Physlib.QFT.AnomalyCancellation.Basic

Given a natural number $n$, this function constructs an `ACCSystemCharges` object which specifies that a gauge theory has $n$ fermion representations. The resulting object defines the dimension of the charge space as $n$.

For an anomaly cancellation system $\chi$ with $n$ charges, the type of charges is defined as the space of functions from the finite index set $\{0, 1, \dots, n-1\}$ to the rational numbers $\mathbb{Q}$. This space represents the assignment of a rational charge $q_i \in \mathbb{Q}$ to each of the $n$ fermion representations in the theory, effectively forming the vector space $\mathbb{Q}^n$.

For an anomaly cancellation system $\chi$, the space of charges $\chi.\text{Charges}$ (which represents the rational charges $q \in \mathbb{Q}^n$ assigned to $n$ fermions) is equipped with the structure of an additive commutative monoid. This structure defines a zero element (where all charges are $0$) and a binary addition operation (the pointwise sum of charges) that is both associative and commutative.

For an anomaly cancellation system $\chi$, the space of charges $\chi.\text{Charges}$ (which represents vectors of rational charges $q \in \mathbb{Q}^n$) is equipped with the structure of a module over the field of rational numbers $\mathbb{Q}$. This structure defines the standard pointwise addition of charge vectors and their scalar multiplication by rational numbers.

For an anomaly cancellation system $\chi$, the space of charges $\chi.\text{Charges}$ (representing vectors of rational charges $q \in \mathbb{Q}^n$) is equipped with the structure of an additive commutative group. This structure defines vector addition, the additive identity (the zero charge vector), and the existence of additive inverses (negation of charges).

For an anomaly cancellation system $\chi$, the space of charges $\chi.\text{Charges}$ (representing the rational charge assignments $q \in \mathbb{Q}^n$ for the $n$ fermion representations in the theory) is a finitely generated module over the field of rational numbers $\mathbb{Q}$.

For a linear anomaly cancellation system $\chi$, let $S$ and $T$ be two solutions in the space of linear solutions $\chi.\text{LinSols}$. If their underlying rational charge assignments in the module of charges $\chi.\text{Charges}$ are equal (that is, $S.\text{val} = T.\text{val}$), then the solutions $S$ and $T$ are equal.

For a linear anomaly cancellation system $\chi$, the set of solutions to the linear constraints, denoted $\chi.\text{LinSols}$, forms an additive commutative monoid. The addition of two solutions $S$ and $T$ is defined as the pointwise sum of their rational charges, and the identity element is the zero charge vector (where all charges are $0$). This structure inherits the properties of associativity and commutativity from the underlying space of rational charges $\mathbb{Q}^n$.

For a linear anomaly cancellation system $\chi$, the set of solutions to the linear constraints, denoted by $\chi.\text{LinSols}$, is equipped with the structure of a module over the field of rational numbers $\mathbb{Q}$. This structure is inherited from the underlying space of charges $\chi.\text{Charges} \cong \mathbb{Q}^n$, such that for any rational scalar $a \in \mathbb{Q}$ and any solution vector $S \in \chi.\text{LinSols}$, the scalar product $a \cdot S$ also satisfies the linear anomaly cancellation conditions.

For a linear anomaly cancellation system $\chi$, the set of solutions to the linear constraints, denoted $\chi.\text{LinSols}$, forms an additive commutative group. This structure is inherited from its $\mathbb{Q}$-module structure, meaning that for any two solutions $S, T \in \chi.\text{LinSols}$, their pointwise sum $S + T$ is a solution, and for every solution $S$, there exists an additive inverse $-S$ in $\chi.\text{LinSols}$ such that $S + (-S) = 0$.

For a linear anomaly cancellation system $\chi$, the map $\text{linSolsIncl}$ is the canonical $\mathbb{Q}$-linear inclusion from the space of solutions $\chi.\text{LinSols}$ into the space of all charges $\chi.\text{Charges} \cong \mathbb{Q}^n$. This map takes a solution vector $S$ and returns its underlying vector of rational charges.

