Physlib.Particles.SuperSymmetry.MSSMNu.AnomalyCancellation.Basic

The definition `MSSMCharges` represents the system of charges for the fermions in the Minimal Supersymmetric Standard Model (MSSM) with three generations and right-handed neutrinos. It is an `ACCSystemCharges` object defined with $20$ distinct fermion species, comprising $18$ matter fermions (three generations of $Q, u^c, d^c, L, e^c,$ and $\nu^c$) and $2$ Higgsinos ($\tilde{H}_u, \tilde{H}_d$).

The definition `MSSMSpecies` represents the charge system for the three generations (or species) of fermions in the Minimal Supersymmetric Standard Model (MSSM). It is defined as an `ACCSystemCharges` object with the number of fermion representations set to 3.

This definition provides an equivalence (a bijection) between the module of charges for the Minimal Supersymmetric Standard Model (MSSM), `MSSMCharges.Charges`, and the space of functions from the disjoint union $\text{Fin}(18) \oplus \text{Fin}(2)$ to the rational numbers $\mathbb{Q}$. In this representation, the first 18 indices correspond to the 18 matter fermion species (comprising three generations of $Q, u^c, d^c, L, e^c,$ and $\nu^c$), while the final two indices correspond to the Higgsinos ($\tilde{H}_u$ and $\tilde{H}_d$).

This definition establishes an equivalence (a bijection) between the space of functions from the disjoint union of two finite sets, $\text{Fin}(18) \oplus \text{Fin}(2)$, to the rational numbers $\mathbb{Q}$ and the product of the spaces of functions $(\text{Fin}(18) \to \mathbb{Q}) \times (\text{Fin}(2) \to \mathbb{Q})$. Specifically, a function $f$ defined on the disjoint union is mapped to the pair $(f \circ \iota_1, f \circ \iota_2)$, where $\iota_1$ and $\iota_2$ are the canonical injections into the disjoint union. Conversely, a pair of functions $(f_1, f_2)$ is mapped to a single function $f$ that acts as $f_1$ on the first component and $f_2$ on the second. In the context of the MSSM, this represents splitting the assignment of rational charges for the Standard Model species and the Higgs sector.

This definition establishes an equivalence (a bijection) between the module of rational charges for the Minimal Supersymmetric Standard Model (MSSM), denoted as `MSSMCharges.Charges`, and the product of two function spaces $(\text{Fin } 18 \to \mathbb{Q}) \times (\text{Fin } 2 \to \mathbb{Q})$. This map decomposes the total assignment of charges for the 20 fermion species into a block of 18 charges, corresponding to the matter fermions (three generations of $Q, u^c, d^c, L, e^c,$ and $\nu^c$), and a block of 2 charges, corresponding to the Higgsinos ($\tilde{H}_u$ and $\tilde{H}_d$).

This definition establishes an equivalence between the space of functions mapping a set of 18 elements to the rational numbers $\mathbb{Q}$ and the space of nested functions mapping a set of 6 elements to a set of 3 elements, which then map to $\mathbb{Q}$. In the context of the Minimal Supersymmetric Standard Model (MSSM), this represents the isomorphism between a flat vector of 18 charges and a structured representation where charges are indexed by 6 distinct particle species and 3 families (generations). Mathematically, it identifies $(\text{Fin } 18 \to \mathbb{Q})$ with $(\text{Fin } 6 \to \text{Fin } 3 \to \mathbb{Q})$.

The definition establishes an equivalence (a bijection) between the module of rational charges for the MSSM, denoted as `MSSMCharges.Charges`, and the product space $(\text{Fin } 6 \to \text{Fin } 3 \to \mathbb{Q}) \times (\text{Fin } 2 \to \mathbb{Q})$. This map decomposes the total assignment of charges for the 20 fermion species by first separating the matter fermions from the Higgsinos, and then further partitioning the 18 matter fermion charges into a structured representation indexed by 6 species and 3 generations (families).

