On Quantum Aspects of 1-Form Symmetries II: Bordism, Invertible Phases, and Anomalies

This paper investigates quantum anomalies of $U(1)$ 1-form symmetries by computing the oriented and spin bordism groups of $K(\mathbb{Z},3)$ up to degree 8, thereby identifying new mixed perturbative and discrete anomalies in 5- and 7-dimensional theories and providing their physical interpretations through invertible phases and bordism invariants.

The Gist

The Big Picture: What is this paper about?

Imagine you are a physicist trying to build a perfect, unbreakable machine (a quantum theory). Usually, these machines work great until you try to turn a specific knob (a symmetry). Sometimes, the machine breaks or behaves strangely when you turn that knob. In physics, we call this a quantum anomaly. It's like a hidden glitch that prevents the laws of physics from working smoothly in certain situations.

This paper focuses on a very specific type of knob: the U(1) 1-form symmetry.

• 0-form symmetry (Normal): Think of this like a light switch. You flip it, and the whole room changes. It acts on individual particles (like electrons).
• 1-form symmetry (This paper): Think of this like a "string" or a "loop" of energy. Instead of acting on a single point, this symmetry acts on entire loops or strings moving through space. The "knob" here is a background field that wraps around these strings.

The authors wanted to map out every possible way this "string symmetry" could glitch (anomaly) in different dimensions (3D, 5D, 7D, etc.). They used a mathematical tool called bordism to do this.

The Mathematical Tool: The "Shape-Checker"

To find these glitches, the authors used a method called bordism.

• The Analogy: Imagine you have a collection of different shapes (manifolds) like spheres, donuts, and weird blobs. You want to know if a specific shape can be smoothly transformed into another without tearing.
• The "Shape-Checker": The authors built a giant catalog (a mathematical group) of all possible shapes that can exist in a universe with this specific "string symmetry."
• The Result: If they find a shape in their catalog that cannot be smoothed out or transformed into nothing, it means there is a glitch (anomaly) in the physics. The catalog tells them exactly what kind of glitch it is and how strong it is.

They calculated this catalog up to 8 dimensions and found two main types of glitches:

1. Smooth Glitches (Perturbative): These are like a car engine that runs slightly rough. You can describe them with standard equations (polynomials).
2. Discrete Glitches (Global/Torsion): These are like a light switch that only works if you flip it an even number of times, but fails if you flip it an odd number of times. You can't describe these with smooth equations; they are "all or nothing" binary errors.

The New Discoveries

The paper found two brand-new types of glitches that hadn't been fully understood before.

1. The 5D "Twisted String" Glitch

In a 5-dimensional world, they found a mixed glitch between the "string symmetry" and the shape of space itself (gravity/diffeomorphisms).

• The Formula: It involves a term called $H_3 \wedge p_1$.
• The Analogy: Imagine a magnetic string (a 1D object) floating in 5D space. In a normal world, you only need to know where the string is and how much "magnetic charge" it has.
• The Twist: Because of this new anomaly, the string carries extra hidden information. It's like the string isn't just a wire; it's a wire wrapped in a specific kind of "ribbon" (a trivialization of a characteristic class).
• The Consequence: To fully describe the string, you don't just need its location; you need to know how this "ribbon" is tied. If you try to remove the string, the way the ribbon was tied matters. This is a higher-dimensional version of how magnetic monopoles in 4D must be fermions (particles that follow specific quantum rules).

2. The 7D "Binary Switch" Glitch

In a 7-dimensional world, they found a purely discrete glitch that is intrinsic to the symmetry itself.

• The Formula: It involves a term called $u \cup Sq_2 u$.
• The Analogy: Imagine a 7D universe where the laws of physics have a "parity check." If you try to perform a certain operation on the string symmetry, the universe might say "No" (giving a -1 phase) or "Yes" (giving a +1 phase) depending on a binary condition.
• The Twist: This glitch is like a secret code. Even if you try to restrict the symmetry to a smaller, simpler version (like a Z2 subgroup), the glitch doesn't disappear. Instead, it transforms into a specific type of error within that smaller group. It's like a virus that mutates but doesn't die when you change the host.

How They Verified It (Top-Down Construction)

The authors didn't just do math on paper; they checked if these glitches could actually appear in real-world theories derived from String Theory.

