Beyond Standard Models and Grand Unifications: anomalies, topological terms, and dynamical constraints via cobordisms

We classify and characterize fully all invertible anomalies and all allowed topo- logical terms related to various Standard Models (SM), Grand Unified Theories (GUT), and Beyond Standard Model (BSM) physics. By all anomalies, we mean the inclusion of (1) perturbative local anomalies captured by perturbative Feynman diagram loop calculations, classified by ℤ free classes, and (2) nonperturbative global anomalies, classified by finite group ℤN torsion classes. Our work built from [31] fuses the math tools of Adams spectral sequence, Thom-Madsen-Tillmann spectra, and Freed-Hopkins theorem. For example, we compute bordism groups ΩdG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\Omega}_d^G $$\end{document} and their invertible topological field theory invariants, which characterize dd topological terms and (d − 1)d anomalies, protected by the following symmetry group G: Spin×SU3×SU2×U1ℤq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{Spin}\times \frac{\mathrm{SU}(3)\times \mathrm{SU}(2)\times \mathrm{U}(1)}{{\mathrm{\mathbb{Z}}}_q} $$\end{document} for SM with q = 1, 2, 3, 6; Spin×Spinnℤ2F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{\mathrm{Spin}\times \mathrm{Spin}(n)}{{\mathrm{\mathbb{Z}}}_2^F} $$\end{document} or Spin × Spin(n) for SO(10) or SO(18) GUT as n = 10, 18; Spin × SU(n) for Georgi-Glashow SU(5) GUT as n=5;Spin×SU4×SU2×SU2ℤq′ℤ2F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ n=5;\frac{\mathrm{Spin}\times \frac{\mathrm{SU}(4)\times \left(\mathrm{SU}(2)\times \mathrm{SU}(2)\right)}{{\mathrm{\mathbb{Z}}}_{q^{\prime }}}}{{\mathrm{\mathbb{Z}}}_2^F} $$\end{document} for Pati-Salam GUT as q′ = 1, 2; and others. For SM with an extra discrete symmetry, we obtain new anomaly matching conditions of ℤ16, ℤ4 and ℤ2 classes beyond the familiar Witten anomaly. Our approach offers an alternative view of all anomaly matching conditions built from the lower-energy (B)SM or GUT, in contrast to high-energy Quantum Gravity or String Theory Landscape v.s. Swampland program, as bottom-up/top-down complements. Symmetries and anomalies provide constraints of kinematics, we further suggest constraints of quantum gauge dynamics, and new predictions of possible extended defects/excitations plus hidden BSM non-perturbative topological sectors.


Physics guide
The world where we reside, to our present knowledge, can be described by quantum theory, gravity theory, and the underlying long-range entanglement. Quantum field theory (QFT), specifically gauge field theory, under the name of Gauge Principle following Maxwell, Hilbert, Weyl [1], Pauli, and others, forms a cornerstone of the fundamental physics. Yang-Mills (YM) gauge theory [2], generalizing the U(1) abelian gauge group to a non-abelian Lie group, has been proven theoretically and experimentally essential to describe the Standard Model (SM) physics [3][4][5].
The SM of particle physics is a gauge theory encoding three of the four known fundamental forces or interactions (the electromagnetic, weak, and strong forces, but without gravity) in the universe. The SM also classifies all experimentally known elementary particles: fermions including three generations of quarks and leptons, while bosons including the electromagnetic force mediator photon γ, the strong force mediator gluon g, the weak force mediator W ± and Z 0 gauge bosons, and Higgs particle; while the graviton has not yet been detected and is not in SM. Physics experiments have confirmed that at a higher energy of SM, the electromagnetic and weak forces are unified into an electroweak interaction sector. Grand Unifications and Grand Unified Theories (GUT) predict that at further higher energy, the strong and the electroweak interactions will be unified into an electroweaknuclear GUT interaction sector. The GUT interaction is characterized by one larger gauge group and its force carrier mediator gauge bosons with a single unified coupling constant. 1 1 Unifying gravity with the GUT interaction gives rise to a Theory of Everything (TOE). However, in our present work, the gravity only plays the role of the background probed fields instead of dynamical gravity. As we will classify and characterize, the background probed gravity also gives new constraints, such as in the gravitational anomaly or the mixed gauge-gravitational anomaly. We however will comment the implications for dynamical gravity such as in Quantum Gravity in section 8.
In our present work, we aim to classify and characterize fully all (invertible) anomalies and all allowed topological terms associated with various Standard Models (SM), Grand Unified Theories (GUT), and Beyond Standard Model physics (BSM) in 4d. 2 Then we will suggest the dynamical constraints on SM, GUT and BSM via non-perturbative statements based on anomalies and topological terms.
By "anomalies" of a theory in physics terminology, physicists may mean one of the following: (1): Classical global symmetry is violated in a quantum theory, such that the classical global symmetry fails to be a quantum global symmetry, e.g. Adler-Bell-Jackiw anomaly [9,10].
(2): Quantum global symmetry is well-defined kinematically. However, there is an obstruction known as '''t Hooft anomaly [11]," to gauge the global symmetry, detectable via coupling the charge operator (i.e., symmetry generators or symmetry defects, which measures the global symmetry charge of charged objects) to background fields. 3 Specifically, we may detect an obstruction to even weakly gauge the symmetry or couple the symmetry to a non-dynamical background probed field (sometimes as background gauge field/connection). '''t Hooft anomaly [11]," is sometimes regarded as a "background gauged anomaly" in condensed matter. Namely, the path integral or partition function Z does not sum over background gauge fields. We only fix a background gauge field and the Z only depends on the background gauge connection as a classical field or as a classical coupling constant.
(3): Quantum global symmetry is well-defined kinematically. However, once we promote the global symmetry to a dynamical local gauge symmetry of the dynamical gauge theory, then the gauge theory becomes ill-defined. Some people call this as a "dynamical gauge anomaly" prohibiting a quantum theory to be well-defined. Namely, the path integral after summing over dynamical gauge fields becomes ill-defined. Therefore, the anomaly-free or anomaly-matching conditions are crucial to avoid the sickness and ill-defineness of quantum gauge theory.
In fact, it is obvious to observe that the anomalies from (3) are descendants of anomalies from (2).
(β). Anomalies from (2) can be related to anomalies from (3) via the ungauging principle. 2 We denote dd means the d spacetime dimensions. The d+1D means the d spatial and 1 time dimensions.
TheDD means theD space dimensions. 3 Throughout our article, we explicitly or implicitly use the modern language of symmetries and higher symmetries of QFTs, introduced in [12].

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Thus our key idea is that if we know the gauge group of a gauge theory (e.g., SM, GUT or BSM), we may identify its ungauged global symmetry group as an internal symmetry group, say G internal via ungauging. 4 To start, we should rewrite the global symmetries of an ungauging theory into the form of where the G spacetime is the spacetime symmetry, the G internal is the internal symmetry, 5 the is a semi-direct product from a "twisted" extension, 6 and the N shared is the shared common normal subgroup symmetry between G spacetime and G internal .
In the later sections of our work, we write down the ungauged global symmetry groups G of SMs, GUTs and BSMs. Then we should determine, classify and characterize all of their associated (invertible) anomalies and topological terms. Moreover, based on the descendant relations between the anomalies from (2) and (3), and the gauging/ungauging principles relate (α) and (β), we thus also determine both: (A). For the ungauged SM, GUT and BSM theories: Invertible 't Hooft anomalies and background probed topological terms associated to a global symmetry group G. This is related to a relation (β).
(B). For the gauged SM, GUT and BSM theories: Dynamical gauge anomalies and dynamical topological terms associated to a gauge group G, descent via the relation (α).

