Anomaly cancellation for two $U(1)$ factors

Technical Summary: Anomaly Cancellation for Two U(1) Factors

Problem Statement
This paper addresses the problem of local anomaly cancellation in four-dimensional gauge quantum field theories where the gauge Lie algebra contains an abelian summand of rank $K \ge 1$. Specifically, the authors focus on the case where the gauge group includes two $U(1)$ factors (rank $K=2$), potentially accompanied by a semisimple summand. The core challenge is solving a system of homogeneous cubic polynomial equations in integers (representing fermion charges) to find all anomaly-free configurations. While the case of a single $U(1)$ factor ($K=1$) has been successfully mapped to finding rational points on a cubic hypersurface, the higher-rank case presents a more formidable algebraic problem involving multiple cubic equations that generally define a projective variety that is neither a cubic nor a hypersurface.

Methodology
The authors reformulate the anomaly cancellation conditions using tools from algebraic and arithmetic geometry.

1. Geometric Reformulation: They demonstrate that solving the anomaly cancellation conditions for a gauge theory with $K$ abelian factors is equivalent to finding $(K-1)$-dimensional projective linear subspaces (e.g., lines for $K=2$) lying on a specific cubic hypersurface $X$. This hypersurface is determined by the data of the semisimple summand and the representations of the Weyl fermions.
2. Fano Varieties: The set of these linear subspaces forms a variety known as the Fano variety of linear subspaces, denoted $F_{K-1}(X)$. The authors utilize the theory of Fano varieties to analyze the structure of the solution space.
3. Case Study ($K=2$): The paper focuses on the rank-2 case, where the problem reduces to finding lines on a cubic hypersurface.
   • Cubic Surfaces: For theories with fewer fermions or specific semisimple representations (e.g., $SU(2) \oplus U(1) \oplus U(1)$), the hypersurface is a cubic surface. The authors employ the Plücker embedding of the Grassmannian to explicitly solve for lines on these surfaces, distinguishing between rational and non-rational solutions.
   • Segre Cubic Primal: For the simplest non-trivial pure $U(1) \oplus U(1)$ case with six fermions, the relevant hypersurface is the Segre cubic primal 3-fold in $\mathbb{P}^4$. The authors analyze the Fano variety of lines on this singular cubic 3-fold using two distinct approaches:
     • Affine Cover: Constructing an affine cover of the Grassmannian $G(1,4)$ to solve quadratic equations defining the lines.
     • Birational Equivalence: Utilizing Richmond's map, a birational equivalence between $\mathbb{P}^3$ and the Segre cubic primal, to map lines in $\mathbb{P}^3$ to lines on the cubic 3-fold.

Key Results

1. Structure of Solutions for $U(1)^2$ (Six Fermions): The Fano variety of lines on the Segre cubic primal is reducible and consists of 21 irreducible components, all of dimension 2:
   • 15 Planar Components: These correspond to lines lying within the 15 non-chiral planes of the Segre cubic primal. They are isomorphic to projective planes and yield non-chiral fermion solutions.
   • 6 Del Pezzo Components: These correspond to chiral solutions. Each is isomorphic to a split del Pezzo surface of degree 5 (a blow-up of $\mathbb{P}^2$ at four rational points).
2. Rationality and Parametrization: All 21 components are rational varieties. This allows for the explicit parametrization of nearly all solutions (a dense open set) using rational parameters. The authors provide explicit birational maps from $\mathbb{P}^2$ to the del Pezzo components, enabling the generation of infinite families of chiral solutions.
3. Distribution of Solutions: The rational points on the del Pezzo components are dense in the Zariski topology and the Euclidean topology. The asymptotic distribution of rational points of bounded height follows Manin's conjecture, scaling as $B(\log B)^4$.
4. Chiral vs. Non-Chiral Intersections: The authors explain the phenomenon where a solution is chiral with respect to the full $U(1) \oplus U(1)$ algebra but non-chiral with respect to individual $U(1)$ factors. This occurs because chiral lines (lying on del Pezzo components) intersect the non-chiral planes. A generic chiral line intersects five distinct planes, yielding 10 specific pairs of points that correspond to such mixed-chirality solutions.
5. Limitations of Previous Methods: The paper clarifies that the "lines through points" method used in prior work for pure $U(1)$ cases does not generalize directly to higher ranks without the geometric framework of Fano varieties, as finding lines through a generic point on a cubic hypersurface over $\mathbb{Q}$ is not guaranteed to yield solutions.

Significance and Claims
The paper claims to provide a complete geometric characterization of the anomaly cancellation conditions for $U(1) \oplus U(1)$ gauge theories with six fermions. By identifying the solution space with the Fano variety of lines on the Segre cubic primal, the authors transform a difficult number-theoretic problem into a tractable geometric one.

• Completeness: The work recovers previous results (such as those in [6]) regarding the existence of six families of solutions but goes further by proving these families correspond to the six del Pezzo components and providing explicit parametrizations for all solutions within them.
• Generality: The formalism is presented as applicable to any number of $U(1)$ summands, though the complexity increases with rank.
• Future Outlook: The authors modestly note that while their methods succeed for the singular Segre cubic primal, they anticipate significant difficulties for smooth cubic hypersurfaces of higher dimension (e.g., $U(1)^2$ with seven fermions). In those cases, the Fano variety of lines is expected to be an irrational variety (e.g., a hyperkähler fourfold), where rational points may not be Zariski dense, suggesting that anomaly-free solutions could be sparse and harder to find.

The paper does not propose new experimental applications but emphasizes the utility of algebraic geometry in systematically classifying and generating consistent quantum field theories.