• They took a 10-dimensional theory (Type IIA String Theory) and "rolled up" extra dimensions to create 5D and 7D worlds.
• They found that the "Twisted String" glitch in 5D appears naturally when you compactify the theory on a specific shape (like a 4-sphere or a complex surface).
• They also found the "Binary Switch" glitch in 7D, though it requires a very specific, twisted setup involving "mod-2" (binary) geometry.

Summary of the "Takeaway"

1. We mapped the glitches: The authors created a complete list of possible errors (anomalies) for "string symmetries" in dimensions up to 7.
2. New Physics in 5D: In 5D, magnetic strings are more complex than we thought; they carry extra topological "ribbons" that must be accounted for.
3. New Physics in 7D: In 7D, there is a binary "yes/no" glitch that is fundamental to the symmetry and doesn't vanish even if you simplify the symmetry group.
4. Real-world connection: These abstract mathematical glitches can actually be derived from high-energy String Theory, suggesting they are real features of the universe's underlying structure.

In short, the paper uses advanced geometry to prove that "string symmetries" have hidden, complex behaviors in higher dimensions that change how we understand magnetic strings and the fabric of spacetime.

Technical Summary

Technical Summary: Quantum Aspects of 1-Form Symmetries II: Bordism, Invertible Phases, and Anomalies

Problem Statement
This work addresses the systematic classification of quantum anomalies associated with continuous $U(1)$ 1-form symmetries in quantum field theories (QFTs). While 0-form symmetries and finite higher-form symmetries have been extensively studied through the lens of bordism and invertible field theories, continuous $U(1)$ 1-form symmetries have lacked a comprehensive treatment from this perspective. The background gauge fields for such symmetries are $U(1)$ gerbe connections (2-form gauge fields), topologically classified by the Eilenberg–Mac Lane space $B^2U(1) = K(\mathbb{Z}, 3)$. The authors aim to compute the relevant bordism groups to classify both perturbative (local) and global (non-perturbative) anomalies for these symmetries in dimensions up to $d=7$.

Methodology
The authors employ the modern framework where anomalies of a $d$-dimensional theory are described by invertible phases in $d+1$ dimensions, classified by the Anderson dual of the bordism spectrum, $(I\mathbb{Z}\Omega^S)_{d+2}(X)$. Here, $X = K(\mathbb{Z}, 3)$ represents the target space for the background fields, and $S$ denotes the spacetime structure ($SO$ for bosonic/oriented theories and $Spin$ for fermionic theories).

The core technical approach involves computing the oriented ($\Omega^{SO}_\bullet$) and spin ($\Omega^{Spin}_\bullet$) bordism groups of $K(\mathbb{Z}, 3)$ up to degree 8. The authors utilize the Atiyah–Hirzebruch Spectral Sequence (AHSS) with the trivial fibration $pt \to K(\mathbb{Z}, 3) \to K(\mathbb{Z}, 3)$.

• AHSS Setup: The $E_2$-page is given by $E_2^{p,q} = H_p(K(\mathbb{Z}, 3); \Omega^S_q(pt))$.
• Extension Problems: A significant portion of the methodology involves resolving extension problems that arise when the spectral sequence converges to the associated graded group but does not immediately reveal the group structure. The authors resolve these using geometric arguments and specific constructions:
  • For $\Omega^{SO}_7(K(\mathbb{Z}, 3))$, they utilize Milnor's construction of exotic spheres (specifically an $S^3$-bundle over $S^4$) to demonstrate a non-trivial extension.
  • For $\Omega^{SO}_8(K(\mathbb{Z}, 3))$, they employ geometric representatives involving products of manifolds like $S^3 \times SU(3)/SO(3)$ and $SU(3)$ to distinguish between possible group structures ($\mathbb{Z}_2 \oplus \mathbb{Z}_2$ vs. $\mathbb{Z}_4$).
  • For spin bordism, they utilize a "suspension trick" relating the bordism groups of $K(\mathbb{Z}, 3)$ to those of $K(\mathbb{Z}, 4)$ and $K(\mathbb{Z}, 2)$, leveraging known results from recent mathematical literature.

Key Contributions and Results

1. Bordism Group Computations:
   The paper provides explicit calculations for $\Omega^{SO/Spin}_n(K(\mathbb{Z}, 3))$ for $n \leq 8$.

