alatc-info-examples.nb

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affine-lie-algebra-tensor-category (alatc) package:
information and examples\
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   StyleBox["\<\"\\nIf Mathematica complains about ill-conditioned matrices, \
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Cell["Installing the package", \
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Cell["\<\
Download the package alatc.wl from github,
https://github.com/ardonne/affine-lie-algebra-tensor-category
and copy it to a directory that mathematica can find.

If one wants the package to be available for the user only, one can copy the \
file to
(the location depends on the operating system, so evaluate the cells to see \
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$UserBaseDirectory/Applications\
\>", "Text",ExpressionUUID->"7c713e66-e77e-4a8d-b17f-aa85d3aef427"],

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If one wants the package to be available for all users, one can copy the file \
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\>", "Text",ExpressionUUID->"08cc6a19-9196-4ea0-be1f-01e7cdf975ca"],

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 "\nMost of the notation used follows the notation",
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 "n the paper on which this package is based:\n\n[1] ",
 StyleBox["Clebsch-Gordan and 6j-coefficients for rank two quantum groups",
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 "\nEddy Ardonne, Joost Slingerland\nJ. Phys. A 43, 395205 (2010)\n\
https://doi.org/10.1088/1751-8113/43/39/395205\n\
https://arxiv.org/abs/1004.5456\n\nInformation about the non-uniform cases \
(which are not (directly) related to WZW theories) can be found in:\n\n[2] \
From Quantum Groups to Unitary Modular Tensor Categories\nEric C. Rowell\n\
Contemporary Mathematics, 413, 215 (2006)\n\
http://dx.doi.org/10.1090/conm/413/07848\nhttps://arxiv.org/abs/math/0503226\n\
\nNote that we use a different definition for q, namely q_ours = q^2_Rowell; \
this has no effect on the F-symbols, etc.. \n\nWhen a certain algebra, rank \
and level allows for non-uniform cases, the program will print this when the \
level is initialized. In these cases, rootfactor can be a multiple of tmax \
(denoted by m in [2]), as long as GCD[rootfactor, (k+g)] == 1. The fusion \
rules for the non-uniform cases differ from those of the uniform cases, even \
at the same \[OpenCurlyQuote]level\[CloseCurlyQuote]). Non-uniform cases are \
possible when:\ntype == \[OpenCurlyDoubleQuote]b\[CloseCurlyDoubleQuote] && \
Mod[k,2] == 0 && k \[GreaterEqual] 1\ntype == \[OpenCurlyDoubleQuote]c\
\[CloseCurlyDoubleQuote] && Mod[k+rank,2] == 0 && k \[GreaterEqual] rank - 1\n\
type == \[OpenCurlyDoubleQuote]f\[CloseCurlyDoubleQuote] && Mod[k,2] == 0 && \
k \[GreaterEqual] 3\ntype == \[OpenCurlyDoubleQuote]g\[CloseCurlyDoubleQuote] \
&& (Mod[k,3] == 0 || Mod[k,3] == 1 ) && k \[GreaterEqual] 2\nOne can obtain \
the possible rootfactors by running possiblerootfactors[type, rank, level], \
which returns a list containing the lists with the possible rootfactors for \
the uniform and non-uniform cases. The latter might be empty.\n\nIn the math \
literature, such as [2], it is common to represent q as q = e^(2 Pi i z/l), \
where 0 < z < l && GCD[z,l] == 1 (using our definition of q). The case is \
uniform if tmax divides l, and non-uniform otherwise. There is a minimum \
value for l, in order to obtain a fusion category, namely for the uniform \
cases (l has to be at least tmax times the dual coxeter number):\ntype == \
\[OpenCurlyDoubleQuote]a\[CloseCurlyDoubleQuote] && l \[GreaterEqual] rank + \
1 \ntype == \[OpenCurlyDoubleQuote]b\[CloseCurlyDoubleQuote] && Mod[l,2] == 0 \
&& l \[GreaterEqual] 4 rank - 2 \ntype == \[OpenCurlyDoubleQuote]c\
\[CloseCurlyDoubleQuote] && Mod[l,2] == 0 && l \[GreaterEqual] 2 rank + 2 \n\
type == \[OpenCurlyDoubleQuote]d\[CloseCurlyDoubleQuote] && l \[GreaterEqual] \
2 rank - 2 \ntype == \[OpenCurlyDoubleQuote]e\[CloseCurlyDoubleQuote] && rank \
== 6 && l \[GreaterEqual] 30\ntype == \[OpenCurlyDoubleQuote]e\
\[CloseCurlyDoubleQuote] && rank == 6 && l \[GreaterEqual] 18\ntype == \
\[OpenCurlyDoubleQuote]e\[CloseCurlyDoubleQuote] && rank == 6 && l \
\[GreaterEqual] 30\ntype == \[OpenCurlyDoubleQuote]f\[CloseCurlyDoubleQuote] \
