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In the standard lore, the baryon asymmetry of the present universe is attributed to the leptogenesis from the sterile right-handed neutrino with heavy Majorana mass decaying into the Standard Model's leptons at the very early universe -- called the Majorana fermion's leptogenesis; while the electroweak sphaleron causes baryogenesis at a later time. In this work, we propose a new mechanism, named topological leptogenesis, to explain the lepton asymmetry. Topological leptogenesis replaces some of the sterile neutrinos by introducing a new gapped topological order sector (whose low-energy exhibits topological quantum field theory with long-range entanglement) that can cancel the baryon minus lepton $({\bf B} - {\bf L})$ mixed gauge-gravitational anomaly of the Standard Model. Then the Beyond-the-Standard-Model dark matter consists of topological quantum matter, such that the gapped non-particle excitations of extended line and surface defect with fractionalization and anyon charges can decay into the Standard Model particles. In addition, gravitational leptogenesis can be regarded as an intermediate step (between Majorana particle leptogenesis and topological non-particle leptogenesis) to mediate such decaying processes from the highly entangled gapped topological order excitations., Comment: Sequel to: arXiv:2006.16996, arXiv:2012.15860, arXiv:2302.14862. The idea of topological leptogenesis has been presented by the author and circulated in his various seminars since the work done in 2020. This is a late follow-up writeup

Built upon the previous work on the 4d anomalies and 5d cobordism invariants (namely, 5d invertible field theory [iTFT] or symmetry-protected topological state [SPTs]) of the Standard Model [SM] gauge theory with compatible (SU(3)$\times$SU(2)$\times$U(1))/$\mathbb{Z}_q$ gauge group for $q=1,2,3,6$, we further enumerate lower-dimensional iTFT / SPTs in 4d, 3d, 2d, and 1d. While the 4d SPTs are interesting gapped phases attachable to the SM, those integer classes of SPTs (either the torsion or the free cobordism class) are not universal and are difficult to detect. However, fractional SPTs are more universal and can be detected by a topological response similar to the Hall conductance under an appropriate symmetry twist background field. We find a nontrivial symmetry-extension from a magnetic 1-symmetry U(1)$_{[1]}^m$ by an electric 1-symmetry $\mathbb{Z}_{6/q,[1]}^e$ introducing the $k \in \mathbb{Z}_{6/q}$ fractionalization class. While we also introduce the background fields for Baryon minus Lepton number $({\bf B}-{\bf L})$ like 0-symmetries of U(1)$_{{\bf Q} - N_c {\bf L}}$, U(1)$_X$, and $\mathbb{Z}_{4,X}$, that introduces two more data $\alpha \in 1,3$ and $\beta \in 1,5$ for the SMs. We present their Hall conductance response data (denoted as $\sigma, \sigma'$, and $\sigma''$ respectively), between 0-symmetry and 1-symmetry background fields, and show their fractional dependence on the SM data $(q,\alpha,\beta, k)$., Comment: Sequel to: arXiv:1910.14668, arXiv:2112.14765, arXiv:2204.08393

Charge conjugation (C), mirror reflection (R), and time reversal (T) symmetries, along with internal symmetries, are essential for massless Majorana and Dirac fermions. These symmetries are sufficient to rule out potential fermion bilinear mass terms, thereby establishing a gapless free fermion fixed point phase, pivotal for symmetric mass generation (SMG) transition. In this work, we systematically study the anomaly of C-R-T-internal symmetry in all spacetime dimensions by analyzing the projective representation (i.e. the fractionalization) of the C-R-T-internal symmetry group in the quantum many-body Hilbert space on the lattice. By discovering the fermion-flavor-number-dependent C-R-T-internal symmetry's anomaly structure, we demonstrate an alternative way to derive the minimal flavor number for SMG, which shows consistency with known results from K\"ahler-Dirac fermion or cobordism classification. Our findings reveal that, in general spatial dimensions, either 8 copies of staggered Majorana fermions or 4 copies of staggered Dirac fermions admit SMG. By directly searching for 4-fermion interactions that form commuting stabilizers respecting all symmetry constraints, we can prove the explicit SMG gapping retained a unique ground state in the codespace. Furthermore, we establish the correspondence between the symmetry operators of staggered fermions and free fermions, which is instrumental in facilitating the analysis of symmetry fractionalization at the field theory level., Comment: 36 pages, 2 figures, 13 tables, sequel to arXiv:2109.15320, arXiv:2312.17126, and arXiv:2412.11958. Motivation: CRT-internal symmetry can rule out fermion bilinear mass terms. Key Result: Examine the fractionalization and anomalies of the CRT-internal symmetry on the lattice and find the minimum flavor number for SMG in all dimensions for both Majorana and Dirac fermions

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., Comment: 54 pages, 39 tables, sequel to arXiv:2109.15320 and arXiv:2312.17126. Include symplectic Majorana in d+1=5,6,7 mod 8, and symplectic Majorana-Weyl in d+1=6 mod 8. Motivation: use CRT-internal symmetries to rule out fermion bilinear masses. Key result: systematically assigned CRT-internal symmetries in all spacetime dimensions for both Majorana and Dirac fermions

We investigate (3+1)d topological orders in fermionic systems with an anomalous $\mathbb{Z}_{2N}^{\mathrm{F}}$ symmetry, where its $\mathbb{Z}_2^{\mathrm{F}}$ subgroup is the fermion parity. Such an anomalous symmetry arises as the discrete subgroup of the chiral U(1) symmetry of $\nu$ copies of Weyl fermions of the same chirality. Guided by the crystalline correspondence principle, we construct (3+1)d symmetry-preserving gapped states on the boundary of a closely related (4+1)d $C_N\times \mathbb{Z}_2^{\mathrm{F}}$ symmetry-protected topological (SPT) state (with $C_N$ being the $N$-fold rotation), whenever it is possible. In particular, for $\nu=N$, we show that the (3+1)d symmetric gapped state admits a topological $\mathbb{Z}_4$ gauge theory description at low energy, and propose that a similar theory saturates the corresponding $\mathbb{Z}_{2N}^\mathrm{F}$ anomaly. For $N\nmid \nu$, our construction cannot produce any topological quantum field theory (TQFT) symmetric gapped state; but for $\nu=N/2$, we find a non-TQFT symmetric gapped state via stacking lower-dimensional (2+1)d non-discrete-gauge-theory topological orders inhomogeneously. For other values of $\nu$, no symmetric gapped state is possible within our construction, which is consistent with the theorem by Cordova-Ohmori., Comment: 15 pages, 3 figures

