Anomaly and Cobordism Constraints Beyond the Standard Model

Anomaly and Cobordism Constraints

Beyond the Standard Model: Topological Force

Juven Wang

Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, USA

Abstract

Standard lore uses perturbative local anomalies to check the kinematic consistency of gauge theories coupled to chiral fermions, such as the Standard Models (SM) of particle physics. In this work, based on a systematic cobordism classification, we examine the constraints from invertible quantum anomalies (including all perturbative local anomalies and all nonperturbative global anomalies) for SM and chiral gauge theories. We also clarify the different uses of these anomalies: including (1) anomaly cancellations of dynamical gauge fields, (2) ’t Hooft anomaly matching conditions of background fields of global symmetries. We apply several 4d anomaly constraints of Z16, Z4, and Z2 classes, beyond the familiar Feynman-graph perturbative Z class local anomalies. As an application, for SU(3)×SU(2)×U(1) Zq SM (with q = 1, 2, 3, 6) and SU(5) Grand Unification with 15n chiral Weyl fermions and with a discrete baryon minus lepton number X = 5(B L) 4Y preserved, we discover a new hidden gapped sector previously unknown to the SM and− Georgi-Glashow− model. The gapped sector at low energy contains either (1) 4d non-invertible topological quantum field theory (TQFT, above the energy gap with heavy fractionalized anyon excitations from 1d particle worldline and 2d string worldsheet, inaccessible directly from Dirac or Majorana mass gap of the 16th Weyl fermions [i.e., right-handed neutrinos], but accessible via a topological quantum phase transition), or (2) 5d invertible TQFT in extra dimensions. Above a higher energy scale, the discrete X becomes dynamically gauged, the entangled Universe in 4d and 5d is mediated by Topological Force. Our model potentially resolves puzzles, surmounting sterile neutrinos and dark matter, in fundamental physics. arXiv:2006.16996v2 [hep-th] 23 Dec 2020

量子異常和配邊約束超越標準模型:新拓撲力 [email protected] http://sns.ias.edu/∼juven/ Contents

1 Introduction 4

2 Overview on Standard Models, Grand Uniﬁcations, and Anomalies 10

2.1 SM and GUT: local Lie algebras to global Lie groups, and representation theory . . . . . 10

2.2 Classiﬁcations of anomalies and their diﬀerent uses ...... 14

3 Dynamical Gauge Anomaly Cancellation 17

3.1 Summary of anomalies from a cobordism theory and Feynman diagrams ...... 19

3 U(1) 2 3 3.2 U(1)Y : 4d local anomaly from 5d CS1 c1(U(1)) and 6d c1(U(1)) ...... 20

2 U(1) 3.3 U(1)Y -SU(2) : 4d local anomaly from 5d CS1 c2(SU(2)) and 6d c1(U(1))c2(SU(2)) ... 21

2 U(1) 3.4 U(1)Y -SU(3)c : 4d local anomaly from 5d CS1 c2(SU(3)) and 6d c1(U(1))c2(SU(3)) ... 21

2 c1(U(1))(σ F F) 3.5 U(1)Y -(gravity) : 4d local anomaly from 5d µ(PD(c1(U(1)))) and 6d 8 − · .... 22

3 1 SU(3) 1 3.6 SU(3)c : 4d local anomaly from 5d 2 CS5 and 6d 2 c3(SU(3)) ...... 23

3.7 Witten SU(2) anomaly: 4d Z2 global anomaly from 5d c2(SU(2))˜η and 6d c2(SU(2))Arf . 23

4 Anomaly Matching for SM and GUT with Extra (B L) Symmetries 25 − 4.1 SM and GUT with extra continuous symmetries and a cobordism theory ...... 25

4.1.1 (B L)-U(1)2 : 4d local anomaly ...... 26 − Y 4.1.2 (B L)-SU(2)2: 4d local anomaly ...... 27 − 4.1.3 (B L)-SU(3)2: 4d local anomaly ...... 27 − c 4.1.4 (B L)-(gravity)2: 4d local anomaly ...... 28 − 4.1.5 (B L)3: 4d local anomaly ...... 29 − 4.1.6 X-SU(5)2: 4d local anomaly ...... 29

4.2 SM and GUT with extra discrete symmetries and a cobordism theory ...... 30

4.2.1 Witten anomaly c2(SU(2))˜η vs c2(SU(2))η0: 4d Z2 vs Z4 global anomalies . . . . . 32

2 4.2.2 ( )c (SU(2)): 4d Z global anomaly ...... 32 AZ2 2 2 4.2.3 ( )c (SU(3)): 4d Z global anomaly ...... 32 AZ2 2 2 2 4.2.4 c1(U(1)) η0: 4d Z4 global anomaly ...... 33

4.2.5 ( )c (SU(5)): 4d Z global anomaly ...... 33 AZ2 2 2 4.2.6 η(PD( )): 4d Z global anomaly ...... 33 AZ2 16

4.3 How to match the anomaly? Preserving or breaking the Z4,X symmetry? ...... 34

5 Beyond Known “Fundamental” Forces: Hidden Topological Force 37

5.1 Hidden 5d invertible TQFT or 5d Symmetry-Protected Topological state (SPTs) . . . . . 37

5.2 Hidden 4d non-invertible TQFT or 4d symmetry-enriched topological order ...... 39

5.2.1 Symmetry extension [Z2] Spin Z4,X Spin F Z4,X and a 4d [Z2] gauge theory 39 → × → ×Z2

5.2.2 4d anomalous symmetric Z2 gauge theory with ν = 2 Z16 preserving Spin F Z4,X 41 ∈ ×Z2 5.3 Gapping Neutrinos: Dirac mass vs Majorana mass vs Topological mass ...... 44

5.3.1 Dirac Mass or Majorana Mass ...... 46

5.3.2 Topological Mass and Topological Energy Gap ...... 47

5.3.3 4d duality for Weyl fermions and “mirror symmetry” ...... 48

5.3.4 Gapping 3, 2, or 1 Weyl spinors / sterile right-handed neutrinos + extra sectors . . 50

6 Ultra Uniﬁcation: Grand Uniﬁcation + Topological Force and Matter 51

6.1 4d-5d Theory: Quantum Gravity and Topological Gravity coupled to TQFT ...... 51

6.2 Braiding statistics and link invariants in 4d or 5d: Quantum communication with gapped hidden sectors from our Standard Model ...... 54

6.3 Topological Force as a new force? ...... 57

6.4 Neutrinos ...... 57

6.5 Dark Matter ...... 58

7 Acknowledgements 59

3 8 Bibliography 59

“Der schwer gefaßte Entschluß. Muß es sein? Es muß sein!” “The heavy decision. Must it be? It must be!” String Quartet No. 16 in F major, op. 135 Ludwig van Beethoven in 1826

1 Introduction

The Universe where we reside, to our contemporary knowledge, is governed by the laws of quantum theory, the information and long-range entanglement, and gravity theory. Quantum ﬁeld theory (QFT), especially gauge ﬁeld theory, under the name of Gauge Principle following Maxwell [1], Hilbert, Weyl [2], Pauli [3], and many pioneers, forms a foundation of fundamental physics. Yang-Mills (YM) gauge theory [4], generalizing the U(1) abelian gauge group to a non-abelian Lie group, has been given credits for theoretically and experimentally essential to describe the Standard Model (SM) physics [5–7].

The SM of particle physics is a chiral gauge theory in 4d1 encoding three of the four known fundamental forces or interactions (the electromagnetic, weak, and strong forces, but without gravity). The SM also classifies all experimentally known elementary particles so far: Fermions include three generations of quarks and leptons. (See Table1 for 15 + 1 = 16 Weyl fermions in SM for each one of the three generations, where the additional 1 Weyl fermion is the sterile right-handed neutrino). Bosons include the vector gauge bosons: one electromagnetic force mediator photon γ, the eight strong force mediator a 0 gluon g (denoted the gauge field A ), the three weak force mediator W ± and Z gauge bosons; and the scalar Higgs particle φH . (See Table2 for 12 gauge boson generators for SM.) While the spin-2 boson, the graviton, has not yet been experimentally verified, and it is not within SM. Physics experiments had confirmed that at a higher energy of SM, the electromagnetic and weak forces are unified into an electroweak interaction sector: Glashow-Salam-Weinberg (GSW) SM [5–7].

Grand Unifications and Grand Unified Theories (GUT) hypothesize that at a further higher energy, the strong and the electroweak interactions will be unified into an electroweak-nuclear GUT interaction sector. The GUT interaction is characterized by one larger gauge group and its force mediator gauge

1We denote dd for the d spacetime dimensions. The d + 1d means the d spatial and 1 time dimensions. The nD means the n space dimensions.

4 bosons with a single uniﬁed coupling constant [8,9]. 2

In this article, we revisit the Glashow-Salam-Weinberg SM [5–7], with four possible gauge groups: SU(3) SU(2) U(1) GSMq × × , q = 1, 2, 3, 6. (1.1) ≡ Zq

SU(3) SU(2) U(1) q = 6 × × We revisit the embedding of this special SM (with a gauge group Zq ) into Georgi- Glashow SU(5) GUT with an SU(5) gauge group [8] (with 24 Lie algebra generators for SU(5) gauge bosons shown in Table2), and Fritzsch-Minkowski SO(10) GUT with a Spin(10) gauge group [9]. Our main motivation to revisit these well-known SM and GUT models, following our prior work [10, 11], is that the recent systematic Freed-Hopkins’ cobordism classiﬁcation of topological invariants [12] can be applied to classify all the invertible quantum anomalies, including

• all perturbative local anomalies, and

• all nonperturbative global anomalies, which can further mathematically and rigorously constrain SM and GUT models. In fact, many earlier works and recent works suggest the cobordism theory is the underlying math structure of invertible quantum anomalies, see a list of selective References [13–26] and an overview therein. By the completion of all invertible anomalies, we mean that it is subject to a given symmetry group

Gspacetime n Ginternal G ( ), (1.2) ≡ Nshared 3 4 where the Gspacetime is the spacetime symmetry, the Ginternal is the internal symmetry, the n is a 5 semi-direct product from a “twisted” extension, and the Nshared is the shared common normal subgroup symmetry between Gspacetime and Ginternal. Ref. [10, 11, 27] applies Thom-Madsen-Tillmann spectra [28, 29], Adams spectral sequence (ASS) [30], and Freed-Hopkins theorem [12] in order to compute, relevant for SM and GUT, the bordism group

G Ωd , (1.3) and a specific version of cobordism group (firstly defined to classify Topological Phases [TP] in [12])

d d ΩG Ω GspacetimenGinternal TPd(G). (1.4) ( ) ≡ Nshared ≡

2Other than the three known fundamental forces, we also have other GUT forces predicted by GUT. However, in our present work, we discover a possible new force based on the global anomaly matching, which we name Topological Force. This Topological Force may be a possible new force not included in the known four fundamental forces nor included in GUT. The Topological Force is not fully explored in the prior particle physics phenomenology literature, at least not rigorously explored based on the cobordism and nonperturbative global anomaly matching framework. On the other hand, when we consider the constraints from the anomaly matching, the gravity mostly plays the role of the gravitational background probed ﬁelds (instead of dynamical gravity), such as in the gravitational anomaly or the mixed gauge-gravitational anomaly. We mostly leave out any dynamical gravity outside our model, so the present work is far from a Theory of Everything (TOE). The only exception that we discuss the inﬂuences of dynamical gravity (or the hypothetical particle: graviton, such as in Quantum Gravity or Topological Gravity) will be in Sec. 6.1. 3 For example, Gspacetime can be the Spin(d), the double cover of Euclidean rotational symmetry SO(d) for the QFT of a spacetime dimension d. 4 SU(3)×SU(2)×U(1) Ginternal GSM ≡ , q = 1, 2, 3, 6. For example, can be the q Zq 5 The “twisted” extension is due to the symmetry extension from Ginternal by Gspacetime. For a trivial extension, the semi-direct product n becomes a direct product ×.

5 For a given G, Ref. [10, 11, 27] ﬁnd out corresponding all possible topological terms and all possible anomalies relevant for SM and GUT. See more mathematical deﬁnitions, details, and References therein our prior work [10, 11].

Along this research direction, other closely related pioneer and beautiful works, by Garcia-Etxebarria- Montero [31] and Davighi-Gripaios-Lohitsiri [32], use a diﬀerent mathematical tool, based on Atiyah- Hirzebruch spectral sequence (AHSS) [33] and Dai-Freed theorem [34], also compute the bordism groups G 6 Ωd and classify possible global anomalies in SM and GUT. As we will see, many SM and GUT with G extra discrete symmetries require the twisted G, whose Ωd is very diﬃcult, if not simply impossible, to G be determined via AHSS alone [31, 32]. But the twisted Ωd can still be relatively easily computed via ASS [10, 37]. Therefore, we will focus on the results obtained in [10,11].

In this work, we have the plans and outline as follows, with topics by sections:

Sec. [2]. Overview on SM, GUT, and Anomalies in Sec.2: We first overview the ingredients of various SM and GUT to set up the stage in Sec. 2.1. Then we comment on the different types of anomalies, and we clarify the different uses of these anomalies: including (1) anomaly cancellations of dynamical gauge fields,7 (2) ’t Hooft anomaly matching conditions [42] of background fields of global symmetries, and others, in Sec. 2.2 and Fig.1. We also describe the perturbative local anomalies vs nonperturbative global anomalies; also bosonic vs fermionic anomalies, etc. These are presented in Sec.2.

Sec. [3]. Dynamical Gauge Anomaly Cancellation in Sec.3: We explicitly show the classiﬁcation of all possible (invertible) anomalies of SM gauge groups by cobordism data in Table3, and explicitly check all anomaly cancellations for perturbative local anomalies vs nonperturbative global anomalies in Sec.3.

Sec. [4]. Anomaly Matching for SM and GUT with Extra Symmetries in Sec.4: It is natural to include additional extra symmetries (such as the B L baryon minus lepton numbers − or X = 5(B L) 4Y symmetry with the hypercharge Y [43], motivated in [31] and [11]) into SM − − and SU(5) GUT. We show the classiﬁcation of all possible (invertible) anomalies of some SM and GUT with extra symmetries by cobordism data in Table4. We explicitly check their anomaly cancellations for perturbative local anomalies vs nonperturbative global anomalies in Sec.4.

Sec. [5]. Beyond Three “Fundamental” Forces: Hidden New Topological Force in Sec.5: We recall that to match the ’t Hooft anomalies of global symmetries, there are several ways:

(a) Symmetry-breaking: (say, discrete or continuous G-symmetry breaking. Explicitly breaking or spontaneously • 6A practical comment is that Adams spectral sequence (ASS) used in Ref. [10–12] turns out to be more powerful than the Atiyah-Hirzebruch spectral sequence (AHSS) used in [31, 32]. • Ref. [10, 11, 35–37] based on Adams spectral sequence (ASS) and Freed-Hopkins theorem [12] includes the more refined data, containing both module and group structure, thus with the advantages of having less differentials. In addition, we can conveniently read and extract the topological terms and co/bordism invariants from the Adams chart and ASS [10]. • Ref. [31, 32] is based on Atiyah-Hirzebruch spectral sequence (AHSS), which includes more differentials and some undetermined extensions. It is also not straightforward to extract the topological terms and co/bordism invariants directly from the AHSS.

7The perturbative parts of calculations are standard on the QFT textbook [38–41]

6 breaking [may give Nambu-Goldstone modes]).

(b) Symmetry-preserving: Degenerate ground states (like the “Lieb-Schultz-Mattis theorem [44, 45],” may host intrinsic • topological orders.8), Gapless, e.g., conformal ﬁeld theory (CFT). There are also novel cases where the anomaly and • symmetry together enforces the robustness of gapless ground states [50–52]. Symmetry-preserving topological quantum ﬁeld theory (TQFT): anomalous symmetry-enriched • topological orders. (c) Symmetry-extension [53]: The symmetry-extension in any dimension is a helpful intermediate stepstone, to construct another earlier scenario: symmetry-preserving TQFT, via gauging the extended-symmetry [53].

We present the possible ways of matching of global anomalies of “SM and GUT with extra symmetries,” in Sec.5. The condensed matter realization of ’t Hooft matching by an anomalous symmetry-preserving gapped TQFT is especially exotic but truly important to us. It is also known as the surface topological order in the condensed matter literature. In a lower dimension, the 2+1d surface topological order is pointed out ﬁrstly by Senthil-Vishwanath [54]. The surface topological order developments are nicely reviewed in [55]. The particular symmetric gapped boundary spin-TQFT construction of fermionic SPTs given in [56] will be especially helpful for our construction.9 It turns out that a certain Z global anomaly for SM with an extra discrete X = 5(B L) 4Y 16 − − symmetry may not be matched by the 15 Weyl fermions per generation. The resolution, other than introducing a sterile right-handed neutrinos (the 16th Weyl fermions), can also be that introducing a new 10 gapped topological sector matching some of the anomaly, but preserves the discrete Z4,X symmetry.

Sec. [6]. Ultra Unification: Grand Unification + Topological Force/Matter in Sec.6: In certain cases of anomaly matching, we require new hidden gapped topological sectors beyond SM and GUT. We may term the unification including SM, Grand Unification plus additional topological sectors with Topological Force and Matter as Ultra Unification. We then suggest possible resolutions to sterile neutrinos, neutrino oscillations, and Dark Matter, in Sec.6.

Notations: Throughout our work, we follow the same notations as [10, 11]. The imaginary number is i √ 1. The TQFT stands for a topological quantum ﬁeld theory, and the iTQFT stands for an ≡ − invertible topological quantum ﬁeld theory. Follow the Freed-Hopkins notation [12], we denote the group G G (G G )/N where N is a common normal subgroup of the groups G and G . We use the 1 ×N 2 ≡ 1 × 2 1 2 standard notation for characteristic classes [61]: ci for the Chern class, en for the Euler class, pi for the Pontryagin class, and wi for the Stiefel-Whitney class. We abbreviate ci(G), en(G), pi(G), and wi(G) for the characteristic classes of the associated vector bundle VG of the principal G bundle (normally

8Topological order is in the sense of Wen’s deﬁnition [46] and References therein. The gapped and gauged topological order in the colloquial sense can have fractionalized excitations such as anyons [47]. Some of the quantum vacua we look for in 4d and 5d may be regarded as a certain version of topological quantum computer [48, 49]. 9Our result (3+1d boundary TQFT of 4+1d bulk) may be regarded as a one higher dimensional fundamental physics realization of the condensed matter systems in Ref. [57, 58] (2+1d boundary TQFT of 3+1d bulk). 10 More precisely, it turns out that as suggested by [59,60], we can only construct a Z4,X -symmetric gapped 4d TQFT to Spin× Z Z2 4 saturate the anomaly of Z16 class from the 5d bordism group Ω5 = Z16, when the anomaly index is an even integer ν ∈ Z16. For example, the SM with three generations of 15 Weyl fermions has the anomaly index ν = −3 ∈ Z16, we can introduce one right-handed neutrino to absorb part of the anomaly ν = 1 ∈ Z16. Then, we also introduce a Z4,X -symmetric gapped 4d TQFT to absorb the remained anomaly ν = 2 ∈ Z16. See more details on the anomaly in Sec.4. See the construction of ν = 2 ∈ Z16 TQFT in Sec. 5.2.2.

7 denoted as ci(VG), en(VG), pi(VG), and wi(VG)). For simplicity, we denote the Stiefel-Whitney class of the tangent bundle TM of a manifold M as w w (TM); if we do not specify w with which bundle, j ≡ j j then it is for TM. The PD is defined as taking the Poincaré dual. All the product notations between cohomology classes are cup product “^”, such as w w w (TM)w (TM) = w (TM) ^ w (TM). 2 3 ≡ 2 3 2 3 All the product notations between a cohomology class and any fermionic topological invariant η (say, A the 4d eta invariant η of Atiyah-Patodi-Singer [13–15]), namely η, are defined as the value of η on A the submanifold of M which represents the Poincaré dual of . Notice that here for η, it is crucial A A to have as a cohomology class, so we can define its Poincaré dual of as PD( ). In other words, A A A the η η(PD( )). Similarly we define the product notations between a cohomology class and other A ≡ A fermionic invariants also via the Poincaré dual PD of cohomology class. Here are some other examples of fermionic invariants appeared in this work: Spin The η˜ is a mod 2 index of 1d Dirac operator as a cobordism invariant of Ω1 = Z2. • Spin Z4 The η is a mod 4 index of 1d Dirac operator as a cobordism invariant of Ω × = Z . • 0 1 4 The Arf invariant [62] is a mod 2 cobordism invariant of ΩSpin = Z . The Arf appears to be the low • 2 2 energy iTQFT of a 1+1d Kitaev fermionic chain [63], whose boundary hosts a single 0+1d real Majorana zero mode on each of open ends. − The Arf-Brown-Kervaire (ABK) invariant [64, 65] is a mod 8 cobordism invariant of ΩPin = Z . The • 2 8 ABK appears to be the low energy iTQFT of a 1+1d Fidkowski-Kitaev fermionic chain [66,67] protected by time reversal and fermion parity symmetry ZT ZF , whose boundary also hosts some number of 0+1d 2 × 2 real Majorana zero modes. For cobordism or cohomology invariants, we may implicitly make a convention that cohomology classes 1 are pulled back to the manifold M. For example, Z2 H (M, Z2) is the first cohomology class on M 1A ∈ F with a Z2 coefficient, as a generator from Z2 H (B(Z4/Z2 ), Z2) of Spin F Z4 pullback to M as the A ∈ ×Z2 former expression.

8 SM fermion F SU(3) SU(2) U(1)Y U(1)YW U(1) ˜ U(1)EM U(1)B L U(1)X Z4,X Z2 SU(5) Spin(10) spinor Y − ﬁeld d¯ 3¯ 1 1/3 2/3 2 1/3 1/3 3 1 1 R − − 5 l 1 2 1/2 1 3 0 or 1 1 3 1 1 L − − − − − − q 3 2 1/6 1/3 1 2/3 or 1/3 1/3 1 1 1 L − 16 u¯R 3¯ 1 2/3 4/3 4 2/3 1/3 1 1 1 10 + − − − − − e¯R = eL 1 1 1 2 6 1 1 1 1 1 ν¯R = νL 1 1 0 0 0 0 1 5 1 1 1

Table 1: We show the quantum numbers of 15 + 1 = 16 left-handed Weyl fermion spinors in each of three generations of matter fields in SM. The 15 of 16 Weyl fermion are 5 10 of SU(5); namely, ⊕ (3, 1, 1/3) (1, 2, 1/2) 5 and (3, 2, 1/6) (3, 1, 2/3) (1, 1, 1) 10 of SU(5). The 1 of 16 is L ⊕ − L ∼ L ⊕ − L ⊕ L ∼ presented neither in the standard GSW SM nor in the SU(5) GUT, but it is within 16 of the SO(10) GUT. The numbers in the Table entries indicate the quantum numbers associated to the representation of the groups given in the top row. We show the first generation of SM fermion matter fields in Table1. There F are 3 generations, triplicating Table1, in SM. All fermions have the fermion parity Z2 representation 1.