For a given index $i \in \{0, 1, \dots, 5\}$ representing one of the six species of matter fermions ($Q, u^c, d^c, L, e^c, \nu^c$), this $\mathbb{Q}$-linear map projects the total charge assignment of the MSSM, $S \in \text{MSSMCharges.Charges}$, onto the charges of the three generations of that specific species. The map is defined by applying the equivalence `toSpecies` to $S$, which decomposes the 20 fermion charges into matter species and Higgsinos, and then extracting the $i$-th species from the matter fermion sector $(\text{Fin } 6 \to \text{Fin } 3 \to \mathbb{Q})$. The result is an element of $\text{MSSMSpecies.Charges}$, which represents the rational charges for three generations.

For any index $i \in \{0, 1, \dots, 5\}$ and any pair $f = (q, h)$ representing rational charges for the six species of matter fermions ($q \in \mathbb{Q}^{6 \times 3}$) and two Higgsinos ($h \in \mathbb{Q}^2$), applying the projection map `toSMSpecies i` to the total MSSM charge vector reconstructed via the inverse equivalence `toSpecies.symm f` yields the charges $q_i \in \mathbb{Q}^3$ assigned to the three generations of the $i$-th species.

The $\mathbb{Q}$-linear map `MSSMCharges.Q` projects the total charge assignment of the MSSM fermions onto the three generations of the left-handed quark doublet $Q$. Given a vector of charges $S \in \text{MSSMCharges.Charges}$ for the 20 fermions in the model, this map extracts the rational charges $(q_1, q_2, q_3) \in \mathbb{Q}^3$ corresponding to the three generations of the $Q$ species. It is defined as the first ($0$-indexed) projection of the matter fermion sector.

The $\mathbb{Q}$-linear map $\text{MSSMCharges.U}$ projects the total charge assignment of the MSSM, $S \in \mathbb{Q}^{20}$, onto the charges of the three generations of the up-type right-handed quark species $u^c$. The result is a vector in $\mathbb{Q}^3$ representing the rational charges assigned to these three fermion generations.

Given the total charge assignment of the MSSM with three generations and right-handed neutrinos, $S \in \mathbb{Q}^{20}$, the map `MSSMCharges.D` is a $\mathbb{Q}$-linear projection that extracts the charges corresponding to the three generations of right-handed down-type quarks ($d^c$). The output is an element of $\mathbb{Q}^3$, representing the rational charges $q \in \mathbb{Q}$ for each of the three families of the $d^c$ species.

The definition `MSSMCharges.L` is a $\mathbb{Q}$-linear map that projects the total charge assignment of the 20 fermion species in the MSSM (comprising matter fermions and Higgsinos) onto the charges of the three generations of left-handed lepton doublets $L$. It takes a total charge vector $S \in \mathbb{Q}^{20}$ and extracts the components corresponding to the $L$ species, resulting in a vector in $\mathbb{Q}^3$ representing the charges $(L_1, L_2, L_3)$.

The function `MSSMCharges.E` is a $\mathbb{Q}$-linear map from the space of MSSM charges $\text{MSSMCharges.Charges} \cong \mathbb{Q}^{20}$ to the space of charges for three fermion generations $\text{MSSMSpecies.Charges} \cong \mathbb{Q}^3$. It projects the total charge assignment onto the charges of the three generations of right-handed charged leptons ($e^c$), which corresponds to the matter species at index 4 in the MSSM charge system.

The function `MSSMCharges.N` is a $\mathbb{Q}$-linear map that extracts the charges of the three generations of right-handed neutrinos $\nu^c$ (often denoted by $N$) from the total MSSM charge assignment. For a given vector of charges $S \in \text{MSSMCharges.Charges} \cong \mathbb{Q}^{20}$, it returns a vector in $\mathbb{Q}^3$ representing the rational charges assigned to the three generations of right-handed neutrinos. This corresponds to the 6th species (index 5) of the matter fermions in the MSSM.

The function `Hd` is a $\mathbb{Q}$-linear map from the space of charges of the MSSM, $\text{MSSMCharges.Charges} \cong \mathbb{Q}^{20}$, to the rational numbers $\mathbb{Q}$. For a given vector of charges $S$, it returns the rational charge assigned to the down-type Higgsino $\tilde{H}_d$, which corresponds to the 19th component (index 18) of the charge system.