Fibonacci string-net condensate, a complex topological state that supports non-Abelian anyon excitations, holds promise for fault-tolerant universal quantum computation. However, its realization by a static-lattice Hamiltonian has remained elusive due to the inherent high-order interactions demanded. Here, we introduce a scalable dynamical string-net preparation (DSNP) approach, suitable even for near-term quantum processors, that can dynamically prepare the state through reconfigurable graphs. DSNP enables the creation and manipulation of the Fibonacci string-net condensate (Fib-SNC). Using a superconducting quantum processor, we couple the DSNP approach with a composite error-mitigation strategy on deep circuits to successfully create, measure, and braid Fibonacci anyons in two spatial dimensions (2D) demonstrating their potential for universal quantum computation. To this end, we measure anyon charges for two species of anyons associated with the doubled topological quantum field theory underlying Fid-SNC, with an average experimental accuracy of 94%. We validate that a scalable 2D braiding operation on a logical qubit encoded on three anyons yields the golden ratio $\phi$ with 98% average accuracy and 8% measurement uncertainty. We further sample the Fib-SNC wavefunction to estimate the chromatic polynomial at $\phi+2$ for various graphs. Given the established computational hardness of the chromatic polynomial, the wavefunction amplitude is classically hard to evaluate. Our results establish the first proof of principle that scalable DSNP can open doors to fault-tolerant universal quantum computation and to classically-hard problems., Comment: 4 pages and 4 figures with Supplemental Materials (47 pages, 18 Figures)

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., Comment: 45 pages, 5 figures, 18 tables, see C-P-T fractionalization in arXiv:2109.15320, and an application of domain wall dimensional reduction in arXiv:2312.14928 v2 update: clarified the definition of Majorana fermions and the choice of Dirac and Majorana mass terms, corrected the statement of CRT theorem, added a new theorem on domain wall dimensional reduction, and some minor revisions

Family Puzzle or Generation Problem demands an explanation of why there are 3 families or generations of quarks and leptons in the Standard Model of particle physics. Here we propose a novel solution -- the multiple of 3 families of 16 Weyl fermions (namely $(N_f=3) \times 16$) in the 3+1d spacetime dimensions are topologically robust due to constraints rooted in profound mathematics (such as Hirzebruch signature and Rokhlin theorems, and cobordism) and derivable in physics (such as chiral edge states, quantized thermal Hall conductance, and gravitational Chern-Simons theory), which holds true even forgetting or getting rid of any global symmetry or gauge structure of the Standard Model. By the dimensional reduction through a sequence of sign-reversing mass domain wall of domain wall and so on, we reduce the Standard Model fermions to obtain the $(N_f=3) \times 16$ multiple of 1+1d Majorana-Weyl fermion with a total chiral central charge $c_-=24$. Effectively via the fermionization-bosonization, the 1+1d theory becomes 3 copies of $c_-=8$ of (E$_8)_1$ conformal field theory, living on the boundary of 3 copies of 2+1d E$_8$ quantum Hall states. Based on the framing anomaly-free $c_- = 0 \mod 24$ modular invariance, the framed bordism and string bordism $\mathbb{Z}_{24}$ class, the 2-framing and $p_1$-structure, the $w_1$-$p_1$ bordism $\mathbb{Z}_3$ class constraints, we derive the family number constraint $N_f \in (\frac{48}{16} =\frac{24}{8}=3) \mathbb{Z}$. The dimensional reduction process, although not necessary, is sufficiently supported by the $\mathbb{Z}_{16}$ class Smith homomorphism. We also comment on the $\frac{45}{15}=3$ relation: the 3 families of 15 Weyl-fermion Standard Model vacuum where the absence of some sterile right-handed neutrinos is fulfilled by additional topological field theories or conformal field theories in Ultra Unification.

We investigate the bilayer nickelates as a platform to realize the symmetric mass generation (SMG) insulator, a featureless Mott insulator that arises due to the Lieb-Schultz-Mattis (LSM) anomaly cancellation in bilayer spin-1/2 lattice systems. Through a single-orbital bilayer square lattice model involving intralayer hopping $t$ and interlayer superexchange interaction $J$, we demonstrate the emergence of high-temperature superconductivity (SC) upon doping the SMG insulator. The SC phase features $s$-wave interlayer spin-singlet pairing and exhibits a crossover between the BCS and BEC limits by tuning the $J/t$ ratio. We estimate the SC transition temperature $T_c$ from both the weak and strong coupling limits at the mean-field level. Our findings offer insights into the experimentally observed decrease in $T_c$ with pressure and the strange metal behavior above $T_c$. Additionally, we propose that both Ni $3d_{z^2}$ and $3d_{x^2-y^2}$ orbitals can exhibit superconductivity in La$_3$Ni$_2$O$_7$ under pressure, but their $T_c$ should vary in opposite ways under doping. This characteristic difference suggests a potential experimental pathway to identify which electronic orbital plays the principal role in the formation of superconductivity in this system., Comment: 11 pages, 5 figures, 2 tables

In the Standard Model, some combination of the baryon $\bf B$ and lepton $\bf L$ number symmetry is free of mixed anomalies with strong and electroweak $su(3) \times su(2) \times u(1)_{\tilde Y}$ gauge forces. However, it can still suffer from a mixed gravitational anomaly, hypothetically pertinent to leptogenesis in the very early universe. This happens when the total "sterile right-handed" neutrino number $n_{\nu_R}$ is not equal to the family number $N_f$. Thus the invertible $\bf B - L$ symmetry current conservation can be violated quantum mechanically by gravitational backgrounds such as gravitational instantons. In specific, we show that a noninvertible categorical $\bf B - L$ generalized symmetry still survives in gravitational backgrounds. In general, we propose a construction of noninvertible symmetry charge operators as topological defects derived from invertible anomalous symmetries that suffer from mixed gravitational anomalies. Examples include the perturbative local and nonperturbative global anomalies classified by $\mathbb{Z}$ and $\mathbb{Z}_{16}$ respectively. For this construction, we utilize the anomaly inflow bulk-boundary correspondence, the 4d Pontryagin class and the gravitational Chern-Simons 3-form, the 3d Witten-Reshetikhin-Turaev-type topological quantum field theory corresponding to a 2d rational conformal field theory with an appropriate rational chiral central charge, and the 4d $\mathbb{Z}_4^{\rm TF}$-time-reversal symmetric topological superconductor with 3d boundary topological order., Comment: 34 pages. v2: refinements with new tables and appendices, including discussions on Atiyah's 2-framing and $p_1$ structures

