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Zheyan Wan

Publications and source records attributed to Zheyan Wan.

At least 19 recordsLinked to original sources
arXiv2026

Fermion Families and Pontryagin Class: Topological Field Theory via Colour Symmetry Extension

Family puzzle asks why the Standard Model (SM) features exactly 3 families of quarks and leptons. Motivated by topological constraints, we study 4d fermionic anomalies with discrete $Z_n$ symmetry, classified by the 5d spin bordism group. We show that only the group-cohomology subclass H$^5(Z_n,U(1))\cong Z_n$ can be canceled by an anomalous $Z_n$-symmetric 4d $Z_n$-gauge topological quantum field theory (TQFT), while beyond-group-cohomology $A_{Z_n}p_1$ involving the Pontryagin class $p_1$ cannot (except $n=2,3$). More generally, we prove that any cocycle $\alpha_d\in$H$^d(Z_n,U(1))$ in odd spacetime dimension $d\ge3$ is trivialised by the symmetry extension $1\to Z_n\to Z_{n^2}\to Z_n\to 1,$ and we construct the corresponding symmetric anomalous boundary TQFT. For $d=5$ and $n=3$, this yields a Spin$\times Z_3$-symmetric 4d $Z_3$-gauge TQFT that cancels the mixed discrete $(\bf B+L)$-gauge-gravitational anomaly of the SM in the absence of 3 "sterile" right-handed neutrinos $\nu_R$. We analyze a generalized SM with $N_c$ colors and $N_f$ families and argue that missing $N_f$ copies of the $\nu_R$ can be naturally replaced by a 4d anomalous $Spin\times_{Z_2^F}Z_{2 N_f,{\bf B + L}}$ symmetric $Z_N$-gauge TQFT under the anomaly cancellation, via a $Z_N$ symmetry extension construction $1\to Z_N\to Spin\times Z_{NN_f}\to Spin\times_{Z_2^F}Z_{2N_f}\to1$ of anomalous topological order. For minimal nonzero $(N,N_f)$, the allowed minimal extensions are $N=1,3,4,12$, depending on divisibility of $N_f$ by 2 and 3. Combining Witten anomaly and other constraints, we prove that $N=N_c=N_f=3$, with 3 families and 3 colors, is the unique minimal solution to match with the color-center baryon-to-quark symmetry extension $1\to Z_{N_c}\to Spin\times_{Z_2^F}Z_{2N_cN_f,{\bf Q}+N_c{\bf L}}\to Spin\times_{Z_2^F}Z_{2 N_f,{\bf B+L}}^F\to1$. We also prove that $A_{Z_3}p_1=0\mod3$ for the mod 3 cohomology class.

hep-th↗
arXiv2025

Anomalous (3+1)d Fermionic Topological Quantum Field Theories via Symmetry Extension

Discrete finite-group global symmetries may suffer from nonperturbative 't-Hooft anomalies. Such global anomalies can be canceled by anomalous symmetry-preserving topological quantum field theories (TQFTs), which contain no local point operators but only extended excitations such as line and surface operators. In this work, we study mixed gauge-gravitational nonperturbative global anomalies of Weyl fermions (or Weyl semimetals in condensed matter) charged under discrete Abelian internal symmetries in four-dimensional spacetime, with spacetime-internal fermionic symmetry $G=$Spin$\times_{\mathbb{Z}_2^{\rm F}}\mathbb{Z}_{2m}^{\rm F}$ or Spin$\times\mathbb{Z}_n$ that contains fermion parity $\mathbb{Z}_{2}^{\rm F}$. We determine the minimal finite gauge group $K$ of anomalous $G$-symmetric TQFTs that can match the fermionic anomaly via the symmetry-extension construction $1 \to K \to G_{\rm Tot} \to G \to 1$, where the anomaly in $G$ is trivialized upon pullback to $G_{\rm Tot}$, computed by Atiyah-Patodi-Singer eta invariant. This allows one to replace a $G$-symmetric four-dimensional Weyl fermion by an anomalous $G$-symmetric discrete-$K$-gauge TQFT as an alternative low-energy theory in the same deformation class. As an application, we show that the four-dimensional Standard Model with 15 Weyl fermions per family, in the absence of a sterile right-handed neutrino $\nu_R$, exhibits mixed gauge-gravitational global anomalies between baryon and lepton number symmetries $({\bf B \pm L})$ and spacetime diffeomorphisms. We identify the corresponding minimal $K$-gauge fermionic TQFT that cancels these anomalies and can be interpreted as a gapped, topologically ordered dark sector replacing missing Weyl fermions via symmetry extension, without invoking conventional Anderson-Higgs symmetry breaking.