The $\mathbb{Q}$-linear map $H_u$ from the space of MSSM charges to the rational numbers $\mathbb{Q}$ assigns to each charge vector $S$ the specific rational charge corresponding to the Higgsino $H_u$. In the vector representation of the 20 fermion species, this corresponds to the component at index 19.

In the Minimal Supersymmetric Standard Model (MSSM) with three generations and right-handed neutrinos, let $S$ and $T$ be two vectors of rational charges (elements of the charge space $\mathbb{Q}^{20}$ representing 18 matter fermions and 2 Higgsinos). Then $S = T$ if and only if the following three conditions are met: 1. For every species index $i \in \{0, \dots, 5\}$ (representing $Q, u^c, d^c, L, e^c, \nu^c$), the projections of the charges onto the three generations of that species are equal, i.e., $\text{toSMSpecies}_i(S) = \text{toSMSpecies}_i(T)$. 2. The charges of the down-type Higgsino $\tilde{H}_d$ are equal, $H_d(S) = H_d(T)$. 3. The charges of the up-type Higgsino $\tilde{H}_u$ are equal, $H_u(S) = H_u(T)$.

For any structured representation of the MSSM charges $f = (M, H) \in (\text{Fin } 6 \to \text{Fin } 3 \to \mathbb{Q}) \times (\text{Fin } 2 \to \mathbb{Q})$, where $M$ represents the charges of the matter fermions and $H$ represents the charges of the two Higgsinos, the down-type Higgsino charge $H_d$ of the reconstructed charge vector $S = \text{toSpecies}^{-1}(f)$ is equal to the first component of the Higgsino charge vector $H(0)$.

Let $f = (g, h)$ be a pair in $(\text{Fin } 6 \to \text{Fin } 3 \to \mathbb{Q}) \times (\text{Fin } 2 \to \mathbb{Q})$ representing the decomposition of fermion charges into matter species and Higgsinos. If we reconstruct the full MSSM charge vector $S$ using the inverse equivalence $\text{toSpecies}^{-1}(f)$, then the charge of the Higgsino $H_u$ for $S$ is equal to $f.2(1)$ (the second component of the Higgsino charge vector $h$).

The $\mathbb{Q}$-linear map `accGrav` computes the gravitational anomaly contribution for a given charge assignment $S$ in the MSSM with three generations and right-handed neutrinos. It is defined as the sum of the charges of all fermion species weighted by their respective dimensions under the $SU(3)_C \times SU(2)_L$ gauge group: $$\text{accGrav}(S) = \sum_{i=1}^3 (6 Q_i + 3 U_i + 3 D_i + 2 L_i + E_i + N_i) + 2 H_d + 2 H_u$$ where $Q_i, U_i, D_i, L_i, E_i,$ and $N_i$ are the charges of the three generations ($i=1, 2, 3$) of quark doublets, up-type singlets, down-type singlets, lepton doublets, charged lepton singlets, and right-handed neutrinos, respectively, and $H_d, H_u$ are the charges of the two Higgsinos.

Let $S$ and $T$ be two charge assignments (vectors in $\mathbb{Q}^{20}$) for the fermions in the MSSM with three generations and right-handed neutrinos. Suppose that for each of the six matter fermion species $j \in \{Q, u^c, d^c, L, e^c, \nu^c\}$, the sum of the charges over the three generations is the same for both $S$ and $T$, i.e., $$\sum_{i=1}^3 (S_j)_i = \sum_{i=1}^3 (T_j)_i$$ Furthermore, suppose the charges of the down-type Higgsino $H_d$ and up-type Higgsino $H_u$ are equal for both assignments ($H_d(S) = H_d(T)$ and $H_u(S) = H_u(T)$). Then the gravitational anomaly contributions are equal: $$\text{accGrav}(S) = \text{accGrav}(T)$$

For a vector of rational charges $S \in \mathbb{Q}^{20}$ assigned to the fermions in the MSSM with three generations and right-handed neutrinos, the function `accSU2` is a $\mathbb{Q}$-linear map that calculates the $SU(2)$ anomaly cancellation condition (specifically the $SU(2)^2 \times U(1)$ anomaly). It is defined as: \[ \text{accSU2}(S) = \sum_{i=1}^{3} (3 Q_i + L_i) + H_d + H_u \] where $Q_i$ and $L_i$ are the charges of the $i$-th generation of quark and lepton doublets respectively, and $H_d, H_u$ are the charges of the down-type and up-type Higgsinos. The factor of $3$ for the quark doublets accounts for the three color degrees of freedom.