We propose that the Fermi surface anomaly of symmetry group $G$ in any dimension is universally classified by $G$-symmetric interacting fermionic symmetry-protected topological (SPT) phases in $(0+1)$-dimensional spacetime. The argument is based on the perspective that the gapless fermions on the Fermi surface can be viewed as the topological boundary modes of Chern insulators in the phase space (position-momentum space). Given the non-commutative nature of the phase space coordinates, we show that the momentum space dimensions should be counted as negative dimensions for SPT classification purposes. Therefore, the classification of phase-space Chern insulators (or, more generally fermionic SPT phases) always reduces to a $(0+1)$-dimensional problem, which can then be answered by the cobordism approach. In addition to the codimension-1 Fermi surface case, we also discuss the codimension-$p$ Fermi surface case briefly. We provide concrete examples to demonstrate the validity of our classification scheme, and make connections to the recent development of Fermi surface symmetric mass generation., Comment: 13 pages + references, 2 figures, 2 tables. Update Tab. II, clarifications to codimension-p Fermi surface, and references added

We propose a novel solution to the Strong CP problem -- to explain why SU(3) strong force has a nearly zero theta angle $\bar\theta_3 \simeq 0$ for the 4d Standard Model (SM). The new ingredient is Symmetric Mass Generation (SMG): symmetry-preserving mass or energy gap can be generated without breaking any symmetry $G$ and without any quadratic mean-field mass deformation as long as $G$ is all perturbative local and nonperturbative global anomaly-free. In our first model, we propose a disordered non-mean-field SMG gap (instead of the ordered Anderson-Higgs induced mass gap) for the $u$ quark (or generally a set of quarks and leptons totally anomaly-free in $G$) generated by multi-fermion interactions or by dynamical disordered mass fields, absorbing $\bar\theta_3$. Another variant of this first model is the SMG gapping a hypothetical hidden full fourth family of SM fermions. In our second model, we have a chiral SM and mirror SM together to respect the Nielsen-Ninomiya fermion-doubling and a parity-reflection $\mathbb{Z}_2^{\rm PR}$ symmetry at high energy, so the $\bar\theta_3 = 0$. Then the SMG lifts only the mirror SM with a large energy gap but leaves the chiral SM at lower energy, which not only "spontaneously" breaks the parity-reflection symmetry maximally but also relates our solution to solve another nonperturbative chiral fermion regularization problem by removing the fermion doubling. The predictive signature of both SMG-based models is that some SM fermions or mirror fermions are highly interacting (beyond the conventional SM Higgs or SM gauge interactions) mediated through hypothetical direct multi-fermion or disordered mass-field interactions., Comment: 30 pages. A companion to arXiv:2207.14813. An overview on Symmetric Mass Generation arXiv:2204.14271

Symmetric mass generation is a novel mechanism to give gapless fermions a mass gap by non-perturbative interactions without generating any fermion bilinear condensation. The previous studies of symmetric mass generation have been limited to Dirac/Weyl/Majorana fermions with zero Fermi volume in the free fermion limit. In this work, we generalize the concept of symmetric mass generation to Fermi liquid with a finite Fermi volume and discuss how to gap out the Fermi surfaces by interactions without breaking the U(1) loop group symmetry or developing topological orders. We provide examples of Fermi surface symmetric mass generation in both (1+1)D and (2+1)D Fermi liquid systems when several Fermi surfaces together cancel the Fermi surface anomaly. However, the U(1) loop group symmetry in these cases is still restrictive enough to rule out all possible fermion bilinear gapping terms, such that a non-perturbative interaction mechanism is the only way to gap out the Fermi surfaces. This symmetric Fermi surface reconstruction is in contrast to the conventional symmetry-breaking mechanism to gap the Fermi surfaces. As a side product, our model provides a pristine 1D lattice regularization for the (1+1)D U(1) symmetric chiral fermion model (e.g., the 3-4-5-0 model) by utilizing a lattice translation symmetry as an emergent U(1) symmetry at low energy. This opens up the opportunity for efficient numerical simulations of chiral fermions in their own dimensions without introducing mirror fermions under the domain wall fermion construction., Comment: 16 pages, 6 figures

We present a model to explain the Standard Model flavor hierarchy. Our model is based on explicit smooth confinement (namely confinement without chiral symmetry breaking in a supersymmetric gauge theory) at an intermediate energy scale, before the electroweak symmetry breaking by the Higgs condensation at lower energy. In our context, the smooth confinement preserves the SU(3) and the chiral SU(2)$_L\times$ U(1)$_Y$ symmetry in a supersymmetric Standard Model, while this internal symmetry becomes dynamically gauged in the end. In contrast to Razamat-Tong's symmetric mass generation model also preserving the $G_{\rm SM} \equiv$ SU(3)$_C\times$ SU(2)$_L\times$ U(1)$_Y$ internal symmetry, our model introduces different matter contents with a different kind of superpotential deformation irrelevant at UV, which further induces Yukawa-Higgs terms marginal at IR, breaking the $G_{\rm SM}$ down to SU(3)$_C$ and the electromagnetic U(1)$_{\rm EM}$ only when Higgs condenses. In our model, the IR fermions in the first and second families are composite of UV fields, while the third family elementary fermions match between UV and IR. The smallness of the first and second family fermion masses is explained by the exponential hierarchy between the cutoff scale and the smooth confinement scale via dimensional transmutation. As a result, our UV Lagrangian only contains the natural parameters close to the order one., Comment: 11 pages, 1 figure (v1); references add, modifications for clarity (v2); matches published version (v3)

