Ultra Unification

Strong, electromagnetic, and weak forces were unified in the Standard Model (SM) with spontaneous gauge symmetry breaking. These forces were further conjectured to be unified in a simple Lie group gauge interaction in the Grand Unification (GUT). In this work, we propose a theory beyond the SM and GUT by adding new gapped Topological Phase Sectors consistent with the nonperturbative global anomaly cancellation and cobordism constraints (especially from the baryon minus lepton number ${\bf B}-{\bf L}$, the electroweak hypercharge $Y$, and the mixed gauge-gravitational anomaly). Gapped Topological Phase Sectors are constructed via symmetry extension, whose low energy contains unitary Lorentz invariant topological quantum field theories (TQFTs): either 3+1d non-invertible TQFT, or 4+1d invertible or non-invertible TQFT (short-range or long-range entangled gapped phase). Alternatively, there could also be right-handed"sterile"neutrinos, gapless unparticle physics, more general interacting conformal field theories, or gravity with topological cobordism constraints, or their combinations to altogether cancel the mixed gauge-gravitational anomaly. We propose that a new high-energy physics frontier beyond the conventional 0d particle physics relies on the new Topological Force and Topological Matter including gapped extended objects (gapped 1d line and 2d surface operators or defects, etc., whose open ends carry deconfined fractionalized particle or anyonic string excitations) or gapless conformal matter. Physical characterizations of these gapped extended objects require the mathematical theories of cohomology, cobordism, or category. Although weaker than the weak force, Topological Force is infinite-range or long-range which does not decay in the distance, and mediates between the linked worldvolume trajectories via fractional or categorical statistical interactions.

"My work is dedicated to you. You can surely feel this in your heart." Intermezzo in A major. Andante teneramente, Six Pieces for Piano, Op. 118 Johannes Brahms in 1893

Introduction and Summary
Unification is a central theme in theoretical physics. In 1864-1865, Maxwell [1] unified the electricity and magnetism into the electrodynamics theory, where the derived electromagnetic wave manifests the light phenomena. In 1961-1967, Glashow-Salam-Weinberg (GSW) [2][3][4][5] made landmark contributions to the electroweak theory of the unified electromagnetic and weak forces between elementary particles, including the prediction of the weak neutral current. The GSW theory together with the strong force [6,7] is now known as the Standard Model (SM), which is verified to be theoretically and experimentally essential to describe the subatomic high energy physics (HEP). In 1974, Georgi-Glashow hypothesized that at a higher energy, the three gauge interactions of the SM would be merged into a single electronuclear force under a simple Lie group gauge theory, known as the Grand Unification or Grand Unified Theory (GUT) [8,9]. The dark gray area schematically shows the possible energy scales for various GUT scenarios, such as the su (5), so(10), . . . , so (18) GUT around 10 16 GeV. The light gray area schematically suggests that the possible energy gap ∆ TQFT for Topological Phase Sector can range from as low energy as the SM, to as high energy to somewhere within the GUT scales (e.g., below the so(10) GUT scale). The dashed lines mean hypothetical unifications that have not yet been confirmed by experiments. Forces are arranged from the strongest to the weakest (horizontally from the left to the right) in the electroweak Higgs vacuum.
In this work, following our previous investigations based on nonperturbative global anomalies and cobordism constraints [10][11][12], 1 we propose an Ultra Unification that a new Topological Force comes into a theme of unification joining with three known fundamental forces and other hypothetical GUT forces. (See Figure 1.) More concretely, there is a new gapped Topological Phase Sector whose underlying dynamical gauge interactions are the Topological Forces. Alternatively, there could also be "right-handed sterile" neutrinos, gapless or more general interacting conformal field theories, or their combinations to altogether cancel the mixed gauge-gravitational anomaly (enumerated in Sec. 2.3). In a modern perspective, we should view the SM and GUT all as effective field theories (EFT) suitable below certain energy scales. Whenever "elementary particles" are mentioned, they only mean to be "elementary field quanta with respect to a given EFT." Likewise, Ultra Unification should be viewed as an EFT which contains SM or GUT but also additional gapped Topological Phase Sectors with low energy Lorentz invariant unitary topological quantum field theories (TQFTs) of Schwarz type (which is the 4d analog of the 3d Chern-Simons-Witten theories [30][31][32]), or additional neutrinos, or additional gapless or conformal sectors. Topological Force and the gapped Topological Phase Sector here have specific physical and mathematical meanings, which we will clarify in Sec. 2.6. Before digging into Topological Phase Sector, we should state the assumptions and the logic that lead to the assertion of Ultra Unification, in Sec. 2. 2 2 Logic to Ultra Unification

Assumptions
Our logic leading to Ultra Unification starts with the three Assumptions mostly given by Nature and broadly confirmed by experiments: Freed-Hopkins theorem [20] to obtain the classification of all invertible quantum anomalies. In addition, there in Ref. [10], we can also fully characterize the dd 't Hooft anomaly [21] of global symmetry G as (d + 1)d cobordism invariants, precisely as topological terms of cohomology classes and fermionic topological invariants. The cobordism invariants can be read from the Adams chart in Ref. [10]. Thus we focus on employing Ref. [10] result. By classifying all the invertible quantum anomalies, we must include • all local anomalies (perturbative anomalies): captured by perturbative Feynman diagram loop calculations, classified by the integer Z classes (the free classes). e.g., Adler-Bell-Jackiw (ABJ) anomalies [22,23], perturbative local gravitational anomalies [24]. Typically the local anomalies are detectable via infinitesimal gauge or diffeomorphism transformations that can be continuously deformed from the identity.
• all global anomalies (nonperturbative anomalies): classified by finite abelian groups as a product of Zn (the torsion classes) for some positive integer n. e.g., Witten SU(2) [25] and the new SU(2) anomalies [26], global gravitational anomalies [27]. Typically the global anomalies are detectable only via large gauge or diffeomorphism transformations that cannot be continuously deformed from the identity.
More examples of dd anomalies characterized by (d + 1)d cobordism invariants can be found in [28,29]. 2 Conventions: We follow the conventions of Ref. [10][11][12]. We denote nd for n-dimensional spacetime. We also follow the modern condensed matter or extreme quantum matter terminology on the interacting phases of quantum matter [33,34]. For example, • Long-range entangled gapped topological phases, whose low energy describes the noninvertible TQFTs, are known as intrinsic topological orders, which include examples of fractional quantum Hall states.
• Short-range entangled gapped topological phases protected by some global symmetry G, whose low energy describes the invertible TQFTs, are known as symmetry-protected topological states (SPTs) [35], which include examples of topological insulators and topological superconductors [36,37].
• By a noninvertible TQFT, it means that the absolute value partition function |Z(M )| = 1 on a generic spacetime manifold M with nontrivial topology (e.g., cycles or homology classes). • By a trivial gapped vacuum with no TQFT or trivial TQFT, it means that the partition function Z(M ) = 1 on any spacetime manifold M with any topology.
We absolutely should distinguish the above beyond-Ginzburg-Landau quantum phases (long-range entangled and shortrange entangled states) from the within-Ginzburg-Landau symmetry-breaking phases (long-range and short-range orders and correlations). See more in Sec. 3.5.
1. Standard Model gauge group G SMq : The Standard Model gauge theory has a local Lie algebra su(3) × su(2) × u(1), but the global structure of Lie group G SMq has four versions: 3 All the quantum numbers of quarks and leptons are compatible with the representations of any version of q = 1, 2, 3, 6. To confirm which version is used by Nature, it requires the experimental tests on extended objects such as 1d line or 2d surface operators (see recent expositions in [38][39][40][41]).
SU (5) and Spin(10) gauge group: Conventionally, people write the Georgi-Glashow model [8] as the su(5) GUT and Fritzsch-Minkowski model [9] as the so(10) GUT because they have the local Lie algebra su(5) and so(10), respectively. However, they have the precise global Lie group SU (5) and Spin(10), respectively. Only q = 6, we are allowed to have the embedding of SM gauge group (5) GUT (see more discussions later in (2.13)).

