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A library for algebra inside the bicategory of Landau-Ginzburg models.
Composition and fusion:
julia> using LandauGinzburgCategories, PolynomialRings
julia> @ring! ℚ[x,y,z]
julia> A = unit_matrix_factorization(x^2, x = y)
2×2 Arrax{ℚ[x,y,z],2}:
0 x^2 + x*y + y^2
-x + y 0
julia> B = unit_matrix_factorization(y^2, y = z)
2×2 Array{ℚ[x,y,z],2}:
0 y^2 + y*z + z^2
-y + z 0
julia> A ⨶ B
4×4 Array{ℚ[x,y,z],2}:
0 0 y^2 + y*z + z^2 x^2 + x*y + y^2
0 0 -x + y y + -z
-y + z x^2 + x*y + y^2 0 0
-x + y -y^2 + -y*z + -z^2 0 0
julia> fuse(A ⨶ B, :y)
2×2 Arrax{ℚ[x,y,z],2}:
0 x^2 + x*z + z^2
-x + z 0
Library of named potentials and of known orbifold equivalences between them:
julia> using LandauGinzburgCategories; LGLib = LandauGinzburgCategories.Library;
The Aₙ
-series of potentials:
julia> LGLib.A₅()
x^6 + y^2
julia> LGLib.A₅(x, y)
x^6 + y^2
julia> LGLib.A(5, x, y)
x^6 + y^2
Exceptional unimodular singularities:
julia> LGLib.E₆(x, y)
x^3 + y^4
Et cetera.
Known orbifold equivalences:
julia> LGLib.orbifold_equivalence(LGLib.A5, LGLib.A2A2)
.....
This library has not been released yet and should therefore be considered alpha-quality software.
If this library has been useful for your work, please cite it as https://arxiv.org/abs/1901.09019.