Let $S$ and $T$ be vectors of rational charges in $\mathbb{Q}^{20}$ assigned to the fermions of the MSSM with three generations and right-handed neutrinos. Suppose that for each of the six fermion species $j \in \{Q, u^c, d^c, L, e^c, \nu^c\}$, the sum of charges over the three generations is the same for $S$ and $T$: \[ \sum_{i=1}^3 S_{j,i} = \sum_{i=1}^3 T_{j,i} \] Additionally, suppose that the charges assigned to the down-type Higgsino $H_d$ and the up-type Higgsino $H_u$ are equal for both assignments ($H_d(S) = H_d(T)$ and $H_u(S) = H_u(T)$). Then the $SU(2)$ anomaly cancellation condition values for $S$ and $T$ are equal, $\text{accSU2}(S) = \text{accSU2}(T)$, where the condition is defined as: \[ \text{accSU2}(S) = \sum_{i=1}^{3} (3 Q_i + L_i) + H_d + H_u \]

The $\mathbb{Q}$-linear map `accSU3` computes the $SU(3)$ anomaly cancellation condition for a charge assignment $S$ in the MSSM with three generations and right-handed neutrinos. The value is given by the sum over the three generations of the charges of the color-carrying fermions: \[ \sum_{i=1}^{3} (2 Q_i + u^c_i + d^c_i) \] where $Q_i$, $u^c_i$, and $d^c_i$ are the rational charges assigned to the $i$-th generation of left-handed quark doublets, right-handed up-type quarks, and right-handed down-type quarks, respectively.

For any two charge assignments $S$ and $T$ in the MSSM with three generations and right-handed neutrinos, if for each of the six species of matter fermions $j \in \{Q, u^c, d^c, L, e^c, \nu^c\}$, the sum of charges over the three generations is the same for $S$ and $T$, i.e., \[ \sum_{i=1}^3 q_{j,i}(S) = \sum_{i=1}^3 q_{j,i}(T) \] where $q_{j,i}$ represents the rational charge of the $i$-th generation of species $j$, then the $SU(3)$ anomaly cancellation condition evaluates to the same value for both assignments: \[ \text{accSU3}(S) = \text{accSU3}(T) \] The $SU(3)$ anomaly condition is defined as $\sum_{i=1}^3 (2 Q_i + u^c_i + d^c_i)$.

The $\mathbb{Q}$-linear map $\text{accYY}$ computes the anomaly cancellation condition (ACC) for the squared hypercharge $Y^2$ in the MSSM with three generations and right-handed neutrinos. For a given assignment of rational charges $S \in \mathbb{Q}^{20}$, the value is defined as: $$\text{accYY}(S) = \sum_{i=1}^3 (Q_i + 8u^c_i + 2d^c_i + 3L_i + 6e^c_i) + 3(H_d + H_u)$$ where $Q_i, u^c_i, d^c_i, L_i,$ and $e^c_i$ represent the rational charges of the $i$-th generation of left-handed quark doublets, right-handed up-type quarks, right-handed down-type quarks, left-handed lepton doublets, and right-handed charged leptons, respectively, while $H_d$ and $H_u$ are the charges of the down-type and up-type Higgsinos.

Let $S$ and $T$ be two assignments of rational charges to the 20 fermion species of the MSSM with three generations and right-handed neutrinos. Suppose that for each of the six matter species $j \in \{Q, u^c, d^c, L, e^c, \nu^c\}$, the sum of the charges over the three generations is equal for $S$ and $T$: $$\sum_{i=1}^3 S_{j,i} = \sum_{i=1}^3 T_{j,i}$$ If, in addition, the charges assigned to the down-type Higgsino $H_d$ and the up-type Higgsino $H_u$ are equal for both assignments ($H_d(S) = H_d(T)$ and $H_u(S) = H_u(T)$), then the anomaly cancellation condition for the squared hypercharge $Y^2$ satisfies $\text{accYY}(S) = \text{accYY}(T)$.