hep-th↗
arXiv2025

Anomaly of 4d Weyl fermions with discrete symmetries

We derive explicit anomaly-index formulas for four-dimensional Weyl fermions charged under the finite symmetries Spin$\times\mathbb Z_n$ and Spin$\times_{\mathbb Z_2^{\mathrm F}}\mathbb Z_{2m}^{\mathrm F}$. The strategy is to start from the standard perturbative anomaly indices for Spin$\times$U(1) and Spin$\times_{\mathbb Z_2^{\mathrm F}}$U(1)$=$Spin$^c$, and then restrict the continuous U(1) symmetry to a finite cyclic subgroup. On the level of invertible field theories this gives natural homomorphisms $$ \mathrm{TP}_5(\mathrm{Spin}\times\mathrm U(1)) \longrightarrow \mathrm{TP}_5(\mathrm{Spin}\times\mathbb Z_n),\quad \mathrm{TP}_5(\mathrm{Spin}^c) \longrightarrow \mathrm{TP}_5(\mathrm{Spin}\times_{\mathbb Z_2^{\mathrm F}}\mathbb Z_{2m}^{\mathrm F}). $$ We compute these maps explicitly by evaluating reduced $\eta$-invariants on geometric representatives of the finite anomaly groups. For Spin$\times\mathbb Z_n$, the relevant backgrounds are the five-dimensional lens-space bundle $X(n;1,1)$ and the product $L(n;1)\times$K3. For Spin$\times_{\mathbb Z_2^{\mathrm F}}\mathbb Z_{2m}^{\mathrm F}$, the relevant backgrounds are $L(m;1,1,1)$ and, depending on the parity of $m$, either $L(m;1)\times$Enriques or $L(m;1)\times$K3. The output is a pair of integer-valued anomaly indices for each finite symmetry. These indices are normalized in the cyclic factors of the finite anomaly group, so they can be used directly in anomaly-cancellation checks for fermions with discrete gauge or global symmetries. They also provide the input for symmetry-extension applications to topologically ordered quantum dark matter: in Standard-Model examples, the indices identify the discrete ${\bf B}+{\bf L}$ mixed gauge--gravitational anomaly left by missing sterile right-handed neutrinos and the finite $K$-gauge topological sector that can match it.

hep-th↗
arXiv2024

Topological Responses of the Standard Model Gauge Group

The local Lie algebra of the Standard Model (SM) is $su(3)\times su(2) \times u(1)$, yet its global gauge group, $G_{{\rm SM}_{\rm q}}=$SU(3)$\times$SU(2)$\times$U(1)/$\mathbb{Z}_{\rm q}$, q$=1,2,3,6$ remains undetermined. Building on previous work on 4d anomalies and 5d cobordism invariants, we classify lower-dimensional invertible field theories (iFTs) or symmetry-protected topological states (SPTs) in 4d, 3d, 2d, and 1d. While the integer SPTs are hard to detect, the fractional SPTs produce measurable topological responses. In particular, the symmetry fractionalization labeled by $k\in\mathbb{Z}_{6/{\rm q}}$ in [arXiv:2411.18160] introduces the symmetry-enriched SM variants, denoted as SM$_{({\rm q},k)}$. We further introduce a new integer $n$ series of baryon-minus-lepton $({\bf B}-{\bf L})$-like U(1) symmetries, $X_n \equiv n (\mathbf{B}-\mathbf{L}) + (1-\frac{n}{N_c})\tilde{Y}$ with electroweak hypercharge $\tilde{Y}$, $n\ge1$, $N_c=3$, where the charge $q_{X_n} = q_{\tilde{Y}} \mod n$. Analyzing the symmetry-enriched SM with 0-form and 1-form symmetries $(G_{[0]}, G_{[1]})$, symmetry-twist group homomorphism $\rho$, and symmetry frationalization obstruction $[\beta]$, their spacetime-internal gauge bundle constraints, and their mixed anomalies, we derive the fractional topological response $\sigma_n({\rm q},k)=\frac{{\rm q}(1-n)\gcd(2,n)}{2n}+\frac{k{\rm q}}{6}\mod1.$ Our $\sigma_n$ response requires more general Spin$^c$ manifolds for odd $n$ and Spin manifolds for even $n$. For a given $n$ (with $n\ge 7$ and $n\ne 10,12,15,30$), $\sigma_n$ uniquely fixes the gauge group parameter q and fractionalization label $k$. Moreover, using pairs such as $(n_1,n_2)=(2,3),(2,5),(3,4),(3,5),(4,5)$, etc. uniquely distinguishes SM$_{({\rm q},k)}$. Our results illuminate the global structure of the SM gauge group via measurable topological responses.