SM boson SU(3) SU(2) U(1)Y U(1)B L U(1)X Z4,X scalar or − vector ﬁeld SM Higgs scalar 1 2 1/2 0 2 2 φH − SM gauge vector bosons (1 + 3 + 8 = 12 generators) photon (γ) 0 0 0 A or A Y 3 + 1 Weak 1,2,3 0 0 0 0 W or W ±,Z Strong (g) 8 0 0 0 Aaτ a BSM Georgi-Glashow gauge vector bosons (2 (3 2) = 12 generators) × × X+ 4 (+ 3 e ) ¯ 0 0 + 1 | | 3 2 5/6 2/3 Y (+ 3 e ) 0 0 1 | | Y− ( 3 e ) 0 0 − 4 | | 3 2 5/6 2/3 X− ( e )− − 0 0 − 3 | |

Table 2: We show quantum numbers of the electroweak Higgs boson φH , and 24 gauge bosons corresponding to 24 Lie algebra su(5) generators of Lie group SU(5). Hereby the representation numbers of gauge fields, we really mean the gauge fields transform in the adjoint of the gauge group. (But the gauge fields are not in the adjoint or other R representations of the gauge group.) Also the readers should not be confused with the symmetry charge X (in the Italic form) and its gauge boson Xg, with the Georgi-Glashow (GG) model gauge boson X (in the boldface text form). The X and Y gauge bosons can carry certain SU(3), SU(2) and U(1) quantum numbers, because the X and Y sit outside the SM F gauge group. All bosons have the fermion parity Z2 representation 0. See also the caption in Table1.

9 2 Overview on Standard Models, Grand Uniﬁcations, and Anomalies

2.1 SM and GUT: local Lie algebras to global Lie groups, and representation theory

We shall ﬁrst overview the local Lie algebra the representation theory of matter ﬁeld contents, and the global Lie groups of of SM and GUT. Then we will be able to be precise about the spacetime symmetry group Gspacetime and internal symmetry group Ginternal relevant for SM and GUT physics,

[I]. The local gauge structure of Standard Model is the Lie algebra u(1) su(2) su(3). This means that the × × Lie algebra valued 1-form gauge fields take values in the Lie algebra generators of u(1) su(2) su(3). × × There are 1 + 3 + 8 = 12 Lie algebra generators. The 1-form gauge fields are the 1-connections of the principals Ginternal-bundles that we should determine. [II]. Fermions are the spinor fields, as the sections of the spinor bundles of the spacetime manifold. For the left-handed Weyl spinor ΨL, it is a doublet spin-1/2 representation of spacetime symmetry group Gspacetime (Minkowski Spin(3, 1) or Euclidean Spin(4)), denoted as

Ψ 2 of Spin(3, 1) = SL(2, C), or Ψ 2 of Spin(4) = SU(2) SU(2) . (2.1) L ∼ L L ∼ L L × R These Spin groups are the double-cover also the universal-cover version of the Lorentz group SO(3, 1)+ or F Euclidean rotation SO(4), extended by the fermion parity Z2 . In the ﬁrst generation of SM, the matter ﬁelds as Weyl spinors ΨL contain: u • The left-handed up and down quarks (u and d) form a doublet in 2 for the SU(2)weak, and they d L are in 3 for the SU(3)strong.

• The right-handed up and down quarks, each forms a singlet, uR and dR, in 1 for the SU(2)weak. They are in 3 for the SU(3)strong. νe • The left-handed electron and neutrino form a doublet in 2 for the SU(2)weak, and they are in e L 1 for the SU(3)strong.

• The right-handed electron forms a singlet eR in 1 for the SU(2)weak, and it is in 1 for the SU(3)strong. There are two more generations of quarks: charm and strange quarks (c and s), and top and bottom quarks (t and b). There are also two more generations of leptons: muon and its neutrino (µ and νµ), and tauon and its neutrino (τ and ντ ). So there are three generations (i.e., families) of quarks and leptons: ! u νe 3color, uR 3color, dR 3color, , eR , (2.2) d L × × × e L ! c νµ 3color, cR 3color, sR 3color, , µR , (2.3) s L × × × µ L ! t ντ 3color, tR 3color, bR 3color, , τR . (2.4) b L × × × τ L In short, for all of them as three generations, we can denote them as: ! u νe 3color, uR 3color, dR 3color, , eR 3 generations. (2.5) d L × × × e L ×

10 In fact, all the following four kinds of SU(3) SU(2) U(1) Ginternal = × × (2.6) Zq with q = 1, 2, 3, 6 are compatible with the above representations of fermion ﬁelds.11 These 15 3 Weyl × spinors can be written in the following more succinct forms of representations for any of the internal symmetry group Ginternal with q = 1, 2, 3, 6: ! (3, 2, 1/6) , (3, 1, 2/3) , (3, 1, 1/3) , (1, 2, 1/2) , (1, 1, 1) 3 generations L R − R − L − R × ! (3, 2, 1/6) , (3, 1, 2/3) , (3, 1, 1/3) , (1, 2, 1/2) , (1, 1, 1) 3 generations. (2.7) ⇒ L − L L − L L ×

The triplet given above is listed by their representations:

(SU(3) representation, SU(2) representation, hypercharge Y ). (2.8)

For example, (3, 2, 1/6) means that 3 in SU(3), 2 in SU(2) and 1/6 for hypercharge. In the second line of (2.7), we transforms the right-handed Weyl spinor ΨR 2R of Spin(3, 1) to its left-handed ΨL ∼ 12 ∼ 2L of Spin(3, 1), while we ﬂip their representation (2.8) to its complex conjugation representation. If we include the right-handed neutrinos (say νeR, νµR, and ντ R), they are all in the representation

(1, 1, 0)R (2.9) with no hypercharge. We can also represent a right-handed neutrino by the left-handed (complex) conjugation version

(1, 1, 0)L. (2.10)

Also the complex scalar Higgs ﬁeld φH is in a representation

(1, 2, 1/2). (2.11)

In the Higgs condensed phase of SM, the conventional Higgs vacuum expectation value (vev) is chosen to 0 be φ = 1 , which vev has a Q = 0. h H i √2 v EM

Note that our hypercharge Y is given conventionally by the relation: Q = T3 + Y where Q is the EM EM 1 1 0 unbroken (not Higgsed) electromagnetic gauge charge and T = is a generator of SU(2)weak. 3 2 0 1 − However, some other conventions are common, we list down three conventions 1 1 Q = T + Y = T + Y = T + Y.˜ (2.12) EM 3 3 2 W 3 6

11 SU(3)×SU(2)×U(1) Ginternal = q = 1, 2, 3, 6 See an excellent exposition on Zq with in a recent work by Tong [68]. 12Note that 2 and 2 are the same representation in SU(2), see, e.g., in the context of Yang-Mills gauge theories with discrete symmetries and cobordism [23].

11 1 ˜ ˜ In the QEM = T3 + 6 Y version, we have the integer quantized Y = 6Y . We can rewrite (2.7) as: ! (3, 2,Y = 1/6) , (3, 1,Y = 2/3) , (3, 1,Y = 1/3) , (1, 2,Y = 1/2) , (1, 1,Y = 1) 3 generations L R − R − L − R × ! = (3, 2,Y = 1/6) , (3, 1,Y = 2/3) , (3, 1,Y = 1/3) , (1, 2,Y = 1/2) , (1, 1,Y = 1) 3 generations L − L L − L L × ! = (3, 2,Y = 1/3) , (3, 1,Y = 4/3) , (3, 1,Y = 2/3) , (1, 2,Y = 1) , (1, 1,Y = 2) 3 generations W L W R W − R W − L W − R × ! = (3, 2,Y = 1/3) , (3, 1,Y = 4/3) , (3, 1,Y = 2/3) , (1, 2,Y = 1) , (1, 1,Y = 2) 3 generations W L W − L W L W − L W L × ! = (3, 2, Y˜ = 1) , (3, 1, Y˜ = 4) , (3, 1, Y˜ = 2) , (1, 2, Y˜ = 3) , (1, 1, Y˜ = 6) 3 generations L R − R − L − R × ! = (3, 2, Y˜ = 1) , (3, 1, Y˜ = 4) , (3, 1, Y˜ = 2) , (1, 2, Y˜ = 3) , (1, 1, Y˜ = 6) 3 generations. (2.13) L − L L − L L ×

The right hand neutrino νR is in a representation

(1, 1,Y = 0) = (1, 1,YW = 0) = (1, 1, Y˜ = 0). (2.14)

times some generation number. Also the complex scalar Higgs ﬁeld φH is in a representation

(1, 2,Y = 1/2) = (1, 2,YW = 1) = (1, 2, Y˜ = 3), (2.15)

where Y˜ = 3YW = 6Y . We organize the above data in Table1.

[III]. If we include the 3 2 + 3 + 3 + 2 + 1 = 15 left-handed Weyl spinors from one single generation, we can × combine them as a multiplet of 5 and 10 left-handed Weyl spinors of SU(5):

(3, 1, 1/3) (1, 2, 1/2) 5 of SU(5), (2.16) L ⊕ − L ∼ (3, 2, 1/6) (3, 1, 2/3) (1, 1, 1) 10 of SU(5). (2.17) L ⊕ − L ⊕ L ∼ Hence these are matter field representations of the SU(5) GUT with a SU(5) gauge group. Other than the electroweak Higgs φH (which φH can be part of a scalar Higgs in the fundamental 5 of SU(5) GUT), we SU(3) SU(2) U(1) φ × × φ may also introduce a different GUT Higgs field GG to break down SU(5) to Z6 . The GG is in the adjoint representation of SU(5) as 5 5 24 = (8, 1,Y = 0) (1, 3,Y = 0) (1, 1,Y = 0) (3, 2,Y = ) (3¯, 2,Y = ). (2.18) ⊕ ⊕ ⊕ −6 ⊕ 6

[IV]. If we include an extra right-handed neutrino plus the 3 2 + 3 + 3 + 2 + 1 = 15 left-handed Weyl spinors × from one single generation, we can combine them as a multiplet of 16 left-handed Weyl spinors:

Ψ 16+ of Spin(10), (2.19) L ∼ + which sits at the 16-dimensional irreducible spinor representation of Spin(10). (In fact, 16 and 16−- dimensional irreducible spinor representations together form a 32-dimensional reducible spinor representation of Spin(10).) Namely, instead of an SO(10) gauge group, we should study the SO(10) GUT with a Spin(10) gauge group.

12 [V]. Lie algebra generators and gauge bosons: We can count the number of Lie algebra generators to represent the local 1-form gauge ﬁeld of gauge bosons. For example, there are 1 + 3 + 8 = 12 independent Lie algebra generators thus gauge bosons for the Lie algebra u(1) su(2) su(3). There are 24 independent × × Lie algebra generators (thus gauge bosons) for the Lie algebra su(5), and 45 Lie algebra generators (thus gauge bosons) for the Lie algebra so(10). In this work, we focus on SM and SU(5) GUT. We list down their quantum numbers in Table2.

[VI]. U(1)B L symmetry and U(1)X symmetry: For SM with Ginternal = GSMq of q = 1, 2, 3, 6, we have a − B L (baryon minus lepton numbers) U(1)B L global symmetry. For SU(5) GUT with Ginternal = SU(5), − − we can have a U(1) symmetry [43], where the X charge is related to B L via X − X 5(B L) 4Y = 5(B L) 2Y , (2.20) ≡ − − − − W X˜ 3X = 5 3(B L) 2 3Y = 5 3(B L) 2Y˜ = 5(B˜ L˜) 2Y,˜ (2.21) ≡ · − − · W · − − − − Y˜ = 3YW = 6Y. (2.22)

Here we have used (2.12).

F [VII]. Z4,X symmetry, Z2 fermion parity, and discrete symmetries: We will learn that it is natural to consider a Z4,X subgroup of U(1)X , when we attempt to embed the SU(5) GUT to SO(10) GUT. In fact, the Z4,X can be regarded as the Z4 center of the Spin(10) gauge group as Z(Spin(10)) = Z = Z ZF [31]. Since Spin(10) is fully dynamically gauged in SO(10) GUT, the Z 4 4,X ⊃ 2 4,X is also dynamically gauged. In summary, we have

U(1)B L can be a global symmetry (maybe anomalous) in the SMs for all q = 1, 2, 3, 6. • − U(1) can be a global symmetry (maybe anomalous) in the SU(5) GUT. • X Z can be chosen to be either a global or a gauge symmetry for the SM and SU(5) GUT. • 4,X But Z4,X is a dynamical gauge symmetry in the SO(10) GUT. ZF fermion parity has a generator ( 1)F with the fermion number F , which can be a global symmetry • 2 − in the SMs (for all q = 1, 2, 3, 6) and in the SU(5) GUT. F But the Z2 is naturally chosen to be a dynamical gauge symmetry in the SO(10) GUT.

For the SO(10) GUT, we may call Z4,X a gauge symmetry colloquially, but we should remind readers that a gauge symmetry is not really a (global) symmetry, but only a gauge redundancy. Note that:

All fermions from leptons and quarks have a Z charge 1, see Table1. • 4,X All gauge bosons have a Z charge 0, but the Higgs have a Z charge 2, see Table2. • 4,X 4,X The ZF gives ( 1) sign to all fermions (additive notation: charge 1 in ZF in Table1). • 2 − 2 The ZF gives (+1) sign to all bosons (additive notation: charge 0 in ZF in Table2). • 2 2

The above facts about Z4,X charges for bosons or fermions (in Table1 and Table2) can be understood from F Z4,X a group extension (as a short exact sequence): 0 Z2 Z4,X (Z2A F ) 0. The fermion parity → → → ≡ Z2 → F F Z2 is a normal subgroup of Z4,X , while the Z2A is only a quotient group. So the fermions, with an odd Z2 F charge, must carry also an odd charge of Z4,X . The bosons, with an even (i.e., no) Z2 charge, must carry 13 also an even charge of Z4,X .

13 X Given the Z4,X charge state |Xi with X = 0, 1, 2, 3 as a representation z . Note that z ∈ U(1) with |z| = 1, where we F embed the normal subgroup Z2 ⊂ Z4,X ⊂ U(1). X • The Z4,X symmetry generator UZ4,X acts on |Xi, which becomes UZ4,X |Xi = i |Xi with z = i. F 2 2 • The subgroup Z2 symmetry generator U F = (UZ ,X ) can also act on |Xi, which becomes U F |Xi = (UZ ,X ) |Xi = Z2 4 Z2 4 2X X F F i |Xi = (−1) |Xi. Thus, we read the fermion parity (−1) , the |1i and |3i has −1 (thus odd 1 in Z2 ), while the |0i and |2i F F A Z4,X has +1 (thus even 0 in Z2 ). This is also consistent with the fact that the group extension 0 → Z2 → Z4,X → Z2 ≡ F → 0 Z2 2 A F requires a nontrivial 2-cocycle H (BZ2 , Z2 ) = Z2 to specify the extension to Z4,X .

13 These extra symmetries are well-motivated in the earlier pioneer works [69–72] and References therein the recent work [31,56,59]. Extra discrete symmetries can be powerful giving rise to new anomaly cancellation constraints. Part of the new ingredients we will survey are the new global anomalies for discrete symmetries of SMs, in Sec.4. Let us discuss how these extra groups can be embedded into the total groups in Remarks [VIII], [IX], [X], and [XI]. [VIII]. We ﬁnd the Lie group embedding for the internal symmetry of GUTs and Standard Models [11,27]: U(1) SU(2) SU(3) SO(10) SU(5) × × . (2.23) ⊃ ⊃ Z6 U(1) SU(2) SU(3) Spin(10) SU(5) × × . (2.24) ⊃ ⊃ Z6

Only q = 6, but not other q = 1, 2, 3, for GSMq can be embedded into Spin(10) nor SU(5). So from the U(1) SU(2) SU(3) 14 × × GUT perspective, it is natural to consider the Standard Model gauge group Z6 . [IX]. We also ﬁnd the following group embedding for the spacetime and internal symmetries of GUTs and Stan- dard Models (Ref. [11,27], see also [37] for the derivations): Spin(d) Spin(10) SU(3) SU(2) U(1) ×F Spin(d) SU(5) Spin(d) × × . (2.25) Z2 ⊃ × ⊃ × Z6 SU(3) SU(2) U(1) Spin(d) Spin(10) Spin(d) SU(5) Spin(d) × × . (2.26) × ⊃ × ⊃ × Z6

c [X]. For an extra U(1)B L or U(1)X symmetry, we need to consider Spin Spin(d) F U(1) structure. We − ≡ ×Z2 ﬁnd the embedding: SU(3) SU(2) U(1) Spinc(d) SU(5) Spinc(d) × × . (2.27) × ⊃ × Z6

[XI]. For an extra Z4,X symmetry, we need to consider Spin(d) Z Z4 = Spin(d) F Z4,X structure. In order × 2 ×Z2 Spin(d) Spin(10) to contain these groups embedded in ×F , it is more naturally to consider: Z2 Spin(d) Spin(10) SU(3) SU(2) U(1) × Spin(d) F Z4,X SU(5) Spin(d) F Z4,X × × . (2.28) F Z2 Z2 Z2 ⊃ × × ⊃ × × Z6

Ref. [11, 27, 31, 32] study the cobordism theory of some of these SM, BSM, and GUT groups. We shall d=5 d=d particular pay attention to d = 5 for ΩG Ω GspacetimenGinternal TPd=5(G), since the 5d cobordism ≡ ( ) ≡ Nshared invariants of TPd=5(G) can classify the 4d invertible anomalies of the total group G. We particularly apply the results in [11], for anomaly constraints and anomaly matchings of BSM physics, in the following Sec.3.

2.2 Classiﬁcations of anomalies and their diﬀerent uses

To start, let us mention diﬀerent concepts of anomalies and their diﬀerent uses. See for example Fig.1 for perturbative local anomalies captured by Feynman-Dyson graphs. Generally, we also have nonperturbative global anomalies not captured by Feynman graphs, but their uses are similar as follows:

14 U(1)×SU(2)×SU(3) In fact, the author believe and declare that GSM ≡ is the correct and natural gauge group of SM. 6 Z6

14 gauge Global sym (dynamical ﬁeld) (Backgrd. ﬁeld)

gauge gauge Global sym Global sym (dynamical field) (dynamical field) (Backgrd. field) (Backgrd. field) (1) (2) gauge Global sym (Backgrd. field) (dynamical field)

gauge gauge Global sym Global sym (dynamical field) (dynamical field) (Backgrd. field) (Backgrd. field) (3) (4)

Figure 1: Examples of perturbative local anomalies of Z classes in 4d that can be captured by the free part d=5 of cobordism group ΩG TPd=5(G) in (1.4), which is in fact descended from the free part of bordism G ≡ group Ωd=6.(1) Dynamical gauge anomaly. (2) ’t Hooft anomaly of background (Backgrd.) fields. (3) Adler-Bell-Jackiw (ABJ [73, 74]) type of anomalies. (4) Anomaly that involves two background fields of global symmetries and one dynamical gauge field.

(1). Dynamical gauge anomaly of Ginternal (e.g., Fig.1( 1)): Anomaly matching cancellation must be zero for a dynamical gauge theory of its group Ginternal.

(2). ’t Hooft anomaly (e.g., Fig.1( 2)): Anomaly matching of background symmetry of G including Ginternal for their background ﬁelds, surprisingly, need not to be zero for a QFT:

( i). If the ’t Hooft anomaly of a QFT of any G background ﬁelds turn out to be exactly zero, this means • that these global symmetries of QFT can be realized local and onsite (or on an n-simplex for a higher generalized global n-symmetry [75]) on a lattice. This also means that we do not need to regularize the QFT living on the boundary of Symmetry-Protected/Enriched Topological states [27, 53, 76–78]. In fact, the QFT may be well-deﬁned as the low energy theory of a quantum local lattice model in its own dimensions, as a high-energy ultraviolet (UV) completion — if it is free from all anomalies, via a cobordism argument [27].

( ii). If the ’t Hooft anomaly of a QFT of any G background ﬁelds turn out to be not zero nor cancelled: • Then this can be regarded as several meanings and implications on QFT dynamics: First, the symmetries of QFT in fact are non-local or non-onsite (or non-on-n-simplex for a higher •

15 generalized global n-symmetry) on a lattice. This means the obstruction of gauging such non-local symmetries, thus this obstruction is equivalent with the definition of ’t Hooft anomaly [42]. Second, the QFT can have a hidden sector of the same dimension. We can match the ’t Hooft • anomaly of QFT with additional sector S0 of the same dimension, but with the S0 sector with the opposite ’t Hooft anomaly. So, the combined system, the QFT and S0, can have a no ’t Hooft anomaly at all. Third, the QFT can have a hidden sector of one higher dimension. In order to have the symmetries • of QFT with ’t Hooft anomaly to be local or onsite instead, we need to append and regularize this dd QFT living on the boundary of (d+1)d Symmetry-Protected/Enriched Topological states (SPTs/SETs) [27,53,76,77]. 15 This is related to the Callan-Harvey anomaly inflow [84] of (d + 1)d bulk and dd boundary with their spacetime dimensions differed by one. Fourth, (some part of) the global symmetry can also be broken spontaneously or explicitly, so that • the ’t Hooft anomaly is matched by symmetry breaking. Fifth, Of course, we can also have a certain combination of the First, Second, Third, and Fourth • scenarios above, in order to make sense of QFT with ’t Hooft anomaly.

(3). Adler-Bell-Jackiw (ABJ [73, 74]) type of anomalies (e.g., Fig.1( 3)): The non-conservations of the global symmetry current is coupled to the background non-dynamical field via the action term ? J A A ∧ J ≡ ddx( µ) with the Hodge dual star ?. The non-conservation of current is proportional to the´ anomaly AµJ ´factor in dd spacetime d(? ) (F )d/2 F F .... J ∝ a ∝ a ∧ a ∧ Here a are dynamical gauge fields; we can also modify the equation appropriate for several different dynamical 1-form or higher-form gauge fields. For d = 4, the ABJ anomaly formula is precisely captured by Fig.1( 3).

