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

Examples

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 sin²(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:

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

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

                Total number of integers on this line (according to wc)
type            type of SM
xbranes         Additional branes: 0=NO 1=YES
udMir           (u,d) mirror pairs   [(3,2)+(3,2*)+(3*,2)+(3*,2*)-3]/2
uMir            u     mirror pairs
dMir            d     mirror pairs
enMir           (e,n) mirror pairs   [(2,-1/2)+(2,1/2)+(2*,1/2)+(2^*,-1/2)-3]/2
eMir            e     mirror pairs
nMir            n     mirror pairs
hNr             Number of Higgs      [(2,-1/2)+(2,1/2)+(2*,1/2)+(2^*,-1/2)/2
udAnom          (u,d) SU(2) anomaly: 3 x [(3,2)+(3*,2)-(3,2^*)-(3^*,2^*)]
enAnom          (e,n) SU(2) anomaly: (2,1/2)+(2,-1/2)-(2*,1/2)-(2^*,-1/2)
hAnom           Higgs SU(2) anomaly: (2,1/2)+(2,-1/2)-(2*,1/2)-(2^*,-1/2)
udAsym          (u,d) asymmetry:     (3,2)+(3^*,2^*)-(3,2^*)-(3^*,2)
enAsym          (e,n) asymmetry:     (2,1/2)-(2,-1/2)+(2*,1/2)-(2^*,-1/2)
hAsym           Higgs asymmetry:     (2,1/2)-(2,-1/2)+(2*,1/2)-(2^*,-1/2)
lq_a            Leptoquarks          [(3,2/3)+(3^*,-2/3)]/2
lq_b            Leptoquarks          [(3,-1/3)+(3^*,1/3)]/2
aS              Symmetric tensors from brane a
aA              AntiSymmetric tensors from brane a
aAdj            Adjoints from brane a
bS              Symmetric tensors from brane b
bA              AntiSymmetric tensors from brane b
bAdj            Adjoints from brane b
cS              Symmetric tensors from brane c
cA              AntiSymmetric tensors from brane c
cAdj            Adjoints from brane c
dS              Symmetric tensors from brane d
dA              AntiSymmetric tensors from brane d
dAdj            Adjoints from brane d
aT              a-brane tension
bT              b-brane tension
cT              c-brane tension
dT              d-brane tension
oT              orientifold tension
Qfrac           Fractional charge matter: 0=NO 1=YES
nrHidden        Total number of hidden gauge groups (NG)
confInd         Confinement indicator;
		0: No frac. charges 1: unconf. 2: confined
aV              a-V
bV              b-V
cV              c-V
dV              d-V
nrHidMat        Total Hidden matter
chirHidMat      Hidden matter chirality
YMass           Y mass     ( x 1000000)
BLMass          B-L mass   ( x 1000000)
BMass           B mass     ( x 1000000)
U2Mass          U(2)-mass  ( x 1000000)
adEquiv         Branes a&d have the same contributions to tadpoles
nrReps          Number of representations
sinSqThetaW     sin^2(theta_w)
alphaS_alphaW   alpha_s/alpha_w
invAlphaY       1/alpha_Y
invAlphaW       1/alpha_W
invAlphaS       1/alpha_S
betaY           b_y as in betafunction: 1/a_y(mu) = 1/a_y(M) + b_y/2pi ln(mu/M)
betaW           b_w as in betafunction, 1/a_y(mu) = 1/a_y(M) + b_y/2pi ln(mu/M)
betaS           b_s from betafunction, 1/a_y(mu) = 1/a_y(M) + b_y/2pi ln(mu/M)
chirBetaY       b_y, counting only chiral d.o.f.
chirBetaW       b_w, counting only chiral d.o.f.
chirBetaS       b_s, counting only chiral d.o.f.
stringScale     String scale [actually ln(M_s/M_z)], assuming all non-chiral
		particles become massive at one scale and the couplings
		flow to the right value at the weak scale.
nonChiralScale  Scale at which non-chiral modes get a mass (see stringScale)
stringCoupling  String Coupling (see stringScale).
h11m            Hodge number h11-
h11p            Hodge number h11+
h21             Hodge number h21
closedVector    Closed string Vector multiplets
closedChiral    Closed string Chiral multiplets
Tensor          Identifier of Minimal Model tensor product
MIPF            Identifier of Modular Invariant Partition Function
Orientifold     Identifier of Orientifold

Syntax description

usage: filtersols [options] file1.msol, file2.msol, ...
Filters lines in .msol files on a supplied condition. If desired, it can
output the lines in an alternate format.

  -c|--condition A condition to filter the data with
  -f|--file	  Get condition from file
  -p|--print	  Pretty print the models (Can be a lot of output!)
  -s|--svml	  Print the models in svml format (Can be a lot of output!)
  -n|--count	  Count number of solutions satisfying condition
  -o|--format	  Specify output format in terms of equations
  --scale	  Output format for plotting Ms/Mnc plots
  --betafunc	  Output format for making rgflow movies
  -a|--accuracy  Set the accuracy while using the ~= operator
  -v|--verbose	  Print debug info
  -h|--help	  Print some info on valid fieldnames and condition syntax
  --fieldnames	  List valid fieldnames

The condition is a series of expressions constructed from one or more
elements separated by algebraic operators:
  ^ * / % 
  - +
These expressions have to be separated by one of the following
logical operators:
  !
  == != ~= >= <= > <
  && ||
Operators on one line have equal precedence, the first line has highest
precedence, and so on. Operators associate to the left, except of course
for the unary !. One can force precedence by using parenthesis. Note that
algebraic operators have higher precedence than the logical ones, even
than the ! (not operator).

The operators have the usual C-meaning, except for ^,% and ~=;
^ is the 'to the power' operator. % is the usual modulus operator, except
that it will convert it's argument to integers first
The only operator that needs some more explanation is ~=; it tests if
two elements are roughly equal. The accuracy with which this is tested is
set by the -a/--accuracy option and defaults to 1e-5

Elements in expressions can either be a literal (like 1 or 3.1415 or 2.8e-4),
a field name or a special function.
Valid names can be found with the --fieldnames option.

At the moment the only special function is REP. The syntax is as follows:
	REP(grp(dim):rep;chr)

Where 'grp' can be U, SO or SP, 'dim' stands for the dimension of the
fundamental representation. You can also enter +dim or -dim to match groups
with greater or smaller 'dim'. If 'dim' is absent all 'dim's will match.
'rep' can be one of V,S,A,Ad; it matches if a vector, symmetric tensor, 
anti-symmetric tensor or adjoint representation is present. Finally, 'chr' 
stands for the chirality; it can be a number between -127 and 128 or just
y or n. Any of these parts, except for 'grp' can be absent. Examples:
  REP(SO(+16))	matches SO(17), SO(18), etc
  REP(U():A;y)	matches any U group with a chiral asymmetric tensor
  REP(SP(18):S)	matches SP(18) with a symmetric tensor

As a default filtersols works as a filter; it output those lines that match
the filter condition supplied with -c/--condition or -f/--file. There are
however a few different other output formats, namely -p/--print, -n/--count
and -o/--format. The last one is the most flexible; it enables you define 
output fields in terms of expressions (in the sense defined above).
For the moment the expressions should be entered separated by commas, the
output will be separated by spaces.