On this page you can search through all our supersymmetric, tadpole-free D=4, N=1 orientifold vacua with a three family chiral fermion spectrum identical to that of the Standard Model. They were constructed in a semi-systematic way by considering orientifolds of all Gepner Models (see Phys.Lett.B609:408-417 and Nucl.Phys.B710:3-57 for more information). Since the publication of these papers all spectra have been re-analysed and checked for the presence of global (Witten) anomalies. A few cases (less than 1%) needed correction. All spectra in this database are now free from global anomalies, and the total number is 211,634, slightly more than reported in these papers.

As explained in referenced articles the standard model gauge group
can be realized in different ways (which we call *types*). In
addition to these factors, the gauge group usually has extra
*hidden* gauge group factors. Chiral states with one leg in the
standard model gauge group are not permitted.

All these models of course have the same *chiral* spectrum for the
standard model gauge group, except for the higgs-sector of which we do
not know how it is realized in nature.

These models then differ in multiplicities of the non-chiral particles,
hidden gauge group, higgs sector coupling constants on the string
scale, and others.

To search for your favorite realization you can use the form below
to filter our set with an condition. Example:
`
type==0 && nrHidden<2
`
You can consult a list of valid field names.
Also much more complicated expressions are possible, see the
syntax description.

The following search string list all models without any mirror fermions and rank-2 tensors
`
udmir==0 && umir==0 && dmir==0 && enmir==0 && emir==0 && nmir==0 &&
aadj==0 && badj==0 && cadj==0 && dadj==0 &&
aa==0 && ba==0 && ca==0 && da==0
&& as==0 && bs==0 && cs==0&& ds==0
`
There are two examples in the set.

To see the 22 models that have exactly the standard model gauge group SU(3) x SU(2) x U(1) type
`type==6 && nrhidden=0`
The first parameter selects a standard model gauge group with a bi-linear
axion coupling to the
B-L gauge
boson, which acquires a mass. The second parameter specifies the absence of
a hidden sector.

This is an example of the output in "summary" mode.

`
Tensor 44622, MIPF 8, Orientifold 0
`
Tensor product, Modular Invariant Partition Function, Orientifold nr.

The latter two are specified according to the internal labelling of the
program kac used to generate the solutions
`
Klein bottle current: 12 Crosscap signs: (-1,-1)
Standard model boundaries: (89,44,45,2)
`
Specifications of the Klein bottle current and the crosscap signs that
define the orientifold, and the labels of the boundary states that
specify the standard model branes. See
Phys.Lett.B495:427-434
for further explanations. The Klein bottle current and boundary labels
are defined in kac
`
Dilaton couplings to SM branes and O-plane:
0.0070459 0.0122039 0.0122039 0.0070459 0.1073896
`
Boundary and crosscap reflection coefficients as defined in formulas
(20) and (21) of Nucl.Phys.B710:3-57
`
alpha_3/alpha_2 = 0.8660246
sin^2(theta_w) = 0.3610368
`
Ratios of strong and weak coupling constants and sin^{2}(ϑ_{w}) at the
string scale, not including any renormalization group running
`
Total number of branes: 9
CP multiplicities: 89:3 91:3 44:2 45:2 2:1 3:1 400:6 38:4 349:2
`
Chan-Paton multiplicities of the solution, in the notation
(label):(multiplicity). Labels defined in kac;
Boundaries and boundary conjugates are shown separately
`
Standard model type: 4
Number of factors in hidden gauge group: 3
Gauge group: U(3) x Sp(2) x Sp(2) x U(1) x Sp(6) x Sp(4) x Sp(2)
Number of representations: 19
3 x (V ,V ,0 ,0 ,0 ,0 ,0 ) chirality 3
3 x (V ,0 ,V ,0 ,0 ,0 ,0 ) chirality -3
3 x (0 ,V ,0 ,V ,0 ,0 ,0 ) chirality 3
3 x (0 ,0 ,V ,V ,0 ,0 ,0 ) chirality -3
2 x (V ,0 ,0 ,V ,0 ,0 ,0 )
2 x (0 ,V ,V ,0 ,0 ,0 ,0 )
2 x (V ,0 ,0 ,0 ,V ,0 ,0 )
2 x (V ,0 ,0 ,0 ,0 ,V ,0 )
2 x (V ,0 ,0 ,0 ,0 ,0 ,V )
1 x (0 ,V ,0 ,0 ,V ,0 ,0 )
1 x (0 ,0 ,V ,0 ,V ,0 ,0 )
2 x (0 ,0 ,0 ,V ,0 ,V ,0 )
1 x (0 ,0 ,0 ,0 ,V ,0 ,V )
2 x (0 ,0 ,0 ,0 ,0 ,V ,V )
2 x (0 ,0 ,0 ,0 ,A ,0 ,0 )
1 x (0 ,0 ,0 ,0 ,S ,0 ,0 )
5 x (0 ,0 ,0 ,0 ,0 ,A ,0 )
5 x (0 ,0 ,0 ,0 ,0 ,S ,0 )
1 x (0 ,0 ,0 ,0 ,0 ,0 ,S )
`
Representations are denoted as `N x (....) chirality M`.
Here N is the total multiplicity for a
representation plus its complex conjugate.
For example `N x (V,V) chirality M` should be interpreted as
(N-M)/2 (V,V) + (N+M)/2 (V*,V*) if (V,V) is complex and as N (V,V) if it
is real (then M is zero).
If `chirality` is not listed
it vanishes. In each Chan-Paton factor the following representations
can occur:

V | Vector |

A | Anti-symmetric tensor |

A | Symmetric tensor |

Ad | Adjoint |

`
Summary:
Higgs: (2,1/2)+(2*,1/2) 2
Non-chiral SM matter (Q,U,D,L,E,N): 0 0 0 0 0 0
Adjoints: 0 0 0 0
Symmetric Tensors: 0 0 0 0
Anti-Symmetric Tensors: 0 0 0 0
Lepto-quarks: (3,-1/3),(3,2/3) 1 0
Non-SM (a,b,c,d) 12 6 6 4
Hidden (Total dimension) 162 (chirality 0)
`
List of non-chiral exotics and Higgs, i.e. all particles that are not quarks or leptons.
This is mostly self-explanatory.
This example has two susy
Higgs pairs, twice as many as the MSSM, and four times as many as
the Standard Model. It has no "exotics" except a single lepto-quark
and some matter in vector representations of hidden sector non-abelian
groups. Representations R with a complex standard model part always occur as R+R*.
Their multiplicities are divided by 2. For example one lepto-quark means a lepto-quark
plus its conjugate. The same convention is used for Standard Model/Hidden matter. Note
that hidden sector complexity is ignored. Anti-symmetric tensors of U(1) groups are listed,
although their massless ground state has dimension 0.

The list of parameters that can be used in the above form is mostly self-explanatory. The input is case-insensitive.

`type` refers to the realization of the
standard model part of the gauge group. The possibilities are

Type | SM part of gauge group | B-L vector boson |

0 | U(3) x Sp(2) x U(1) x U(1) | Massless |

1 | U(3) x U(2) x U(1) x U(1) | Massless |

2 | U(3) x Sp(2) x O(2) x U(1) | Massless |

3 | U(3) x U(2) x O(2) x U(1) | Massless |

4 | U(3) x Sp(2) x Sp(2) x U(1) | Massless |

5 | U(3) x U(2) x Sp(2) x U(1) | Massless |

6 | U(3) x Sp(2) x U(1) x U(1) | Massive |

7 | U(3) x U(2) x U(1) x U(1) | Massive |

Note that for a gauge group of type 2 through 5 the gauge boson coupling to B-L is necessarily massless. Examples of type 7 were not found.