A variant discrete Baryon minus Lepton number (B − L) is preserved at high energy:
We hypothesize a discrete X symmetry, which is a modified version of (B − L) number up to some electroweak hypercharge Y [42] is preserved (preserved at least at a higher energy): 6 The importance of this discrete symmetry Z 4,X (as a mod 4 symmetry of U(1) X ), in the context of global anomalies for SM and GUT is emphasized by Garcia-Etxebarria-Montero [14].
It is easy to check that (e.g., see the Table 1 and 2 of [11]): • all the particles from5 of su(5) GUT has a U(1) X charge −3.
• all the particles from 10 of su(5) GUT has a U(1) X charge +1. 3 We denote the lower-case su, so, . . . for the Lie algebra and the upper-case SU, SO, . . . for the Lie group. 4 Here Weyl fermions are spacetime Weyl spinors, which is 2L of Spin(1, 3) = SL(2, C) with a complex representation in the Lorentz signature. On the other hand, the Weyl spinor is 2L of Spin(4) = SU(2)L × SU(2)R with a pseudoreal representation in the Euclidean signature. 5 However, the so(10) GUT requires 16 Weyl fermions per generation due to the fermions sit at the 16 of the Spin(10). This fact is used to argue the possibility of topological quantum phase transition between the energy scale of the 15n Weyl-fermion su(5) GUT and 16n Weyl-fermion so(10) GUT in Ref. [12]. 6 Follow [10][11][12], we choose the convention that the U(1)EM electromagnetic charge is QEM = T3 + Y . The U(1)EM is the unbroken (not Higgsed) electromagnetic gauge symmetry and T3 = 1 • the singlet right-handed neutrino (if any) is in 1 of su(5) GUT with a U(1) X charge +5.
• all the fermions of SM has a Z 4,X charge +1.
• the electroweak Higgs φ has a U(1) X charge −2, thus a Z 4,X charge +2. The Z 4,X also contains the fermion parity Z F 2 (whose operator (−1) F gives (−1) to all fermions) as a normal subgroup (so Z 4,X generator square X 2 = (−1) F as the fermion parity): By looking at these consistent Z 4,X quantum number of SM particles, it is natural to hypothesize the discrete X symmetry plays an important role at a higher energy above the SM energy scale.
In Sec. 2.2, we review the anomaly and cobordism constraints given in [10][11][12]. Readers can freely skip the technical discussions on anomalies, and directly go to the final logic step lead to Ultra Unification in Sec. 2.3.

Anomaly and Cobordism Constraints
Based on the three mild and widely accepted assumptions listed in Sec. 2.1, we then impose the constraints from all invertible quantum anomalies via the cobordism calculation on SM and GUT models. The purpose is to check the consistency of the 15n Weyl fermion SM and GUT models: Check: Perturbative local and nonperturbative global anomalies classified via cobordism.
The classification of dd 't Hooft anomalies of global symmetries G is equivalent to the classification of (d + 1)d invertible TQFTs with G-symmetry defined on a G-structure manifold, 7 given by the cobordism group data Ω d G ≡ TP d (G) defined in Freed-Hopkins [20]. 8 The symmetry contains the spacetime symmetry G spacetime and the internal symmetry G internal . 9 Our perspective is that: • We can treat the spacetime-internal G as a global symmetry, and we view the anomaly associated with G as 't Hooft anomalies [21] of G symmetry.
• Then, we can ask all obstructions to dynamically gauging the G internal as a gauge group, which give rise to all the dynamical gauge anomaly cancellation conditions that any consistent gauge theory must obey.
To proceed, we follow the results of [10,11,43], the relevant total spacetime-internal symmetry G for the SM q is G = Spin(d)× Z F 2 Z 4,X ×G SMq with q = 1, 2, 3, 6, and for the su . Only the q = 6 case of SM 6 can be embedded into the su(5) GUT. 7 For the QFT setup, we only require the category of smooth, differentiable, and triangulable manifolds. 8 Let us compare the cobordism group Ω d G ≡ TP d (G) defined in Freed-Hopkins [20] and the more familiar bordism group Ω G d . Here the cobordism group Ω d G ≡ TP d (G) not only contains Hom(Ω G,tors d , U(1)) (the Pontryagin dual of the torsion subgroup (= tors) of the bordism group Ω G d ), but also contains the integer Z classes (the free part) descended from the free part of the bordism group Ω G,free d+1 of one higher dimension. In other words, • The classification of (d − 1)d nonperturbative global anomalies can be read from the torsion part (the finite subgroup part) of cobordism group Ω d,tors G ≡ TP tors d (G). It can also be read from the torsion part of the bordism group Ω G,tors d data.
• The classification of (d − 1)d perturbative local anomalies can be read from the free part (the Z classes) of cobordism group Ω d,free G ≡ TP free d (G), also from the free part of the bordism group Ω G,free d+1 data. In this work, we concern the most for (d − 1) = 4 and d = 5. 9 • The Gspacetime is the spacetime symmetry, such as the spacetime rotational symmetry SO ≡ SO(d) or the fermionic graded spacetime rotational Spin group symmetry Spin ≡ Spin(d).
• The G internal is the internal symmetry, such as G internal in the SM as GSM q ≡ SU(3)×SU(2)×U(1) Zq with q = 1, 2, 3, 6. We also have G internal = SU(5) in the su(5) GUT and G internal = Spin (10) in the so (10) or Spin(10) GUT. The N shared is the shared common normal subgroup symmetry between Gspacetime and G internal . The "semi-direct product " extension is due to a group extension from G internal by Gspacetime. For a trivial extension, the semi-direct " " becomes a direct product "×."

Standard Models
Ref. [10] considers the classification of G-anomalies in 4d given by the d=5 cobordism group Ω d G ≡ TP d (G) for all Standard Models of SM q with an extra discrete Z 4,X symmetry: Here we summarize the anomaly classification for the Standard Model G SMq ≡ SU(3)×SU(2)×U(1) Y Zq obtained in [10,11,43]. Below we write the 4d anomalies in terms of the 5d cobordism invariants or invertible topological quantum field theories (iTQFTs). For perturbative local 4d anomalies, we can also write them customarily as the 6d anomaly polynomials, and their cubic terms of gauge or gravitational couplings in the one-loop triangle Feynman diagram. Here is the list of classifications of anomalies:  (2)) and 6d c 1 (U(1))c 2 (SU(2)). 10 Here we follow the conventions of [10][11][12]28]: • We can characterize anomalies via (perturbative) local anomalies or (nonperturbative) global anomalies.
• We can also characterize anomalies via their induced fields: pure gauge anomalies, mixed gauge-gravity anomalies, or gravitational anomalies (those violate the general covariance under coordinate reparametrization; i.e. diffeomorphism).
• The cj(G) is the jth Chern class of the associated vector bundle of the principal G-bundle.
• We will use CS V 2n−1 to denote the Chern-Simons (2n − 1)-form for the Chern class (if V is a complex vector bundle) or the Pontryagin class (if V is a real vector bundle). The relation between the Chern-Simons form and the Chern class is cn(V ) = dCS V 2n−1 where the d is the exterior differential and the cn(V ) is regarded as a closed differential form in de Rham cohomology. The wj(T M ) is the j-th Stiefel-Whitney class of spacetime tangent bundle T M of the base manifold M .
• The PD is defined as the Poincaré dual. We define the product notation cη between a cohomology class c and a fermionic invariant η via the Poincaré dual PD of cohomology class, thus cη ≡ η(PD(c)). We use the notation for the cup product between cohomology classes. We often make the cup product and the Poincaré dual PD implicit.  . The Arf appears to be the low energy iTQFT of a 1+1d Kitaev fermionic chain [46], whose boundary hosts a single 0+1d real Majorana zero mode on each of open ends. • We use the notation "∼" to indicate the two sides are equal in that dimension up to a total derivative term.
• Because of the Z4,X ⊃ Z F 2 , we have a short exact sequence 0 → Z F 2 → Z4,X → Below we also write down the 4d anomalies in terms of the 5d cobordism invariants or iTQFTs, or the 6d anomaly polynomials: 1. SU(5) 3 : 4d Z class local gauge anomaly. It is given by a 5d cobordism invariant 1 2 CS SU(5) 5 , or more precisely (5)): 4d Z 2 class global gauge anomaly. It is given by a 5d cobordism invariant (A Z 2 )c 2 (SU(5)) which detects a mixed anomaly between the Z 4,X Z F 2 gauge field and SU(5) gauge field.
3. η(PD(A Z 2 )): 4d Z 16 global mixed gauge-gravity anomaly is given by a 5d cobordism invariant η(PD(A Z 2 )) which detects a mixed anomaly between the Z 4,X Z F 2 gauge field and the gravity. This is again the same Z 16 global anomaly from Ω

Consequences lead to Ultra Unification
Ref. [11,12] checked all the above local and global anomalies enlisted in Sec. 2.2 vanished for the SM q with q = 1, 2, 3, 6 and for the su(5) GUT with 15 Weyl fermions per generation, except the 4d Z 16 class global anomaly (the mixed gauge-gravitational anomaly probed by the discrete Z 4,X symmetry and fermionic spacetime rotational symmetry Spin(d) background fields) may not be completely canceled. 11 The cobordism invariant for any ν ∈ Z 16 corresponds to a 5d iTQFT partition function Given a G ⊇ Spin × Z F 2 Z 4,X structure, the cohomology class A Z 2 ∈ H 1 (M, Z 2 ) is the generator from

