 + 2 Y_{l_L}^3+ Y_{\bar{e}_R}^3 + Y_{\bar{\nu}_R}^3=0,\cr ({\boldsymbol{B}-L})\text{-}u(1)_{Y}^2 &:& (2 Y_{q_L}^2 -Y_{\bar{u}_R}^2 -Y_{\bar{d}_R}^2) -(2 Y_{l_L}^2-Y_{\bar{e}_R}^2)=0, \end{align}\] and the solution when \(Y_{\nu_{{\rm I},R}}=0\) is given by \[\begin{align} \label{eq:hypercharge} (Y_{\bar{d}_R}, Y_{{l}_L}, Y_{q_L}, Y_{\bar{u}_R} , Y_{\bar{e}_R}, Y_{\nu_{{\rm I},R}} ) = h \times ( N_c-1 , -N_c, 1, - (N_c+1) , 2 N_c , 0) \end{align}\tag{17}\] At \(h=1\), \(N_c=3\), we get \[\begin{align} (Y_{\bar{d}_R}, Y_{{l}_L}, Y_{q_L}, Y_{\bar{u}_R} , Y_{\bar{e}_R}, Y_{\nu_{{\rm I},R}} ) = ( 2 , -3, 1, - 4 , 6 , 0). \end{align}\] So \(N_c=3\) typically goes as \[\begin{gather} ({\psi_L})_{\rm I} = ( \bar{d}_R \oplus {l}_L \oplus q_L \oplus \bar{u}_R \oplus \bar{e}_R )_{\rm I} \oplus n_{\nu_{{\rm I},R}} {\bar{\nu}_{{\rm I},R}} \sim \big((\overline{\boldsymbol{N}}_c,{\boldsymbol{1}})_{2} \oplus ({\boldsymbol{1}},{\boldsymbol{2}})_{-3} \oplus ({\boldsymbol{N}}_c,{\boldsymbol{2}})_{1} \oplus (\overline{\boldsymbol{N}}_c,{\boldsymbol{1}})_{-4} \oplus ({\boldsymbol{1}},{\boldsymbol{1}})_{6} \big)_{\rm I} \oplus n_{\nu_{{\rm I},R}} {({\boldsymbol{1}},{\boldsymbol{1}})_{0}} \end{gather}\]

The total number of Weyl fermions in \(N_f\) family for the whole multiplet of eq. (16 ) is \[\begin{align} N_f (4 N_c + 3) + \sum_{{\rm I}} n_{\nu_{{\rm I},R}}. \end{align}\] For \(N_f = N_c = 3\), this total number becomes \(3 \cdot 15 + \sum_{{\rm I}} n_{\nu_{{\rm I},R}}\).

Now the Witten SU(2) anomaly free [68] demands that the total number of 2 dimensional representation of SU(2) Weyl fermions need to be an even integer: \[\begin{align} \label{eq:Witten} &&N_f(N_c+1) \in 2 {\mathbb{Z}} \text{ for Witten SU(2) anomaly free,} \cr && \text{ so either } \left\{ \begin{array}{l} N_f \in {\mathbb{Z}}_{\text{odd}}, N_c \in {\mathbb{Z}}_{\text{odd}}, {\text{ thus baryon is a fermion}}.\\ N_f \in {\mathbb{Z}}_{\text{even}}, N_c \in {\mathbb{Z}}, {\text{ thus baryon can be a fermion (N_c \in {\mathbb{Z}}_{\text{odd}}) or a boson (N_c \in {\mathbb{Z}}_{\text{even}})}}. \end{array} \right. \end{align}\tag{18}\]

Again, we treat the anti-particle of right-handed neutrino \(\bar{\nu}_R\) as the left-handed particle, whose quantum numbers (here discrete charges of abelian global symmetries) are given by: \[\begin{align} \begin{tabular}{| c | c | c | c | c | c | c | c | c | } \hline & {\rm U}(1)_{\boldsymbol{L}} & {\rm U}(1)_{{\boldsymbol{B}}-{\boldsymbol{L}}} & {\mathbb{Z}}_{2 N_f ,{{\boldsymbol{B}} + {\boldsymbol{L}}}}^{\rm F} & {\mathbb{Z}}_2^{\rm F} & {\mathbb{Z}}_{2 N_c N_f, {\boldsymbol{Q}} + 3 {\boldsymbol{L}}}^{\rm F} \\ \hline \hline \bar{\nu}_R & -1 & 1 & -1 & 1 & -N_c \\ \hline \end{tabular} \end{align}\] Moreover, only when \(N_f\) and 2 are coprime, namely their greatest common divisor is \(\gcd(N_f,2)=1\), such as \(N_f=3,5,7,\dots\), then we further have \({\mathbb{Z}}_{2 N_f}^{\rm F}= {\mathbb{Z}}_{2}^{\rm F}\times {\mathbb{Z}}_{N_f}\), such that \(\bar{\nu}_R\) has a well-defined \({\mathbb{Z}}_{N_f ,{{\boldsymbol{B}} + {\boldsymbol{L}}}}\) charge \(-1\): \[\begin{align} \begin{tabular}{| c | c | } \hline &{\mathbb{Z}}_{2N_f ,{{\boldsymbol{B}} + {\boldsymbol{L}}}}^{\rm F} ={\mathbb{Z}}_{2}^{\rm F}\times {\mathbb{Z}}_{N_f ,{{\boldsymbol{B}} + {\boldsymbol{L}}}} \\ \hline \hline \bar{\nu}_R & -1 \sim 1 \cdot -1 \\ \hline \end{tabular}, \quad \gcd(N_f,2)=1. \end{align}\] Furthermore, only when \(N_c N_f\) and 2 are coprime, namely their greatest common divisor is \(\gcd(N_c N_f,2)=1\), then we further have \({\mathbb{Z}}_{2 N_cN_f}^{\rm F}= {\mathbb{Z}}_{2}^{\rm F}\times {\mathbb{Z}}_{N_c N_f}\), such that \(\bar{\nu}_R\) has a well-defined \({\mathbb{Z}}_{N_cN_f, {{\boldsymbol{Q}} + N_c{\boldsymbol{L}}}= N_c({{\boldsymbol{B}} + {\boldsymbol{L}}}) }\) charge \(-N_c\): \[\begin{align} \begin{tabular}{| c | c | } \hline &{\mathbb{Z}}_{2 N_c N_f ,{{\boldsymbol{Q}} + N_c{\boldsymbol{L}}}}^{\rm F} ={\mathbb{Z}}_{2}^{\rm F}\times {\mathbb{Z}}_{N_c N_f ,{{\boldsymbol{Q}} + N_c{\boldsymbol{L}}}} \\ \hline \hline \bar{\nu}_R & -N_c \sim 1 \cdot -N_c \\ \hline \end{tabular} , \quad \gcd(N_c N_f,2)=1. \end{align}\] For a generic \(N_f\)-family SM missing some \(\bar{\nu}_R\), we have to determine its anomaly index in \(\Omega_5^{{\rm Spin} \times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2N_f ,{{\boldsymbol{B}} + {\boldsymbol{L}}}}^{\rm F}}\), which we require the following bordism group classification (let \(N_f={2^{p} \cdot 3^r \cdot s}\)) \[\begin{align} \label{eq:bordism-group} &&\Omega_5^{{\rm Spin} \times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{2^{p+1} \cdot 3^r \cdot s }}} \cong \Omega_5^{{\rm Spin} \times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{2^{p+1} }}} \oplus \tilde{\Omega}_5^{\rm SO}(\mathrm{B}{\mathbb{Z}}_{3^r \cdot s} ) \cr &&= {\mathbb{Z}}_{2^{p+3}}\oplus {\mathbb{Z}}_{2^{p-1}}\oplus {\mathbb{Z}}_{3^{r+1}}\oplus {\mathbb{Z}}_{3^{r-1}}\oplus {\mathbb{Z}}_s \oplus {\mathbb{Z}}_s,\quad p\geqslant 1,\quad r\geqslant 1,\quad 2\nmid s,\quad 3\nmid s. \end{align}\tag{19}\] Here \(\tilde{\Omega}_5^{\rm SO}(\mathrm{B}G):=\Omega_5^{\rm SO}(\mathrm{B}G)/\Omega_5^{\rm SO}\) is the reduced bordism group, modding out the \(\Omega_5^{\rm SO}=\Omega_5^{\rm SO}(pt)\).

Now we ask two general questions relevant for high-energy phenomenology for \(G={\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2 N_f,{{\boldsymbol{B}} + {\boldsymbol{L}}}}\) symmetry:

1. For a single charge \(q=-1 \in {\mathbb{Z}}_{2 N_f,{{\boldsymbol{B}} + {\boldsymbol{L}}}} \subset {\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2 N_f,{{\boldsymbol{B}} + {\boldsymbol{L}}}}\)-symmetry \(\bar{\nu}_R\), can there exists a symmetric-gapped 4d TQFT matching the \(\bar{\nu}_R\)’s symmetry and anomaly in the full \({\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2 N_f,{{\boldsymbol{B}} + {\boldsymbol{L}}}}\)?

   The answer to this question is the same as asking in the case of a single charge \(q=1 \in {\mathbb{Z}}_{2 N_f} \subset {\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2 N_f}\)-symmetry Weyl fermion, up to a \(-1\) sign of the chosen basis. The answer is no, there exists no such symmetric-gapped 4d TQFT matching \(q=1\) or \(-1\) Weyl fermion’s anomaly in general.

2. For \(N_f\) copies of charge \(q=-1 \in {\mathbb{Z}}_{2 N_f,{{\boldsymbol{B}} + {\boldsymbol{L}}}} \subset {\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2 N_f,{{\boldsymbol{B}} + {\boldsymbol{L}}}}\)-symmetry \(\bar{\nu}_R\), can there exists a symmetric-gapped 4d TQFT matching the \(\bar{\nu}_R\)’s symmetry and anomaly in the full \({\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2 N_f,{{\boldsymbol{B}} + {\boldsymbol{L}}}}\)?

   The answer to this question is the same as asking in the case of \(N_f\) copies of charge \(q=1 \in {\mathbb{Z}}_{2 N_f} \subset {\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2 N_f}\)-symmetry Weyl fermions, up to a \(-1\) sign of the chosen basis. The answer is in general yes, there exists such symmetric-gapped 4d TQFT matching \(N_f\) copies of \(q=1\) or \(-1\) Weyl fermion’s anomaly in general.

   But there is a refined question: Is this \(N_f\) copies of Weyl fermion anomaly within a group cohomology (GC) class or beyond a group cohomology (BGC) class?

   1. For group cohomology (GC) class, Ref. [] shows that there always exists a symmetric anomalous gapped boundary with a finite abelian gauge group as the low-energy TQFT (here in 4d) to cancel the GC class

   2. For beyond a group cohomology (BGC) class, Ref. [] cannot show that a symmetric anomalous gapped boundary with a finite abelian gauge group exists or not. But we are able to determine what are the minimal finite group \(K\) symmetry extension that can trivialize the anomaly in \(G\) via pulling back through \(1\to K \to G_{\rm Tot}\to G \to 1\) to the anomaly-free in \(G_{\rm Tot}\).

Due to Witten’s SU(2) anomaly free constraint in eq. (18 ), we summarize the results in two cases, eq. (20 ) and eq. (21 ). Here we determine the minimal finite abelian \(K\)-gauge group extension for the generalized SM with \(N_f\) family number (in the column) and \(N_c\) color number (in the row). The symmetry-extension trivialization via \(K\) means that we can replace \(N_f\) copies of \(\bar{\nu}_R\) by \(K\)-gauge symmetric-gapped 4d TQFT (namely with a gauge group \(K\)). The color index \(N_c\) does not directly affect the minimal \(K\)-gauge TQFT. But for a certain appropriate \(N_c\), when \(K={\mathbb{Z}}_{N_c}\), there is an interesting interplay between \(N_c\) color and \(N_f\) family.

1. The \(N_f \in {\mathbb{Z}}_{\text{odd}}\) and \(N_c \in {\mathbb{Z}}_{\text{odd}}\) case gives rise to the following relation in a table: \[\begin{align} \label{eq:Nfodd} \begin{tabular}{| c | c | c | c | c | c | c | c | c | c |} \hline K-group extension & N_f=1 & \textcolor{white}{N_f=2} & N_f=3 & \textcolor{white}{N_f=4} & N_f=5 & \textcolor{white}{N_f=6} & N_f=7 & \textcolor{white}{N_f=8} & N_f=9 \\ \hline \hline & & & & & & & & &\\ N_c=3 & No & & \textcolor{black}{{\mathbb{Z}}_{3=N_c}} GC & & Trivial & & Trivial & & {{\mathbb{Z}}_{3}} GC (9 \bar{\nu}_{R}) \\ & & & & & & & &&\\ N_c=5 & No & & {\mathbb{Z}}_3 GC & & Trivial & & Trivial && \\ & & & & &&& && \\ N_c=7 & No & & {\mathbb{Z}}_3 GC & & Trivial & & Trivial && \\ & & & & &&& && \\ N_c=9 & No & & {\mathbb{Z}}_3 GC & & Trivial & & Trivial && {{\mathbb{Z}}_{9}} GC (3 \bar{\nu}_{R}) \\ \hline \end{tabular}. \end{align}\tag{20}\]

   1. For \(N_f=1\), there is no anomaly. We consider the SM missing \(N_f=1\) \(\bar{\nu}_{R}\), thus “No” means “No anomaly” (even for a single Weyl fermion) and “No TQFT.”