The definition `MSSMACCs.quadBiLin` defines a symmetric bilinear form $B: V \times V \to \mathbb{Q}$ on the space of charges $V = \text{MSSMCharges.Charges} \cong \mathbb{Q}^{20}$ for the Minimal Supersymmetric Standard Model (MSSM) with three generations and right-handed neutrinos. Given two charge assignments $S, T \in \mathbb{Q}^{20}$, the value is computed as: \[ B(S, T) = \sum_{i=1}^3 \left( Q_i(S)Q_i(T) - 2 U_i(S)U_i(T) + D_i(S)D_i(T) - L_i(S)L_i(T) + E_i(S)E_i(T) \right) - H_d(S)H_d(T) + H_u(S)H_u(T) \] where: - $Q_i, U_i, D_i, L_i, E_i$ represent the rational charges of the $i$-th generation of left-handed quark doublets, right-handed up-type quarks, right-handed down-type quarks, left-handed lepton doublets, and right-handed charged leptons, respectively. - $H_d$ and $H_u$ represent the rational charges assigned to the down-type and up-type Higgsinos. This bilinear form is the symmetric mapping associated with the quadratic anomaly cancellation condition (ACC).

The definition `accQuad` represents the quadratic anomaly cancellation condition (ACC) for the Minimal Supersymmetric Standard Model (MSSM) with three generations and right-handed neutrinos. It is a homogeneous quadratic map $f: V \to \mathbb{Q}$ from the space of rational charges $V \cong \mathbb{Q}^{20}$ to the rational numbers. For a given charge assignment $S \in V$, the value is computed as: \[ f(S) = \sum_{i=1}^3 \left( Q_i(S)^2 - 2 U_i(S)^2 + D_i(S)^2 - L_i(S)^2 + E_i(S)^2 \right) - H_d(S)^2 + H_u(S)^2 \] where $Q_i, U_i, D_i, L_i, E_i$ denote the rational charges of the $i$-th generation of left-handed quark doublets, right-handed up-type quarks, right-handed down-type quarks, left-handed lepton doublets, and right-handed charged leptons, respectively, and $H_d, H_u$ represent the charges assigned to the down-type and up-type Higgsinos.

Let $S$ and $T$ be two assignments of rational charges for the 20 fermion species in the MSSM (representing three generations of the species $Q, u^c, d^c, L, e^c, \nu^c$ and the two Higgsinos $H_d, H_u$). Suppose that for every matter species $j \in \{Q, u^c, d^c, L, e^c, \nu^c\}$, the sum of the squares of the charges across its three generations is identical for both assignments: $$\sum_{i=1}^3 q_{j,i}(S)^2 = \sum_{i=1}^3 q_{j,i}(T)^2$$ where $q_{j,i}$ denotes the charge of the $i$-th generation of species $j$. If the charges assigned to the Higgsinos are also equal, $H_d(S) = H_d(T)$ and $H_u(S) = H_u(T)$, then the quadratic anomaly cancellation condition $f_{\text{quad}}$ evaluates to the same value for both assignments: $$f_{\text{quad}}(S) = f_{\text{quad}}(T)$$

For a triple of charge assignments $(S_1, S_2, S_3)$, where each $S_k \in \mathbb{Q}^{20}$ represents the rational charges of the 20 fermion species in the MSSM with three generations and right-handed neutrinos, this function computes the symmetric trilinear sum: $$\begin{aligned} f(S_1, S_2, S_3) = \sum_{i=1}^3 \Big( &6 Q_i(S_1) Q_i(S_2) Q_i(S_3) + 3 u^c_i(S_1) u^c_i(S_2) u^c_i(S_3) + 3 d^c_i(S_1) d^c_i(S_2) d^c_i(S_3) \\ &+ 2 L_i(S_1) L_i(S_2) L_i(S_3) + e^c_i(S_1) e^c_i(S_2) e^c_i(S_3) + \nu^c_i(S_1) \nu^c_i(S_2) \nu^c_i(S_3) \Big) \\ &+ 2 \tilde{H}_d(S_1) \tilde{H}_d(S_2) \tilde{H}_d(S_3) + 2 \tilde{H}_u(S_1) \tilde{H}_u(S_2) \tilde{H}_u(S_3) \end{aligned}$$ where $Q_i, u^c_i, d^c_i, L_i, e^c_i, \nu^c_i$ denote the $\mathbb{Q}$-linear projections onto the $i$-th generation of the respective matter fermion species, and $\tilde{H}_d, \tilde{H}_u$ denote the projections onto the down-type and up-type Higgsinos. This function serves as the underlying map for the symmetric trilinear form used to define the cubic anomaly cancellation condition (ACC).

