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Non-commutative Hodge structures: Towards matching categorical and geometric examples

The subject of the present work is the de Rham part of non-commutative Hodge structures on the periodic cyclic homology of differential graded algebras and categories. We discuss explicit formulas for the corresponding connection on the periodic cyclic homology viewed as a bundle over the punctured formal disk. Our main result says that for the category of matrix factorizations of a polynomial the formulas reproduce, up to a certain shift, a well-known connection on the associated twisted de Rham cohomology which plays a central role in the geometric approach to the Hodge theory of isolated singularities.


Introduction
The study of various generalizations of the classical Hodge theory has evolved into a vast branch of mathematics; a comprehensive overview of relevant notions and references can be found in [22]. The present work is in the framework of the Hodge theory of categories, as outlined in [1,2,12,15,16]. We refer the reader to these sources for an introduction to the circle of ideas surrounding the subject, as well as motivation and references. Our main goal is to present a piece of evidence (see Theorem  The periodic cyclic homology of a differential graded (dg) category is well-known to be a direct generalization of the de Rham cohomology of a space, and it is natural to expect it to carry a Hodge-like structure. This expectation has been converted into a precise conjecture in [12,Section 2.2.2], at least in the case of the so-called homologically smooth and proper dg categories which are to be thought of as analogs of smooth and proper varieties. According to the conjecture, the cyclic homology of such a category can be endowed with a non-commutative Hodge structure. Despite the terminology, non-commutative Hodge structures are defined without referring to non-commutative mathematics; they are a subject of their own, with a number of geometric applications [8,10,12].
The definition of a non-commutative Hodge structure involves two sets of data called the "de Rham data" and the "Betti data" in [12]. Roughly, the former generalizes the Hodge filtration of a classical Hodge structure (cf. section 3.5) while the latter is an analog of the Q-structure. The main obstacle to proving the aforementioned conjecture seems to have been the fact that the periodic cyclic homology of, say, a C-linear dg category does not carry an obvious Q-structure in general. More on this can be found in [12,Section 2.2.6] and [11,Section 8]. The present work concerns the de Rham part of the sought-for non-commutative Hodge structure on the cyclic homology whose origin has been understood for some time now.
The de Rham data of a non-commutative Hodge structure can be defined as a pair (H, ∇) where H is a Z/2-graded free C{u}-module of finite rank and ∇ is a meromorphic connection on H with a pole of order at most 2 at the origin. There is a formal counterpart of such data in which the ring C{u} of convergent series is replaced with the ring C [[u]] of formal series. We will be considering only the formal analog of the de Rham data due to the very nature of our main example, the cyclic homology.
In what follows, the term "bundle on the formal (resp. punctured formal) disk" is used as a synonym of "free C [[u]]-module of finite rank (resp. finite-dimensional C((u))-vector space)".
Accordingly, we will speak of "bundles with connection (on the formal disk)" instead of "the de Rham data of non-commutative Hodge structures".
A general way of producing such bundles with connections was outlined in [12,Section 4.2.4].
Let us reproduce this idea here using three examples.
Let X be a compact Kähler manifold and A(X) the Z/2-graded space of complex C ∞ -forms on X, with the Z/2-grading given by the parity of the degree of differential forms. Consider the Z/2-graded C [[u]]-linear complex (A(X) [[u]],∂ + u∂) where A(X) [[u]] stands for the space of formal power series with coefficients in A(X). By the classical Hodge theory, its cohomology H * (A(X) [[u]],∂ + u∂) is a vector bundle on the formal disk of rank dim C H DR (X). It carries a meromorphic connection defined as follows. First, define ∇ X : A(X)((u)) → A(X)((u)) by , Γ ′ ∈ End(A(X)), Γ ′ | A p,q (X) = q − p and then observe that (1.1) [∇ X ,∂ + u∂] = 1 2u (∂ + u∂) Therefore, ∇ X induces a connection on the cohomology H * (A(X)((u)),∂ + u∂), the latter being the restriction of the vector bundle H * (A(X) [[u]],∂ + u∂) to the punctured formal disk.