An analogous "Strong CP problem" is identified in a toy model in 2-dimensional spacetime: a general 1+1d abelian U(1) anomaly-free chiral fermion and chiral gauge theory with a generic theta instanton term $\frac{\theta}{2 \pi} \int F$. The theta term alone violates the charge-conjugation-time-reversal CT and the parity P discrete symmetries. The analogous puzzle here is the CT or P problem in 1+1d: Why can the $\bar{\theta}$ angle (including the effect of $\theta$ and the complex phase of a mass matrix) be zero or small for a natural reason? We show that this CT or P problem can be solved by a Symmetric Mass Generation mechanism (SMG, namely generating a mass or energy gap while preserving an anomaly-free symmetry). This 1+1d toy model mimics several features of the 3+1d Standard Model: chiral matter content, confinement, and Anderson-Higgs-induced mass by Yukawa-Higgs term. One solution replaces some chiral fermion's Higgs-induced mean-field mass with SMG-induced non-mean-field mass. Another solution enriches this toy model by introducing several new physics beyond the Standard Model: a parity-reflection PR discrete symmetry maps between the chiral and mirror fermions as fermion doubling localized on two domain walls at high energy, and SMG dynamically generates mass to the mirror fermion while still preserving the anomaly-free chiral symmetry at an intermediate energy scale, much before the Higgs mechanism generates mass to the chiral fermion at lower energy. Without loss of generality, an arguably simplest 1+1d U(1) symmetric anomaly-free chiral fermion/gauge theory (e.g., Weyl fermions with $3_L$-$4_L$-$5_R$-$0_R$ U(1) charges) is demonstrated. As an analogy to the superfluid-insulator or order-disorder quantum phase transition, in contrast to the Peccei-Quinn solution sitting in the (quasi-long-range-order) superfluid phase, our solution is in the SMG insulator disordered phase., Comment: 28 pages. Two classes of solutions, without or with the mirror fermion sector. A companion to arXiv:2212.14036. Related models: arXiv:1307.7480, arXiv:1807.05998, arXiv:2202.12355. v3: More misprints corrected. New refinement and clarification upon the Physical Review D version

The most well-known mechanism for fermions to acquire a mass is the Nambu-Goldstone-Anderson-Higgs mechanism, i.e. after a spontaneous symmetry breaking, a bosonic field that couples to the fermion mass term condenses, which grants a mass gap for the fermionic excitation. In the last few years, it was gradually understood that there is a new mechanism of mass generation for fermions without involving any symmetry breaking within an anomaly-free symmetry group. This new mechanism is generally referred to as the "Symmetric Mass Generation (SMG)." It is realized that the SMG has deep connections with interacting topological insulator/superconductors, symmetry-protected topological states, perturbative local and non-perturbative global anomaly cancellations, and deconfined quantum criticality. It has strong implications for the lattice regularization of chiral gauge theories. This article defines the SMG, summarizes current numerical results, introduces a novel unifying theoretical framework (including the parton-Higgs and the s-confinement mechanisms, as well as the symmetry-extension construction), and overviews various features and applications of SMG., Comment: 33 pages. An invited overview of Symmetric Mass Generation. Introduce a novel unifying theoretical framework. We thank Cenke Xu for initiating this project

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 $\mathbb{Z}_2^{\rm F}$., Comment: 6 pages. Sequel to arXiv:2112.14765. v2: refinement. Related talks: https://www.youtube.com/results?search_query=cobordism+deformation+standard+model+crticiality. The original title "Proton Stability: From the Standard Model to Ultra Unification" is adjusted in PRD

This article overviews the recent developments in applying the idea of deconfined quantum criticality in condensed matter physics to understand quantum phase transitions among grand unified theories in high energy physics in the 4-dimensional spacetime. In particular, dictated by a mod 2 class nonperturbative global mixed gauge-gravitational anomaly, there can be a gapless deconfined quantum critical region between Georgi-Glashow and Pati-Salam models -- not only the Standard Model occurs as a neighbor phase, but also beyond the Standard Model phenomena (such as deconfined dark gauge force and fractionalized/fragmentary fermionic partons) emerge near the critical region., Comment: 9 pages. An overview on arXiv:2106.16248. An invited contribution to Professor Chen-Ning Yang Centenary 100 anniversary Festschrift

Lattice regularization of chiral fermions has been a long-standing problem in physics. In this work, we present the density matrix renormalization group (DMRG) simulation of the 3-4-5-0 model of (1+1)D chiral fermions with an anomaly-free chiral U(1) symmetry, which contains two left-moving and two right-moving fermions carrying U(1) charges 3,4 and 5,0, respectively. Following the Wang-Wen chiral fermion model, we realize the chiral fermions and their mirror partners on the opposite boundaries of a thin strip of (2+1)D lattice model of multi-layer Chern insulator, whose finite-width implies the quantum system is effectively (1+1)D. By introducing carefully designed two sets of six-fermion local interactions to the mirror sector only, we demonstrate that the mirror fermions can be gapped out by the interaction beyond a critical strength without breaking the chiral U(1) symmetry, via the symmetric mass generation (SMG) mechanism. We show that the interaction-driven gapping transition is in the Berezinskii-Kosterlitz-Thouless (BKT) universality class. We determine the evolution of Luttinger parameters before the transition, which confirms that the transition happens exactly at the point when the interaction term becomes marginal. As the mirror sector is gapped after the transition, we check that the fermions in the light chiral fermion sector remain gapless, which provides the desired lattice regularization of chiral fermions., Comment: 5 + 3 pages, Symmetric Mass Generation of the chiral fermion model arXiv:1307.7480

'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 $\eta$ 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 $(N_f,n_{\nu_{R}},$ 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., Comment: 6 pages. Sequel to arXiv:1910.14668, arXiv:2006.16996, arXiv:2008.06499, arXiv:2012.15860, arXiv:2106.16248, arXiv:2111.10369, arXiv:2202.13498, arXiv:2204.08393. Related prior talks: https://www.youtube.com/results?search_query=cobordism+deformation+standard+model+ultra+unification+crticiality. v4: Phys. Rev. D (Letter)

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., Comment: 34 pages two columns (many appendices on the toolkits of cochain, cocycle, cohomology, characteristic class, cobordism, anomalies, and higher dimensional bosonization with emergent fermions). This work includes not only unpublished work in 2018 after arXiv:1810.00844, but also many new results. v2: Phys. Rev. B accepted