hep-th↗
arXiv2024

C-R-T Fractionalization in the First Quantized Hamiltonian Theory

Recent research has revealed that the CRT symmetry for fermions exhibits a fractionalization distinct from the $\mathbb{Z}_2^{\mathcal{C}}\times\mathbb{Z}_2^{\mathcal{R}}\times\mathbb{Z}_2^{\mathcal{T}}$ for scalar bosons. In fact, the CRT symmetry for fermions can be extended by internal symmetries such as fermion parity, thereby forming a group extension of the $\mathbb{Z}_2$ direct product. Conventionally, a Majorana fermion is defined by one Dirac fermion with trivial charge conjugation. However, when the spacetime dimension $d+1=5,6,7\bmod8$, the real dimension of Majorana fermion (dim$_{\mathbb{R}}\chi_{\mathcal{C}\ell(d,0)}$) aligns with the real dimension of Dirac fermion (dim$_{\mathbb{R}}\psi_{\mathcal{C}\ell(d)}$), rather than being half, which necessitates the introduction of a symplectic Majorana fermion, defined by two Dirac fermions with trivial charge conjugation. To include these two types of Majorana fermions, we embed the theory in $n_{\mathbb{R}}$ and define the Majorana fermion field as a representation of the real Clifford algebra with 8-fold periodicity. Within the Hamiltonian formalism, we identify the 8-fold CRT-internal symmetry groups across general dimensions. Similarly, Dirac fermion field is defined as a representation of the complex Clifford algebra with 2-fold periodicity. Interestingly, we discover that the CRT-internal symmetry groups exhibit an 8-fold periodicity that is distinct from that of the complex Clifford algebra. In certain dimensions where distinct mass terms can span a mass manifold, the CRT-internal symmetries can act non-trivially upon this mass manifold. Employing domain wall reduction method, we are able to elucidate the relationships between symmetries across different dimensions.

cond-mat.str-el↗
arXiv2024

A matrix solution to any polygon equation

In this article, we construct matrices associated to Pachner $\frac{n-1}{2}$-$\frac{n-1}{2}$ moves for odd $n$ and matrices associated to Pachner $(\frac{n}{2}-1)$-$\frac{n}{2}$ moves for even $n$. The entries of these matrices are rational functions of formal variables in a field. We prove that these matrices satisfy the $n$-gon equation for any $n$.

math-ph↗
arXiv2023

C-R-T Fractionalization, Fermions, and Mod 8 Periodicity

Charge conjugation (C), mirror reflection (R), time reversal (T), and fermion parity $(-1)^{\rm F}$ are basic discrete spacetime and internal symmetries of the Dirac fermions. In this article, we determine the group, called the C-R-T fractionalization, which is a group extension of $\mathbb{Z}_2^{\rm C}\times\mathbb{Z}_2^{\rm R}\times\mathbb{Z}_2^{\rm T}$ by the fermion parity $\mathbb{Z}_2^{\rm F}$, and its extension class in all spacetime dimensions $d$, for a single-particle fermion theory. For Dirac fermions, with the canonical CRT symmetry $\mathbb{Z}_2^{\rm CRT}$, the C-R-T fractionalization has two possibilities that only depend on spacetime dimensions $d$ modulo 8, which are order-16 nonabelian groups, including the famous Pauli group. For Majorana fermions, we determine the R-T fractionalization in all spacetime dimensions $d=0,1,2,3,4\mod8$, an order-8 abelian or nonabelian group. For Weyl fermions, we determine the C or T fractionalization in all even spacetime dimensions $d$, which is an order-4 abelian group. We only have an order-2 $\mathbb{Z}_2^{\rm F}$ group for Majorana-Weyl fermions. We determine the maximal number of linearly independent Dirac and Majorana mass terms and construct them explicitly. We also discuss how the conventional Dirac and Majorana mass terms break the symmetries C, R, or T. We study the domain wall dimensional reduction of the fermions and their C-R-T fractionalization: from $d$-dim Dirac to $(d-1)$-dim Dirac or Weyl and from $d$-dim Majorana to $(d-1)$-dim Majorana or Majorana-Weyl.