For our second example, let us take a polynomial w = w(y 1 , . . . , y k ) on Y = C k such that (1.2) w(0, . . . , 0) = ∂ 1 w(0, . . . , 0) = . . . = ∂ k w(0, . . . , 0) = 0 We will assume that the origin is the only critical point of w. To produce a bundle with connection associated with w we will mimic the above construction for Kähler manifolds. Namely, let Ω(Y ) be the Z/2-graded space of (holomorphic) differential forms with polynomial coefficients on Y . Consider the so-called twisted de Rham complex (Ω(Y ) [[u]], −dw + ud) where d is the (holomorphic) de Rham differential and dw is the operator of wedge multiplication with dw (as before, we view it as a Z/2-graded C [[u]]-linear complex). It is a classical fact that The latter example is closely related to the subject of the Gauss-Manin systems and the Brieskorn lattices (see section 5) and through that to the study of generalized Hodge structures and Frobenius manifolds associated with isolated singularities (see [9,23] for review and references).
Our last example is algebraic. Let A be a differential Z/2-graded algebra. There are two possible options at this point: either we work with those dg algebras for which the torsion submodules do not occur (conjecturally, this is so for any homologically smooth and proper dg algebra [16]), or we replace the cohomology of (C(A) [[u]], b + uB) by its image in the cohomology of (C(A)((u)), b + uB), the periodic cyclic homology of A. Choosing the first option is analogous to working with compact Kähler manifolds only. Then, as we discussed previously, there is a chance that the resulting bundles with connections can be promoted to fullfledged non-commutative Hodge structures. We will keep considering arbitrary dg algebras or, more precisely, dg algebras with finite-dimensional periodic cyclic homology; the latter condition guarantees that we are still dealing with finite rank vector bundles on the formal disk.
The periodic cyclic homology, viewed as a bundle on the punctured formal disk, should carry a connection by a general argument ([16, Section 11.5],[12, Section 2.2.5]) involving the socalled non-commutative Gauss-Manin connection [7] (along the parameter of a one-parameter deformation of A). In order to get a connection at the level of the periodic cyclic complex one should, perhaps, repeat the same argument but for a refined version of the non-commutative Gauss-Manin connection obtained in [28] (see also [5]). We hope to return to this idea on another occasion. At the moment, borrowing an idea from [16,Section 11.5], we can write out a connection ∇ A on the periodic cyclic complex of A satisfying the same property as in our geometric examples above: An explicit formula for ∇ A can be found in section 3.3 (Proposition 3.4). The origin of the formula is easy to explain. The reader familiar with the non-commutative differential calculus will notice that ∇ A is built by combining a grading operator on C(A) with the non-commutative Cartan homotopy operator (cf. [7,Section 2]) corresponding to the differential d of A. Then the relation (1.5) for this connection is a slight variation on the non-commutative Cartan homotopy formula [7, (2.1)] (or, rather, a very special case of the latter which goes back to [19]).
Since ∇ A is written in terms of some basic operations on the Hochschild complex, it can be easily generalized to define a connection ∇ D on the cyclic complex of an arbitrary differential Z/2-graded category D.
Notice that when the Z/2-grading on A can be lifted to a Z-grading, there is another obvious connection on the cyclic complex satisfying (1.5), namely where Γ ′ is the corresponding Z-grading operator on C(A). It turns out that ∇ A and ∇ A gr coincide at the cohomological level (see 1 section 3.4). This observation shows that the de Rham part of non-commutative Hodge structures on the periodic cyclic homology is not very interesting in the case of Z-graded categories (in other words, the connections are not essential for developing the Hodge theory in the Z-graded setting; for more on this, see section 3.5). On the other hand, this same observation implies that our first geometric example is a special case of the algebraic one, at least, when X is a smooth projective variety. Let us present the argument in a very sketchy manner since this is known, and is not the main point of the paper anyway.
