Anomalies in family unification models from bordism classification

This paper employs bordism classification and the Atiyah-Hirzebruch spectral sequence to demonstrate the absence of global sigma model and gauged anomalies in four-dimensional $\mathcal{N}=1$ supersymmetric family unification models based on the exceptional group $E_7$.

The Gist

Imagine you are an architect trying to build a skyscraper that represents the entire universe. You have blueprints for the laws of physics, but there's a catch: if you make even one tiny calculation error, the whole building could collapse into a paradox. In the world of theoretical physics, these "calculation errors" are called anomalies.

This paper, written by Tsubasa Sugeno and Hiroki Wada, is like a rigorous safety inspection of two specific, ambitious blueprints for a "Family Unification" model. These models try to explain why we have three generations of particles (like three copies of the same family: electrons, muons, and taus) using a very complex mathematical shape called a coset space built from a giant symmetry group called $E_7$.

Here is the breakdown of their work using everyday analogies:

1. The Problem: The "Glitch" in the System

In physics, particles are like dancers on a stage. Sometimes, the rules of the dance (symmetries) work perfectly. But sometimes, when you try to combine quantum mechanics with these rules, a "glitch" appears.

• Perturbative Anomalies: These are like obvious mistakes in the choreography. If you count the steps, they don't add up. Physicists have known how to fix these for a long time by adding more dancers (particles) to balance the count.
• Global Anomalies: These are the tricky, invisible glitches. They don't show up in a simple count. They are like a hidden structural flaw in the stage itself that only reveals itself if you try to rotate the stage in a specific, weird way. If this flaw exists, the entire theory is impossible, no matter how many dancers you add.

2. The Tool: The "Topological Map"

To find these invisible glitches, the authors use a mathematical tool called Bordism Classification.

• The Analogy: Imagine you are trying to figure out if a specific shape (the universe's target space) has a hidden hole or twist that makes it unstable. Instead of looking at the shape directly, you try to build a "shadow" of it in a higher dimension.
• If you can build a solid, continuous shadow without any tears or gaps, the shape is safe. If you cannot build the shadow without it tearing apart, the shape has a "global anomaly."
• The authors used a complex mathematical machine called the Atiyah-Hirzebruch spectral sequence to check these shadows. Think of this machine as a high-powered X-ray that can see through layers of complexity to find the hidden structural flaws.

3. The Two Blueprints: $E_7/G$ and $E_7/H$

The paper focuses on two specific designs for the universe:

• Model A ($E_7/G$): A model where the "family" of particles arises from a specific arrangement of symmetries.
• Model B ($E_7/H$): A slightly different arrangement, which some theorists think might be connected to F-theory (a branch of string theory that tries to unify all forces).

In both models, the three generations of quarks and leptons (the building blocks of matter) naturally pop out as "superpartners" of the fields defining the shape of the universe. This is a beautiful feature, but the authors had to ask: "Is the foundation strong enough?"

4. The Investigation: Checking for Global Anomalies

The authors performed a deep dive into the math:

1. The Target Space: They looked at the shape of the universe in these models ($E_7/G$ and $E_7/H$).
2. The Gauge Symmetry: In these models, they "gauge" the symmetries (meaning they turn the internal symmetries into forces, like electromagnetism). This is like locking the doors of the building to make it secure.
3. The Calculation: They calculated the "bordism groups" (the mathematical safety certificates) for these shapes.

The Result:

• Good News: They found that the "torsion part" of these groups (the part that encodes global anomalies) is zero.
• Translation: The X-ray showed no hidden cracks. The "shadow" of the universe in these models can be built perfectly. There are no global anomalies. The models are topologically safe.

5. The Catch: Perturbative Anomalies Still Exist

While the global structure is safe, the authors also checked for the perturbative (counting) anomalies.