   • Oriented Case: The authors resolve a non-trivial extension in degree 7, proving $\Omega^{SO}_7(K(\mathbb{Z}, 3)) \cong \mathbb{Z}$. In degree 8, they show $\Omega^{SO}_8(K(\mathbb{Z}, 3)) \cong \mathbb{Z}_2 \oplus \mathbb{Z}_2$.
   • Spin Case: The results align with recent independent computations, confirming $\Omega^{Spin}_7(K(\mathbb{Z}, 3)) \cong \mathbb{Z}$ and $\Omega^{Spin}_8(K(\mathbb{Z}, 3)) \cong \mathbb{Z}_2 \oplus \mathbb{Z}_2$.
   • Invertible Phases: Using the universal coefficient theorem, the authors derive the groups of invertible phases $(I\mathbb{Z}\Omega^S)_{d+2}(K(\mathbb{Z}, 3))$, which classify the anomaly theories.

2. Identification of Anomalies:
   The bordism invariants are mapped to specific anomaly terms:

   • 5d Theories ($d=5$): The free part of the bordism group yields a mixed perturbative anomaly between the $U(1)$ 1-form symmetry and spacetime diffeomorphisms.
     • Oriented: The anomaly polynomial is generated by $H_3 \wedge p_1$, where $H_3$ is the field strength of the background 2-form gauge field and $p_1$ is the first Pontryagin class.
     • Spin: The normalization is refined to $\frac{1}{4}H_3 \wedge p_1$.
   • 7d Theories ($d=7$): The torsion part of the bordism group reveals new discrete anomalies intrinsic to the continuous $U(1)$ 1-form symmetry.
     • Pure 1-form Anomaly: A $\mathbb{Z}_2$-valued anomaly generated by $u \smile Sq_2 u$, where $u$ is the mod-2 reduction of the universal degree-3 class.
     • Mixed Anomaly (Oriented only): An additional anomaly $u \smile w_2 \smile w_3$ on non-spin manifolds.

3. Physical Interpretations and Boundary Realizations:

   • Magnetic Strings in 5d: The $H_3 \wedge p_1$ anomaly implies that magnetic strings in 5d theories carry additional topological data. Specifically, the string worldsheet requires a choice of trivialization of the characteristic class ($p_1$ or $\frac{1}{2}p_1$) on the boundary of its tubular neighborhood. This trivialization forms a torsor classified by $H^1(\Sigma, \mathbb{Z})$, refining the description of the string beyond just its charge and embedding.
   • Discrete $\theta$-angles in 7d: The $u \smile Sq_2 u$ anomaly is interpreted as a discrete $\theta$-angle ($\theta = 0, \pi$) for generalized Maxwell-type theories. The authors discuss its restriction to the $\mathbb{Z}_2 \subset U(1)$ subgroup, finding that the anomaly does not trivialize but maps to an order-two element within the intrinsic $\mathbb{Z}_8$ anomaly group of the $\mathbb{Z}_2$ 1-form symmetry (an example of anomaly interplay).
   • Maxwell Theory Phases: The framework reproduces known phases of 4d Maxwell theory (including mixed electric-magnetic anomalies) and extends the classification to 5d and 7d Maxwell theories, identifying new mixed anomalies involving magnetic dual symmetries.

4. Top-Down Constructions:
   The authors provide string theory realizations for these anomalies:

   • The $H_3 \wedge p_1$ term arises from the dimensional reduction of Type IIA superstring theory on a compact 4-manifold $L_4$, specifically from the topological coupling $B_2 \wedge X_8$. The coefficient depends on the signature of $L_4$.
   • The $u \smile Sq_2 u$ term is constructed from a mod-2 topological term in 10d IIA theory reduced on a manifold $L_2$ (e.g., $\mathbb{R}P^2$) with specific torsional cohomology properties.

Significance
The paper claims to provide the first systematic bordism-based classification of anomalies for continuous $U(1)$ 1-form symmetries. By resolving extension problems geometrically, it clarifies the structure of the anomaly groups in dimensions 5 and 7, distinguishing between perturbative anomalies (captured by free bordism invariants) and global/discrete anomalies (captured by torsion invariants). The work establishes a concrete link between abstract bordism invariants and physical phenomena, such as the topological structure of magnetic strings and the existence of discrete $\theta$-angles in higher-dimensional gauge theories. Furthermore, it demonstrates how these anomalies can be realized via top-down string theory constructions, validating the mathematical classification with physical models. The authors note that while the current work focuses on oriented and spin structures without time-reversal symmetry, the framework is extendable to include time-reversal and non-Abelian gerbes.