The claim is that there is an isomorphism of bundles on the formal disk (here par dg X stands for the dg category of perfect complexes 2 on X) which induces an isomorphism of connections 3 (H * (A(X)((u)),∂ + u∂), ∇ X ) ∼ = (H * (C(par dg X)((u)), b + uB), ∇ par dg X ) To see this, note first that the category par dg X is Z-graded and therefore we can replace ∇ par dg X with ∇ par dg X gr . Furthermore, since the left-hand sides are known to be free C[[u]]-modules. The right-hand sides can be endowed with meromorphic connections given by the same formulas that define ∇ X and ∇ par dg X gr and the isomorphisms above can be easily chosen so that they preserve the connections. To conclude the argument, we observe that the two Γ ′ s -in the definitions of ∇ X and ∇ par dg X gr , respectively -match under a well-known isomorphism The goal of the present paper is to explain why our second geometric example is a special case of the algebraic one. Namely, we will prove 1 If we knew that ∇ A was the connection that the aforementioned argument produced then this statement would also follow from the argument.
2 See [14, Section 5.3] and references therein for a discussion of the cyclic homology of this dg category. 3 Let us use the same notation for connections at the level of complexes and at the level of cohomology.
which induces an isomorphism of connections Here MF(w) stands for the dg category of matrix factorizations of w.
Let us omit the definition of MF(w) since we will not need it. We will use as a black box a result of [6] which says that this dg category is (quasi-)equivalent to the category of dg modules over a certain differential Z/2-graded algebra, A w . This result allows us to replace the cyclic complex of the category MF(w) with the much smaller cyclic complex of A w . The dg algebra is actually quite simple, so let us reproduce its definition here.
As a Z/2-graded algebra, it is the tensor product space of polynomials in k odd variables θ 1 , . . . , θ k . The only part of A w that depends on w is the differential. The differential depends, in fact, on more than just w itself 4 : one also needs to pick a decomposition of w of the form w = y 1 w 1 + . . . + y k w k Then the differential on A w is simply the commutator with D(w) := i y i ∂ ∂θ i + w i θ i . Clearly, D(w) 2 = w and so d w = [D(w), −] squares to 0.
The second key ingredient used in the proof of Theorem 1.1 is an explicit quasi-isomorphism from the Hochschild complex of MF(w) to the complex (Ω(Y ), −dw) constructed recently in [26] by combining results of loc.cit. with some facts obtained earlier in [4,6]. We will need only the composition of this quasi-isomorphism with the embedding of the Hochschild complexes The composition is still a quasi-isomorphism, by the aforementioned result of [6].
It is worthwhile to emphasize that the results from [6,26] we need (and, as a consequence, is the local ring at the origin, and w satisfies (1.2) and has an isolated singularity at the origin. For the sake of simplicity we will stick to the polynomial algebra and will not attempt to present the results in their "most natural" generality. 4 Thus, our notation Aw is a bit misleading.
To conclude, let us point out that Theorem 1.1 has some implications for the classical singularity theory (not related to non-commutative Hodge structures). These will be discussed briefly in section 5.
Organization of the paper. The main body of the paper comprises four sections.
In section 2 we develop some simple convenient formalism of mixed complexes with connections. It can be viewed as a step toward defining the derived category of mixed complexes with connections but we do not pursue this idea due to lack of motivation.
In section 3 we recall the definition of the mixed cyclic complexes of dg algebras, both unital and non-unital, and then move on to describing connections on these complexes. We present several equivalent sets of formulas for the connections and discuss separately the case of differential Z-graded algebras. The section is concluded with an explanation of why the language of connections is suitable for developing the Hodge theory in the categorical framework.
In section 4 we explain the proof of Theorem 1.1.
Section 5 is devoted to some applications of Theorem 1.1.
In order to keep the main body of the text as short as possible, we have collected all the proofs in several appendices which occupy most of the paper.
Conventions. All complexes and dg algebras in this paper are Z/2-graded, unless we explicitly say otherwise. That is, our complexes are pairs (C, d), where C is a Z/2-graded vector space C = C even ⊕ C odd and d is an odd operator such that d 2 = 0, and our dg algebras are such complexes equipped with a compatible multiplication.
