Physlib.StringTheory.FTheory.SU5.Fluxes.NoExotics.Completeness

In an $SU(5)$ F-theory model, let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the $\mathbf{\bar{5}}$ representation matter curves, where $M$ is the chirality flux and $N$ is the hypercharge flux. If $F$ satisfies the `NoExotics` condition (which requires the total chiral indices for the $D$ and $L$ representations to be exactly 3 with no anti-chiral states) and the `HasNoZero` condition (meaning $(0, 0) \notin F$), then every individual flux $x \in F$ must belong to the following set of 15 possible flux pairs: $$\{(0, 1), (0, 2), (0, 3), (1, -1), (1, 0), (1, 1), (1, 2), (2, -2), (2, -1), (2, 0), (2, 1), (3, -3), (3, -2), (3, -1), (3, 0) \}.$$ This set contains all non-zero pairs $(M, N)$ that satisfy $0 \le M \le 3$ and $0 \le M + N \le 3$.

In an $SU(5)$ F-theory model, let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the $\mathbf{\bar{5}}$ representation matter curves. Suppose $F$ satisfies the `NoExotics` condition (the total chiral indices for $D$ and $L$ representations are exactly 3, with no anti-chiral states) and the `HasNoZero` condition (containing no zero flux $(0, 0)$). Let $S$ be a sub-multiset of $F$ with cardinality $n+1$. Suppose $Y$ is a finite set of multisets that contains all sub-multisets of $F$ of cardinality $n$. Let $X$ be a finite set of multisets such that for any flux pair $a$ in the set of 15 allowed pairs: $$\mathcal{A} = \{(0, 1), (0, 2), (0, 3), (1, -1), (1, 0), (1, 1), (1, 2), (2, -2), (2, -1), (2, 0), (2, 1), (3, -3), (3, -2), (3, -1), (3, 0) \}$$ and any multiset $y \in Y$, if the multiset union $\{a\} \cup y$ satisfies: 1. $\sum_{(M, N) \in \{a\} \cup y} (M + N) \le 3$ 2. $\sum_{(M, N) \in \{a\} \cup y} M \le 3$ then $\{a\} \cup y \in X$. Under these conditions, the sub-multiset $S$ is an element of $X$.

For a given natural number $n$, this function returns the finite set of all multisets of flux pairs $(M, N) \in \mathbb{Z}^2$ with cardinality $n$ that can be formed as subsets of a 5-bar flux configuration in $SU(5)$ F-theory satisfying the "no exotics" and "no zero" conditions. The definition explicitly enumerates valid multisets for $0 \le n \le 6$ and returns the empty set for $n \ge 7$, indicating that no valid configuration exists with cardinality greater than 6.

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the $\mathbf{\bar{5}}$ representation matter curves in an $SU(5)$ F-theory model. Suppose $F$ satisfies the `NoExotics` condition (ensuring exactly three generations of $L$ and $D$ with no anti-chiral states) and the `HasNoZero` condition (containing no zero flux $(0, 0)$). Then, for any sub-multiset $S \subseteq F$, $S$ is an element of the set of allowed subsets of cardinality $|S|$, denoted as `noExoticsSubsets(|S|)`.

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the $\mathbf{\bar{5}}$ representation matter curves in an $SU(5)$ F-theory model. If $F$ satisfies the `NoExotics` condition (ensuring exactly three generations of $L$ and $D$ representations with no anti-chiral states) and the `HasNoZero` condition (containing no zero flux $(0, 0)$), then the cardinality of the multiset $F$ is at most 6.

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the $\mathbf{\bar{5}}$ representation matter curves in an $SU(5)$ F-theory model. If $F$ satisfies the `NoExotics` condition (ensuring exactly three generations of $L$ and $D$ representations with no anti-chiral states) and the `HasNoZero` condition (stating that the zero flux $(0, 0)$ is not in $F$), then the cardinality of the multiset $F$ is an element of the set $\{0, 1, 2, 3, 4, 5, 6\}$.

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the $\mathbf{\bar{5}}$ representation matter curves in an $SU(5)$ F-theory model. If $F$ satisfies the `NoExotics` condition (ensuring exactly three generations of $L = (1, 2)_{-1/2}$ and $D = (\bar{3}, 1)_{1/3}$ representations with no anti-chiral states) and the `HasNoZero` condition (stating that the zero flux $(0, 0)$ is not an element of $F$), then $F$ is an element of the multiset `elemsNoExotics` containing all such valid configurations.

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the $\mathbf{\bar{5}}$ (5-bar) representation matter curves in an $SU(5)$ F-theory model. The configuration $F$ satisfies the `NoExotics` condition (ensuring exactly three generations of chiral lepton doublets $L = (1, 2)_{-1/2}$ and down-type quarks $D = (\bar{3}, 1)_{1/3}$ with no corresponding anti-chiral states) and the `HasNoZero` condition (stating that the zero flux $(0, 0)$ is not an element of $F$) if and only if $F$ is a member of the pre-defined multiset `elemsNoExotics`.

In an $SU(5)$ F-theory model, let $F$ be a multiset of flux pairs $\langle M, N \rangle \in \mathbb{Z}^2$ associated with the 10-dimensional representation matter curves. If the flux configuration $F$ satisfies the condition of having no chiral exotics (`NoExotics`) and contains no zero fluxes (`HasNoZero`), then every individual flux pair $x \in F$ must belong to the set $\{\langle 1, -1 \rangle, \langle 1, 0 \rangle, \langle 1, 1 \rangle, \langle 2, -1 \rangle, \langle 2, 0 \rangle, \langle 2, 1 \rangle, \langle 3, 0 \rangle\}$.

