Physlib.StringTheory.FTheory.SU5.Charges.OfRationalSection

This inductive type represents the set of distinct geometric configurations of the zero-section $\sigma_0$ relative to the additional rational section $\sigma_1$ within the codimension-one fiber (specifically an $I_5$ fiber) of an $SU(5)$ F-theory model. These configurations characterize how sections intersect the fiber components and are used to determine the allowed $U(1)$ charges for matter fields in the $10$ and $\bar{5}$ representations.

This instance provides a procedure to determine whether two codimension-one configurations of sections $\sigma_0$ and $\sigma_1$ in an $SU(5)$ F-theory model are identical. Specifically, for any two configurations $c_1, c_2 \in \text{CodimensionOneConfig}$, it is decidable whether $c_1 = c_2$.

The set of distinct geometric configurations of the zero-section $\sigma_0$ and the additional rational section $\sigma_1$ in the codimension-one fiber of an $SU(5)$ F-theory model, represented by the type $\text{CodimensionOneConfig}$, is finite. This finite set is explicitly given by the collection of configurations $\{\text{same}, \text{nearestNeighbor}, \text{nextToNearestNeighbor}\}$.

This function assigns a finite set of integers representing the allowed $U(1)$ charges for matter fields in the $\mathbf{\bar{5}}$ representation of $SU(5)$ F-theory, based on the codimension-one configuration of the sections $\sigma_0$ and $\sigma_1$. For a given configuration $c \in \text{CodimensionOneConfig}$, the set of allowed charges is defined as: - If $c$ is `same`, the set is $\{-3, -2, -1, 0, 1, 2, 3\}$. - If $c$ is `nearestNeighbor`, the set is $\{-14, -9, -4, 1, 6, 11\}$. - If $c$ is `nextToNearestNeighbor`, the set is $\{-13, -8, -3, 2, 7, 12\}$.

Given a codimension-one configuration $I$ of the sections $\sigma_0$ and $\sigma_1$ in an $SU(5)$ F-theory model, this function returns the finite set of allowed $U(1)$ charges for matter fields in the $10$ representation. The sets are defined based on the configuration $I$ as follows: - If $I = \text{same}$, the set of allowed charges is $\{-3, -2, -1, 0, 1, 2, 3\}$. - If $I = \text{nearestNeighbor}$, the set of allowed charges is $\{-12, -7, -2, 3, 8, 13\}$. - If $I = \text{nextToNearestNeighbor}$, the set of allowed charges is $\{-9, -4, 1, 6, 11\}$.

For any codimension-one configuration $I$ of the sections $\sigma_0$ and $\sigma_1$ in an $SU(5)$ F-theory model, the set of allowed $U(1)$ charges for matter fields in the $\mathbf{\bar{5}}$ representation is finite.

For any codimension-one configuration $I$ of the sections $\sigma_0$ and $\sigma_1$ in an $SU(5)$ F-theory model, the set of allowed $U(1)$ charges for matter fields in the $10$ representation is finite.