In sections 2.1-3.4 we are working over a ground field k whose characteristic is not equal to 2. Starting from section 3.5 the ground field is C. We will be also considering complexes over the field of formal Laurent series but it will be clear from the context (or an explicit comment) which of the fields is important at a given moment.
Acknowledgements. I would like to thank Y. Soibelman for explaining some basics of the categorical Hodge theory to me several years ago. I am also grateful to M. Herbst, C. Hertling, D. Murfet, A. Polishchuk, H. Ruddat, E. Scheidegger, C. Sevenheck, and K. Wendland for inspiring discussions on matrix factorizations and Hodge theory. Special thanks are due to C. Sevenheck for a number of interesting comments on the subject matter of this paper, to the anonymous referee for very valuable suggestions on how the exposition could be improved, and to both of them for spotting several typos.

Formalism of mixed complexes and u-connections
plex and B is an odd operator on C that anti-commutes with b and squares to 0. In the Z/2 setting, the mixed complexes are merely double complexes but the use of the term "mixed" suggests the analogy with the mixed complexes in the conventional Z-graded setting [17], and also emphasizes the fact that b and B play different roles.
Recall that we are using u to denote a formal (even) variable, playing the role of a coordinate on the formal disk, and V ((u)) to denote the k((u))-vector space of formal Laurent series with coefficients in V where V is any k-vector space. Given a mixed complex (C, b, B), we will call the k((u))-linear complex (C((u)), b + uB) the u-totalization of (C, b, B).
As it should be clear from the Introduction, the mixed complexes and their u-totalizations are used as means to construct vector bundles over the formal disk. With this in mind, we will restrict ourselves to mixed complexes of finite type, i.e. those complexes whose u-totalizations have finite-dimensional cohomology over k((u)).
A morphism of mixed complexes is defined as a morphism of the underlying ordinary complexes that commutes with the B-operators. We will use also the following weaker notion 5 : a u-morphism of mixed complexes is a k((u))-linear morphism of the corresponding u-totalizations (as complexes). Any u-morphism f (u) : (C, b, B) → (C ′ , b ′ , B ′ ) can be written in the form where the coefficients f i : C → C ′ are even operators.
The reader, we hope, understands how to extend this terminology to other notions used in the study of complexes, such as quasi-isomorphisms, homotopy equivalences, etc. For instance, two u-morphisms f (u) and g(u), with the same domain and range, will be called u-homotopic if there is an odd k((u))-linear operator H(u) : C((u)) → C ′ ((u)) (a u-homotopy) such that Again, each u-homotopy has the form This is a version of the S-morphisms [17].
where k ∈ Z and H i : C → C ′ are odd operators. Accordingly, two mixed complexes (C, b, B) and (C ′ , b ′ , B ′ ) will be called u-homotopy equivalent if there are u-morphisms such that ι(u)p(u) and p(u)ι(u) are u-homotopic to id C ′ and id C , respectively.
It is a standard fact that for complexes of vector spaces there is not much difference between homotopy equivalences and quasi-isomorphisms. This fact applies to our setting since we are working with complexes of vector spaces over k((u)): satisfying the properties: is a k((u))-linear operator from C((u)) to itself) and The equality (2.2) implies that ∇ descends to the cohomology of the u-totalization, thereby giving rise to a connection on the corresponding bundle on the punctured formal disk.
Remark 2.2. Notice that since ∇ does not commute with the differential, a u-connection on a mixed complex is not a Z/2-graded complex of D-modules on the punctured formal disk, in the conventional sense. However, with any mixed complex with a u-connection one can associate a Z-graded (unbounded) complex of D-modules as follows. Let (C, b, B, ∇) be as above and D n = D n (C, b, B, ∇) be the sequence of D-modules given by D n = C even ((u)), ∇| Ceven ((u)) n 2 , n even C odd ((u)), ∇| C odd ((u)) n 2 , n odd where (V ((u)), ∇) n 2 := (V ((u)), ∇ − n 2 · 1 u ), a Tate twist of (V ((u)), ∇) (cf. [12, Section 2.1.7]). Then by (2.2) the differential b + uB : D n → D n+1 is a morphism of D-modules for all n. We will return to this construction in section 3.5; apart from that section, it will not be used in what follows.