(4). Anomaly that involves two background ﬁelds of global symmetries and one dynamical gauge ﬁeld (e.g., Fig.1( 4)).

Indeed it is obvious to observe that the anomalies( 1) are tighten to anomalies( 2).

(α). Anomalies from( 1) can be related to anomalies from( 2) via the gauging principle.

(β). Anomalies from( 2) can be related to anomalies from( 1) via the ungauging principle, or via gauging the higher symmetries [75].

Thus if we learn the gauge group of a gauge theory (e.g., SM, GUT or BSM), we may identify its ungauged 16 global symmetry group as an internal symmetry group, say Ginternal via ungauging.

Now let us discuss the classiﬁcations of anomalies. By “all invertible quantum anomalies” obtain from a cobordism classiﬁcation, we mean the inclusion of:

15Symmetry-Protected Topological states (SPTs) are the short-range entangled states as a generalization of the free non- interacting topological insulators and superconductors [79–83] with interactions. Symmetry-Enriched Topological states (SETs) are topological ordered states enriched by global symmetries. The enthusiastic readers can overview the condensed matter terminology bridging to QFT in Ref. [23] for QFT theorists, or the excellent condensed matter reviews [46, 55]. 16By gauging or ungauging, also depending on the representation of the matter ﬁelds that couple to the gauge theory, we may gain or lose symmetries or higher symmetries [75]. It will soon become clear, for our purpose, we need to primarily focus on the ordinary (0-form) internal global symmetries and their anomalies.

16 (i). Perturbative local anomalies captured by perturbative Feynman graph loop calculations, classiﬁed by the integer group Z classes, or the free classes in mathematics. Some selective examples from QFT or gravity are:

(1): Perturbative fermionic anomalies from chiral fermions with U(1) symmetry, originated from Adler- Bell-Jackiw (ABJ) anomalies [73, 74] with Z classes. (2): Perturbative bosonic anomalies from bosonic systems with U(1) symmetry with Z classes [10]. (3): Perturbative gravitational anomalies [85].

(ii). Non-perturbative global anomalies, classiﬁed by a product of ﬁnite groups such as Zn, or the torsion classes in mathematics. Some selective examples from QFT or gravity are:

(1): An SU(2) anomaly of Witten in 4d or in 5d [86] with a Z2 class, which is a gauge anomaly.

(2): A new SU(2) anomaly in 4d or in 5d [87] with a diﬀerent Z2 class, which is a mixed gauge-gravity anomaly.

(3): Higher ’t Hooft anomalies of Z2 class for a pure 4d SU(2) YM theory with a second-Chern-class topological term [88–90] (or the so-called SU(2)θ=π YM): The higher anomaly involves a discrete 0-form time-reversal symmetry and a 1-form center Z2-symmetry. The first anomaly is discovered in [88]; later the anomaly is refined via a mathematical well-defined 5d bordism invariant as its topological term, with additional new Z2 class anomalies found for Lorentz symmetry-enriched four siblings of Yang-Mills gauge theories [89,90]. (4): Global gravitational anomalies [91] detected by exotic spheres. (5): Bosonic anomalies: Many types of bosonic anomalies in diverse dimensions are global anomalies [19, 92–95]. These bosonic anomalies only require bosonic degrees of freedom, but without the requirement of chiral fermions. Many such bosonic anomalies are related to group cohomology or generalized group cohomology theory, living on the boundary of SPTs [76], closely related to Dijkgraaf-Witten topological gauge theories [96].

Our present work explore global anomalies (generically not captured by Feynman graphs). In order to help the readers to digest the physical meanings of global anomalies, we should imagine the computation of anomalies on generic curved spacetime manifolds with mixed gauge and gravitational background probes, in the cobordism theory setting (as in Fig.2).

3 Dynamical Gauge Anomaly Cancellation

In this section, we explicitly check the dynamical gauge anomaly cancellations of various SMs with four gauge group GSM (SU(3) SU(2) U(1))/Z with q = 1, 2, 3, 6. The cobordism classiﬁcations of q ≡ × × q GSMq ’s 4d anomalies are done in [11, 32]. The 4d anomalies can be written as 5d cobordism invariants, which are 5d invertible TQFT (iTQFT). These 5d cobordism invariants/iTQFT are derived in [11], we summarized the classiﬁcations and invariants in Table3.

Moreover, as Ref. [32] points out correctly already, we will show that dynamical gauge anomaly

cancellation indeed holds to be true for all GSMq with q = 1, 2, 3, 6. Ref. [32] has not written down the 5d cobordism invariants nor explicit anomaly polynomials. However, by simply looking at the group classiﬁcations of anomalies, Ref. [32] argues that there is only the famous Witten SU(2) nonperturbative global anomaly of Z2 class [86] , other anomalies all are perturbative local anomalies captured by

17 Figure 2: General nonperturbative global anomalies not captured by Feynman graphs can still be characterized by generic curved manifolds with mixed gauge and gravitational background probes. The ﬁgure shows a bordism between manifolds. Here M and M 0 are two closed d-manifolds, N is a compact d + 1- manifold whose boundary is the disjoint union of M and M , namely ∂N = M M . If there are 0 t 0 additional G-structures on these manifolds, then the G-structure on N is required to be compatible with the G-structures on M and M 0. If there are additional maps from these manifolds to a ﬁxed topological space, then the maps are also required to be compatible with each other. If these conditions are obeyed, then M and M 0 are called bordant equivalence and N is called a bordism between M and M 0.

Feynman graphs. So why do we bother to do the calculations again, if Ref. [32] has found all dynamical

gauge anomalies cancel for GSMq ? There are multiple reasons: • First, we will show that the 5d cobordism invariants obtained in [11] indeed match with the anomaly polynomials. For perturbative local anomalies of Z classes, see Fig.3, we indeed can show that the 5d cobordism invariants map to some one-loop Feynman graph calculations known in the standard QFT textbooks [38–41]. (In contrast, Ref. [32] focus on global anomalies, and pays less attention on perturbative local anomalies of Z classes. Follow Ref. [11], we will fill in this gap by considering all local and all global anomalies.) We should work through all these correspondences carefully to gain a solid confidence for our understanding of cobordism classifications of anomalies.

• Second, we can learn how to translate the cobordism data from math into the anomalies in physics. For example, in Table3, we see that the 5d cobordism invariant c2(SU(2))˜η in fact captures the 4d boundary theory has the Witten SU(2) nonperturbative global anomaly [86].

• Third, the most important thing, we will discover new phenomena later when we include the additional discrete symmetries into SM and GUT in Sec.4. In fact, we will discover entirely new physics that previously have never been ﬁgured out in the past.

Notations: Throughout our work, we write the three SU(2) Lie algebra generator σa of the rank-2 a b matrix of fundamental representation satisfying Tr[ σ σ ] = 1 δ with a, b 1, 2, 3 . We write the 2 2 2 ab ∈ { } eight SU(3) Lie algebra generator τ a of the the rank-3 matrix of fundamental representation satisfying a b Tr[ τ τ ] = 1 δ with a, b 1, 2,..., 8 . 2 2 2 ab ∈ { }

18 3.1 Summary of anomalies from a cobordism theory and Feynman diagrams

Cobordism group TP (G) with GSM (SU(3) SU(2) U(1))/Z with q = 1, 2, 3, 6 d q ≡ × × q classes cobordism invariants

G = Spin GSM × 1 U(1) 2 U(1) SU(2) 5 µ(PD(c1(U(1)))), CS1 c1(U(1)) , CS1 c2(SU(2)) c1(U(1))CS3 , Z Z SU(3) ∼ 5d 2 U(1) SU(3) CS × CS c (SU(3)) c (U(1))CS , 5 , c (SU(2))˜η 1 2 ∼ 1 3 2 2

G = Spin GSM × 2 U(2) U(2) U(2) 2 CS1 c2(U(2)) c1(U(2))CS3 5 µ(PD(c1(U(2)))), CS1 c1(U(2)) , 2 2 , 5d Z SU(3)∼ U(2) SU(3) CS CS c (SU(3)) c (U(2))CS , 5 1 2 ∼ 1 3 2

G = Spin GSM × 3 U(3) 2 U(3) SU(2) 5 µ(PD(c1(U(3)))), CS1 c1(U(3)) , CS1 c2(SU(2)) c1(U(3))CS3 , Z Z U(3) U(3) U(3) U(3) ∼ 5d 2 CS c (U(3))+CS c (U(3))CS +CS U(3) × 1 2 5 1 3 5 , CS , c (SU(2))˜η 2 ∼ 2 5 2

G = Spin GSM × 6 U(3) U(2) U(3) 2 CS1 c2(U(2)) c1(U(3))CS3 5 µ(PD(c1(U(2)))) µ(PD(c1(U(3)))), CS1 c1(U(3)) , 2 2 , 5d Z ∼ U(3) U(3) U(3) U(3) ∼ CS c (U(3))+CS c (U(3))CS +CS U(3) 1 2 5 1 3 5 , CS 2 ∼ 2 5 Table 3: The 4d anomalies can be written as 5d cobordism invariants of Ωd=5 TP (G), which are G ≡ d=5 5d iTQFTs. These 5d cobordism invariants/iTQFTs are derived in [11]. We summarized the group classifications of 4d anomalies and their 5d cobordism invariants. The anomaly classification of Z5 means G that there are 5 perturbative local anomalies (of Z classes descended from the 6d bordism group Ωd=6), precisely match 5 perturbative one-loop triangle Feynman diagrams in Fig.3. The anomaly classification of Z2 means that there is a 1 nonperturbative global anomaly, which turns out to be Witten SU(2) anomaly [86]. The cj(G) is the jth Chern class of the associated vector bundle of the principal G-bundle. 4 3 3 σ F F 4 The µ is the 3d Rokhlin invariant. If ∂M = M , then µ(M ) = ( −8 · )(M ), thus µ(PD(c1(U(1)))) is c1(U(1))(σ F F) 4 related to 8 − · . Here is the intersection form of M . The F is the characteristic 2-surface [97] 4 · 4 in a 4-manifold M , it obeys the condition F x = x x mod 2 for all x H2(M , Z). By the Freedman- σ F F 4 · 4 · ∈ Kirby theorem, we have ( −8 · )(M ) = Arf(M , F) mod 2. The PD is defined as the Poincaré dual. The Arf is a 2d Arf invariant, whose condensed matter realization is the 1+1d Kitaev fermionic chain [63]. The η˜ is a mod 2 index of 1d Dirac operator. In Sec. 3.7 and Ref. [11], we propose that the Z2 class 5d cobordism invariant c2(SU(2))˜η corresponds to the 4d Witten SU(2) anomaly [86]. See our notational conventions in Sec. 1 and Sec. 1.2.4 of Ref. [11]. The symbol “ ” here means the equivalent rewriting of ∼ cobordism invariants on a closed 5-manifold.

19 U(1)Y gauge U(1)Y gauge U(1)Y gauge

U(1)Y gauge U(1)Y gauge SU(2) gauge SU(2) gauge SU(3)c gauge SU(3)c gauge (i) (ii) (iii)

U(1)Y gauge SU(3)c gauge

gravity gravity SU(3)c gauge SU(3)c gauge (iv) (v)

Figure 3: Examples of dynamical gauge anomaly cancellations in SM. In fact, the 5 perturbative local anomalies from perturbative one-loop triangle Feynman diagrams precisely match anomaly classiﬁcation of Z5 obtained from the cobordism group calculations in Table3 and Ref. [11].

3 U(1) 2 3 3.2 U(1)Y : 4d local anomaly from 5d CS1 c1(U(1)) and 6d c1(U(1))

We read from Ref. [11] and Table3 for the Z class of the 5d cobordism invariants of the following: U(1) 2 U(2) 2 U(3) 2 5d CS1 c1(U(1)) for GSM1 , 5d CS1 c1(U(2)) for GSM2 and GSM6 , while 5d CS1 c1(U(3)) for GSM3

and GSM6 . These 5d cobordism invariants correspond to the 4d perturbative local anomalies captured by the one-loop Feynman graph:

. (3.1)

3 U(1) 2 Without losing generality, we focus on the 4d cubic anomaly (U(1)Y ) from 5d CS1 c1(U(1)) , which 3 also descends from 6d c1(U(1)) of bordism group Ω6 in Ref. [11]. Plug in data in Sec. 2.1, it is standard to check the anomaly (3.1) vanishes,17 X X Tr[(Yˆ )3] = (Y )3 (Y )3 q qL − qR q qL,qR

17When we switch between from the L-chiral fermion to the R-chiral fermion (anti-chiral fermion), there could be an additional minus sign.

20 1 3 3 3 3 3 3 = δ Ngeneration N 2 (1/6) + ( 2/3) + (1/3) + 2 ( 1/2) + (1) + (0) 2 ab · c · · − · − = Ngeneration ( N + 3) (1/4), (3.2) · − c ·

which is 0 when Nc = 3 for 3 colors as we have. The Ngeneration (or Nfamily) counts the number of generations (same as families).

2 U(1) 3.3 U(1)Y -SU(2) : 4d local anomaly from 5d CS1 c2(SU(2)) and 6d c1(U(1))c2(SU(2))

We read from Ref. [11] and Table3 for a Z class of 5d cobordism invariants of the following: U(2) U(2) U(1) CS1 c2(U(2)) c1(U(2))CS3 5d CS1 c2(SU(2)) for GSM1 , 5d 2 2 for GSM2 , ∼ U(3) U(2) U(3) SU(2) CS1 c2(U(2)) c1(U(3))CS3 5d CS c (SU(2)) c (U(3))CS for GSM , and 5d for GSM . These 1 2 ∼ 1 3 3 2 ∼ 2 6 5d cobordism invariants correspond to the 4d perturbative local anomalies captured by the one-loop Feynman graph:

U(1)Y gauge

SU(2) gauge SU(2) gauge . (3.3)

2 U(1) Without losing generality, we focus on the 4d anomaly U(1)Y -SU(2) from 5d CS1 c2(SU(2)), which also descends from 6d c1(U(1))c2(SU(2)) of bordism group Ω6 in Ref. [11]. Plug in data in Sec. 2.1, we check the anomaly (3.3) vanishes, X 1 X Tr[Yˆ σaσb] = δ ( (Y ) (Y )) q 2 ab qL − qR q qL,qR 1 = δ Ngeneration N 2 (1/6) + 2 ( 1/2) 2 ab · c · · · − 1 = δ Ngeneration (N /3 1), (3.4) 2 ab · c −

which is 0 when Nc = 3.

2 U(1) 3.4 U(1)Y -SU(3)c : 4d local anomaly from 5d CS1 c2(SU(3)) and 6d c1(U(1))c2(SU(3))

We read from Ref. [11] and Table3 for a Z class of 5d cobordism invariants of the following: U(1) SU(3) U(2) SU(3) 5d CS1 c2(SU(3)) c1(U(1))CS3 for GSM1 , 5d CS1 c2(SU(3)) c1(U(2))CS3 for GSM2 , 5d U(3) U(3) ∼ U(3) U(3) U(3) ∼ U(3) U(3) U(3) CS1 c2(U(3))+CS5 c1(U(3))CS3 +CS5 CS1 c2(U(3))+CS5 c1(U(3))CS3 +CS5 2 2 for GSM3 , and 5d 2 2 for ∼ U(3) SU(3) ∼ GSM6 . (Note that part of the additional contribution from CS5 or CS5 will be separately discussed

21 later in Sec. 3.6 and (3.9).) These 5d cobordism invariants correspond to the 4d perturbative local anomalies captured by the one-loop Feynman graph:

U(1)Y gauge

SU(3)c gauge SU(3)c gauge . (3.5)

2 Without losing generality, we focus on the 4d anomaly U(1)Y -SU(3)c from 5d U(1) SU(3) CS c (SU(3)) c (U(1))CS , which also descends from 6d c (U(1))c (SU(3)) of bordism 1 2 ∼ 1 3 1 2 group Ω6 in Ref. [11]. Plug in data in Sec. 2.1, we check the anomaly (3.6) vanishes, X X 1 X Tr[Yˆ τ aτ b] = Tr[Yˆ τ aτ b] Tr[Yˆ τ aτ b] = δ ( (Y ) (Y )) q qL − qR 2 ab qL − qR q qL,qR qL,qR 1 = δ Ngeneration N 2 (1/6) + ( 2/3) + (1/3) 2 ab · c · · − 1 = δ Ngeneration N 0 = 0. (3.6) 2 ab · c ·

2 c1(U(1))(σ F F) − · 3.5 U(1)Y -(gravity) : 4d local anomaly from 5d µ(PD(c1(U(1)))) and 6d 8

We read from Ref. [11] and Table3 for a Z class of 5d cobordism invariants of the following:

5d µ(PD(c1(U(1)))) for GSM1 , 5d µ(PD(c1(U(2)))) for GSM2 and GSM6 , while 5d µ(PD(c1(U(3)))) for

GSM3 and GSM6 . These 5d cobordism invariants correspond to the 4d perturbative local anomalies captured by the one-loop Feynman graph:

U(1)Y gauge

gravity gravity . (3.7)

2 Without losing generality, we focus on the 4d anomaly U(1)Y -gravity of U(1)Y -gravitational anomaly c1(U(1))(σ F F) from 5d µ(PD(c1(U(1)))), which also descends from 6d 8 − · of bordism group Ω6 in Ref. [11]. Plug in data in Sec. 2.1, we check the anomaly (3.7) vanishes, X X Tr[Yˆ ] = (Y ) (Y ) q qL − qR q qL,qR

22 = Ngeneration N 2 (1/6) + ( 2/3) + (1/3) + 2 ( 1/2) + (1) + (0) · c · · − · − = Ngeneration (0 N + 0) = 0. (3.8) · · c We remark that if we view the gravity as dynamical ﬁelds, then (3.7) checks the dynamical gauge anomaly cancellation of the Fig.1( 1) and Remark( 1); if we view the gravity as background probe ﬁelds, then (3.7) checks the anomaly cancellation of the type of Fig.1( 4) and Remark( 4).

3 1 SU(3) 1 3.6 SU(3)c : 4d local anomaly from 5d 2 CS5 and 6d 2 c3(SU(3))

We read from Ref. [11] and Table3 for a Z class of 5d cobordism invariants of the following: 1 SU(3) U(3) 5d 2 CS5 for GSM1 and GSM2 , and 5d CS5 for GSM3 and GSM6 . (Note that part of the contributions U(3) from CS5 also occur in Sec. 3.4.) These 5d cobordism invariants correspond to the 4d perturbative local anomalies captured by the one-loop Feynman graph:

SU(3)c gauge

SU(3)c gauge SU(3)c gauge . (3.9)

3 1 SU(3) Without losing generality, we focus on the 4d anomaly SU(3)c from 5d 2 CS5 , which also descends 1 from 6d 2 c3(SU(3)) of bordism group Ω6 in Ref. [11]. Plug in data in Sec. 2.1, we check the anomaly (3.9) vanishes. In fact, in the context of SM physics, even without checking explicitly, it is clear that this 3 SU(3)c anomaly (3.9) must vanish, since the color SU(3)c is vector gauge theory not chiral gauge theory respect to the color charge. We recall that only U(1)Y and SU(2)weak are chiral gauge theories in SM.

Readers may ask what happen to the SU(2)3 anomaly by replacing the gauge ﬁelds in (3.9) to SU(2), 3 since SU(2)weak is chiral? The answer is that SU(2) anomaly does not exist thus must vanish, because there is no such corresponding 5d cobordism invariant found in Ref. [11] and Table3. In fact, for SU(2) and SO(N) group, all representations have zero 4d perturbative local anomalies, thus they must have none of Z classes of 5d cobordism invariants, agreed with Ref. [11].

3.7 Witten SU(2) anomaly: 4d Z2 global anomaly from 5d c2(SU(2))˜η and 6d c2(SU(2))Arf

The old SU(2) anomaly of Witten in 4d [86] is summarized in [87] for the context we need. We read from Ref. [11] and Table3 for a Z2 class of 5d cobordism invariant and suggest the 4d SU(2) anomaly 18 corresponds to 5d c2(SU(2))˜η, and descends from 6d c2(SU(2))Arf of bordism group Ω6 in Ref. [11].

18As explained in the Notation in the end of Sec.1: The η˜ is a mod 2 index of 1d Dirac operator as a cobordism invariant Spin Spin of Ω1 = Z2. The Arf invariant [62] is a mod 2 cobordism invariant of Ω2 = Z2, whose realization is the 1+1d Kitaev fermionic chain [63].

23 1. 5d c2(SU(2))˜η and 4d Witten SU(2) anomaly: This 4d anomaly is a mod 2 index of Z2 class counts the spin-2r + 1/2 (or 4r + 2 in the dimension of representation) Weyl spinor as fermion doublet under SU(2) [87]. From Sec. 2.1, there are four of spin-2r + 1/2 fermions from (3, 2, 1/6)L and (1, 2, 1/2) , multiplied by Ngeneration. So we check overall the Witten anomaly vanishes in SM: − L

(even number) of mod 2 = 0. | 2

2. Näively, 5d c2(SU(2))˜η only presents for GSM1 and GSM3 , but not for GSM2 and GSM6 . Readers may wonder how does Witten anomaly vanish for SM of q = 2, 6? Ref. [98] explains nicely and accurately that the Witten anomaly mutates from a Z class global • 2 anomaly into a subclass of perturbative local Z class when we changes the SU(2) group to the U(2) group, namely for q = 2, 6. Ref. [11] gives a formal explanation as follows: The difference between q = 1 and q = 2 case • is parallel to the difference between q = 3 and q = 6 case. So without losing generality, we focus on the difference between q = 1 and q = 2 cases. The Witten anomaly from c2(SU(2))˜η U(1) SU(2) of Z2 for q = 1 vanishes in q = 2, while in contrast the CS1 c2(SU(2)) c1(U(1))CS3 of U(2) U(2) ∼ U(1) Z for q = 1 becomes 1 CS c (U(2)) 1 c (U(2))CS in q = 2. The CS c (SU(2)) in 2 1 2 ∼ 2 1 3 1 2 5d comes from c1(U(1))c2(U(2)) in 6d. Here the c1(U(2))c2(U(2)) mod 2 is the quotient Z2, the 1 c1(U(2))c2(U(2)) is the total Z, and the 2 c1(U(2))c2(U(2)) is the normal Z in the short exact 2 sequence 0 Z Z Z 0. Let us express the levels of those cobordism invariants as → → → 2 → kq=2, kq=1, and kq0 =1 respectively. Indeed we can also understand the short exact sequence as the 2 classes of the levels: 0 k Z k Z k Z 0. There is yet another way to → q=1 ∈ → q=2 ∈ → q0 =1 ∈ 2 → explain why c1(U(2))c2(U(2)) mod 2 vanishes for q = 2: Via Wu formula, the c1(U(2))c2(U(2)) = Sq2c (U(2)) = (w + w2)c (U(2)) = 0 mod 2 on the Spin 6-manifolds. In short, the k Z of 2 2 1 2 q=2 ∈ local anomalies now also carry information of the k Z of the Witten SU(2) global anomaly. q0 =1 ∈ 2

In summary of Sec.3, by checking ﬁve Z classes of local anomalies and one Z2 class of Witten SU(2) global anomaly, we have shown that the for SM with GSMq of q = 1, 2, 3, 6 are indeed free from all dynamical gauge anomalies, thus dynamical gauge anomaly cancellation holds.

