⚛️ phenomenology

Gauging the Standard Model 1-form symmetry via gravitational instantons

This paper demonstrates that gravitational instantons, specifically Eguchi-Hanson geometries, induce quantized fluxes that enforce global boundary conditions on fermion wavefunctions and, upon summation over flux sectors in the path integral, gauge the Standard Model's $\mathbb{Z}_6^{(1)}$ 1-form symmetry, thereby proving it cannot persist as an exact global symmetry while simultaneously inducing exponentially suppressed baryon- and lepton-number violating processes.

Original authors: Mohamed M. Anber

Published 2026-02-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mohamed M. Anber

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, complex video game. In this game, the "Standard Model" is the rulebook that tells all the particles (like electrons and quarks) how to behave and interact. For a long time, physicists thought this rulebook had a hidden "cheat code" or a global symmetry—a special rule that said, "No matter what happens, certain things must always stay exactly the same."

This paper investigates whether that cheat code actually works when you introduce the "gravity" engine into the game. The authors, led by Mohamed M. Anber, argue that gravity breaks this cheat code, turning a rigid global rule into a flexible, local one.

Here is the breakdown using simple analogies:

1. The Setting: The "Eguchi-Hanson" Bubble

To test their theory, the authors don't look at our flat, boring universe. Instead, they imagine a special, curved pocket of space called an Eguchi-Hanson (EH) instanton.

The Analogy: Think of normal space as a flat sheet of paper. The EH instanton is like taking that paper and folding it into a specific, smooth shape that looks like a donut with a hole in the middle, but the hole is actually a tiny, smooth sphere (called a "bolt") rather than a sharp edge.
The Key Feature: This shape has a unique property: it can hold a "magnetic flux" (a kind of invisible field) right in that central sphere without the shape of the space changing or collapsing. It's like threading a glowing string through the center of a donut without squishing the donut.

2. The Problem: The "Global" Cheat Code

The Standard Model has a symmetry called $Z_6^{(1)}$ .

The Analogy: Imagine a rule that says, "You can only paint the walls of the universe in groups of 6 specific colors." This is a global rule; it applies everywhere at once, no matter where you are.
The Conflict: In quantum gravity, the "No Global Symmetries" rule is a famous law. It says that in a universe with gravity, you can't have rules that apply everywhere perfectly. Everything must be local. If you try to enforce a global rule, gravity should find a way to break it.

3. The Solution: "Gauging" the Symmetry

The authors show that the EH instanton provides the mechanism to break this global rule.

The Mechanism: They place the Standard Model particles inside this "donut" space and thread the glowing string (flux) through the center.
The Twist: Because of the shape of the donut and the string, the particles (fermions) have to behave in a very specific way to stay "whole." If they don't, they tear apart.
The Result: To keep the particles from tearing, the universe is forced to allow fractional amounts of the "colors" (fluxes). Instead of just whole groups of 6, you can have 1/6th or 2/6ths.
The "Gauging": By allowing these fractional amounts and summing over all possible ways to thread the string, the rigid "Global Rule" (Cheat Code) is destroyed. It is replaced by a Gauge Symmetry.
- Simple Translation: The rule changes from "You must have 6 colors" to "You can have any combination of colors, as long as the total adds up correctly locally." The cheat code is gone; the game is now fair and dynamic.

4. The Consequence: Proton Decay (The "Forbidden" Process)

When you break a symmetry, you usually allow things to happen that were previously forbidden.

The Analogy: Imagine a bank vault that used to be locked with a single, unbreakable key (the global symmetry). Now, because the lock is broken (gauged), the vault can be opened, but only with a very specific, incredibly difficult combination.
The Physics: This setup allows for Baryon and Lepton number violation. In plain English, this means protons (which make up matter) could theoretically decay into other particles.
The Catch: The authors calculate that while this can happen, the probability is astronomically small. It's like trying to win the lottery every second for a billion years. The "cost" (energy) to thread the string through the donut is so high that nature almost never does it. So, while the symmetry is broken, we don't see our universe falling apart.

5. The "Entangled" State

The paper also offers a mind-bending interpretation of what this space represents in the quantum world.