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Therefore, by determining all (d−1)d anomalies, we also determine all dd topological terms, and vice versa. 8 [II]. What tools are we using to classify and characterize all anomalies and all topological terms?
The short answer is based on the Freed-Hopkin's theorem [19] and our prior work [31][32][33][34]. 9 • topological invariants vs. geometric invariants: We say that a cobordism invariant is topological if it can be defined purely using topological data, such as a cohomology class. While we say that a cobordism invariant is geometric if it can be defined purely using geometric data like metrics, connections, and curvatures. These two definitions have no conflict, a cobordism invariant can be both topological and geometric.
• topological invariants vs. topological terms vs. iTQFTs: In physics, loosely speaking, for co/bordism invariants of a cobordism theory, people sometimes interchangeably use topological invariants, topological terms, and iTQFTs for the same thing. The G-iTQFTs with a global symmetry G (obtained from the cobordism invariants of cobordism group for manifolds with a G-structure) describes the low energy physics of the short-range entangle SPTs in condensed matter. However, in the general context, topological invariants and topological terms may not need to be invertible. 8 There are however a disclaimer and some caveats: (a) By all anomalies and all topological terms, their classifications and characterizations depend on the category of manifolds that can detect them. The categories of manifolds can be: TOP (topological manifolds), PL (piecewise linear manifolds), or DIFF (differentiable thus equivalently smooth manifolds), etc. These categories are different, and they are related by TOP ⊇ PL ⊇ DIFF.
Since the SM, GUT and BSM are given by continuum QFT data, in this work, we only focus on the DIFF manifolds and their associated all possible anomalies and topological terms. However, if we refine the data of QFT later in the future to include PL or TOP data from PL or TOP manifolds, we may also need to refine the corresponding SM, GUT and BSM. Thus, we will have a new set of so-called all anomalies and all topological terms. The tools we use in either case would be a certain version of cobordism theory suitable for a specific category of manifolds. See more in [27].
(b) By anomalies and topological terms for some SM, GUT and BSM theories in our work, we either mean (A). Invertible 't Hooft anomalies and background probed topological terms for the ungauged SM, GUT and BSM, or (B). Dynamical gauge anomalies and dynamical topological terms for the gauged SM, GUT and BSM.
But after gauging G-symmetry of (A), for the gauged SM, GUT and BSM in (B), there could be additional new higher 't Hooft anomalies associated to the higher symmetries (depending on the group representations of the matter fields) whose charged objects are dynamical extend objects (e.g. 1-lines, 2-surfaces, etc.). In this present work, we do not discuss these additionally gained new higher 't Hooft anomalies after dynamically gauging, but will leave them for future work [27]. Examples of such higher 't Hooft anomalies can be found in [16-18, 28, 29] and references therein.
(c) Another possible loop hole is that we do only focus on invertible anomalies captured by invertible topological quantum field theories (iTQFTs), we do not study non-invertible anomalies (e.g. [30]). Different experts and different research fields may regard and define anomalies in different ways.

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The long answer is follows. Based on the Freed-Hopkin's theorem [19] and an extended generalization that we propose [31,34], there exists a one-to-one correspondence between "the invertible topological quantum field theories (iTQFTs) with symmetry (including higher symmetries or generalized global symmetries [12])" and "a cobordism group." In condensed matter physics, this means that there is a relation from iTQFT to "the symmetric invertible topological order (iTO, see a review [21]) with symmetry (including higher symmetries) or symmetry-protected topological state (SPTs, see [21,[36][37][38]) that can be regularized on a lattice in its own dimensions." More precisely, it is a one-to-one correspondence (isomorphism " ∼ =") between the following two well-defined "mathematical objects" (which turn out to be abelian groups): Deformation classes of the reflection positive invertible d-dimensional extended topological field theories (iTQFT) with symmetry group We shall explain the notation above: M T G is the Madsen-Tillmann-Thom spectrum [39] of the group G, Σ is the suspension, IZ is the Anderson dual spectrum, and tors means the torsion group by taking only the finite group sector. The right hand side is the torsion subgroup of homotopy classes of maps from a Madsen-Tillmann-Thom spectrum (M T G) to a suspension shift (Σ d+1 ) of the Anderson dual to the sphere spectrum (IZ).
In other words, we classify the deformation classes of symmetric iTQFTs and also symmetric invertible topological orders (iTOs), via this particular cobordism group defined as follows by classifying the cobordant relations of smooth, differentiable and triangulable manifolds with a stable G-structure, via associating them to the homotopy groups of Thom-Madsen-Tillmann spectra [39,40], given by a theorem in ref. [19]. Ref. [19] introduced TP which means the abbreviation of "Topological Phases" classifying the above symmetric iTQFT, where our notations follow [19] and [31]. (For an introduction of the mathematical background and mathematical notations explained for physicists, the readers can consult the appendix A of [35] or [31].) Now let us pause for a moment to trace back some recent history of relating these anomalies/topological terms to a cobordism theory. The d dimensional 't Hooft
Recently, the iTQFTs and SPTs are found to be systematically classified by a powerful cobordism theory of Freed-Hopkins [19], following the earlier framework of Thom-Madsen-Tillmann spectra [39,40].
A new ingredient in our work [31,34] is a generalization of the calculations and the cobordism theory of Freed-Hopkins [19] involving higher symmetries: instead of the ordinary group G or ordinary classifying space BG, we consider a generalized cobordism theory studying spacetime manifolds endorsed with G spacetime structure, with an additional higher group G (i.e., generalized as principal-G bundles) and higher classifying spaces BG. 12 [III]. What do we mean by classifications and characterizations?
• By classification, we mean that given certain physics theories or phenomena (here, higher-iTQFT and higher quantum anomalies), given a spacetime dimensions (here d + 1d for higher-iTQFT or dd for higher quantum anomalies), and their spacetime G spacetime -structure and the internal higher global symmetry G internal , we compute how many classes (a number to count them) there are? Also, we aim to determine the mathematical structures of classes (i.e. here group structure as for (co)bordism groups: would the classes be a finite group Z N or an infinite group Z or others, etc.).
• By characterization, we mean that we formulate their mathematical invariants (here, we mean the bordism invariants) to fully describe or capture their mathematical essences and physics properties. Hopefully, one may further compute their physical observables from mathematical invariants. 10 We abbreviate both Symmetry-Protected Topological state and Symmetry-Protected Topological states as SPTs. We also abbreviate both Symmetry-Enriched Topologically ordered state and Symmetry-Enriched Topologically ordered states as SETs. 11 For example, the interacting versions of 10 Cartan symmetry classes of fermionic superconductors/insulators classifications (e.g. [47] in 4d or 3+1D) from condensed matter can be captured precisely by bordism invariants as invertible TQFTs [35]. 12 Although most of our results in this article focus on the ordinary symmetry group, our framework does allow us to consider possible higher symmetries and higher anomalies [31,34].