`Qfrac` can be used to require the absence (or presence) of
matter from strings stretching between the standard model and the hidden
branes. Such matter is always fractionally charged, but may be confined
by the hidden gauge group. This may be specified by the `confInd`
parameter.`aV` ... `dV` specifies the number of such
fractionally charged particles for the a...d branes separately.

The 211,634 models are distinghuished by the value of at least one of the first 34 parameters listed below, from "type" to "dT", plus the tensor product and MIPF from which they originate. Note that no information regarding the hidden sector is used, apart from its presence. The orientifold choice is not used to distinguish models, but tensor product and MIPF are. There are a few cases where different MIPFs and/or tensor products yield identical spectra.

Number if MIPFs considered: 5403

Number of Orientifolds: 49304 (33012 of which have non-zero tension)

(Note: the number of orientifolds was incorrectly stated as 49322 in Nucl.Phys.B710:3-57) because a few orientifolds were inadvertently double counted)

Number of tensor products searched: 161 out of 168

(Not done: five non-chiral K3 x T2 compactifications (1,1,1,2,3,18), (1,1,1,2,6,6), (1,1,1,3,3,8), (1,1,1,2,2,2,2), (1,1,1,1,1,1,2,2), and (1,5,42,922) and (1,5,43,628) because of memory and CPU time limitations)

Number of MIPFs searched: 5311 out of 5403

Number of orientifolds searched: 32784 out of the 33012 non-zero tension cases

Number of MIPFs searched: 161 out of 168

In four cases, namely (1,5,47,292), (1,5,54,166), (1,5,82,82) and (1,6,24,310) the search was limited to types 0 and 1 because of CPU time limitations.

The following table summarizes all the numbers. Column 2 gives the total number of brane configurations present in all chiral Gepner models, column 3 the number of these that we actually checked, column 4 the number of those configurations that agreed with one of the eight standard model types, and the last column indicates the lower bound on the number of cases where the tadpole conditions could be solved. This is only a bound, because in many cases the set of equations was too large to obtain a conclusive answer (the number of solutions is in fact slightly larger than quoted in Nucl.Phys.B710:3-57 because of results obtained after publication). In addition some more solutions were found in a re-analysis needed for checking global anomalies, which are not included in these numbers (but are included in the set of models accessible through this webpage). Note that the numbers in the table count solutions prior to comparison. After comparing them the number of distinct ones reduces to about 200,000.

Combination (types) | Total number | Number searched | SMs found | Tadpoles solved |

USUU (0+6) | 187648179869355108 | 187171389940312068 | 1096682+49794 | 215846+4468 |

UUUU (1+7)) | 42766246654184825664 | 42730101309436185264 | 131704+1306 | 1280 + 0 |

USOU (2) | 35594807811446520 | 21498035622653976 | 9474494 | 431633 |

UUOU (3) | 2579563256116048068 | 720412912488220932 | 16891580 | 12533 |

USSU (4) | 4486269786712304 | 2792296847030752 | 16227372 | 978200 |

UUSU (5) | 187648179869355108 | 90192673747778532 | 1178970 | 5682 |

Total | 45761187347637742772 | 43752168618082181524 | 45051902 | 1649642 |

Combination (types) | SM fraction | Tadpole solution fraction |

USUU (0+6) | 5.9 x 10^{-12}+2.7 x 10^{-13} | 2.7 x 10^{-13}+2.4 x 10^{-14} |

UUUU (1+7)) | 3.1 x 10^{-15}+3.1 x 10^{-17} | 3.0 x 10^{-17} + 0 |

USOU (2) | 1.1 x 10^{-10} | 4.8 x 10^{-12} |

UUOU (3) | 2.3 x 10^{-11} | 1.7 x 10^{-14} |

USSU (4) | 5.8 x 10^{-9} | 3.5 x 10^{-10} |

UUSU (5) | 1.3 x 10^{-11} | 6.3 x 10^{-14} |

Total | 1.0 x 10^{-12} | 3.8 x 10^{-14} |