   2. For \(N_f=3\), there is a \(\Omega_5^{{\rm Spin} \times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{6}^{\rm F}}}= \Omega_5^{{\rm Spin} \times {{\mathbb{Z}}_{3}}}={\mathbb{Z}}_9\) class anomaly. We consider the SM missing \(N_f=3\) \(\bar{\nu}_{R}\), there “\({\mathbb{Z}}_3\) GC” means \(K={\mathbb{Z}}_3\)-gauge TQFT can match the group cohomology \({\mathbb{Z}}_3\) subclass anomaly. When \(N_c=3\), we have a \(K={\mathbb{Z}}_{N_c=3}\)-gauge TQFT that the \(K\)-extension matches an intriguing \({\mathbb{Z}}_{N_c=3}\)-color extension, because surprisingly \({N_c} = {N_f=3}\) in this case.

   3. For \(N_f=5\), \(N_f=7\), or other \(N_f\) such that \(2 \nmid N_f\) and \(3 \nmid N_f\), there is a \(\Omega_5^{{\rm Spin} \times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{2 N_f}}}= \Omega_5^{{\rm Spin} \times {{\mathbb{Z}}_{N_f}}}={\mathbb{Z}}_{N_f} \oplus {\mathbb{Z}}_{N_f}\) class anomaly. Although a generic number of \(\bar{\nu}_{R}\) can contribute an anomaly, when we consider the SM missing \(N_f\) \(\bar{\nu}_{R}\), the total anomaly class is trivial, thus we write “Trivial” in this case and there is also no need to have any 4d TQFT to cancel the anomaly.

   4. For \(N_f=9\), there is a \(\Omega_5^{{\rm Spin} \times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{18}^{\rm F}}}= \Omega_5^{{\rm Spin} \times {{\mathbb{Z}}_{9}}}= {\mathbb{Z}}_{27} \oplus {\mathbb{Z}}_3\) class anomaly.

      We consider the SM missing \(N_f=9\) \(\bar{\nu}_{R}\), there “\({\mathbb{Z}}_3\) GC” means \(K={\mathbb{Z}}_3\)-gauge TQFT can match the group cohomology \({\mathbb{Z}}_3\) subclass anomaly. When \(N_c=3\), we have a \(K={\mathbb{Z}}_{N_c=3}\)-gauge TQFT that the \(K\)-extension matches a \({\mathbb{Z}}_{N_c=3}\)-color extension, but \({N_c=3} \neq {N_f=9}\) in this case.

      Instead if we consider the (\(N_f=9\))-SM missing \(3\) \(\bar{\nu}_{R}\), there “\({\mathbb{Z}}_9\) GC” means \(K={\mathbb{Z}}_9\)-gauge TQFT can match the group cohomology \({\mathbb{Z}}_9\) subclass anomaly. When \(N_c=9\), we have a \(K={\mathbb{Z}}_{N_c=9}\)-gauge TQFT that the \(K\)-extension matches a \({\mathbb{Z}}_{N_c=9}\)-color extension, although \({N_c} ={N_f=9}\) in this case, we need to have 6 extra \(\bar{\nu}_{R}\) added into the SM.

      Thus we show that \({N_c} = {N_f=3}\) case is more natural in terms of the \({\mathbb{Z}}_{N_c}\)-color extension.

      While \(N_f=9\) or higher 3-torsions \[N_f=3^r, r \geq 2,\] for the (\(N_f=3^r\))-SM missing \(N_f=3^r\) copies of sterile neutrinos \(\bar{\nu}_{R}\), there we also have “\({\mathbb{Z}}_3\) GC” means \(K={\mathbb{Z}}_3\)-gauge TQFT can match the group cohomology \({\mathbb{Z}}_3\) subclass anomaly. When \(N_c=3\), we have a \(K={\mathbb{Z}}_{N_c=3}\)-gauge TQFT that the \(K\)-extension matches a \({\mathbb{Z}}_{N_c=3}\)-color extension, but \({N_c =3} \neq {N_f=3^r}\), with \(r \geq 2\) in this case.

      Here we need to quote our results derived in Appendices 6 and 11, the \(k=3^r\) anomaly of the 4d Weyl fermion for \(k=3 \in {\mathbb{Z}}_{3^{r+1}} \subset \Omega_5^{\rm Spin}(\mathrm{B}{\mathbb{Z}}_{3^r } ) \cong {\mathbb{Z}}_{3^{r+1}}\oplus {\mathbb{Z}}_{3^{r-1}}\) eq. (37 ) with \({\rm Spin}\times{\mathbb{Z}}_{3^r}\) symmetry can be trivialized by a \({\mathbb{Z}}_3\) extension.

2. The \(N_f \in {\mathbb{Z}}_{\text{even}}\) and \(N_c \in {\mathbb{Z}}\) case gives rise to the following \(K\)-extension: \[\begin{align} \label{eq:Nfeven} \begin{tabular}{| c | c | c | c | c | c | c | c |} \hline K-group extension & \textcolor{white}{N_f=1} & N_f=2 & \textcolor{white}{N_f=3} & N_f=4 & \textcolor{white}{N_f=5} & N_f=6 & \textcolor{white}{N_f=7} \\ \hline \hline N_c=2 & & {{\mathbb{Z}}_{4}} BGC & & {{\mathbb{Z}}_{4}} BGC & & {{\mathbb{Z}}_{12}} BGC &\\ N_c=3 & & {{\mathbb{Z}}_{4}} BGC & & {{\mathbb{Z}}_{4}} BGC & & {{\mathbb{Z}}_{12}} BGC & \\ N_c=4 & & \textcolor{black}{{\mathbb{Z}}_{4=N_c}} BGC & & \textcolor{black}{{\mathbb{Z}}_{4=N_c}} BGC & & {{\mathbb{Z}}_{12}} BGC & \\ N_c=5 & & {{\mathbb{Z}}_{4}} BGC & & {{\mathbb{Z}}_{4}} BGC & & {{\mathbb{Z}}_{12}} BGC & \\ N_c=6 & & {{\mathbb{Z}}_{4}} BGC & & {{\mathbb{Z}}_{4}} BGC & & {{\mathbb{Z}}_{12}} BGC & \\ N_c=7 & & {{\mathbb{Z}}_{4}} BGC & & {{\mathbb{Z}}_{4}} BGC & & {{\mathbb{Z}}_{12}} BGC & \\ \dots & & \dots & & \dots & & \dots & \\ N_c=12 & & {{\mathbb{Z}}_{4}} BGC & & {{\mathbb{Z}}_{4}} BGC & & \textcolor{black}{{\mathbb{Z}}_{12=N_c}} BGC &\\ \hline \end{tabular}. \end{align}\tag{21}\]

   1. For \(N_f=2\), there is a \(\Omega_5^{{\rm Spin} \times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{4}^{\rm F}}}= {\mathbb{Z}}_{16}\) class anomaly. We consider the SM missing \(N_f=2\) \(\bar{\nu}_{R}\), there “\({\mathbb{Z}}_4\) BGC” means \(K={\mathbb{Z}}_4\)-gauge TQFT can match the beyond-the-group-cohomology (BGC) \({\mathbb{Z}}_8\) subclass anomaly. When \(N_c=4\), we have a \(K={\mathbb{Z}}_{N_c=4}\)-gauge TQFT that the \(K\)-extension matches a \({\mathbb{Z}}_{N_c=4}\)-color extension, but \({N_c=4} \neq {N_f=2}\) in this case.

   2. For \(N_f=4\), there is a \(\Omega_5^{{\rm Spin} \times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{8}^{\rm F}}}= {\mathbb{Z}}_{32} \oplus {\mathbb{Z}}_2\) class anomaly. We consider the SM missing \(N_f=4\) \(\bar{\nu}_{R}\), there “\({\mathbb{Z}}_4\) BGC” means \(K={\mathbb{Z}}_4\)-gauge TQFT can match the beyond-the-group-cohomology (BGC) \({\mathbb{Z}}_8\) subclass anomaly. When \(N_c=4\), we have a \(K={\mathbb{Z}}_{N_c=4}\)-gauge TQFT that the \(K\)-extension matches a \({\mathbb{Z}}_{N_c=4}\)-color extension, so \({N_c} = {N_f=4}\) in this case.

   3. For \(N_f=6\), there is a \(\Omega_5^{{\rm Spin} \times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{12}^{\rm F}}}= {\mathbb{Z}}_{16} \oplus {\mathbb{Z}}_9\) class anomaly. We consider the SM missing \(N_f=6\) \(\bar{\nu}_{R}\), there “\({\mathbb{Z}}_{12}\) BGC” means \({\mathbb{Z}}_4\)-gauge TQFT can match the beyond-the-group-cohomology (BGC) \({\mathbb{Z}}_8 \in {\mathbb{Z}}_{16}\) subclass anomaly and an additional \({\mathbb{Z}}_3\)-gauge TQFT can match the group-cohomology (GC) \({\mathbb{Z}}_3 \in {\mathbb{Z}}_{9}\) subclass anomaly. When \(N_c={12}\), we have a \(K={\mathbb{Z}}_{N_c=12}\)-gauge TQFT that the \(K\)-extension matches a \({\mathbb{Z}}_{N_c=12}\)-color extension, but \({N_c=12} \neq {N_f=6}\) in this case.

   4. For \(N_f \in {\mathbb{Z}}_{\text{even}}\), we can go through similar discussions like the above.

To conclude, for the general \(N_f\) family and general \(N_c\) color Standard Model (SM), we find the following case of \(\mathbb{Z}_{N_c}\) color symmetry-extension \[\begin{align} 1 \to \mathbb{Z}_{N_c}\to {\rm Spin}\times \mathbb{Z}_{N_cN_f}\to {\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2 N_f} \to 1 \end{align}\] can trivialize the \(N_f\) copies of sterile neutrinos \(\bar{\nu}_{R}\) (and also its corresponding SM complement) such that the following three cases are the most intriguing: \[\begin{align} \left\{ \begin{array}{l} N_c=3,\; N_f=3^r s, \;r \geqslant 1,\; 2\nmid s, \;3\nmid s, {\text{ the group-cohomology anomaly. Baryon is a fermion}}.\\ N_c=4,\; N_f=2^p s, \;p\geqslant 1,\; 2\nmid s,\; 3\nmid s, {\text{ beyond-the-group-cohomology anomaly. Baryon is a boson}}.\\ N_c=12,\; N_f=2^p 3^r s,\; p\geqslant 1, \;r\geqslant 1,\; 2\nmid s,\; 3\nmid s, {\text{ beyond-the-group-cohomology anomaly. Baryon is a boson}}.\\ N_c=1,\; N_f, \; 2\nmid N_f,\; 3\nmid N_f, {\text{ no anomaly. Baryon does not exist.}} \end{array} \right. \end{align}\] The final case is in fact one less interesting case, when \(2\nmid N_f\) and \(3\nmid N_f\), such that no color extension is required because there is no anomaly for missing \(N_f\) of \(\nu_R\) neutrinos. Thus there is no color gauge group \(N_c=1\), and no QCD. We shall exclude the final case because no baryon is not a physical situation in the real world.

We can prove the following theorem, which is a restatement of Theorem 1.

In summary, if we restrict to the more familiar generalized SM with baryon as a fermion, and if we require that \(N_c\) and \(N_f\) are minimal, then \(N_c=N_f=3\) emerges as the unique case for constructing a 4d anomalous \({\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2 N_f,{{\boldsymbol{B}} + {\boldsymbol{L}}}}\)-symmetric gapped topological order with low-energy TQFT, such that the missing \(N_f\) copies of the sterile neutrinos \(\bar{\nu}_{R}\) can be naturally replaced by a 4d \({\mathbb{Z}}_{N_c}\)-gauge fermionic TQFT.

Finally, we remark that our scenario shall be different from the Dark Dimension [69] scenario involving:

• Higgs mechanism on the \({{\boldsymbol{B}} - {\boldsymbol{L}}}\) gauge field sector — note that the Higgs mechanism involves the symmetry-breaking mechanism.

• 3 right-handed neutrinos propagate in the 5th Dark Dimension.

In our case and in parallel work [12], [35], we emphasize:

• Symmetry-extension construction of anomalous topological order is beyond the Anderson-Higgs mechanism, different from the symmetry-breaking mechanism.

• Massive 4+1d Dirac fermion with a relative sign \(\pm 1\) of mass flips can give rise to 4+1d invertible topological field theory (iTFT)/ Symmetry-Protected Topological states (SPTs) in the 4+1d bulk. However, the anomalous topological order lives on 3+1d, which can be attached to the boundary of 4+1d bulk; or simply attached to the 3+1d SM without the need of the 4+1d bulk at all.