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Since some of readers are still with us reading this sentence (after we answer the three questions [I], [II] and [III]), we believe that these readers decide to be interested in understanding our results.
Here we concern theories of 4d SMs, GUTs and BSMs and their anomalies and topological terms. Their 4d 't Hooft anomalies captured by 5d iTQFTs. These 5d iTQFTs or bordism invariants are defined on the dd manifolds (d = 5). In fact, in our work, we present all (d − 1)d 't Hooft anomalies captured by dd iTQFTs, for d = 1, 2, 3, 4, 5, associated with various SMs, GUTs and BSM (ungauged) symmetries. The manifold generators for the bordism groups are actually the closed dd manifolds. We should clarify that although there are 't Hooft anomalies for (d − 1)d QFTs (so G internal may not be gauge-able on the boundary), the SPTs/topological invariants defined in the closed dd actually have G internal always gauge-able in that dd. This is related to the fact that the bulk dd SPTs in condensed matter physics has an onsite local internal G internal -symmetry (or on-n-simplexsymmetry as a generalization for higher-SPTs), thus this G internal must be gauge-able. By gauging the topological terms, this idea has been used to study the vacua of YM gauge theories coupling to dynamically gauged SPT terms (like the orbifold techniques in string theory, but here we generalize this thinking to any dimension), for example, in [35] and references therein. There are other uses and interpretations of our cobordism theory data that we will explain in section 8.
We should emphasize that several recent pursuits are also along the fusions between the non-perturbative physics of SMs, GUTs, and BSMs via a cobordism theory: • Garcia-Etxebarria-Montero [50] studies global anomalies of some SMs and GUTs model via a Dai-Freed theorem and Atiyah-Hirzebruch spectral sequence (AHSS) [25].
• Wang-Wen [51], independently, studies the non-perturbative definitions (e.g. on a lattice) and the global anomalies of SO(10) GUTs or SO(18) GUTs via Ω Spin(5)×Spin(n) Z F 2 5 with n = 10, 18 and SU(5) GUTs via Ω Spin 5 (BSU (5)). They ask what are all allowed thus all possible anomalies for a fermionic theory with Spin(d)×Spin(10) Under the interaction effects, the answer turns out to be a Z 2 class (or a mod 2 class) global anomaly captured by the 5d iTQFT: where w j (T M ) is the j-th-Stiefel-Whitney class for the tangent bundle of 5D spacetime M 5 . We note that on a M 5 , we have a Spin(d=5)×Spin(n) connection -a mixed gravity-gauge connection, rather than the pure gravitational Spin(d = 5) connection, such that w 2 (T M ) = w 2 (V SO(n) ) ≡ w 2 (SO(n)) and w 3 (T M ) = w 3 (V SO(n) ) ≡ 13 Before dynamically gauging Spin(10), SO(10)-GUT is one kind of such theory: an Spin(10)-chiral gauge theory with fermions in the half-integer (iso)spin-representation. So we may call this ungauged theory as SO(10)-GUT chiral fermion theory.

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w 3 (SO(n)), where w j (V SO(n) ) := w j (SO(n)) is the j-th-Stiefel-Whitney class for the associated vector bundle of an SO(n) gauge bundle. The M 5 can be a non-spin manifold. This is the same global anomaly known as "a new SU(2) anomaly" studied in ref. [15]. But ref. [51] and [15] show explicitly, since if the SO(10) GUT chiral fermion theory is free from "the new SU(2) anomaly [15]" (which indeed is true), then the SO(10) GUT chiral fermion theory contains no anomaly at all. Thus this SO (10) GUT is all anomaly-free [15,51]." This leads to a possible non-perturbative construction of SO(10) GUTs on the lattice proposed in [52,53], rooted in the idea of gapping the mirror chiral fermions of Eichten-Preskill [54]. However, we will not pursue this idea of [51] nor [52,53] (such as the lattice regularization) further in this work, but leave this for a future exploration [55]. 14 • Wan-Wang [31][32][33][34] attempts to classify all invertible local or global anomalies and the invertible higher-anomalies, based on a generalized cobordism group classification of invertible TQFTs and invertible higher-TQFTs in one higher dimensions via Adams spectral sequence. Ref. [31] computes the cobordism classification relevant for perturbative anomalies of chiral fermions (e.g. originated from Adler-Bell-Jackiw [9,10]) or chiral bosons with U(1) symmetry in any even spacetime dimensions; they also compute the cobordism classification for the non-perturbative global anomalies such as Witten anomaly [14] and the new SU(2) anomaly [15] in 4d and 5d. Ref. [31] also obtains the cobordism classification relevant for higher 't Hooft anomalies for a pure 4d SU(N) YM theory with a second-Chern-class θ = π topological term [16][17][18].
• McNamara-Vafa [61] studies the cobordism classes and the constraints on the Quantum Gravity or String Theory Landscape v.s. Swampland. QFT must satisfy some consistent criteria in order to be part of a consistent theory of Quantum Gravity. Those QFT not obeying those criteria are quoted to reside in Swampland.
• Kaidi-Parra-Martinez-Tachikawa [63,64] studies the possible fermionic SPTs (or invertible spin TQFTs) on the worldsheet of the string as the Gliozzi-Scherk-Olive (GSO) projections in the superstring theory. Their approach, based on the various relevant bordism groups, also shows the relationship to the K-theoretic classification of D-branes.
• Freed-Hopkins [65] studies a global anomaly cancellation involving the time-reversal symmetry relevant for the 11-dimensional M theory.

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The outline of our article is the following. We consider the following models/theories, their co/bordism groups, TP groups, topological terms and anomalies, written in terms of iTQFTs.

Zq
) with q = 1, 2, 3, 6. The Lie algebra of SM is known to be su(3) × su(2) × u(1), but it is known that the global structure of gauge group allows a quotient group of which is well-explained, for example, in [66]. 15
2. Grand Unified Theories (GUT) in section 4, section 5 and section 6: Here the Z F 2 is the well-known fermion parity symmetry, which acts on any fermionic operator Ψ by such that the operation is (−) N F where N F is the fermion number. We provide an overview how this data can be used to dynamically constrain SMs, GUTs and BSMs in Conclusions in section 8. After all these physics stories and inputs, and especially thanks to the readers are still staying in tune with us, now let us reward the readers by introducing some mathematics preliminary in section 1.2.