• The Finding: Just like a building might have a solid foundation but a leaky roof, these models still have counting errors. The number of particles doesn't balance perfectly on its own.
• The Fix: To fix this, you need to add extra "dancers" (additional particles) to the model. The authors showed that you can fix the counting errors by adding specific types of particles, though it requires a bit of fine-tuning. Interestingly, some of these errors might be fixed by a mechanism called the Green-Schwarz mechanism, which is like a self-repairing patch in the fabric of spacetime.

6. Why This Matters

This paper is a "sanity check" for some very fancy theories of everything.

• For String Theory: Since the $E_7/H$ model might be realizable in F-theory (a string theory framework), knowing it has no global anomalies is crucial. If it had a global anomaly, it would mean that specific version of string theory is broken.
• For Physics: It tells us that these models are viable candidates for describing our universe. They aren't ruled out by these deep, topological laws. However, it also tells us that if we want to use them, we must be careful to include the right extra particles to fix the counting errors.

Summary

Think of the authors as structural engineers inspecting two futuristic skyscrapers designed to house the laws of physics.

1. They used a super-advanced scanner (Bordism Classification) to check for invisible, catastrophic cracks (Global Anomalies).
2. Verdict: The buildings are structurally sound! No hidden cracks found.
3. Caveat: The plumbing (particle counts) is a bit off. You need to install some extra pipes (add more particles) to make the water flow correctly.
4. Conclusion: These blueprints are safe to build, provided you follow the instructions to fix the plumbing.

Technical Summary

1. Problem Statement

The paper addresses the consistency of Family Unification Models, specifically those based on four-dimensional $N=1$ supersymmetric nonlinear sigma models with target spaces constructed from the exceptional Lie group $E_7$ and its subgroups. In these models, the three generations of quarks and leptons arise naturally as superpartners of the sigma model fields.

The central problem is the potential existence of quantum anomalies that render the theory inconsistent. While perturbative anomalies (associated with infinitesimal gauge transformations) have been studied in these models, the authors investigate global anomalies (associated with large gauge transformations and topological obstructions). The specific goals are:

1. To determine if global sigma model anomalies exist for the target spaces $E_7/G$ and $E_7/H$.
2. To analyze additional anomalies arising when the symmetries of the coset spaces ($G$ and $H$) are gauged to form a consistent gauge theory.
3. To establish the conditions under which perturbative anomalies can be cancelled by introducing additional matter fields, respecting the global structure of the gauge groups.

The models under study are defined by the coset spaces:

• $E_7/G$ model: $G = \frac{SU(5) \times SU(3) \times U(1)}{\mathbb{Z}_5 \times \mathbb{Z}_3}$
• $E_7/H$ model: $H = \frac{SU(5) \times U(1)^3}{\mathbb{Z}_5 \times \mathbb{Z}_3}$

2. Methodology

The authors employ the modern framework of invertible field theories and bordism classification to analyze anomalies.

• Bordism Classification: Anomalies in a $d$-dimensional theory are classified by the Anderson dual of the spin bordism group of the target space (or the Borel construction for gauged theories), denoted as $(I\mathbb{Z}\Omega^{\text{Spin}})^{d+2}(X)$.
  • Perturbative Anomalies: Encoded in the free part of the bordism group $\Omega^{\text{Spin}}_{d+2}(X)$.
  • Global Anomalies: Encoded in the torsion part of the bordism group $\Omega^{\text{Spin}}_{d+1}(X)$.
• Computational Tools:
  • Atiyah-Hirzebruch Spectral Sequence (AHSS): Used to explicitly compute the spin bordism groups $\Omega^{\text{Spin}}_*(X)$ for the target spaces and their Borel constructions.
  • Serre Spectral Sequence: Used to compute the cohomology rings $H^*(BG; \mathbb{Z})$ and $H^*(E_7/G; \mathbb{Z})$ required as input for the AHSS.
  • Splitting Principle: Used to compute characteristic classes (Chern characters) for perturbative anomaly polynomials.
• Global Structure Analysis: The authors explicitly derive the constraints on linear representations of $G$ and $H$ imposed by their non-trivial global structures (the $\mathbb{Z}_5 \times \mathbb{Z}_3$ quotients). This is crucial for determining which additional fermions can be consistently introduced to cancel anomalies.

