Necessary conditions for the existence of Morita Contexts in the bicategory of Landau-Ginzburg Models

A Morita context also called pre-equivalence data [4], is a generalization of Morita equivalence between categories of modules. Two rings $T$ and $S$ are called Morita equivalent if the categories of left $T-$Modules ($T-$Mod) and of left $S-$Modules $(S-$Mod) are equivalent. The prototype of Morita equivalent rings is provided by a ring $T$ and the ring of $n\times n$ matrices over $T$ (for details, see corollary 22.6 of [2]).
It is evident that if two rings are isomorphic, they are Morita equivalent. However, the converse is not true in general. There exist rings that are not isomorphic, yet are Morita equivalent (cf. p. 470 of [22]). In fact, it suffices to take a ring $T$ and the ring of $n\times n$ matrices over $T$. However, there is a partial converse which holds. Indeed, if two rings are Morita equivalent, then their centers are isomorphic. In particular, if the rings are commutative, then they are isomorphic ([22], p.494). Because of this result, Morita equivalence is interesting solely in the situation of noncommutative rings. For more details on Morita equivalence, see [2]. This notion can be generalized using the notion of Morita contexts.
Morita contexts were first introduced in the bicategory of unitary rings and bimodules as $6$-tuples $(A,B,M,N,\phi,\psi)$; where $A$ and $B$ are rings, $M$ is an A-B-bimodule, $N$ is a B-A-bimodule and $\phi:M\otimes N\rightarrow A$ and $\psi:N\otimes M\rightarrow B$ are homomorphisms satisfying $\phi\otimes M=M\otimes\psi$ and $N\otimes\phi=\psi\otimes N$. For a characterization of Morita context in this bicategory, see theorem 5.4 of [25]. Bass (cf. chap. 2, section 4.4 of [3]) proved that the Morita context $(A,B,M,N,\phi,\psi)$ is a Morita equivalence if and only if $M$ is both projective and a generator in the category of $A-$modules. Recall (cf. [22]) that if $A$ is a ring, an $A$-module $P$ is projective if for every surjective $A$-linear map $f:M\rightarrow N$ and every $A$-linear map $g:P\rightarrow N$ there is a unique $A$-linear map $h:P\rightarrow M$ such that $g=fh$.
Morita contexts were recently studied in many bicategories [25]. But they have not yet been studied in the bicategory $\mathcal{LG}_{K}$ of Landau-Ginzburg models (section 2.2 of [6]) over a commutative ring $K$ whose intricate construction ([8]) is reminiscent of, but more complex than that of the bicategory of associative algebras and bimodules. In this paper, we study the concept of Morita context in the bicategory of Landau-Ginzburg models. A Landau-Ginzburg model is a model in solid state physics for superconductivity. $\mathcal{LG}_{K}$ possesses adjoints (also called duals, cf. [8]) and this helps in explaining a certain duality that exists in the setting of Landau-Ginzburg models in terms of some specified relations (cf. page 1 of [8]). The objects of $\mathcal{LG}_{K}$ are potentials which are polynomials satisfying some conditions (definition 2.4 p.8 of [8]). But unlike [8], we do not restrict ourselves to potentials. It turns out that the authors of [8] used potentials to suit their purposes because even if we take the objects of $\mathcal{LG}_{K}$ to be polynomials rather than potentials and then apply the construction of $\mathcal{LG}_{K}$ given in [8], we obtain virtually the same bicategory except that we now have more objects. Thus in this paper, the objects of $\mathcal{LG}_{K}$ are simply polynomials.
There are many reasons for studying the notion of Morita context. The first reason is that it generalizes the very important notion of Morita equivalence. Another reason is that it is used to prove some celebrated results. For example, the Morita context which has been introduced in [23] was used since to prove Wedderburn theorem on the structure of simple rings [3]. Morita contexts were also used in [1] to obtain various results: Goldie’s theorem ([16], [17]) on the ring of quotients of semi-prime rings and as a specialization, Wedderburn’s structure theorems of semi-simple Artinian rings were obtained. Other applications, though sometimes not stated in an explicit form, can be found in various places (e.g. [18], p.75]).
In this paper, we use properties of matrix factorizations to give necessary conditions to obtain a Morita context between two objects of the bicategory $\mathcal{LG}_{K}$. Thus, our first main result is the following theorem.

Theorem A.
Let

• •

  $(R,f)$ and $(S,g)$ be two objects of $\mathcal{LG}_{K}$.

• •

  $X\in\mathcal{LG}_{K}((R,f),(S,g))=hmf(R\otimes_{K}S,g-f)$ i.e., $X:(R,f)\rightarrow(S,g)$ is a finite rank matrix factorization of $g-f$.

