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Anomaly and Cobordism Constraints Beyond Grand Unification: Energy Hierarchy

Juven Wang

TL;DR

This work extends cobordism-based anomaly constraints to energy scales above the traditional $su(5)$ GUT, probing $so(10)$ and $so(18)$ unifications with a discrete $\mathbb{Z}_{4,X}$ symmetry. It analyzes how a hidden 4d TQFT or 5d invertible TQFT can nonperturbatively match a mod 16 mixed gauge-gravitational anomaly, enabling anomaly matching without breaking $\mathbb{Z}_{4,X}$ or introducing new fermions. The paper develops a detailed representation-theoretic framework, evaluates KW-type gapping scenarios across successive GUT breakings, and highlights the possibility of Topological Mass and TQFT-driven dynamics (collectively termed Ultra Unification) at scales between traditional GUT thresholds. It emphasizes that anomaly data constrain, but do not uniquely fix, high-energy dynamics, suggesting multiple viable pathways—including topological phase transitions and symmetry-preserving mass gaps—to realize a consistent high-energy completion. Overall, it posits a cohesive picture in which Grand Unification coexists with topological sectors that can account for sterile neutrinos and possibly dark matter while maintaining anomaly consistency across the energy hierarchy.

Abstract

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.

Anomaly and Cobordism Constraints Beyond Grand Unification: Energy Hierarchy

TL;DR

This work extends cobordism-based anomaly constraints to energy scales above the traditional GUT, probing and unifications with a discrete symmetry. It analyzes how a hidden 4d TQFT or 5d invertible TQFT can nonperturbatively match a mod 16 mixed gauge-gravitational anomaly, enabling anomaly matching without breaking or introducing new fermions. The paper develops a detailed representation-theoretic framework, evaluates KW-type gapping scenarios across successive GUT breakings, and highlights the possibility of Topological Mass and TQFT-driven dynamics (collectively termed Ultra Unification) at scales between traditional GUT thresholds. It emphasizes that anomaly data constrain, but do not uniquely fix, high-energy dynamics, suggesting multiple viable pathways—including topological phase transitions and symmetry-preserving mass gaps—to realize a consistent high-energy completion. Overall, it posits a cohesive picture in which Grand Unification coexists with topological sectors that can account for sterile neutrinos and possibly dark matter while maintaining anomaly consistency across the energy hierarchy.

Abstract

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 Grand Unified Theory (GUT) with 15n chiral Weyl fermions and a discrete symmetry of . This class global anomaly is a mixed gauge-gravitational anomaly between the discrete 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 GUT, the discrete can be either a global symmetry or gauged. In the GUT, the 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 GUT to GUT and 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 -symmetric topological superconductor), possible topological quantum phase transitions, and Ultra Unification that includes GUT with new topological sectors.

Paper Structure

This paper contains 22 sections, 34 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Gauge Group Embedding and Representation Hierarchy
  3. Spacetime and internal symmetry group embedding
  4. Decomposition: Lie algebras to Lie groups, and representation theory
  5. Representation of spacetime symmetry groups
  6. Representation of internal/gauge symmetry groups
  7. Energy and Mass Hierarchy: Local and Global Anomaly Constraints
  8. $so(18)$ GUT without mirror fermion doubling and the energy scale $\Delta_{\mathrm{KW}.so(18)}$
  9. With an additional discrete $X=5({\bf B}- {\bf L})-4Y$ symmetry
  10. Kinematics vs Dynamics
  11. Conclusion:
  12. Energy hierarchy, possible dynamics, and topological quantum phase transitions
  13. Energy Scale of Ultra Unification: Grand Unification $+$ Topological Force and Matter
  14. Dynamical Gauge Anomaly Cancellation
  15. SU(5)$^3$ for $su(5)$ GUT: 4d local anomaly from 5d $\frac{1}{2}{\text{CS}_5^{{\rm SU}(5)}}$ and 6d $\frac{1}{2}{c_3({\rm SU}(5))}$
  16. ...and 7 more sections

Figures (4)