5
= Z 16 ×. . . , we use the Madsen-Tillmann (M T ) spectra [19] , where ∧ is the smash product and the Σ denotes a suspension. 12 Although the Spin × Z F 2 Z 4,X gauge field A Spin× Z F 2 Z 4,X denoted as A Z 4 in brief is not an ordinary abelian gauge field (see footnote 10), the Thom spectra M ( suggests that the cobordism invariant depends on the A Z 2 gauge field (instead of the A Z 4 gauge field) in the cohomology class H 1 (M, . The η(PD(A Z 2 )) is the value of APS eta invariant η ∈ Z 16 on the Poincaré dual (PD) submanifold of the cohomology class A Z 2 . This PD takes ∩A Z 2 (of the cohomology H 1 ) from 5d to 4d 11 We should briefly compare the perspectives of Garcia-Etxebarria-Montero [14], Davighi-Gripaios-Lohitsiri [16], and our previous result [11,12], and those with Wen [15] or with Wan [10]. In terms of the relevancy to the 4d anomalies of SM and su(5) GUT given the spacetime-internal symmetry G, • Garcia-Etxebarria-Montero [14] checked: G = Spin ×Z 2 Z4, Spin × SU(n), Spin × Spin(n).
Thus, given by the starting assumption in Sec. 2.1, only Wan-Wang [10,43] contains the complete anomaly classification data for G = Spin ×Z 2 Z4 × GSM q and Spin ×Z 2 Z4 × SU(5) that we need for completing the argument. So we must have to employ the results of Ref. [10]. Although Ref. [14,16] checked several global anomalies, but they do not exhaust checking all anomalies that we need for G = Spin ×Z 2 Z4 × GSM q and Spin ×Z 2 Z4 × SU(5).
Indeed, specifically for G = Spin ×Z 2 Z4 × GSM q and Spin ×Z 2 Z4 × SU (5), only Ref. [10,43] exhausted the cobordism classifications for all their possible anomalies, and completed the anomaly cancellation checks on these groups. 12 For a pointed topological space X , the Σ denotes a suspension ΣX = S 1 ∧ X = (S 1 × X )/(S 1 ∨ X ) where ∧ and ∨ are smash product and wedge sum (a one point union) of pointed topological spaces respectively.
(of the homology H 4 ). The APS eta invariant η ≡ η Pin + ∈ Z 16 is the cobordism invariant of the bordism group Ω Pin + 4 = Z 16 . The notation "| M 5 " means the evaluation on this invariant on a 5-manifold M 5 . Note We summarize the consequences and implications of this Z 16 anomaly non-vanishing for 15n Weyl fermions in Sec. 2.3.
• Ref. [14] used the existence of Z 16 global anomaly and its anomaly cancellation to verify the conventional lore: the 16 Weyl fermions per generation scenario, by introducing a right-handed neutrino per generation.
• Ref. [11,12] take the Z 16 anomaly as a secrete entrance to find hidden new sectors beyond the Standard Model, given the fact the HEP experiments only have detected 15 Weyl fermions per generation thus far.
Consequences Follow Sec. 2.1 and 2.2, the Z 16 anomaly index for both the SM q (q = 1, 2, 3, 6) and the su (5) GUT is the (15 = −1 mod 16) per generation, and (N gen = 3). Given the right-handed neutrino number n ν j,R (each ν j,R has Z 4,X charge 1) for the j-th generation, and the new hidden sectors' anomaly index ν new sectors , we have the following anomaly cancellation condition to cancel the Z 16 anomaly: The question is: How to cancel the Z 16 anomaly? We enlist as many Scenarios as possible below.
1. Standard Lore: We can introduce the right-handed neutrino (the 16th Weyl fermion) number n ν j,R = 1 for each generation (so n ν e,R = n ν µ,R = n ν τ,R = 1 for electron, muon, and tau neutrinos). In this case, there is no new hidden sector.
(1b). Dirac mass: Z 4,X is also preserved by the Yukawa-Higgs-Dirac Lagrangian, but Z 4,X is spontaneously broken by the Higgs condensate to give a Dirac mass gap.
2. Proposals in Ref. [11,12]: Ref. [11,12] proposed other novel ways to cancel the Z 16 anomaly. Consequently, we can introduce new hidden sectors beyond the SM and the su(5) GUT: (2a). Z 4,X -symmetry-preserving anomalous gapped 4d topological quantum field theory (TQFT). 13 We also call the finite energy gap for the first excitation(s) above the ground state sectors of this 4d TQFT as Topological mass gap. The underlying quantum system has a 4d intrinsic topological order. We name the anomaly index ν 4d for this anomalous 4d TQFT.
(2b). Z 4,X -symmetry-preserving 5d invertible topological quantum field theory (iTQFT) given by the 5d cobordism invariant in (2.8). The underlying quantum system has a 5d symmetry-protected topological state (SPTs) with an extra bulk 5th dimension whose 4d boundary can live the 4d Standard Model world. We name the anomaly index ν 5d to specify the boundary 4d anomaly of this 5d iTQFT.
]-gauged TQFT and 4d boundary [Z 4,X ]-gauged TQFT. Overall the spacetime rotational symmetry is the fermionic Spin group Spin(d) graded the bosonic rotation special orthogonal group SO(d) by the fermion parity Overall the spacetime rotational symmetry is the bosonic special orthogonal group SO(d). If the diffeomorphism symmetry of SO(d) or Spin(d) is further dynamically gauged, the outcome new sector may be a gravity theory or a topological gravity theory.
(2f). Z 4,X -symmetry-preserving or Z 4,X -symmetry-breaking gapless phase, e.g., extra massless theories, free or interacting conformal field theories (CFTs). The interacting CFT with scale invariant gapless energy spectrum is also related to unparticle physics [57] in the high-energy phenomenology community.
Scenarios (2a) gives rise to Scenario (2d). The underlying quantum systems in Scenario (2a) and (2c) have the symmetry-enriched topologically ordered state (SETs) in a condensed matter terminology.
We name the combination of the above anomaly cancellation scenarios, including the standard lore (right-handed neutrinos in Scenario 1) and the new proposals (all enlisted in Scenario 2) beyond the SM and the GUT, as the Ultra Unification. Although introducing additional CFTs (Scenario (2f)) to cancel the anomaly is equally fascinating, we instead mostly focus on introducing the gapped Topological Phase Sector due to high-energy physics phenomenology (HEP-PH) constraints (see a summary in [11]). A central theme of Ultra Unification suggesting a new HEP frontier is that Ultra Unification: HEP-PH provides Gapped Extended Objects or Gapless Conformal Objects beyond Particle Physics.
• These extended operators are heavy in the sense that they sit at the energy scale above the TQFT energy gap ∆ TQFT (so at or above the scale of E excited in (2.10)).
• These extended operators are heavy in the sense that they have Topological mass and they can interact with dynamical gravity. So these gapped extended objects may be the Dark Matter candidate. If the ∆ TQFT is large, then the gapped extended objects are heavy Dark Matter candidates, in contrast the gapless conformal objects are light Dark Matter candidates.
• There are fractionalized anyonic excitations at the open ends of topological operators (1d line, 2d surface, 3d brane, etc.). In other words, the particle 1d worldline is the 1d line topological operator. The anyonic string 2d worldsheet is the 2d surface topological operator. 14 In summary, based on the anomaly cancellation and cobordism constraints, we propose that the SM and Georgi-Glashow su(5) GUT (with 15 Weyl fermions per generation, and with a discrete baryon minus lepton number Z 4,X preserved) contains a new hidden sector that can be a linear combination of above Scenarios [11,12]. In particular, we can focus on the new hidden sectors given by a 4d TQFT (with the anomaly index ν 4d ) and a 5d iTQFT (with the 4d boundary's anomaly index ν 5d ), so (2.9) becomes (2.12)