• •

  $Y\in\mathcal{LG}_{K}((S,g),(R,f))=hmf(S\otimes_{K}R,f-g)$ i.e., $Y:(S,g)\rightarrow(R,f)$ is a finite rank matrix factorization of $f-g$.
  such that $X\otimes Y$ and $Y\otimes X$ are finite rank matrix factorizations.

• •

  $\Delta_{f}\in\mathcal{LG}_{K}((R,f),(R,f))=hmf(R\otimes_{K}R,f\otimes id-id\otimes f)$ i.e., $\Delta_{f}:(R,f)\rightarrow(R,f)$ is a finite rank matrix factorization of $f\otimes id-id\otimes f$.

• •

  $\Delta_{g}\in\mathcal{LG}_{K}((S,g),(S,g))=hmf(R\otimes_{K}S,g\otimes id-id\otimes g)$ i.e., $\Delta_{g}:(S,g)\rightarrow(S,g)$ is a finite rank matrix factorization of $g\otimes id-id\otimes g$.

• •

  $\eta:X\otimes_{R}Y\rightarrow 1_{(R,f)}=\Delta_{f}$ and let $(P,Q)$ and $(R,T)$ be pairs of matrices representing respectively the finite rank matrix factorizations $X\otimes Y$ and $\Delta_{f}$.

• •

  $\rho:Y\otimes_{S}X\rightarrow 1_{(S,g)}=\Delta_{g}$ and let $(P^{\prime},Q^{\prime})$ and $(R^{\prime},T^{\prime})$ be pairs of matrices representing respectively the finite rank matrix factorizations $Y\otimes X$ and $\Delta_{g}$.

Then: A necessary condition for $\Gamma=(X,Y,\eta,\rho)$ to be a Morita Context is

$$\begin{cases}\eta^{1}=0,\;and\;\eta^{0}Q=0,\\ \rho^{1}=0,\;and\;\rho^{0}Q^{\prime}=0.\end{cases}$$

where for ease of notation, we wrote $\rho^{i}$ and $\eta^{i}$ respectively for the matrices of $\rho^{i}$ and $\eta^{i}$, $i=0,1$.

Next, thanks to a celebrated result (due to Schur [26], [27]) on determinants of block matrices, we observe that if $X$ and $X^{\prime}$ are respectively two matrix factorizations of two arbitrary polynomials, then the determinants of the four matrices appearing in the Yoshino tensor products $X\widehat{\otimes}X^{\prime}=(P,Q)$ and $X^{\prime}\widehat{\otimes}X=(P^{\prime},Q^{\prime})$ are all equal.
Moreover, when we translate this in $\mathcal{LG}_{K}$ where a $1-$morphism is a matrix factorization of the difference of two polynomials, we find out that those four determinants are all equal to zero. So our second main result is stated as follows:

Theorem B.
Let $(R,f)$ and $(S,g)$ be two objects in $\mathcal{LG}_{K}$ and let

$$X:(R,f)\rightarrow(S,g)\,\,and\,\,X^{\prime}:(S,g)\rightarrow(R,f)$$

be 1-morphisms in $\mathcal{LG}_{K}$. If $X\otimes X^{\prime}=(P,Q)$ and $X^{\prime}\otimes X=(P^{\prime},Q^{\prime})$, then

$$det(P)=det(Q)=det(P^{\prime})=det(Q^{\prime})=0$$

Thanks to this result we conclude that the necessary conditions earlier stated are not sufficient.
This paper is organized as follows: In the next section, we review the notion of matrix factorization. In section 3, we recall properties of matrix factorizations. Section 4 is a recall of the definition of the bicategory of Landau-Ginzburg models. The notion of Morita contexts in $\mathcal{LG}_{K}$ is discussed in section 5. Finally, we discuss further problems in the last section.

In this section, we first recall the definition of a matrix factorization and describe the category of matrix factorizations of a power series $f$. Next, the definition of Yoshino’s tensor product is recalled.

In 1980, Eisenbud came up with an approach of factoring both reducible and irreducible polynomials in $R$ using matrices. For instance, the polynomial $f=x^{2}+y^{2}$ is irreducible over the real numbers but can be factorized as follows:

$$\begin{bmatrix}x&-y\\ y&x\end{bmatrix}\begin{bmatrix}x&y\\ -y&x\end{bmatrix}=(x^{2}+y^{2})\begin{bmatrix}1&0\\ 0&1\end{bmatrix}=fI_{2}$$

We say that $(\begin{bmatrix}x&-y\\ y&x\end{bmatrix},\begin{bmatrix}x&y\\ -y&x\end{bmatrix})$ is a $2\times 2$ matrix factorization of $f$.