As we have checked, the GSMq is a healthy chiral gauge theory by its own with a dynamical gauge 19 group GSMq . However, what if we include additional global symmetries or gauge sectors? Such as the B L or X 5(B L) 4Y ? This motivates us to explore further in the next section Sec.4. − ≡ − −

24 Global sym B L Global sym B L Global sym B L − − (Backgrd. field) (Backgrd. field) (Backgrd. field)−

U(1)Y gauge U(1)Y gauge SU(2) gauge SU(2) gauge SU(3)c gauge SU(3)c gauge (i) (ii) (iii) Global sym B L Global sym X Global sym B L − (Backgrd. field)− (Backgrd. field) (Backgrd. field)

gravity gravity B L B L SU(5) gauge SU(5) gauge (iv) (v) − − (vi)

Figure 4: Examples of anomaly constraint for SM (or GUT) with extra symmetries such as B L or − X 5(B L) 4Y . We only show perturbative local anomalies from perturbative one-loop triangle ≡ − − Feynman diagrams discussed in (4.2). We will explore nonperturbative global anomalies (not captured by Feynman diagrams) in later sections. Assume the gravity contributes as background ﬁeld: If B L or X is not gauged, (i), (ii), (iii), and (vi) are ABJ anomalies of Fig.1( 3) and Remark( 3); • − (iv) and (v) are ’t Hooft anomalies of Fig.1( 2) and Remark( 2). If B L or X is gauged, (i)-(iv), (vi) are dynamical gauge anomalies of Fig.1( 1) and Remark( 1); • − (v) is an anomaly of Fig.1( 4) and Remark( 4). If these anomalies are not matched, we can still saturate the anomalies by proposing new sectors appending to the QFT; we will explore those new physics in Sec.5.

4 Anomaly Matching for SM and GUT with Extra (B L) Symmetries −

4.1 SM and GUT with extra continuous symmetries and a cobordism theory

In Sec. 2.1 Remark [X], for SM and SU(5) GUT with an extra U(1)B L or U(1)X symmetry, we need to − consider Spinc Spin(d) U(1) structure. We ﬁnd the embedding: ≡ ×Z2 SU(3) SU(2) U(1) Spinc(d) SU(5) Spinc(d) × × . × ⊃ × Z6 Ref. [32] checks that the 5d bordism group:

Spinc SU(3) SU(2) U(1) Ω5 ( × × ) = 0, (4.1) Zq

19 It has been long sought that GSMq is a healthy chiral gauge theory by its own with a dynamical gauge group GSMq free from all dynamical gauge anomalies. However, it is only until very recently by the systematic computations of cobordism groups in [31] (which checks q = 1), [98] and [11] (which the two papers checks q = 1, 2, 3, 6) completing the full checks

on GSMq . Without a cobordism classiﬁcation of anomalies, previous literature either only check perturbative anomalies, or may still miss additional global anomaly constraints (as we shall see new anomaly constraints in Sec.4).

25 which means no global anomalies. Ref. [37] computes the following cobordism groups TP5 and bordism groups Ω6:

c 4 Spinc 4 TP5(Spin SU(5)) = Z , Ω6 (SU(5)) = Z . × (4.2) c SU(3) SU(2) U(1) 11 Spinc SU(3) SU(2) U(1) 11 TP5(Spin × × ) = Z , Ω ( × × ) = Z . × Zq 6 Zq

Ref. [37] ﬁnds that these TP5 and Ω6 only contain Z classes, thus they only correspond to 4d local anomalies captured by the one-loop Feynman graph shown in Fig.4.

We emphasize that if U(1)B L or U(1)X is free of all anomalies, then we can dynamically gauge this − symmetry. In that case, we should regard the corresponding gauge field as a Spinc connection instead of the familiar U(1) gauge fields, since the original theory requires to be defined on Spinc manifolds. c (When we mention gauge fields for gauging U(1)B L or U(1)X , what we really have in mind is the Spin − connection.) The U(1)B L or U(1)X is free of all anomalies if they are free from perturbative local − anomalies given by (4.2), since they do not have global anomalies.

Are all the perturbative local anomalies canceled for these SM and GUT with extra continuous symmetries? The baryon and lepton local currents densities are 1 jµ = (¯q γµq +u ¯ γµu + d¯ γµd ), jµ = (¯l γµl +e ¯ γµe + N ν¯ γµν ). (4.3) B 3 L L R R R R L L L R R νR R R

Here NνR is the number of right-handed neutrinos in one generation whose representation is given in (2.9): In the standard GSW SM, we have N = 0. • νR In the SU(5) GUT, it is common to have N = 0. • νR In the SO(10) GUT, we have N = 1. • νR We used to believe that there are no perturbative local anomalies for an additional U(1) if this U(1) is the U(1)B L or U(1)X . Let us check explicitly in the next subsections. We should pay attention on the − anomaly cancellation and its dependence on NνR = 0 or 1. What we will check is the conservation of the U(1)B L current, − d ? (j j ) = ∂ (jµ jµ)ddx, (4.4) B − L µ B − L by taking into account all possible anomaly contributions from cobordism considerations.

4.1.1 (B L)-U(1)2 : 4d local anomaly − Y

Plug in data from Table1 to check the 4d local anomaly of (B L)-U(1)2 : − Y Global sym B L (Backgrd. ﬁeld)−

U(1)Y gauge U(1)Y gauge , (4.5)

26 we ﬁnd the anomaly factor contributes to 20 2 2 2 j : Ngeneration (N /3) 2 (1/6) ( 2/3) (1/3) = Ngeneration (N /3) (1/2), B · c · · − − − · c · 2 2 j : Ngeneration 2 ( 1/2) + 1 = Ngeneration (1/2), (4.6) L · · − · such that d ? j = 0 and d ? j = 0, but d ? (j j ) = 0 if N = 3 as the color number it is. B 6 L 6 B − L c

4.1.2 (B L)-SU(2)2: 4d local anomaly −

Plug in data from Table1 to check the 4d local anomaly of (B L)-SU(2)2: − Global sym B L (Backgrd. ﬁeld)−

SU(2) gauge SU(2) gauge (4.7)

we ﬁnd the anomaly factor contributes to

σa σb j : Ngeneration (N /3) Tr[ ] = Ngeneration (N /3) (δ /2), B · c · 2 2 · c · ab σa σb j : Ngeneration Tr[ ] = Ngeneration (δ /2), (4.8) L · 2 2 · ab such that d ? j = 0 and d ? j = 0, but d ? (j j ) = 0 if N = 3 as the color number it is. B 6 L 6 B − L c

4.1.3 (B L)-SU(3)2: 4d local anomaly − c

Plug in data from Table1 to check the 4d local anomaly of (B L)-SU(3)2: − c Global sym B L (Backgrd. ﬁeld)−

SU(3)c gauge SU(3)c gauge (4.9)

20The minus sign can be interpreted either from the anti-quark of the right-handed R-chiral fermion (anti-chiral fermion).

27 we ﬁnd the anomaly factor contributes to

τ a τ b j : Ngeneration (2 1 1)Tr[ ] = Ngeneration 0 (δ /2) = 0, B · − − 2 2 · · ab j : Ngeneration 0 = 0, (4.10) L · such that d ? j = d ? j = d ? (j j ) = 0. B L B − L

4.1.4 (B L)-(gravity)2: 4d local anomaly −

Plug in data from Table1 to check the 4d local anomaly of (B L)-(gravity)2: − Global sym B L (Backgrd. ﬁeld)−

gravity gravity (4.11) we ﬁnd the anomaly factor contributes to j : Ngeneration (N /3) 2 1 1 = 0. (4.12) B · c · − − j : Ngeneration 2 1 N = Ngeneration (1 N ). (4.13) L · − − νR · − νR

It turns out that d ? j = 0 but d ? j = 0 unless N = 1. Same for d ? (j j ) = 0 only if N = 1. B L 6 νR B − L νR Perturbative anomaly seems to suggest one right-handed neutrino N = 1 to saturate the (B L) νR − current non-conservation. Are there other ways to saturate this ABJ type anomaly? When the (B L) − becomes a discrete symmetry, we will be able to resolve the anomaly matching issue with other novel possibilities in Sec.5.

28 4.1.5 (B L)3: 4d local anomaly −

Plug in data from Table1 to check the 4d local anomaly of (B L)3: − Global sym B L (Backgrd. ﬁeld)−

B L B L − − (4.14) we ﬁnd the anomaly factor contributes to 3 j : Ngeneration N (1/3) 2 1 1 = 0. (4.15) B · c · · − − 3 j : Ngeneration (1) 2 1 N = Ngeneration (1 N ) (4.16) L · · − − νR · − νR It turns out that d ? j = 0 but d ? j = 0 unless N = 1. Same for d ? (j j ) = 0 only if B L 6 νR B − L NνR = 1. Perturbative anomaly seems to suggest one right-handed neutrino NνR = 1 to saturate the (B L) current non-conservation. Are there other ways to saturate this ABJ type anomaly? When the − (B L) becomes a discrete symmetry, we will resolve the anomaly matching with other novel possibilities − in Sec.5.

4.1.6 X-SU(5)2: 4d local anomaly

Recall in Sec. 2.1, the U(1)B L is not a proper symmetry of SU(5) GUT. The “baryon minus lepton − number symmetry” of SU(5) GUT is U(1)X . Plug in data from Table1 to check 4d local anomaly of X-SU(5)2:

Global sym X (Backgrd. ﬁeld)

SU(5) gauge SU(5) gauge (4.17) we ﬁnd the anomaly factor contributed from the representation R of fermions in SU(5) as the anti- fundamental R = 5¯ and anti-symmetric R = 10, from the 15 Weyl fermions 5¯ 10 in one generation. ⊕ Let us check the X current conservation or violation by ABJ type anomaly: X X d ? (j ) X Tr [F F ] X c (SU(5)). (4.18) X ∝ R · R SU(5) ∧ SU(5) ∝ R · 2 R R

29 Here c2(SU(5)) is the second Chern class of SU(5), which is also related to the 4d instanton number of SU(5) gauge bundle. For 5¯ 10 with Ngeneration, from Table1, we get the U(1) charges for ⊕ X

X¯ = 3,X = 1, 5 − 10 so21 d ? (j ) Ngeneration X¯Tr¯[F F ] + X Tr [F F ] = Ngeneration 0 = 0 (4.21) X ∝ 5 5 ∧ 10 10 ∧ · vanishes. We conﬁrm that the U(1)X symmetry is ABJ anomaly free at least perturbatively in SU(5) GUT.

4.2 SM and GUT with extra discrete symmetries and a cobordism theory

In the subsection, we aim to digest better how robust are the anomalies from Sec. 4.1.4 and Sec. 4.1.5 that seems only to be matched with a right-handed neutrino (the 16th Weyl spinor) per generation. These anomalies are not dynamical gauge anomalies if (B L) and X are only global symmetries — namely − the theory is only suﬀered from ’t Hooft anomaly which only results in nonlocal or non-onsite (B L) − symmetry. However, the (B L) and X have to be gauged in the SO(10) GUT. If fact, the discrete − Z U(1) acts as the Z center of Spin(10), 4,X ⊂ X 4 Z = Z(Spin(10)) Spin(10) (4.22) 4,X ⊂

Spin Z Z4 × 2 needs to be dynamically gauged in the SO(10) GUT. This fact motivates Ref. [31] to use the Ω5 = Z16 to argue the 16 chiral fermions for the 4d GUT in one generation. This fact also motivates Ref. [11,37] to check the following cobordism groups TP5(G) with G Spin F Z4,X summarized in Table4. ⊃ ×Z2 21 To evaluate the c2 or the instanton number in diﬀerent representations, R1 and R2, we use the fact that

TrR1 [F ∧ F ]/TrR2 [F ∧ F ] = (C2(R1)d(R1))/(C2(R2)d(R2)), (4.19)

here C2(R) and d(R) are respectively the quadratic Casimir and the dimension of an irreducible representation R. For the representation R of SU(N), we have

R d C2 Fundamental NN 2 − 1 . (4.20) Antisymmetric N(N − 1)/2 2(N + 1)(N − 2)

For SU(5) with N = 5, we get Tr10[F ∧ F ] = (N − 2)Tr5¯[F ∧ F ] = 3Tr5¯[F ∧ F ].

30 Cobordism group TP (G) with GSM (SU(3) SU(2) U(1))/Z and q = 1, 2, 3, 6 d q ≡ × × q dd classes cobordism invariants

G = Spin Z GSM ×Z2 4 × 1 U(1) 2 U(1) SU(2) µ(PD(c1(U(1)))), CS1 c1(U(1)) , CS1 c2(SU(3)) c1(U(1))CS3 , 5 2 2CSSU(3)∼ CSSU(3) Z Z Z Z U(1) SU(3) ( Z2 ) 3 + 5 5d 2 4 16 CS c (SU(3)) c (U(1))CS , A , × × × 1 2 ∼ 1 3 2 ( )c (SU(3)), c (SU(2))η , c (U(1))2η , η(PD( )) AZ2 2 2 0 1 0 AZ2

G = Spin Z GSM ×Z2 4 × 2 2 U(2) U(2) 2 U(2) U(2) U(2) ( Z ) CS +CS c2(U(2)) ( Z ) CS +c1(U(2))CS 2 A 2 3 1 A 2 3 3 µ(PD(c1(U(2)))), CS1 c1(U(2)) , 2 2 , SU(3)∼ SU(3) 5 2 ( )2CS +CS 5d Z Z2 Z4 Z16 U(2) SU(3) Z2 3 5 × × × CS c2(SU(3)) c1(U(2))CS , A , 1 ∼ 3 2 ( )c (SU(3)), ( )c (U(2)), c (U(2))2η , η(PD( )) AZ2 2 AZ2 2 1 0 AZ2

G = Spin Z GSM ×Z2 4 × 3 2 U(3) U(3) SU(2) µ(PD(c1(U(3)))), c1(U(3)) CS1 , CS1 c2(SU(2)) c1(U(3))CS3 , 5 2 2CSU(3) CSU(3) CSU(3) 2CSU(3) CSU(3)∼ CSU(3) Z Z Z Z ( Z2 ) 3 + 1 c2(U(3))+ 5 ( Z2 ) 3 +c1(U(3)) 3 + 5 U(3) 5d 2 4 16 A A , CS , × × × 2 ∼ 2 5 ( )c (U(3)), c (SU(2))η , c (U(3))2η , η(PD( )) AZ2 2 2 0 1 0 AZ2

G = Spin Z GSM ×Z2 4 × 6 2 U(2) U(3) 2 U(2) U(2) U(3) ( Z ) CS +CS c2(U(2)) ( Z ) CS +c1(U(3))CS 2 A 2 3 1 A 2 3 3 µ(PD(c1(U(3)))), c1(U(3)) CS1 , 2 2 , 5 2 2 U(3) U(3) U(3) 2 U(3) U(3)∼ U(3) 5d Z Z Z Z ( Z ) CS3 +CS1 c2(U(3))+CS5 ( Z ) CS3 +c1(U(3))CS3 +CS5 U(3) × 2 × 4 × 16 A 2 A 2 , CS , 2 ∼ 2 5 ( )c (U(3)), ( )c (U(2)), c (U(3))2η , η(PD( )) AZ2 2 AZ2 2 1 0 AZ2 G = Spin Z SU(5) ×Z2 4 × 2 SU(3) SU(3) ( Z ) CS +CS 5d Z Z Z A 2 3 5 , ( )c (SU(5)), η(PD( )) × 2 × 16 2 AZ2 2 AZ2 1 Table 4: Our setup follows Table3 and Ref. [11, 37]. The Z2 H (M, Z2) is the generator from 1 F A ∈ H (B(Z4/Z2 ), Z2) of Spin F Z4. The η0 is a Z4 valued 1d eta invariant which is the extension of a ×Z2 quotient by the normal 1d η˜. So ( ) ( ) mod 2 is the quotient, while Z U(1) . The AZ2 AZ2 ≡ AZ4 4,X ⊂ X η(PD( )) is the value of η Z on the Poincaré dual (PD) submanifold of . AZ2 ∈ 16 AZ2

We aim to initiate a new approach on matching the nonperturbative global Z16 anomaly (descended from the perturbative local Z anomalies of Sec. 4.1.4 and Sec. 4.1.5) for the missing neutrinos.

The Z5 classes perturbative local anomalies are the same results parallel to Sec. 3.1 and Table3. So in the following subsections, we only focus on checking global anomaly cancellations of Zn classes. Generically Feynman diagrams cannot characterize global anomalies (so we do not present Feynman diagrams below). But we can characterize global anomalies by generic curved manifolds as in Fig.2 with gauge, gravity, or mixed gauge-gravity background ﬁelds.

31 4.2.1 Witten anomaly c2(SU(2))˜η vs c2(SU(2))η0: 4d Z2 vs Z4 global anomalies

Follow Sec. 3.7, the old SU(2) anomaly of Witten in 4d is a mod 2 class. It is a Z2 global anomaly given by 5d c2(SU(2))˜η and 6d c2(SU(2))Arf in [11] and Table4. The c2(SU(2))˜η as Witten SU(2) anomaly can be contributed by the 4d fermion doublet 2 under the SU(2) representation (or the (iso-)spin-2r + 1/2 of SU(2) Weyl spinor for some non-negative integer r Z 0). So to check the anomaly cancellation, we ∈ ≥ count the fermion doublet 2 under SU(2). There are four of 2 of SU(2) fermions from (3, 2, Y˜ = 1)L and (1, 1, Y˜ = 6)L. The anomaly from c2(SU(2))˜η counts the number of 4d Weyl spinors of SU(2) fundamental 2 mod 2. From (2.13) for Ngeneration, we have:

Ngeneration (3 + 1) = 0 mod 2. (4.23) ·

There is also an extended Z4 global anomaly from the 5d cobordism invariant c2(SU(2))η0 in [11] and Table4 counting the number of 4d Weyl spinors of SU(2) fundamental 2 mod 4.22 From (2.13) for Ngeneration, we have:

Ngeneration (3 + 1) = 0 mod 4. (4.24) ·

Therefore, we have checked no global SU(2) anomaly of Z2 or Z4 classes for SM with a Spin or Spin F Z4 ×Z2 structure.

4.2.2 ( )c (SU(2)): 4d Z global anomaly AZ2 2 2

The 4d Z global anomaly from the 5d cobordism invariant ( )c (SU(2)), in [11] and Table4, counts 2 AZ2 2 the number of 4d left-handed Weyl spinors of SU(2) fundamental 2 mod 2. Here ( Z2 ) ( Z4 ) mod 2, 1 1 F A ≡ A where Z2 H (M, Z2) is the generator from H (B(Z4/Z2 ), Z2) of Spin F Z4. So ( Z2 ) ( Z4 ) A ∈ ×Z2 A ≡ A mod 2 is the quotient, while Z U(1) . The ( )c (SU(2)) counts the number of 4d Weyl spinors 4,X ⊂ X AZ2 2 of SU(2) fundamental 2 mod 2. From (2.13) for Ngeneration, we have:

Ngeneration (3 + 1) = 0 mod 2. (4.25) · Thus the anomaly vanishes. We have no obstruction to gauge the Z by making dynamical at least 4 AZ4 from this anomaly cancellation.

4.2.3 ( )c (SU(3)): 4d Z global anomaly AZ2 2 2

The 4d Z global anomaly from the 5d cobordism invariant ( )c (SU(3)), in [11] and Table4, counts 2 AZ2 2 the number of 4d left-handed Weyl spinors of SU(3) fundamental 3 mod 2. From (2.13), we count (3, 2, Y˜ = 1) , (3, 1, Y˜ = 4) , and (3, 1, Y˜ = 2) with 3 generations. For Ngeneration, we have: L − L L

Ngeneration (2 1 1) = 0 mod 4. (4.26) · − − Thus there is no anomaly. We have no obstruction to gauge the Z by making dynamical at least 4 AZ4 from this anomaly cancellation.

22As explained in the Notation in the end of Sec.1: The η0 is a mod 4 index of 1d Dirac operator as a cobordism invariant Spin×Z4 of Ω1 = Z4.

32 2 4.2.4 c1(U(1)) η0: 4d Z4 global anomaly

2 The 4d Z4 global anomaly from a 5d cobordism invariant c1(U(1)) η0 (in Ref. [11] and Table4) counts 2 ˜ the sum of 4d Weyl spinors’ (U(1) charge) mod 4. Let us apply Y for U(1)Y˜ charge from (2.13) with Ngeneration. For each generation, we not only have the sum of U(1) charge:

3 2 1 + 3 1 ( 4) + 3 1 2 + 1 2 ( 3) + 1 1 6 = 0 mod 4. (4.27) · · · · − · · · · − · · but also have the sum of (U(1) charge)2:

3 2 12 + 3 1 ( 4)2 + 3 1 22 + 1 2 ( 3)2 + 1 2 (6)2 = 3 4 13 = 0 mod 4. (4.28) · · · · − · · · · − · · · · 2 The c1(U(1)) also counts the U(1) instanton number up to a proportional factor. Thus there is no anomaly. We have no obstruction to gauge the Z by making the dynamical at least from this 4 AZ4 anomaly cancellation.

4.2.5 ( )c (SU(5)): 4d Z global anomaly AZ2 2 2

Similar to (4.18), we consider the discrete 4d Z2 global anomaly from the 5d cobordism invariant ( )c (SU(5)) in [11, 37] and Table4. Here c (SU(5)) is the second Chern class of SU(5), which is AZ2 2 2 also related to the 4d instanton number of SU(5) gauge bundle. For 5¯ 10 with Ngeneration, from Table ⊕ 1, we get the Z4,X charges for

X¯ = 3 = 1 mod 4,X = 1 mod 4. 5 − 10 By footnote 21, we compute the anomaly factor Ngeneration (X5¯ mod 4)Tr5¯[F F ] + (X10 mod 4)Tr10[F F ] ∧ ∧ = Ngeneration 1 1 + 1 3 = Ngeneration 4 = 0 mod 4. (4.29) · · · · This certainly vanishes for the mod 2 anomaly for SU(5) GUT.