5 Acknowledgment↩︎

JW thanks Dan Freed, Pavel Putrov, Constantin Teleman, and Matthew Yu for the discussions on the related topics. ZW is supported by the NSFC Grant No. 12405001. JW is supported by LIMS and Ben Delo Fellowshop. JW would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Diving Deeper into Defects: On the Intersection of Field Theory, Quantum Matter, and Mathematics, where work on this paper was undertaken. This work was supported by EPSRC grant EP/Z000580/1. JW also thanks Simons Foundation Collaboration on Global Categorical Symmetries Annual Meetings in 2024 and 2025, where this work is discussed and performed during the meetings.

6 Any cocycle \(\alpha_d \in \operatorname{H}^d({\mathbb{Z}}_n,{\rm U}(1))\) is trivialized by the symmetry extension \(1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\to{\mathbb{Z}}_n\to 1\) for odd \(d\geqslant 3\) and any \(n\geqslant 2\)↩︎

In this appendix, we show that any cocycle \(\alpha\in \operatorname{H}^d({\mathbb{Z}}_n,{\rm U}(1))\) is trivialized by the symmetry extension \(1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\to{\mathbb{Z}}_n\to1\) for odd \(d\geqslant 3\) and any \(n\geqslant 2\).

We consider the Lyndon-Hochschild-Serre (LHS) spectral sequence \[\begin{align} \label{eq:SSS} E_2^{p,q}=\operatorname{H}^p({\mathbb{Z}}_n,\operatorname{H}^q({\mathbb{Z}}_n,{\rm U}(1)))\Rightarrow \operatorname{H}^{p+q}({\mathbb{Z}}_{n^2},{\rm U}(1)) \end{align}\tag{22}\] associated with the extension \(1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\to{\mathbb{Z}}_n\to1\).

Since \[\begin{align} \operatorname{H}^d({\mathbb{Z}}_n,{\rm U}(1))=\left\{\begin{array}{lll}{\mathbb{Z}}_n&d\text{ odd}\\0&d\text{ even}>0\\{\rm U}(1)&d=0\end{array}\right. \end{align}\] and \[\begin{align} \operatorname{H}^d({\mathbb{Z}}_n,{\mathbb{Z}}_n)={\mathbb{Z}}_n\;\;\;\forall d\geqslant 0, \end{align}\] the \(E_2\) page of the LHS spectral sequence 22 is shown in Fig. 1.

None

The differentials in the LHS spectral sequence 22 are \[\begin{align} d_r^{p,q}:E_r^{p,q}\to E_r^{p+r,q-r+1}\text{ for }r\geqslant 2, \end{align}\] and the pages \(E_r\) are defined inductively from \(E_2\) by \[\begin{align} E_{r+1}^{p,q}=\frac{\text{Ker }d_r^{p,q}}{\text{Im }d_r^{p-r,q+r-1}}. \end{align}\] The differentials \(d_r\) vanish and the pages \(E_r\) stabilize for sufficiently large \(r\geqslant N\). The page \(E_N\) is denoted \(E_{\infty}\).

The homomorphism \(\operatorname{H}^d({\mathbb{Z}}_n,{\rm U}(1))\to \operatorname{H}^d({\mathbb{Z}}_{n^2},{\rm U}(1))\) induced from the extension \(1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\to{\mathbb{Z}}_n\to1\) is the composition \[\begin{align} \label{eq:composition} E_2^{d,0}=\operatorname{H}^d({\mathbb{Z}}_n,{\rm U}(1))\twoheadrightarrow E_{\infty}^{d,0}\hookrightarrow \operatorname{H}^d({\mathbb{Z}}_{n^2},{\rm U}(1)). \end{align}\tag{23}\]

For \(p+q=d\), there is a filtration \[\begin{align} F^{-1}=0\subset F^0\subset F^1\subset\cdots\subset F^d=\operatorname{H}^d({\mathbb{Z}}_{n^2},{\rm U}(1)) \end{align}\] of \(\operatorname{H}^d({\mathbb{Z}}_{n^2},{\rm U}(1))\) with \[\begin{align} F^q/F^{q-1}=E_{\infty}^{d-q,q}. \end{align}\]

We will show that \(E_{\infty}^{d,0}=0\) for odd \(d\geqslant 3\) and any \(n\geqslant 2\), hence by 23 , any cocycle \(\alpha\in \operatorname{H}^d({\mathbb{Z}}_n,{\rm U}(1))\) is trivialized by the extension \(1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\to{\mathbb{Z}}_n\to1\) for odd \(d\geqslant 3\) and any \(n\geqslant 2\).

Since \(\operatorname{H}^2({\mathbb{Z}}_{n^2},{\rm U}(1))=0\), the group \(E_2^{1,1}={\mathbb{Z}}_n\) is eliminated by the differential \[d_2^{1,1}:E_2^{1,1}\longrightarrow E_2^{3,0},\] so it does not survive to the \(E_3\)–page. Hence \(E_{\infty}^{3,0}=E_3^{3,0}=0\).

Since \(\operatorname{H}^4({\mathbb{Z}}_{n^2},{\rm U}(1))=0\), the groups \(E_2^{1,3}={\mathbb{Z}}_n\) and \(E_2^{3,1}={\mathbb{Z}}_n\) are removed on some page by differentials. On the other hand, because \(\operatorname{H}^3({\mathbb{Z}}_{n^2},{\rm U}(1))={\mathbb{Z}}_{n^2}\) and \(E_2^{0,3}=E_2^{2,1}={\mathbb{Z}}_n\), and since \(E_2^{3,0}={\mathbb{Z}}_n\) is eliminated by the differential \(d_2^{1,1}:E_2^{1,1}\to E_2^{3,0}\), the groups \(E_2^{0,3}={\mathbb{Z}}_n\) and \(E_2^{2,1}={\mathbb{Z}}_n\) survive to the \(E_{\infty}\)–page.

Therefore \(E_2^{3,1}={\mathbb{Z}}_n\) cannot be the target of any nonzero differential; instead it supports the differential \[d_2^{3,1}:E_2^{3,1}\longrightarrow E_2^{5,0},\] and so does not persist to \(E_{\infty}\). Hence \(E_{\infty}^{5,0}=E_3^{5,0}=0\).

Similarly, \(E_2^{1,3}=E_3^{1,3}={\mathbb{Z}}_n\) cannot be the target of any nonzero differential; it is the source of the differential \[d_3:E_3^{1,3}\longrightarrow E_3^{4,1},\] and therefore \(E_{\infty}^{4,1}=E_4^{4,1}=0\).

Since \(\operatorname{H}^6({\mathbb{Z}}_{n^2},{\rm U}(1))=0\), the group \(E_2^{5,1}={\mathbb{Z}}_n\) is eliminated by some differential. On the other hand, because \(\operatorname{H}^5({\mathbb{Z}}_{n^2},{\rm U}(1))={\mathbb{Z}}_{n^2}\) and \(E_2^{0,5}=E_2^{2,3}={\mathbb{Z}}_n\), while \(E_2^{4,1}={\mathbb{Z}}_n\) and \(E_2^{5,0}\) are removed by the differentials \(d_3\) and \(d_2\) respectively, the groups \(E_2^{0,5}={\mathbb{Z}}_n\) and \(E_2^{2,3}={\mathbb{Z}}_n\) survive to \(E_{\infty}\). Hence \(E_2^{5,1}={\mathbb{Z}}_n\) cannot be the target of a nonzero differential; instead it supports \[d_2^{5,1}:E_2^{5,1}\longrightarrow E_2^{7,0},\] and therefore \(E_{\infty}^{7,0}=E_3^{7,0}=0\).

In fact, one can show in general that \(E_2^{d,0}\) is eliminated by the differential \[d_2^{d-2,1}:E_2^{d-2,1}\longrightarrow E_2^{d,0},\] so \(E_{\infty}^{d,0}=E_3^{d,0}=0\) for every odd \(d\geqslant 3\) and any \(n\geqslant 2\).

The cohomology ring \(\operatorname{H}^*({\mathbb{Z}}_n,{\mathbb{Z}}_n)\) is generated by \(x\in \operatorname{H}^1({\mathbb{Z}}_n,{\mathbb{Z}}_n)\) and \(y\in \operatorname{H}^2({\mathbb{Z}}_n,{\mathbb{Z}}_n)\) and they satisfy the relation \(x^2=0\) for odd \(n\) and \(x^2=\frac{n}{2}y\) for even \(n\) [70]. In particular, for \(n=2\), \(x^2=y\) and \(\operatorname{H}^*({\mathbb{Z}}_2,{\mathbb{Z}}_2)\) is generated by \(x\in \operatorname{H}^1({\mathbb{Z}}_2,{\mathbb{Z}}_2)\). The generator of \(E_2^{1,1}={\mathbb{Z}}_n\) is \(x\) and the generator of \(E_2^{2,1}={\mathbb{Z}}_n\) is \(y\). Since we have shown that \(d_2(x)\) is non-trivial and \(d_2(y)=0\), the differentials are derivations, and \(\smile y: \operatorname{H}^m({\mathbb{Z}}_n,{\rm U}(1))\to \operatorname{H}^{m+2}({\mathbb{Z}}_n,{\rm U}(1))\) is an isomorphism for odd \(m\)¹¹, for \(n=2\), \(d_2(x^k)=kd_2(x)x^{k-1}\) is non-trivial for odd \(k\), and for \(n>2\), \(d_2(xy^k)=d_2(x)y^k\) is non-trivial for all \(k\geqslant 0\). Hence \(E_{\infty}^{d,0}=E_3^{d,0}=0\) for odd \(d\geqslant 3\) and any \(n\geqslant 2\).

7 \(A_{{\mathbb{Z}}_n}p_1\) cannot be trivialized by any finite group extension except for \(n=2\) and \(n=3\)↩︎

In this appendix, we show that \(A_{{\mathbb{Z}}_n}p_1\) cannot be trivialized by any finite group extension except for \(n=2\) and \(n=3\) where \(A_{{\mathbb{Z}}_n}\) is the generator of \(\operatorname{H}^1({\mathbb{Z}}_n,{\rm U}(1))\), here \(A_{{\mathbb{Z}}_n}p_1\) is defined on any dimensional manifold \(M\) with \({\rm Spin}\times_{{\mathbb{Z}}_2^{{\rm F}}}{\mathbb{Z}}_{2n}\) structure and \(p_1=p_1(TM)\).

For any group extension \[\begin{align} 1\to K\to H\to G={\mathbb{Z}}_n\to 1, \end{align}\] we have a similar LHS spectral sequence \[\begin{align} E_2^{p,q}=\operatorname{H}^p({\mathbb{Z}}_n,\operatorname{H}^q(K,{\rm U}(1)))\Rightarrow\operatorname{H}^{p+q}(H,{\rm U}(1)). \end{align}\] For degree reasons, there are no differentials from or to \(E_2^{1,0}=\operatorname{H}^1({\mathbb{Z}}_n,{\rm U}(1))\), so \(E_2^{1,0}=\operatorname{H}^1({\mathbb{Z}}_n,{\rm U}(1))\) survives to the \(E_{\infty}\) page. Therefore, \(A_{{\mathbb{Z}}_n}\in\operatorname{H}^1({\mathbb{Z}}_n,{\rm U}(1))\) cannot be trivialized by any group extension.

7.1 \({\rm Spin}\times_{{\mathbb{Z}}_2^{{\rm F}}}{\mathbb{Z}}_{2n}\) structure and symmetry extension trivialization↩︎

Now we consider the availability of the symmetry extension trivialization of \(A_{{\mathbb{Z}}_n}p_1\) on the manifolds with \({\rm Spin}\times_{{\mathbb{Z}}_2^{{\rm F}}}{\mathbb{Z}}_{2n}\) structure. Let us explain the exceptional cases \(n=2\) and \(n=3\):

• For \(n=3\), we have \(A_{{\mathbb{Z}}_3}p_1=0\mod 3\) [45], [46] (see Appendix 13 for the proof) where \(p_1\) is the first Pontryagin class. For other \(n\), \(A_{{\mathbb{Z}}_n}p_1\not\equiv0\mod n\).

• For \(n=2\) and \({\rm Spin}\times_{{\mathbb{Z}}_2^{{\rm F}}}{\mathbb{Z}}_4\) structure, we consider the symmetry extension \[\begin{align} \label{eq:extension-Spin-Z4} 1\to{\mathbb{Z}}_2^{{\rm F}}\to{\rm Spin}\times{\mathbb{Z}}_4\xrightarrow{f}{\rm Spin}\times_{{\mathbb{Z}}_2^{{\rm F}}}{\mathbb{Z}}_4\to1. \end{align}\tag{24}\] Since \(\operatorname{H}^4(\mathrm{B}{\rm Spin},{\mathbb{Z}})\) is generated by \(\lambda=\frac{p_1}{2}\), \(f^*(p_1)=2\lambda\). Also, \(f^*(A_{{\mathbb{Z}}_2})=2A_{{\mathbb{Z}}_4}\). Hence \(f^*(A_{{\mathbb{Z}}_2}p_1)=2A_{{\mathbb{Z}}_4}\cdot2\lambda=0\mod4\). Hence, \(A_{{\mathbb{Z}}_2}p_1\) can be trivialized by the finite group extension 24 .