Let $F$ be a multiset of flux pairs $\langle M, N \rangle \in \mathbb{Z}^2$ representing matter curves in the 10-dimensional representation of an $SU(5)$ F-theory model. Suppose $F$ satisfies the condition of having no chiral exotics (`NoExotics`) and contains no zero fluxes (`HasNoZero`). Let $n$ be a natural number and $S$ be a sub-multiset of $F$ with cardinality $|S| = n + 1$. Suppose $Y$ is a finite set containing all sub-multisets of $F$ with cardinality $n$. Furthermore, let $X$ be a finite set of multisets such that for any flux pair $a \in \{\langle 1, -1 \rangle, \langle 1, 0 \rangle, \langle 1, 1 \rangle, \langle 2, -1 \rangle, \langle 2, 0 \rangle, \langle 2, 1 \rangle, \langle 3, 0 \rangle\}$ and any multiset $y \in Y$, the multiset $a \cup y$ is an element of $X$ if the following three physical constraints (representing the chiral indices of the $Q$, $U$, and $E$ representations) are satisfied: 1. $\sum_{\langle M, N \rangle \in \{a\} \cup y} M \leq 3$ 2. $\sum_{\langle M, N \rangle \in \{a\} \cup y} (M - N) \leq 3$ 3. $\sum_{\langle M, N \rangle \in \{a\} \cup y} (M + N) \leq 3$ Then it holds that $S \in X$.

For a given natural number $n$, this definition returns the finite set of all multisets of cardinality $n$ that can be formed as sub-multisets of $10$-dimensional representation flux configurations in $SU(5)$ F-theory that satisfy the `NoExotics` and `HasNoZero` conditions. A flux is represented as a pair $\langle M, N \rangle \in \mathbb{Z}^2$. The definition proceeds by cases on $n$: - If $n=0$, the result is the set containing the empty multiset: $\{\emptyset\}$. - If $n=1$, the result is the set of singleton multisets: $\{\{\langle 1, -1 \rangle\}, \{\langle 1, 0 \rangle\}, \{\langle 1, 1 \rangle\}, \{\langle 2, -1 \rangle\}, \{\langle 2, 0 \rangle\}, \{\langle 2, 1 \rangle\}, \{\langle 3, 0 \rangle\}\}$. - If $n=2$, the result is the set of multisets: $\{\{\langle 1, -1 \rangle, \langle 1, 0 \rangle\}, \{\langle 1, -1 \rangle, \langle 1, 1 \rangle\}, \{\langle 1, -1 \rangle, \langle 2, 1 \rangle\}, \{\langle 1, 0 \rangle, \langle 1, 0 \rangle\}, \{\langle 1, 0 \rangle, \langle 1, 1 \rangle\}, \{\langle 1, 0 \rangle, \langle 2, 0 \rangle\}, \{\langle 1, 1 \rangle, \langle 2, -1 \rangle\}\}$. - If $n=3$, the result is the set of multisets: $\{\{\langle 1, -1 \rangle, \langle 1, 0 \rangle, \langle 1, 1 \rangle\}, \{\langle 1, 0 \rangle, \langle 1, 0 \rangle, \langle 1, 0 \rangle\}\}$. - If $n \geq 4$, the result is the empty set $\emptyset$.

For any multiset $F$ of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with 10-dimensional matter curves in an $SU(5)$ F-theory model, if $F$ satisfies the `NoExotics` condition (ensuring no chiral exotics in the MSSM spectrum) and the `HasNoZero` condition (containing no $(0, 0)$ flux pairs), then any sub-multiset $S \subseteq F$ is a member of the finite set of allowed 10d flux sub-multisets of cardinality $|S|$, denoted by `noExoticsSubsets(|S|)`.

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with 10-dimensional representation matter curves in an $SU(5)$ F-theory model (of type `FluxesTen`). If $F$ satisfies the `NoExotics` condition (ensuring there are no exotic chiral matter representations and exactly 3 generations of $Q$, $U$, and $E$ in the MSSM spectrum) and the `HasNoZero` condition (ensuring that the zero flux $(0, 0)$ is not an element of $F$), then the cardinality of the multiset $F$ is at most 3, i.e., $|F| \le 3$.

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with 10-dimensional matter curves in an $SU(5)$ F-theory model. If $F$ satisfies the `NoExotics` condition (ensuring no exotic chiral matter and exactly 3 generations of $Q$, $U$, and $E$) and the `HasNoZero` condition (ensuring the zero flux $(0, 0)$ is not in $F$), then the cardinality $|F|$ of the multiset is an element of the set $\{0, 1, 2, 3\}$.

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ representing the $10$-dimensional representation matter curves in an $SU(5)$ F-theory model. If $F$ satisfies the `NoExotics` condition (ensuring no chiral exotics and exactly three generations of the representations $Q$, $U$, and $E$) and the `HasNoZero` condition (ensuring the zero flux $(0, 0)$ is not in $F$), then $F$ is an element of the multiset `elemsNoExotics`.

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the 10-dimensional representation matter curves in an $SU(5)$ F-theory model. $F$ satisfies the `NoExotics` condition (which ensures that the spectrum contains exactly three generations of the Standard Model representations $Q$, $U$, and $E$ with no anti-chiral exotics) and the `HasNoZero` condition (which states that the zero flux $(0, 0)$ is not an element of $F$) if and only if $F$ is an element of the multiset `elemsNoExotics`.