CICY Coxeter DB | 2322

CICY2322

  • rank 2
  • I₂(4)
  • finite
Favorable Kähler-favProduct
h1,1h^{1,1}
8
h2,1h^{2,1}
24
χ\chi
−32
ambient factors
8
polynomials
10
iso-flops
2
Coxeter rank
2
Coxeter group
I₂(4)
Configuration matrix
8×10 configuration
X2322=[P11100000000P10010010000P10001001000P10000100100P10000000011P21000100010P20101000010P40010011101]−328,24X_{2322} = \left[\begin{array}{c|cccccccccc} \mathbb{P}^{1} & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbb{P}^{1} & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ \mathbb{P}^{1} & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ \mathbb{P}^{1} & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ \mathbb{P}^{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ \mathbb{P}^{2} & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\ \mathbb{P}^{2} & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \\ \mathbb{P}^{4} & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 \end{array}\right]^{8,24}_{-32}
Second Chern class
c2(X)⋅Di=(2424242424363652)Tc_2(X)\cdot D_i = \begin{pmatrix} 24 & 24 & 24 & 24 & 24 & 36 & 36 & 52 \end{pmatrix}^{T}
Coxeter diagram Gallery →
4
Coxeter matrix
(1441)\begin{pmatrix} 1 & 4 \\ 4 & 1 \end{pmatrix}
Iso-flop reflections Kähler representation
M^1\hat{M}_1: row 2, Type 1a
M^1=(100000000−1000000001000000001000000001000000001000000001001000001)\hat{M}_1 = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}
M^2\hat{M}_2: row 8, Type 1b
M^2=(100000000100000200100001000100010000100100000100000000100000000−1)\hat{M}_2 = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \end{pmatrix}

Database record

Mathematica 
<|Num -> 2322, H11 -> 8, H21 -> 24, C2 -> {24, 24, 24, 24, 24, 36, 36, 52}, Conf -> {{1, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {1, 0, 0, 0, 1, 0, 0, 0, 1, 0}, {0, 1, 0, 1, 0, 0, 0, 0, 1, 0}, {0, 0, 1, 0, 0, 1, 1, 1, 0, 1}}, Favour -> True, KahlerPos -> True, IsProduct -> False, IsoFlopRows -> {{2, "Type 1a"}, {8, "Type 1b"}}, KahlerRefGens -> {{{1, 0, 0, 0, 0, 0, 0, 0}, {0, -1, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0}, {0, 1, 0, 0, 0, 0, 0, 1}}, {{1, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 2}, {0, 0, 1, 0, 0, 0, 0, 1}, {0, 0, 0, 1, 0, 0, 0, 1}, {0, 0, 0, 0, 1, 0, 0, 1}, {0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, -1}}}, CoxeterMat -> {{1, 4}, {4, 1}}|>
Plain text 
Num           : 2322
H11           : 8
H21           : 24
C2            : {24, 24, 24, 24, 24, 36, 36, 52}
Conf          : {{1, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {1, 0, 0, 0, 1, 0, 0, 0, 1, 0}, {0, 1, 0, 1, 0, 0, 0, 0, 1, 0}, {0, 0, 1, 0, 0, 1, 1, 1, 0, 1}}
Favour        : True
KahlerPos     : True
IsProduct     : False
IsoFlopRows   : {{2, "Type 1a"}, {8, "Type 1b"}}
KahlerRefGens : {{{1, 0, 0, 0, 0, 0, 0, 0}, {0, -1, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0}, {0, 1, 0, 0, 0, 0, 0, 1}}, {{1, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 2}, {0, 0, 1, 0, 0, 0, 0, 1}, {0, 0, 0, 1, 0, 0, 0, 1}, {0, 0, 0, 0, 1, 0, 0, 1}, {0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, -1}}}
CoxeterMat    : {{1, 4}, {4, 1}}
27 sibling models share the Coxeter group I₂(4) · filter Explorer · random sibling 100313121620162716891702190819101920206423752683270128202821285728713114322435243622424342744342434655285552
Conventions and glossary

Conventions

Configuration matrix
Rows are the ambient \(\mathbb{P}^{n_i}\) factors; columns indicate the homogeneous degrees of the defining polynomials. The superscript is \((h^{1,1},\,h^{2,1})\) and the subscript is the Euler characteristic \(\chi = 2(h^{1,1} - h^{2,1})\).
Second Chern class
The intersection numbers \(c_2(X)\cdot D_i\) in the favorable divisor basis \(\{D_i\}_{i \in \{ 1, \dotsc, h^{1,1} \}}\).

Glossary

Iso-flop (isomorphic flop)
A flop \(X \dashrightarrow X'\) between diffeomorphic families of Calabi–Yau threefolds.
Kähler-favorable
A favorable CICY whose Kähler cone directly descends from that of the ambient space.
Iso-flop row: Type 1a
A configuration matrix row of the form \(\left[\begin{array}{c|cccccc} \mathbb{P}^{n} & 1 & \cdots & 1 & 0 & \cdots & 0 \end{array}\right]\) whose charge vectors over the ones columns all coincide. The flop across the corresponding Kähler cone wall is always an iso-flop.
Iso-flop row: Type 1b
A configuration matrix row of the same form as Type 1a but without the coincident charge vectors. The flop across the corresponding Kähler cone wall is typically (but not necessarily) not an iso-flop.
Iso-flop row: Type 2
A configuration matrix row of the form \(\left[\begin{array}{c|ccccccc} \mathbb{P}^{n} & 2 & 1 & \cdots & 1 & 0 & \cdots & 0 \end{array}\right]\), including the case with no ones. The flop across the corresponding Kähler cone wall is always an iso-flop.
Parabolic (P) vs. hyperbolic (H)
Both mark a Coxeter-matrix entry of infinite order: the product \(Q_{ij} = \hat{M}_i \hat{M}_j\) has \(\operatorname{ord}(Q_{ij}) = \infty\). They distinguish two inequivalent types of faithful representation: parabolic (P) and hyperbolic (H). See Section 4.1 of the companion paper for details.
Non-Kähler-favorable (NonKahlerPos)
A CICY whose Kähler cone does not directly descend from that of the ambient space. The Coxeter symmetry is left undetermined for such models.

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