Multiplicative Tensor Product of Matrix Factorizations and Some Applications

• Main Author: Fomatati, Yves Baudelaire
• Other Authors: Blute, Richard
• Format: Others
• Language: en
• Published: Université d'Ottawa / University of Ottawa 2019
• Subjects: Matrix Factorizations; Multiplicative Tensor Product; Applications; Algorithm for Matrix Factorization of Polynomials
• Online Access: http://hdl.handle.net/10393/39913; http://dx.doi.org/10.20381/ruor-24152

An n × n matrix factorization of a polynomial f is a pair of n × n matrices (P, Q) such that PQ = f In, where In is the n × n identity matrix. In this dissertation, we study matrix factorizations of an arbitrary element in a given unital ring. This study is motivated on the one hand by the construction of the unit object in the bicategory LGK of Landau-Ginzburg models (of great utility in quantum physics) whose 1−cells are matrix factorizations of polynomials over a commutative ring K, and on the other hand by the existing tensor product of matrix factorizations b⊗. We observe that the pair of n × n matrices that appear in the matrix factorization of an element in a unital ring is not unique. Next, we propose a new operation on matrix factorizations denoted e⊗ which is such that if X is a matrix factorization of an element f in a unital ring (e.g. the power series ring K[[x1, ..., xr]] f) and Y is a matrix factorization of an element g in a unital ring (e.g. g ∈ K[[y1, ..., ys]]), then Xe⊗Y is a matrix factorization of f g in a certain unital ring (e.g. in case f ∈ K[[x1, ..., xr]] and g ∈ K[[y1, ..., ys]], then f g ∈ K[[x1, ..., xr , y1, ..., ys]]). e⊗ is called the multiplicative tensor product of X and Y. After proving that this product is bifunctorial, many of its properties are also stated and proved. Furthermore, if MF(1) denotes the category of matrix factorizations of the constant power series 1, we define the concept of one-step connected category and prove that there is a one-step connected subcategory of (MF(1),e⊗) which is semi-unital semi-monoidal. We also define the concept of right pseudo-monoidal category which generalizes the notion of monoidal category and we prove that (MF(1),e⊗) is an example of this concept. Furthermore, we define a summand-reducible polynomial to be one that can be written in the form f = t1 + · · · + ts + g11 · · · g1m1 + · · · + gl1 · · · glml under some specified conditions where each tk is a monomial and each gji is a sum of monomials. We then use b⊗ and e⊗ to improve the standard method for matrix factorization of polynomials on this class and we prove that if pji is the number of monomials in gji, then there is an improved version of the standard method for factoring f which produces factorizations of size 2 Qm1 i=1 p1i+···+ Qml i=1 pli−( Pm1 i=1 p1i+···+ Pml i=1 pli) times smaller than the size one would normally obtain with the standard method. Moreover, details are given to elucidate the intricate construction of the unit object of LGK. Thereafter, a proof of the naturality of the right and left unit maps of LGK with respect to 2−morphisms is presented. We also prove that there is no direct inverse for these (right and left) unit maps, thereby justifying the fact that their inverses are found only up to homotopy. Finally, some properties of matrix factorizations are exploited to state and prove a necessary condition to obtain a Morita context in LGK.