Prior work [arXiv:2106.16248] shows that the Standard Model (SM) naturally arises near a gapless quantum critical region between Georgi-Glashow (GG) $su(5)$ and Pati-Salam (PS) $su(4) \times su(2) \times su(2)$ models of quantum vacua (in a phase diagram or moduli space), by implementing a modified $so(10)$ Grand Unification (GUT) with a Spin(10) gauge group plus a new discrete Wess-Zumino Witten term matching a 4d nonperturbative global mixed gauge-gravity $w_2 w_3$ anomaly. In this work, we include Barr's flipped $su(5)$ model into the quantum landscape, showing these four GUT-like models arise near the quantum criticality near SM. Highlights include: First, we find the precise GG or flipped $u(5)$ gauge group requires to redefine a Lie group U(5)$_{\hat q}$ with $\hat q=2$ or $3$ (instead of non-isomorphic analog $\hat q=1$ or 4), and different $\hat q$ are related by multiple covering. Second, we show that the GG and flipped $u(5)$ are two different symmetry-breaking vacua of the same order parameter separated by a first-order Landau-Ginzburg transition. We also show that analogous 3+1d deconfined quantum criticalities, both between GG and PS, and between the flipped $u(5)$ and PS, are beyond Landau-Ginzburg paradigm. Third, topological quantum criticality occurs by tuning between the 15n vs 16n scenarios. Fourth, we explore the generalized higher global symmetries in the SM and GUTs. Gauging the $\mathbb{Z}_2$ flip symmetry between GG and flipped $u(5)$ models, leads to a potential categorical higher symmetry that is a non-invertible global symmetry: within a gauge sector $(u(1) \times u(1)) \rtimes \mathbb{Z}_2$, the fusion rule of 2d topological surface operator splits. However, the un-Higgs Spin(10) at UV retracts this categorical symmetry., Comment: 65 pages. Sequel to arXiv:2106.16248. Dedicate to the anniversaries of Subir Sachdev (60), Xiao-Gang Wen (60), Edward Witten (70), Shing-Tung Yau (72). Sec 5 on Generalized Higher Symmetries of Standard Models and Grand Unifications, Sec 6 on Potential Categorical Symmetry. v2: misprints corrected

Discrete spacetime symmetries of parity P or reflection R, and time-reversal T, act naively as $\mathbb{Z}_2$-involutions in the passive transformation on the spacetime coordinates; but together with a charge conjugation C, the total C-P-R-T symmetries have enriched active transformations on fields in representations of the spacetime-internal symmetry groups of quantum field theories (QFTs). In this work, we derive that these symmetries can be further fractionalized, especially in the presence of the fermion parity $(-1)^{\rm F}$. We elaborate on examples including relativistic Lorentz invariant QFTs (e.g., spin-1/2 Dirac or Majorana spinor fermion theories) and nonrelativistic quantum many-body systems (involving Majorana zero modes), and comment on applications to spin-1 Maxwell electromagnetism (QED) or interacting Yang-Mills (QCD) gauge theories. We discover various C-P-R-T-$(-1)^{\rm F}$ group structures, e.g., Dirac spinor is in a projective representation of $\mathbb{Z}_2^{\rm C}\times \mathbb{Z}_2^{\rm P} \times \mathbb{Z}_2^{\rm T}$ but in an (anti)linear representation of an order-16 nonabelian finite group, as the central product between an order-8 dihedral (generated by C and P) or quaternion group and an order-4 group generated by T with T$^2=(-1)^{\rm F}$. The general theme may be coined as C-P-T or C-R-T fractionalization., Comment: 8 pages. Heredity of setups: Spinor theories follow any standard QFT textbook. The 0+1d Majorana zero modes analysis follows arXiv:2011.13921, arXiv:2011.12320. Special thanks to Shing-Tung Yau on "Can C-P-T symmetries be fractionalized more than involutions?" v4: refinement

Standard lore views our 4d quantum vacuum governed by one of the candidate Standard Models (SMs), while lifting towards some Grand Unification-like structure (GUT) at higher energy scales. In contrast, in our work, we introduce an alternative view that the SM arises from various neighbor vacua competition in a quantum phase diagram. In general, we regard the SM arising near the gapless quantum criticality (either critical points or critical regions) between the competing neighbor vacua. In particular, we demonstrate how the $su(3)\times su(2)\times u(1)$ SM with 16n Weyl fermions arises near the quantum criticality between the GUT competition of Georgi-Glashow (GG) $su(5)$ and Pati-Salam (PS) $su(4)\times su(2)\times su(2)$. We propose two enveloping toy models. Model I is a conventional $so(10)$ GUT with a Spin(10) gauge group plus GUT-Higgs potential inducing various Higgs transitions. Model II modifies Model I plus a 4d discrete torsion Wess-Zumino-Witten-like term built from GUT-Higgs field (that matches a nonperturbative global mixed gauge-gravity anomaly captured by a 5d invertible topological field theory $w_2w_3$), which manifests a Beyond-Landau-Ginzburg criticality between GG and PS models, with extra Beyond-the-Standard-Model (BSM) excitations emerging near a quantum critical region. If the internal symmetries were treated as global symmetries, we show a gapless 4d deconfined quantum criticality with new BSM fractionalized fragmentary excitations of Color-Flavor separation, and gauge enhancement including a Dark Gauge force sector, altogether requiring a double fermionic Spin structure named DSpin. If the internal symmetries are dynamically gauged, we show a 4d boundary criticality such that only appropriately gauge enhanced dynamical GUT gauge fields propagate into an extra-dimensional 5d bulk. The phenomena may be regarded as SM deformation or morphogenesis., Comment: 65 pages. Primers: https://www.youtube.com/results?search_query=ultra+unification+quantum+crticiality+deformation+standard+model. Dedicate to Subir Sachdev (60), Xiao-Gang Wen (60), Edward Witten (70), Shing-Tung Yau (72). v3: Add more figures, motivations, comments on proton decay, non-perturbative effects on perturbatively irrelevant deformations. Sequel: arXiv:2111.10369, arXiv:2112.14765

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., Comment: 38 pages. Sequel to: arXiv:1910.14668, arXiv:2006.16996, arXiv:2008.06499. Supplementary: arXiv:1809.11171, arXiv:1705.06728. Physical Review journal version accepted subtitle: Unified model beyond grand unification. v3: refinement, and more clarifications on math/physics and neutrino masses. Related talks: https://www.youtube.com/results?search_query=Ultra+Unification+Quantum+Criticality+Wang