There is also another Z class local anomalies for SU(5) GUT captured by 5d cobordism invariants 2 SU(3) SU(3) ( Z ) CS +CS A 2 3 5 2 , we can easily check that SU(5) GUT is free from any local anomaly given by another 5d cobordism invariant [99].

4.2.6 η(PD( )): 4d Z global anomaly AZ2 16

The 4d Z global anomaly given by a 5d cobordism invariant η(PD( )), in [11] and Table4, counts 16 AZ2 the number mod 16 of 4d left-handed Weyl spinors (Ψ 2 of Spin(3, 1) or Ψ 2 of Spin(4) = L ∼ L L ∼ L SU(2) SU(2) ). From (2.13) with Ngeneration (e.g., 3 generations), for each generation, we have: L × R 3 2 + 3 1 + 3 1 + 1 2 + 1 1 = 15 = 1 mod 16. (4.30) · · · · · − For 1 generation, we need to saturates the anomaly:

ν = 1 mod 16. (4.31) −

33 For 3 generations, we need ! 3 3 2 + 3 1 + 3 1 + 1 2 + 1 1 = 45 = 3 mod 16. (4.32) · · · · · −

Therefore we need to saturates the anomaly:

ν = 3 mod 16. (4.33) −

For Ngeneration generations, we need to saturates the anomaly:

ν = Ngeneration mod 16. (4.34) − This anomaly can be canceled by adding new degrees of freedom

ν = Ngeneration (N = 1) mod 16. (4.35) · νR The anomaly matching in this Sec. 4.2.6 seems to be matched with a right-handed neutrino (the 16th Weyl spinor) per generation, similar to Sec. 4.1.4 and Sec. 4.1.5. This also shows the robustness of Sec. 4.1.4 and Sec. 4.1.5 even if we break down U(1)B L or U(1)X down to Z4,B L or to Z4,X . Again − − this Z4 as the center Z(Spin(10)) of Spin(10) is important for the SO(10) GUT.

Are there other ways to match the anomaly other than introducing the right-handed neutrino (the 16th Weyl spinor) per generation? Let us explore the alternatives in the next subsection.

4.3 How to match the anomaly? Preserving or breaking the Z4,X symmetry?

Let us summarize what we learn from the anomaly computation and matching in the previous sections. We have shown that all anomalies presented in Table3 and Table4 can be cancelled, except the additional anomaly constraint from Sec. 4.1.4, Sec. 4.1.5, and Sec. 4.2.6 may not be matched unless we obey

Ngeneration (1 N + hidden sector ) = 0, from local anomalies of Sec. 4.1.4 and 4.1.5. · − νR (4.36) Ngeneration (1 N + hidden sector ) = 0 mod 16, from a global anomaly of Sec. 4.2.6. · − νR

So the above näively suggests we need the right-handed neutrino (the 16th Weyl spinor) NνR = 1 per generation. However, by demanding only a discrete Z4,X instead of a continuous U(1)X symmetry, the local Z class anomaly becomes a global Z16 class anomaly. If so, we do have diﬀerent ways to match the anomaly, other than introducing the right-handed neutrino (the 16th Weyl spinor) NνR = 1. Can we match the anomaly by additional new hidden sectors not yet discovered in SM or in SU(5) Georgi- Glashow GUT? Let us focus on the Z4,X symmetry for the sake of thinking Z4,X = Z(Spin(10)) is gauged in the SO(10) GUT eventually. Let us enumerate the possibilities to match the anomaly (4.36):

1. Anomaly matched by a right-handed neutrino (the 16th Weyl spinor) NνR = 1: For the right-handed neutrino νR to be massless (or gapless) while preserving the Z4,X symmetry, we need to have ν to be a complex Weyl spinor (with a Z charge 1 mod 4), instead of a real R 4,X − Majorana spinor, in order to have the Z4,X symmetry transformation manifest:

ν exp( 2πi/4) ν = ( i) ν . (4.37) R → − R − R

34 Since this is a sterile neutrino with a trivial representation of SM gauge group (2.9), (1, 1, 0)R, we can rotate it to left-handed Weyl spinor ν¯R = νL with (1, 1, 0)L and ﬂips the Z4,X representation to its complex conjugation:

ν exp(2πi/4) ν = (i) ν . (4.38) L → L L

What can the low energy dynamics of the νR be?

(1). The Z4,X preserving massless neutrino: If the sterile neutrino νR remains massless, we can match the anomaly (4.36) by a symmetric gapless low energy theory with an action on a 4d spacetime M 4:

µ µ ν¯R(iσ ∂µ)νR, or equivalently ν¯L(iσ ¯ ∂µ)νL, (4.39) ˆM 4 ˆM 4 with σ¯ (1, ~σ). Note that ν has a Z charge 1 and ν¯ has a Z charge 1, so the action ≡ L 4,X L 4,X − preserves the Z4,X symmetry.

(2). The Z4,X preserving Dirac mass term, but the Z4,X is spontaneously broken by the electroweak Higgs: The Yukawa-Higgs-Dirac term preserves the Z4,X symmetry:

ν¯RφH νL +ν ¯LφH νR, (4.40) ˆM 4 because ν (or ν¯ ) has a Z charge 1, ν (or ν¯ ) has a Z charge 1, and φ has a Z L R 4,X R L 4,X − H 4,X charge 2. However, when the electroweak symmetry breaking sets in (at a lower energy around 246 GeV), the Higgs condenses, φ = 0, and spontaneously breaks the Z symmetry. h H i 6 4,X (3). The Z4,X explicit breaking Majorana mass term: If the sterile neutrino νR becomes gapped by Majorana mass mMaj, the spacetime spinor is in a real Majorana representation. Then, the Z4,X symmetry in (4.37) and (4.38) is explicitly broken. We can still match the anomaly (4.36) by a Z4,X -symmetry-breaking gapped theory with an action on a 4d spacetime M 4:

T µ imMaj T 2 2 χ (iσ ¯ µ)χ + (χ σ χ + χ†σ χ∗), (4.41) ˆM 4=∂M 5 ∇ 2 χ where we have written the 4-component Majorana spinor as ΨMaj = 2 with the trans- iσ χ∗ pose T, complex conjugate , and complex conjugate transpose . ∗ † See more discussions on Dirac or Majorana masses for three generations of SM fermions in Sec. 5.3.1.

2. Anomaly matched by new additional or hidden sectors beyond SM: Let us hypothesize many scenarios with diﬀerent low energy dynamics following Ref. [11]’s Sec. 8.2:

(1). Z4,X -symmetry-preserving anomalous gapless, free or interacting, 4d CFT.

(2). Z4,X -symmetry-breaking gapless, free or interacting, 4d CFT.

(3). Z4,X -symmetry-preserving anomalous long-range entangled gapped 4d TQFT.

(4). Z4,X -symmetry-breaking gapped 4d TQFT.

(5). Z4,X -Symmetry-Protected Topological state (SPTs) in 5d as a short-range entangled gapped phase captured by a 5d cobordism invariant 2πi exp ( Ngeneration) η PD( ) . (4.42) 16 · − · AZ2

35 (6). Z4,X -gauged-(Symmetry)-Enriched Topological state (SETs) in 5d coupled to gravity.

All the above theories are unitary by their own. We propose that all the above scenarios and their proper linear combinations, conventional or exotic, if existing, can saturate the anomaly (4.36). Based on the contemporary knowledge of SM physics and experimental hints, Scenario (1) and (2) seem less practical, because it is less likely to have any new gapless or interacting • CFT that we do not observe below the TeV energy scale.23 Also if there is spontaneous symmetry- breaking (SSB), for U(1)X SSB, we expect to observe new Goldstone boson modes; for Z4,X SSB, we may observe different vacua or domain walls between different vacua. This shall be falsifiable in the experiments. Scenario (3) is exotic but very interesting, which we discover new insights into the neutrino physics • and Dark Matter. Since the Z4,X -symmetry is a discrete chiral symmetry, and the gapped 4d TQFT can be regarded as a gapped phase. The anomalous Z4,X -symmetry preserving gapped 4d TQFT can be regarded as a gapped phase without Z4,X -chiral symmetry breaking. However, the 4d TQFT has additional generalized global symmetries [75], known as the higher-symmetries, whose charged objects are extended objects (1d lines, or 2d surfaces, etc.).

- The higher-symmetries of the 4d TQFT can be spontaneously broken. In this sense, the spontaneously symmetry breaking (SSB) of higher-symmetries imply that the 4d TQFT is deconﬁned. Below the topological order energy gap, we have the low energy 4d TQFT. Above the topological order energy gap, the 4d TQFT hosts deconﬁned excitations of fractionalized particles (from breaking 1d worldline with open 0d ends) or fractionalized strings (from breaking 2d worldsheet with open 1d boundaries).

- If the higher-symmetries of the 4d TQFT have no SSB, then this leads to a gapped conﬁned phase without Z4,X -chiral symmetry breaking. However, eventually in a quantum gravity theory, all the global symmetries must be either gauged or broken. Thus we should not have the option of SSB or symmetry- preserving. Preferably, in a quantum gravity theory, we can have either of the following options: (1) the higher-symmetries of TQFT are dynamically gauged due to a higher energy theory (above GUT), (2) the higher-symmetries of TQFT are explicitly broken.

Moreover, Scenario (3) can give rise to Scenario (4), if we construct such an anomalous Z4,X -symmetry- preserving TQFT ﬁrst, we can break some of the symmetry to obtain the symmetry-breaking gapped TQFT. We can also construct a mixed symmetry-extension and symmetry-breaking TQFT following Ref. [53]. So we should focus on Scenario (3) explored in Sec. 5.2. Scenario (5) implies that our 4d SM lives on the boundary of 5d Z -SPTs given by 5d η(PD( )). • 4,X AZ2 If Scenario (5) describes our universe, we discover at least an extra dimension from the 5d theory. We explore this 5d SPT or invertible TQFT theory in Sec. 5.1. Scenario (6) Above certain higher energy scale, the Z may be dynamically gauged such as in Z = • 4,X 4,X Z(Spin(10)) of the SO(10) GUT, then the 4d and 5d bulk are fully gauged and entangled together. The 5d bulk is in fact a 5d Symmetry-Enriched Topologically ordered state (SETs) that can be coupled to dynamical gravity. We explore this 5d SETs coupled to dynamical gravity theory in Sec. 6.1.

23However, if the hidden new gapless sectors are fully decoupled from SM forces, they may be weakly interacting massless or gapless matter account for the Dark Matter sector.

36 5 Beyond Known “Fundamental” Forces: Hidden Topological Force

Follow Sec. 4.3, now we propose new Scenarios, beyond SM and beyond Georgi-Glashow (GG) SU(5) GUT, to match the Z16 global anomalies from Sec. 4.2.6. We focus on the Scenario (3) (thus also Scenario (4)) for a hidden 4d non-invertible TQFT in Sec. 5.2, and for a 5d SPTs or 5d SETs coupled to gravity in Scenario (5) and (6) in Sec. 6.1.

5.1 Hidden 5d invertible TQFT or 5d Symmetry-Protected Topological state (SPTs)

We can saturate the missing anomaly of ν = Ngeneration mod 16 in (4.34) by a 5d invertible TQFT (or − a 5d SPTs protected by G = Spin Z Ginternal-symmetry in a condensed matter language) with a ×Z2 4 × 5d partition function: 2πi Z5d-iTQFT = exp ( Ngeneration) η PD( Z2 ) . (5.1) 16 · − · A M 5 Here we define the eta invariant η Z , and η PD( ) Z ∈ 16 AZ2 ∈ 16 slightly different by a proportional factor in the math literature. Let us overview quickly what we need about the APS eta invariant η [13–15,22,100]. Let us consider the 4d η invariant and then relate to our 24 4d η invariant and 5d η PD( ) invariant. AZ2 On an Euclidean signature curved spacetime manifold M d, the path integral of a massive 5d Dirac fermion spinor ψ with a mass m coupled to a background gauge field A or a background gravity of a metric gµν, the Euclidean path integral is ¯ ¯ 4 p ¯ [ ψ][ ψ] exp ( SE) = [ ψ][ ψ] exp d xE det g ψ(D/ A + m)ψ = det(D/ A + m), (5.2) ˆ D D − ˆ D D − ˆM d

µ µ0 µ where locally D/ e 0 γ (∂ + iω iA ), e 0 is a vielbein, with a spin connection ω , and a gauge A ≡ µ µ µ − µ µ µ ˜ λν λ ν connection Aµ for a gauge bundle of a group G. More explicitly, the components are ωµ = iωµ [γ , γ ]/8 a a a ˜ and Aµ = AµT with T generators of the Lie algebra Lie(G). We also specify the transition functions that relate fields ψ and ψ¯ (two independent Grassmann fields) and A on different patches in order to d globally define the Dirac operator D/ A on M . The transition function also leaves the local expression of the partition function invariant. If we define m to be a trivial gapped vacua without any → ∞ topological feature, then the m gapped vacua can host a nontrivial iTQFT or SPTs in dd. Follow → −∞ the notations in [23], the dd partition function of nontrivial iTQFT or SPTs can be defined via a ratio25

det(D/ A m ) Y iλ m 1 X s Zdd-iTQFT[A] = lim − | | lim − | | = (N0 + lim (sign λ) λ − ) (5.3) m det(D/ + m ) ≡ m iλ + m 2 s 0+ · | | | |→∞ A | |→∞ λ → λ=0 | | | | 6 24The 4d η invariant has condensed matter realizations. It is known as the class DIII topological superconductor [TSC] in the free fermion limit with a Z classiﬁcation [81–83]. The ν = 1 of this TSC is the Balian-Werthamer (BW) state of the B phase of 3He liquid (3He B) [101]. A 2+1d single Majorana fermion can live on the surface protected by time reversal T symmetry in the non-interacting free system. Including the time-reversal Z4 symmetry preserving interactions (preserving + the Pin structure), the 4d η invariant is reduced from a free Z class to a torsion Z16 class. 25 Q Q λ The det(−iD/ A) = λ. One can regularize it by det(−iD/ A) = via a Pauli-Villars regulator of mass M. λ P λ λ+ i M The APS η-invariant [13–15] is a regularization of λ(sign λ).

37 as a regularized sum of eigenvalues of Dirac operator. Here D/ is anti-Hermitian, so iD/ is Hermitian. A − A The λ are eigenvalues of iD/ so λ R are real. − A ∈ N are the number of the operator iD/ ’s zero modes. Depending on the underlying G-structure of 0 − A manifolds and iD/ (such as G = Pin+ [22], G = Pinc [100, 102], or G = Pin+ SU(2) [23]), the 4d − A ×Z2 iTQFT/SPT partition functions are respectively:

 2π i 1 ν η + η + Z ν Z ν exp( 2 Pin ), with Pin 8 , 16. m  · · ∈ 1 ∈ | |→∞ c c Z4d-iTQFT exp(2πi ν ηPin ), with ηPin 8 Z, ν Z8. (5.4) −−−−−→ · · ∈ 1 ∈  ν η + η + Z ν Z exp(2πi Pin Z SU(2)), with Pin Z SU(2) 4 , 4. · · × 2 × 2 ∈ ∈

However, in our work, we adjust above deﬁnitions to make η + Z via η + = 8η + mod 16, so: Pin ∈ 16 Pin Pin

ν m 2πi | |→∞ Z4d-iTQFT exp( ν ηPin+ ), with ηPin+ Z16, ν Z16. (5.5) −−−−−→ 16 · · M 4 ∈ ∈

Similarly, based on the Smith homomorphism [31,56,59, 103] 26

Z Spin Z Z4 ∩A 2 + Ω × 2 = Z (generated by RP5) ΩPin = Z (generated by RP4), (5.8) 5 16 −−−−→ 4 16 we characterize the 5d iTQFT whose manifold generator for bordism group is RP5 at ν = 1 below. The cobordism invariant for any ν corresponds to a 5d iTQFT partition function

ν m 2πi | |→∞ Z5d-iTQFT exp( ν η(PD( Z2 )) ), with η = ηPin+ Z16, ν Z16. (5.9) −−−−−→ 16 · · A M 5 ∈ ∈

1 Given a G Spin F Z4 structure, the cohomology class Z H (M, Z2) is the generator from ⊇ ×Z2 A 2 ∈ H1(B(Z /ZF ), Z ). So ( ) ( ) mod 2 is the quotient for the Z gauge ﬁeld , which is the 4 2 2 AZ2 ≡ AZ4 4 AZ4 background ﬁeld for the symmetry Z U(1) . The η(PD( )) is the value of η Z on the 4,X ⊂ X AZ2 ∈ 16 Poincaré dual (PD) submanifold of the cohomology class . This PD is the precise meaning of (5.8) AZ2 taking from 5d to 4d. ∩AZ2

Thus, to summarize, the full 5d iTQFT and the 4d SM or SU(5) GG GUT model action SSM or GUT with Ngeneration together can make the anomaly (4.34) matched via the full 5d-4d coupled partition

26In fact, there are more Smith homomorphisms in any dimension. Related discussions along these maps are abundant, see [103] and also [20, 37, 56]:

+ Pin Spin×Z2 7 Ω8 = Z32 × Z2 → Ω7 = Z16 generated by RP , − Spin×Z2 Pin 6 Ω7 = Z16 → Ω6 = Z16 generated by RP , − Spin× Z Pin Z2 4 5 Ω6 = Z16 → Ω5 = Z16 generated by RP , Spin× Z + Z2 4 Pin 4 Ω5 = Z16 → Ω4 = Z16 generated by RP , + (5.6) Pin Spin×Z2 3 Ω4 = Z16 → Ω3 = Z8 generated by RP , − Spin×Z2 Pin 2 Ω3 = Z8 → Ω2 = Z8 generated by RP , − Spin× Z Pin Z2 4 1 1 1 Ω2 = Z8 → Ω1 = Z4 generated by RP = S /Z2 = S , Spin× Z + Z2 4 Pin 0 Ω1 = Z4 → Ω0 = Z2 generated by RP = point.

1 n 1 n 4 5 Notice that H (RP , Z2) = Z2, for example by using n = 5, if we choose AZ2 for H (RP , Z2) = Z2, then RP and RP detect all ν ∈ Z16: 2π i 2π i exp( ν η| ) = exp( ν). 16 M4=RP4 16 2π i 2π i exp( ν η PD(AZ ) ) = exp( ν). (5.7) 16 2 M5=RP5 16

38 function

2πi ¯ Z5d-4d = exp( ν η(PD( )) 5 ) [ ψ][ ψ][ A] exp (iSSM or GUT 4 ). (5.10) 16 Z2 M ˆ M · · A | ν= Ngeneration · D D D ··· | − 5 4 with a bulk-boundary correspondence ∂M = M . The full Z5d-4d is gauge invariant, in particular also invariant under the background Z4,X transformation (at least at the higher-energy of GUT scale). This concludes the 5d SPTs coupled to 4d SM or GUT in the Scenario (5).

5.2 Hidden 4d non-invertible TQFT or 4d symmetry-enriched topological order

Now we explore the Scenario (3) (thus also Scenario (4)) for a hidden 4d non-invertible TQFT in Sec. 5.2 to match the missing anomaly. In fact to construct such an anomalous symmetry-preserving 4d TQFT (here we preserve Z in Spin Z structure), we take the inspirations from the quantum condensed 4,X ×Z2 4 matter phenomenon in one lower dimension, known as the anomalous symmetry-enriched boundary topological order in 3d (2+1d spacetime) living on the boundary of 4d SPTs (3+1d spacetime). In condensed matter thinking, it can be understood as saturating the ’t Hooft anomaly by a symmetric gapped sector, without breaking any symmetry breaking and without gapless modes, via smearing the ’t Hooft anomaly and anomalous symmetry into the long-range entanglement. In the context of 3d boundary and 4d bulk, a novel surface topological order was ﬁrstly pointed out by Vishwanath-Senthil in an insightful work [54]. Later on many people follow up on developing the surface topological order constructions (see overviews in [46,55] and [53]). We need to generalize this condensed matter idea to ﬁnd an anomalous symmetric 4d TQFT living on the boundary of 5d fermionic SPTs (5.9). In particular, we require the symmetry-extension method [53] and the spin-TQFT construction from a previous work [56].

5.2.1 Symmetry extension [Z2] Spin Z4,X Spin F Z4,X and a 4d [Z2] gauge theory → × → ×Z2

First, we can rewrite the 5d iTQFT partition function (5.9) on a 5d manifold M 5 in terms of

ν 2πi Z5d-iTQFT = exp( ν η(PD( Z2 )) ) 16 · · A M 5

2πi p1(TM) 3 4 5 = exp( ν 8 (PD( Z2 )) + 4 Arf(PD(( Z2 ) )) + 2 η˜(PD(( Z2 ) )) + ( Z2 ) ) 16 · · · 48 A · A · A A M 5

2πi σ 3 4 5 = exp( ν 8 (PD( Z2 )) + 4 Arf(PD(( Z2 ) )) + 2 η˜(PD(( Z2 ) )) + ( Z2 ) ), (5.11) 16 · · · 16 A · A · A A M 5

for a generic ν = Ngeneration Z . Here are some explanations on the mathematical inputs of (5.11) − ∈ 16 (also partly explained in the Notation in the end of Sec.1): The p (TM) is the ﬁrst Pontryagin class of spacetime tangent bundle TM of the manifold M. Via the • 1 Hirzebruch signature theorem, we have

1 4 p1(TM) = σ(Σ ) = σ 3 ˆΣ4

4 4 1 on a 4-manifold Σ , where σ is the signature of Σ . So in (5.11), we evaluate the 3 Σ4 p1(TM) = σ on 4 5 the Poincaré dual Σ manifold of the ( Z2 ) cohomology class within the M . ´ A Spin The η˜ is a mod 2 index of 1d Dirac operator as a cobordism invariant of Ω = Z . • 1 2

39 The Arf invariant [62] is a mod 2 cobordism invariant of ΩSpin = Z , whose realization is the 1+1d • 2 2 Kitaev fermionic chain [63]. 5 The ( Z2 ) is a mod 2 class purely bosonic topological invariant, which corresponds to a 5d bosonic • A 5 SPT phase given by the group cohomology class data H (BZ2, U(1)) = Z2, which is also one of the Z2 SO generators in Ω5 (BZ2).