To define morphisms of mixed complexes with u-connections, we need ) be mixed complexes with u-connections. Then for any u-morphism f (u) of the mixed complexes Proof is given in section A.1.
We will say that f (u) defines a morphism of mixed complexes with u-connections, or simply a morphism of u-connections, if (2.4) is u-homotopic to 0.
As the next claim shows, being a morphism of u-connections is stable under passing to uhomotopic u-morphisms. is a morphism of u-connections then so is g(u).
Proof is given in section A.2.
The following statement follows immediately from the definitions: Proposition 2.5. The composition of morphisms of u-connections is a morphism of u-connections.
We will say that two mixed complexes with u-connections (C, b, B, ∇) and (C ′ , b ′ , B ′ , ∇ ′ ) (or simply two u-connections ∇ and ∇ ′ ) are homotopy gauge equivalent if there is a u-homotopy equivalence (2.1) such that p(u) and ι(u) are morphisms of u-connections.
By Proposition 2.5, homotopy gauge equivalence is an equivalence relation.
is a u-quasi-isomorphism and a morphism of u-connections then ∇ and ∇ ′ are homotopy gauge equivalent.
Proof is given in section A.3.
Homotopy gauge equivalence can be refined as follows: We may require it to be furnished by regular u-homotopy equivalences, in the sense of the previous section. This is similar to the notion of holomorphic equivalence in the classical theory of connections. In all the examples we will consider in the remaining sections, the u-homotopy equivalences will be regular, and so in all the statements we will formulate homotopy gauge equivalence can be understood in the above stronger sense.
A very special instance of homotopy gauge equivalence is the case when one u-connection is obtained from another one, living on the same mixed complex, by adding a u-endomorphism that is u-homotopic to 0 (in this case the two u-connections are homotopy gauge equivalent with respect to the identity endomorphism of the mixed complex). Since there is not any actual "gauging" in this cases, we will simply say that two such u-connections are equal up to a u-homotopy.
We will conclude this section with the following result which allows one to transfer u-connections using u-homotopy equivalences.
Proposition 2.7. Given a u-homotopy equivalence (2.1) and a u-connection ∇ ′ = d du + A ′ (u) on (C ′ , b ′ , B ′ ), there exists a u-connection ∇ on (C, b, B) that is homotopy gauge equivalent to ∇ ′ with respect to p(u) and ι(u). Explicitly, one can set Proof is given in section A.4.
3. u-Connections on the cyclic complexes of dg algebras 3.1. The cyclic complexes of unital dg algebras. Let A be a (not necessarily unital) dg algebra and sA stand for A with the reversed Z/2-grading. Given a ∈ A, the corresponding element of sA will be denoted by sa. The parity of a will be denoted by |a|; thus, |sa| = |a| − 1. Let A ⊗ sA ⊗n equipped with the induced Z/2-grading. We will write the elements of A ⊗ sA ⊗n as a 0 [a 1 |a 2 | . . . |a n ] (i.e. a 0 [a 1 |a 2 | . . . |a n ] = a 0 ⊗ sa 1 ⊗ sa 2 ⊗ . . . ⊗ sa n ), or simply a 0 , if n = 0.
Throughout the paper, we will use the following convention: for an operator T from C(A) to anywhere we will write T n+1 for the restriction of T onto A ⊗ sA ⊗n . Let τ denote the cyclic permutation on C(A): One can easily see that τ n+1 n+1 = 1. Set Explicitly, Then the Hochschild differential b is defined as That b squares to 0 can be deduced form the formulas we write out in section B. The cohomology of (C(A), b) is called the Hochschild homology of A.
Let us recall now the definition of the mixed complex that computes the cyclic homology of (unital) dg algebras. The underlying ordinary complex is (C(A), b), and B is defined as follows.
Assume A is unital. Set Clearly, B is an odd operator that squares to 0. Again, the formulas from section B imply that B anti-commutes with both b(δ) and b(µ). The cohomology of the u-totalization of this mixed complex is called the periodic cyclic homology of A.