&& Mod[l,2] == 0 && l \[GreaterEqual] 18\ntype == \[OpenCurlyDoubleQuote]g\
\[CloseCurlyDoubleQuote] && Mod[l,3] == 0 && l  \[GreaterEqual] 12\nFor the \
non-uniform cases, the constraint on l is:\ntype == \[OpenCurlyDoubleQuote]b\
\[CloseCurlyDoubleQuote] && Mod[l,2] == 1 && l \[GreaterEqual] 2 rank + 1\n\
type == \[OpenCurlyDoubleQuote]c\[CloseCurlyDoubleQuote] && Mod[l,2] == 1 && \
l \[GreaterEqual] 2 rank + 1\ntype == \[OpenCurlyDoubleQuote]f\
\[CloseCurlyDoubleQuote] && Mod[l,2] == 1 && l \[GreaterEqual] 13\ntype == \
\[OpenCurlyDoubleQuote]g\[CloseCurlyDoubleQuote] && (Mod[l,3] == 1 || \
Mod[l,3] == 2) && l \[GreaterEqual] 7\nOne can initialize the algebra, rank \
and root of unity using the values of l and z as follows:\ninitializelz [\
\[OpenCurlyDoubleQuote]x\[CloseCurlyDoubleQuote], rank, l, z]\n\nFor the \
non-uniform cases, it is customary to introduce a so-called \
\[OpenCurlyQuote]fractional level\[CloseCurlyQuote] k_frac, such that tmax \
(k_frac + g) == (k+g), so k_frac == (k+g)/tmax - g, where \[OpenCurlyQuote]k\
\[CloseCurlyQuote] is the integer \[OpenCurlyQuote]level\[CloseCurlyQuote] as \
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(M)TCs associated with affine Lie algebras. As such, the labels of the \
\[OpenCurlyQuote]particles\[CloseCurlyQuote] in the (M)TC stem from the \
affine Lie algebras, that is, we use the Dynkin labels of the highest weights \
of the admissible representations, which are lists of length \
\[OpenCurlyQuote]rank\[CloseCurlyQuote], the rank of the associated Lie \
algebra. \n\nThe F-symbols are calculated by first calculating the qCG \
coefficients, as explained in [1]. The qCG coefficients are labelled by three \
highest weights (j1, j2, j3), three weights (m1,m2,m3) within the respected \
highest weight representation. To distinguish the weights in case of a weight \
multiplicity, we use an additional integer for each weight. Similarly, j3 can \
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\[CloseCurlyQuote]s and m\[CloseCurlyQuote]s are lists of length \
\[OpenCurlyQuote]rank\[CloseCurlyQuote], while the n\[CloseCurlyQuote]s and v \
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\[CloseCurlyQuote], or highest weight representations. Traditionally, one \
uses a,b,c,d,e,f, which is why we switch to this notation from now on and \
abandon the j\[CloseCurlyQuote]s. Apart from the six \
\[OpenCurlyQuote]particles\[CloseCurlyQuote], there are four vertices to be \
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notation for the F-symbols:\nFsym[a,b,c,d,e,f,{v1,v2,v3,v4}]\n\nThe fusion \
matrices are obtained by running:\nNmat[a]\n\nThe number of vertices of type \
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a x b, that is, ",
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calculated using the method of [1], section 6. This does not give all \
R-symbols, so the remaining ones are calculated by explicitly solving the \
hexagon equations. The notation for the R-symbols follows the one for the \
F-symbols:\nRsym[a,b,c,{v1,v2}]\n\nApart from the F- and R-symbols, the F- \
and R-matrices are also provided:\nFmat[a,b,c,d]\nRmat[a,b,c]\n\nIf the \
R-matrices are not diagonal, they can be diagonalized by running \
diagonalizermatrices[]. This will change both F- and R-symbols, which is why \
the pentagon and hexagon equations are checked again. One can revert back to \
the original F- and R-symbols by running undiagonalizermatrices[].\n\nWe \
should note that the F-matrices satisfy (in the chosen gauge, and provided \
that one does not (have to) diagonalize the R-matrices) the relation F . F^T \
= F^T . F = 1 (here, one only transposes, even if the F-symbols are \
imaginary). So, one can easily obtain the `inverse\[CloseCurlyQuote] \
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F-symbols are either real, or purely imaginary.\n\nVariables that are set by \
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      "True", " ", "if", " ", "the", " ", "theory", " ", "is", " ", 
       "unitary"}], ",", " ", 
      RowBox[{"False", " ", "otherwise"}]}]},
    {"modular", 
     RowBox[{"boolean", ",", " ", 
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  ExpressionUUID->"bb8878f9-ce84-4ff7-9b9f-8d81d210167b"],
 "\n\n(*) The T-matrix is defined ",
 StyleBox["without",
  FontWeight->"Bold",
  FontSlant->"Italic"],