Below we ask whether we can construct a fully gauge-invariant 5d-4d coupled partition function preserving the Spin Z structure: ×Z2 4 ν Z5d-iTQFT Z4d-TQFT . (5.12)

Preserving the Spin Z structure means that under the spacetime background transformation and the ×Z2 4 background gauge transformation, the 5d-4d coupled partition function is still fully gauge invariant. AZ2

1. When ν is odd, such as ν = 1, 3, 5, 7, Z , Ref. [59] suggested that the symmetry-extension method · · · ∈ 16 [53] cannot help to construct a symmetry-gapped TQFT. Furthermore, Cordova-Ohmori [60] proves that a Spin Z Z4 × 2 symmetry-preserving gapped TQFT phase is impossible for this odd ν Z16 anomaly from Ω5 = ∈ Spin Z Z4 Z . The general statement in [60] is that given an anomaly index ν Ω × 2 = Z , we can at most 16 ∈ 5 16 construct a fully symmetric gapped TQFT if and only if

4 2ν. | Namely, 4 has to be a divisor of 2ν. Apparently, the 4 2ν is true only when ν is even, while the 4 2ν | | is false when ν is odd.

Since ν = Ngeneration, the case of ν = 1 (for a single generation) and ν = 3 (for three generations) are − particularly important for the high energy physics phenomenology. This means that we are not able to directly construct any 4d symmetric gapped TQFT that explicitly matches the same Z16 anomaly for one right-handed neutrino (ν = 1) or three right-handed neutrinos (ν = 3).

2. When ν is even, such as ν = 2, 4, 6, 8, Z , Ref. [59] suggested that the symmetry-extension · · · ∈ 16 method [53] can trivialize the ’t Hooft anomaly. Furthermore, Cordova-Ohmori [60] shows that there is no obstruction to construct a symmetry-preserving gapped TQFT phase for any even νeven Z . We ∈ 16 can verify the claim by rewriting (5.11) in terms of the ( νeven ) Z index with a 2d Arf-Brown-Kervaire 2 ∈ 8 (ABK) invariant:

ν even 2πi 2πi νeven 3 Z5d-iTQFT = exp( νeven η(PD( Z2 )) ) = exp( ( ) ABK(PD(( Z2 ) )) ) 16 · · A M 5 8 · 2 · A M 5

2πi νeven 3 4 5 = exp( ( ) 4 Arf(PD(( Z2 ) )) + 2 η˜(PD(( Z2 ) )) + ( Z2 ) ). (5.13) 8 · 2 · · A · A A M 5

Notice that (5.13) can become trivialized if we can trivialize the ( )2 factor. In fact, the topological AZ2 term ( )2 can be trivialized by the symmetry extension [53] AZ2 Z Z Z 4,X 0 2 4,X F 0. → → → Z2 →

2 2 Z4,X Namely, the cocycle topological term ( Z2 ) in H (B( F ), U(1)) becomes a coboundary once we lifting A Z2 Z4,X 2 the F -gauge ﬁeld Z2 to a Z4,X gauge ﬁeld in H (BZ4,X , U(1)). So this suggests that the following Z2 A

40 symmetry extension for the spacetime-internal symmetry27

1 [Z2] Spin Z4,X Spin F Z4,X 1. (5.18) → → × → ×Z2 → can fully trivialize any even νeven Z cobordism invariant given in (5.13). The [Z ] means that we ∈ 16 2 can gauge the normal subgroup [Z ] in the total group Spin Z . This symmetry extension (5.18) also 2 × 4,X means that a 4d [Z2] gauge theory can preserve the Spin F Z4,X symmetry while can also saturate the ×Z2 even νeven Z anomaly. This 4d [Z ] gauge theory is the anomalous symmetric gapped non-invertible ∈ 16 2 TQFT that we aim for in the Scenario (3).

5.2.2 4d anomalous symmetric Z2 gauge theory with ν = 2 Z16 preserving Spin F Z4,X ∈ ×Z2

For example, when νeven = 2 Z , we have (5.12) with the input of 5d bulk iTQFT (5.13), then we can ∈ 16 explicitly construct the partition function on a 5d manifold M 5 with a 4d boundary M 4 ∂M 5 as,28 ≡

νeven=2 Z d-iTQFT Z d- -gauge theory 5 · 4 Z2 X 2π i ABK(c PD( 3)) = e 8 ∪ A c ∂0−1(∂[PD( 3)]) ∈ A 1 X 2 2π i ABK PD0 M4 b(δa+ ) 8 (c ( a)) 4 ( 1) A e ∪ A (5.19) π0(M ) ´ · 2| | 1 4 − · · a C (M ,Z2), ∈ 2 4 b C (M ,Z2) ∈ Here we write the mod 2 cohomology class Z gauge ﬁeld as H1(M 5, Z ). We generalize 2 A ≡ AZ2 ∈ 2 the boundary TQFT construction in the Section 8 of [56]. Here are some remarks about this 5d bulk iTQFT-4d boundary TQFT partition function (5.19):

27 In fact, the recent works [35,104,105] suggest that many cobordism invariants given in (5.6) can be trivialized via a Z2 extension. For example, the following group extension to EPin(d) via

− 1 → Z2 → EPin(d) → Pin (d) → 1 (5.14)

Pin− Pin− can trivialize the cobordism invariants when ν = 4 ∈ Z8 = Ω2 at d = 2, and for any ν ∈ Z16 = Ω6 at d = 6. The EPin(d) is ﬁrstly introduced in [35]. The following extension to Spin(d) × Z4 via

1 → Z2 → Spin(d) × Z4 → Spin(d) × Z2 → 1 (5.15)

Spin×Z2 Spin×Z2 can trivialize the cobordism invariants for any even νeven ∈ Z8 = Ω3 at d = 3 and for any ν ∈ Z16 = Ω7 at d = 7. The following group extension to EPin(d) via

+ 1 → Z2 → EPin(d) → Pin (d) → 1 (5.16)

Pin+ can trivialize the cobordism invariants for any even νeven ∈ Z16 = Ω4 at d = 4. The following extension to Spin(d) × Z4 via

1 → Z2 → Spin(d) × Z4 → Spin(d) ×Z2 Z4 → 1 (5.17)

Spin× Z Z2 4 can trivialize the cobordism invariants for any even νeven ∈ Z16 = Ω5 at d = 5. This is similar to what Ref. [106] suggested that if the anomaly can be matched by a symmetric gapped boundary TQFT, then a Z2 gauge theory almost always work. 28Follow the Notation in the end of Sec.1, we use the ^ notation for the cup product between cohomology classes, or between a cohomology class and a fermionic topological invariant. We use the ∪ notation for the surgery gluing the boundaries of two manifolds within relative homology classes. So the M1 ∪ M2 means gluing the boundary ∂M1 = ∂M2 such that the common orientation of ∂M2 is the reverse of ∂M2.

41 1). The (PD( 3)) is a 2d manifold taking the Poincaré dual (PD) of 3-cocycle 3 in the M 5; but the 2d A A manifold (PD( 3)) may touch the the 4d boundary ∂M 5 = M 4. The 1d boundary ∂(PD( 3)) can be A A regarded as the 1d intersection between the 2d (PD( 3)) and the M 4. A 2). More precisely, for a Spin Z manifold M 5 with a boundary, we have used the Poincaré-Lefschetz ×Z2 4 duality for a manifold with boundaries:

= 2 H3(M 5, Z ) ∼ H (M 5,M 4, Z ) PD( 3). (5.20) A ∈ 2 −→ 2 2 3 A

3). For any pair (S, S0), where S0 is a subspace of S, the short exact sequence of chain complexes

0 C (S0) C (S) C (S, S0) 0 (5.21) → ∗ → ∗ → ∗ → induces a long exact sequence of homology groups

∂ Hn(S0) Hn(S) Hn(S, S0) Hn 1(S0) . (5.22) · · · → → → −→ − → · · ·

Here Cn(S, S0) := Cn(S)/Cn(S0), the Hn(S, S0) is the relative homology group, and ∂ is the boundary map. Take (S, S ) = (PD( 3), ∂PD( 3)), we denote the boundary map by ∂: • 0 A A ∂ H (PD( 3), ∂PD( 3)) H (∂PD( 3)). (5.23) 2 A A −→ 1 A Here PD( 3) is not a closed 2-manifold, but which has a boundary closed 1-manifold ∂PD( 3). A A Take (S, S ) = (M 4 = ∂M 5, ∂PD( 3)), we denote another boundary map by ∂ : • 0 A 1 ∂ H (M 4, ∂PD( 3)) 1 H (∂PD( 3)). (5.24) 2 A −→ 1 A Both M 4 = ∂M 5 and ∂PD( 3) are closed manifolds, of 4d and 1d, respectively. A 4 4). Now, the c is deﬁned as a 2d surface living on the boundary M . The ∂1c uses the boundary map (5.24)’s ∂ of c on the M 4. We can compensate the 2-surface PD( 3) potentially with a 1-boundary, by gluing it 1 A with c to make a closed 2-surface. To do so, we require both PD( 3) and c share the same 1d boundary. A 1 3 5). The c ∂ − (∂[PD( )]) also means ∈ 1 A ∂ c = ∂[PD( 3)] = [∂PD( 3)]. (5.25) 1 A A This is exactly the requirement that the bulk 2-surface PD( 3) (living in M 5) and the boundary 2-surface A c (living on M 4) share the same 1d boundary. The [PD( 3)] means the fundamental class and the relative homology class of the 2d manifold PD( 3). • A A The ∂[PD( 3)] = [∂PD( 3)] means the boundary of the fundamental class (via the boundary ∂ map • A A in (5.23)) is equivalent to the fundamental class of the boundary ∂ of PD( 3). Beware that the two ∂ A operations in ∂[PD( 3)] = [∂PD( 3)] have diﬀerent meanings. A A 6). Note that when M 5 is a closed manifold with no boundary M 4 = ∂M 5 = thus c = , then the term ∅ ∅ ν =2 X 2π i ABK(c PD( 3)) 2π i ABK(PD( 3)) even e 8 ∪ A is equivalently reduced to e 8 A = Z5d-iTQFT . −1 3 c ∂1 (∂[PD( )]) ∈ A

This is a satisfactory consistent check, consistent with the 5d bulk-only iTQFT at νeven = 2 in (5.13).

42 1 4 2 4 7). The a C (M , Z2) means that a is a 1-cochain, and the b C (M , Z2) means that b is a 2-cochain. 2 ∈ 4 b(δa+ ) 2 ∈ The factor ( 1) M A = exp(iπ 4 b(δa + )) gives the weight of the 4d Z gauge theory. The − ´ M A 2 bδa term is the BF theory written in´ the mod 2 class.29 The path integral sums over these distinct cochain classes. The variation of b gives the equation of motion (δa + 2) = 0 mod 2. In the path integral, we can A integrate out b to give the same constraint (δa + 2) = 0 mod 2. This is precisely the trivialization of A the second cohomology class,

2 = ( )2 = δa mod 2 (5.26) A AZ2 so the 2-cocycle becomes a 2-coboundary which splits to a 1-cochain a. This is exactly the condition 2 imposed by the symmetry extension (5.18): 1 [Z2] Spin Z4,X Spin F Z4,X 1, where the Z2 → → ×2 → 2 × → A term in Spin F Z4,X becomes trivialized as a coboundary = δa (so = 0 in terms of a cohomology ×Z2 A A or cocycle class) in Spin Z . × 4,X 1 4 8). The 4 factor mod out the gauge redundancy for the boundary 4d Z2 gauge theory. The π0(M ) 2|π0(M )| is the zeroth homotopy group of M 4, namely the set of all path components of M 4. Thus, we only sum over the gauge inequivalent classes in the path integral summation.

3 5 9). The ABK(c PD( )) is deﬁned on a 2d manifold with Pin− structure. Recall the M has the Spin Z2 Z4 ∪ 5 A × structure. If M is closed, then there is a natural Smith map (5.6) to induce the 2d Pin− structure on the closed surface via PD( 3). However, the M 5 has a boundary M 4, so PD( 3) may not be closed — A 3 A 30 the previously constructed closed 2-surface (c PD( )) is meant to induce a 2d Pin− structure. Then ∪ A we compute the ABK on this closed 2-surface (c PD( 3)). ∪ A 10). Let us explain the other term ABK(c PD0( a)) of the 4d boundary TQFT in (5.19). The PD0 is the ∪ A Poincaré dual on M 4 = ∂M 5. Since the fundamental classes of M 5 and M 4 = ∂M 5 are related by

∂ [M 5] H (M 5,M 4, Z ) H (M 4, Z ) [M 4] = [∂M 5]. (5.27) ∈ 5 2 −→ 4 2 3 Here we have the following relations:31

PD = [M 5] , 4 ∩ 5 5 PD0 = [M ] = [∂M ] = ∂[M ] , (5.28) ∩ ∩ 3 ∩ 3 ∂0PD0( a) = PD0(δ( a)) = PD0( ) = ∂[PD( )] = ∂ c. (5.29) A A A A 1 The cap product here is to define PD homology class, so PD( ) = [M 5] . • ∩ A ∩ A Here we use [M 4] = [∂M 5] = ∂[M 5]: the fundamental class of boundary of M 5 gives the boundary of • fundamental class. The (5.28)’s first equality ∂ PD0( a) = PD0(δ( a)) uses the coboundary operator δ on the cohomology • 0 A A class a. A 29 Z exp(i 2 bda + 1 bA2)) b The continuum QFT version of this 2 gauge theory is M4 2π π2 , where the integration over a closed 2-cycle, b, can be nπ with some integer n ∈ Z. The a and A´ integration over a closed 1-cycle, a and A, can be nπ with some integer‚ n ∈ Z. ¸ ¸ 30We do not yet know whether it is always possible to induce a unique 2d Pin− structure on (c∪PD(A3)) for any possible pair of data (M 5,M 4 = ∂M 5) given any Pin− M 5. However, we claim that it is possible to find some suitable M 4 so that the 2d (c ∪ PD(A3)) has Pin− induced, thus in this sense the ABK(c ∪ PD(A3)) is defined. 31 0 Let us clarify the notations: ∂, ∂ , and ∂1. The boundary notation ∂ may mean as (1) taking the boundary, or (2) in the boundary map of relative homology class in (5.23). It should be also clear to the readers that • the ∂ is associated with the operations on objects living in the bulk M 5 or ending on the boundary M 4, • while the ∂0 is associated with the operations on objects living on the boundary M 4 alone. The ∂1 is defined as another boundary map in (5.24).

43 3 The (5.28)’s second equality PD0(δ( a)) = PD0( ) uses the condition δ = 0 and the trivialization • A A A condition (5.26): δa = 2. A 3 3 The (5.28)’s third equality PD0( ) = ∂[PD( )], we use “the naturality of the cap product”. There are • A A natural pushforward and pullback maps on homology and cohomology, related by the projection formula, also known as “the naturality of the cap product.” The (5.28)’s last equality ∂[PD( 3)] = ∂ c is based on (5.25). Importantly, as a satisfactory consistency • A 1 check, this also shows that the 2-surface c obeys: 3 the c in ABK(c PD( )) is the same c in ABK(c PD0( a)). (5.30) ∪ A ∪ A Similarly as before, the union (c PD0( a)) is a closed 2-surface and we induce a Pin− structure on • ∪ A 4 32 this 2-surface. So we can compute the ABK on this closed 2-surface (c PD0( a)) on the M . ∪ A

In summary, we have constructed the 4d Z2-gauge theory in (5.19) preserving the (Spin F Z4,X )- ×Z2 structure, namely it is a (Spin F Z4,X )-symmetric TQFT but with νeven = 2 Z16 anomaly. This can ×Z2 ∈ be used to compensate the anomaly ν = Ngeneration mod 16 with Ngeneration = 2, two generations of − missing right-handed neutrinos. We could not however directly construct the symmetric gapped TQFT for ν is odd (thus symmetric TQFTs not possible for Ngeneration = 1 or 3), due to the obstruction found in [59, 60].

However, we can still compensate the ν = Ngeneration = 3 mod 16 anomaly by: − −

1. Appending the symmetric gapped 4d TQFT with the anomaly ν = 2 Z , plus an additional one ∈ 16 generation of right-handed neutrino ν with the anomaly ν = 1 Z . This single right-handed R ∈ 16 neutrino νR can pair with any of the three generation of left-handed neutrino νL to give a Dirac mass via the Yukawa-Higgs term. The quantum interference between the three generation of neutrinos (three νL and one νR) in the TQFT background (of ν = 2) may give rise to the neutrino oscillations observed in the phenomenology (see Sec. 6.4). 2. Appending the symmetric gapped 4d TQFT with the anomaly ν = 2 Z , plus an additional 5d ∈ 16 iTQFT in an extra dimension with the anomaly ν = 1 Z . ∈ 16

Either of the above combinations still complete a scenario for the full anomaly matching. By the same logic, we can also propose many more other possible full anomaly matching scenarios, such as the linear combinations of scenarios given in Sec. 4.3.

5.3 Gapping Neutrinos: Dirac mass vs Majorana mass vs Topological mass

It should be clear that the 4d TQFT in Sec. 5.2 with an index νeven mod 16 in (4.34) cannot be directly access from gapping the free fermion theory (νeven mod 16 number of Weyl fermions or Majorana fermions). In order to access the 4d TQFT, we have to:

1. Go to a higher-energy mother effective field theory (EFT) at a deeper UV (ultraviolet), such that the mother EFT can land at IR (infrared) with two different types of low energy physics: 4d free fermions on one side, and the 4d TQFT on another side. 32Similar to Footnote 30, we do not yet know whether it is always possible to induce a unique 2d Pin− structure on (c ∪ PD0(Aa)) on any M 4 given any Pin− M 5. However, we claim that it is possible to find suitable some M 4 so that the 2d Pin− is induced, thus in this sense the ABK(c ∪ PD0(Aa)) is defined.

44 2. Access via the dual fermionic vortex zero mode bound state condensation and go through a 4d topological quantum phase transition [107, 108] or “4d mirror symmetry [109]” (i.e., a duality of QFTs in the version of 4d spacetime). We comment more about the 4d duality in Sec. 5.3.3.

Before then, let us clarify the meanings of Mass or Energy Gap.

The masses for left-handed neutrinos (thus also right-handed antineutrinos) are experimentally estimated to be very small, nearly million times smaller than the electron mass [110] m , m , m < 1eV m 0.51MeV. (5.31) νe νµ ντ e ' More precisely, the flavor states (νe, νµ, ντ ) are superpositions of the different mass eigenstates (ν1, ν2, ν3). So a flavor state should be weighed and averaged over different masses of the different mass eigenstates. The current experiments show that the sum of the masses of the three neutrinos should also be below about 1eV. These mass bounds hold for neutrinos regardless being Dirac fermion or Majorana fermion particles. These mass bounds only apply to left-handed neutrinos (thus also right-handed antineutrinos, or its own anti-particle if a neutrino is a Majorana fermion). But we do not yet know about the mass, or mass bound, or the existence of right-handed neutrinos (called sterile neutrinos because they are in the trivial representation (1, 1, 0)R in (2.9) and do not interact with any of three SM forces). They also can be regarded as the left-handed (1, 1, 0)L with complex conjugation on the representation.

Let us overview the two known mass generation mechanism and propose a new third mechanism (a topological mass or energy gap mechanism to gap the neutrinos):33

1. Dirac mass mechanism [111,112]: Dirac mass and Anderson-Higgs mechanism are believed to give other Standard Model particles their masses. 2. Majorana mass mechanism [113]: This requires that the neutrino and antineutrino to be the same particle. If a neutrino is indeed a Majorana fermion, then lepton-number violating processes such as neutrino-less double beta decay would be allowed. The neutrino-less double beta decay is not allowed if neutrinos are Dirac fermions. 3. Topological mass mechanism (or Topological Energy Gap mechanism): This describes the gapped excitations of interacting systems (typical many-body quantum matter). The underlying low energy theory can be either invertible TQFT (invertible or SPT phases), or non-invertible TQFT (topological order). The energy gap is induced by the interaction mechanism [27, 66, 67, 78, 114–119]. The typical quadratic mass term breaks the chiral symmetry; but the interaction-induced energy gap can be fully symmetric without any chiral symmetry breaking [120].

Is the mass or energy gap necessarily associated with a single particle picture (only deﬁned if there is a free limit)? Or can the mass gap associated with the whole quantum system in a strongly correlated interacting limit? It turns out that a free particle mass may be suitable for Dirac mass 1 and Majorana mass 2 mechanisms; but a free particle mass may not be appropriate for the Topological mass 3 mechanism. 33By a gap, we mean giving a mass gap or an energy gap to the system’s energy spectrum, which is the eigenenergy values of the quantum Hamiltonian of the system. (E.g. Solving a big matrix eigenvalues in the linear algebra.) The mass gap usually already assumes the free particle descriptions exist. However, in the interacting systems such as CFT or strongly-correlated many-body quantum matter, we may not always be able to ﬁnd a suitable free particle description. In the later case, we can still have an energy gap from the interacting or many-body physics. The energy gap can be a generalization of the particle mass gap for the interacting or many-body systems. After all, Einstein had told us long ago the energy is the mass: ∆E ∼ (∆m)c2.

45 5.3.1 Dirac Mass or Majorana Mass

In more details, we write Dirac mass (1) and Majorana mass (2) as follows. By the 2-component Weyl spinor notations in 4d, the undotted indices are for left-handed spinor ﬁelds, the dotted indices are for right-handed spinor ﬁelds. We write the 4-component spinors (such as Dirac or Majorana) in terms of two 2-component Weyl spinors:34 χα α α α˙ Dirac spinor ΨD α˙ . ΨD (ζ , χ†α˙ ). Dirac mass: ΨDΨD = ζ χα + χα†˙ ζ† = ζχ + (ζχ)†. (5.32) ≡ ζ† ≡ χα α Majorana spinor ΨM α˙ . ΨM (χ , χα†˙ ). Majorana mass: ΨMΨM = χχ + h.c. = χχ + (χχ)†. (5.33) ≡ χ† ≡

If the neutrino mass is generated by Dirac mass (1) like other Standard Model fermions, then the mass term needs to be an SU(2) singlet. The left-handed neutrino from the SU(2) doublet lLν would have the Yukawa interactions with the SU(2) doublet Higgs φH and an SU(2) singlet right-handed neutrino χν.

Another mechanism is that neutrino mass can be generated by a Majorana mass (2), which would require the neutrino and antineutrino to be the same particle (particle as its anti-particle).