hep-th↗
arXiv2023

Topological complexity of finding flex points on cubic plane curves

We prove a lower bound for the topological complexity, in the sense of Smale, of the problem of finding a flex point on a cubic plane curve. The key is to bound the Schwarz genus of a cover associated to this problem. We also show that our lower bound for the complexity is close to be the best possible.

math.GT↗
arXiv2023

The photography method: solving pentagon equation

In the present paper, we consider two applications of the pentagon equation. The first deals with actions of flips on edges of triangulations labelled by rational functions in some variables. The second can be formulated as a system of linear equations with variables corresponding to triangles of a triangulation. The general method says that if there is some general {\em data} (say, edge lengths or areas) associated with {\em states} (say, triangulations) and a general {\em data transformation rule} (say, how lengths or areas are changed under flips) then after returning to the initial state we recover the initial data.

math.GT↗
arXiv2022

Quantum 4d Yang-Mills Theory and Time-Reversal Symmetric 5d Higher-Gauge Topological Field Theory

We explore 4d Yang-Mills gauge theories (YM) living as boundary conditions of 5d gapped short/long-range entangled (SRE/LRE) topological states. Specifically, we explore 4d time-reversal symmetric pure YM of an SU(2) gauge group with a second-Chern-class topological term at $θ=π$ (SU(2)$_{θ=π}$ YM), by turning on background fields for both the time-reversal (i.e., on unorientable manifolds) and 1-form center global symmetry. We find four classes of time-reversal and Lorentz symmetry-enriched SU(2)$_{θ=π}$ YM, labeled by $(K_1, K_2)$: $K_1=0,1$ specifies Kramers singlet/doublet Wilson line and new mixed higher 't Hooft anomalies; $K_2=0,1$ specifies boson/fermionic Wilson line and a new Wess-Zumino-Witten-like counterterm. Higher anomalies indicate that to realize all higher $n$-global symmetries locally on $n$-simplices, the 4d theory becomes a boundary of a 5d higher-symmetry-protected topological state (SPTs, as an invertible topological quantum field theory (TQFT) or a cobordism invariant in math, or as a 5d higher-symmetric interacting topological superconductor in condensed matter). By dynamically gauging the 1-form symmetry, we transform a 5d bulk SRE SPTs into an LRE symmetry-enriched topologically ordered state (SETs); thus we obtain the 4d SO(3)$_{θ=π}$ YM-5d LRE-higher-SETs coupled system with higher-form gauge fields. We further derive new exotic anyonic statistics of extended objects such as 2-worldsheet of strings and 3-worldvolume of branes, physically characterizing the 5d SETs. We discover triple and quadruple link invariants associated with the 5d higher-gauge TQFTs, hinting at a relation between non-supersymmetric 4d pure YM and topological links in 5d. We provide 4d-5d lattice simplicial complex regularizations and bridge to 4d quantum spin liquids. We constrain gauge dynamics by higher anomalies and a higher symmetry-extension method.