Symmetry Breaking vs Symmetry Extension: Dirac or Majorana masses vs Topological mass
The distinctions between Dirac mass, Majorana mass, and Topological mass are already explored in Ref. [11]. They represent the Scenarios (1b), (1c), and (2a) respectively in Sec. 2.3. Here we summarize their essences that: • Symmetry breaking: Dirac mass and Majorana mass are induced by symmetry breaking -either global symmetry breaking or gauge symmetry breaking, for example via the Anderson-Higgs mechanism or through Yukawa-Higgs term. More precisely, we start from a symmetry group (specifically here an internal symmetry, global or gauged) G, and we break G down to an appropriate subgroup G sub ⊆ G to induce quadratic mass term for matter fields. Mathematically we write an injective homomorphism ι: For example, in Anderson-Higgs mechanism, for a Bardeen-Cooper-Schrieffer type Z 2 -gauged superconductor, we have G sub = Z 2 and G = U(1) electromagnetic gauge group. For the SM electroweak Higgs mechanism, we have 3) and G = G SMq ≡ SU(3)×SU(2)×U(1) EM Zq with q = 1, 2, 3, 6 and the appropriate greatest common divisor (gcd).
• Symmetry extension: Topological mass as the energy gap above a TQFT with 't Hooft anomaly (of a spacetime-internal symmetry G) can be induced by symmetry extension [56]. The symmetry extension mechanism extended the original Hilbert space (with nonperturbative global anomalies) to an enlarged Hilbert space by adding extra degrees of freedom to the original quantum system (see [66,67] for explicit quantum Hamiltonian lattice constructions). The enlarged Hilbert space is meant to trivialize the 't Hooft-anomaly in G in an extendedG. The pullback r * of the map can be understood as part of the group extension in an exact sequence [56], while in general this can be generalized as the fibrations of their classifying spaces and higher classifying spaces [68][69][70]. In many simplified cases, we have a surjective homomorphism r in a short-exact sequence of group extension: where G =G N normal becomes a quotient group ofG whose normal subgroup is N normal . The G-anomaly becoming anomaly-free in the extendedG requires the essential use of algebraic topology criteria, such as the Lydon-Hochschild-Serre spectral sequence method [56]. We will explain further details in Sec. 3.2.1.

Gauging a discrete Baryon B, Lepton L, and Electroweak Hypercharge Y
We provide some more logical motivations why we should preserve Z 4,X and dynamically gauge Z 4,X with X ≡ 5(B − L) − 4Y , at a higher energy (Scenario (2c) and (2d) in Sec. 2.3): 1. First, as stated before, the Z 4,X is a good global symmetry read from the quantum numbers of SM particles and SM path integral kinematically. It is a global symmetry that has not yet been dynamically gauged in the G SMq nor in the SU(5) of the su (5) GUT.
2. The Z 4,X = Z(Spin (10)) sits at the center subgroup Z 4 of the Spin(10) for the so (10) GUT [14]. Thus the Z 4,X must be dynamically gauged, if the so (10) GUT is a correct path to unification at a higher energy.
It is natural to consider the following group embedding from the GUT to the SM [10][11][12]: • It is worthwhile mentioning that the Z 16 global anomaly (occurred in the cobordism group for the So the Z 4,X can be an anomalous symmetry in the SM q and the su (5) GUT, but the Z 4,X is an anomaly-free symmetry in the so (10) GUT. 15 • The 15n Weyl fermion SM q or the 15n Weyl fermion su(5) GUT alone may have a Z 16 global anomaly, while they can become anomaly-free at a higher-energy 16n Weyl fermion so (10) GUT [15]. This fact motivates Ref. [12] to propose an analogous concept of topological quantum phase transition happens between two energy scales: (1) above the energy scale of the SM q or the su (5)  3. Global symmetry must be gauged or broken in quantum gravity. If for the above reasons, we ask the Z 4,X -symmetry to be preserved, then the Z 4,X must be dynamically gauged at a higher energy for the sake of quantum gravity. 16 15 The anomalous symmetry means a non-onsite symmetry in the condensed matter terminology, that cannot be realized acting only locally on a 0-simplex (a point). The anomaly-free symmetry means an onsite symmetry in the condensed matter terminology, that acts only locally on a 0-simplex (a 0d point), which can be easily gauged by coupling to dynamical variables living on 1-simplices (1d line segments). 16 String theory landscape and swampland program develops the similar concepts of the use of cobordism for quantum gravity, see [71] and References therein. This can be understood as the deformation classes of quantum gravity. The deformation classes of quantum field theory is also proposed by Seiberg in [72]. In our context, we propose that the whole quantum system including the low energy SM plus additional hidden sectors, must correspond to the trivial group element 0 class in TP d=5 (Spin × Z F 2 Z4,X × GSM q ). Similarly, the whole quantum system including the su(5) GUT plus additional hidden sectors, must correspond to the trivial group element 0 class in TP d=5 (Spin × Z F 2 Z4,X × SU (5)).
• Anomaly matching: To take a step back, for a usual quantum field theory, one can try to match the index ν of 't Hooft anomaly of ultraviolet high energy (UV) with infrared low energy (IR). The index ν is a renormalization group (RG) flow invariant but possibly can be nonzero. This is the anomaly matching of the index ν between UV and IR theories.
• Anomaly cancellation: Here in contrast, in our case, we consider the whole quantum system (low energy and high energy) into account, due to our assumption that the Z4,X is preserved thus gauged at the quantum gravity scale, we must have the system anomaly matched to a trivial group element with the total index ν = 0 in the cobordism class (similar to [71]). We may also quote this cancellation as the anomaly matching to zero.
If we ignore the dynamical gravity, we can make some comments about the UV completion of Ultra Unification with a localtensor product Hilbert space (namely, as a regularized quantum lattice model): • If the Z4,X ⊇ Z F 2 is treated as a global internal symmetry (thus not dynamically gauged), the Ultra Unification requires a fermionic Hilbert space with local gauge-invariant fermionic operators such that the Z4,X ⊇ Z F 2 acts onsite in the 4d-5d coupled system. Namely, the system can be defined on a manifold with a fermionic Spin × Z F 2 Z4,X -structure with its Z4,X generator square X 2 = (−1) F as the fermion parity.
• If the Z4,X ⊇ Z F 2 is dynamically gauged at higher energy (as it should be), the Ultra Unification requires a bosonic Hilbert space with only local gauge-invariant bosonic operators in the 4d-5d coupled system, without any global symmetry. Namely, the system can be defined on a manifold with a bosonic SO-structure.
Overall, to have the Ultra Unification applied to our Universe's quantum vacuum, it is convenient to regard the Z4,X looks like a global symmetry (thus a fermionic Hilbert space) at a lower energy around SM scales, but the Z4,X is eventually dynamically gauged at higher energy (thus eventually a bosonic Hilbert space at the deep UV completion). Moreover, since the Spin × Z F 2 Z4,X gauge field is not an ordinary abelian discrete gauge field but with the extra constraint w2(T M ) = A 2 Z 2 , we may require to sum over the Spin × Z F 2 Z4,X -gauge bundle and Spin × Z F 2 Z4,X -gauge field AZ 4 altogether properly.

Topological Phase Sector and Topological Force
Topological Force and the gapped Topological Phase Sector have specific physical and mathematical meanings in our context. We should clarify what they are, and then what they are not in the next: 1. Topological Phase Sector: In Sec. 2.3 and (2.12), we propose a Topological Phase Sector beyond the Standard Model (BSM) includes an appropriate linear combination of the following theories (selecting the best scenario to fit into the HEP phenomenology): (a). 4d long-range entangled gapped topological phase with an energy gap (named the gap ∆ TQFT ) whose low energy physics is characterized by a 4d noninvertible topological quantum field theories. This 4d TQFT is a Schwarz type unitary TQFT (which is the 4d analog of the 3d Chern-Simons-Witten theories [31,32]).
• This 4d TQFT has a 't Hooft anomaly [21] of a global symmetry G. We named the anomaly index ν 4d for this 4d TQFT.
• Proper mathematical tools to study this 4d TQFT requires the category or higher category theories.
(b). 5d short-range entangled gapped topological phase with an energy gap whose low energy physics is characterized by a 5d invertible topological quantum field theory (iTQFT). This 5d iTQFT is also a unitary TQFT. But the iTQFT is nontrivial distinct from a trivial gapped vacuum only in the presence of a global symmetry G, see Footnote 2. A G-symmetric iTQFT is mathematically given by a G-cobordism invariant, classified by an appropriate cobordism group Ω d G ≡ TP d (G), defined in the Freed-Hopkins classification of invertible topological phases (TP) [20].
• The boundary of this 5d iTQFT has a 4d 't Hooft anomaly of a global symmetry G. We named the anomaly index ν 5d for this 5d iTQFT.
• Proper mathematical tools to study this 4d TQFT requires characteristic classes, cohomology, and cobordism theories.
(c). 5d long-range entangled gapped topological phase with an energy gap whose low energy physics is characterized by a 5d TQFT. This is the case when the discrete X symmetry is dynamically gauged, stated in Scenario (2c) and (2d) in Sec. 2.3.