3.2.
The cyclic complexes of non-unital dg algebras. Notice that when A is non-unital, the definition of B from the previous section does not make sense. To fix the problem, one replaces the Hochschild chain complex with a quasi-isomorphic one, which contains a replacement for the unit. The aim of the present section is to recall the definition of this new complex.
Let A be a dg algebra, with or without unit. Denote by A + the dg algebra obtained from A by adjoining a unit: Consider the following Z/2-graded vector space: The operators δ The vector space C e (A) admits the following decomposition: With (3.4) in mind, we can represent the elements of C e (A) by column-vectors with two components, the first component being an element of C(A) and the second one an element of C + (A). Accordingly, operators on C e (A) can be represented by 2 × 2 matrices. For example, the differential b e (δ) preserves the decomposition (3.4) and therefore can be represented by the The operator µ (0) , on the other hand, preserves C(A) but maps C + (A) to C(A); it is then represented by the matrix Note that, formally, the two copies of b(δ) in the first matrix (or µ (0) in the second matrix) represent different operators, since they have different domains/ranges.
In this notation, the differential b e (µ) is easily seen to correspond to the following upper- Then, in our matrix notation, B e : C e (A) → C e (A) is given by Let A be a unital dg algebra. Consider the following u-morphisms (the operator hµ (0) in the last line maps e[a 0 |a 1 |a 2 | . . . |a n ] to 1[a 0 |a 1 |a 2 | . . . |a n ]). 7 We extend our previous convention regarding subscripts as follows: For an operator T from C e (A) to anywhere we will write Tn+1 for the restriction of T onto (A ⊗ sA ⊗n ) ⊕ (ke ⊗ sA ⊗n ). Proposition 3.1. ι(u) and p(u) establish a u-homotopy equivalence between (C(A), b, B) and Proof is given in section C.1.
The last result in this section relates the cyclic complexes of a dg algebra A and its opposite Consider the following isomorphism of Z/2-graded vector spaces: Proof is based on the following properties of Φ, which are easy to verify using the definitions of the maps involved: It follows from these properties that which implies the statement.
To conclude our discussion of the cyclic complexes, let us point out a variation on the definition of (C(A), b, B) and (C e (A), b e , B e ) which we will use in section 4. Namely, in the definition of C(A) (or C e (A)) one can replace the direct sums by the direct products: (and a similar version for C e (A) which we will denote by C Π, e (A)). Then all the above definitions (e.g. those of the differentials b, B) and conclusions (e.g. Proposition 3.1) are easily checked to hold true in the new setting. We refer the reader to [18] for a detailed study of these versions of the Hochschild complexes.

Canonical u-connections on the cyclic complexes. Let us omit a detailed discussion
on what the adjective "canonical" in the title refers to. In plain words, by a canonical uconnection one should understand a u-connection given by a universal formula, i.e. a formula that works simultaneously for all dg algebras. Such a u-connection can be extended to the cyclic complexes of dg categories thereby giving rise to a functor from the category of (small klinear) dg categories to that of mixed complexes with connections which has various reasonable properties (i.e. Morita equivalent dg categories correspond to homotopically gauge equivalent u-connections, etc.).
Observe that any canonical u-connection has a "twin" obtained by means of the morphism Φ from the previous section (see Proposition 3.2). Namely, suppose is a canonical u-connection. The following statement is an immediate consequence of Proposition 3.2.
is also a canonical u-connection.
Let us call this u-connection dual to the original one. The rest of this section is devoted to one example of a canonical u-connection. As we will show, it is self-dual (up to a u-homotopy).
Let A be a (not necessarily unital) dg algebra. Consider the following operators on C e (A): Proposition 3.4.
is a u-connection on (C e (A), b e , B e ); its dual is given by Proof is given in section C.2.
Proposition 3.5. ∇ and ∇ • are equal up to a u-homotopy.
Proof is given in section C.3.
Let us point out two corollaries of Proposition 3.4. To formulate the first corollary, we need two more operators on C e (A): Proof is given in section C.4.