Next, for \({\rm Spin}\times_{{\mathbb{Z}}_2^{{\rm F}}}{\mathbb{Z}}_{2n}\) structure, let us explain the remaining cases.

• For odd \(n>3\), we note that \({\rm Spin}\times_{{\mathbb{Z}}_2^{{\rm F}}}{\mathbb{Z}}_{2n}={\rm Spin}\times{\mathbb{Z}}_n\). Because \(\pi_1({\rm Spin})=0\), there are no finite covers of \({\rm Spin}\). Since \(\operatorname{H}^4(\mathrm{B}{\rm Spin},{\mathbb{Z}})\) is generated by \(\lambda=\frac{p_1}{2}\), \(A_{{\mathbb{Z}}_n}p_1=A_{{\mathbb{Z}}_n}\cdot2\lambda\). For odd \(n>3\), because \(2 \neq 0 \mod n\), so \(A_{{\mathbb{Z}}_n}p_1=A_{{\mathbb{Z}}_n}\cdot2\lambda\) cannot be trivialized by any finite group extension.

• For even \(n>2\) and \({\rm Spin}\times_{{\mathbb{Z}}_2^{{\rm F}}}{\mathbb{Z}}_{2n}\) structure, we consider the symmetry extension \[\begin{align} \label{eq:extension-Spin-Z2n} 1\to{\mathbb{Z}}_2^{{\rm F}}\to{\rm Spin}\times{\mathbb{Z}}_{2n}\xrightarrow{g}{\rm Spin}\times_{{\mathbb{Z}}_2^{{\rm F}}}{\mathbb{Z}}_{2n}\to1. \end{align}\tag{25}\] Since \(\operatorname{H}^4(\mathrm{B}{\rm Spin},{\mathbb{Z}})\) is generated by \(\lambda=\frac{p_1}{2}\), \(g^*(p_1)=2\lambda\). Also, \(g^*(A_{{\mathbb{Z}}_n})=2A_{{\mathbb{Z}}_{2n}}\). Hence \(g^*(A_{{\mathbb{Z}}_n}p_1)=2A_{{\mathbb{Z}}_{2n}}\cdot 2\lambda\). For even \(n>2\), because \(2\cdot2 \neq 0 \mod 2n\), so \(A_{{\mathbb{Z}}_n}p_1\) cannot be trivialized by the finite group extension 25 . Therefore, \(A_{{\mathbb{Z}}_n}p_1\) cannot be trivialized by any finite group extension.

Therefore, except for \(n=2\) and \(n=3\), \(A_{{\mathbb{Z}}_n}p_1\) cannot be trivialized by any finite group extension.¹²

7.2 \(\mathop{\mathrm{SO}}\times {\mathbb{Z}}_{n}\) structure and symmetry extension trivialization↩︎

Now we consider the availability of the symmetry extension trivialization of \(A_{{\mathbb{Z}}_n}p_1\) on the manifolds with \(\mathop{\mathrm{SO}}\times {\mathbb{Z}}_{n}\) structure, instead of the \({\rm Spin}\times_{{\mathbb{Z}}_2^{{\rm F}}}{\mathbb{Z}}_{2n}\) structure.

Let us explain the two exceptional cases, \(n=3\) and \(n=2\):

• For \(n=3\), on manifolds with the \(\mathop{\mathrm{SO}}\times {\mathbb{Z}}_3\) structure, we have \(A_{{\mathbb{Z}}_3}p_1=0\mod 3\) [45], [46] (see Appendix 13 for the proof) where \(p_1\) is the first Pontryagin class. For other \(n\), \(A_{{\mathbb{Z}}_n}p_1\not\equiv0\mod n\). Hence, except for \(n=3\), \(A_{{\mathbb{Z}}_n}p_1\) can only be trivialized by trivializing \(p_1\mod n\).

• For \(n=2\), on manifolds with the \(\mathop{\mathrm{SO}}\times {\mathbb{Z}}_2\) structure, since \(p_1=w_2^2\mod2\) where \(w_2\) is the second Stiefel-Whitney class, \(p_1\mod2\) can be trivialized by trivializing \(w_2\) via the symmetry extension, because \(w_2 =0\) on the pullback spin manifold, \[\begin{align} 1\to{\mathbb{Z}}_2^{{\rm F}}\to{\rm Spin}\xrightarrow{f}\mathop{\mathrm{SO}}\to1. \end{align}\] So \(A_{{\mathbb{Z}}_2}p_1= A_{{\mathbb{Z}}_2} w_2^2\mod2\) can be trivialized by a fermion parity \({\mathbb{Z}}_2^{{\rm F}}\) group extension on the pullback manifold with \({\rm Spin}\times {\mathbb{Z}}_2\) structure.

• For odd \(n\) and even \(n>2\): Because \(\pi_1(\mathop{\mathrm{SO}})={\mathbb{Z}}_2\), the only connected finite covers of \(\mathop{\mathrm{SO}}\) are \({\rm Spin}\) and \(\mathop{\mathrm{SO}}\). Since \(\operatorname{H}^4(\mathrm{B}{\rm Spin},{\mathbb{Z}})\) is generated by \(\lambda=\frac{p_1}{2}\), \(f^*(p_1\mod n)=2\lambda\mod n\). For odd \(n\) and even \(n>2\), because \(2 \neq 0 \mod n\), so \(p_1\mod n\) cannot be trivialized by any finite group extension.

Therefore, except for \(n=2\) and \(n=3\), \(A_{{\mathbb{Z}}_n}p_1\) cannot be trivialized by any finite group extension.

8 Explicit \((d-1)\)-cochain \(\tilde{\beta}_{d-1}\) that splits the \(d\)-cocycle \(\tilde{\alpha}_d=\delta\tilde{\beta}_{d-1}\) as a coboundary in \(\operatorname{H}^d({\mathbb{Z}}_{n^2},{\rm U}(1))\) by the symmetry extension \(1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\to{\mathbb{Z}}_n\to1\) for any odd \(d\geqslant 3\) and any \(n\geqslant 2\)↩︎

In this appendix, we find an explicit \((d-1)\)-cochain \(\tilde{\beta}_{d-1}\) that splits the \(d\)-cocycle \(\tilde{\alpha}_d=\delta\tilde{\beta}_{d-1}\) as a coboundary in \(\operatorname{H}^d({\mathbb{Z}}_{n^2},{\rm U}(1))\) by the symmetry extension \(1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\to{\mathbb{Z}}_n\to1\) for any odd \(d\geqslant 3\) and any \(n\geqslant 2\). Our strategy follows Ref. []’s symmetry-extension approach to trivialize a cocycle in terms of coboundary, especially in Ref. []’s Appendices.

8.1 \(d= 3\) and any \(n\geqslant 2\): Find \(\tilde{\beta}_2\) such that \(\tilde{\alpha}_3=\delta\tilde{\beta}_2\)↩︎

Explicitly, any 3-cocycle \(\alpha_3\in \operatorname{H}^3({\mathbb{Z}}_n,{\rm U}(1))\) has the form [60], [71] \[\begin{align} \alpha_3(g_1,g_2,g_3)=\zeta_n^{g_1 [\frac{g_2+g_3}{n}]} \end{align}\] where \(g_i\in{\mathbb{Z}}_n\) for \(i=1,2,3\), \(\zeta_n\) is an \(n\)-th root of unity (for example, \(\zeta_n = \exp(\frac{2 \pi \mathrm{i}}{n})\)), and \([\frac{p}{q}]\) denotes the integer part of \(\frac{p}{q}\).

We can find an explict 2-cochain \(\tilde{\beta}_2\in C^2({\mathbb{Z}}_{n^2},{\rm U}(1))\) such that \(\tilde{\alpha}_3=\delta\tilde{\beta}_2\) where \(\tilde{\alpha}_3=f^*\alpha_3\in \operatorname{H}^3({\mathbb{Z}}_{n^2},{\rm U}(1))\) and \(f\) is the map in the extension \(1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\xrightarrow{f}{\mathbb{Z}}_n\to1\). Namely, \[\begin{align} \tilde{\alpha}_3(h_1,h_2,h_3)=\delta\tilde{\beta}_2(h_1,h_2,h_3)=\frac{\tilde{\beta}_2(h_2,h_3)\tilde{\beta}_2(h_1,h_2h_3)}{\tilde{\beta}_2(h_1h_2,h_3)\tilde{\beta}_2(h_1,h_2)} \end{align}\] where \(h_i=(g_i,k_i)\in{\mathbb{Z}}_{n^2}\) for \(i=1,2,3\) and \[\begin{align} \label{eq:composition-law} (g_1,k_1)\cdot(g_2,k_2)=(g_1+g_2,k_1+k_2+[\frac{g_1+g_2}{n}]). \end{align}\tag{26}\] Here, \(\phi(g_1,g_2)=[\frac{g_1+g_2}{n}]\) is the 2-cocycle in \(\operatorname{H}^2({\mathbb{Z}}_n,{\mathbb{Z}}_n)\) classifying the extension \(1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\to{\mathbb{Z}}_n\to1\). Note that for \(n=2\), \([\frac{g_1+g_2}{2}]=g_1g_2\mod 2\), so the composition law 26 agrees with that in [1] for \(n=2\).

Explicitly, the 2-cochain \(\tilde{\beta}_2\in C^2({\mathbb{Z}}_{n^2},{\rm U}(1))\) is given by [1] \[\begin{align} \label{eq:2-cochain} \tilde{\beta}_2(h_1,h_2)=\zeta_n^{g_1k_2}. \end{align}\tag{27}\] Then \[\begin{align} \delta\tilde{\beta}_2(h_1,h_2,h_3)=\zeta_n^{g_2k_3+g_1(k_2+k_3+[\frac{g_2+g_3}{n}])-(g_1+g_2)k_3-g_1k_2}=\zeta_n^{g_1[\frac{g_2+g_3}{n}]}=\alpha_3(g_1,g_2,g_3)=\tilde{\alpha}_3(h_1,h_2,h_3). \end{align}\] Therefore, \(\tilde{\alpha}_3=\delta\tilde{\beta}_2\).

This 2-cochain \(\tilde{\beta}_2\) can also be constructed from the LHS spectral sequence method, see Appendix 6 and [1]. In Appendix 6, we prove that \(\operatorname{H}^3({\mathbb{Z}}_n,{\rm U}(1))={\mathbb{Z}}_n\) is eliminated by the differential \(d_2^{1,1}:E_2^{1,1}\to E_2^{3,0}\) where \[\begin{align} E_2^{1,1}=\operatorname{H}^1s{\mathbb{Z}}_n,\operatorname{H}^1({\mathbb{Z}}_n,{\rm U}(1)))=\operatorname{H}^1({\mathbb{Z}}_n,{\mathbb{Z}}_n)={\mathbb{Z}}_n \end{align}\] and \[\begin{align} E_2^{3,0}=\operatorname{H}^3({\mathbb{Z}}_n,\operatorname{H}^0({\mathbb{Z}}_n,{\rm U}(1)))=\operatorname{H}^3({\mathbb{Z}}_n,{\rm U}(1))={\mathbb{Z}}_n. \end{align}\] Since any 1-cocycle \(\alpha_1\in\operatorname{H}^1({\mathbb{Z}}_n,{\rm U}(1))\) has the form [60], [71] \[\begin{align} \alpha_1(g)=\zeta_n^{g} \end{align}\] where \(g\in{\mathbb{Z}}_n\), the 2-cochain 27 can be expressed as \[\begin{align} \tilde{\beta}_2(h_1,h_2)=(\alpha_1(k_2))^{g_1}. \end{align}\]

8.2 \(d= 5\) and any \(n\geqslant 2\): Find \(\tilde{\beta}_4\) such that \(\tilde{\alpha}_5=\delta\tilde{\beta}_4\)↩︎

Explicitly, any 5-cocycle \(\alpha_5\in \operatorname{H}^5({\mathbb{Z}}_n,{\rm U}(1))\) has the form [60], [71] \[\begin{align} \alpha_5(g_1,g_2,g_3,g_4,g_5)=\zeta_n^{g_1 [\frac{g_2+g_3}{n}] [\frac{g_4+g_5}{n}]} \end{align}\] where \(g_i\in{\mathbb{Z}}_n\) for \(i=1,2,3,4,5\), \(\zeta_n\) is an \(n\)-th root of unity, and \([\frac{p}{q}]\) denotes the integer part of \(\frac{p}{q}\).