 " the factor Exp[-2 Pi I/24 centralcharge], because at best, we obtain the \
central charge modulo 8, not 24. This is not a problem when checking the \
relation Exp[-3*2Pi I/24 centralcharge] (S.T)^3 = S^2 = C.\n\nThe unitarity \
is checked by checking if the quantum dimensions are all positive. It is \
conjectured that if this is the case, the theory is unitary. A warning is \
generated if the quantum dimensions are all positive, but the F-matrices are \
not all unitary (one should check if the F-matrices can be made unitary). \
Conversely, if the F-matrices are all unitary, but the quantum dimensions are \
not all positive, it is likely that there exists another pivotal structure, \
such that the quantum dimensions are all positive, even though the theory \
itself is not unitary. One can calculate all possible (spherical) pivotal \
structures by running  possiblepivotalstructures[] \
(possiblesphericalpivotalstructures[]) after calculating the F-symbols. Note \
that only one of these possible pivotal structures is actually realized by \
the selected quantum group. One can calculated the modular data associated \
with any valid ",
 StyleBox["spherical",
  FontWeight->"Regular",
  FontSlant->"Italic"],
 " pivotal structure by running calculatemodulardata[pivotalstructure]. If \
the selected pivotal structure differs from the one coming from the selected \
quantum group, the calculated modular data is not the one coming from the \
quantum group. \n \nWhen the theory is not unitary, the overall sign of the \
total quantum dimension is not determined because of the sign ambiguity of \
the square root, when calculating the total quantum dimension.  This leads to \
an overall sign ambiguity in the S-matrix, accompanied by the central charge \
being defined modulo 4 instead of eight. This sign is chosen to be positive, \
which in turn fixes the central charge modulo eight.\nThe variable \
centralcharge is set to the appropriate rational number if the theory is \
modular. Otherwise, it is set to a Missing[] object."
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Cell[TextData[{
 StyleBox["The exact representation",
  FontSize->18,
  FontWeight->"Bold"],
 "\n\nIt was observed that the numerically obtained F-symbols can be turned \
into exact expressions as follows. We represent q as q = e^(2 Pi i z/l), with \
z and l co-prime. Then, both the F- and R-symbols takes the form\nExp[I Pi \
\[Alpha]] Sqrt[ Sum[ a[i] Cos[2 Pi z/l]^i , {i, 0 , EulerPhi[l]/2 - 1 }] ],\n\
where \[Alpha] and the a[i] are rational. We note that for the F-symbols, \
\[Alpha] \[Element] {0, 1/2, 1, -1/2}. As stated above, one should not \
diagonalize the R-matrices if they are not diagonal, because this also \
changes the F-symbols, which might lead to F-symbols which can not be \
expressed in the exact form given above (one can \
\[OpenCurlyQuote]undiagonalize\[CloseCurlyQuote] the R-matrices by running \
undiagonalizermatrices[]).\n\nThe coefficients a[i] are obtained by making \
use of the Lenstra-Lenstra-Lov\[AAcute]sz integer relation algorithm, \
implemented in mathematica as LatticeReduce[]. After numerically calculating \
the F-symbols (R-symbols), one can obtain the exact form of these symbols by \
running findexactfsymbols[] (findexactrsymbols[]). findexactfsymbols[] \
obtains the exact form of the F-symbols, and checks the pentagon equations \
using a precision of 2*precision (with a default for precision = 100). \
findexactrsymbols[] obtains the exact form of the R-symbols, and, provided \
that the exact form of the F-symbols was obtained, checks the hexagon \
equations using a precision of 2*precision. The exact form of the F- and \
R-symbols can be obtained via Fsymexact[] and Rsymexact[]. The format is {\
\[Alpha], {a[0], a[1], a[2], ... }}.\n\nIf one calculates the exact form of \
the R-symbols before calculating the exact form of the F-symbols, the hexagon \
equations can not be checked using the exact form of the R-symbols, because \
the exact form of the F-symbols is also needed. However, one can check both \
the pentagon and hexagon equations numerically but using the exact form of \
the F- and R-symbols \[OpenCurlyQuote]by hand\[CloseCurlyQuote] at a later \
stage,  by using the commands checkpentagonexactformnumerically[] and \
checkhexagonexactformnumerically[].\n\nFinally, one can obtain the exact form \