Topological Force:
In the context of Sec. 2.3, Topological Force is a discrete gauge force mediated between the linked worldvolume trajectories (1d worldlines, 2d worldsheets from gapped extended operators) via fractional or categorical statistical interactions (See Sec. 5 and 6 of [11]).
• Bosonic finite group gauge theory: The conventional discrete gauge theories are bosonic types of finite group gauge theories [73,74]. Bosonic types mean that their ultraviolet (UV) completion only requires a local tensor product Hilbert space of local (gauge-invariant) bosonic operators; the UV completion does not require local (gauge-invariant) fermionic operators. The underlying TQFT does not require the spin structures and can be defined on non-Spin manifolds (such as the oriented SO structures). The TQFTs are known as bosonic or non-Spin TQFTs.
• Fermionic finite group gauge theory: The discrete gauge theories for our gapped Topological Phase Sectors for the beyond SM hidden sector are fermionic types of finite group gauge theories [52,69].
Fermionic types mean that their UV completion must require local (gauge-invariant) fermionic operators. The underlying TQFT requires the additional spin structures defined on Spin manifolds. The TQFTs are known as Spin TQFTs. In fact the 4d TQFT in Sec. 2.3 requires the Spin × Z F 2 Z 4,X structure and can be defined on the Spin × Z F 2 Z 4,X manifolds (including both Spin manifolds and some non-Spin manifolds).
We should emphasize that our Topological Phase Sector and Topological Force are not the kinds of Chern class topological terms which are already summed over in the continuous Lie group gauge theory. Namely, our Topological Phase Sector and Topological Force are not the followings: -The θ-term with or without a dynamical θ-axion [75,76], well-known as θF ∧ F or θFF in the particle physics, is in fact related to the second Chern class c 2 (V G ) and the square of the first Chern class c 1 (V G ) of the associated vector bundle of the gauge group G: (2.14) In particular, here we consider G as the U(N) or SU(N) gauge group, so we can define the Chern characteristic classes associated with complex vector bundles. The V G is the associated vector bundle of the principal G bundle. This θ-term is a topological term, but it is summed over as a weighted factor to define a Yang-Mills gauge theory partition function [38,77,78]. This θ-term does not define a quantum system or a quantum phase of matter by itself, distinct from our 4d TQFT (with intrinsic topological order) and 5d iTQFT (with SPTs) as certain unitary quantum phases of matter by themselves.
-The instantons [79,80] or the sphalerons [81], are also not the Topological Phase Sector and Topological Force in our context. Instantons and sphalerons are again the objects with nontrivial Chern class integrated over the spacetime manifold. Those objects are already defined as part of the SM and GUT continuous group gauge theories.
As we will mention in Sec. 3.1, we can also include (1) the θ-term with or without a dynamical θ-axion,

Ultra Unification Path Integral
In this section, we provide the functional path integral (i.e., partition function) Z UU of Ultra Unification, which includes the standard paradigm of the Standard Model path integral Z SM or the Georgi-Glashow su(5) GUT path integral Z GUT in Sec. 3.1. Then we provide the Topological Phase Sector TQFT path integral Z TQFT ≡ Z 5d-iTQFT · Z 4d-TQFT in Sec. 3. (3.1) The . . . depends on the details of which variant versions of SM that we look at (e.g., adding axions or not). In the schematic way, we have the action S: But more precisely we really need more details in the Lagrangian L with Weyl fermions, with S ≡´Ld 4 x: Here come some Remarks: 1. Yang-Mills gauge theory [82] has the action S YM =´Tr(F ∧ F) and Lagrangian L YM = − 1 4 F a µν F aµν . The F is the Lie algebra valued field strength curvature 2-form F = dA − igA ∧ A, with its Hodge dual F , all written in differential forms. In the trace "Tr" we pick up a Lie algebra representation R whose Lie algebra generators T a labeled by "a." We have also the subindex I = 1, 2, 3 to specify the SM Lie algebra sectors u(1), su(2), or su (3). 2 . The θ-term and dynamical θ-axion: We can also introduce the Chern class topological θ-term S θ-Chern = −´θ 8π 2 g 2 Tr(F ∧ F ) and L θ-Chern = − θ 64π 2 g 2 µνµ ν F a µν F a µ ν . Given a U(N) or SU(N) bundle V G and its field strength F , the first and second Chern classes are given by c 1 (V G ) = TrF 2π and c 2 ( . If a dynamical θ-axion [75,76] is introduced, it requires a summation of the compact θ in the path integral measure´[Dθ]. There is an overall constant that can be absorbed into the field A and coupling g redefinition. 18 The path integral´[DA] for continuous Lie group gauge field theory (here U(1), SU(N), U(N) for the SM and su(5) GUT), really means (1) the summation of all inequivalent principal gauge bundles P A , and then (2) the summation of all inequivalent gauge connectionsÃ (under a given specific principal gauge bundles P A ), whereÃ is a (hopefully globally defined physically) 1-form gauge connection. So we physically define: [DA] · · · ≡ gauge bundle P Aˆ[ DÃ] . . . . 18 For example, by redefining A → A = 1 g A and F → F = 1 g F , then F = dA − i A ∧ A and F a µν = ∂µA a ν − ∂ν A a µ + f bca A b µ A c ν . Then we can also write 2. Dirac fermion theory has S Dirac =´ψ(i / D A )ψ d 4 x and L Dirac =ψ(i / D)ψ with the Dirac spinor ψ defined as a section of the spinor bundles. Theψ(i / D)ψ is an inner product in the complex vector space with the Dirac operator / D A as a natural linear operator in the vector space. The path integral´[Dψ][Dψ] is (1) the summation of all inequivalent spinor bundles, and then (2) the summation of all inequivalent sections (as spinors) of spinor bundles (under a given specific spinor bundle). We requires the spin geometry and spin manifold, in particular we require the Spin × Z 2 Z 4 = Spin × Z F 2 Z 4,X structure.
In fact preferably we present not in the Dirac spinor basis, but we present all of (3.3) in the Weyl spinor basis (below).

Weyl fermion theory has
Weyl spinor bundle splits the representation of the Dirac spinor bundle. Weyl spinor again is defined as the section of Weyl spinor bundle. The Weyl spacetime spinor is in 2 L of Spin(1, 3) = SL(2, C) with a complex representation in the Lorentz signature, or 2 L of Spin(4) = SU(2) L × SU(2) R with a pseudoreal representation in the Euclidean signature. We also write the analogous right-handed Weyl fermion theory. The σ µ andσ µ are the standard spacetime spinor rotational su(2) Lie algebra generators. We will emphasize and illuminate the meanings of covariant derivative D µ,A,A Z 4 altogether in Remark 5.

Higgs theory has S Higgs
x with gauged kinetic and potential terms. The Higgs field bundle is typically a trivial complex line bundle. 19 The Higgs scalar field is the section of a field bundle. The electroweak Higgs is in complex value C and also in 2 of SU(2) gauge field. Again by doing summation´[Dφ] we (1) sum over the field bundles, and (2) sum over the section of each field bundle. We illuminate the meanings of covariant derivative D µ,A,A Z 4 altogether in Remark 5.

Covariant derivative operator D µ,A,A Z 4 in (3.3) is defined as:
Placed on a curved spacetime (with a non-dynamical metric, only with background gravity) requires a covariant derivative ∇ µ , and a spin connection for the spinors. Comments about the term g q R A in a differential form (e.g., quantum numbers read from Table 1 in [11]): The ς a and τ a are the rank-2 and rank-3 Lie algebra generator matrix representations for su(2) and su (3) respectively. The D µ,A,A Z 4 acting on ψ L contains the su(2) gauge field. The D µ,A,A Z 4 acting on ψ R does not contain the su(2) gauge field, because the su(2) weak interaction is a maximally parity violating chiral gauge theory. The D µ,A,A Z 4 acting only on quarks (both ψ L and ψ R ) contain the su(3) gauge field. The D µ,A,A Z 4 acting on φ contains the su(2) × u(1) gauge field.
Comments about the term q X A Z 4 (e.g., quantum numbers read from Table 1

and 2 in [11]):
• The D µ,A,A Z 4 acts on all left-handed SM Weyl fermion ψ L via its A Z 4 charge q X = 1.
• The D µ,A,A Z 4 acts on all right-handed SM Weyl fermion ψ R via its A Z 4 charge q X = −1.
• The D µ,A,A Z 4 acts on the electroweak Higgs φ via its A Z 4 charge q X = 2.
The subtle part is that A Z 4 should be treated as a cohomology class, such as a cohomology or cochain gauge field. The A Z 4 ∈ H 1 (M, Z 4 ) is the generator from H 1 (BZ 4,X , Z 4 ). In physics, for the continuum QFT theorists who prefer to think Z 4,X ⊂ U(1) X as a continuum gauge field breaking down to a discrete Z 4,X , we can introduce an extra Z 4 charge new Higgs field ϕ and its potential V(ϕ): There are extra superconductivity-like term φ 2 ϕ † + (φ † ) 2 ϕ does not break the Z 4 . Their Z 4 or U(1) transformations are: In the ϕ = 0 condensed Higgs phase as a discrete Z 4 gauge theory, which we can dualize the theory as a level-4 BF theory [83]. The formulation starts from adding´[Dϕ] in the path integral measure, and it ends with a 2-form B and 1-form gauge field A Z 4 But more precisely, we really should formulate in terms of a cohomology/cochain TQFT and taking care of the Spin × Z F 2 Z 4,X structure, which we will do in Sec. 3.2 (also in Sec. 5 of [11]).
6. Yukawa-Higgs-Dirac term has S Yukawa-Higgs-Dirac =´L Yukawa-Higgs-Dirac d 4 x =´(ψ † L φψ R + h.c.) d 4 x. In this case, we pair the ψ † L 's2 of SU(2) with the φ's 2 of SU(2), and vice versa pair ψ L with φ † to get an SU(2) singlet. The right-handed ψ R here is (meant to be) an SU (2) singlet. This Yukawa-Higgs-Dirac term at the kinetic level also preserves the Z 4,X , although the Higgs vacuum expectation value (vev) breaks the Z 4,X dynamically.

