Abstract
We classify and characterize fully all invertible anomalies and all allowed topo- logical terms related to various Standard Models (SM), Grand Unified Theories (GUT), and Beyond Standard Model (BSM) physics. By all anomalies, we mean the inclusion of (1) perturbative local anomalies captured by perturbative Feynman diagram loop calculations, classified by ℤ free classes, and (2) nonperturbative global anomalies, classified by finite group ℤN torsion classes. Our work built from [31] fuses the math tools of Adams spectral sequence, Thom-Madsen-Tillmann spectra, and Freed-Hopkins theorem. For example, we compute bordism groups {Omega}_d^G and their invertible topological field theory invariants, which characterize dd topological terms and (d − 1)d anomalies, protected by the following symmetry group G: mathrm{Spin}times frac{mathrm{SU}(3)times mathrm{SU}(2)times mathrm{U}(1)}{{mathrm{mathbb{Z}}}_q} for SM with q = 1, 2, 3, 6; frac{mathrm{Spin}times mathrm{Spin}(n)}{{mathrm{mathbb{Z}}}_2^F} or Spin × Spin(n) for SO(10) or SO(18) GUT as n = 10, 18; Spin × SU(n) for Georgi-Glashow SU(5) GUT as n=5;frac{mathrm{Spin}times frac{mathrm{SU}(4)times left(mathrm{SU}(2)times mathrm{SU}(2)right)}{{mathrm{mathbb{Z}}}_{q^{prime }}}}{{mathrm{mathbb{Z}}}_2^F} for Pati-Salam GUT as q′ = 1, 2; and others. For SM with an extra discrete symmetry, we obtain new anomaly matching conditions of ℤ16, ℤ4 and ℤ2 classes beyond the familiar Witten anomaly. Our approach offers an alternative view of all anomaly matching conditions built from the lower-energy (B)SM or GUT, in contrast to high-energy Quantum Gravity or String Theory Landscape v.s. Swampland program, as bottom-up/top-down complements. Symmetries and anomalies provide constraints of kinematics, we further suggest constraints of quantum gauge dynamics, and new predictions of possible extended defects/excitations plus hidden BSM non-perturbative topological sectors.
Highlights
1.1 Physics guideThe world where we reside, to our present knowledge, can be described by quantum theory, gravity theory, and the underlying long-range entanglement
Since the Standard Models (SM), Grand Unified Theories (GUT) and Beyond Standard Model (BSM) are given by continuum Quantum field theory (QFT) data, in this work, we only focus on the DIFF manifolds and their associated all possible anomalies and topological terms
Based on the Freed-Hopkin’s theorem [19] and an extended generalization that we propose [31, 34], there exists a one-to-one correspondence between “the invertible topological quantum field theories with symmetry” and “a cobordism group.”. This means that there is a relation from iTQFT to “the symmetric invertible topological order with symmetry or symmetry-protected topological state (SPTs, see [21, 36,37,38]) that can be regularized on a lattice in its own dimensions.”
Summary
The world where we reside, to our present knowledge, can be described by quantum theory, gravity theory, and the underlying long-range entanglement. For the ungauged SM, GUT and BSM theories: Invertible ’t Hooft anomalies and background probed topological terms associated to a global symmetry group G. This means that there is a relation from iTQFT to “the symmetric invertible topological order (iTO, see a review [21]) with symmetry (including higher symmetries) or symmetry-protected topological state (SPTs, see [21, 36,37,38]) that can be regularized on a lattice in its own dimensions.” It is a one-to-one correspondence (isomorphism “∼=”) between the following two well-defined “mathematical objects” (which turn out to be abelian groups): Deformation classes of the reflection positive invertible d-dimensional extended topological field theories (iTQFT) with symmetry group.
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