We can find an explict 4-cochain \(\tilde{\beta}_4\in C^4({\mathbb{Z}}_{n^2},{\rm U}(1))\) such that \(\tilde{\alpha}_5=\delta\tilde{\beta}_4\) where \(\tilde{\alpha}_5=f^*\alpha_5\in \operatorname{H}^5({\mathbb{Z}}_{n^2},{\rm U}(1))\) and \(f\) is the map in the extension \(1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\xrightarrow{f}{\mathbb{Z}}_n\to1\). Namely, \[\begin{align} \tilde{\alpha}_5(h_1,h_2,h_3,h_4,h_5)=\delta\tilde{\beta}_4(h_1,h_2,h_3,h_4,h_5)=\frac{\tilde{\beta}_4(h_2,h_3,h_4,h_5)\tilde{\beta}_4(h_1,h_2h_3,h_4,h_5)\tilde{\beta}_4(h_1,h_2,h_3,h_4h_5)}{\tilde{\beta}_4(h_1h_2,h_3,h_4,h_5)\tilde{\beta}_4(h_1,h_2,h_3h_4,h_5)\tilde{\beta}_4(h_1,h_2,h_3,h_4)} \end{align}\] where \(h_i=(g_i,k_i)\in{\mathbb{Z}}_{n^2}\) for \(i=1,2,3,4,5\) and \[\begin{align} (g_1,k_1)\cdot(g_2,k_2)=(g_1+g_2,k_1+k_2+[\frac{g_1+g_2}{n}]). \end{align}\] Here, \[\begin{align} \phi(g_1,g_2)=[\frac{g_1+g_2}{n}] \end{align}\] is the 2-cocycle in \(\operatorname{H}^2({\mathbb{Z}}_n,{\mathbb{Z}}_n)\) classifying the extension \(1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\to{\mathbb{Z}}_n\to1\). Note that for \(n=2\), \([\frac{g_1+g_2}{2}]=g_1g_2\mod 2\).

Explicitly, the 4-cochain \(\tilde{\beta}_4\in C^4({\mathbb{Z}}_{n^2},{\rm U}(1))\) is given by \[\begin{align} \tilde{\beta}_4(h_1,h_2,h_3,h_4)=\zeta_n^{g_1k_2[\frac{g_3+g_4}{n}]}. \end{align}\] In fact, if we write \(\tilde{\beta}_4=\zeta_n^{\tilde{\gamma}_4}\) and \(\tilde{\beta}_2=\zeta_n^{\tilde{\gamma}_2}\), then \(\tilde{\gamma}_4=\tilde{\gamma}_2\smile \phi\). Then \[\begin{align} \delta\tilde{\beta}_4(h_1,h_2,h_3,h_4,h_5)=\zeta_n^X. \end{align}\] We check that \[\begin{align} X&=&g_2k_3[\frac{g_4+g_5}{n}]+g_1(k_2+k_3+[\frac{g_2+g_3}{n}])[\frac{g_4+g_5}{n}]+g_1k_2[\frac{g_3+(g_4+g_5)\mod n}{n}]\cr &&-(g_1+g_2)k_3[\frac{g_4+g_5}{n}]-g_1k_2[\frac{(g_3+g_4)\mod n+g_5}{n}]-g_1k_2[\frac{g_3+g_4}{n}]\cr &=&g_1[\frac{g_2+g_3}{n}][\frac{g_4+g_5}{n}]. \end{align}\] Here, we have used the fact that \[\begin{align} [\frac{g_4+g_5}{n}]+[\frac{g_3+(g_4+g_5)\mod n}{n}]=[\frac{g_3+g_4+g_5}{n}]=[\frac{(g_3+g_4)\mod n+g_5}{n}]+[\frac{g_3+g_4}{n}]. \end{align}\] This is in fact the cocycle condition for \(\phi\). Hence \[\begin{align} \delta\tilde{\beta}_4(h_1,h_2,h_3,h_4,h_5)=\zeta_n^X=\zeta_n^{g_1[\frac{g_2+g_3}{n}][\frac{g_4+g_5}{n}]}=\alpha_5(g_1,g_2,g_3,g_4,g_5)=\tilde{\alpha}_5(h_1,h_2,h_3,h_4,h_5). \end{align}\] Therefore, \(\tilde{\alpha}_5=\delta\tilde{\beta}_4\).

8.3 Any odd \(d\geqslant 3\) and any \(n\geqslant 2\): Find \(\tilde{\beta}_{d-1}\) such that \(\tilde{\alpha}_d=\delta\tilde{\beta}_{d-1}\)↩︎

Explicitly, for odd \(d\geqslant 3\), any \(d\)-cocycle \(\alpha_d\in \operatorname{H}^d({\mathbb{Z}}_n,{\rm U}(1))\) has the form [71] \[\begin{align} \alpha_d(g_1,g_2,\dots,g_d)=\zeta_n^{ g_1 [\frac{g_2+g_3}{n}] [\frac{g_4+g_5}{n}] \cdots [\frac{g_{d-1}+g_d}{n}]} \end{align}\] where \(g_i\in{\mathbb{Z}}_n\) for \(i=1,2,\dots,d\), \(\zeta_n\) is an \(n\)-th root of unity, and \([\frac{p}{q}]\) denotes the integer part of \(\frac{p}{q}\).

We can find an explict \((d-1)\)-cochain \(\tilde{\beta}_{d-1}\in C^{d-1}({\mathbb{Z}}_{n^2},{\rm U}(1))\) such that \(\tilde{\alpha}_d=\delta\tilde{\beta}_{d-1}\) where \(\tilde{\alpha}_d=f^*\alpha_d\in \operatorname{H}^d({\mathbb{Z}}_{n^2},{\rm U}(1))\) and \(f\) is the map in the extension \(1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\xrightarrow{f}{\mathbb{Z}}_n\to1\). Namely, \[\begin{align} \tilde{\alpha}_d(h_1,h_2,\dots,h_d)&=&\delta\tilde{\beta}_{d-1}(h_1,h_2,\dots,h_d)\cr &=&\frac{\tilde{\beta}_{d-1}(h_2,h_3,\dots,h_d)\tilde{\beta}_{d-1}(h_1,h_2h_3,h_4,\dots,h_d)\cdots\tilde{\beta}_{d-1}(h_1,h_2,\dots,h_{d-2},h_{d-1}h_d)}{\tilde{\beta}_{d-1}(h_1h_2,h_3,h_4,\dots,h_d)\tilde{\beta}_{d-1}(h_1,h_2,h_3h_4,h_5,\dots,h_d)\cdots\tilde{\beta}_{d-1}(h_1,h_2,\dots,h_{d-1})} \end{align}\] where \(h_i=(g_i,k_i)\in{\mathbb{Z}}_{n^2}\) for \(i=1,2,\dots,d\) and \[\begin{align} (g_1,k_1)\cdot(g_2,k_2)=(g_1+g_2,k_1+k_2+[\frac{g_1+g_2}{n}]). \end{align}\] Here, \(\phi(g_1,g_2)=[\frac{g_1+g_2}{n}]\) is the 2-cocycle in \(\operatorname{H}^2({\mathbb{Z}}_n,{\mathbb{Z}}_n)\) classifying the extension \(1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\to{\mathbb{Z}}_n\to1\). Note that for \(n=2\), \([\frac{g_1+g_2}{2}]=g_1g_2\mod 2\).

Explicitly, the \((d-1)\)-cochain \(\tilde{\beta}_{d-1}\in C^{d-1}({\mathbb{Z}}_{n^2},{\rm U}(1))\) is given by \[\begin{align} \tilde{\beta}_{d-1}(h_1,h_2,\dots,h_{d-1})=\zeta_n^{g_1k_2[\frac{g_3+g_4}{n}]\cdots[\frac{g_{d-2}+g_{d-1}}{n}]}. \end{align}\] If we write \(\tilde{\beta}_{d-1}=\zeta_n^{\tilde{\gamma}_{d-1}}\) and \(\tilde{\beta}_2=\zeta_n^{\tilde{\gamma}_2}\), then \[\begin{align} \tilde{\gamma}_{d-1}=\tilde{\gamma}_2\smile \phi^{\frac{d-3}{2}}. \end{align}\] If we write \(\alpha_d=\zeta_n^{\epsilon_d}\) and \(\alpha_3=\zeta_n^{\epsilon_3}\), then \[\begin{align} \epsilon_d=\epsilon_3\smile \phi^{\frac{d-3}{2}}. \end{align}\] We have shown that \(\delta \tilde{\gamma}_2(h_1,h_2,h_3)=\epsilon_3(g_1,g_2,g_3)\) and we have \(\delta\phi=0\). Hence \[\begin{align} \delta\tilde{\beta}_{d-1}(h_1,h_2,\dots,h_d) &=&\zeta_n^{\delta\tilde{\gamma}_{d-1}}(h_1,h_2,\dots,h_d)\cr &=&\zeta_n^{\delta(\tilde{\gamma}_2\smile \phi^{\frac{d-3}{2}})}(h_1,h_2,\dots,h_d)\cr &=&\zeta_n^{\epsilon_3\smile \phi^{\frac{d-3}{2}}}(g_1,g_2,\dots,g_d)\cr &=&\zeta_n^{\epsilon_d}(g_1,g_2,\dots,g_d)\cr &=&\alpha_d(g_1,g_2,\dots,g_d)\cr &=&\tilde{\alpha}_d(h_1,h_2,\dots,h_d). \end{align}\] Therefore, \(\tilde{\alpha}_d=\delta\tilde{\beta}_{d-1}\).

9 \(d\)d-bulk/\((d-1)\)d-boundary coupled invertible topological field theory /symmetric anomalous gapped TQFT by the symmetry extension \(1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\to{\mathbb{Z}}_n\to 1\) for any odd \(d\geqslant 3\) and any \(n\geqslant 2\)↩︎

In Appendix 6, we prove that the group cohomology class (\(\operatorname{H}^d({\mathbb{Z}}_n,{\rm U}(1))\cong {\mathbb{Z}}_n\)) can be canceled by anomalous \(G\)-symmetric \(K={\mathbb{Z}}_n\)-gauge \((d-1)\)d TQFTs for odd \(d\geqslant 3\) and any \(n\geqslant 2\), via the appropriate symmetry-extension construction [1] of \[\begin{align} 1\to K \to G_{\rm Tot}\to G \to 1. \end{align}\] as \[\begin{align} \label{eq:Zn-extension-general} 1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\to{\mathbb{Z}}_n\to 1. \end{align}\tag{28}\]

More generally and mathematically, in this work, for odd \(d\geqslant 3\) and any \(n\geqslant 2\), we prove that any group cohomology cocycle \[\begin{align} \alpha_d \in \operatorname{H}^d({\mathbb{Z}}_n,{\rm U}(1)) \cong {\mathbb{Z}}_n \end{align}\] is trivialized by the group extension as eq. (28 )’s \(1\to{\mathbb{Z}}_n\to{\mathbb{Z}}_{n^2}\to{\mathbb{Z}}_n\to 1\) [1].

In Appendix 8, we find an explicit \((d-1)\)-cochain \(\beta_{d-1}\) that splits the \(d\)-cocycle \(\alpha_d\) by that extension for odd \(d\geqslant 3\) and any \(n\geqslant 2\). Namely, \(\alpha_d=\delta \beta_{d-1}\) holds when pulling back the quotient \({\mathbb{Z}}_n\) to the total \({\mathbb{Z}}_{n^2}\) group, from the cocycle \(\alpha_d\) in \(\operatorname{H}^d({\mathbb{Z}}_n,{\rm U}(1))\) to the coboundary \[\begin{align} \text{\tilde{\alpha}_d=\delta \tilde{\beta}_{d-1} in \operatorname{H}^d({\mathbb{Z}}_{n^2},{\rm U}(1)).} \end{align}\]