Yukawa-Higgs-Majorana term with Weyl fermion:
We can add Yukawa-Higgs-Majorana term for Weyl fermions. For example, for the left-handed ψ L , we can add a dimension-5 operator: Again the σ 2 is from the σ µ of the spacetime spinor rotational su(2) Lie algebra generators. Renormalizability is not an issue because we are concerned the effective field theory. For the right-handed ψ R , we can add a dimension-3 operator for some Majorana mass coupling M : x, which breaks the lepton number conservation. However, in either cases, both Yukawa-Higgs-Majorana terms above break the Z 4,X explicitly. So they are not encouragingly favored if we pursue the Z 4,Xpreserving theory at least at higher energy.
We have presented above the Standard Model coupled to a discrete X gauge field in the path integral (3.1), the action (3.2), and the Lagrangian (3.3). Below we can quickly modify a few terms to obtain the su(5) Grand Unification coupled to a discrete X gauge field.

The su(5) Grand Unification Path Integral coupled to X
We have the su(5) GUT path integral coupled to X: (3.8) We should write all5 and 10 of the SU(5) as the left-handed Weyl fermions ψ L , so there are 15 Weyl fermions ψ L per generation. In Sec. 2.3, we may or may not introduce the right-handed neutrinos here denoted as χ R . In the schematic way, we have the action: We are left now only with an SU(5) gauge field whose 1-form connection written as: We require the T a as the rank-5 and rank-10 Lie algebra generator matrix representations for su (5) to couple to5 and 10 of SU(5) respectively. Yukawa-Higgs pairs the appropriate ψ L and ψ L Weyl fermions. We may or may not introduce the Majorana mass terms to χ R in the . . . , while the consequences are already discussed (which break the Z 4,X explicitly) in the Remark 7. The discussions about this path integral (3.8) directly follow the above Remarks 1-7, so we should not repeat. • If the Z 4,X is only coupled to a background gauge field, this only means the system has 't Hooft anomaly under the Z 4,X anomalous symmetry and the spacetime (Spin group) coordinate reparametrization transformations (e.g., the Euclidean rotation or Lorentz boost part of diffeomorphism).

Topological
• If the Z 4,X is dynamically gauged and preserved at high energy, then we must append a new sector to make the whole theory well-defined.
1. The 5d iTQFT partition function is given by (2.8): 2. We propose the full gauge-invariant path integral, invariant under the mixed gauge-gravity transformation (i.e., gauge-diffeomorphism) of Spin × Z F 2 Z 4,X structure and free from its Z 16 global anomaly as follows. The SM version employs (3.1) into: (3.12) The GUT version employs (3.8) into: Below we ask whether we can construct a fully gauge-diffeomorphism invariant 5d-4d coupled partition function preserving the Spin × Z 2 Z 4 structure: (3.14) Preserving the Spin × Z 2 Z 4 structure means that under the spacetime coordinate background transformation (i.e., diffeomorphism) and the A Z 4 background gauge transformation, the 5d-4d coupled partition function is still fully gauge-diffeomorphism invariant.
First, we can rewrite the 5d iTQFT partition function (3.11) on a 5d manifold M 5 into for a generic ν = −N generation ∈ Z 16 . = Z 2 , whose quantum matter realization is the 1+1d Kitaev fermionic chain [46] whose each open end hosts a 0+1d Majorana zero mode.
• The (A Z 2 ) 5 is a mod 2 class purely bosonic topological invariant, which corresponds to a 5d bosonic SPT phase given by the group cohomology class data H 5 (BZ 2 , U(1)) = Z 2 , which is also one of the Z 2 generators in Ω SO 5 (BZ 2 ).
Since ν = −N generation , 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 Z 16 anomaly for one right-handed neutrino (ν = 1) or three right-handed neutrinos (ν = 3) alone.
with a 2d Arf-Brown-Kervaire (ABK) invariant which is also known as the Pin − -structure Z 8 -class of iTQFT of the 1+1d Fidkowski-Kitaev fermionic chain [89,90] with a time reversal T 2 = +1 symmetry. Notice that (3.16) can become trivialized if we can trivialize the (A Z 2 ) 3 factor. In fact, the (A Z 2 ) 3 can be trivialized by the symmetry extension [56], written in terms of the group extension of a short exact sequence: Namely, the 2-cocycle topological term , U(1)) becomes a coboundary once we lifting the Z 4,X Z F 2 -gauge field A Z 2 to a Z 4,X gauge field in H 3 (BZ 4,X , U(1)). So this suggests that the following symmetry extension for the spacetime-internal symmetry, written in terms of the group extension of a short exact sequence: 21 (3.17) can fully trivialize any even ν even ∈ Z 16 cobordism invariant given in (3.16). The [Z 2 ] means that we can gauge the anomaly-free normal subgroup [Z 2 ] in the total group Spin × Z 4,X . This symmetry extension (3.17) also means that a 4d [Z 2 ] gauge theory preserves the Spin × Z F 2 Z 4,X symmetry while also saturates the even ν even ∈ Z 16 anomaly. This 4d [Z 2 ] gauge theory is the anomalous symmetric gapped non-invertible TQFT (with 't Hooft anomaly of Spin × Z F 2 Z 4,X -symmetry) desired in the Scenario (2a).
Since a symmetric anomalous 4d TQFT only exists with even ν even ∈ Z 16 , below we formulate the path integral Z of the root phase ν 4d = 2. We generalize the boundary TQFT construction in the Section 8 of [52]. With ν even = 2 ∈ Z 16 , we have (3.14) with the input of 5d bulk iTQFT (3.16), then we can explicitly construct the partition function on a 5d manifold M 5 with a 4d boundary M 4 ≡ ∂M 5 as, 22 2). More precisely, for a Spin × Z 2 Z 4 manifold M 5 with a boundary, we have used the Poincaré-Lefschetz duality for a manifold with boundaries:

19)
21 See more discussions in Sec. 5 of [11], and in [87]. 22 We use the notation for the cup product between cohomology classes, or between a cohomology class and a fermionic topological invariant (paired via a Poincaré dual PD). We often make the cup product and the Poincaré dual PD implicit. 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.
3). For any pair (S, S ), where S is a subspace of S, the short exact sequence of chain complexes 0 → C * (S ) → C * (S) → C * (S, S ) → 0, (3.20) with C n (S, S ) ≡ C n (S)/C n (S ), induces a long exact sequence of homology groups Here H n (S, S ) is the relative homology group, and ∂ is the boundary map.
• Take (S, S ) = (M 4 = ∂M 5 , ∂PD(A 3 )), we denote another boundary map by ∂ 1 : This is a satisfactory consistent check, consistent with the 5d bulk-only iTQFT at ν even = 2 in (3.16). 7). The a ∈ C 1 (M 4 , Z 2 ) means that a is a 1-cochain, and the b ∈ C 2 (M 4 , Z 2 ) means that b is a 2-cochain. The factor (−1)´M 4 a(δb+A 3 ) = exp(iπ´M 4 a(δb + A 3 )) gives the weight of the 4d Z 2 gauge theory. The aδb term is the level-2 BF theory written in the mod 2 class. 23 The δb is a coboundary operator δ acting on b. The path integral sums over these distinct cochain classes. 23 The continuum QFT version of this Z2 gauge theory is exp( i´M 4 2 2π a db + 1 π 3 aA 3 )), where the b 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. Another alternative possibility of 4d TQFT of (3.18) can be The variation of a gives the equation of motion (δb+A 3 ) = 0 mod 2. In the path integral, we can integrate out a to give the same constraint (δb + A 3 ) = 0 mod 2. This is precisely the trivialization of the second cohomology class, so the 3-cocycle becomes a 3-coboundary which splits to a 2-cochain b. This exactly matches the condition imposed by the symmetry extension (3.17):  (3.29) 24 We do not yet know whether it is always possible to induce a unique 2d Pin − structure on (c ∪ PD(A 3 )) for any possible pair of data (M 5 , M 4 = ∂M 5 ) given any M 5 with Spin ×Z 2 Z4 structure. However, we claim that it is possible to find some suitable M 4 so that the 2d (c ∪ PD(A 3 )) has Pin − induced, thus in this sense the ABK(c ∪ PD(A 3 )) is defined. For physics purposes, it is enough that we can firstly focus on studying the theory on these types of (M 5 , M 4 = ∂M 5 ). 25 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 (3.22). 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 ∂ is associated with the operations on objects living on the boundary M 4 alone.
• The ∂1 is defined as another boundary map in (3.23).
• As before, the union (c ∪ PD (b)) is a closed 2-surface and we induce a Pin − structure on this 2-surface. So we can compute the ABK on this closed 2-surface (c ∪ PD (b)) on the M 4 . 26 In summary, we have constructed the 4d fermionic discrete gauge theory in (3.18) preserving the (Spin × Z F 2 Z 4,X )-structure, namely it is a (Spin × Z F 2 Z 4,X )-symmetric TQFT but with ν even = 2 ∈ Z 16 anomaly. This can be used to compensate the anomaly ν = −N generation mod 16 with N generation = 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 N generation = 1 or 3), due to the obstruction found in [51,88].
Now we have derived an Ultra Unification path integral in (3.30) and (3.31), including 4d SM (3.1), 4d GUT (3.8), 5d iTQFT (3.15) and 4d TQFT (3.18), comprising many Scenarios and their linear combinations enlisted in Sec. 2.3: (1a), (1b), (1c), (2a), and (2b). Then we can dynamically gauge the appropriate bulkboundary global symmetries, promoting the theory to a bulk gauge theory in Scenarios (2c) and (2d), or break some of the (global or gauge) symmetries to (2e). 26 Similar to Footnote 24, we can find some suitable M 4 so that the 2d Pin − is induced, thus in this sense the ABK(c∪PD (b)) is defined. 27 The nonperturbative global anomaly cancellation constraint (−(Ngen = 3) + nν e,R + nν µ,R + nν τ,R + ν 4d,even − ν 5d ) = 0 mod 16 provides the capacity for many kinds of the Ultra Unification model building. For example, the 4d Z4,X -symmetry preserving TQFT sector can take the index ν 4d,even = 0, 2, 4, . . . for any even integer. There are many (perhaps countably infinite) types of TQFTs for each index ν 4d,even . But to be more economic, we can ask for the minimum degrees of freedom required by a TQFT for any given index ν 4d,even . Also for HEP phenomenological purposes, by taking account of the experimentally observed neutrino mass eigenstates splitting: • if one use the conventional quadratic mass mechanism to generate the observed neutrino masses, one may propose to have at least two generations of right-handed neutrinos, which means that a possible phenomenological input nν e,R +nν µ,R +nν τ,R ≥ 2. A viable Ultra Unification candidate can be, for example, nν e,R +nν µ,R +nν τ,R = 2, ν 4d,even = 2, and ν 5d = 1, which saturates the anomaly cancellation and some phenomenological constraints.
• if we use the interacting topological mass ∆TQFT and its topological defect energy subgap ∆ sub to account for the observed neutrino masses (see Sec. 3.5 and Fig. 9 for illustrations), then we may have different choices of the number of right-handed neutrinos, written as ( j=e,µ,τ,... nν j,R ) in (2.9). So far, we mainly use the cobordism theory to study the invertible anomalies and invertible topological field theories, and we also use the cohomology data to construct non-invertible topological quantum field theories. However, once the discrete symmetries (such as Z4,X ) are dynamically gauged, it is more natural to use the mathematical category or higher category theories to characterize the Topological Phase Sectors.

General Principle
In summary, we propose a general principle behind the Ultra Unification: 1. We start with a QFT in general as an effective field theory (EFT) given some full spacetime-internal symmetry G, say in a D dimensional (Dd) spacetime. 2. We check the anomaly and cobordism constraint given by G via computing Ω D+1 G ≡ TP D+1 (G). 3. We check the anomaly index of the Dd QFT/EFT constrained by cobordism Ω D+1 G ≡ TP D+1 (G). 4. If all anomalies are canceled, then we do not require any new hidden sector to define Dd QFT/EFT. 5. If some anomalies are not canceled, we have two perspectives, either regarding the G-symmetry as a global symmetry with 't Hooft anomaly at a lower energy; or regarding the G-symmetry and its anomaly still persist at a higher energy (thus we can further assume the G symmetry is dynamically gauged at a higher energy due to "no global symmetry in quantum gravity reasonings"), then either (1) we need to break some symmetry out of G to eliminate the anomaly, or (2) we extend the symmetry G to an appropriateG to trivialize nonperturbative global anomalies in Ω D+1 G ≡ TP D+1 (G), or (3) we propose new hidden sectors appending to Dd QFT/EFT, with a schematic path integral (say if we add Dd-TQFT or CFT and (D+1)d-iTQFT onto the original theory): (3.33) Here A is a generic G-symmetry background field, that is also to be dynamically gauged at a higher energy.

Detect Topological Phase Sectors and the Essence of Ultra Unification
Detect Topological Forces By looking at Fig. 1, in the Higgs vacuum where our Standard Model effective field theory (SM EFT) resides in, we have only detected the Strong, Electromagnetic, and Weak in the subatomic physics. The GUT forces are weaker than the Weak force, and the Topological Force is further weaker than the GUT and Weak forces. So how could we experimentally detect Topological Forces?
Notice that the gravity is further weaker than all other forces. (So how could we experimentally detect gravity?) But the gravity has accumulative effects that only have the gravitational attractions. Without doubt, the gravity has been detected by everyone and by all astrophysics and cosmology observations. The gravity had been detected first in the human history among all the forces! Similarly, although Topological Force is also weak (but stronger than the gravity), Topological Force is infinite-range or long-range which does not decay in the long distance, and mediates between the linked worldline/worldsheet/worldvolume trajectories of the charged (point-like or extended) objects via fractional or categorical anyonic statistical interactions. So in principle, we may have already experienced Topological Force in our daily life, in a previously scientifically unnoticed way. 28 28 For example, if the Z4,X is dynamically gauged, there is a dynamical discrete gauge Wilson line connecting all SM fermions living in 4d SM or GUT (e.g., quarks and electrons in our body). Namely, the SM fermions can live at the open ends of the Z4,X gauged Wilson line. Thus there could be long-distance topological interactions and communications between the Z4,X -gauge charged objects.
Moreover, the Z4,X gauged Wilson line as a AZ 4 gauge field on the 4d theory can be leaked into the 5d bulk theory as a AZ 2 gauge field. In the 5d bulk, a nontrivial link configuration can be charged under the other end of AZ 2 [11].
It is worthwhile to emphasize that the new sectors that we propose (the 4d gapped phase with TQFT, the 4d gapless phase with CFT, and the 5d gapped phase with iTQFT or TQFT) are invisible to SM gauge forces (su(3) × su(2) × u(1)) also invisible to the su(5) GUT forces. However, these new sectors are detectable via Topological Forces (i.e., statistical interaction via discrete gauge forces). These new sectors are also detectable via the gravitations.

Neutrino Oscillations and Dark Matter
In fact, Ref. [11] had proposed that the Topological Force may cause (thus be detected by) the phenomena of neutrino oscillations. We can consider the Majorana zero modes of the vortices in the 4d TQFT defects. The left-handed neutrinos (confirmed by the experiments) are nearly gapless/massless. When the lefthanded neutrinos traveling through the 4d TQFT defects, we may observe nearly gapless neutrino flavor oscillations interfering with the Majorana zero modes trapped by the vortices/vortex strings/monopoles in the 4d TQFT defects (see Sec. 3.5 and Fig. 9 for illustrations). On the other hand, the gapped heavy excitations (point or extended objects) of Topological Phase Sector may be a significant contribution to Dark Matter [91].
If the TQFT energy gap ∆ TQFT is large, the heavy excitations above the ∆ TQFT may contribute the heavy Dark Matter . In contrast, if the ∆ TQFT is small compared to the SM's particle masses, or if the new hidden sector contains CFT, then the new sector contributes the light Dark Matter .

