'; ?> RCFT orientifolds with Standard Model Spectrum

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.

Filter form

Two output formats are provided. The first only gives the number of answers, the second lists all the spectra satisfying the search criteria. Be warned that output can be very large and take up to a minute to compile; at the moment we have 211,634 models in the database, which means you can generate hunderds of MBs of output!


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.

Output description

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 sin2w) 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:

AAnti-symmetric tensor
ASymmetric tensor

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.

Search criteria

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

Types of standard model gauge groups
TypeSM part of gauge groupB-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.

Distinct Models

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.

List of valid field names

Syntax description

Some Statistics

Number of N=2 tensor products: 168
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 Gepner combinations 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.

Statistics of brane configurations
Combination (types) Total number Number searched SMs found Tadpoles solved
USUU (0+6) 187648179869355108 187171389940312068 1096682+49794215846+4468
UUUU (1+7)) 42766246654184825664 42730101309436185264 131704+1306 1280 + 0
USOU (2) 35594807811446520 21498035622653976 9474494431633
UUOU (3) 2579563256116048068 720412912488220932 1689158012533
USSU (4) 4486269786712304 279229684703075216227372978200
UUSU (5) 187648179869355108 9019267374777853211789705682
Total 45761187347637742772 43752168618082181524 450519021649642

Percentages of brane configurations
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}
Total1.0 x 10^{-12}3.8 x 10^{-14}

array("pipe", "r"), 1 => array("pipe", "w"), 2 => array("pipe", "w") ); $process = proc_open($proc, $descriptorspec, $pipes); if (is_resource($process)) { while (!feof($pipes[1])) { echo fgets($pipes[1], 1024); } while (!feof($pipes[2])) { echo fgets($pipes[2], 1024); } fclose($pipes[1]); fclose($pipes[2]); //$return_value = proc_close($process); //echo "command returned $return_value\n"; } /* $fp=popen($proc,"r"); while(!feof($fp) ){ strlen($ret=fread($fp,1024)) ){ echo $ret; } */ } function passthru_xhtml_safe($cmd){ $str=""; $fp=popen($cmd,"r"); while( strlen($ret=fread($fp,1024)) ){ $str.=$ret; } echo htmlspecialchars($str); } ?>