Physlib.StringTheory.FTheory.SU5.Fluxes.NoExotics.Elems

This multiset consists of the 31 distinct configurations of flux pairs $(M, N) \in \mathbb{Z} \times \mathbb{Z}$ for the $\mathbf{\bar{5}}$ matter curves in an $SU(5)$ F-theory model that satisfy the conditions of having no chiral exotics (`NoExotics`) and no zero fluxes (`HasNoZero`). Each configuration is represented as a multiset of integer pairs, where $M$ is the chirality flux and $N$ is the hypercharge flux for each curve.

The multiset of flux configurations for the $\mathbf{\bar{5}}$ matter curves in an $SU(5)$ F-theory model that satisfy the conditions of having no chiral exotics (`NoExotics`) and no zero fluxes (`HasNoZero`) has a cardinality of 31.

The multiset of configurations of flux pairs $(M, N) \in \mathbb{Z} \times \mathbb{Z}$ for the $\mathbf{\bar{5}}$ matter curves in an $SU(5)$ F-theory model that satisfy the conditions of having no chiral exotics (`NoExotics`) and no zero fluxes (`HasNoZero`) contains no duplicate elements.

Let $F$ be a configuration of flux pairs $(M, N) \in \mathbb{Z} \times \mathbb{Z}$ for the $\mathbf{\bar{5}}$ matter curves in an $SU(5)$ F-theory model. If $F$ belongs to the multiset `elemsNoExotics`, then $F$ satisfies the `NoExotics` condition. This means that the configuration results in exactly three generations of chiral lepton doublets $L = (1, 2)_{-1/2}$ and down-type quarks $D = (\bar{3}, 1)_{1/3}$, with no anti-chiral representations of either.

Let $F$ be a configuration of flux pairs $(M, N) \in \mathbb{Z} \times \mathbb{Z}$ for the $\mathbf{\bar{5}}$ matter curves in an $SU(5)$ F-theory model. If $F$ belongs to the set of configurations that satisfy the no chiral exotics and no zero flux conditions (`elemsNoExotics`), then the number of distinct flux pairs in $F$ is at most 4.

Let $F$ be a configuration of flux pairs $(M, N) \in \mathbb{Z} \times \mathbb{Z}$ for the $\mathbf{\bar{5}}$ matter curves in an $SU(5)$ F-theory model. If $F$ belongs to the set of configurations that satisfy the no chiral exotics and no zero flux conditions (`elemsNoExotics`), then the total number of flux pairs in $F$ (the cardinality of the multiset) is at most 6.

Let $F$ be a configuration of flux pairs $(M, N) \in \mathbb{Z} \times \mathbb{Z}$ for the $\mathbf{\bar{5}}$ matter curves in an $SU(5)$ F-theory model. If $F$ is an element of the multiset of configurations that satisfy the conditions of having no chiral exotics and no zero fluxes (`elemsNoExotics`), then the sum of all flux pairs in $F$ is equal to $(3, 0)$.

Let $F$ be a configuration of flux pairs $(M, N) \in \mathbb{Z} \times \mathbb{Z}$ associated with the $\mathbf{\bar{5}}$ (5-bar) representation matter curves in an $SU(5)$ F-theory model. If $F$ is an element of the multiset `elemsNoExotics` (which consists of configurations satisfying the conditions of having no chiral exotics and no zero fluxes), then $F$ satisfies the `HasNoZero` property, meaning the zero flux pair $(0, 0)$ is not an element of $F$.

Let $F$ be a configuration of fluxes of the $\mathbf{\bar{5}}$ matter curves in the set of configurations with no chiral exotics and no zero fluxes ($F \in \text{elemsNoExotics}$). For any sub-multiset $S \subseteq F$, let $(M_x, N_x)$ denote the chirality flux $M$ and hypercharge flux $N$ of each element $x$ in $S$. The sum of the flux pairs in $S$ satisfies the following identity: $$\sum_{x \in S} (|M_x|, -|M_x|) + \sum_{x \in S} (0, |M_x + N_x|) = \sum_{x \in S} (M_x, N_x)$$ where the addition and summation of flux pairs are performed component-wise.

The multiset `elemsNoExotics` contains the six configurations of flux pairs $(M, N) \in \mathbb{Z}^2$ for the 10-dimensional representation matter curves in an $SU(5)$ F-theory model that satisfy the "no chiral exotics" (`NoExotics`) and "no zero flux" (`HasNoZero`) conditions. Each configuration is represented as a multiset of flux pairs $(M, N)$, where $M$ is the chirality flux and $N$ is the hypercharge flux. The six configurations are: 1. $\{(1, 0), (1, 0), (1, 0)\}$ 2. $\{(1, 1), (1, -1), (1, 0)\}$ 3. $\{(1, 0), (2, 0)\}$ 4. $\{(1, -1), (2, 1)\}$ 5. $\{(1, 1), (2, -1)\}$ 6. $\{(3, 0)\}$

The multiset `elemsNoExotics` contains the configurations of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the 10-dimensional representation matter curves in an $SU(5)$ F-theory model that satisfy the "no chiral exotics" and "no zero flux" conditions. The cardinality of this multiset is 6.

The multiset `elemsNoExotics`, which contains the configurations of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the 10-dimensional representation matter curves in an $SU(5)$ F-theory model that satisfy the "no chiral exotics" and "no zero flux" conditions, contains no duplicate elements.

For any configuration of fluxes $F$ for the 10-dimensional representation matter curves in an $SU(5)$ F-theory model, if $F$ is an element of the multiset `elemsNoExotics` (the collection of configurations satisfying both the no chiral exotics and no zero flux conditions), then $F$ satisfies the condition for no chiral exotics (`NoExotics`).

For any configuration of fluxes $F$ for the 10-dimensional representation matter curves in an $SU(5)$ F-theory model, if $F$ belongs to the collection of flux configurations that satisfy the "no chiral exotics" and "no zero flux" conditions (denoted as `elemsNoExotics`), then the number of distinct flux pairs $(M, N) \in \mathbb{Z}^2$ in $F$ is at most 3.

For any configuration of fluxes $F$ for the 10-dimensional representation matter curves in an $SU(5)$ F-theory model, if $F$ belongs to the collection of configurations satisfying the "no chiral exotics" and "no zero flux" conditions (denoted as `elemsNoExotics`), then the sum of all flux pairs $(M, N) \in \mathbb{Z}^2$ in the multiset $F$ is equal to $(3, 0)$, where $M$ is the chirality flux and $N$ is the hypercharge flux.

For any configuration of flux pairs $F$ for the 10-dimensional representation matter curves in an $SU(5)$ F-theory model, if $F$ belongs to the collection `elemsNoExotics` (the set of flux configurations satisfying "no chiral exotics" and "no zero flux" conditions), then $F$ satisfies the "no zero flux" property, meaning that the zero flux pair $(0, 0)$ is not an element of the multiset $F$.