The Essence of Ultra Unification
Finally, we come to the essence of Ultra Unification. What is unified after all? We have united the Strong, Electromagnetic, Weak, GUT forces, and Topological forces into the same theory in the Ultra Unification QFT/TQFT path integral (that this theory can also be coupled to the curved spacetime geometry and gravity, at least well-defined in a background non-dynamical way). See Fig. 2,  Fig. 3, Fig. 4, and Fig. 5 for illustrations.
However, the Grand Unification [8,9] united the three gauge interactions of the SM into a single electronuclear force under a simple Lie group gauge theory. Do we have any equivalent statement to also unite Strong, Electromagnetic, Weak, GUT forces and Topological forces into a single force at a high enough energy? We believe that the definite answer relies on studying the details of analogous topological quantum phase transitions [12,91] and the parent effective field theory that describes the phase transition and neighborhood phases, such as those explored in 4d [70,[92][93][94][95]. The underlying mathematical structure suggests a 4d version of particle-vortex duality, S-duality, T-duality, or mirror symmetry. See Fig. 6, Fig. 7, and Fig. 8 for illustrations.

Summary of Ultra Unification and Quantum Matter in Drawings
Let us summarize what we have done in this work in drawings.
• Fig. 2: We have started from the Nature given Standard Model (SM) quarks and leptons, and their quantum numbers (see Tables in [11]), in three generations.
• Fig. 3: We have included gauge forces and various Higgs for SM and Grand Unification (GUT).
• Fig. 4: After the essential check of the anomaly and cobordism constraints, the detection of the Z 16 global anomaly for 15n Weyl fermion SM and GUT implies that we can choose (as one of many options) to realize our 4d world living on an extra-dimensional 5d invertible TQFT (a 5d topological superconductor, mathematically a 5d cobordism invariant). Here Fig. 4 may be understood as a one-brane 4d world with an extra-large fifth dimension.
In a metaphysics sense, perhaps the long-distance Topological Force might be related to some of the unexplained mysterious phenomena. In any case, every phenomenon and every law of Nature should be explained by mathematics and physics principles.
• Fig. 5: Ultra Unification (UU) incorporates the SM, GUT, and Topological forces into the same theory (that this theory can also be coupled to curved spacetime geometry and gravity in a background non-dynamical way). (1) The upper left has the 4d SM and GUT. (2) The upper right has the 4d noninvertible TQFT. (3) The bottom has the 5d invertible TQFT (alternatively 5d non-invertible TQFT if we dynamically gauge the full Z 4,X ). The three sectors can communicate with each other mediated via the dynamical Z 4,X gauge forces. Here Fig. 5 may be understood as a multi-branes or two-brane 4d world with an extra fifth dimension. The issues of mirror fermion doubling [96] on the mirror world, depending on the precise anomaly index on the mirror sector, may be fully trivially gapped (if anomaly-free), may contain a mirror chiral gauge theory or unparticle conformal field theory, or may be topological order gapped with a low energy TQFT. The issues of gapping mirror fermions are tackled in the past starting from Eichten-Preskill [97] and by many recent works [15,[98][99][100][101][102][103][104][105][106][107][108].
• Fig. 6: In Quantum Matter terminology, we show that SM and GUT belong to a framework of a continuous gauge field theory, Anderson-Higgs (global or gauge) symmetry-breaking mass, and Ginzburg-Landau paradigm. In contrast, the new sectors that we introduce are beyond Ginzburg-Landau paradigm. The new sectors include a fermionic discrete gauge theory, symmetry-extension topological mass, and modern issues on symmetry, topology, nonperturbative interactions, and short/long-range entanglements.
• Fig. 7 and Fig. 8: In general, our SM vacuum (with possible BSM corrections that we denote SM * ) may live in a landscape of quantum vacua (e.g., see recent works [109,110]). Possible quantum vacua tuning parameters may be the GUT-Higgs potential or other QFT parameters that can induce quantum phase transitions from SM to neighbor GUT vacua in an SM deformation class [72,111]. UU may not only be just a higher-energy effective field theory of SM * , but also provide a parent effective field theory to go between SM, SM * , or different GUT vacua, from the left-hand sided 15n-Weyl-fermion model plus a 4d TQFT, a 4d CFT, or a 5d iTQFT, to the right-hand sided 16n-Weyl-fermion model in 4d, via a topological quantum phase transition through an energy-gap-closing gapless quantum critical region.
• Fig. 9: UU provides alternative new ways to explain the mass of the neutrinos. The traditional seesaw mechanism argues that the right-handed sterile neutrinos have a Majorana mass m M , while the lefthanded paired with right-handed neutrinos get a Dirac mass m D ; the mass eigenstate allows a small mass m 2 D m M m D ; these quadratic masses break the Z 4,X symmetry. In contrast, UU replaces some of sterile neutrinos with a TQFT sector within the same anomaly index. TQFT energy gap ∆ TQFT is the topological mass gap. The ∆ TQFT may sit at any energy scale below GUT or Planck scale M GUT,Pl . The symmetric TQFT can preserve Z 4,X but still give a topological mass; the TQFT may also have Z 4,X -topological defects which trap the zero modes. (Proliferating the topological defects restores the Z 4,X -symmetry driving to the symmetric TQFT phase.) The nearly gapless lefthanded neutrinos (ν e,L , ν µ,L , ν τ,L ) travel in waves and interact with TQFT defect's zero modes. Quantum interference between left-handed neutrinos and zero modes possibly causes neutrino oscillations. The energy spectrum near the defect (e.g., vortex points/strings) has some energy subgap ∆ sub      3) The bottom has the 5d invertible TQFT (alternatively 5d non-invertible TQFT if we dynamically gauge the Z 4,X or Z 4,X /Z F 2 ). The three sectors can communicate with each other mediated via the dynamical discrete gauge forces. Figure 6: Ultra Unification summarized in terms of Quantum Matter terminology. We can trade some even number of right-handed neutrinos (the 16th Weyl fermions, possibly with symmetry-breaking masses) for the symmetry-extension topological order sector in 3+1d, via a topological quantum phase transition. Once the Z 4,X is dynamically gauged (e.g., at higher energy), the 3+1d theory and 4+1d bulk all are coupled and correlated with each other via the topological Z 4,X -gauge force. In a colloquial sense, our Standard Model world may live with the neighbors of 3+1d intrinsic topological order or 3+1d unparticle CFTs, and also live on the boundary of some medium of 4+1d topological quantum computer.

15n
Weyl fermions in 4d + 4d TQFT or CFT, or 5d iTQFT, with interactions · · · · · · · · · · · · · · · Topological Quantum Phase Transition Quantum Critical Region Quantum Vacua tuning parameters · · · · · · · · · · · · · · · · 16n Weyl fermions in 4d with interactions   [109,110]). Possible quantum vacua tuning parameters may be the GUT-Higgs potential or other QFT deformation parameters that can induce quantum phase transition from SM to neighbor GUT vacua in an SM deformation class [72,111]. In Ref. [109,110]'s viewpoint, Ultra Unification may not only be just a higher-energy effective field theory of SM * , but also provide a parent effective field theory to go between SM, SM * , or different GUT vacua, from the left-hand sided 15n-Weyl-fermion model plus 4d TQFT/CFT or 5d iTQFT, to the right-hand sided 16n-Weyl-fermion model in 4d, tuning through a gapless topological quantum critical region (the schematic gray region).
Standard Model's "nearly massless" left-handed neutrinos ν j,L travel and interact with topological defects ν e,L ν µ,L ν τ,L zero energy modes trapped at a topological defect Ultra Unification replaces some generation of "right-handed sterile neutrinos ν j,R " with a Z 4,X -symmetry-preserving TQFT (or CFT) sector which may allow Z 4,X -topological defects locally  Figure 9: Ultra Unification (UU) also provides alternative new possible ways to explain the mass of the neutrinos. The traditional seesaw mechanism argues that the right-handed sterile neutrinos have a Majorana mass m M , while the left-handed paired with right-handed neutrinos get a Dirac mass m D ; the mass eigenstate allows a small mass m 2 D m M m D ; these quadratic masses break the Z 4,X symmetry. In contrast, UU replaces some of sterile neutrinos to a TQFT sector with the same anomaly index, which can have a topological mass of TQFT energy gap ∆ TQFT . The ∆ TQFT may sit at any energy scale below GUT or Planck scale M GUT,Pl . The symmetric TQFT can preserve Z 4,X but still give a topological mass; the TQFT may also have Z 4,X -topological defects which trap the zero modes. (Proliferating the topological defects restores the Z 4,X -symmetry driving to the symmetric TQFT phase.) The nearly gapless left-handed neutrinos (ν e,L , ν µ,L , ν τ,L ) travel in waves and interact with TQFT defect's zero modes, which quantum interference possibly causes neutrino oscillations. The energy spectrum near the defect (such as vortex points/strings) has energy subgap ∆ sub