Definition of bordism groups
We assume that the readers are familiar with the basic algebraic topology, such as (co)homology and homotopy. To fill in the gap between this knowledge and the mathematical tools used in this article, we introduce some prerequisites in this subsection.
First, we introduce the notion of bordism group. To define the bordism group, we need to define a tangential structure which involves the notion of classifying spaces. For any JHEP07(2020)062 Lie group G, the BG is the classifying space of principal G-bundles, namely the homotopy classes of maps X → BG are in one-to-one correspondence with the isomorphism classes of principal G-bundles over X for any topological space X. For any abelian group G, the iterated classifying space B 2 G is also the Eilenberg-MacLane space K(G, 2). For any abelian group G, more generally, we have the B d G = K(G, d).
To define tangential structure, we require the orthogonal group O(n) since a tangential structure involves a fibration over the classifying space of O(n). There is a natural inclusion O(n) → O(n+1). Let O denote the colimit O := colim n→∞ O(n). The inclusions R q → R q+1 induce the closed inclusions of Grassmannian spaces Gr n (R q ) → Gr n (R q+1 ). The colimit colim q→∞ Gr n (R q ) is the classifying space BO(n). There are closed inclusions Gr n (R q ) → Gr n+1 (R q+1 ) obtained by sending W → R ⊕ W where we write R q+1 = R ⊕ R q . These induce maps BO(n) → BO(n + 1), and we define BO := colim n→∞ BO(n). (1.8) The BO is a classifying space for the infinite orthogonal group O. An n-dimensional tangential structure is a topological space BG(n) and a fibration π(n) : BG(n) → BO(n). A stable tangential structure is a topological space BG and a fibration π : BG → BO. It gives rise to an n-dimensional tangential structure for each n ∈ Z ≥0 by letting π(n) : BG(n) → BO(n) be the fiber product (also called pullback) Here R n−m is the trivial real vector bundle of dimension n − m. A G-structure on M is a family of coherent G(n)-structures for n sufficiently large. Here coherent G(n)-structures for n sufficiently large means that the composite of the G(l)-structure on M and the inclusion map BG(l) → BG(n): M → BG(l) → BG(n) is exactly the G(n)-structure on M for l < n where l and n are sufficiently large. Notice that an n-dimensional tangential structure induces an l-dimensional tangential structure for all l < n by taking the fiber product (also called pullback) (1.11) See [69] for more details.
JHEP07(2020)062 For any structure group G and any topological space X, we define the bordism group as the set of equivalence classes: Here a G-structure on a manifold is a G-structure on the stable tangent bundle of this manifold. In this article, we will focus on the case when X is a classifying space. In most cases, X is just a point.
If Ω G d (X) = G 1 × G 2 × · · · × G r where G i are cyclic groups, then the group homomorphisms ϕ i : Ω G d (X) → G i are called bordism invariants, and they form a complete set of bordism invariants if ϕ = (ϕ 1 , ϕ 2 , . . . , ϕ r ) : In this article, we will compute the bordism groups Ω G d for G = Spin×G or Spin× Z 2 G for some groups G . So we first clarify these notations here. Spin (or Spin(d)) is a nontrivial extension of the group SO (or SO(d)) by Z 2 , namely there are short exact sequences of groups 14)

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where SO is the colimit of SO(d), and SO(d) is the special orthogonal group. The Spin is the colimit of Spin(d). The Spin × Z 2 G is the quotient group of Spin × G by the diagonal central subgroup Z 2 .

Spectra
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. Now we recall some basic notions regarding spectrum, see [31,69] for more details. A prespectrum T • is a sequence {T n } n∈Z ≥0 of pointed spaces and the maps s n : ΣT n → T n+1 .
An Ω-prespectrum is a prespectrum T • such that the adjoints t q : T n → ΩT n+1 of the structure maps are weak homotopy equivalences. Here Ω means the loop space, its meaning is different from the bordism notation Ω. A spectrum is a prespectrum T • such that the adjoints t n : T n → ΩT n+1 of the structure maps are homeomorphisms. For example, let X be a pointed space, T n = Σ n X for n ≥ 0, then T • is a prespectrum. In particular, if T n = S n , then T • is a prespectrum. Let G be an abelian group, T n = K(G, n) be the Eilenberg-MacLane space, then T • is an Ω-prespectrum. In general, these examples are not spectra. But we can always construct a spectrum from a prespectrum using spectrification.
Let T • be a prespectrum, define (LT ) n to be the colimit of Namely, (LT ) n = colim l→∞ Ω l T n+l , then (LT ) • is a spectrum, called the spectrification of T • . For example, if T n = S n , then (LT ) • is a spectrum S (the sphere spectrum). Let G be an abelian group, T q = K(G, n) the Eilenberg-MacLane space, then (LT ) • is a spectrum HG (the Eilenberg-MacLane spectrum). Next we define the homotopy groups and cohomology rings of prespectra. Let T • be a prespectrum, we define the (stable) homotopy group π d T • to be the colimit of If M • and N • are two prespectra, and N • is an Ω-prespectrum, then for any integer k, the abelian group of homotopy classes of maps from M • to N • of degree −k is defined as follows: a map from M • to N • of degree −k is a sequence of maps M n → N n+k such that the following diagram commutes Thom spaces satisfy where V → X and W → Y are real vector bundles, R n is the trivial real vector bundle of dimension n, and X + is the disjoint union of X and a point. Let G be a group with a group homomorphism ρ : G → O. The Madsen-Tillmann spectrum M T G is the colimit of Σ n M T G(n), where the virtual Thom spectrum M T G(n) = Thom(BG(n); −V n ) is the spectrification of the prespectrum whose (n + q)-th space is Thom(BG(n, n + q); Q q ) where BG(n, n + q) is the pullback BG(n, n + q) and there is a direct sum R n+q = V n ⊕ Q q of vector bundles over Gr n (R n+q ) and, by pullback, over BG(n, n + q) where R n+q is the trivial real vector bundle of dimension n + q.
In other words, M T G = Thom(BG; −V ) where V is the colimit of V n − n.

Adams spectral sequence
Adams spectral sequence is a mathematical tool to compute the homotopy groups of spectra [70]. In particular, the homotopy group of the Madsen-Tillmann spectrum M T G [39] JHEP07(2020)062 t − s s E s,t r E s+r,t+r−1 r d r Figure 2. The E r page of Adams spectral sequence.
is the bordism group Ω G d . We use Adams spectral sequence to compute several bordism groups related to Standard Models (SM), Grand Unified Theories (GUT) and beyond. We also compute the group TP d (G) classifying the topological phases (i.e., the topological terms in QFT or the topological phases of quantum matter) based on the computation of bordism groups and a short exact sequence. See [19,31,35] for primers. We will call the group TP d (G) the cobordism group. The relation between Ω G d and TP d (G) is like that between homology group and cohomology group, as we will see later.
The Adams spectral sequence shows [70]: where the Ext denotes the extension functor. Note that this extension functor has 2 upper indices. It is different from but similar to the usual extension functor. The index s refers to the degree of resolution, and the index t is the internal degree of the graded module. Here A p = [HZ p , HZ p ] − * is the mod p Steenrod algebra which consists of mod p cohomology operations [71], and Y is any spectrum. In particular, the mod 2 Steenrod algebra A 2 is generated by the Steenrod squares Sq i : H * (−, Z 2 ) → H * +i (−, Z 2 ). By the Yoneda lemma, the H * (Y, Z p ) = [Y, HZ p ] − * is automatically an A p -module whose internal degree t is given by the * . The (π t−s (Y )) ∧ p is the p-completion of the (t − s)-th homotopy group of the spectrum Y . For any finitely generated abelian group G, the abelian group is the ring of p-adic integers. Here the G is meant to be substituted by a homotopy group π t−s (Y ) in (1.18). Here are some explanations and inputs: (1). Here the double-arrow "⇒" means "convergent to." The E 2 page are groups Ext s,t with double indices (s, t), we reindex the bidegree by (t−s, s). Inductively, see figure 2, there are differentials d r in E r page which are arrows from (t − s, s) to (t − s − 1, s + r) (that is, Ext s,t → Ext s+r,t+r−1 ). Take Kerd r /Imd r at each (t − s, s), then we get the E r+1 page. Finally E r page equals E r+1 page (there are no differentials) for r ≥ N , we call this E N page as the E ∞ page, we can read the result π d at t − s = d. See further details discussed in ref. [31]'s section 2.3.