The Analogy: Imagine the universe is split into two identical halves, Left and Right. The EH instanton acts like a bridge connecting them.
The Quantum State: The authors suggest that the path integral (the math used to calculate probabilities) describes a transition from a state where the Left and Right halves are entangled (like two dice that always roll the same number, no matter how far apart they are) to the empty vacuum.
Why it matters: This suggests that the "breaking" of the symmetry is a result of the deep quantum connection between different parts of space.

Summary

In short, this paper uses a fancy mathematical shape (the Eguchi-Hanson instanton) to prove a fundamental point: Gravity cannot tolerate rigid, global rules.

It shows that the Standard Model's "global symmetry" is actually just a disguise for a "gauge symmetry" that allows for fractional fluxes. While this theoretically allows for the decay of matter (protons), the process is so suppressed by the weakness of the forces involved that we won't see it happening anytime soon.

The Big Takeaway: Gravity is the ultimate "rule breaker." It ensures that no symmetry in the universe is truly global; everything is local, dynamic, and subject to the subtle, entangled nature of spacetime itself.

1. Problem Statement

The Standard Model (SM) of particle physics, in its current formulation, possesses a global electric $\mathbb{Z}^{(1)}_6$ 1-form symmetry. According to the Swampland program and general principles of quantum gravity, exact global symmetries (including higher-form symmetries) are believed to be absent in a consistent theory of quantum gravity. They must either be explicitly broken by new physics or gauged (promoted to a local symmetry).

While mechanisms like black hole evaporation or stringy branes are known to violate global symmetries, the specific mechanism by which the SM's $\mathbb{Z}^{(1)}_6$ symmetry is gauged in a semiclassical gravitational context remains an open question. This paper investigates whether Eguchi-Hanson (EH) gravitational instantons can provide a semiclassical mechanism to gauge this symmetry, induce fractional topological charges, and generate baryon/lepton-number violating processes.

2. Methodology

The author employs a semiclassical Euclidean path integral approach, treating the EH geometry as a fixed classical background without backreaction from quantum fluctuations of the metric itself. The methodology involves:

Geometry: Utilizing the Eguchi-Hanson metric, an Asymptotically Locally Euclidean (ALE) space that is Ricci-flat and self-dual. The space contains a non-contractible 2-cycle (a "bolt") at a finite radius $r=a$ and has an asymptotic boundary $\partial(EH) \cong \mathbb{R}P^3$ (real projective 3-space), which is $S^3/\mathbb{Z}_2$ .
Gauge Fields: Introducing background gauge fields valued in the Cartan subalgebra of the SM gauge group $SU(3) \times SU(2) \times U(1)_Y$ . These fields are constructed to be self-dual (or anti-self-dual) so that their stress-energy tensor vanishes, ensuring they do not backreact on the EH metric.
Flux Quantization: Turning on fractional fluxes through the bolt $S^2$ to represent the background of the $\mathbb{Z}^{(1)}_N$ 1-form symmetry. This requires careful handling of flux quantization conditions to ensure consistency with the Dirac quantization condition when combined with the $U(1)$ hypercharge sector.
Fermion Analysis: Solving the Dirac equation explicitly in this background to find fermion zero modes. Crucially, the analysis enforces global consistency conditions (boundary conditions) for spinors. Because the boundary is $\mathbb{R}P^3$ , spinors must be well-defined under the antipodal identification $(x, y, z, t) \sim -(x, y, z, t)$ . This imposes specific boundary conditions on the fermion wavefunctions near the bolt and at infinity.
Index Theorem: Verifying the count of zero modes using the Atiyah-Patodi-Singer (APS) index theorem, which accounts for the non-trivial boundary topology and the $\eta$ -invariant.
Path Integral Interpretation: Interpreting the Euclidean path integral in the EH background as a transition amplitude from an entangled initial state (correlating "left" and "right" halves of space due to the antipodal identification) to the vacuum.

3. Key Contributions

A. Fractional Fluxes and Topological Charge

The paper demonstrates that EH instantons support quantized fluxes localized on the bolt $S^2$ .

For a $U(1)$ field, the flux is $\int_{S^2} F = -2\pi C$ ( $C \in \mathbb{Z}$ ).
When embedded into $SU(N)$ to represent the $\mathbb{Z}^{(1)}_N$ symmetry, the flux becomes fractional: $\int_{S^2} F \in 2\pi \mathbb{Z}_N$ .
These configurations induce fractional topological charges (instanton numbers). For example, the $U(1)$ topological charge is $Q = C^2/4$ , and for $SU(N)$, $Q \propto m^2(1 - 1/N)$ .