hep-th↗
arXiv2022

Proton Stability: From the Standard Model to Beyond Grand Unification

A proton is known for its longevity, but what is its lifetime? While many Grand Unified Theories predict the proton decay with a finite lifetime, we show that the Standard Model (SM) and some versions of Ultra Unification (which replace sterile neutrinos with new exotic gapped/gapless sectors, e.g., topological or conformal field theory under global anomaly cancellation constraints) with a discrete baryon plus lepton symmetry permit a stable proton. For the 4d SM with $N_f$ families of 15 or 16 Weyl fermions, in addition to the continuous baryon minus lepton U(1)$_{\bf B - L}$ symmetry, there is also a compatible discrete baryon plus lepton $\mathbb{Z}_{2N_f, \bf B + L}$ symmetry. The $\mathbb{Z}_{2N_f, \bf B + L}$ is discrete due to the ABJ anomaly under the BPST SU(2) instanton. Although both U(1)$_{\bf B - L}$ and $\mathbb{Z}_{2N_f, \bf B + L}$ symmetries are anomaly-free under the dynamical SM gauge field, it is important to check whether they have mixed anomalies with the gravitational background field and higher symmetries (whose charged objects are Wilson electric or 't Hooft magnetic line operators) of SM. We can also replace the U(1)$_{\bf B - L}$ with a discrete variant $\mathbb{Z}_{4,X}$ for $X \equiv 5({\bf B - L})-\frac{2}{3} {\tilde Y}$ of electroweak hypercharge ${\tilde Y}$. We explore a systematic classification of candidate perturbative local and nonperturbative global anomalies of the 4d SM, including all these gauge and gravitational backgrounds, via a cobordism theory, which controls the SM's deformation class. We discuss the proton stability of the SM and Ultra Unification in the presence of discrete ${\bf B + L}$ symmetry protection, in particular (U(1)$_{\bf B - L} \times \mathbb{Z}_{2N_f,\bf B + L})/{\mathbb{Z}_2^{\rm F}}$ or $(\mathbb{Z}_{4,X} \times \mathbb{Z}_{2N_f, \bf B + L})/{\mathbb{Z}_2^{\rm F}}$ symmetry with the fermion parity $\mathbb{Z}_2^{\rm F}$.

hep-ph↗
arXiv2022

Cobordism and Deformation Class of the Standard Model

't Hooft anomalies of quantum field theories (QFTs) with an invertible global symmetry G (including spacetime and internal symmetries) in a $d$d spacetime are known to be classified by a $d+1$d cobordism group TP$_{d+1}$(G), whose group generator is a $d+1$d cobordism invariant written as an invertible topological field theory (iTFT) Z$_{d+1}$. The deformation class of QFT is recently proposed to be specified by its symmetry G and an iTFT Z$_{d+1}$. Seemingly different QFTs of the same deformation class can be deformed to each other via quantum phase transitions. In this work, we ask which deformation class controls the 4d ungauged or gauged (SU(3)$\times$SU(2)$\times$U(1))/$\mathbb{Z}_q$ Standard Model (SM) for $q=1,2,3,6$ with a continuous or discrete $(\bf{B}-\bf{L})$ symmetry. We show that the answer contains some combination of 5d iTFTs: two $\mathbb{Z}$ classes associated with $(\bf{B}-\bf{L})^3$ and $(\bf{B}-\bf{L})$-(gravity)$^2$ 4d perturbative local anomalies, a mod 16 class Atiyah-Patodi-Singer $η$ invariant and a mod 2 class Stiefel-Whitney $w_2w_3$ invariant associated with 4d nonperturbative global anomalies, and additional $\mathbb{Z}_3\times\mathbb{Z}_2$ classes involving higher symmetries whose charged objects are Wilson electric or 't Hooft magnetic line operators. Out of $\mathbb{Z}$ classes of local anomalies and 24576 classes of global anomalies, we pin down a deformation class of SM labeled by $(N_f,n_{ν_{R}},$ p$',q)$, the family and "right-handed sterile" neutrino numbers, magnetic monopole datum, and mod $q$ relation. Grand Unifications and Ultra Unification that replaces sterile neutrinos with new exotic gapped/gapless sectors (e.g., topological or conformal field theory) or gravitational sectors with topological or cobordism constraints, all reside in an SM deformation class. Neighbor phases/transitions/critical regions near SM exhibit beyond SM phenomena.

hep-th↗
arXiv2022

3+1d Boundaries with Gravitational Anomaly of 4+1d Invertible Topological Order for Branch-Independent Bosonic Systems

