degreeMatrix -- charge matrix Q whose rows generate ker A

Usage:

Q = degreeMatrix A

Q = degreeMatrix T
Inputs:
- A, a matrix, a $d \times N$ integer configuration matrix
- T, an instance of the type Triangulation,
Outputs:
- a matrix, an integer matrix of shape $(N - d) \times N$ whose rows are a $\mathbb{Z}$-basis of $\ker A$

Description

The charge matrix is computed as transpose syz A. In the GLSM/Calabi-Yau context the rows of $Q$ are the U(1) charges of the chiral fields under the gauge symmetries. Mathematically, $Q$ provides explicit coordinates on the quotient $\mathbb{R}^N / \mathrm{rowspan}(A)$, which is where the secondary cone naturally lives.

For $T = $ triangulation(A, tri), degreeMatrix T uses the stored configuration matrix matrix T (which is $A$ auto-homogenized to one extra row of $1$'s when $A$ was supplied as a point set).

Users who want a reduced basis (for instance, an LLL-reduced one with smaller integer entries) can compute it themselves from syz A and pass it to secondaryCone via the DegreeMatrix option.

i1 : A = transpose matrix {{0,0},{1,0},{0,1},{1,1}}

o1 = | 0 1 0 1 |
     | 0 0 1 1 |

              2       4
o1 : Matrix ZZ  <-- ZZ

i2 : T = triangulation(A, {{0,1,3},{0,2,3}})

o2 = triangulation {{0, 1, 3}, {0, 2, 3}}

o2 : Triangulation

i3 : degreeMatrix T

o3 = | 1 -1 -1 1 |

              1       4
o3 : Matrix ZZ  <-- ZZ

Ways to use degreeMatrix:

degreeMatrix(Matrix)
degreeMatrix(Triangulation)

For the programmer

The object degreeMatrix is a method function.

The source of this document is in Triangulations.m2:2264:0.

degreeMatrix -- charge matrix Q whose rows generate ker A

Description

See also

Ways to use degreeMatrix:

For the programmer