Let $f$ be the symmetric trilinear form used to define the cubic anomaly cancellation condition (ACC) for the MSSM with three generations and right-handed neutrinos. For any rational number $a \in \mathbb{Q}$ and any charge assignments $S, T, R \in \mathbb{Q}^{20}$ (representing the rational charges of the 20 fermion species in the model), the function satisfies the scalar homogeneity property in its first argument: $$f(a \cdot S, T, R) = a \cdot f(S, T, R)$$ where $a \cdot S$ denotes the scalar multiplication of the charge vector $S$ by $a$.

For any charge vectors $S, T, R, L \in \mathbb{Q}^{20}$ representing the 20 fermion species in the MSSM with three generations and right-handed neutrinos, the symmetric trilinear function $f$ (defined as `cubeTriLinToFun`), which characterizes the cubic anomaly cancellation condition, satisfies the additivity property in its first argument: $$f(S + T, R, L) = f(S, R, L) + f(T, R, L)$$ where $S + T$ denotes the pointwise addition of rational charges across all species.

For any three assignments of rational charges $S, T, R \in \mathbb{Q}^{20}$ for the 20 fermion species in the MSSM with three generations and right-handed neutrinos, the trilinear function $f(S, T, R)$ used to define the cubic anomaly cancellation condition (ACC) satisfies the symmetry property $f(S, T, R) = f(T, S, R)$.

Let $S, T, R \in \mathbb{Q}^{20}$ denote three rational charge assignments for the 20 fermion species in the MSSM with three generations and right-handed neutrinos (comprising $Q, u^c, d^c, L, e^c, \nu^c$ and the Higgsinos $\tilde{H}_u, \tilde{H}_d$). The function $f(S, T, R)$, which defines the symmetric trilinear form for the cubic anomaly cancellation condition (ACC), is symmetric in its second and third arguments: $$f(S, T, R) = f(S, R, T)$$

The definition `cubeTriLin` is the symmetric trilinear form $f: V \times V \times V \to \mathbb{Q}$ on the space of charges $V \cong \mathbb{Q}^{20}$ for the MSSM with three generations and right-handed neutrinos. For a triple of charge assignments $(S_1, S_2, S_3)$, it is defined by the following sum: $$\begin{aligned} f(S_1, S_2, S_3) = \sum_{i=1}^3 \Big( &6 Q_i(S_1) Q_i(S_2) Q_i(S_3) + 3 u^c_i(S_1) u^c_i(S_2) u^c_i(S_3) + 3 d^c_i(S_1) d^c_i(S_2) d^c_i(S_3) \\ &+ 2 L_i(S_1) L_i(S_2) L_i(S_3) + e^c_i(S_1) e^c_i(S_2) e^c_i(S_3) + \nu^c_i(S_1) \nu^c_i(S_2) \nu^c_i(S_3) \Big) \\ &+ 2 \tilde{H}_d(S_1) \tilde{H}_d(S_2) \tilde{H}_d(S_3) + 2 \tilde{H}_u(S_1) \tilde{H}_u(S_2) \tilde{H}_u(S_3) \end{aligned}$$ where $Q_i, u^c_i, d^c_i, L_i, e^c_i, \nu^c_i$ represent the rational charges of the three generations of matter fermions, and $\tilde{H}_d, \tilde{H}_u$ represent the charges of the down-type and up-type Higgsinos. This symmetric trilinear form provides the mathematical structure required to define the cubic anomaly cancellation condition (ACC).