When $n=1$, we get a $1$ $\times$ $1$ matrix factorization of $f$, i.e., $f=[g][h]$ which is simply a factorization of $f$ in the classical sense. But in case $f$ is not reducible, this is not interesting, that’s why we will mostly consider $n>1$.
The original definition of a matrix factorization was given by Eisenbud [13] as follows: a matrix factorization of an element $x$ in a ring $A$ (with unity) is an ordered pair of maps of free $A-$modules $\phi:F\rightarrow G$ and $\psi:G\rightarrow F$ s.t., $\phi\psi=x\cdot 1_{G}$ and $\psi\phi=x\cdot 1_{F}$. Though this definition is valid for any arbitrary ring (with unity), in order to effectively study matrix factorizations, it is important to restrict oneself to specific rings. Working with specific rings makes it possible to easily give examples and it also allows one to carry out computations in a well-defined framework. Yoshino [29] restricted himself to matrix factorizations of power series. In this section, we will restrict ourselves to matrix factorizations of a polynomial.

We now propose a simple straightforward algorithm to obtain an $n\times n$ matrix factorization from one that is $m\times m$, where $m<n$.

Simple straightforward algorithm: Let $(P,Q)$ be an $m\times m$ matrix factorizations of a power series $f$. Suppose we want an $n\times n$ matrix factorization of $f$, where $n>m$.
Let $(i,j)$ stand for the entry in the $i^{th}$ row and $j^{th}$ column.

• •

  Turn $P$ and $Q$ into $n\times n$ matrices by filling them with zeroes everywhere except at entries $(i,j)$ where $1\leq i\leq m$ and $1\leq j\leq m$.
  Then for all entries $(k,k)$ with $k>m$, Either:

  1. 1.

     In $P$, replace the diagonal elements (which are zeroes) with $f$
     And

  2. 2.

     In $Q$, replace the diagonal elements (which are zeroes) with 1

  Or:
  Interchange the roles of $P$ and $Q$ in steps $1.$ and $2.$ above.

It is evident that this simple algorithm works not only for polynomials but also for any element in a unital ring.

The standard algorithm to factor polynomials using matrices is found in [10] and for an improved version see [14] and [15].

We will now state and prove some properties of matrix factorizations of polynomials. Some of them will be used when studying the notion of Morita Context between two objects in the bicategory $\mathcal{LG}_{K}$.
In this section, except otherwise stated $R=K[x_{1},\cdots,x_{n}]=K[x]$. All statements and proofs in this section are taken from [10] except for proposition 3.2. They are reproduced here (sometimes with slight modifications) for the purposes of completeness.

Since $P$ is invertible over $\mathcal{F}$, the unique $Q$ such that $PQ=fI_{n}$ is $Q=P^{-1}fI_{n}=\frac{f}{det(P)}adj(P)$, where $adj(P)$ is the adjoint of the matrix $P$.

Now, $adj(P)$ is a matrix over $R=K[x_{1},\cdots,x_{n}]$. So, if $det(P)$ divides $f$, then $Q$ will also be a matrix over $R$. However, it is possible for $Q$ not to have entries in $R$, and therefore $P$ will not appear in any matrix factorization of $f$ over $R$. We now prove that matrices appearing in a matrix factorization commute with each other.

We state and prove another property of matrix factorizations thanks to which we can conclude that an $n\times n$ matrix factorization of a polynomial $f$ is not unique.

Before we proceed, it is worthwhile stating some well-known facts in the literature, see notes on tensor products by Conrad (for points 1. and 2. below see theorem 4.9, example 4.11 of [9], point 3. simply generalizes point 2.).

We quickly recall the construction of the bicategory $\mathcal{LG}_{K}$ of Landau-Ginzburg models. In definition 5.5 of [14], a $B-$category is defined to be a structure having all requirements of a bicategory but without necessarily satisfying the unit morphisms requirement. In order to easily recall the construction of $\mathcal{LG}_{K}$, we first construct a $B-$category which we call $B-Fac$ and then we proceed to the unit construction. The objects of $B-Fac$ are polynomials $f$ denoted by pairs $(R,f)$ where $f\in R=K[x]$. Here we do not impose restrictions on the objects of our bicategory as was originally done in [8], where the authors instead consider potentials (Definition 2.4, p.8 of [8]). This generalization we do at the level of the objects does not pose a problem in the construction of $\mathcal{LG}_{K}$. The end result is simply a bicategory which has more objects than the original $\mathcal{LG}_{K}$ defined in [8], we still call it $\mathcal{LG}_{K}$.
We first recall the notion of linear factorizations which is an ingredient for the construction of $\mathcal{LG}_{K}$.