We study bosonic systems on a spacetime lattice defined by path integrals of commuting fields. We introduce branch-independent bosonic (BIB) systems, whose path integral is independent of the branch structure of the spacetime simplicial complex, even for a spacetime with boundaries. In contrast, a generic lattice bosonic (GLB) system's path integral may depend on the branch structure. We find the invertible topological order characterized by the Stiefel-Whitney cocycle (e.g., 4+1d w$_2$w$_3$) to be nontrivial for BIB systems, but this topological order and a trivial gapped tensor product state belong to the same phase for GLB systems. The invertible topological orders in GLB systems are not classified by the oriented cobordism. The branch dependence on a lattice may be related to the orthonormal frame of smooth manifolds and the framing anomaly of continuum field theories. The branch structure on a discretized lattice may be related to a frame structure on a smooth manifold that trivializes any Stiefel-Whitney classes. We construct BIB systems to realize the w$_2$w$_3$ topological order, and its 3+1d gapped or gapless boundaries. A 3+1d $\mathbb{Z}_2$ gauge theory with (1) fermionic $\mathbb{Z}_2$ gauge charge particle trivializes w$_2$ and (2) fermionic $\mathbb{Z}_2$ gauge flux line trivializes w$_3$. In particular, if the flux loop's worldsheet is unorientable, then an orientation-reversal 1d worldline corresponds to a fermion worldline carrying no $\mathbb{Z}_2$ gauge charge. Spin and Spin$^c$ structures trivialize the w$_2$w$_3$ global pure gravitational anomaly to zero (which helps to construct 3+1d $\mathbb{Z}_2$ and all-fermion U(1) gauge theories), but the Spin$^h$ and Spin$\times_{\mathbb{Z}_2}$Spin$(n \geq 3)$ structures modify the w$_2$w$_3$ into a global mixed gauge-gravitational anomaly, which helps to constrain Grand Unifications (e.g., $n=10,18$) or construct new models.

cond-mat.str-el↗
arXiv2021

Studying complex manifolds by using groups $G_{n}^{k}$ and $Γ_{n}^{k}$

In the present paper, we study several complex manifolds by using the following idea. First, we construct a certain moduli space and study the fundamental group of this space. This fundamental group is naturally mapped to the groups $G_{n}^{k}$ and $Γ_{n}^{k}$. This is the step towards "complexification" of the $G_{n}^{k}$ and $Γ_{n}^{k}$ approach first developed in \cite{2019arXiv190508049M}.

math.GT↗
arXiv2020

Higher Anomalies, Higher Symmetries, and Cobordisms II: Lorentz Symmetry Extension and Enriched Bosonic/Fermionic Quantum Gauge Theory

We systematically study Lorentz symmetry extensions in quantum field theories (QFTs) and their 't Hooft anomalies via cobordism. The total symmetry $G'$ can be expressed in terms of the extension of Lorentz symmetry $G_L$ by an internal global symmetry $G$ as $1 \to G \to G' \to G_L \to 1$. By enumerating all possible $G_L$ and symmetry extensions, other than the familiar SO/Spin/O/Pin$^{\pm}$ groups, we introduce a new EPin group (in contrast to DPin), and provide natural physical interpretations to exotic groups E($d$), EPin($d$), (SU(2)$\times$E(d))/$\mathbb{Z}_2$, (SU(2)$\times$EPin(d))/$\mathbb{Z}_2^{\pm}$, etc. By Adams spectral sequence, we systematically classify all possible $d$d Symmetry Protected Topological states (SPTs as invertible TQFTs) and $(d-1)$d 't Hooft anomalies of QFTs by co/bordism groups and invariants in $d\leq 5$. We further gauge the internal $G$, and study Lorentz symmetry-enriched Yang-Mills theory with discrete theta terms given by gauged SPTs. We not only enlist familiar bosonic Yang-Mills but also discover new fermionic Yang-Mills theories (when $G_L$ contains a graded fermion parity $\mathbb{Z}_2^F$), applicable to bosonic (e.g., Quantum Spin Liquids) or fermionic (e.g., electrons) condensed matter systems. For a pure gauge theory, there is a one form symmetry $I_{[1]}$ associated with the center of the gauge group $G$. We further study the anomalies of the emergent symmetry $I_{[1]}\times G_L$ by higher cobordism invariants as well as QFT analysis. We focus on the simply connected $G=$SU(2) and briefly comment on non-simply connected $G=$SO(3), U(1), other simple Lie groups, and Standard Model gauge groups (SU(3)$\times$SU(2)$\times$U(1))/$\mathbb{Z}_q$. We comment on SPTs protected by Lorentz symmetry, and the symmetry-extended trivialization for their boundary states.

hep-th↗
arXiv2020

Beyond Standard Models and Grand Unifications: Anomalies, Topological Terms, and Dynamical Constraints via Cobordisms