For any linear anomaly cancellation system $\chi$, the canonical $\mathbb{Q}$-linear inclusion map $\text{linSolsIncl} : \chi.\text{LinSols} \to \chi.\text{Charges}$, which maps a solution to the linear anomaly cancellation conditions to its underlying vector of rational charges in $\mathbb{Q}^n$, is injective.

For an anomaly cancellation system $\chi$ with linear and quadratic conditions, let $S$ and $T$ be two solutions in the space of quadratic solutions $\chi.\text{QuadSols}$. If their underlying rational charge assignments in the module of charges $\chi.\text{Charges}$ are equal (that is, $S.\text{val} = T.\text{val}$), then the solutions $S$ and $T$ are equal.

For a quadratic anomaly cancellation system $\chi$, the set of solutions to both the linear and quadratic constraints, denoted by $\chi.\text{QuadSols}$, is equipped with a scalar multiplication action (a `MulAction`) by the rational numbers $\mathbb{Q}$. This means that for any rational scalar $a \in \mathbb{Q}$ and any solution $S \in \chi.\text{QuadSols}$, the pointwise product $a \cdot S$ also satisfies the linear and quadratic anomaly cancellation conditions, making it an element of $\chi.\text{QuadSols}$.

For a quadratic anomaly cancellation system $\chi$, the function $\chi.\text{quadSolsInclLinSols}$ is the $\mathbb{Q}$-equivariant inclusion map from the set of solutions satisfying both linear and quadratic constraints, $\chi.\text{QuadSols}$, into the set of solutions satisfying only the linear constraints, $\chi.\text{LinSols}$. This map sends a solution $s \in \chi.\text{QuadSols}$ to itself viewed as an element of $\chi.\text{LinSols}$, and it commutes with scalar multiplication by rational numbers $q \in \mathbb{Q}$.

For a quadratic anomaly cancellation system $\chi$, the inclusion map $\chi.\text{quadSolsInclLinSols}$ that embeds the set of quadratic solutions $\chi.\text{QuadSols}$ into the space of linear solutions $\chi.\text{LinSols}$ is injective.

For a quadratic anomaly cancellation system $\chi$, if the number of quadratic equations is zero ($\chi.\text{numberQuadratic} = 0$), this definition provides the $\mathbb{Q}$-equivariant inclusion map from the space of linear solutions $\chi.\text{LinSols}$ to the space of solutions satisfying both linear and quadratic conditions $\chi.\text{QuadSols}$. Since the set of quadratic constraints is empty when $\chi.\text{numberQuadratic} = 0$, every vector in the $\mathbb{Q}$-module $\chi.\text{LinSols}$ vacuously satisfies the quadratic anomaly cancellation conditions.

For a quadratic anomaly cancellation system $\chi$, the map $\text{quadSolsIncl}$ is the $\mathbb{Q}$-equivariant inclusion from the space of quadratic solutions $\chi.\text{QuadSols}$ into the space of all charges $\chi.\text{Charges} \cong \mathbb{Q}^n$. This map is defined by composing the inclusion of solutions satisfying both linear and quadratic constraints into the space of linear solutions ($\chi.\text{LinSols}$) with the canonical inclusion of linear solutions into the full space of rational charges. The map preserves scalar multiplication by rational numbers $q \in \mathbb{Q}$.

For a quadratic anomaly cancellation system $\chi$, the inclusion map $\chi.\text{quadSolsIncl} : \chi.\text{QuadSols} \to \chi.\text{Charges}$, which maps each solution satisfying both linear and quadratic anomaly cancellation conditions to its corresponding vector of rational charges in $\mathbb{Q}^n$, is injective.

For an anomaly cancellation system $\chi$, let $S$ and $T$ be two solutions in the space of full solutions $\chi.\text{Sols}$ (satisfying the linear, quadratic, and cubic conditions). If their underlying rational charge assignments in the module of charges $\chi.\text{Charges}$ are equal (that is, $S.\text{val} = T.\text{val}$), then the solutions $S$ and $T$ are equal.