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(2). In Adams spectral sequence, we consider Ext s,t R (L, Z p ). In this article, we will consider the algebra R = A p or A 2 (1) for p = 2, and L is an R-module. The A 2 (1) is the subalgebra of A 2 generated by the Steenrod squares Sq 1 and Sq 2 . Ext groups are defined by firstly taking a projective R-resolution P • of L, then secondly computing the (co)homology group of the (co)chain complex Hom(P • , Z p ). Here a projective Rresolution P • is an exact sequence of R-modules · · · → P s → P s−1 → · · · → P 0 → L where P s is a projective R-module for s ≥ 0. An R-module is projective if and only if it is a direct summand of a free R-module.
For Y = M T G, where M T G is the Madsen-Tillmann spectrum of a group G, the Adams spectral sequence (1.18) shows: The last equality is by the generalized Pontryagin-Thom isomorphism, we have an equality between the d-th bordism group of G given by Ω G d and the d-th homotopy group of M T G given by The T in M T G means that the G-structures are on the stable tangent bundles instead of stable normal bundles. For Spin, the Madsen-Tillmann spectrum M T Spin = M Spin is equivalent to the Thom spectrum. We also compute the cobordism group of topological phases (TP) defined in [19] as Here IZ is the Anderson dual spectrum. By [19], the torsion part of TP d (G) classifies deformation classes of reflection positive invertible d-dimensional extended topological field theories with a symmetry group G(d).
Here G(d) means the d-dimensional spacetime version of the group G. The TP d (G) and the bordism group Ω G d are related by a short exact sequence This short exact sequence is very similar to the universal coefficient theorem relating homology group and cohomology group. It is split, since Ext 1 (Ω G d , Z) is always torsion, Hom(Ω G d+1 , Z) is always free, and Ext 1 (Z, Z n ) = 0. So we can directly derive the group TP d (G) from the data of Ω G d and Ω G d+1 . We will use the (1. (1.23)

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The ∧ is the smash product. 16 The (BG ) + is the disjoint union of the classifying space BG and a point. 17 By the generalized Pontryagin-Thom isomorphism, for any structure group G and any topological space X, we have The reduced cohomology H * (X, Z 2 ) is an A 2 (1)-module whose internal degree t is given by the * .

Lemma 1 (Lemma 11 of [31]). Given a short exact sequence of
then for any t, there is a long exact sequence We can compute the E 2 page of A 2 (1)-module based on Lemma 1. More precisely, in order to compute Ext s,t A 2 (1) (L 2 , Z 2 ), we find a short exact sequence of A 2 (1)-modules then we apply Lemma 1 to compute Ext s,t . Our strategy is choosing L 1 to be the direct sum of suspensions of Z 2 on which Sq 1 and Sq 2 act trivially, then we take L 3 to be the quotient of L 2 by is undetermined, then we take L 3 to be the new L 2 and repeat this procedure. We can use this procedure again and again until Ext s,t A 2 (1) (L 2 , Z 2 ) is determined. 16 Smash product between a spectrum M• and a topological space X is a spectrum whose n-th space is Mn ∧ X which is the ordinary smash product of topological spaces. 17 For a topological space X, it is a standard convention to denote that X+ as the disjoint union of X and a point. Note that the reduced cohomology of X+ is exactly the ordinary cohomology of X. JHEP07(2020)062

Characteristic classes
Throughout the article, we use the standard notation for characteristic classes: w i for the Stiefel-Whitney class, c i for the Chern class, p i for the Pontryagin class, and e n for the Euler class. Note that the Euler class only appears in the total dimension of the vector bundle. We use the notation w i (G), c i (G), p i (G), and e n (G) to denote the characteristic classes of the associated vector bundle of the principal G bundle (normally denoted as , and e n (V G )). For simplicity, we may denote the Stiefel-Whitney class of the tangent bundle T M as w j ≡ w j (T M ); if we do not specify w j with which bundle, then we really mean T M . 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). Note The relation between the Chern-Simons form and the Chern class is where the d is the exterior differential and the c n (V ) is regarded as a closed differential form in de Rham cohomology. There is also another kind of Chern-Simons form for Euler class e 2n (V ) [74], we denote it by CS The relations between Pontryagin class, Euler class and Stiefel-Whitney class are By the Hirzebruch signature theorem, the relation between the signature and the first Pontryagin class of a 4-manifold M is For our other conventions, see section 1 of [31]. For instance, all the product notations between cohomology classes are cup product, such as ; all the product notations between a cohomology class x andη (or Arf, ABK, etc.), namely xη means the value ofη (or Arf, ABK, etc) on the submanifold of M which represents the Poincaré dual of x. In other words, the xη :=η(PD(x)).
In the definition of Ω Spin×G Our convention is that all bordism invariants which are cohomology classes of certain classifying spaces are pulled back to M via the maps in the definition of bordism groups. For example, there is an a involved in the bordism invariants of Ω in the definition of bordism group. So a is implicitly also a cohomology class of H 1 (M, Z 2 ). JHEP07(2020)062

Lie algebra to Lie group and the representation theory
To justify the spacetime symmetry group G spacetime and internal symmetry group G internal relevant for Standard Model physics, we shall first review the Lie algebra of Standard Models, to the representation theory of matter field contents, and to the Lie groups of Standard Models.
[I]. The local gauge structure of Standard Model is given by the Lie algebra u(1) × su(2) × su (3). This means that the Lie algebra valued 1-form gauge fields take values in u(1) × su(2) × su (3). The 1-form gauge fields are the 1-connections of the principals G internal -bundles that we should determine.
[II]. Fermions as the spinor fields. A spinor field is a section of the spinor bundle. For the left-handed Weyl spinor Ψ L , it is a doublet spin-1/2 representation of spacetime symmetry group G spacetime (Minkowski Spin (3,1) or Euclidean Spin (4)), denoted as On the other hand, the matter fields as Weyl spinors Ψ L contain: • The left-handed up and down quarks (u and d) form a doublet u d L in 2 for the SU(2) weak , and they are in 3 for the SU(3) strong .
• The right-handed up and down quarks, each forms a singlet u R and d R in 1 for the SU(2) weak . They are in 3 for the SU(3) strong .
• The left-handed electron and neutrino form a doublet ν e e L in 2 for the SU(2) weak , and they are in 1 for the SU(3) strong .
• The right-handed electron forms a singlet in 1 for the SU(2) weak , and it is in 1 for the SU(3) strong .
There are two more families of quarks: charm and strange quarks (c and s), and top and bottom quarks (t and b). There are also two more families of leptons: muon and its neutrino (µ and ν µ ), and tauon and its neutrino (τ and ν τ ). So there are three families (i.e., generations) of quarks and leptons:

