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.
| Identifer | oai:union.ndltd.org:uottawa.ca/oai:ruor.uottawa.ca:10393/39913 |
| Date | 03 December 2019 |
| Creators | Fomatati, Yves Baudelaire |
| Contributors | Blute, Richard |
| Publisher | Université d'Ottawa / University of Ottawa |
| Source Sets | Université d’Ottawa |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
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