Given an anomaly cancellation system $\chi$ and a charge assignment $S \in \chi.\text{Charges}$, the predicate $\text{IsSolution}(\chi, S)$ holds if $S$ satisfies the full set of anomaly cancellation conditions (linear, quadratic, and cubic). Formally, this is defined by the existence of an element $sol$ in the space of solutions $\chi.\text{Sols}$ such that its underlying value in the module of charges is equal to $S$.

For an anomaly cancellation system $\chi$, the set of solutions $\chi.\text{Sols}$ to the full set of conditions (linear, quadratic, and cubic) is equipped with a scalar multiplication action (a `MulAction`) by the rational numbers $\mathbb{Q}$. Given a rational number $a \in \mathbb{Q}$ and a solution $S \in \chi.\text{Sols}$, the product $a \cdot S$ is also a solution. This is well-defined because the linear, quadratic, and cubic anomaly cancellation conditions are homogeneous; specifically, the cubic part of the action is verified by the fact that if $S$ satisfies the cubic condition, then $a \cdot S$ satisfies it as the cubic form scales by $a^3$.

For an anomaly cancellation system $\chi$, this is the equivariant inclusion map from the set of solutions $\chi.\text{Sols}$ (which satisfy the full set of linear, quadratic, and cubic conditions) to the set of solutions $\chi.\text{QuadSols}$ (which satisfy the linear and quadratic conditions). The map commutes with the scalar multiplication action of the rational numbers $\mathbb{Q}$.

For an anomaly cancellation system $\chi$, the inclusion map $\chi.\text{solsInclQuadSols} : \chi.\text{Sols} \to \chi.\text{QuadSols}$, which maps solutions satisfying the full set of (linear, quadratic, and cubic) conditions to the space of solutions satisfying only the linear and quadratic conditions, is injective.

For an anomaly cancellation system $\chi$, this is the $\mathbb{Q}$-equivariant inclusion map from the set of full solutions $\chi.\text{Sols}$ (which satisfy the linear, quadratic, and cubic conditions) to the set of solutions $\chi.\text{LinSols}$ (which satisfy only the linear conditions). This map is defined by the composition of the inclusion $\chi.\text{Sols} \to \chi.\text{QuadSols}$ and the inclusion $\chi.\text{QuadSols} \to \chi.\text{LinSols}$.

For an anomaly cancellation system $\chi$, the inclusion map $\chi.\text{solsInclLinSols} : \chi.\text{Sols} \to \chi.\text{LinSols}$, which maps solutions satisfying the full set of (linear, quadratic, and cubic) conditions to the set of solutions satisfying only the linear conditions, is injective.

For an anomaly cancellation system $\chi$, this is the $\mathbb{Q}$-equivariant inclusion map from the space of solutions $\chi.\text{Sols}$ into the space of all charges $\chi.\text{Charges} \cong \mathbb{Q}^n$. The set $\chi.\text{Sols}$ consists of the rational charge allocations that satisfy the complete set of linear, quadratic, and cubic anomaly cancellation conditions. The map is defined by the composition of the inclusion of full solutions into the space of quadratic solutions ($\chi.\text{QuadSols}$) and the inclusion of quadratic solutions into the space of charges. It preserves scalar multiplication by rational numbers $a \in \mathbb{Q}$.

For an anomaly cancellation system $\chi$, the inclusion map $\chi.\text{solsIncl} : \chi.\text{Sols} \to \chi.\text{Charges}$, which maps the set of rational charge allocations satisfying the full set of linear, quadratic, and cubic anomaly cancellation conditions into the space of all possible charges, is injective.

Given three anomaly cancellation systems $\chi, \eta,$ and $\varepsilon$, and two morphisms $f: \chi \to \eta$ and $g: \eta \to \varepsilon$, their composition $g \circ f: \chi \to \varepsilon$ is defined as the morphism whose: 1. Linear map on the space of charges is the composition of the linear maps of the individual morphisms, $g.\text{charges} \circ f.\text{charges}$. 2. Map between the solution spaces is the composition of the maps $g.\text{anomalyFree} \circ f.\text{anomalyFree}$. The resulting map satisfies the required commutativity condition for morphisms between anomaly cancellation systems.