Physlib.QFT.AnomalyCancellation.GroupActions

For an anomaly cancellation condition (ACC) system $\chi$ and a group action $G$ defined on it, this definition provides the group structure (including multiplication, identity, and inverse) for the underlying group $G.\text{group}$ associated with the action.

For an anomaly cancellation system $\chi$ and a group action $G$ on that system, this definition specifies the $\mathbb{Q}$-linear map from the space of linear solutions $\chi.\text{LinSols}$ to itself induced by a group element $g \in G.\text{group}$. Specifically, for a solution $S \in \chi.\text{LinSols}$, the map returns the charge vector $g \cdot S$ (determined by the representation $G.\text{rep}$), which is guaranteed to remain within the space of linear solutions because the linear anomaly equations are invariant under the group action.

For an anomaly cancellation condition (ACC) system $\chi$ and a group action $G$ on that system, this definition constructs the representation of the group $G.\text{group}$ on the $\mathbb{Q}$-vector space of linear solutions $\chi.\text{LinSols}$. This representation maps each group element $g \in G.\text{group}$ to the linear operator defined by $G.\text{linSolMap } g$, which acts on a solution vector $S \in \chi.\text{LinSols}$ via the representation $G.\text{rep}$. Since the linear anomaly equations are invariant under the group action, the transformation of a linear solution remains a linear solution.

For an anomaly cancellation condition (ACC) system $\chi$ and a group action $G$ on the system, let $\iota: \chi.\text{LinSols} \hookrightarrow \chi.\text{Charges}$ be the inclusion map from the $\mathbb{Q}$-vector space of linear solutions into the space of charges $\mathbb{Q}^n$. For any group element $g \in G.\text{group}$ and any linear solution $S \in \chi.\text{LinSols}$, the representation of the group action on the space of linear solutions $G.\text{linSolRep}$ commutes with the representation on the space of charges $G.\text{rep}$ via the inclusion $\iota$: $$\iota(G.\text{linSolRep}(g, S)) = G.\text{rep}(g, \iota(S)).$$ This indicates that the inclusion of linear solutions into the charge space is an equivariant map with respect to the group action.

For an anomaly cancellation condition (ACC) system $\chi$ and a group action $G$ defined on it, this definition constructs a multiplicative group action of the group $G.\text{group}$ on the set $\chi.\text{QuadSols}$ of solutions that satisfy both the linear and quadratic anomaly equations. The action of an element $g \in G.\text{group}$ on a quadratic solution $S$ is given by applying the representation $G.\text{rep}$ to the underlying charge vector; because the group action $G$ is defined such that the quadratic anomaly equations are invariant, the resulting vector remains within $\chi.\text{QuadSols}$.

Let $\chi$ be an anomaly cancellation condition (ACC) system and $G$ be a group action defined on it. Let $\chi.\text{QuadSols}$ denote the set of solutions satisfying both linear and quadratic anomaly equations, and $\chi.\text{LinSols}$ denote the $\mathbb{Q}$-vector space of solutions satisfying the linear equations. Let $\iota : \chi.\text{QuadSols} \hookrightarrow \chi.\text{LinSols}$ be the natural inclusion map. For any group element $g \in G.\text{group}$ and any quadratic solution $S \in \chi.\text{QuadSols}$, the action of $g$ commutes with the inclusion $\iota$: $$\iota(g \cdot S) = g \cdot \iota(S)$$ where the left-hand side represents the group action on quadratic solutions and the right-hand side represents the group representation acting on linear solutions. This identifies the inclusion map as an equivariant map with respect to the group action.

Let $\chi$ be an anomaly cancellation condition (ACC) system and $G$ be a group action defined on it. Let $\chi.\text{QuadSols}$ denote the set of solutions satisfying both the linear and quadratic anomaly equations, and let $\chi.\text{Charges}$ denote the $\mathbb{Q}$-vector space of rational charges (typically $\mathbb{Q}^n$). Let $\iota : \chi.\text{QuadSols} \hookrightarrow \chi.\text{Charges}$ be the natural inclusion map. For any group element $g \in G.\text{group}$ and any quadratic solution $S \in \chi.\text{QuadSols}$, the inclusion map $\iota$ is equivariant with respect to the group action: $$\iota(g \cdot S) = \rho(g)(\iota(S))$$ where the left-hand side represents the group action on the space of quadratic solutions and the right-hand side represents the group representation $\rho$ acting on the space of charges.

For an anomaly cancellation condition (ACC) system $\chi$ and a group action $G$ defined on it, this definition constructs a multiplicative group action of the group $G.\text{group}$ on the set $\chi.\text{Sols}$ of solutions that satisfy the full system of anomaly equations (linear, quadratic, and cubic). For any element $g \in G.\text{group}$ and any solution $S \in \chi.\text{Sols}$, the action $g \cdot S$ is obtained by applying the representation $G.\text{rep}$ to the underlying charge vector of $S$; the resulting vector remains a solution because the group action is defined to leave the anomaly equations invariant.

Let $\chi$ be an anomaly cancellation condition (ACC) system and $G$ be a group action on it. Let $\chi.\text{Sols}$ denote the set of solutions satisfying the full system of anomaly equations (linear, quadratic, and cubic), and let $\chi.\text{QuadSols}$ denote the set of solutions satisfying only the linear and quadratic equations. Let $\iota: \chi.\text{Sols} \to \chi.\text{QuadSols}$ be the equivariant inclusion map between these solution spaces. For any group element $g \in G.\text{group}$ and any solution $S \in \chi.\text{Sols}$, the action of $g$ commutes with the inclusion map: $$\iota(g \cdot S) = g \cdot \iota(S)$$ where $g \cdot S$ is the action of the group on the space of full solutions and $g \cdot \iota(S)$ is the action of the group on the space of quadratic solutions.

Let $\chi$ be an anomaly cancellation condition (ACC) system and $G$ be a group action on it. Let $\chi.\text{Sols}$ be the set of solutions satisfying the full system of anomaly equations (linear, quadratic, and cubic), and let $\chi.\text{LinSols}$ be the set of solutions satisfying the linear equations. Let $\iota: \chi.\text{Sols} \to \chi.\text{LinSols}$ be the inclusion map between these solution spaces. For any group element $g \in G.\text{group}$ and any solution $S \in \chi.\text{Sols}$, the action of $g$ commutes with the inclusion map: $$\iota(g \cdot S) = g \cdot \iota(S)$$ where $g \cdot S$ is the action of the group on the space of full solutions and $g \cdot \iota(S)$ is the action of the group representation on the space of linear solutions.

Let $\chi$ be an anomaly cancellation condition (ACC) system and $G$ be a group action on it. Let $\chi.\text{Sols}$ denote the set of charge allocations satisfying the full system of anomaly equations (linear, quadratic, and cubic), and let $\chi.\text{Charges}$ be the vector space of all charges (isomorphic to $\mathbb{Q}^n$). Let $\iota: \chi.\text{Sols} \hookrightarrow \chi.\text{Charges}$ be the inclusion map that views a solution as a charge vector. For any group element $g \in G.\text{group}$ and any solution $S \in \chi.\text{Sols}$, the following holds: $$\iota(g \cdot S) = \rho(g)(\iota(S))$$ where $g \cdot S$ is the action of the group on the solution space and $\rho(g)$ is the linear representation of the group acting on the space of charges.