Since we are dealing with a $\mathbb{Z}_{2}-$grading and $d$ is odd, we can also say $d$ is a degree one map. $d$ is called a twisted differential in [7]. $d$ is actually a pair of maps $(d^{0},d^{1})$ that we may depict as follows:

and the stated condition on them is: $d^{0}\circ d^{1}=f\cdot id_{X^{1}}$ and $d^{1}\circ d^{0}=f\cdot id_{X^{0}}$.
If $X$ is a free $R-$module, then the pair $(X,d)$ is called a matrix factorization and we often refer to it by $X$ without explicitly mentioning the differential $d$.

Concretely (see page 19 of [21]), $\phi$ is a pair of maps and such that the following diagram commutes:

Let ($R=K[x],f$) and ($S=K[y],g$) be elements of $B-Fac$. The small category $B-Fac((R,f),(S,g))$ is defined as follows:

$$B-Fac((R,f),(S,g)):=hmf(R\otimes S,1_{R}\otimes g-f\otimes 1_{S})^{\omega}=hmf(K[x,y],g-f)^{\omega}$$

viz. a 1-morphism between two polynomials $f$ and $g$ is a matrix factorization of $g-f$.
Then given two composable $1-$cells $X\in B-Fac((R,f),(S,g))$ and $Y\in B-Fac((S,g),(T,h))$, we define their composition using Yoshino’s tensor product as discussed in subsection 2.2. $Y\circ X:=Y\otimes_{S}X$ $\in\,HMF(R\otimes_{K}T,1_{R}\otimes h-f\otimes 1_{T})$ which is a $\mathbb{Z}_{2}-$graded module, where:

$$(Y\otimes_{S}X)^{0}=(Y^{0}\otimes_{S}X^{0})\oplus(Y^{1}\otimes_{S}X^{1})\;\;and\;\;(Y\otimes_{S}X)^{1}=(Y^{0}\otimes_{S}X^{1})\oplus(Y^{1}\otimes_{S}X^{0}),$$

the differential ([8]) is

$$d_{Y\otimes X}=d_{Y}\otimes 1+1\otimes d_{X},$$

where the tensor product is taken over $S$ and $1\otimes d_{X}$ has the usual Koszul signs when applied to elements. That is;

$$d_{Y\otimes X}(y,x)=d_{Y}(y)\otimes x+(-1)^{i}y\otimes d_{X}(x)\,\,\,\,\,\,\,\,\,$$

(see   p.28 [21]) where $y\in Y^{i}$.
By remark 2.1.8 on p.29 of [6], $Y\otimes_{S}X$ is a free module of infinite rank over $R\otimes_{K}T$. However, the argument of Section 12 of [12] shows that it is naturally isomorphic to a direct summand in the homotopy category of something finite-rank. So, we may define $Y\circ X:=Y\otimes_{S}X$ $\in\,hmf(R\otimes_{K}T,1_{R}\otimes h-f\otimes 1_{T})^{\omega}=B-Fac((R,f),(T,h))$.

We now define the tensor product of morphisms of matrix factorizations. Let $X_{1}$, $X_{2}$ be objects of $B-Fac((R,f),(S,g))$ and $Y_{1}$, $Y_{2}$ be objects of $B-Fac((S,g),(T,h))$. Let $\alpha:X_{1}\rightarrow X_{2}$ and $\beta:Y_{1}\rightarrow Y_{2}$ be two morphisms, then we define their tensor product in the obvious way $\beta\otimes\alpha:Y_{1}\otimes X_{1}\rightarrow Y_{2}\otimes X_{2}$ in $B-Fac((R,f),(T,h))$.
With the above data, the composition (bi-)functor is entirely determined in our B-category:

$\star_{(R,f),(S,g),(T,h)}:B-Fac((R,f),(S,g))$ x $B-Fac((S,g),(T,h))\rightarrow B-Fac((R,f),(T,h))$

$$(X,Y)\mapsto Y\otimes_{S}X$$

The definition of the associativity morphism is easy to state. In fact, for $X\in B-Fac((R,f),(S,g))$, $Y\in B-Fac((S,g),(T,h))$ and $Z\in B-Fac((T,h),(P,r))$, the associator is the 2-isomorphism

$$a_{Z,Y,X}:Z\otimes(Y\otimes X)\rightarrow(Z\otimes Y)\otimes X$$

given by the usual formula

$$z\otimes(y\otimes x)\rightarrow(z\otimes y)\otimes x$$

where $x\in X$, $y\in Y$ and $z\in Z$.