More explicitly, for the \(\alpha_d\) given by \[\begin{align} \alpha_d= \exp \big( \mathrm{i}\frac{2 \pi}{n} \int_{M^d} ( {A_{{\mathbb{Z}}_n}} (\beta_{(n,n)} {A_{{\mathbb{Z}}_n}})^{\frac{d-1}{2}} ) \big), \end{align}\] we have \(\beta_{d-1}\) with \(\alpha_d= \delta \beta_{d-1}\), obtained in Appendix 8, which suggests a construction of the \(d\)d iTFT on the bulk \(d\)-manifold \(M^d\) and the \((d-1)\)d noninvertible TQFT on the \((d-1)\)d boundary \(M^{d-1}= \partial M^d\) with dynamical 1-cochain gauge field \(a_{{\mathbb{Z}}_n} \in C^1(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n)\) and \((d-3)\)-cochain (dual) gauge field \(b_{{\mathbb{Z}}_n} \in C^{d-3}(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n)\), such that the full \(d\)d/\((d-1)\)d coupled path integral is given by \[\begin{align} \label{eq:dd-d-1d} &&\exp \big( \mathrm{i}\frac{2 \pi}{n} \int_{M^d} ( {A_{{\mathbb{Z}}_n}} (\beta_{(n,n)} {A_{{\mathbb{Z}}_n}})^{\frac{d-1}{2}} ) \big) \cdot \cr && \cdot \sum_{ \substack{ a_{{\mathbb{Z}}_n} \in C^1(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n) \\ b_{{\mathbb{Z}}_n} \in C^{d-3}(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n) }}\exp \big( \mathrm{i}\frac{2 \pi}{n} \int_{M^{d-1}=\partial M^d} ( b_{{\mathbb{Z}}_n} \mathrm{d}a_{{\mathbb{Z}}_n} - b_{{\mathbb{Z}}_n} \beta_{(n,n)} {A_{{\mathbb{Z}}_n}} - a_{{\mathbb{Z}}_n} {A_{{\mathbb{Z}}_n}} (\beta_{(n,n)} {A_{{\mathbb{Z}}_n}})^{\frac{d-3}{2}} ) \big) \cr &=&\exp \big( \mathrm{i}\frac{2 \pi}{n} \int_{M^d} ( {A_{{\mathbb{Z}}_n}} (\beta_{(n,n)} {A_{{\mathbb{Z}}_n}})^{\frac{d-1}{2}} ) \big) \cdot \cr && \cdot\sum_{ \substack{ a_{{\mathbb{Z}}_n} \in C^1(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n) \\ b_{{\mathbb{Z}}_n} \in C^{d-3}(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n) }}\exp \big( \mathrm{i}\frac{2 \pi}{n} \int_{M^{d-1}=\partial M^d}( a_{{\mathbb{Z}}_n}(\mathrm{d}b_{{\mathbb{Z}}_n}- {A_{{\mathbb{Z}}_n}} (\beta_{(n,n)} {A_{{\mathbb{Z}}_n}})^{\frac{d-3}{2}})-b_{{\mathbb{Z}}_n}\beta_{(n,n)} {A_{{\mathbb{Z}}_n}} ) \big). \end{align}\tag{29}\] This \(d\)d/\((d-1)\)d coupled path integral analogously matches the discrete cocycle forms or cochain forms (e.g., [60]) derived in Appendix 8, as the \(d\)-cocycle \[\alpha_d(g_1,g_2,\dots,g_d)=\zeta_n^{g_1 [\frac{g_2+g_3}{n}]\cdots [\frac{g_{d-1}+g_d}{n}]}\] and the \((d-1)\)-cochain \[\tilde{\beta}_{d-1}(h_1,h_2,\dots,h_{d-1})=\zeta_n^{g_1k_2[\frac{g_3+g_4}{n}]\cdots[\frac{g_{d-2}+g_{d-1}}{n}]},\] where \(\zeta_n\) is an \(n\)-th root of unity such as \(\zeta_n = \exp(\frac{2 \pi \mathrm{i}}{n})\), with variables \(g \in {\mathbb{Z}}_n\) and \(k \in {\mathbb{Z}}_n\).

The \(d\)d bulk partition function on a \(d\)d manifold with a \((d-1)\)d boundary is not gauge-invariant, but the full \(d\)d/\((d-1)\)d coupled path integral eq. (29 ) is gauge-invariant under \[\begin{align} \label{eq:gauge-transformation-general} && A_{{\mathbb{Z}}_n} \mapsto A_{{\mathbb{Z}}_n} + \mathrm{d}\lambda_{0,{\mathbb{Z}}_n}, \cr && a_{{\mathbb{Z}}_n} \mapsto a_{{\mathbb{Z}}_n} + \mathrm{d}\mu_{0,{\mathbb{Z}}_n}, \cr && b_{{\mathbb{Z}}_n} \mapsto b_{{\mathbb{Z}}_n} +\lambda_{0,{\mathbb{Z}}_n}(\beta_{(n,n)}A_{{\mathbb{Z}}_n})^{\frac{d-3}{2}} + \mathrm{d}\mu_{d-4,{\mathbb{Z}}_n}, \cr && A_{{\mathbb{Z}}_n} \in\operatorname{H}^1(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n)= {\mathbb{Z}}_n,\cr && a_{{\mathbb{Z}}_n} \in C^1(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n),\cr && b_{{\mathbb{Z}}_n} \in C^{d-3}(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n),\cr &&\lambda_{0,{\mathbb{Z}}_n} \in C^0(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n),\cr &&\mu_{0,{\mathbb{Z}}_n} \in C^0(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n),\cr &&\mu_{d-4,{\mathbb{Z}}_n} \in C^{d-4}(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n). \end{align}\tag{30}\] Here, \(C^k(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n)\) is the group of \({\mathbb{Z}}_n\)-valued \(k\)-cochains of the classifying space \(\mathrm{B}{\mathbb{Z}}_n\). The cohomology group \(\operatorname{H}^k(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n)\) is defined as the quotient group \(Z^k(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n)/B^k(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n)\) where \(Z^k(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n)\) is the group of \({\mathbb{Z}}_n\)-valued \(k\)-cocycles of the classifying space \(\mathrm{B}{\mathbb{Z}}_n\) and \(B^k(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n)\) is the group of \({\mathbb{Z}}_n\)-valued \(k\)-coboundaries of the classifying space \(\mathrm{B}{\mathbb{Z}}_n\).

Below we check that 29 is gauge-invariant under 30 . Because \(\beta_{(n,n)} \mathrm{d}\lambda_{0,{\mathbb{Z}}_n}=0\), \(\beta_{(n,n)} {A_{{\mathbb{Z}}_n}}\) and \(\mathrm{d}b_{{\mathbb{Z}}_n}- {A_{{\mathbb{Z}}_n}} (\beta_{(n,n)} {A_{{\mathbb{Z}}_n}})^{\frac{d-3}{2}}\) are gauge-invariant under the gauge transformation 30 , hence 29 transforms under 30 as \[\begin{align} && \exp \big( \mathrm{i}\frac{2 \pi}{n} \int_{M^d} ( {A_{{\mathbb{Z}}_n}} (\beta_{(n,n)} {A_{{\mathbb{Z}}_n}})^{\frac{d-1}{2}} ) \big) \cdot \cr && \cdot\sum_{ \substack{ a_{{\mathbb{Z}}_n} \in C^1(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n) \\ b_{{\mathbb{Z}}_n} \in C^{d-3}(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n) }}\exp \big( \mathrm{i}\frac{2 \pi}{n} \int_{M^{d-1}=\partial M^d}( a_{{\mathbb{Z}}_n}(\mathrm{d}b_{{\mathbb{Z}}_n}- {A_{{\mathbb{Z}}_n}} (\beta_{(n,n)} {A_{{\mathbb{Z}}_n}})^{\frac{d-3}{2}})-b_{{\mathbb{Z}}_n}\beta_{(n,n)} {A_{{\mathbb{Z}}_n}} ) \big)\cr &\mapsto& \exp \big( \mathrm{i}\frac{2 \pi}{n} \int_{M^d} ( ({A_{{\mathbb{Z}}_n}}+\mathrm{d}\lambda_{0,{\mathbb{Z}}_n}) (\beta_{(n,n)} {A_{{\mathbb{Z}}_n}})^{\frac{d-1}{2}} ) \big) \cdot \cr && \cdot\sum_{ \substack{ a_{{\mathbb{Z}}_n} \in C^1(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n) \\ b_{{\mathbb{Z}}_n} \in C^{d-3}(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n) }}\exp \big( \mathrm{i}\frac{2 \pi}{n} \int_{M^{d-1}=\partial M^d}( (a_{{\mathbb{Z}}_n}+\mathrm{d}\mu_{0,{\mathbb{Z}}_n})(\mathrm{d}b_{{\mathbb{Z}}_n}- {A_{{\mathbb{Z}}_n}} (\beta_{(n,n)} {A_{{\mathbb{Z}}_n}})^{\frac{d-3}{2}})\cr &&-(b_{{\mathbb{Z}}_n}+\lambda_{0,{\mathbb{Z}}_n}(\beta_{(n,n)}A_{{\mathbb{Z}}_n})^{\frac{d-3}{2}} + \mathrm{d}\mu_{d-4,{\mathbb{Z}}_n})\beta_{(n,n)} {A_{{\mathbb{Z}}_n}} ) \big). \end{align}\] Since by the Stokes theorem, we have \[\begin{align} \int_{M^{d-1}=\partial M^d}(\mathrm{d}\mu_{0,{\mathbb{Z}}_n})(\mathrm{d}b_{{\mathbb{Z}}_n}- {A_{{\mathbb{Z}}_n}} (\beta_{(n,n)} {A_{{\mathbb{Z}}_n}})^{\frac{d-3}{2}})=0, \end{align}\] \[\begin{align} \int_{M^{d-1}=\partial M^d}(\mathrm{d}\mu_{d-4,{\mathbb{Z}}_n})\beta_{(n,n)} {A_{{\mathbb{Z}}_n}}=0, \end{align}\] and \[\begin{align} \int_{M^{d-1}=\partial M^d}\lambda_{0,{\mathbb{Z}}_n}(\beta_{(n,n)}A_{{\mathbb{Z}}_n})^{\frac{d-1}{2}}=\int_{M^d}(\mathrm{d}\lambda_{0,{\mathbb{Z}}_n})(\beta_{(n,n)}A_{{\mathbb{Z}}_n})^{\frac{d-1}{2}}, \end{align}\] 29 is gauge-invariant under 30 .

10 3+1d Nonperturbative Global Anomaly in \({\rm Spin}\times {{\mathbb{Z}}_{n}}\) for integer \(n\) with \(2 \nmid n\) and \(3 \nmid n\)↩︎

In this appendix, we explore the 3+1d nonperturbative global anomaly for a Weyl fermion in \({\rm Spin}\times {{\mathbb{Z}}_{n}}\) for integer \(n\) with \(2 \nmid n\) and \(3 \nmid n\), compared with the fact that \[\begin{align} && \Omega_5^{\rm Spin}(\mathrm{B}{\mathbb{Z}}_{n} ) \cong \tilde{\Omega}_5^{\rm SO}(\mathrm{B}{\mathbb{Z}}_{n} ) \cong {\mathbb{Z}}_n \oplus {\mathbb{Z}}_n, \quad 2\nmid n, \quad 3\nmid n. \end{align}\] Here \(\tilde{\Omega}_5^{\rm SO}(\mathrm{B}G):=\Omega_5^{\rm SO}(\mathrm{B}G)/\Omega_5^{\rm SO}\) is the reduced bordism group, modding out the \(\Omega_5^{\rm SO}=\Omega_5^{\rm SO}(pt)\).

The perturbative local anomaly of U(1) charge \(q=1\) left-handed Weyl fermion of \({\rm Spin}\times {\rm U}(1)\) symmetry in 3+1d or 4d is captured by a 5d invertible field theory (iTFT) with the anomaly index \(k=1\): \[\exp( \mathrm{i}k \int_{M^5} A \frac{c_1^2}{6}-A \frac{p_1}{24}).\] Now we redefine the U(1) gauge field \(A\) as a \({\mathbb{Z}}_n\) gauge field \(A_{{\mathbb{Z}}_n} \in\operatorname{H}^1(\mathrm{B}{\mathbb{Z}}_n,{\mathbb{Z}}_n)= {\mathbb{Z}}_n\) with the following replacement: \[\begin{align} A &\mapsto& \frac{2 \pi}{n} A_{{\mathbb{Z}}_n}.\cr c_1 = \frac{\mathrm{d}A}{2 \pi} &\mapsto& \frac{\mathrm{d}A_{{\mathbb{Z}}_n}}{n} \equiv \beta_{(n,n) } A_{{\mathbb{Z}}_n}. \end{align}\] The \(\beta_{(n,m)}: \operatorname{H}^*(-,{\mathbb{Z}}_m) \mapsto \operatorname{H}^{*+1}(-,{\mathbb{Z}}_n)\) is the Bockstein homomorphism associated with the extension \({\mathbb{Z}}_n \stackrel{\cdot m}{\to} {\mathbb{Z}}_{nm} \to {\mathbb{Z}}_m\). Thus we get the 5d topological invariant of the \({\rm Spin}\times {\mathbb{Z}}_n\) as: \[\begin{align} \label{eq:Spin-Zn} &&\exp \big( \mathrm{i}2 \pi k \int_{M^5} ( \frac{1}{6n} {A_{{\mathbb{Z}}_n}} (\beta_{(n,n)} {A_{{\mathbb{Z}}_n}}) (\beta_{(n,n)} {A_{{\mathbb{Z}}_n}}) -\frac{1}{ 24n} {A_{{\mathbb{Z}}_n}} p_1 ) \big). \end{align}\tag{31}\] Since the anomaly of 4d Weyl fermion with symmetry \({\rm Spin}\times{\mathbb{Z}}_n\) does not contain 2-torsion and 3-torsion for \(2\nmid n\) and \(3\nmid n\) [10], [28]–[30], [58], [59], we can regard \(6=2\cdot3\) and \(24=2^3\cdot3\) as invertible in \({\mathbb{Z}}_n\) for \(2\nmid n\) and \(3\nmid n\).

