Physlib.QFT.QED.AnomalyCancellation.Basic

For a given natural number $n$, which represents the number of Weyl fermions in a pure $U(1)$ gauge theory, this definition specifies the vector space of charges $\mathbb{Q}^n$. Each element in this space corresponds to the charges $(x_1, x_2, \dots, x_n)$ assigned to the $n$ fermions.

For a pure $U(1)$ gauge theory with $n$ Weyl fermions represented by a vector of rational charges $S = (x_1, x_2, \dots, x_n) \in \mathbb{Q}^n$, this $\mathbb{Q}$-linear map calculates the gravitational anomaly, defined as the sum of the charges $\sum_{i=1}^n x_i$.

In a pure $U(1)$ gauge theory with $n$ Weyl fermions, where the space of charges is represented by the vector space $\mathbb{Q}^n$, this definition constructs the symmetric trilinear form $\mathcal{A}: \mathbb{Q}^n \times \mathbb{Q}^n \times \mathbb{Q}^n \to \mathbb{Q}$. For three vectors of rational charges $x, y, z \in \mathbb{Q}^n$, the form is defined as the sum of the pointwise products of their components: \[ \mathcal{A}(x, y, z) = \sum_{i=1}^{n} x_i y_i z_i \] This symmetric trilinear form is used to characterize the cubic anomaly cancellation condition, which corresponds to the case where $x = y = z$ (i.e., $\sum x_i^3 = 0$).

In a pure $U(1)$ gauge theory with $n$ Weyl fermions, where the charges are represented by a vector $x = (x_1, x_2, \dots, x_n) \in \mathbb{Q}^n$, this definition specifies the homogeneous cubic form that maps the charge vector $x$ to the sum of the cubes of its components: \[ \sum_{i=1}^{n} x_i^3 \] This cubic form is derived from the associated symmetric trilinear form $\mathcal{A}(x, y, z) = \sum_{i=1}^n x_i y_i z_i$ by evaluating it at $x = y = z$. It characterizes the cubic anomaly cancellation condition (ACC) for the theory.

In a pure $U(1)$ gauge theory with $n$ Weyl fermions, let $S = (x_1, x_2, \dots, x_n) \in \mathbb{Q}^n$ be the vector of rational charges assigned to the fermions. The cubic anomaly cancellation condition (ACC) form, denoted as $\text{accCube}_n$, evaluated at the charge vector $S$, is equal to the sum of the cubes of its components: \[ \text{accCube}_n(S) = \sum_{i=1}^{n} x_i^3 \]

For a given natural number $n$, this defines the Anomaly Cancellation Condition (ACC) system for a pure $U(1)$ gauge theory with $n$ Weyl fermions. The theory is characterized by a vector of rational charges $x = (x_1, x_2, \dots, x_n) \in \mathbb{Q}^n$. The system consists of: - One linear ACC, defined by the gravitational anomaly: $\sum_{i=1}^n x_i = 0$. - Zero quadratic ACCs. - One cubic ACC, defined by the gauge anomaly: $\sum_{i=1}^n x_i^3 = 0$.

For natural numbers $n$ and $m$, if $n = m$, there is a linear isomorphism (linear equivalence over $\mathbb{Q}$) between the space of rational charges for a pure $U(1)$ gauge theory with $n$ fermions and the space of charges for a theory with $m$ fermions. This map identifies the charge vectors in $\mathbb{Q}^n$ and $\mathbb{Q}^m$ by reindexing the components according to the equality of the number of fermions.

Consider a pure $U(1)$ gauge theory with $n$ Weyl fermions carrying rational charges $x_1, x_2, \dots, x_n$. If these charges satisfy the linear anomaly cancellation conditions (ACCs) of the theory, then the sum of the charges is zero, i.e., $$\sum_{i=1}^n x_i = 0$$

For a pure $U(1)$ gauge theory with $n$ Weyl fermions, let $S = (x_1, x_2, \dots, x_n) \in \mathbb{Q}^n$ be a vector of rational charges that satisfies the anomaly cancellation conditions (ACCs). Then, the cubic anomaly cancellation condition is satisfied, meaning the sum of the cubes of the charges is zero: \[ \sum_{i=1}^{n} x_i^3 = 0 \]

Consider a pure $U(1)$ gauge theory with $n+1$ Weyl fermions carrying rational charges $x_1, x_2, \dots, x_{n+1}$. If these charges satisfy the linear anomaly cancellation condition (ACC), which requires that the sum of all charges is zero ($\sum_{i=1}^{n+1} x_i = 0$), then the last charge $x_{n+1}$ is equal to the negation of the sum of the other charges: $$x_{n+1} = - \sum_{i=1}^n x_i$$

Consider a pure $U(1)$ gauge theory with $n+1$ Weyl fermions. Let $S = (x_1, x_2, \dots, x_{n+1})$ and $T = (y_1, y_2, \dots, y_{n+1})$ be two vectors of rational charges that satisfy the linear anomaly cancellation condition (ACC), which requires that the sum of the charges is zero ($\sum_{i=1}^{n+1} x_i = 0$ and $\sum_{i=1}^{n+1} y_i = 0$). If the first $n$ charges of both solutions are identical, such that $x_i = y_i$ for all $i \in \{1, \dots, n\}$, then the two solutions are equal, $S = T$.

In a pure $U(1)$ gauge theory with $n$ fermions, let $f$ be a sequence of $k$ charge vectors, where each $f(i) \in \mathbb{Q}^n$ represents the rational charges of the $n$ fermions. For any fermion index $j \in \{0, \dots, n-1\}$, the $j$-th component of the sum of these charge vectors is equal to the sum of the $j$-th components of the individual vectors: $$\left( \sum_{i=0}^{k-1} f(i) \right)_j = \sum_{i=0}^{k-1} f(i)_j$$

Consider a pure $U(1)$ gauge theory with $n$ Weyl fermions. Let $f$ be a collection of $k$ charge vectors in $\mathbb{Q}^n$, where each vector $f(i)$ satisfies the linear Anomaly Cancellation Condition (ACC) $\sum_{l=1}^n (f(i))_l = 0$. For any fermion index $j \in \{1, \dots, n\}$, the $j$-th charge of the sum of these solutions is equal to the sum of the $j$-th charges of each individual solution: $$\left( \sum_{i=1}^k f(i) \right)_j = \sum_{i=1}^k (f(i))_j$$