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In short, for all of them as three families, we can denote them as: In fact, all the following four kinds of with q = 1, 2, 3, 6 are compatible with the above representations of fermion fields. These Weyl spinors can be written in the following more succinct forms of representations for any of the internal symmetry group G internal with q = 1, 2, 3, 6: (1.42) The triplet given above is listed by their (SU(3) representation, SU (2) representation, hypercharge Y ). 18 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, we transforms the right-handed Weyl spinor Ψ R ∼ 2 R of Spin(3, 1) to its left-handed Ψ L ∼ 2 L of Spin(3, 1), while we flip their (SU(3) representation, SU(2) representation, hypercharge) to its (complex) conjugation representations. 19 If we include the right-handed neutrinos (say ν eR , ν µ R , and ν τ R ), they are all in the representation (1, 1, 0) R with no hypercharge. We can also represent a right-handed neutrino by the left-handed (complex) conjugation version (1, 1, 0) L .
[V]. We find the following Lie group embedding for the internal symmetry of GUTs and Standard Models: The other Z q for q = 1, 2, 3 cannot be embedded into Spin(10) nor SU (5). So from the GUT perspective, it is natural to consider the Standard Model gauge group .
[VI]. We find the following group embedding for the spacetime and internal symmetries of GUTs and Standard Models (see also [34] for the derivations): We shall study the cobordism theory of these SM, BSM, and GUT groups in the following subsections.
For the dimension d = t − s < 8, since there is no odd torsion, 21 for M T G = M Spin ∧ (B(SU(3) × SU(2) × U(1))) + , for the dimension d = t− s < 8, by (1.27), we have the Adams spectral sequence and We also have the Wu formula 22 By Künneth formula, we have Here only in this subsection, c i is the Chern class of SU(3) bundle, and c i is the Chern class of SU(2) bundle, and c i is the Chern class of U(1) bundle. (1)), Z 2 ) below degree 6 and the E 2 page are shown in figure 3, 4. Here we have used the correspondence between A 2 (1)-module structure and the E 2 page shown in figure 35 and 37.
In Adams chart, the horizontal axis labels the integer degree d = t − s and the vertical axis labels the integer degree s. The differential d s,t r : E s,t r → E s+r,t+r−1 r is an arrow starting at the bidegree (t − s, s) with direction (−1, r). E s,t r+1 := Kerd s,t r Imd s−r,t−r+1 r for r ≥ 2. There exists N such that E N +k = E N stabilized for k > 0, we denote the stabilized page E ∞ := E N .
To read the result from the Adams chart in figure 4, we look at the stabilized E ∞ page, one dot indicates a finite group Z p , a vertical finite line segment connecting n dots indicates a finite group Z p n . But when n = ∞, the infinite line connecting infinite dots indicates a Z. Here p is given by the mod p Adams spectral sequence in (1.19). Here in figure 4, p = 2, we can read from the Adams chart Ω 21 By computation using the mod p Adams spectral sequence for an odd prime p, we find there is no odd torsion. 22 There is another Wu formula which will be used later: Z2) and ui is the Wu class.

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Bordism group (2)), , c 2 (SU(2))Arf is the signature of manifold. F is the characteristic 2-surface [75] in a 4-manifold M 4 , it satisfies the condition , c 2 (SU(2))η Table 2. Topological phase classification (≡ TP) as a cobordism group, following table 1.η is a mod 2 index of 1d Dirac operator. Arf is a 2d Arf invariant. c i (G) is the Chern class of the associated vector bundle of the principal G-bundle. CS V 2n−1 or CS G 2n−1 is the Chern-Simons form of the vector bundle V or the associated vector bundle of the principal G-bundle. The PD is the Poincaré dual. The T M is the spacetime tangent bundle. The µ is the 3d Rokhlin invariant.
Here the c 2 (SU(2))η in 5d captures the Witten anomaly in 4d. See appendix B for comment on the difference in 5d between table 2 and 4. We use the notation "∼" to indicate the two sides are equal in that dimension up to a total derivative term.
For the dimension d = t − s < 8, since there is no odd torsion (see footnote 21), by (1.27), we have the Adams spectral sequence Here only in this subsection, c i is the Chern class of SU(3) bundle, and c i is the Chern class of U(2) bundle. (2)), Z 2 ) below degree 6 and the E 2 page are shown in figure 5, 6. Here we have used the correspondence between A 2 (1)-module structure and the E 2 page shown in figure 35 and 37.
Thus we obtain the bordism group Ω In table 4, note that CS and CS up to a total derivative term (vanishing on a closed 5-manifold). See footnote 23.
We have the Adams spectral sequence By Künneth formula, we have Here only in this subsection, c i is the Chern class of U(3) bundle, and c i is the Chern class of SU(2) bundle. The Adams chart of Ext s,t figure 8. There is no differential since the arrow of the differential d r is of bidegree (−1, r), while all lines are of interval 2 at degree t − s.
So there is actually no 3-torsion in Ω For the dimension d = t − s < 8, since there is no odd torsion (see footnote 21), by (1.27), we have the Adams spectral sequence (2.14) By Künneth formula, we have
In table 8, note that CS and CS up to a total derivative term (vanishing on a closed 5-manifold). See footnote 23.
• Refs. [50,62] is based on Atiyah-Hirzebruch spectral sequence (AHSS), which includes only the group structure, but with the disadvantage of having more differentials and some undetermined extensions. It is also not known or difficult, if not impossible, to extract the iTQFT data directly (namely, co/bordism invariants) from the AHSS. Therefore, refs. [50,62] cannot provide the explicit iTQFT data from the AHSS calculations.
By (1.25) .20), Ω for some topological space X , but X is not the disjoint union of a topological space X and a point, while by (1.24), Ω Spin d (X) = π d (M Spin ∧ X + ) where X + is the disjoint union of X and a point.
Specifically, in ref. [62], using Atiyah-Hirzebruch spectral sequence, the authors compute the cobordism groups Ω Spin )) = e(Z 3 , e(Z 3 , Z 4 )) where e(Q, N) is the group extension of Q by N. So the group e(Q, N) fits into the short exact sequence 0 → N → e(Q, N) → Q → 0 but it may not be uniquely determined.
In contrast, our results are more refined and can uniquely determine the extension in this case. Our result from Adams spectral sequence demonstrates that each step of extensions is nontrivial, while the trivial extension yields 3-torsion. Using Adams spectral sequence, we find that there is no 3-torsion for the Γ = Z 6 case. So we also provide the solutions to the extension problems in ref. [62], given by the nontrivial extension Z → Z → Z 3 . We obtained the precise answer Zq with q = 6. Furthermore, inspired by the SMs with additional discrete symmetries (see some of the earlier work [76][77][78][79] and references therein [50,80]) and motivated by a version of Smith homomorphism map between 5d and 4d bordism groups [81], we find the following group embedding for the spacetime and internal symmetries for GUTs and the SMs with additional discrete symmetries (see also [34] for the derivations): We shall study the cobordism theory of the SM groups, Spin × Z 2 Z 4 × SU(3)×SU(2)×U(1) Zq , with q = 1, 2, 3, 6 in the following subsections. For SM with this discrete symmetry, we obtain new anomaly matching conditions of Z 16 , Z 4 and Z 2 classes beyond the familiar Witten anomaly. Depend on whether this Z 4 is a global symmetry or a gauge symmetry, we shall interpret some of the 4d anomalies obtained from the 5d cobordism groups below as 't Hooft anomalies, and some of others as a dynamical gauge anomalies.

