Physlib.QFT.QED.AnomalyCancellation.LowDim.Three

Consider a pure $U(1)$ gauge theory with 3 fermions carrying rational charges $x_0, x_1, x_2$. Suppose these charges satisfy the linear anomaly cancellation condition $x_0 + x_1 + x_2 = 0$. Then the product of the charges scaled by three is zero, $3x_0 x_1 x_2 = 0$, if and only if the cubic anomaly cancellation condition is satisfied: $$\sum_{i=0}^2 x_i^3 = 0$$

Consider a pure $U(1)$ gauge theory with 3 fermions having rational charges $x_0, x_1, x_2$. Suppose the charges satisfy the linear anomaly cancellation condition $x_0 + x_1 + x_2 = 0$. Then the cubic anomaly cancellation condition is satisfied: $$\sum_{i=0}^2 x_i^3 = 0$$ if and only if at least one of the charges is zero ($x_0 = 0$, $x_1 = 0$, or $x_2 = 0$).

Consider a pure $U(1)$ gauge theory with 3 fermions having rational charges $x_0, x_1, x_2$. If these charges satisfy the anomaly cancellation conditions, namely the linear gravitational condition $x_0 + x_1 + x_2 = 0$ and the cubic gauge condition $x_0^3 + x_1^3 + x_2^3 = 0$, then at least one of the charges must be zero ($x_0 = 0$, $x_1 = 0$, or $x_2 = 0$).

For a pure $U(1)$ gauge theory with 3 fermions, let $S = (x_0, x_1, x_2) \in \mathbb{Q}^3$ be a linear solution satisfying the gravitational anomaly cancellation condition $\sum_{i=0}^2 x_i = 0$. Given a proof $h_S$ that at least one of the charges is zero (i.e., $x_0 = 0 \lor x_1 = 0 \lor x_2 = 0$), this function constructs a full solution to the anomaly cancellation system. A full solution is a vector that satisfies both the linear condition $\sum_{i=0}^2 x_i = 0$ and the cubic gauge anomaly cancellation condition $\sum_{i=0}^2 x_i^3 = 0$. This construction relies on the property that for linear solutions in this system, the cubic condition is satisfied if and only if one of the charges is zero.

For any solution $S = (x_0, x_1, x_2) \in \mathbb{Q}^3$ to the anomaly cancellation conditions of a pure $U(1)$ gauge theory with 3 fermions—which requires $x_0 + x_1 + x_2 = 0$ and $x_0^3 + x_1^3 + x_2^3 = 0$—there exists a linear solution $T$ (satisfying $x_0 + x_1 + x_2 = 0$) and a proof $h_T$ that at least one of its charges is zero ($x_0 = 0$, $x_1 = 0$, or $x_2 = 0$), such that the mapping of this linear solution to a full solution, denoted by `solOfLinear T hT`, is equal to $S$.