We classify and characterize all invertible anomalies and all allowed topological 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 $\mathbb{Z}$ free classes, and (2) non-perturbative/global anomalies, classified by finite group $\mathbb{Z}_N$ torsion classes. Our work built from [arXiv:1812.11967] fuses the math tools of Adams spectral sequence, Thom-Madsen-Tillmann spectra, and Freed-Hopkins theorem. For example, we compute bordism groups $Ω^{G}_d$ and their invertible topological field theory invariants, which characterize $d$d topological terms and $(d-1)$d anomalies, protected by the following symmetry group $G$: $Spin\times \frac{SU(3)\times SU(2)\times U(1)}{\mathbb{Z}_q}$ for SM with $q=1,2,3,6$; $\frac{Spin \times Spin(n)}{\mathbb{Z}_2^F}$ or $Spin \times Spin(n)$ for SO(10) or SO(18) GUT as $n=10, 18$; $Spin \times SU(n)$ for Georgi-Glashow SU(5) GUT as $n=5$; $\frac{Spin\times \frac{SU(4)\times(SU(2)\times SU(2))}{\mathbb{Z}_{q'}}}{\mathbb{Z}_2^F}$ for Pati-Salam GUT as $q'=1,2$; and others. For SM with an extra discrete symmetry, we obtain new anomaly matching conditions of $\mathbb{Z}_{16}$, $\mathbb{Z}_{4}$, and $\mathbb{Z}_{2}$ classes in 4d 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.

hep-th↗
arXiv2020

Extension of elementary $p$-groups and its application in classification of groups of prime exponent

Let $p$ be a prime number and $\mathbb{Z}_p=\mathbb{Z}/p\mathbb{Z}$. We study finite groups with abelian derived subgroup and exponent $p$ in terms of group extension data and their matrix presentations. We show a one-to-one correspondence between the following two sets: (i) the isoclasses of class 2 groups of exponent $p$ and order $p^{m+n}$ and with derived subgroup $\mathbb{Z}_p^n$, and (ii) the set $\text{Gr}(n,\text{AS}_m(\mathbb{Z}_p))/\text{GL}_m(\mathbb{Z}_p)$ of orbits of $\text{Gr}(n,\text{AS}_m(\mathbb{Z}_p))$ under the congruence action by $\text{GL}_m(\mathbb{Z}_p)$, where $\text{Gr}(n,\text{AS}_m(\mathbb{Z}_p))$ is the set of $n$-dimensional subspaces of anti-symmetric matrices of order $m$ over $\mathbb{Z}_p$. We give a description of the orbit spaces $\text{Gr}(2, \text{AS}_m(\mathbb{Z}_p))/\text{GL}_m(\mathbb{Z}_p)$ for all $m$ and $p$ by applying the theory of pencils of anti-symmetric matrices. Based on this, we show complete sets of representatives of orbits of $\text{Gr}(3,\text{AS}_4(\mathbb{Z}_3))/\text{GL}_4(\mathbb{Z}_3)$, $\text{Gr}(4, \text{AS}_4(\mathbb{Z}_3))/\text{GL}_4(\mathbb{Z}_3)$ and $\text{Gr}(3, \text{AS}_5(\mathbb{Z}_3))/\text{GL}_5(\mathbb{Z}_3)$. As a consequence, we obtain a classification of corresponding class 2 groups of exponent $p$. In particular, we recover the classification of groups with exponent 3 and order $\le 3^8$.

math.GR↗
arXiv2019

Higher Anomalies, Higher Symmetries, and Cobordisms III: QCD Matter Phases Anew

We explore QCD$_4$ quark matter, the $μ$-T (chemical potential-temperature) phase diagram, possible 't Hooft anomalies, and topological terms, via non-perturbative tools of cobordism theory and higher anomaly matching. We focus on quarks in 3-color and 3-flavor on bi-fundamentals of SU(3), then analyze the continuous and discrete global symmetries and pay careful attention to finite group sectors. We input constraints from $T=CP$ or $CT$ time-reversal symmetries, implementing QCD on unorientable spacetimes and distinct topology. Examined phases include the high T QGP (quark-gluon plasma/liquid), the low T ChSB (chiral symmetry breaking), 2SC (2-color superconductivity) and CFL (3-color-flavor locking superconductivity) at high density. We introduce a possibly useful but only approximate higher anomaly, involving discrete 0-form axial and 1-form mixed chiral-flavor-locked center symmetries, matched by the above four QCD phases. We also enlist as much as possible, but without identifying all of, 't Hooft anomalies and topological terms relevant to Symmetry Protected/Enriched Topological states (SPTs/SETs) of gauged SU(2) or SU(3) QCD$_d$-like matter theories in general in any spacetime dimensions $d=2,3,4,5$ via cobordism.

hep-th↗