We show how 1+1-dimensional fermionic symmetry-protected topological states (SPTs, i.e. nontrivial short-range entangled gapped phases of quantum matter whose boundary exhibits 't Hooft anomaly and whose bulk cannot be deformed into a trivial tensor product state under finite-depth local unitary transformations only in the presence of global symmetries), indeed can be unwound to a trivial state by enlarging the Hilbert space via adding extra degrees of freedom and suitably extending the global symmetries. The extended projective global symmetry on the boundary can become supersymmetric in a specific sense, i.e., it contains group elements that do not commute with the fermion number parity $(-1)^F$, while the anti-unitary time-reversal symmetry becomes fractionalized. This also means we can uplift and remove certain exotic fermionic anomalies (e.g., "parity" anomaly in time-reversal or reflection symmetry) via appropriate supersymmetry extensions in terms of group extensions. We work out explicit examples for multi-layers of 1+1d Majorana fermion chains, then comment on models with Sachdev-Ye-Kitaev (SYK) interactions, intrinsic fermionic gapless SPTs protected by supersymmetry, and generalizations to higher spacetime dimensions via a cobordism theory., Comment: 61 pages. Sequel to arXiv:1804.11236, arXiv:1705.06728

We prove that the boundaries of all non-trivial 1+1 dimensional intrinsically fermionic symmetry-protected-topological phases, protected by finite on-site symmetries (unitary or anti-unitary), are supersymmetric quantum mechanical systems. This supersymmetry does not require any fine-tuning of the underlying Hamiltonian, arises entirely as a consequence of the boundary 't Hooft anomaly that classifies the phase and is related to a `Bose-Fermi' degeneracy different in nature from other well known degeneracies such as Kramers doublets., Comment: 5 pages, 2 figures. Minor changes, close to published version

A recent work [2006.16996] suggests that a 4d nonperturbative global anomaly of mod 16 class hinting a possible new hidden gapped topological sector beyond the Standard Model (SM) and Georgi-Glashow $su(5)$ Grand Unified Theory (GUT) with 15n chiral Weyl fermions and a discrete $\mathbb{Z}_{4,X}$ symmetry of $X=5({\bf B- L})-4Y$. This $\mathbb{Z}_{16}$ class global anomaly is a mixed gauge-gravitational anomaly between the discrete $X$ and spacetime backgrounds. The new topological sector has a GUT scale high energy gap, below its low energy encodes either a 4d noninvertible topological quantum field theory (TQFT), or a 5d short-range entangled invertible TQFT, or their combinations. This hidden topological sector provides the 't Hooft anomaly matching of the missing sterile right-handed neutrinos (3 generations of 16th Weyl fermions), and possibly also accounts for the Dark Matter sector. In the SM and $su(5)$ GUT, the discrete $X$ can be either a global symmetry or gauged. In the $so(10)$ GUT, the $X$ must become gauged, the 5d TQFT becomes noninvertible and long-range entangled (which can couple to dynamical gravity). In this work, we further examine the anomaly and cobordism constraints at higher energy scales above the $su(5)$ GUT to $so(10)$ GUT and $so(18)$ GUT (with Spin(10) and Spin(18) gauge groups precisely). We also find [2006.16996]'s proposal on new hidden gapped topological sectors can be consistent with anomaly matching under the energy/mass hierarchy. Novel ingredients along tuning the energy include various energy scales of anomaly-free symmetric mass generation (i.e., Kitaev-Wen mechanism), the Topological Mass/Energy Gap from anomalous symmetric topological order (attachable to a 5d $\mathbb{Z}_{4,X}$-symmetric topological superconductor), possible topological quantum phase transitions, and Ultra Unification that includes GUT with new topological sectors., Comment: 50 pages. Primers: arXiv:2006.16996, arXiv:1910.14668, arXiv:1810.00844, arXiv:1809.11171

Standard lore uses local anomalies to check the kinematic consistency of gauge theories coupled to chiral fermions, e.g. Standard Models (SM). Based on a systematic cobordism classification, we examine constraints from invertible quantum anomalies (including all perturbative local and nonperturbative global anomalies) for gauge theories. We also clarify the different uses of these anomalies: including (1) anomaly cancellations of dynamical gauge fields, (2) 't Hooft anomaly matching conditions of background fields of global symmetries, and others. We apply several 4d $\mathbb{Z}_{n}$ anomaly constraints of $n=16,4,2$ classes, beyond the familiar Feynman-graph perturbative $\mathbb{Z}$ class local anomalies. As an application, for (SU(3)$\times$SU(2)$\times$U(1))/$\mathbb{Z}_q$ SM (with $q=1,2,3,6$) and SU(5) Grand Unification with 15n chiral Weyl fermions and with a discrete baryon minus lepton number $X=5({\bf B}- {\bf L})-4Y$ preserved, we discover a new hidden gapped sector previously unknown to the SM and Georgi-Glashow model. The gapped sector at low energy contains either (1) 4d non-invertible topological quantum field theory (TQFT, above the energy gap with heavy fractionalized anyon excitations from 1d particle worldline and 2d string worldsheet, inaccessible directly from Dirac or Majorana mass gap of the 16th Weyl fermions [i.e., right-handed neutrinos], but accessible via a topological quantum phase transition), or (2) 5d invertible TQFT in extra dimensions. Above a higher energy scale, the discrete $X$ becomes dynamically gauged, the entangled Universe in 4d and 5d is mediated by Topological Force. Our model potentially resolves puzzles, surmounting sterile neutrinos and dark matter, in fundamental physics., Comment: 68 pages. Sequel: arXiv:2008.06499, arXiv:1910.14668. Primers: arXiv:1812.11967, arXiv:1711.11587, arXiv:1705.06728, arXiv:1612.09298. v2: Significant revision. Correct statements on 4d TQFTs in Sec 5. We focus only on the even class such as a 2 mod 16 symmetric anomalous 4d TQFT. We can also introduce 5d invertible TQFTs and right-hand neutrinos