of the modular data (provided the latter is calculated numerically first) by \
running findexactmodulardata[]. This sets the following variables (with \
content that should be clear from the table above)\n{FPdimlistexact, \
qdimlistexact, pivotlistexact, thetalistexact, FSlistexact, smatexact, \
tmatexact, cmatexact}.\n\nWe note that for some \[OpenCurlyQuote]larger\
\[CloseCurlyQuote] cases, it can happen that the default precision = 100, is \
not enough to obtain the exact form of the F- and R-symbols correctly. In \
this case, the check of the pentagon and hexagon equations using the exact \
form of the F- and R-symbols fails (that is, the maximum deviation is of the \
order of or higher than 10^(-precision), while the check is done with \
2*precision. If this happens, which for instance occurs for g_2 at level 3, \
one should increase the precision. For the case g_2 at level 3, precision = \
150 suffices.\n\nIt turns out that the phase of the S-matrix elements is not \
always of the form Exp[i  Pi \[Alpha]], with \[Alpha] a rational number. In \
these cases, the program tries to find a different exact representation of \
the phase, using a different integer number field (which is inspired by the \
rational coefficients a[i] for the absolute value of the given S-matrix \
element). It turns out that one can describe \[Alpha] in the following way.\n\
\[Alpha] = +/- ArcCos[{\[Beta], {b[0], b[1], b[2], ...}]/Pi, where {\[Beta], \
{b[0], b[1], b[2], ...} describes the argument using an exact representation \
similar to the one above, i.e., including the square root, but typically \
using a different number field than the one used for the F- and R-symbols \
(that is, using a different l, and z = 1); finally \[Beta] is 0 or 1. \
Concretely, in the case \[Alpha] is not a rational number, we represent the \
S-matrix element in the following way\n{{\[Gamma], {\[Beta], {b[0], b[1], \
b[2], ...}}, p}, {a[0], a[1], a[2], ...}}, with \[Gamma] = 1,-1 and p is a \
positive integer. This represents the S-matrix element as follows\nExp[I Pi \
\[Alpha]] Sqrt[ Sum[ a[i] Cos[2 Pi z/l]^i , {i, 0 , EulerPhi[l]/2 - 1 }]], \
where \[Alpha] is given by\n\[Alpha] = \[Gamma]/Pi ArcCos[ (-1)^\[Beta] \
Sqrt[Sum[b[i] Cos[2 Pi/p]^i ,  {i, 0, EulerPhi[p]/2 - 1}] ] ]\nIf it turns \
out that the (phase of the) S-matrix element can not be described exactly in \
this way either, \[Alpha] will be given as a Real with 200 digits precision \
(this is the default precision for the when loading the data from file, see \
below). No cases where this is necessary are known so far."
}], "Text",ExpressionUUID->"ed24c6c4-aae2-4af4-b48d-1093180a9187"],

Cell[TextData[{
 StyleBox["Saving and loading the data",
  FontSize->18,
  FontWeight->"Bold"],
 "\n\nOnce the exact form of the F-symbols, R-symbols and the modular data is \
calculated, one can save this data by using the command savecurrentdata[]. \
This saves the tensor category data in its exact form, but not, for instance, \
the q-Clebsch-Gordan coefficients, which are not obtained exactly. This data \
is stored (provided no errors occurred while calculating the data)  in the \
directory\nFileNameJoin[{$UserBaseDirectory, \
\[OpenCurlyDoubleQuote]ApplicationData\[CloseCurlyDoubleQuote], \
\[OpenCurlyDoubleQuote]alatc\[CloseCurlyDoubleQuote]}]\n(which is created if \
necessary).\nAt a later stage, this data can be loaded again by running\n\
loaddata[type, rank, level, rootfactor]\nwhich loads the exact data, and \
initializes the system, such that it is in a state equivalent to the state it \
would be in, if one had just calculated the exact form of the F- and \
R-symbols and the modular data (this pertains to the tensor category data, \
but not to most of the Lie-algebra data and the q-Clebsch-Gordan \
coefficients). The numerical values of the F- and R-symbols and the modular \
data, as given by FSym[] etc., are evaluated with 200 digits precision.\n\n\
One can also load the data by making use of the l and z values (see above) by \
running\nloaddatalz[type, rank, l, z]"
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Now, the same non-uniform case, but doing the initialization with the \
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An example with the exact form of the F- and R-symbols and modular data\
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