B. Fermion Zero Modes and Boundary Conditions

A central technical achievement is the explicit solution of the Dirac equation in the presence of these fractional fluxes.

Global Well-Definedness: The requirement that fermion wavefunctions be continuous along contractible paths connecting identified points on the $\mathbb{R}P^3$ boundary imposes strict boundary conditions.
Selection Rules: The boundary condition near the bolt depends on the parity of the flux integer $C$ $C$ .
- If $C$ is even, fermions must have integer total angular momentum $j$ .
- If $C$ is odd, fermions must have half-integer $j$ .
Mode Counting: The number of normalizable zero modes is calculated explicitly and confirmed via the APS index theorem. The result depends on the parity of the flux integers and the representation of the fermion. For a $U(1)$ flux $C=2p$ , the index is $p^2$ ; for $C=2p+1$ , it is $p(p+1)$ .

C. Gauging the $\mathbb{Z}^{(1)}_6$ Symmetry

The paper argues that summing over all possible fractional flux sectors $(m_{(2)}, m_{(3)}, C)$ in the path integral effectively gauges the global $\mathbb{Z}^{(1)}_6$ symmetry.

In the semiclassical limit, the EH instanton acts as a saddle point mediating a transition between an entangled state of two spatial halves and the vacuum.
Summing over all flux sectors removes the global symmetry constraint, promoting it to a dynamical gauge symmetry. This provides a concrete realization of the "no global symmetries" conjecture in quantum gravity.

D. Baryon and Lepton Number Violation

The presence of these fractional fluxes induces non-zero fermion zero modes, leading to effective 't Hooft vertices that violate baryon ( $B$ ) and lepton ( $L$ ) numbers.

The amplitude for these processes is proportional to $e^{-S_{inst}}$ , where the action $S$ is dominated by the gauge field terms.
Crucially, the violation is exponentially suppressed due to the smallness of the $U(1)$ hypercharge coupling constant ( $g_1$ ).
The paper calculates specific examples (e.g., minimal fluxes $m_{(2)}=m_{(3)}=1, C=0$ ) showing $\Delta B = \Delta L = -3$ with an amplitude $\sim 10^{-5268}$ , rendering these effects phenomenologically irrelevant at current energies but theoretically significant.

4. Key Results

Existence of Solutions: Self-dual gauge fields with fractional fluxes can be consistently defined on EH space without backreacting on the geometry.
Zero Mode Spectrum: The number of fermion zero modes is non-zero only when specific fractional fluxes are turned on. The count is given by:
- $I = p^2$ for even flux $C=2p$ .
- $I = p(p+1)$ for odd flux $C=2p+1$ .
- These results match the APS index theorem predictions, including the correct $\eta$ -invariant contributions from the $\mathbb{R}P^3$ boundary.
Symmetry Gauging: Summing over the discrete flux sectors in the path integral dynamically gauges the SM $\mathbb{Z}^{(1)}_6$ symmetry, eliminating it as an exact global symmetry in the semiclassical limit.
Suppressed Violation: While $B$ and $L$ violating processes are induced, their rates are suppressed by factors of $e^{-1/g_Y^2}$ , making them unobservable in current experiments but confirming the theoretical mechanism.

5. Significance

Resolution of the Global Symmetry Problem: The paper provides a concrete, non-perturbative mechanism within semiclassical gravity (using EH instantons) that explains how the SM's exact global 1-form symmetry is gauged, satisfying the Swampland constraints.
New Non-Perturbative Effects: It identifies a new class of gravitational instanton effects that induce fractional topological charges and fermion zero modes, distinct from standard BPST instantons in flat space.
Entanglement and Geometry: The interpretation of the EH instanton as mediating a transition from an entangled state of spatial halves offers a novel perspective on the role of topology and entanglement in quantum gravity path integrals.
Phenomenological Constraints: While the induced $B/L$ violation is too small to be observed, the framework sets strict consistency conditions on the SM gauge group structure and the quantization of axion couplings (related to the Chern-Simons terms) in the presence of fractional fluxes.

In summary, the paper establishes that Eguchi-Hanson gravitational instantons serve as a viable semiclassical arena where the Standard Model's global 1-form symmetry is dynamically gauged through the summation over fractional flux sectors, leading to consistent, albeit highly suppressed, baryon and lepton number violation.