We propose an effective field theory (EFT) of fractional quantum Hall systems near the filling fraction $\nu=5/2$ that flows to pertinent IR candidate phases, including non-abelian Pfaffian, anti-Pfaffian, and particle-hole Pfaffian states (Pf, APf, and PHPf). Our EFT has a 2+1$d$ O(2)$_{2,L}$ Chern-Simons gauge theory coupled to four Majorana fermions by a discrete charge conjugation gauge field, with Gross-Neveu-Yukawa-Higgs terms. Including deformations via a Higgs condensate and fermion mass terms, we can map out a phase diagram with tunable parameters, reproducing the prediction of the recently-proposed percolation picture and its gapless topological quantum phase transitions. Our EFT captures known features of both gapless and gapped sectors of time-reversal-breaking domain walls between Pf and APf phases. Moreover, we find that Pf$\mid$APf domain walls have higher tension than domain walls in the PHPf phase. Then the former, if formed, may transition to the energetically-favored PHPf domain walls; this could, in turn, help further induce a bulk transition to PHPf., Comment: 72 pages, 10 figures, 5 tables. Many appendices. Add Sec 2.2, new remarks in Conclusion, and other clarifications. Accepted by Physical Review Research

The existence of quantum non-liquid states and fracton orders, both gapped and gapless states, challenges our understanding of phases of entangled matter. We generalize the cellular topological states to liquid or non-liquid cellular states. We propose a mechanism to construct more general non-abelian states by gluing gauge-symmetry-breaking vs gauge-symmetry-extension interfaces as extended defects in a cellular network, including defects of higher-symmetries, in any dimension. Our approach also naturally incorporates the anyonic particle/string condensations and composite string (related to particle-string or p-string)/membrane condensations. This approach shows gluing the familiar extended topological quantum field theory or conformal field theory data via topology, geometry, and renormalization consistency criteria (via certain modified group cohomology or cobordism theory data) in a tensor network can still guide us to analyze the non-liquid states. (Part of the abelian construction can be understood from the K-matrix Chern-Simons theory approach and the coupled-layer-by-junction constructions.) This approach may also lead us toward a unifying framework for quantum systems of both higher-symmetries and sub-system/sub-dimensional symmetries., Comment: 45 pages. Subtitle: Gluing Gauge-(Higher)-Symmetry-Breaking vs -Extension Interfacial Defects. This work relates to but also generalizes the construction in arXiv:2002.02433. v3: refinement and figures added. Accepted by Phys. Rev. Research

By gauging a higher-moment polynomial degree global symmetry and a discrete charge conjugation (i.e., particle-hole) symmetry coupled to matter fields (two symmetries mutually non-commutative), we derive a new class of higher-rank tensor non-abelian gauge field theory with dynamically gauged fractonic matter fields: Non-abelian gauged fractons interact with a hybrid class of higher-rank (symmetric or generic non-symmetric) tensor gauge theory and anti-symmetric tensor topological field theory, generalizing [arXiv:1909.13879, 1911.01804]. We also apply a quantum phase transition similar to that between insulator v.s. superfluid/superconductivity (U(1) symmetry disordered phase described by a topological gauge theory or a disordered Sigma model v.s. U(1) global/gauge symmetry-breaking ordered phase described by a Sigma model with a U(1) target space underlying Goldstone modes): We can regard our tensor gauge theories as disordered phases, and we transient to their new ordered phases by deriving new Sigma models in continuum field theories. While one low energy theory is captured by degrees of freedom of rotor or scalar modes, another side of low energy theory has vortices and superfluids -- we explore non-abelian vortices (two types of vortices mutually interacting non-commutatively beyond an ordinary group structure) and their Cauchy-Riemann relation., Comment: 28 pages. Sequel to: arXiv:1909.13879, arXiv:1911.01804. Dedicated to the 70 year anniversary of Ginzburg-Landau theory [Zh. Ekaper. Teoret. Fiz. 20, 1064 (1950)] in 2020. Accepted by Physical Review Research

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., Comment: Sequel to: arXiv:1711.11587, arXiv:1812.11967, arXiv:1904.00994 arXiv:1910.14668, arXiv:1912.13514. Accepted by Annals of Mathematical Sciences and Applications, Vol. 5, No. 2 (2020)

We explore QCD$_4$ quark matter, the $\mu$-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., Comment: Sequel to: arXiv:1711.11587, arXiv:1812.11967 and arXiv:1910.14668

We explore the low energy dynamics of the four siblings of Lorentz symmetry enriched SU(2) Yang-Mills theory with a theta term at $\theta=\pi$ in $(3+1)$d. Due to a mixed anomaly between time reversal symmetry and the center symmetry, the low energy dynamics must be nontrivial. We focus on two possible scenarios: 1) time reversal symmetry is spontaneously broken by the two confining vacua, and 2) the low energy theory is described by a U(1) Maxwell gauge theory (e.g. U(1) spin liquid in condensed matter) which is deconfined and gapless while preserving time reversal symmetry. In the first scenario, we first identify the global symmetry on the time reversal domain wall, where time reversal symmetry in the bulk induces a $\mathbb{Z}_2$ unitary symmetry on the domain wall. We discuss how the Lorentz symmetry and the unitary $\mathbb{Z}_2$ symmetry enrich the domain wall theory. In the second scenario, we relate the symmetry enrichments of the SU(2) Yang-Mills to that of the U(1) Maxwell gauge theory. This further opens up the possibility that SU(2) QCD with large and odd flavors of fermions could be a direct second order phase transition between two phases of U(1) gauge theories as well as between a U(1) gauge theory and a trivial vacuum (e.g. a trivial paramagnet), where the gauge group is enhanced to be non-Abelian at and only at the transition. We characterize these transitions, and name them as Gauge Enhanced Quantum Critical Points., Comment: 39 pages, 1 figure, Refinements and Refs added

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 $\Omega^{G}_d$ 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., Comment: 91 pages. v4: Refinement and Refs added. More clarifications with new subsections and appendices on the same cobordism group results. Part of the work is reported in Workshop on Lattice for Beyond the Standard Model physics on May 2-3, 2019 at Syracuse University