The anomaly ν = Ngeneration mod 16 in (4.34) dictates that in the free fermion limit, we need − 35 to add ν = Ngeneration = 3 mod 16 right-handed spinors to SM to match the anomaly. Denote the

right-handed spinor of sterile neutrinos of 3 generations (νe, νµ, ντ ) as (χνe , χνµ , χντ ), we can write down the free quadratic non-interacting action where σ¯ (1, ~σ), ≡     χνe µ µ µ 1 χ† iσ ¯ ∂µχνe + χ† iσ ¯ ∂µχνµ + χ† iσ ¯ ∂µχντ + (χνe , χνµ , χντ )  M  χνµ  + h.c. . (5.34) νe νµ ντ 2 χντ

Along the rank-3 mass matrix M, its diagonalized elements represent the Majorana mass (5.33).

More generally, we can pair ν = Ngeneration = 3 right-handed Weyl spinors (sterile neutrinos) with

34 00 Here we use the 2-component Weyl spinor notation. Not to confuse the “µ in muon neutrino index νµ with the spacetime index µ. For two general left-handed Weyl spinors, say χ and χ0 (where each component is a Grassman number 0 0 with anti-commutation properties χαχβ = −χβ χα), we have

0 α 0 αβ 0 βα 0 T 2 0 αβ 0 βα 0 0 χχ ≡ χ χα ≡ χβ χα = − χβ χα = −(χ )β (iσ )βαχα = − χαχβ = χαχβ ≡ χ χ,

0 † α 0 † 0 † α † 0† †α˙ 0† † 0† 2 † 0† 2 ∗ † 0†α˙ † 0† 0 † (χχ ) = (χ χ ) = (χ ) (χ ) = (χ )(χ ) ≡ χ χ = (χ )(iσ ) ˙ χ = (χ )(iσ ) ˙ χ = (χ )(χ ) ≡ χ χ = (χ χ) . α α α˙ α˙ α˙ β β˙ α˙ α˙ β β˙ α˙ ˙ ˙ 0 1 αβ = βα = − = = α˙ β = βα˙ = − = = (iσ2) = χα ≡ αβ χ We use the convention αβ βα α˙ β˙ β˙α˙ αβ −1 0 , also β αβ α † †α˙ αβ˙ † † αβ˙ 2 † 2 ∗ (χ ) = (χ ) = ( χ ) = (χ ) = (iσ ) ˙ χ = (iσ ) ˙ χ and β β α˙ β β˙ α˙ β β˙ . The Hermitian conjugate switches the two SU(2) Lie algebras in the Lie algebra of the Lorentz group Spin(3, 1) = SL(2, C). Spin(4) = Spin(3) × Spin(3) = SU(2)L × SU(2)R. The h.c. is Hermitian complex conjugate. 35 This anomaly ν = −Ngeneration mod 16 in (4.34) also rules out many BSM introducing more than Ngeneration = 3 mod 16 of sterile neutrinos or right-handed Weyl spinors.

46 each other and with the 3 left-handed Weyl spinors via another free non-interacting matrix term action:

 φH  lL h i νe φH |hφ i| 3 3  h H i  lLνµ φ  !  |h H i|  3 0 MDirac φH 1 φH lLν h i  ( l , l , l , χ , χ , χ ) τ φH + h.c. . Lνe Lνµ Lντ h i ν†e ν†µ ν†τ  |h i|  (5.35) 2 φH 3 MDirac MS   |h i|  χν†e     χν†µ 

χν†τ

There is a rank-6 mass matrix. Here the Dirac mass scale, for example, is given by Higgs vev MDirac ∼ φH . We remind that the ﬂavor states can be superpositions of the diﬀerent mass eigenstates. In above, |h i| we rewrite the three generations of SU(2) doublet 2 of left-handed neutrino lLνe , lLνµ , lLντ pair with φH the SU(2) doublet 2 Higgs as lL , lL , lL h i . The usual seesaw mechanism [121] sets the scale νe νµ ντ φH |h i| 2 MDirac MS MDirac | | MDirac of . So the three mass eigenstates have mass MS (for the observed 3 | | | | ' | | | | neutrinos much smaller other Dirac mass of MeV or GeV scales), while the other three mass eigenstates have mass MS which can set to be the GUT scale (thus too heavy yet to be detected). ' | | We should emphasize that by having Dirac mass or Majorana mass to any of the Weyl fermion spinors, this quadratic mass would break the Z4,X symmetry ((4.37) and (4.38) assigned to the complex Weyl fermions). More precisely, as explained in Sec. 4.3, the Dirac mass term preserves the Z4,X symmetry, but the Z symmetry is spontaneously broken by the electroweak Higgs condensate φ = 0. In contrast, 4,X h H i 6 the Majorana mass term explicitly breaks the Z4,X symmetry.

Is it necessary to break Z4,X symmetry in order to saturate the anomaly by a gapped theory? No, we do not have to break the Z4,X symmetry if we introduce a Topological Mass/Energy Gap for Weyl fermions, see Sec. 5.3.2.

5.3.2 Topological Mass and Topological Energy Gap

Now we introduce Topological mass and Topological energy gap mechanism. The concept of free particle mass may not be appropriate here, so we should digest the mechanism from the interacting theory viewpoint. We may colloquially call any of the following mass gaps as the topological energy gap:

(1). Interaction-induced mass or interaction-induced energy gap: This idea has been used to classify the topological phases of interacting quantum matter. Given the symmetry G (including the Gspacetime and Ginternal) of the system, are the two ground states (i.e., two vevs) deformable to each other by preserving the symmetry G? This is the key question for the community studying the classification of Symmetry-Protected Topological state (SPTs). Fidkowski-Kitaev [66, 67] had shown that 1+1d Z classification [81–83] of T 2 = +1 topological • superconductor with ZT ZF symmetry can be reduced to Z class. Fidkowski-Kitaev may be the 2 × 2 8 first example of showing the interaction can produce the energy gap between 8 Majorana fermions in 1+1d without breaking the original symmetry. (See recent discussions along QFT reviewed in [118,122].)

Kitaev [21] and Fidkowski-Chen-Vishwanath (FCV) [57] suggested that 3+1d Z classiﬁcation of • T 2 = ( 1)F topological superconductor (TSC) with ZT ZF symmetry can be reduced to Z class. − 4 ⊃ 2 16

47 This implies that the 16 number of 2+1d Majorana fermions on the boundary of 3+1d TSC can open up an energy gap without introducing any free quadratic mass: neither Majorana nor Dirac masses. Wen [78], and the author and Wen [27, 114, 119], suggest that all the G-anomaly-free gapless theory • can be fully gapped without breaking G-symmetry. This idea includes introducing a random disorder new Higgs ﬁeld [78]; or introducing the proper-designed direct non-perturbative interactions [27, 114], (which are usually irrelevant or marginal operator deformations viewed from IR QFTs) or introducing a symmetric gapped boundary [119]. Many of such examples are applied to construct chiral fermion or chiral gauge theories on the lattice. This approach is pursued also by You-BenTov-Xu [115, 116]. BenTov and Zee [117,118] names this mechanism as Kitaev-Wen mechanism, or the mass without mass. In the lattice gauge theory, gapping the mirror fermions dated back to Eichten-Preskill [123]; a recent work by Kikukawa also suggested the same conclusion that the SO(10) GUT mirror fermion can be gapped [124]. This deformation of G-anomaly-free theory is consistent with Seiberg’s conjecture that the deformation classes of QFTs is constrained by symmetry and anomaly [125].

(2). Vortex condensation: This is an approach commonly used in condensed matter literature for 2+1d strongly-correlated systems. The idea is that the symmetry-breaking defects (such as vortices) may trap the zero modes and which carry nontrivial quantum number. The question is to ﬁnd which number of vortex zero modes with what kind of symmetry assignment, can the vortices be proliferated to restore the broken global symmetry — this would drive the quantum phase transition between the symmetry- breaking phase and the symmetry-restoring phase. This approach, called the vortex condensation, has been used to construct 2+1d surface topological orders, see the condensed matter review [55].

(3). Symmetry-extension and symmetry-preserving gapped topological order/TQFT: As mentioned brieﬂy in Sec. 5.2.1, the symmetry-extension mechanism [53] and higher-symmetry extension generalization [51] are a rather exotic mechanism. We trivialize the anomaly and introduce a gapped phase, not by breaking G to its subgroup Gsub G, but by extending it to a larger group Gtotal which ⊂ can be regarded as a ﬁbration of the original group G as a quotient group. (See the down-to-earth lattice constructions in any dimension [53] and in 1+1d bulk [104,126]) This mechanism is a useful intermediate step stone, to construct a symmetry-preserving TQFT, via gauging the extended-symmetry [53]. This approach is applicable to bosonic systems [53,127] and fermionic systems [56,104,128] in any dimension. In contrast to the well-known gapped phase saturate the anomaly via symmetry breaking (either global symmetry breaking or Anderson-Higgs mechanism for gauge theory), this approach is based on symmetry extension (thus beyond symmetry breaking and Anderson-Higgs mechanism).

The Topological mass and Topological energy gap mechanism (including (1), (2) and (3)), in fact, is obviously beyond the familiar Dirac, Majorana mass, and seesaw mechanism [121]. In a colloquial sense, we do not have a Higgs ﬁeld φ breaking the symmetry and gives vev φ = 0. In certain case, H h H i 6 we can consider a new randomly disordered Higgs ﬁeld φh such that [27,78]

φ = 0, φ 2 = 0. (5.36) h hi h| h| i 6 So Topological mass/energy gap is a quantum behavior beyond the mean ﬁeld quadratic semiclassical theory, beyond Anderson-Higgs mechanism, and beyond Landau-Ginzburg symmetry-breaking paradigm.

5.3.3 4d duality for Weyl fermions and “mirror symmetry”

Before apply the Topological Mass from Sec. 5.3.2 to neutrino physics furthur, we like to introduce a potential helpful supersymmetry (SUSY) duality in 4d known as Seiberg duality [129] studied in = 1 N

48 theory and supersymmetric quantum chromodynamics (SQCD). Seiberg duality is an = 1 electric N magnetic duality in SUSY non-abelian gauge theories with the weak-strong duality. On the left-hand side of the duality may have quarks and gluons; on the right-hand side dual theory, they become the solitons (such as nonabelian magnetic monopoles) of the elementary fields. When the left-hand side theory is Higgsed by an expectation value of a squark, the right-hand side dual theory’s is confined. Massless glueballs, baryons, and magnetic monopoles in the confined strongly coupled description in the left-hand side theory becomes some weakly coupled elementary quarks in the right-hand side dual Higgs description.

Schematically, there is an IR duality between left-hand side (LHS) and right-hand side (RHS) under a renormalization group (RG) ﬂow for = 1: N

SU(Nc) gauge theory with Nf chiral and antichiral multiplets Q, Q˜ in color fundamental Nc, N¯c IR duality SU(N N ) gauge theory with q and q˜ in color fundamental N N , N N and meson M. ←−−−−−→ f − c f − c f − c (5.37) Include the representations of chiral multiplet/superfields, we have the relations: 4d = 1 Seiberg duality LHS: SQCD RHS: dual theory N color gauge group SU(N ) SU(N N ) c f − c Same global internal symmetries SU(N ) SU(N ) U(1) U(1) f L × f R × B × R Chiral multiplet/superfields: Nf Nc N Q :(N , 1, 1, − ) q : (1,N , 1 , c ) (5.38) f N f N − Nc N Representation R f f − f Nf Nc N Q˜ : (1, N¯ , 1, − )q ˜ :(N¯ , 1, 1 , c ) f N f N Nc N − f f − f 2(N Nc) M :(N , N¯ , 0, f − ) f f Nf We are particularly interested in the case when the RHS flows to free Weyl spinors, which means that we can choose as simple as N = 2 and N = 3, and = 1: c f N IR duality SU(N = 2) with N = 3 chiral and anti-chiral multiplets Q and Q˜ 15 Weyl spinors. (5.39) c f ←−−−−−→ For Weyl spinor counting we have Nf = 3 chiral multiplets and Nf = 3 anti-chiral multiplets, each is the fundamental or anti fundamental 2 or 2¯ of SU(2), thus they contribute 2 2 3 = 12 Weyl spinors. There · · is also a vector multiplet which sits at the adjoint representation 3 of SU(2), this contributes another 3 Weyl spinors. So in total we have 2 2 3 + 3 = 15 Weyl spinors. The N = N + 1 = 3 is interesting · · f c because it sits at the lower boundary (3Nc/2 = Nf ) of 3Nc/2 < Nf < 3Nc, where the origin of the moduli space is an interacting CFT and non-abelian Coulomb phase. Also this case we have Nf = Nc +1 thus two moduli spaces are identical but the interpretations of the singularity at the origin are different — massless particles can be regarded as, either strongly coupled mesons and baryons on LHS, or weakly interacting or free quarks on RHS.

This duality helps as 15 mod 16 = 1 mod 16 so to cancel the anomaly (4.31) as ν = N − − νR mod 16 = 1 mod 16 in one generation of SM. One way to simplify 15 Weyl spinors to 1 Weyl spinor − on RHS would be that adding 1 Weyl spinor on both sides in the trivial representation, and adding nonperturbative deformations to gap the RHS completely:

SU(Nc = 2) with Nf = 3 chiral and anti-chiral multiplets Q and Q˜ + deformations IR duality (gapping 16 Weyl spinors via nonperturbative interacting deformations) ←−−−−−→ + ( 1) Weyl spinors (in the complex conjugate representation). (5.40) −

49 We leave details of this construction in an upcoming work [130]. In the following subsections, we can argue several phenomenon of gapping Weyl spinors based on this proposed duality (5.40). The hope is that we can access the 4d TQFT from the dual fermionic vortex zero mode bound state condensation via the topological quantum phase transition [107,108] or “4d mirror symmetry [109] description” (i.e., a duality of QFTs in the version of 4d spacetime).

In a general colloquial sense, this duality is also related to the particle-vortex duality [131,132]. The renown supersymmetric version of duality in 3+1d includes, for example, the = 2 Seiberg-Witten N theory [133], and the = 1 Seiberg duality [129], and many other theories (see a review [134]). A N recent ongoing pursuit, along a fermionic non-supersymmetric version of particle-vortex duality in 3+1d, includes [51,135–139]. 36

5.3.4 Gapping 3, 2, or 1 Weyl spinors / sterile right-handed neutrinos + extra sectors

In Sec. 5.3.1, the 4d anomaly ν = Ngeneration mod 16 in (4.34) dictates that in the free particle limit, we − need to add ν = Ngeneration = 3 mod 16 right-handed spinors to match the anomaly. But as mentioned, in Sec. 5.2, there is an obstruction to obtain a Topological Mass: there does not exist any 4d noninvertible TQFT to saturate the ν = Ngeneration = 3 mod 16 anomaly, shown by [59,60].

In order to gap 3 Weyl spinors in a topological fashion, we can obtain a 4d νeven = 2-TQFT, with a left-over right-handed neutrino (which introduces Dirac/Majorana masses):

4d (ν = 3) massless right-handed neutrinos 4d topological quantum phase transition −−−−−−−−−−−−−−−−−−−−−−−−−→ 4d (νeven = 2)-TQFT + 1 right-handed neutrino gives Dirac/Majorana masses to the other 3 of SM’s left-handed neutrinos. (5.41) We can go through a quantum phase transition by using 3 times of (5.40). We may access the 4d noninvertible TQFT from the LHS. On the other hand, we do not gain the expectation values for the quadratic mass to two generations of fermions:     M11 M12 M13 0 0 0 mean field vev M21 M22 M23 quadratic mass term vev 0 0 0  . (5.42) −−−−−−−−−→ ∝ ν M M M 0 0 # φ or MMajorana 31 32 33 h H i Schematically, on the left-hand side of (5.42), we may include the quadratic fermion pairing term with the Mij stands for possible highly-order interaction (or momentum-dependent pairing) terms. On the right- hand side, we ask what are the quadratic mass at the mean-field level. Two generations have the zero ν mean-field mass. But one generation has left with a Dirac mass (# φ ) or Majorana mass (MMajorana). h H i If we aim to preserve the Z4,X symmetry at the Lagrangian level, then we should choose the Dirac mass term instead of Majorana mass term.

However, as mentioned in Sec. 4.3, Yukawa-Higgs-Dirac mass term spontaneously breaks the Z4,X , while the Majorana mass explicitly breaks the Z4,X . Neither scenario involving Dirac or Majorana mass

36There were also a fermionic version in 2+1d of particle-vortex duality proposed in [140–142] and formalized in [143–145]. They have achieved great success on understanding condensed matter phenomena in 2+1d. They may provide insightful guidelines to the 3+1d, namely 4d duality construction.

50 preserves the Z4,X at the dynamical level. Nonetheless, we can preserve the Z4,X at the dynamical level if we trade a single generation of 4d right-handed neutrino to a 5d (ν = 1)-iTQFT. So we can consider the Z4,X -symmetric topological quantum phase transition:

4d (ν = 2) massless right-handed neutrinos + 5d (ν = 1)-iTQFT 4d topological quantum phase transition −−−−−−−−−−−−−−−−−−−−−−−−−→ 4d (νeven = 2)-TQFT + 5d (ν = 1)-iTQFT. (5.43)

To complete a story for high-energy phenomenology, we can add “(4d SM with (Ngeneration = 3) 15 × Weyl fermions)” on both sides of topological quantum phase transitions in (5.41) and (5.43), in order to apply to SM and BSM physics.

6 Ultra Uniﬁcation: Grand Uniﬁcation + Topological Force and Matter

In the previous sections, we had shown that in order to match a nonperturbative Z16 global anomaly 37 (Sec. 4.2) but still preserve Z4,X of SM and SU(5) GUT Georgi-Glashow (GG) model with only 15 Weyl fermions per generation, we can introduce a new hidden gapped sector appending to the SM and GUT: • Topological Mass/Energy Gap (Sec. 5.2 and Sec. 5.3) to gap the 16th Weyl fermions (right-handed neutrinos). The outcome low energy gives rise to a 4d non-invertible TQFT.

• 5d invertible TQFT (iTQFT) with one extra dimension (Sec. 5.1, known as 5d SPTs in a quantum condensed matter analogy).

Overall, we consider the combinations of the above two solutions plus additional scenarios enlisted in Sec. 4.3. In either cases, we require new hidden gapped topological sectors beyond SM and GUT. We name the unification including SM, Grand Unification, plus additional unitary gapped topological sectors with Topological Force and Matter as Ultra Unification.

6.1 4d-5d Theory: Quantum Gravity and Topological Gravity coupled to TQFT

I]: The combinations of solutions from Sec. 4.3, including adding 4d non-invertible TQFT and 5d invertible TQFT, mean that we can propose a new schematic partition function / path integral, generalizing (5.10) to

2πi ¯ Z5d-iTQFT/[ Z4 ] = exp( ν5d η(PD( Z2 )) M 5 ) [ ψ][ ψ][ A][ φH ][ a][ b] 4d-QFT A 16 · · A | · ˆ D D D D D D ···

¯ (ν4d) exp(i S4d-SM/GUT[ψ, ψ, A, φH ,..., Z4 ] 4 + i S4d-TQFT[a, b, . . . , Z2 ] ) . M M 4 A A ν ν = Ngeneration 5d− 4d − (6.1)

37 By preserving Z4,X , we mean the Z4,X -symmetry is preserved at some higher energy scale above the electroweak scale. Of course, the energy scale lower such as the Higgs scale 125 GeV, the usual Yukawa-Higgs-Dirac mass terms would spontaneously break the Z4,X .

51 Partition function depends on the ( = mod 2) background ﬁelds. The anomaly (4.34) is now AZ2 AZ4 matched by:

ν = ν ν = Ngeneration mod 16. (6.2) 5d − 4d − The S4d-SM/GUT is the 4d SM or GUT action. The ψ, ψ,¯ A, φH are SM and GUT quantum fields, where ψ, ψ¯ are the 15 or 16 Weyl spinor fermion fields, the A are 12 or 24 gauge bosons given by gauge group (ν4d) Lie algebra generators, and φH is the electroweak Higgs (we can also add GUT Higgs). The S4d-TQFT is a 4d noninvertible TQFT outlined in Sec. 5.2 (in particular, ν = ν = 2,... ). The a and b 4d even ± (and possibly others fields from fermionic invariants such as the ABK invariant) are TQFT gauge fields (locally differential 1-form a and 2-form b, as anti-symmetric tensor gauge connections). II]: Many different perspectives guide to an understanding that at high enough energy scale, every global symmetry should be either dynamically gauged or explicitly broken [146–149]. By every global symmetry, F we include the internal symmetry Ginternal, the spacetime Gspacetime, the fermion parity Z2 , the time reversal symmetry ZT or ZT ZF , and so on. Below we discuss the consequences on gauging some of 2 4 ⊃ 2 the symmetries on the boundary or in the bulk. Our notations on the bulk and boundary symmetries associated with the group extension follow the conventions of [53,127]:

(a) Gauge the Z on the 4d: We can gauge a H1(M, Z ) on the boundary which sits at the normal 2 ∈ 2 subgroup [Z2] of the symmetry extension (5.18):

1 [Z2]bdry gauge Spin Z4,X (Spin F Z4,X )Bulk/bdry sym 1, (6.3) → → × → ×Z2 → while the 5d bulk and 4d boundary both keep the global symmetry Spin F Z4,X . In this case, there ×Z2 is a 4d Z2 gauge theory living on the boundary of a 5d iTQFT. Z4,X (b) Gauge the F in 5d and gauge the Z4,X on the 4d: We can also dynamically gauge Z2 Z2 A ∈ 1 F H (M, Z4,X /Z2 ) which sits at the normal subgroup Z2 of the following short exact sequence respect to the bulk:

Z4,X 1 [ ]Bulk gauge Spin F Z4,X (Spin)Bulk sym 1. (6.4) F Z2 → Z2 → × → →

Z4,X If we also gauge the [ F ] out of the 5d bulk symmetry Spin ZF Z4,X , then the 5d bulk short-range Z2 × 2 entangled gapped iTQFT can become a 5d bulk long-range entangled gapped Z2 gauge TQFT with a fermionic rotational Spin(d) symmetry deﬁnable on manifolds with a Spin structure. This is due to gauging the normal subgroup of the above group extension (6.4). Z4,X Once the [ F ] is dynamically gauged in 5d, it also implies that the [Z4,X ] should be dynamically gauged Z2 on the 4d boundary theory. (It is natural that the Z4,X is gauged above some GUT scale, especially given that the Z is a gauge subgroup Z = Z(Spin(10)) Spin(10) of the SO(10) GUT, see more 4,X 4,X ⊂ discussions in [99].) By doing so, we need to sum over H1(M, Z ) for a given spacetime topology AZ4 ∈ 4,X and geometry. We can gauge the normal subgroup Z out of the 4d boundary symmetry Spin Z 4,X × 4,X in the group extension,

1 [Z ]bdry gauge Spin Z (Spin) 1. (6.5) → 4,X → × 4,X → Bulk/bdry sym → F Z4,X F (c) Gauge the Z2 F in 5d and gauge the Z2 Z4,X on the 4d: Other than gauging Z2 × Z2 × A ∈ 1 F F H (M, Z4,X /Z2 ), we can also gauge the fermion parity Z2 . So we gauge the normal subgroup out of the following short exact sequence respect to the bulk:

F Z4,X 1 [Z ]Bulk gauge Spin F Z4,X (SO)Bulk sym 1. (6.6) 2 F Z2 → × Z2 → × → →

52 Then the 5d bulk becomes a long-range entangled gapped TQFT with a bosonic SO(d = 5) symmetry deﬁned on 5d manifolds with a special orthogonal group SO structure. For the boundary theory, we gauge the normal subgroup out of the following short exact sequence respect to the boundary:

F 1 [Z Z ]bdry gauge Spin Z (SO) 1. (6.7) → 2 × 4,X → × 4,X → Bulk/bdry sym → F By gauging the fermion parity Z2 , this means the UV completion at a higher energy scale should be a bosonic system, presumably emerged from a local tensor product Hilbert space — this is analogous to the “It from Qubit” and the quantum condensed matter view [27]. (d) Gauge the Z with the spacetime Spin and SO symmetry: We can also sum over 4,X AZ4 ∈ H1(M, Z ) together with all spacetime M topology, geometry. Let us consider the case the 4,X ∈ ∀ ∀ spacetime is a closed M = M 5 without boundary. But summing over all spacetime M certainly diverges, one needs to make sense of the partition function by regularization. To deal with such a path integral is a challenging problem similar to topological quantum gravity. We are not be able to solve it for now. In summary, a schematic path integral says:

X X 2πi exp( ν η(PD( )) 5 ) 16 5d Z2 M 1 F · · A | M topology, geometry Z H (M,Z /Z ) ∈∀ ∀ A 2 ∈ 4,X 2 regularize X 2πi exp( ν η(PD( )) 5 ). (6.8) −−−−−−→ 16 · 5d · AZ2 |M M topo/geo , ∈ { } A ∈H1(M,Z /ZF ) Z2 4,X 2

A few thoughts can help to attack this challenging problem on regularization of summing over spacetimes. • In 2d topological gravity, we can sum over conformal structures. This can be a ﬁnite dimensional integral for a ﬁxed topology.