The second corollary concerns a canonical u-connection on C(A) in the case when A has unit. Consider the following operators on C(A): Corollary 3.7.
is a u-connection on (C(A), b, B), homotopy gauge equivalent to ∇.
Proof is given in section C.5.
In section 4, we will apply the above results to the dg algebra A w defined in the Introduction.
In that example, it will be important for us to work with the version of the cyclic complex defined at the end of section 3.2, namely, C Π, e (A). Let us therefore emphasize the obvious fact that the above u-connections extend to this complex.

3.4.
On the case of differential Z-graded algebras. Let us assume that A is Z-graded or, more precisely, that its Z/2-grading can be lifted to a Z-grading.
Let us denote the Z-degree of an elements a ∈ A by deg(a). Clearly, the Z/2-grading on the unitalization A e of A also lifts to a Z-grading, if we set deg(e) = 0. Therefore, the space C e (A) has a natural Z-grading determined by the following operator With respect to this grading b e and B e have degrees 1 and −1, respectively, i.e.
This shows that Proposition 3.8. The u-connections ∇ and ∇ gr are equal up to a u-homotopy.
Proof is given in section C.6.
An important Hodge theoretic implication of this result will be explained in the next section.
3.5. Non-commutative Hodge filtrations. One of the goals of the present work is to contribute to the idea that the canonical u-connections form the correct framework for developing the Hodge theory of dg algebras and dg categories (at least, over the field of complex numbers).
However, in the case of differential Z-graded categories the Hodge theory can be formulated in an alternative way, with no mention of connections (see, for instance, [11]). Proposition 3.8 implies that the two approaches are equivalent, and the aim of this section is to explain this in more detail.
Let us start by recalling a description of the periodic cyclic homology of a differential Z-graded category as a 2-periodic sequence of filtered vector spaces [11,Sections 2,3] where the filtrations are thought of as generalizing the classical Hodge filtration. The ground field in this section is C.
Let D be a differential Z-graded category. Then the Hochschild complex of D carries a Z-grading 8 (cf. section 3.4), and one can consider a refined version of the u-totalization of (C(D), b, B), namely, (C • (D)((u)) gr , b + uB) where u is a formal variable of degree 2 and "gr" means that we look at the subspace in C • (D)((u)) spanned by homogeneous series. The cohomology groups HP • (D) of the resulting Z-graded complex are known as the periodic cyclic homology groups of D. Clearly, all the groups HP 2k (D) resp. HP 2k+1 (D) are isomorphic to each other.
In the example D = par dg X (see the Introduction) where X is a smooth projective complex variety we have [13,29]: For simplicity, we will assume from now on that the periodic cyclic homology groups of our categories are finite-dimensional.
The spaces HP • (D) come equipped with canonical decreasing filtrations which generalize the Hodge filtration. These filtrations are defined via an analog of the Hodge-to-de Rham spectral sequence (cf. [11]), in complete agreement with the classical picture. Namely, the complex is what we need. It follows from the definition that This is the aforementioned 2-periodicity property.
To see the connection to the classical case, note that the above filtration on the cyclic complex gives rise to a spectral sequence with E 1 = HH • (D)((u)) gr where HH • is the Hochschild homology. Given that when D = par dg X (cf. [13,29]), this spectral sequence is a close relative of the classical Hodgeto-de Rham spectral sequence. In fact, in the above example, the non-commutative Hodge filtration on, say, HP 0 (par dg X) transforms under the isomorphism (3.12) into Note that on each H 2p DR (X) it differs from the usual Hodge filtration by a shift. Let us explain now how the canonical u-connection ∇ D gives rise, under certain conditions, to such a 2-periodic sequence of filtered vector spaces even when D is not necessarily Z-graded.