With a view toward a fracton theory in condensed matter, we introduce a higher-moment polynomial degree-p global symmetry, acting on complex scalar/vector/tensor fields (e.g., ordinary or vector global symmetry for p$=0$ and p$=1$ respectively). We relate this higher-moment global symmetry of $n$-dimensional space, to a lower degree (either ordinary or higher-moment, e.g., degree-(p-$\ell$)) subdimensional or subsystem global symmetry on layers of $(n-\ell)$-submanifolds. These submanifolds are algebraic affine varieties (i.e., solutions of polynomials). The structure of layers of submanifolds as subvarieties can be studied via mathematical tools of embedding, foliation, and algebraic geometry. We also generalize Noether's theorem for this higher-moment polynomial global symmetry. We can promote the higher-moment global symmetry to a local symmetry, and derive a new family of higher-rank-m symmetric tensor gauge theory by gauging, with m = p$+1$. By further gauging a discrete $\mathbb{Z}_2^C$ charge conjugation (particle-hole) symmetry, we derive a new general class of rank-m tensor non-abelian gauge field theory (the gauge structure is non-commutative thus non-abelian but not an ordinary group): a hybrid class of (symmetric or non-symmetric) higher-rank-m tensor gauge theory and anti-symmetric tensor topological field theory, generalizing [arXiv:1909.13879], interplaying between gapless and gapped sectors., Comment: 23 pages. Sequel to arXiv:1909.13879, arXiv:1912.13485. Dedicated to 90 years of Gauge Principle since Hermann Weyl. v2: clarifications added

We classify subsystem symmetry-protected topological (SSPT) phases in $3+1$D protected by planar subsystem symmetries, which are dual to abelian fracton topological orders. We distinguish between weak SSPTs, which can be constructed by stacking $2+1$D SPTs, and strong SSPTs, which cannot. We identify signatures of strong phases, and show by explicit construction that such phases exist. A classification of strong phases is presented for an arbitrary finite abelian group. Finally, we show that fracton orders realizable via $p$-string condensation are dual to weak SSPTs, while strong SSPTs do not admit such a realization., Comment: 21 pages, 7 figures

We formulate a new class of tensor gauge field theories in any dimension that is a hybrid class between symmetric higher-rank tensor gauge theory (i.e., higher-spin gauge theory) and anti-symmetric tensor topological field theory. Our theory describes a mixed unitary phase interplaying between gapless and gapped topological order phases (which can live with or without Euclidean, Poincar\'e or anisotropic symmetry, at least in ultraviolet high or intermediate energy field theory, but not yet to a lattice cutoff scale). The "gauge structure" can be compact, continuous, abelian or non-abelian. Our theory sits outside the paradigm of Maxwell electromagnetic theory in 1865 and Yang-Mills isospin/color theory in 1954. We discuss its local gauge transformation in terms of the ungauged vector-like or tensor-like higher-moment global symmetry. The non-abelian gauge structure is caused by gauging the non-commutative symmetries: a higher-moment symmetry and a charge conjugation (particle-hole) symmetry. Vector global symmetries along time direction may exhibit time crystals. We explore the relation of these long-range entangled matters to a non-abelian generalization of Fracton order in condensed matter, a field theory formulation of foliation, the spacetime embedding and Embeddon that we newly introduce, and possible fundamental physics applications to dark matter or dark energy., Comment: 40 pages, 8 figures, 4 tables. v3: Refinement, Refs added, and minor corrections. See also arXiv:1911.01804. Our new theory is dedicated to 90 years of Gauge Principle since Hermann Weyl and 65 years of Yang-Mills 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 $\theta=\pi$ (SU(2)$_{\theta=\pi}$ 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)$_{\theta=\pi}$ 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)$_{\theta=\pi}$ 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., Comment: 87 pages. Sec 8 discusses both symmetry-extended TQFTs and symmetry-breaking TQFTs, agreeing with Cordova-Ohmori arXiv:1910.04962. See Primers: arXiv:1812.11968, arXiv:1812.11967, arXiv:1711.11587, arXiv:1705.06728 and Refs therein, also https://video.ias.edu/highenergy/2019/0419-JuvenWang. v4: More technical misprints/typos fixed

To formulate the universal constraints of quantum statistics data of generic long-range entangled quantum systems, we introduce the geometric-topology surgery theory on spacetime manifolds where quantum systems reside, cutting and gluing the associated quantum amplitudes, specifically in 2+1 and 3+1 spacetime dimensions. First, we introduce the fusion data for worldline and worldsheet operators capable of creating anyonic excitations of particles and strings, well-defined in gapped states of matter with intrinsic topological orders. Second, we introduce the braiding statistics data of particles and strings, such as the geometric Berry matrices for particle-string Aharonov-Bohm, 3-string, 4-string, or multi-string adiabatic loop braiding process, encoded by submanifold linkings, in the closed spacetime 3-manifolds and 4-manifolds. Third, we derive new `quantum surgery' formulas and constraints, analogous to Verlinde formula associating fusion and braiding statistics data via spacetime surgery, essential for defining the theory of topological orders, 3d and 4d TQFTs and potentially correlated to bootstrap boundary physics such as gapless modes, extended defects, 2d and 3d conformal field theories or quantum anomalies. This article is meant to be an extended and further detailed elaboration of our previous work [arXiv:1602.05951] and Chapter 6 of [arXiv:1602.05569]. Our theory applies to general quantum theories and quantum mechanical systems, also applicable to, but not necessarily requiring the quantum field theory description., Comment: 35 pages, 3d and 4d figures, 3 tables. An extended sequel and further detailed elaboration of [arXiv:1602.05951] and Chapter 6 of Thesis [arXiv:1602.05569] in 2015

We formulate a family of spin Topological Quantum Filed Theories (spin-TQFTs) as fermionic generalization of bosonic Dijkgraaf-Witten TQFTs. They are obtained by gauging $G$-equivariant invertible spin-TQFTs, or, in physics language, gauging the interacting fermionic Symmetry Protected Topological states (SPTs) with a finite group $G$ symmetry. We use the fact that the latter are classified by Pontryagin duals to spin-bordism groups of the classifying space $BG$. We also consider unoriented analogues, that is $G$-equivariant invertible pin$^\pm$-TQFTs (fermionic time-reversal-SPTs) and their gauging. We compute these groups for various examples of abelian $G$ using Adams spectral sequence and describe all corresponding TQFTs via certain bordism invariants in dimensions 3, 4, and other. This gives explicit formulas for the partition functions of spin-TQFTs on closed manifolds with possible extended operators inserted. The results also provide explicit classification of 't Hooft anomalies of fermionic QFTs with finite abelian group symmetries in one dimension lower. We construct new anomalous boundary deconfined spin-TQFTs (surface fermionic topological orders). We explore SPT and SET (symmetry enriched topologically ordered) states, and crystalline SPTs protected by space-group (e.g. translation $\mathbb{Z}$) or point-group (e.g. reflection, inversion or rotation $C_m$) symmetries, via the layer-stacking construction., Comment: 81 pages, 28 figures