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We have a homotopy pullback square where a is the generator of H 1 (BZ 2 , Z 2 ). By [83], since there is a homotopy pullback square which is equivalent to the homotopy pullback square where 2ξ : BZ 2 → BSO is twice the sign representation, the final identification is by [84].
The E 2 page is shown in figure 18. Here we have used the correspondence between A 2 (1)-module structure and the E 2 page shown in figure 37, 38 and 39.
Thus we obtain the bordism group Ω
The E 2 page is shown in figure 20. Here we have used the correspondence between A 2 (1)-module structure and the E 2 page shown in figure 37, 38 and 39.
Thus we obtain the bordism group Ω

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Cobordism group  In table 14, note that CS up to a total derivative term (vanishing on a closed 5-manifold). See footnote 23. (2)), the Madsen-Tillmann spectrum M T G of the group G is
The E 2 page is shown in figure 22. Here we have used the correspondence between A 2 (1)-module structure and the E 2 page shown in figure 37, 38

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Bordism group d Ω Here · is the intersection form of M 4 . By the Freedman-Kirby theorem,

Pati-Salam models
Now we consider the co/bordism classes relevant for Pati-Salam GUT models [8]. There are actually two different cases for modding out different discrete normal subgroups.

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By definition, the Madsen-Tillmann spectrum M T G of the group G is M T G = Thom(BG; −V ), where V is the induced virtual bundle (of dimension 0) by the map BG → BO.

SO(10), SO(18) and SO(n) Grand Unifications
Now we consider the co/bordism classes relevant for Fritzsch-Minkowski SO(10) GUT [7]. There are actually two cases, depending on whether the gauged SO(10) GUT allows gaugeinvariant fermions, or whether the gauged SO(10) GUT only allows gauge-invariant bosons.  (SU(2)))) 5  bundle. Here w 2 = w 2 = w 2 , w 3 = w 3 = w 3 .η is a mod 2 index of 1d Dirac operator. Arf is a 2d Arf invariant. Since Sq 2 w 4 = w 2 w 4 + w 6 = (w 2 + w 2 1 )w 4 = w 2 w 4 on oriented 6-manifold by Wu formula, we have e 6 = w 6 = 0 mod 2. On a 4-manifold, the oriented bundle of rank 6 splits as the direct sum of an oriented bundle of rank 4 and a trivial plane bundle, the Euler class e 4 is the Euler class of the subbundle of rank 4. Here f : Ω
The E 2 page is shown in figure 28. Here we have used the correspondence between A 2 (1)-module structure and the E 2 page shown in figure 35 and 44.
Thus we obtain the bordism group Ω  We have H * (BSO(n), Z 2 ) = Z 2 [w 2 , w 3 , . . . , w n ]. (5.10) Here in this subsection, w i is the Stiefel-Whitney class of SO(n) bundle. We also have the Wu formula For n ≥ 7, the A 2 (1)-module structure of H * (BSO(n), Z 2 ) below degree 6 is shown in figure 29.
The E 2 page is shown in figure 30. Here we have used the correspondence between A 2 (1)-module structure and the E 2 page shown in figure 35, 36     Here w 2 = w 2 (SO(n)).

Interpretations of the Z classes of co/bordism invariants
(1). The bordism generator of bordism group Ω SO 4 = Z is the complex projective space CP 2 , CP 2 and their connected-sum manifolds, whose bordism invariant is the signature σ = 1 3 M 3 p 1 (T M ) related to Pontryagin class of tangent spacetime T M of manifold M . Since σ(CP 2 ) = 1 and all other SO-manifolds have quantized signatures, we can define a so-called θ-term whose partition function on non-spin manifolds with a compact periodic θ ∈ [0, 2π), with a different θ specifying a different theory on non-spin manifolds.
The cobordism generator of cobordism group TP 3 (SO) = Z is the 3d gravitational Chern-Simons (CS) theory given by a partition function here k ∈ Z and ω is a connection of the tangent bundle of an oriented nonspin manifold M . The exp(i k 3 CS grav ) on the non-spin M 3 can be defined by extending the 3-manifold M 3 as a boundary of 4-manifold M 4 with a stable tangential SO structure. The extension from 3d to 4d is doable, thanks to Ω SO When k = 1, the 3d gravitational CS theory is the gravitational background field theory, which can probe the dynamical internal 3d CS gauge theory of gauge group JHEP07(2020)062 (E 8 ) 1 . Namely, by integrating out the internal gauge fields of (E 8 ) 1 , we should obtain back the 3d gravitational CS background field theory. Also this bulk internal field theory can be equivalently written as at least three CS descriptions: • First, the 3d (E 8 ) 1 CS theory with a gauge group E 8 at the level 1.
• Second, a SO(16) 1 CS ⊗ (a spin TQFT {1, f } with only a trivial line operator 1 and a single fermionic line f ).
• Third, a rank-8 K E 8 matrix abelian CS theory Z = [Da] exp( with an E 8 Cartan matrix Note that all these internal 3d CS theories can have a 2d boundary CFT with a chiral central charge c − = 8, namely associated with a E 8 chiral boson theory with 8 chiral modes each with c − = 1. This is a rank-8 K-matrix complex chiral boson theory whose K = K E 8 . Alternatively, this CFT can also be formed by gapless 16 multiple of chiral Majorana-Weyl modes, each mode with a c − = 1/2.

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(3). Next let us consider the bordism group Ω SO 4 (BSU(2)) = Z 2 and Ω Spin 4 (BSU(2)) = Z 2 for the internal symmetry SU (2). The two bordism invariants of Ω SO 4 (BSU(2)) = Z 2 are the signature σ and the second Chern class c 2 (V SU(2) ) of the SU(2) bundle. The two bordism invariants of Ω Spin 4 (BSU(2)) = Z 2 are σ 16 where σ is the signature and the second Chern class c 2 (V SU(2) ) of the SU(2) bundle. We had already discussed the invariants related to signature σ. Let us focus on the other, but common, bordism invariant c 2 (V SU(2) ) of Ω SO 4 (BSU(2)) and Ω Spin 4 (BSU (2)). This, in fact, specifies the so-called θ-term of particle physics whose partition function contains on both non-spin and spin manifolds with a compact periodic θ ∈ [0, 2π), because the instanton number i θ 8π 2 M 4 Tr(F ∧ F) for the SU(2) gauge bundle on 4-manifolds is in Z value. Here the curvature 2-form is µ T α is Lie algebra valued 1-form and we choose the Lie algebra [T α , T β ] = f αβγ T γ with an anti-Hermitian T α and a structure constant f αβγ . 26 26 We warn the readers that mathematicians and physicists may prefer different conventions. For a vector bundle V : (a) Mathematicians may choose the curvature 2-form as µ T α is Lie algebra g valued 1-form. The Lie algebra can be [T α , T β ] = f αβγ T γ with an anti-Hermitian T α and a structure constant f αβγ . For some variable X, the following polynomial expansion computes the Chern classes of the vector bundle V : The I is an identity matrix whose rank is equal to the rank of the matrix representation of Lie algebra g. We use det( e M ) = e Tr(M) and take M = log(I + X( i F 2π )). In that case, ) is related to exp( i k 4π M 3 Tr(A dA + 2 3 A 3 )). (b) Physicists may choose the curvature 2-form as µ T α is Lie algebra valued 1-form. The Lie algebra can be [T α , T β ] = i f αβγ T γ with a Hermitian T α and a structure constant f αβγ . For some variable X, the following polynomial JHEP07(2020)062 Also TP 3 (SO×SU(2)) = Z 2 and TP 3 (Spin×SU(2)) = Z 2 . We had discussed one cobordism generator associated to a multiple of 3d CS grav . There is another cobordism generator CS SU(2) 3 common to both TP 3 (SO×SU(2)) and TP 3 (Spin×SU (2)). Here CS SU(2) 3 is the Chern-Simons 3-form of the SU(2) gauge bundle given by Here A is the connection of the SU(2) gauge bundle. Similarly, the exp(ik(2π) M 3 CS SU (2)  3 ) on the spin M 3 can be defined by extending the 3-manifold M 3 as a boundary of 4-manifold M 4 with a principal SU(2) structure, thanks to.
We can interpret the Z class co/bordism invariants obtained in our tables in section 2-6 in a similar manner. In this next subsection, we enumerate the interpretations of the 4d anomalies (and 5d cobordism invariants) for various SM, BSM, and GUT obtained in section 2-6.

