Graded matrix factorisations‌ Sample Clauses

Graded matrix factorisations‌. All the known MFs presented so far have quasi-homogeneous matrix entries, as do the MFs generated from exact sequences. Moreover they are graded. We will now define what being graded means and how it relates to R-charge and weight of variables. From the last subsection we know we have assigned weights to each variable such that the potential W is quasi homogeneous. We can therefore write. W (λ|x1|x1, ..., λ|xk|xm) = λ|W|W (x1, ..., xm) , λ ∈ C× := C \ {0} . (4.4) ∈ | | The weight of W (note this was previously called d) is set to 2, i.e. W = 2. The central charge c(W ) for a polynomial W C[x1, ..., xm] is then computed using the weights of the variables in the (quasi-homogeneous) polynomial according to the formula Σ m c(V ) = 3 (1 − |xi|) , (4.5) i=1 and is a rational number. Sometimes it is more convenient to work with cˆ = c/3. From our study of invertibility of matrix elements in previous sections we know that W ∈ m2 ⟨ ⟩ where m = x1, ..., xm is the maximal ideal of C[x] . We call a rank N MF Q of a potential W graded if there exists a diagonal 9 matrix called the grading matrix of Q such that U(λ)Q(λ|x1|x1, ..., λ|xk|xk)U —1(λ) = λQ(x1, ..., xk) (4.6) where U(λ) = diag(λg1 , ..., λg2N ) . (4.7) We say that Q has R-charge 1 since the power of λ in front of Q is 1. This makes sense as we require Q2 = W I2N , and the natural weight of the potential used in the LG Lagrangian is two (the correct R-charge from the point of view of supersymmetry). In this sense the R-charge of a graded matrix is analogous to the weight of a monomial (or polynomial). It is easily seen that the grading relates to the weights of the matrix elements in the following way. If we denote the weight of Qij by wij we can read off from the equation above that wij = gj − gi + 1 . (4.8) More generally, for a graded matrix of R-charge R this becomes wirj = gj − gi + R . (4.9) One can instantly see that the difference in weights between any two columns or rows is the same for all matrix elements in those columns or rows. It is also clear that any even morphism with matrix elements along the leading diagonal must have the same weight as the R-charge of that matrix. This severely restricts the possibilities for the quantum dimension formula and value of central charge for the two potentials. Note that any ma- trix element of any weight can be a zero entry. These simple relations only hold with a diagonal U, and will be very useful in setting up a search algorithm outlined later. We can use t...