Physlib.StringTheory.FTheory.SU5.Charges.AnomalyFree

For a given charge spectrum $c$ over a commutative ring $\mathcal{Z}$ in the $SU(5) \times U(1)$ F-theory model, the property `IsAnomalyFree` holds if there exists a configuration of quanta $x$ in the multiset of lifts $\text{liftCharge}(c)$ that satisfies the linear anomaly cancellation condition. Specifically, there must exist an $x$ such that: $$q_{H_d} - q_{H_u} + \sum_{i} q_i N_i + \sum_{a} q_a N_a = 0$$ where $q_{H_d}$ and $q_{H_u}$ are the $U(1)$ charges of the Higgs fields, and the sums represent the contributions from the $\overline{\mathbf{5}}$ and $\mathbf{10}$ representations with charges $q$ and fluxes (multiplicities) $N$. The lifting process ensures that the resulting quanta $x$ have no chiral exotics and no zero fluxes.

For a charge spectrum $c$ over a commutative ring $\mathcal{Z}$ with decidable equality in the $SU(5) \times U(1)$ F-theory model, the proposition `IsAnomalyFree c` is decidable. This property holds if there exists a configuration of quanta $x$ in the finite multiset of lifts $\text{liftCharge}(c)$ that satisfies the linear anomaly cancellation condition: $$q_{H_d} - q_{H_u} + \sum_{i} q_i N_i + \sum_{a} q_a N_a = 0$$ where $q_{H_d}$ and $q_{H_u}$ are the $U(1)$ charges of the Higgs fields, and the sums represent the contributions from the $\overline{\mathbf{5}}$ and $\mathbf{10}$ representations with charges $q$ and multiplicities (fluxes) $N$.

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be commutative rings, and let $f: \mathcal{Z} \to \mathcal{Z}_1$ be a ring homomorphism. If a charge spectrum $c$ defined over $\mathcal{Z}$ is anomaly-free, then the charge spectrum obtained by mapping all its charges under $f$ is also anomaly-free. A charge spectrum is anomaly-free if there exists a configuration of quanta $x$ (a lift of the spectrum) that satisfies the linear anomaly cancellation condition: $$q_{H_d} - q_{H_u} + \sum_{i} q_i N_i + \sum_{a} q_a N_a = 0$$ where $q_{H_d}, q_{H_u}$ are Higgs charges and $N$ are the fluxes associated with the representation charges $q$.

For a given codimension-one configuration $I$ of sections $\sigma_0$ and $\sigma_1$ in an $SU(5)$ F-theory model, the multiset of viable $U(1)$ charge spectra (denoted $\text{viableCharges}(I)$) that satisfy the anomaly-free condition $\text{IsAnomalyFree}$ is given by the following classification: 1. If $I = \text{same}$, the anomaly-free viable charge spectra are: $$\{\langle 2, -2, \{-3, -1\}, \{-1\}\rangle, \langle 2, -2, \{-1, 1\}, \{-1\}\rangle, \langle -2, 2, \{-1, 1\}, \{1\}\rangle, \langle -2, 2, \{1, 3\}, \{1\}\rangle\}$$ 2. If $I = \text{nearestNeighbor}$, the anomaly-free viable charge spectra are: $$\{\langle -4, -14, \{6, 11\}, \{-7\}\rangle, \langle 6, -14, \{-9, 1\}, \{-7\}\rangle, \langle 6, -14, \{1, 11\}, \{-7\}\rangle, \langle -14, 6, \{1, 11\}, \{3\}\rangle\}$$ 3. If $I = \text{nextToNearestNeighbor}$, the anomaly-free viable charge spectra are: $$\{\langle 2, 12, \{-13, -8\}, \{6\}\rangle\}$$ A charge spectrum is `IsAnomalyFree` if it can be lifted to a configuration of quanta that satisfies the linear anomaly cancellation condition $q_{H_d} - q_{H_u} + \sum_{i} q_i N_i + \sum_{a} q_a N_a = 0$.