Let $V \cong \mathbb{Q}^{20}$ be the space of rational charges for the 20 fermion species in the Minimal Supersymmetric Standard Model (MSSM) with three generations and right-handed neutrinos. The function `accCube` is a homogeneous cubic map $f: V \to \mathbb{Q}$ that represents the cubic anomaly cancellation condition (ACC). For a given charge assignment $S \in V$, the map is defined by evaluating the symmetric trilinear form $\tau$ (given by `cubeTriLin`) on the diagonal $f(S) = \tau(S, S, S)$, resulting in: $$\begin{aligned} f(S) = \sum_{i=1}^3 \Big( &6 Q_i(S)^3 + 3 (u^c_i(S))^3 + 3 (d^c_i(S))^3 + 2 L_i(S)^3 + (e^c_i(S))^3 + (\nu^c_i(S))^3 \Big) \\ &+ 2 \tilde{H}_d(S)^3 + 2 \tilde{H}_u(S)^3 \end{aligned}$$ where $Q_i, u^c_i, d^c_i, L_i, e^c_i, \nu^c_i$ represent the rational charges of the three generations of matter fermions, and $\tilde{H}_d, \tilde{H}_u$ represent the charges of the down-type and up-type Higgsinos. The map satisfies the homogeneity property $f(a \cdot S) = a^3 f(S)$ for any $a \in \mathbb{Q}$.

Let $S$ and $T$ be two charge assignments (vectors in $\mathbb{Q}^{20}$) for the fermions in the MSSM with three generations and right-handed neutrinos. Suppose that for each of the six matter species $j \in \{Q, u^c, d^c, L, e^c, \nu^c\}$, the sum of the cubes of the charges across the three generations is identical for $S$ and $T$: $$\sum_{i=1}^3 q_{j,i}(S)^3 = \sum_{i=1}^3 q_{j,i}(T)^3$$ where $q_{j,i}(S)$ denotes the rational charge of the $i$-th generation of the $j$-th species under assignment $S$. Furthermore, suppose the charges assigned to the down-type and up-type Higgsinos are the same for both assignments ($H_d(S) = H_d(T)$ and $H_u(S) = H_u(T)$). Then the value of the cubic anomaly cancellation condition is the same for both assignments: $$\text{accCube}(S) = \text{accCube}(T)$$

The anomaly cancellation system `MSSMACC` for the Minimal Supersymmetric Standard Model (MSSM) with three generations and right-handed neutrinos is defined as an `ACCSystem` over the space of rational charges $V \cong \mathbb{Q}^{20}$. This system encompasses the following set of conditions: 1. **Four linear conditions**: These include the gravitational anomaly $\text{accGrav}$, the $SU(2)$ gauge anomaly $\text{accSU2}$, the $SU(3)$ gauge anomaly $\text{accSU3}$, and the hypercharge anomaly $\text{accYY}$. 2. **One quadratic condition**: Represented by the homogeneous quadratic map $\text{accQuad}$. 3. **One cubic condition**: Represented by the homogeneous cubic map $\text{accCube}$. The charge space $V$ assigns a rational number to each of the 20 fermion species in the theory, specifically the three generations of quark doublets $Q_i$, up-type singlets $u^c_i$, down-type singlets $d^c_i$, lepton doublets $L_i$, charged lepton singlets $e^c_i$, and right-handed neutrinos $\nu^c_i$, as well as the two Higgsinos $\tilde{H}_d$ and $\tilde{H}_u$.

In the anomaly cancellation system for the MSSM with three generations and right-handed neutrinos, let $S$ be a charge assignment belonging to the space of quadratic solutions ($\text{QuadSols}$). Then the quadratic anomaly cancellation condition $\text{accQuad}$ evaluated at $S$ is zero: $$\text{accQuad}(S) = 0$$ where $\text{accQuad}$ is the homogeneous quadratic map defined over the space of rational charges $V \cong \mathbb{Q}^{20}$.