Interpretations of the 4d anomalies of SM and GUT from co/bordism invariants: beyond the Witten anomaly
We can interpret the classifications of all possible 4d perturbative local and nonperturbative global invertible anomalies for SM, BSM, and GUTs obtained in section 2-6 as follows: expansion computes the Chern classes of the vector bundle V : In that case, JHEP07(2020)062 1. In section 2, we find that their 5d cobordism invariants contain the following: • A Z 2 class from the well-known Witten anomaly c 2 (SU(2))η for q = 1, 3. This familiar Witten SU(2) anomaly disappears from the global anomaly and can be cancelled by a local anomaly for q = 2, 6 with an internal U(2) symmetry group, see [86] and more discussions below.
• Other Z 4 classes from Chern-Simons 5 form like terms from the gauge bundles E of gauge group SU(3)×SU(2)×U(1)
2. In section 3, we find that their 5d cobordism invariants contain the following: • The Z 5 classes are similar to section 2. Again one of a Z class is from the more exotic µ(PD(c 1 (U(3)))). The other four Z 4 class are from Chern-Simons 5 form like terms from the gauge bundles E.
The difference between q = 3 and q = 6 cases in section 3 is that the global Witten SU(2) anomaly of Z 2 class in q = 3 becomes a local anomaly of Z class in q = 6. In short, the situations of q = 3 and q = 6 are exactly parallel to the situations of q = 1 and q = 2 discussed above.
What we have found here is consistent with the fact that the Witten anomaly as a global non-perturbative anomaly becomes a local perturbative anomaly, whenever we change from q = 1 to q = 2, or change from q = 3 to q = 6, which trades the SU(2) internal symmetry for the U(2) internal symmetry [86].

Conclusions, and explorations on non-perturbative and topological sectors of BSM
We have explored the cobordism theory relevant for SM and GUTs in section 2-6. We also have interpreted the classifications of various 4d perturbative local and non-perturbative global invertible anomalies for SM and GUTs in section 7. Below we summarize the results of co/bordism groups and conclude with further implications.
We find that there is only 2-torsion in these bordism groups, and the bordism groups We use the 3d Rokhlin invariant and Chern-Simons forms to express the topological terms.

Zq
) and the topological terms for 0 ≤ d ≤ 5.
We find that there is only 2-torsion in these bordism groups, and the bordism groups Ω We also use the 3d Rokhlin invariant and Chern-Simons forms to express the topological terms. Compared to section 2, there are new bordism invariants. For example, in 1d,η becomes η which is Z 4 valued. In 4d, the σ 16 becomes σ−F·F 8 where F is the characteristic 2-surface [75] in a 4-manifold M 4 , it satisfies the condition F · x = x · x mod 2 for all x ∈ H 2 (M 4 , Z). Here · is the intersection form of We also find that Euler class appears in the bordism invariants and the Chern-Simons form for Euler class appears in the topological terms.
We find that for n ≥ 7, Ω We also find that Euler class appears in the bordism invariants of Ω

Constraints on quantum dynamics
We have discussed and summarized potential anomalies and topological terms in SM, GUT and BSM. There are actually two versions of anomalies we are speaking of: one is the Anomaly (2) for the ungauged SM, GUT and BSM with the G is simply a global symmetry. Another is the Anomaly (3) for the gauged SM, GUT and BSM with the G internal dynamically gauged inside G. In more details, I. For ungauged SM, GUT and BSM, we can use the 't Hooft anomaly (Anomaly (2)) to the gappability of these models' matter field sectors. For ungauged SM, GUT and BSM, we can simply have matter field contents (e.g. fermions: quarks and leptons) without dynamical gauge fields. In fact, refs. [51][52][53] use the all anomaly free conditions to support that a non-perturbative definition (lattice regularization) of SO (10) GUT is doable, by checking. By anomaly free, there exists non-perturbative interactions for gapping the mirror world chiral fermions [55]. This mirror-fermion gapping can help to get rid of the mirror world chiral fermion doublers, surpassing the Nielsen-Ninomiya fermion doubling theorem (which is only true for the free non-interacting systems). The fact that all anomaly free gapless theories can be deformed to a fully gapped trivial vacuum is also consistent with the concept of Seiberg's deformation class [87]. More details can be found in [55].
II. For gauged SM, GUT and BSM, we can use the dynamical gauge anomaly matching conditions (Anomaly (3)) to rule out inconsistent theories. Importantly, depending on the matter contents and their representations in G internal , we may gain or loss some global symmetries. For example, for fermions in the adjoint representation, we can have a 1-form center symmetry for the gauge theory. Thus, we should beware potentially additional new higher 't Hooft anomalies (see [31,34]) for gauged SM, GUT and BSM can help us to constrain quantum dynamics (also more discussions below).
We use the path integral and the action to understand the basic kinematics and the global symmetry of the QFTs. We can apply the spacetime geometric topology properties to constrain QFTs, such as doing the spacetime surgery for QFTs [88][89][90][91]. We can also determine the anomalies of QFTs at UV. However, given the potentially complete anomalies, we can constrain the IR dynamics by UV-IR anomaly matching. The consequence of anomaly matching implies that the IR theories with 't Hooft anomalies in G-symmetry must be matched by at least one of the following dynamics scenarios: 1. Symmetry-breaking: • (say, discrete or continuous G-symmetry breaking).

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which means the cancellation of cobordism classes via new additional symmetry defects. Namely, the kernel in Ω QG k of this map is mapped to a trivial class in Ω QG+defects k .
(2). Symmetry is gauged: Mathematically, this suggests a map which means the cancellation of cobordism classes via additional new gauge sectors.
Namely, the cokernel in Ω QG k of this map is mapped reversely to a trivial class in Ω QG+gauge sectors k .
We leave the further detailed explorations of the above predictions in upcoming works [27,34,82].

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A The correspondence between A 2 (1)-module structure and the E 2 page In this appendix, we list the correspondence between A 2 (1)-module L and its E 2 page used in our computation before.

B Comment on the difference between Standard Models
There is a Z 2 valued bordism invariant of Ω The A 2 (1)-module structure of H * (B(SU(2) × SU(2) × SU(4)), Z 2 ) below degree 6 and the E 2 page are shown in figure 50, 51. Here we have used the correspondence between A 2 (1)-module structure and the E 2 page shown in figure 35 and 37.
We find that the red part in figure 26 is related to the red part in figure 51. More precisely, the image of the red part in figure 51 under the group homomorphism f is exactly the red part in figure 26. So we can express the three bordism invariants corresponding