  • Figure 1: Energy and Mass Hierarchy contemporarily confirmed in the Standard Model: In the figure, the breaking structure and hierarchy structure always concern the global Lie group: $\frac{{\rm SU}(3)\times {\rm SU}(2)\times {\rm U}(1)}{\mathbb{Z}_q}, etc$. However, we simply denote the local Lie algebra such as $su(3) \times su(2) \times u(1), etc.$, only for the abbreviation brevity and only to be consistent with the notations of early physics literature. The present work address possible the energy hierarchy in the gray region (with a question mark ?) around the GUT scale above the SM scale and below the Planck scale. See the new proposal in Fig. \ref{['fig:energy-hierarchy-lie-algebra-1']} and Fig. \ref{['fig:energy-hierarchy-lie-algebra-2']}.
  • Figure 2: The full spacetime-internal symmetry $G={\frac{{G_{\text{spacetime} }} \ltimes {{G}_{\text{internal}} }}{{N_{\text{shared}}}}}$ (the precise global symmetry before gauging the ${{G}_{\text{internal}} }$) for the hierarchy starting from the $so(18)$ GUT with ${{{\rm Spin} \times_{\mathbb{Z}_2^F} {\rm Spin}(18)}}$, which can be placed on non-spin manifolds. Note that ${{\rm Spin}(6)={\rm SU}(4)}$$\supset$${{\rm Spin}(5)={\rm Sp}(2)={\rm U}{\rm Sp}(4)}$ and recall $G_{\text{SM}_q}$$\equiv$$\frac{{\rm SU}(3)_{\text{strong}}\times {\rm SU}(2)_{\text{weak}}\times {\rm U}(1)_Y}{\mathbb{Z}_q}$. The subscript "3-Family" means there are 3 families (or 3 generations) of matter fields, e.g., quarks and leptons. Here the arrow from $G_1 \to G_2$ means particularly that $G_1 \supseteq G_2$ contains the later as a subgroup. This shows the web of full symmetry group embedding, similar to Table 4 of 1711.11587GPW. We have computed the cobordism group $\mathrm{TP}_d(G)$ of these spacetime-internal symmetry group $G$ in Ref. WanWangv2.
  • Figure 3: The full spacetime-internal symmetry $G={\frac{{G_{\text{spacetime} }} \ltimes {{G}_{\text{internal}} }}{{N_{\text{shared}}}}}$ (the precise global symmetry before gauging the ${{G}_{\text{internal}} }$) for the hierarchy starting from the $so(18)$ GUT with ${{{\rm Spin} \times_{} {\rm Spin}(18)}}$, which can be placed on spin manifolds. Also we follow the notations/explanations of Fig. \ref{['table:sym-web-1']}'s caption. We have computed the cobordism group $\mathrm{TP}_d(G)$ of these spacetime-internal symmetry group $G$ in Ref. WanWangv2.
  • Figure 6: The full spacetime-internal symmetry $G={\frac{{G_{\text{spacetime} }} \ltimes {{G}_{\text{internal}} }}{{N_{\text{shared}}}}}$ (the precise global symmetry before gauging the ${{G}_{\text{internal}} }$) for the hierarchy starting from the $so(18)$ GUT with ${{{\rm Spin} \times_{\mathbb{Z}_2} {\rm Spin}(18)}}$, which can be placed on non-spin manifolds. The setup is similar to Fig. \ref{['table:sym-web-1']}, but now we include the additional discrete symmetry sector $\mathbb{Z}_{4,{X}} = Z({\rm Spin}(10)) =Z({\rm Spin}(18))$ sitting at the center $Z({{G}_{\text{internal}} })$ normal subgroup of ${{G}_{\text{internal}} }={\rm Spin}(10)$ and ${\rm Spin}(18)$. We follow the notations/explanations of Fig. \ref{['table:sym-web-1']}'s caption. We compute the cobordism group $\mathrm{TP}_d(G)$ of these spacetime-internal symmetry group $G$ in Ref. WanWangv2. The arrow $G_1 \to G_2$ (with the condition $G_1 \supseteq G_2$) shows that a possible breaking process. We explore the two possible breaking patterns on the left-hand side (l.h.s) and right-hand side (r.h.s), with their possible energy hierarchy shown in Fig. \ref{['fig:energy-hierarchy-lie-algebra-1']} and Fig. \ref{['fig:energy-hierarchy-lie-algebra-2']}. Some of the arrows have a subtitle "possible TQFT generated," which means that a noninvertible TQFT may be generated to match the 4d anomaly, especially from the cobordism group $\mathrm{TP}_5({{\rm Spin} \times_{\mathbb{Z}_2^F} \mathbb{Z}_{4,X}}) =\mathbb{Z}_{16}$. The l.h.s breaking pattern suggests (at least) three possible breaking steps to generate a possible TQFT. In particular, thanks to mathematical and phenomenological constraints, the l.h.s step ${\rm Spin} \times_{\mathbb{Z}_2^F} \mathbb{Z}_{4,X} \times {\rm SU}(5) \times {\rm Spin}(5) \to ({\rm Spin} \times_{\mathbb{Z}_2^F} \mathbb{Z}_{4,X} \times {\rm SU}(5))_{\text{3-Family}}$ and the r.h.s step $({\rm Spin} \times_{\mathbb{Z}_2^F} {\rm Spin}(10) )_{\text{3-Family}} \to ({\rm Spin} \times_{\mathbb{Z}_2^F} \mathbb{Z}_{4,X} \times {\rm SU}(5))_{\text{3-Family}}$, these two steps seem to be the most promising energy scale denoted $\Delta_{\text{TQFT}}$ to generate a TQFT with a gap size $\Delta_{\text{TQFT}}$. A possible interpretation of topological quantum phase transition(s) around this energy scale $M_{su(5) \times \mathbb{Z}_{4,X}\;\text{3-Family}}$ is given in Table \ref{['table:scenarios']}. We also enlist other sequences of possible energy scales analogous to Kitaev-Wen (KW) mechanism, gapping the fully anomaly-free extra matter. We denote these KW-type energy scales as $\Delta_{\text{KW}}$. See Fig. \ref{['fig:energy-hierarchy-lie-algebra-1']} and Fig. \ref{['fig:energy-hierarchy-lie-algebra-2']}.