Given a charge assignment $S \in \mathbb{Q}^{20}$ for the MSSM with three generations and right-handed neutrinos, this function constructs an anomaly-free solution (an element of the solution space `MSSMACC.Sols`) provided that $S$ satisfies all six anomaly cancellation conditions: - The gravitational anomaly $\text{accGrav}(S) = 0$ - The $SU(2)$ gauge anomaly $\text{accSU2}(S) = 0$ - The $SU(3)$ gauge anomaly $\text{accSU3}(S) = 0$ - The hypercharge anomaly $\text{accYY}(S) = 0$ - The quadratic anomaly $\text{accQuad}(S) = 0$ - The cubic anomaly $\text{accCube}(S) = 0$

Let $S \in \mathbb{Q}^{20}$ be a rational charge assignment for the 20 fermion species in the MSSM with three generations and right-handed neutrinos. Suppose $S$ satisfies the six anomaly cancellation conditions (ACCs): - The gravitational anomaly: $\text{accGrav}(S) = 0$ - The $SU(2)$ gauge anomaly: $\text{accSU2}(S) = 0$ - The $SU(3)$ gauge anomaly: $\text{accSU3}(S) = 0$ - The hypercharge anomaly: $\text{accYY}(S) = 0$ - The quadratic anomaly: $\text{accQuad}(S) = 0$ - The cubic anomaly: $\text{accCube}(S) = 0$ Then the underlying charge vector of the anomaly-free solution object constructed from $S$ (denoted by `AnomalyFreeMk`) is equal to $S$.

Let $S$ be a charge assignment in the MSSM charge space $V \cong \mathbb{Q}^{20}$ that satisfies the linear anomaly cancellation conditions (gravitational, $SU(2)$, $SU(3)$, and hypercharge). If $S$ additionally satisfies the quadratic anomaly cancellation condition $\text{accQuad}(S) = 0$, then $S$ is a quadratic solution (a solution satisfying both the linear and quadratic conditions) of the MSSM anomaly cancellation system.

Let $S$ be a linear solution to the anomaly cancellation system for the MSSM with three generations and right-handed neutrinos (i.e., $S \in \text{MSSMACC.LinSols}$), meaning it satisfies the four linear conditions: gravitational, $SU(2)$, $SU(3)$, and hypercharge anomalies. Given proofs that $S$ also satisfies the quadratic condition $\text{accQuad}(S) = 0$ and the cubic condition $\text{accCube}(S) = 0$, this construction produces a full solution $S \in \text{MSSMACC.Sols}$ that satisfies all six anomaly cancellation conditions.

Given a charge assignment $S$ in the space of MSSM charges $V \cong \mathbb{Q}^{20}$ that already satisfies the linear and quadratic anomaly cancellation conditions (a member of `MSSMACC.QuadSols`), and a proof $h_{\text{cube}}$ that $S$ also satisfies the cubic anomaly cancellation condition $\text{accCube}(S) = 0$, this definition constructs a complete solution to the full anomaly cancellation system (a member of `MSSMACC.Sols`).

In the Minimal Supersymmetric Standard Model (MSSM) with three generations and right-handed neutrinos, let $S$ be a rational charge assignment in the vector space $V \cong \mathbb{Q}^{20}$ that satisfies the linear and quadratic anomaly cancellation conditions. If $S$ also satisfies the cubic anomaly cancellation condition $\text{accCube}(S) = 0$, then the underlying charge vector of the full anomaly-free solution constructed from $S$ is identical to $S$ itself.

The symmetric bilinear map $\text{MSSMACC.dot}$ defines a dot product on the vector space of charges for the MSSM with three generations and right-handed neutrinos, $\text{MSSMCharges.Charges} \cong \mathbb{Q}^{20}$. For two charge assignment vectors $S, T \in \mathbb{Q}^{20}$, the value is computed as: $$S \cdot T = \sum_{i=1}^3 (Q_i(S)Q_i(T) + u^c_i(S)u^c_i(T) + d^c_i(S)d^c_i(T) + L_i(S)L_i(T) + e^c_i(S)e^c_i(T) + \nu^c_i(S)\nu^c_i(T)) + H_d(S)H_d(T) + H_u(S)H_u(T)$$ where $Q_i, u^c_i, d^c_i, L_i, e^c_i, \nu^c_i$ represent the charges of the $i$-th generation of the six matter fermion species, and $H_d, H_u$ represent the charges of the two Higgsinos.