In fact, for \(2\nmid n\) and \(3\nmid n\), there exists an integer \(x_n\) such that \(x_n=1\mod n\) and \(24|x_n\). Then we can rewrite 31 as \[\begin{align} \label{eq:D5} \exp \big( \mathrm{i}\frac{2 \pi k }{n} \int_{M^5} ( \frac{x_n}{6} {A_{{\mathbb{Z}}_n}} (\beta_{(n,n)} {A_{{\mathbb{Z}}_n}}) (\beta_{(n,n)} {A_{{\mathbb{Z}}_n}}) -\frac{x_n}{ 24} {A_{{\mathbb{Z}}_n}} p_1 ) \big). \end{align}\tag{32}\] Since \(\gcd(n,x_n)=1\), \(\gcd(n,\frac{x_n}{6})=1\), and \(\gcd(n,\frac{x_n}{24})=1\), the first term and the second term individually in eq. (32 ), as well as the combined two terms in eq. (32 ), all generate a \({\mathbb{Z}}_n\) class. We expect eq. (32 ) as a schematic way to write one of the two cobordism invariant generators of \(\Omega_5^{\rm Spin}(\mathrm{B}{\mathbb{Z}}_{n} ) \cong {\mathbb{Z}}_n \oplus {\mathbb{Z}}_n,\) with \(2\nmid n\) and \(3\nmid n.\)

11 3+1d Nonperturbative Global Anomaly in \({\rm Spin}\times {\mathbb{Z}}_{3^r} = {\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{2 \cdot 3^r}^{\rm F}}\)↩︎

In this appendix, we explore the 3+1d nonperturbative global anomaly for a Weyl fermion in \({\rm Spin}\times {\mathbb{Z}}_{3^r} = {\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{2 \cdot 3^r}^{\rm F}}\), compared with the fact that \[\begin{align} \label{eq:E1} &&\Omega_5^{{\rm Spin} \times {\mathbb{Z}}_{3^r}} \cong \Omega_5^{{\rm Spin} \times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{2 \cdot 3^r }}} \cong \tilde{\Omega}_5^{\rm SO}(\mathrm{B}{\mathbb{Z}}_{3^r } ) \cr &&= {\mathbb{Z}}_{3^{r+1}}\oplus {\mathbb{Z}}_{3^{r-1}} . \end{align}\tag{33}\] Here \(\tilde{\Omega}_5^{\rm SO}(\mathrm{B}G):=\Omega_5^{\rm SO}(\mathrm{B}G)/\Omega_5^{\rm SO}\) is the reduced bordism group, modding out the \(\Omega_5^{\rm SO}=\Omega_5^{\rm SO}(pt)\).

We also prove that the \(k=3^r\) anomaly of the 4d Weyl fermion with \({\rm Spin}\times{\mathbb{Z}}_{3^r}\) symmetry can be trivialized by a \({\mathbb{Z}}_3\) extension.

The perturbative local anomaly of U(1) charge \(q=1\) left-handed Weyl fermion of \({\rm Spin}\times {\rm U}(1)\) symmetry in 3+1d or 4d is captured by a 5d invertible field theory (iTFT) with the anomaly index \(k=1\): \[\exp( \mathrm{i}k \int_{M^5} A \frac{c_1^2}{6}-A \frac{p_1}{24}).\] Now we redefine the U(1) gauge field \(A\) as a \({\mathbb{Z}}_{3^r}\) gauge field \(A_{{\mathbb{Z}}_{3^r}} \in\operatorname{H}^1(\mathrm{B}{\mathbb{Z}}_{3^r},{\mathbb{Z}}_{3^r})= {\mathbb{Z}}_{3^r}\) with the following replacement: \[\begin{align} A &\mapsto& \frac{2 \pi}{3^r} A_{{\mathbb{Z}}_{3^r}}.\cr c_1 = \frac{\mathrm{d}A}{2 \pi} &\mapsto& \frac{\mathrm{d}A_{{\mathbb{Z}}_{3^r}}}{3^r} \equiv \beta_{(3^r,3^r) } A_{{\mathbb{Z}}_{3^r}}. \end{align}\] The \(\beta_{(n,m)}: \operatorname{H}^*(-,{\mathbb{Z}}_m) \mapsto \operatorname{H}^{*+1}(-,{\mathbb{Z}}_n)\) is the Bockstein homomorphism associated with the extension \({\mathbb{Z}}_n \stackrel{\cdot m}{\to} {\mathbb{Z}}_{nm} \to {\mathbb{Z}}_m\). Thus we get the 5d topological invariant of the \({\rm Spin}\times {\mathbb{Z}}_{3^r}\) as: \[\begin{align} \label{eq:Spin-Z3r} &&\exp \big( \mathrm{i}2 \pi k \int_{M^5} ( \frac{1}{2\cdot 3^{r+1}} {A_{{\mathbb{Z}}_{3^r}}} (\beta_{(3^r,3^r)} {A_{{\mathbb{Z}}_{3^r}}}) (\beta_{(3^r,3^r)} {A_{{\mathbb{Z}}_{3^r}}}) -\frac{1}{ 8\cdot 3^{r+1}} {A_{{\mathbb{Z}}_{3^r}}} p_1 ) \big). \end{align}\tag{34}\] Since the anomaly of 4d Weyl fermion with symmetry \({\rm Spin}\times{\mathbb{Z}}_{3^r}\) contains only 3-torsion [10], [28]–[30], [58], [59], we can regard \(2\) and \(8\) as invertible in \({\mathbb{Z}}_{3^{r+1}}\).

In fact, there exists an integer \(y_r\) such that \(y_r=1\mod 3^{r+1}\) and \(8|y_r\). Then we can rewrite 34 as \[\begin{align} \label{eq:E5} \exp \big( \mathrm{i}\frac{2 \pi k }{3^{r+1}} \int_{M^5} ( \frac{y_r}{2} {A_{{\mathbb{Z}}_{3^r}}} (\beta_{(3^r,3^r)} {A_{{\mathbb{Z}}_{3^r}}}) (\beta_{(3^r,3^r)} {A_{{\mathbb{Z}}_{3^r}}}) -\frac{y_r}{ 8} {A_{{\mathbb{Z}}_{3^r}}} p_1 ) \big). \end{align}\tag{35}\] Since \(\gcd(3^{r+1},y_r)=1\), \(\gcd(3^{r+1},\frac{y_r}{2})=1\), and \(\gcd(3^{r+1},\frac{y_r}{8})=1\), the first term and second term in eq. (35 ) individually generate a \({\mathbb{Z}}_{3^r}\) class.

Since \(A_{{\mathbb{Z}}_{3^r}}=A_{{\mathbb{Z}}_3}\mod3\) and \(A_{{\mathbb{Z}}_3}p_1=0\mod3\) [45], [46] (see Appendix 13 for the proof), we have \(A_{{\mathbb{Z}}_{3^r}}p_1=0\mod3\) and we can rewrite eq. (35 ) as \[\begin{align} \label{eq:E6} \exp \big( \mathrm{i}\frac{2 \pi k }{3^{r+1}} \int_{M^5} ( \frac{y_r}{2} {A_{{\mathbb{Z}}_{3^r}}} (\beta_{(3^r,3^r)} {A_{{\mathbb{Z}}_{3^r}}}) (\beta_{(3^r,3^r)} {A_{{\mathbb{Z}}_{3^r}}}) -3\cdot\frac{y_r}{ 8} \frac{{A_{{\mathbb{Z}}_{3^r}}} p_1}{3} ) \big). \end{align}\tag{36}\] Since \(\frac{{A_{{\mathbb{Z}}_{3^r}}} p_1}{3}\) generates a \({\mathbb{Z}}_{3^r}\) class and the \(k=3^r\) anomaly of the 4d Weyl fermion with \({\rm Spin}\times{\mathbb{Z}}_{3^r}\) symmetry is \[\begin{align} \label{eq:E7} \exp(\mathrm{i}\frac{2\pi}{3}\int_{M^5}\frac{y_r}{2}{A_{{\mathbb{Z}}_{3^r}}} (\beta_{(3^r,3^r)} {A_{{\mathbb{Z}}_{3^r}}}) (\beta_{(3^r,3^r)} {A_{{\mathbb{Z}}_{3^r}}})). \end{align}\tag{37}\] Since \({A_{{\mathbb{Z}}_{3^r}}} (\beta_{(3^r,3^r)} {A_{{\mathbb{Z}}_{3^r}}}) (\beta_{(3^r,3^r)} {A_{{\mathbb{Z}}_{3^r}}})={A_{{\mathbb{Z}}_{3}}} (\beta_{(3,3)} {A_{{\mathbb{Z}}_{3}}}) (\beta_{(3,3)} {A_{{\mathbb{Z}}_{3}}})\mod3\), this term generates a \({\mathbb{Z}}_3\) class. Therefore, the combined two terms in eq. (36 ) generate a \({\mathbb{Z}}_{3^{r+1}}\) class. We expect eq. (36 ) as a schematic way to write the first one of the two cobordism invariant generators of \(\Omega_5^{\rm Spin}(\mathrm{B}{\mathbb{Z}}_{3^r} ) \cong {\mathbb{Z}}_{3^{r+1}} \oplus {\mathbb{Z}}_{3^{r-1}}\).

By the results in Appendix 6, the \(k=3^r\) anomaly eq. (37 ) of the 4d Weyl fermion with \({\rm Spin}\times{\mathbb{Z}}_{3^r}\) symmetry, namely \(k=3^r \in {\mathbb{Z}}_{3^{r+1}} \subset \Omega_5^{\rm Spin}(\mathrm{B}{\mathbb{Z}}_{3^r } ) \cong {\mathbb{Z}}_{3^{r+1}}\oplus {\mathbb{Z}}_{3^{r-1}}\), can be trivialized by a \({\mathbb{Z}}_3\) extension.

12 3+1d Nonperturbative Global Anomaly in \({\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{2 \cdot 2^p}^{\rm F}}\)↩︎

In this appendix, we explore the 3+1d nonperturbative global anomaly for a Weyl fermion in \({\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{2 \cdot 2^p}^{\rm F}}\), compared with the fact that \[\begin{align} \Omega_5^{{\rm Spin} \times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{2 \cdot 2^p }}} = {\mathbb{Z}}_{2^{p+3}}\oplus {\mathbb{Z}}_{2^{p-1}} . \end{align}\]

The perturbative local anomaly of U(1) charge \(q=1\) left-handed Weyl fermion of \({\rm Spin}^c={\rm Spin}\times_{{\mathbb{Z}}_2^{{\rm F}}} {\rm U}(1)\) symmetry in 3+1d or 4d is captured by a 5d invertible field theory (iTFT) with the anomaly index \(k=1\): \[\exp( \mathrm{i}k \int_{M^5} A' \frac{(2c_1)^2}{48}-A' \frac{p_1}{48}).\] Here, \(c_1'=2c_1=\frac{\mathrm{d}A'}{2\pi}\) is the first Chern class of the \({\rm U}(1)/{\mathbb{Z}}_2^{{\rm F}}\) bundle. Now we redefine the U(1) gauge field \(A'\) as a \({\mathbb{Z}}_{2^p}\) gauge field \(A_{{\mathbb{Z}}_{2^p}} \in\operatorname{H}^1(\mathrm{B}{\mathbb{Z}}_{2^p},{\mathbb{Z}}_{2^p})= {\mathbb{Z}}_{2^p}\) with the following replacement: \[\begin{align} A' &\mapsto& \frac{2 \pi}{2^p} A_{{\mathbb{Z}}_{2^p}}.\cr c_1' = \frac{\mathrm{d}A'}{2 \pi} &\mapsto& \frac{\mathrm{d}A_{{\mathbb{Z}}_{2^p}}}{2^p} \equiv \beta_{(2^p,2^p) } A_{{\mathbb{Z}}_{2^p}}. \end{align}\] The \(\beta_{(n,m)}: \operatorname{H}^*(-,{\mathbb{Z}}_m) \mapsto \operatorname{H}^{*+1}(-,{\mathbb{Z}}_n)\) is the Bockstein homomorphism associated with the extension \({\mathbb{Z}}_n \stackrel{\cdot m}{\to} {\mathbb{Z}}_{nm} \to {\mathbb{Z}}_m\). Thus we get the 5d topological invariant of the \({\rm Spin}\times_{{\mathbb{Z}}_2^{{\rm F}}} {\mathbb{Z}}_{2\cdot 2^p}\) as: \[\begin{align} \label{eq:Spin-Z2p} &&\exp \big( \mathrm{i}2 \pi k \int_{M^5} ( \frac{1}{3\cdot 2^{p+4}} {A_{{\mathbb{Z}}_{2^p}}} (\beta_{(2^p,2^p)} {A_{{\mathbb{Z}}_{2^p}}}) (\beta_{(2^p,2^p)} {A_{{\mathbb{Z}}_{2^p}}}) -\frac{1}{ 3\cdot 2^{p+4}} {A_{{\mathbb{Z}}_{2^p}}} p_1 ) \big). \end{align}\tag{38}\] Since the anomaly of 4d Weyl fermion with symmetry \({\rm Spin}\times_{{\mathbb{Z}}_2^{{\rm F}}}{\mathbb{Z}}_{2\cdot 2^p}\) contains only 2-torsion [10], [28]–[30], [58], [59], we can regard \(3\) as invertible in \({\mathbb{Z}}_{2^{p+3}}\).