• One can simplify the problem to sum over diﬀerent topologies given by the 5d η(PD( Z2 )) invariant 5 A with M of Spin Z Z4 = Spin F Z4,X structure × 2 ×Z2 • Recent work by Dijkgraaf and Witten on 2d topological gravity provides a guide to the analogous 2d partition function [150] related to the 2d Arf invariant. The theory (6.8) may be regarded as a 5d gravity (dynamical, quantum, and topological gravity) related to the 5d η(PD( )) invariant. AZ2 III]: Suppose we ﬁnd a way to regularize the path integral, then we can consider the theory (6.8) with boundary, where we can place the 4d SM theory. We thus rewrite (6.1) as the fully gauged version of a schematic path integral:

X 2πi ¯ Z5d-TQFT.QG/ = exp( ν5d η(PD( Z2 )) M 5 ) [ Z4 ][ ψ][ ψ][ A][ φH ][ a][ b] 4d-QFT 16 · · A | ·ˆ DA D D D D D D ··· M topo/geo , ∈{ } A ∈H1(M,Z /ZF ) Z2 4 2

¯ (ν4d) exp(i S4d-SM/GUT[ψ, ψ, A, φH ,..., Z4 ] 4 + i S4d-TQFT[a, b, . . . , Z2 ] ) . M M 4 A A ν ν = Ngeneration 5d− 4d − (6.9)

Recall we gauge as it is the Z gauge ﬁeld and mod 2 = . This is a 4d and 5d coupled AZ4 4,X AZ4 AZ2 fully gauged quantum system — where the 4d has SM, GUT and noninvertible TQFT, and the 5d can allow a certain gravity (dynamical, quantum and topological) coupled to the 5d TQFT. The 1-form gauge ﬁeld (obtained by gauging the 0-form Z symmetry) can mediate between the 5d bulk, and 4d AZ4 4,X boundary (the 5d bulk and the 4d SM and BSM gapped topological sectors).

53 6.2 Braiding statistics and link invariants in 4d or 5d: Quantum communication with gapped hidden sectors from our Standard Model

Follow Sec. 6.1, above a higher energy scale (way above SM and above SU(5) GUT but around SO(10) GUT), the discrete X = 5(B L) 4Y symmetry (the Z symmetry) becomes dynamically gauged, − − 4,X with a dynamical gauge vector boson mediator Xg, mathematically associated with a 1-cochain gauge ﬁeld .38 So the entangled Universe in 4d and 5d may be described by (6.9). AZ4

Once the Z4,X is dynamically gauged, the 4d and 5d gauged TQFT sectors become noninvertible TQFTs. In fact such a medium can be regarded as some version of topological quantum computer [48,49] with intrinsic topological orders [46].

There is an immediate question: Do we have any way to communicate or interact with objects living in 4d or 5d TQFTs? Could we communicate with those objects within SM particles that we human beings and creatures are made of? Another way to phrase this question is: Can we detect the Topological Force?

For example in the TQFTs of (5.19), we have some gauge invariant extended line and surface operators: 1 1 a + , b + 2 , (6.10) ˛ 2AZ2 ‹ 2AZ2 invariant under the gauge transformation: Z2 Z2 + δλ, a a + λδλ, and b b + λδλδλ, where 1 A → A → → λ is a 0-cochain valued in H (M, Z2). Since the extended a line and b surface operators do not end on the SM particles, nor do those a and b extended operators interact with SM gauge forces, it seems that we SM particles cannot communicate with the TQFT at least näively. At least, we SM particles cannot directly interact with the 4d TQFT sector via the SM gauge forces.

But in fact, the answer is YES, we SM particles can indeed communicate with the 4d TQFT sector, only via the gauge ﬁeld. Moreover, we SM particles can also communicate with the 5d TQFT sector AZ2 via the gauge ﬁeld. The idea is that the TQFT path integral (5.19) schematically AZ4

2 2πi [ Z2 ] [ b][ a] exp iπ b(δa + Z2 ) + ABK(c PD0( Z2 a)) . (6.11) ˆ DA ˆ D D ˆM 4 A 8 ∪ A contains a term with a schematic coupling: 1 π (b ^ δa + δ ^ + b ^ ^ ) + (ABK ^ ^ a). (6.12) ˆ B AZ2 AZ2 AZ2 4 AZ2

For the purpose which will become clear soon, we introduce a dynamical 2-cochain ﬁeld which is B natural if the is dynamically gauged, with ^ δ = δ ^ on a closed 4-manifold. Again AZ2 B AZ2 B AZ2 the schematic cup product term (ABK ^ ^ a), coupling between a fermionic invariant ABK and AZ2 the cohomology class Aa, is formally ABK(c PD0( a)) deﬁned in Sec. 5.2.2. We denote PD0 as the ∪ AZ2 Poincaré dual on the 4-manifold M 4 = ∂M 5. So what are the experimental designs for us to communicate with “TQFT sectors”?39

38 Again this is a discrete Z4,X or U(1)X gauge theory’s gauge boson Xg, different from the SU(5) GUT’s continuous Z4,X nonabelian Lie group’s gauge boson X and Y. We have been denoted Xg’s 1-cochain gauge field as AZ2 for the case of F Z2 symmetry. We have been denoted Xg’s 1-cochain gauge field as AZ4 for the case of Z4,X symmetry. 39In fact, previous works on braiding statistics and link invariants, such as the multi-string 3-loop braiding process and other exotic braiding process ( [151–153] and [154–157]), provide helpful guidelines to these topological link designs, shown in Fig.5.

54 The action term (ABK ^ ^ a) in (6.12) prompts us to design a conﬁguration in Fig.5. AZ2

To motivate the discussion, the fermionic topological term (ABK ^ ^ a) ABK(PD0( a)) is AZ2 ∼ AZ2 similar to the fermionic topological invariant (Arf ^ a ^ a ) = ABK(PD0(a ^ a )) = ABK(PD0(a ) 1 2 1 2 1 ∩ PD0(a2)) studied in [56]. The (Arf ^ a1 ^ a2) corresponds to a Z2 torsion class cobordism invariant in 2 Spin (Z4) the subgroup of Ω × = (Z )2 Z . The a and a are the mod 2 classes of 1-cochain of two distinct 4 4 × 2 1 2 Z4 gauge fields. Only this Z2 generator is intrinsically fermionic. Ref. [56] shows that the Sato-Levine invariant [158], Arf(V(1) V(2)), can detect a certain link configuration. The link configuration L is a ∪ 2 2 2 2 disjoint union of two surface links Σ(1) and Σ(2), so L = Σ(1) Σ(2). The “ ” means the disjoint union. t t 3 3 The L is a semi-boundary link [158], so that, by definition, there exists Seifert volumes V(1) and V(2) for each link component ∂V 3 = Σ2 , such that the intersections V 3 Σ2 = and Σ2 V 3 = are (j) (j) (1) ∩ (2) ∅ (1) ∩ (2) ∅ empty.

To design a detectable link configuration for the fermionic topological term (ABK ^ Z2 ^ a) 2 2 A ∼ ABK(PD0( a)), we start from finding appropriate two 2-surfaces Σ and Σ to be linked, see Fig.5. AZ2 (1) (2) We have a 2-cochain field dual to the 1-cochain which is dynamically gauged. In this case, the B AZ2 2-surface Σ2 fundamental class can be paired with a 2-surface operator from the cohomology class (1) B field, + .... And another 2-surface Σ2 fundamental class can be paired with a 2-surface operator L1 B (2) ‚ b b + ...... from the cohomology class field, L2 with extra terms to make the 2-surface operator gauge invariant [130]. ‚

Let us explain how to design a semi-boundary two-component link L = Σ2 Σ2 (in Fig.5 (b), this (1) t (2) link conﬁguration is only an example, there could be other conﬁgurations trapping also a Z4,X charge):

3 1. First, we can start from thinking of a Whitehead link ` = `1 `2 in a 3-ball D shown in Fig.5 (a). In 1 3 t Fig.5 (a), the `1 and `2 are both S circles linked in a D .

2. Then, we make the ` circle spun around another S10 circle along the S10 of D3 S10 outside of the 1 × D3 sphere. (So the readers can imagine this S10 rotation of D3 going outside of the paper.) If M 4 is a sphere S4, it can be chosen as S4 = (D3 S10) (S2 D2), gluing the (D3 S10) with another × ∪ × × (S2 D2) along their common boundary (S2 S10). So this spun ` gives us a 2-torus that we call this × × 1 2-surface Σ2 = ` S10. We place a cohomology class field on the ` . (1) ∼ 1 × B 1 3. We also make the `2 rotating along the blue dashed arrow, circling around the `1 component. We call this 2-surface Σ2 = ` S1. We place a cohomology class b field on the ` . (2) ∼ 2 × 2 4. This semi-boundary two-component 2-surface link L = Σ2 Σ2 in Fig.5 (b), with and b field (1) t (2) B inserted, is a nontrivial link configuration which detects the fermionic topological term (ABK ^ ^ AZ2 a).

5. We can also replace the 2-cochain field to its dual 1-cochain field. In the sense, we can interpret B AZ2 the coupling π ^ δ + b ^ + 1 (ABK ^ a) + ... as the gauge field coupled to a AZ2 B AZ2 4 AZ2 current (with´ a Hodge dual ?), say in a schematic differential form: JZ2

? + .... (6.13) ˆ AZ2 ∧ JZ2

The sits at the 2-surface Σ2 , so the Hodge dual of the current ? δ + ... sits at the Seifert B (1) JZ2 ∝ B

55 1 Z2 or Z4 gauge ﬁeld on I A A ( extendable to a 5d bulk) AZ2 ( can end on SM particles in 4d) AZ4

4d spacetime

? or ? on V 3 JZ2 JZ4 (1)

2 Σ(2) 2 Σ(1)

4d noninvertible TQFT extended operators form a spun Whitehead link carrying a discrete Z -charge of X = 5(B L) 4Y `2 4,X − −

`1 (a) (b)

Figure 5: (a) The Whitehead link formed by a disjoint union of two-component 1-loops, ` = ` ` , 1 t 2 is detectable by the Sato-Levine invariant. (See related QFTs explored in [56, 155].) (b) A schematic nontrivial link conﬁguration in a 4d spacetime (inside the large black circle) that can carry an odd Z4,X Z4,X charge (also an odd F charge) measured by a codimension 1 operator ? Z2 (from the 5d bulk Z2 J perspective) or ? (from the 4d boundary perspective). The Z charge is trapped non-locally within JZ4 4,X the surface link. The surface link L = Σ2 Σ2 contains two surfaces obtained from a spun version of (1) t (2) 3 2 10 Whitehead link. Let a D ball contain the Whitehead link `. The Σ(1) rotates the `1 along an S of D3 S10 outside the D3. The Σ2 rotates the ` along the blue dashed arrow around the ` circle within × (2) 2 1 the D3. From a 4d boundary theory perspective, we have the along a 1-line I1 (drawn in the red AZ4 dashed) Poincaré dual (PD) to the ? . From a 5d bulk theory perspective, we have the along a JZ4 AZ2 1-line I1 PD to the ? . JZ2

3 1 3-volume V(1). If so, we learn that the 1-cochain gauge field Z2 can sit along a 1-line I , Poincaré dual 3 A 1 (PD) to the Seifert volume V(1), drawn along the red dashed line I in Fig.5 (b). 6. The gauge field in (6.13) is actually the dynamical gauge field living in the 5d bulk TQFT perspec- AZ2 tive; in fact, on the 4d boundary, the gauge field and the a gauge field can be combined together AZ2 to become a Z gauge field: . The relation between the boundary a H1(M, Z ), the boundary 4 AZ4 ∈ 2 H1(M, Z ), and the bulk H1(M, Z /ZF ) gauge fields can be understood by a group AZ4 ∈ 4,X AZ2 ∈ 4,X 2 extension (following (6.4) and (6.5)):

Z4,X 0 [Z ]bdry gauge [Z ]bdry gauge [ ]Bulk gauge 0. 2 a 4,X Z4 F Z2 (6.14) → → A → Z2 A → Recall that = mod 2, similarly = mod 2. Thus we have the coupling term eqn. (6.13) AZ2 AZ4 JZ2 JZ4

56 in 5d, but we have the following term on the 4d boundary:

? + .... (6.15) ˆ AZ4 ∧ JZ4

7. Due to (6.12), we can view the spun version of Whitehead link as a source of an odd Z4,X charge (also Z4,X an odd F charge), wrapped inside a current ? Z4 in a 3-volume (or ? Z2 in a 4-volume) [56,130]. Z2 J J

In fact, the odd Z4,X charge qX of the topological link means that the dynamically gauged Wilson line with a line integral exp(iq ) can mediate between a topological link on one end (p ) and any X AZ4 1 other object carrying a nontrivial´ Z4,X charges on the other end (p2).

So we only need to look for all SM and GUT particles carrying the Z4,X charges (especially the odd Z4,X charge). We now look at Table1 and2, all the left-handed Weyl spinors carry Z4,X charge +1 (and the right-handed Weyl spinors carry a complex conjugate of Z charge 1 = 3 mod 4). 4,X − The electroweak Higgs φH carries an even Z4,X charge 2 mod 4. So, yes, in fact, all SM Weyl spinors carry odd Z4,X charge, thus, all SM Weyl spinors, say ψSM, can be the other end of the line integral exp(iq ) of Z . All the odd number of bound states (like proton and neutrons in our body, also X AZ4 4,X electrons)´ presumably can carry the odd Z4,X . In summary, we may have a schematic communication between a SM particle ψSM and a topological link via a line operator p 2 exp(iπ b + ... ) exp(iqX ) ψSM(p ), (6.16) ‹ · ˆ AZ4 · 2 a link L formed around p1 p1 where is the discrete Z gauge ﬁeld. A candidate link L is shown in Fig.5 (b). The Z discrete AZ4 4,X 4,X gauge boson is the mediator of Topological Force. Topological Force has eﬀects of long-range remote AZ4 interactions but only through braiding and fusion statistics, thus Topological Force is presumably weaker than the GUT forces, strong, electroweak forces. In summary, if either the 4d or 5d gapped topological sectors exist, our SM world and the gapped topological sector can be mediated and communicated by Topological Force.

6.3 Topological Force as a new force?

We had mentioned that under some assumptions about the discrete Z4,X symmetry unbroken at high energy and only 15 Weyl fermions per generation, it is natural to include a nonperturbative topological sector of 4d TQFT, 5d iTQFT, or 5d TQFT, appending to the SM or the GUT. The Topological Force derived here is not within three known Fundamental Forces or GUT forces. So we may view the Topological Force as a new force. Our model in Sec. 6.1 suggests that perhaps the Topological Force can be a new force in the model (6.1). But the Topological Force may mix with the gravity (the Fourth force) in the model (6.9), when the Z4,X is gauged and the geometry/topology are summed but regularized. It is not clear how the gravity in Sec. 6.1 has anything to do with Einstein gravity. It may be interesting to explore this direction in the future.

6.4 Neutrinos

Neutrino oscillations may also be explained if we consider the Majorana zero modes of the vortices in the 4d TQFT defects in Sec.5. The left-handed neutrinos (conﬁrmed in the experiments) are nearly

57 gapless/ massless. When neutrinos traveling through the 4d TQFT defects, we may observe nearly gapless left-handed neutrino ﬂavor oscillations interfering with the Majorana zero modes trapped by the vortices in the 4d TQFT defects.

A possible scenario is to arrange an even ν4d anomaly index for the 4d TQFT ν = 2, 4, Z , 4d,even ± ± · · · ∈ 16 so there is an additional internal rotational symmetry within the 4d TQFT. For example, given a 40 ν4d,even = 2 of 4d TQFT, there is an additional U(1) internal rotational symmetry. We can imagine a mother EFT with a ν = 2 Z anomaly as a deeper UV theory which lands to two possible 4d,even ∈ 16 IR theories: One side has two right-handed neutrinos, a free fermion theory with a ν = 2 Z anomaly. • 4d,even ∈ 16 Another side has a 4d TQFT also with a ν = 2 Z anomaly. • 4d,even ∈ 16 As a dual description of TQFT on the free fermion side, we have the internal rotational U(1) or O(2) symmetry realized as an internal ﬂavor rotation between two right-handed neutrinos (as 3+1d Weyl or Majorana fermions). The TQFT defects are U(1) symmetry breaking vortices (or vortex strings). The vortices can trap Majorana zero modes below the vortex energy subgap.

SM and GUT Models with right-handed neutrino, 4d TQFT, and 5d iTQFT: We may still need to introduce at least a right-handed neutrino in order to give a conventional Dirac mass to the three left-handed neutrinos. To match the ν = Ngeneration = 3 Z anomaly, we had considered the − − ∈ 16 scenario in (5.41):  4d SM or GUT with ν = 3 15 Weyl fermions.  • ×  4d (νeven = 2)-TQFT. • (6.17) 4d ν = 1 right-handed neutrino gives Dirac/Majorana masses  •  to the other 3 of SM’s left-handed neutrinos. If we want to introduce also a 5d iTQFT but still keep a 4d TQFT and a right-handed neutrino (in order to give a conventional Dirac mass to left-handed neutrinos), we can also propose another scenario:  4d SM or GUT with ν = 3 15 Weyl fermions.  • ×  4d (νeven = 4)-TQFT or (νeven = 2 + k)-TQFT.  • 5d (ν = 2)-iTQFT or (ν = k)-iTQFT. (6.18) • − −  4d ν = 1 right-handed neutrino gives Dirac/Majorana masses  •  to the other 3 of SM’s left-handed neutrinos.

We have a generic k Zeven. The above models, (6.17) and (6.18), all have a 0 mod 16 index, thus free ∈ from the Z16 anomaly.

Right-handed sterile neutrinos are not sterile to Z4,X gauge ﬁeld: We should emphasize again that although the right-handed neutrinos are sterile to SM forces, they are not sterile to the Z4,X gauge ﬁeld , because they carry the odd Z charge. AZ4 4,X

6.5 Dark Matter

Dark Matter: In the heavy sector of TQFTs that we described in Sec.5, the surface b is the worldsheet 40We emphasize this additional U(1) is an extra symmetry, distinct from the (B − L), the X = 5(B − L) − 4Y , the SM, or the GUT gauge groups. In one lower dimension, for a 3d boundary of a 4d bulk iTQFT given by the 4d APS η invariant Pin+ Pin+ of Ω4 = Z16, Ref. [58] introduces a similar extra U(1) symmetry when ν = ±2 ∈ Z16 of Ω4 .

58 of heavy string excitations. Those are new heavy objects whose mass is in the scale of a topological order energy gap, ∆TQFT, possibly around the GUT scales (for example, suggested in [99], the ∆TQFT can be in between the SU(5) GUT and the SO(10) GUT scales). So these new heavy higher-dimensional extended objects may be a suitable candidate account for the abundant Dark Matter in the Unvierse.

Dark Energy: We have very little to say about the Dark Energy. Except that in the gravity version (sum over geometry and topology; dynamical, quantum, and topological gravity) of partition function in (6.9), it may be worthwhile to investigate the underlying energy stored in this partition function. It seems that 4d and 5d topological sectors can be very heavy and store a much huge amount of energy than what we knew of from our Standard Model world.

7 Acknowledgements

JW is grateful to his previous collaborators for fruitful past researches as helpful precursors for the present work. JW appreciates Email correspondences with Robert Gompf, Miguel Montero, Kantaro Ohmori, Pavel Putrov, Ryan Thorngren, Zheyan Wan, and Yunqin Zheng; and the mental support from Shing- Tung Yau.41 JW thanks the participants of Quantum Matter in Mathematics and Physics program at Harvard University CMSA for the enlightening atmosphere. Part of this work is presented by JW in the workshop Lattice for Beyond the Standard Model physics 2019 at Syracuse University (May 2-3, 2019) [160], also at Higher Structures and Field Theory at Erwin Schrödinger Institute in Wien (August 4, 2020) [161], and at Harvard University Particle Physics Lunch (November 30, 2020) and String Lunch (December 4, 2020). JW was supported by NSF Grant PHY-1606531. This work is also supported by NSF Grant DMS-1607871 “Analysis, Geometry and Mathematical Physics” and Center for Mathematical Sciences and Applications at Harvard University.

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