3. Key Contributions and Results

A. Absence of Global Anomalies

The most significant result of the paper is the rigorous proof that no global anomalies exist in these family unification models, both before and after gauging.

• Untwisted Models: The authors computed the 5-dimensional spin bordism groups of the target spaces:
  • $\Omega^{\text{Spin}}_5(E_7/G) = 0$
  • $\Omega^{\text{Spin}}_5(E_7/H) = 0$
  • Conclusion: There are no global sigma model anomalies in the $E_7/G$ and $E_7/H$ models.
• Gauged Models: They further computed the bordism groups for the Borel constructions $(E_7/G)_G$ and $(E_7/H)_H$ (representing the theory with the symmetry $G$ or $H$ gauged):
  • $\Omega^{\text{Spin}}_5((E_7/G)_G) = 0$
  • $\Omega^{\text{Spin}}_5((E_7/H)_H) = 0$
  • Conclusion: Gauging the isotropy subgroups $G$ and $H$ does not introduce new global anomalies.

B. Perturbative Anomalies and Cancellation

While global anomalies are absent, the models suffer from perturbative sigma model anomalies (encoded in the free part of $\Omega^{\text{Spin}}_6$).

• Anomaly Polynomials: The authors derived the explicit third Chern character (anomaly polynomial) for the fermions arising from the sigma model fields.
  • For $E_7/G$, the anomaly is non-zero.
  • For $E_7/H$, the anomaly is non-zero.
• Cancellation Attempts:
  • The authors analyzed the possibility of cancelling these anomalies by introducing additional chiral multiplets.
  • They found that for the $E_7/G$ model, simple candidates (like complex conjugate representations) make the theory non-chiral (phenomenologically undesirable).
  • They investigated specific sets of additional multiplets proposed in literature (e.g., one $(\bar{5}, 1)$ and five $(1, 3)$ multiplets). They showed that while the $b_1 c'_2$ terms can be cancelled by tuning charges, the $b_1^3$ term cannot be cancelled by integer charges alone without introducing non-standard degrees of freedom.
  • They noted that the $b_1^3$ term might be cancelled via the Green-Schwarz mechanism in four dimensions, relaxing the strict cancellation condition.

C. Topological Data

The paper provides extensive new topological data essential for future research in this field:

• Explicit computation of cohomology rings $H^*(BG; \mathbb{Z})$ and $H^*(BH; \mathbb{Z})$ up to degree 6.
• Explicit computation of cohomology rings $H^*(E_7/G; \mathbb{Z})$ and $H^*(E_7/H; \mathbb{Z})$ up to degree 6.
• Detailed derivation of the differentials in the Serre and Atiyah-Hirzebruch spectral sequences for these specific spaces.

4. Significance

• Validation of Model Viability: The demonstration that global anomalies are absent is a critical consistency check. It confirms that these family unification models are not ruled out by topological obstructions, a common failure point for BSM (Beyond Standard Model) theories.
• Rigorous Framework: The paper moves beyond heuristic checks of global anomalies (often based on $\pi_4(G)$ or similar homotopy groups) to a complete classification using bordism theory, which is the modern standard for anomaly analysis.
• Phenomenological Constraints: By explicitly deriving the constraints on representations due to the global structure of $G$ and H, the paper clarifies the landscape of allowed matter content. This is vital for constructing realistic models where the three generations of fermions emerge naturally.
• Connection to String Theory: The authors highlight that the $E_7/H$ model has potential realizations in F-theory. The absence of global anomalies is a prerequisite for any theory to be embedded in string theory. The results suggest that if the $E_7/H$ model is realized in F-theory, the perturbative anomalies must be cancelled by stringy mechanisms (e.g., anomaly inflow or Green-Schwarz terms), providing a target for future string phenomenology research.

In summary, Sugeno and Wada have provided a definitive topological analysis of $E_7$-based family unification models, proving their global consistency while clarifying the specific conditions required to cancel perturbative anomalies.