The bundle constraint for the \({\rm Spin}\times_{{\mathbb{Z}}_2^{{\rm F}}} {\mathbb{Z}}_{2\cdot 2^p}\) structure is \[\begin{align} w_2=\beta_{(2^p,2^p)} {A_{{\mathbb{Z}}_{2^p}}}\mod2 \end{align}\] where \(w_2\) is the second Stiefel-Whitney class of the tangent bundle \(TM\). Therefore, \[\begin{align} {A_{{\mathbb{Z}}_{2^p}}} (\beta_{(2^p,2^p)} {A_{{\mathbb{Z}}_{2^p}}}) (\beta_{(2^p,2^p)} {A_{{\mathbb{Z}}_{2^p}}}={A_{{\mathbb{Z}}_{2^p}}} w_2^2={A_{{\mathbb{Z}}_{2^p}}} p_1\mod2. \end{align}\] So we can rewrite 38 as \[\begin{align} \label{eq:Spin-Z2p-rewrite} &&\exp \big( \mathrm{i}2 \pi k \frac{1}{3\cdot 2^{p+3}} \int_{M^5} \frac{1}{2}( {A_{{\mathbb{Z}}_{2^p}}} (\beta_{(2^p,2^p)} {A_{{\mathbb{Z}}_{2^p}}}) (\beta_{(2^p,2^p)} {A_{{\mathbb{Z}}_{2^p}}}) - {A_{{\mathbb{Z}}_{2^p}}} p_1 ) \big). \end{align}\tag{39}\] We expect eq. (39 ) as a schematic way to write the first one of the two cobordism invariant generators of \(\Omega_5^{{\rm Spin} \times_{{\mathbb{Z}}_2^{\rm F}} {{\mathbb{Z}}_{2 \cdot 2^p }}} = {\mathbb{Z}}_{2^{p+3}}\oplus {\mathbb{Z}}_{2^{p-1}}\).

13 Proof of \(A_{{\mathbb{Z}}_3}p_1=0\mod3\)↩︎

In this appendix, we prove a general fact which implies \(A_{{\mathbb{Z}}_3}p_1=0\mod3\) [45], [46] and explain why 3 is special. Since \(\operatorname{H}^1({\mathbb{Z}}_n,{\rm U}(1))\cong \operatorname{H}^1({\mathbb{Z}}_n,{\mathbb{Z}}_n)\), we can regard \(A_{{\mathbb{Z}}_n}\) as the generator of \(\operatorname{H}^1({\mathbb{Z}}_n,{\mathbb{Z}}_n)\). The cup product \(A_{{\mathbb{Z}}_n}p_1\) is a mod \(n\) cohomology class on a manifold \(M\) when \(A_{{\mathbb{Z}}_n}\) is pulled back to \(M\).

13.1 A general fact about the mod \(q\) Steenrod reduced power \(P_q^r\)↩︎

Let \(P_q^r:\operatorname{H}^i(-,{\mathbb{Z}}_q)\to \operatorname{H}^{i+2(q-1)r}(-,{\mathbb{Z}}_q)\) be the mod \(q\) Steenrod reduced power where \(q\) is an odd prime.

On an oriented \(n\)-manifold \(M^n\), by the Poincaré duality, there exists \(s_q^r\in \operatorname{H}^{2(q-1)r}(M^n,{\mathbb{Z}}_q)\) such that \[\begin{align} P_q^r(x)=s_q^r\smile x,\quad \forall x\in \operatorname{H}^{n-2(q-1)r}(M^n,{\mathbb{Z}}_q). \end{align}\] Let \(P_q:=\sum_{r=0}^{\infty}P_q^r\) be the mod \(q\) total Steenrod reduced power, and \(s_q:=\sum_{r=0}^{\infty}s_q^r\). We prove the following theorem [72] as its proof is hard to find in the literature.

We mimic the proof for \(\mathrm{Sq}(v)=w\) given in [15] to prove the above theorem. Here, \(\mathrm{Sq}\) is the total Steenrod square, \(v\) is the total mod 2 Wu class, and \(w\) is the total Stiefel-Whitney class.

13.2 \(A_{{\mathbb{Z}}_3}p_1=0\mod3\)↩︎

By the above theorem, we have \(p_1=s_3^1\mod3\), hence \[\begin{align} A_{{\mathbb{Z}}_3}p_1=A_{{\mathbb{Z}}_3}s_3^1=P_3^1(A_{{\mathbb{Z}}_3})=0\mod3 \end{align}\] since \(\deg(A_{{\mathbb{Z}}_3})=1\) and \(P_3^r(x)=0\) if \(\deg(x)<2r\).

For odd prime \(q>3\), \(\deg(s_q^1)=2(q-1)>4=\deg(p_1)\), so there is no similar result for odd prime \(q>3\).

References↩︎

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1. wanzheyan@bimsa.cn↩︎

2. jw@lims.ac.uk↩︎

3. styau@tsinghua.edu.cn↩︎

4. To clarify, whenever we say “fermionic” in fermionic anomalies and fermionic TQFTs, we mean that there are gauge-invariant fermions such that fermion parity \({\mathbb{Z}}_2^{\rm F}\) is part of the internal global symmetry, namely the Spin group is part of the spacetime-internal symmetry group. So fermionic TQFTs mean the TQFTs that has the Spin group symmetry such as spin TQFTs; in contrast with bosonic TQFTs that are non-spin TQFTs.↩︎

5. Note that readers may wonder whether the alternative extension works? \[1 \to {\mathbb{Z}}_{N_c}\to {\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2N_cN_f}\to {\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2 N_f,{\boldsymbol{B} +L}}^{\rm F}\to 1.\] \(\bullet\) When \(N_f\) and \(N_c\) are postive odd integers, this extension coincides with eq. (4 ), because \({\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2N_cN_f}= {\rm Spin}\times {\mathbb{Z}}_{N_cN_f}\).
   \(\bullet\) When \(N_f\) and \(N_c\) are positive even integers, then the above alternative extension does not exist. Because in \({\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2m}\), the charge should be odd, but charge \(q=1\) in \({\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2N_f,{\boldsymbol{B}+L}}^{{\rm F}}\) becomes charge \(q=N_c\) in \({\rm Spin}\times_{{\mathbb{Z}}_2^{\rm F}} {\mathbb{Z}}_{2N_cN_f}\), which is impossible.↩︎

6. We can read this statement also as: \[\begin{gather} \textit{ The anomaly of the N_f-family Standard Model missing all sterile right-handed neutrinos } \\ \textit{with a spacetime-internal symmetry {\rm Spin}\times_{{\mathbb{Z}}_2^{{\rm F}}}{\mathbb{Z}}_{2N_f}, is trivialized by a minimal {\mathbb{Z}}_{N_c}-extension.}\nonumber\\ \end{gather}\] For \(n_{\nu_R}\) the number of types of right-handed neutrinos, so we have \(n_{\nu_R}=0\) to make the anomaly index of the SM (missing all sterile right-handed neutrinos) as \(- N_f + n_{\nu_R}= - N_f\).↩︎

7. The real Euclidean rotational Lie algebra \[so(4, \mathbb{R}) \equiv so(4) \cong so(3) \oplus so(3) \cong su(2) \oplus su(2)\] and the real Lorentz Lie algebra \[so(1,3, \mathbb{R}) \equiv so(1,3) \cong sl(2,\mathbb{C})\] are both the real forms of the same complexified Lie algebra \[so(4,\mathbb{C}) \cong (su(2)\oplus su(2))\otimes_{\mathbb{R}} \mathbb{C}\cong sl(2,\mathbb{C}) \oplus sl(2,\mathbb{C}),\] the Euclidean \(so(4, \mathbb{R})\) and the Lorentz \(so(1,3, \mathbb{R})\) are related by the Wick rotation of their time coordinates \(t_{\rm E} \mapsto \mathrm{i}t\). So the Euclidean Lie algebra and the Lorentz Lie algebra have the same complexification.

   Some comments:

   1. The complexification of a real vector space \(V\) is denoted as \(V \otimes_{\mathbb{R}} \mathbb{C}\).

   2. If \(V\) happens to be a complex vector space, then the complexification of the underlying real vector space of a complex vector space \(V\), namely \(V \otimes_{\mathbb{R}} \mathbb{C}\), is the direct sum of two copies of \(V\), namely \(V \otimes_{\mathbb{R}} \mathbb{C}=V \oplus V\).

   3. Note that \(su(2)\) is a real vector space (not closed under the scalar multiplication by \(\mathrm{i}\)), \(sl(2,\mathbb{C})\) is a complex vector space (closed under the scalar multiplication by \(\mathrm{i}\)). So \(sl(2,\mathbb{C}) \cong su(2) \otimes_{\mathbb{R}} \mathbb{C}\) is the complexification of \(su(2)\), and the complexification of the underlying real vector space of a complex vector space \(sl(2,\mathbb{C})\) is \(sl(2,\mathbb{C})\otimes_{\mathbb{R}}\mathbb{C}\cong sl(2,\mathbb{C}) \oplus sl(2,\mathbb{C}) \cong (su(2) \otimes_{\mathbb{R}} \mathbb{C}) \oplus (su(2) \otimes_{\mathbb{R}} \mathbb{C}) =(su(2) \oplus su(2) ) \otimes_{\mathbb{R}} \mathbb{C}\), although \(sl(2,\mathbb{C}) \not\cong su(2) \oplus su(2)\).

   ↩︎

8. We label \(n_6 \in {\mathbb{Z}}_6 = {\mathbb{Z}}_6^{\rm F}\supset {\mathbb{Z}}_2^{\rm F}\) in terms of a doublet \((n_2^{\rm F}, n_3) \in {\mathbb{Z}}_2^{\rm F}\times {\mathbb{Z}}_3\), such that the bosons have \(n_2^{\rm F}=0\) and the fermions have \(n_2^{\rm F}=1\). In addition, without loss of generality, we assign the charge \(q=1 \in {\mathbb{Z}}_6^{\rm F}\) fermion to the \((n_2^{\rm F}, n_3)=(1,1) \in {\mathbb{Z}}_2^{\rm F}\times {\mathbb{Z}}_3\). This constrains the map as \(n_6 = 3 n_2^{\rm F}- 2 n_3\), so \(n_6 = n_3 \mod 3\).
   We label \(n_{18} \in {\mathbb{Z}}_{18} = {\mathbb{Z}}_{18}^{\rm F}\supset {\mathbb{Z}}_2^{\rm F}\) in terms of a doublet \((n_2^{\rm F}, n_9) \in {\mathbb{Z}}_2^{\rm F}\times {\mathbb{Z}}_9\), such that the bosons have \(n_2^{\rm F}=0\) and the fermions have \(n_2^{\rm F}=1\). In addition, without loss of generality, we assign the charge \(q=1 \in {\mathbb{Z}}_{18}^{\rm F}\) fermion to the \(( n_2^{\rm F}, n_9)=(1,1) \in {\mathbb{Z}}_2^{\rm F}\times {\mathbb{Z}}_9\). This constrains the map as \(n_{18} = 9 n_2^{\rm F}- 8 n_9\), so \(n_{18} = n_9 \mod 9\).↩︎

9. For \(n=3\) as \(\operatorname{H}^5({\mathbb{Z}}_3,{\rm U}(1))\cong {\mathbb{Z}}_3\), that generator is the anomaly of three right-handed neutrinos \(3\nu_R\), which has the anomaly index \(3 \in \Omega_5^{\rm Spin \times {\mathbb{Z}}_3} \cong \Omega_5^{\rm Spin}(B{\mathbb{Z}}_3) \cong {\mathbb{Z}}_9\). That is also the anomaly of the SM missing the \(3\nu_R\), up to a negative sign \(-1\) for the anomaly index.↩︎

10. Ref. [61]’s Section 7 constructs low dimensional coupled bulk–boundary TQFTs via symmetry extension; examples include a \((2+1)\)d bulk with a \((1+1)\)d boundary and a \((3+1)\)d bulk with a \((2+1)\)d boundary. Our approach is similar to Ref. [61]’s Sec. 7.↩︎

11. This is because \(\operatorname{H}^m({\mathbb{Z}}_n,{\rm U}(1))=\operatorname{H}^{m+1}({\mathbb{Z}}_n,{\mathbb{Z}})\), while \(\operatorname{H}^*({\mathbb{Z}}_n,{\mathbb{Z}})\) is periodic of period 2, and \(\operatorname{H}^2({\mathbb{Z}}_n,{\mathbb{Z}})=\operatorname{H}^2({\mathbb{Z}}_n,{\mathbb{Z}}_n)\) by the universal coefficient theorem.↩︎

12. However, \(p_1\) or \(\frac{p_1}{2}\) is trivialized by a higher group extension \[\begin{align} 1\to \mathrm{B}^2{\mathbb{Z}}\to{\rm String}\to{\rm Spin}\to1. \end{align}\] Here \(p_1\) is integer \({\mathbb{Z}}\) valued for SO manifolds, while \(\frac{p_1}{2}\) is integer \({\mathbb{Z}}\) valued for Spin manifolds because \(p_1=w_2^2 =0 \mod 2\).↩︎

13. Cartan’s formula usually applies to the cup product. The cross product \(a\times b\) is defined as \(\pi_1^*a\smile \pi_2^*b\) where \(\pi_i\) is the projection from \(M\times M\) onto its \(i\)-th factor. Then \[\begin{align} P_q(a\times b)=P_q(\pi_1^*a\smile \pi_2^*b)=P_q(\pi_1^*a)\smile P_q(\pi_2^*b)=\pi_1^*P_q(a)\smile \pi_2^*P_q(b)=P_q(a)\times P_q(b). \end{align}\]↩︎