We will begin by recalling briefly some basic facts regarding regular singular connections in one variable (see [20] and [21, Section 2.2] for a detailed treatment and a brief reminder, respectively). Given a finite-dimensional vector space G over C((u)) equipped with a regular singular connection ∇ there is a canonical decreasing filtration of G by free C[[u]]-submodules V α G determined by certain simple properties which we will not recall here (for instance, u∇ − α is nilpotent on Gr α V ( G)). In the most general case, it is indexed by C (equipped with a proper total order) but we will assume for simplicity that α ∈ R. Furthermore, there is a functor ψ from the category of triples ( G, ∇, G 0 ), where G and ∇ are as above and G 0 ⊂ G is a lattice, to the category of finite-dimensional filtered vector spaces (over C) defined as follows: Let D be a dg category with the additional property that the connection ∇ D induces a regular singular connection on the cohomology of (C(D)((u)), b + uB). Then, using (2.3), we obtain a , b+uB) under the canonical map. Applying (3.14) to the resulting triples, we obtain a sequence of filtered vector spaces One can check easily that these filtered vector spaces satisfy (3.13).
When D is Z-graded, the regularity assumption is satisfied by Proposition 3.8. Moreover, it is easy to see that the above construction, when applied to ∇ D gr , reproduces the periodic cyclic homology groups with their non-commutative Hodge filtrations.
The previous discussion explains why the canonical u-connection should be viewed as a true generalization of the Hodge filtration. Note, however, that in the Z-graded setting the canonical u-connection does not give us any essential information beyond the filtered periodic cyclic homology. In this sense the u-connections are not needed for developing the Hodge theory in the Z-graded case.
As we will see in section 5, where the example D = MF(w) is treated in some more detail, the canonical u-connections, in general, carry more information than the filtrations.

Mixed complexes and u-connections associated with polynomials
We remind the reader that from now on the ground field is C. In this section, Y , w, A w etc.
have the same meaning as in the Introduction, so we will not repeat their definitions. We will be also using all the other definitions and notation related to that part of the Introduction. 4.1. Mixed complexes associated with w. As we mentioned in the Introduction, there is a quasi-isomorphism, constructed in [26], from the Hochschild complex of the algebra A w to the complex (Ω(Y ), −dw). The purpose of this section is to show that the quasi-isomorphism can be promoted to a u-quasi-isomorphism of the corresponding mixed complexes. This requires working with the extended versions, (C e (A w ), b e , B e ) and (C Π, e (A w ), b e , B e ), of the mixed cyclic complexes. Apart from that, the only addition to the original construction of [26] we need is the following well-known fact: are purely even or purely odd (i.e. sit in one degree only) then f is a u-quasi-isomorphism.
Sketch of the Proof. The conditions of the Lemma imply, by a simple inductive argument, The induced mapf on the cohomology is a morphism of free k[[u]]-modules whose reduction modulo u is an isomorphism, that is,f is itself an isomorphism. To conclude the proof, it remains to use the exactness of the Observe that the operators extend to C Π, e (A w ). Also note that b e (w) preserves the image of The second ingredient we need is a map str : C Π, e (A w ) → C Π, e (C[Y ]) defined as follows 9 . Pick a basis {v 1 , . . . , v 2 n } in V consisting of even and odd vectors and denote by E ij , i, j = 1, . . . 2 n , the operators on V given by the elementary 2 n × 2 n matrices in the above basis. Then  e) The composition ǫ · str · exp(−b e (D(w))) :

Recall that
Proof is given in section D.2.

4.2.
Comparing connections. Consider the following operators on C e (A w ): n+1 0 Note that the operators extend to C Π, e (A w ).
Proof is given in section D.5.  Let w be a polynomial on Y = C k satisfying (1.2) and having an isolated singularity at the origin. For the time being, we will treat w as the germ of an analytic function, as it is common in singularity theory.
One of the basic objects associated with w is the Gauss-Manin system [3]. Roughly, it is a regular holonomic D-module on |t| ≪ 1 which extends the flat holomorphic bundle associated with the local system 0<|t|≪1 H k−1 (w −1 (t), C) (herew is the corresponding Milnor fibration).
The Gauss-Manin system, together with a certain subspace known as the Brieskorn lattice, encodes various important characteristics and numerical invariants of the singularity such as the complex monodromy, the Steenbrink-Varchenko Hodge numbers, the spectrum, etc. (for a detailed treatment of the subject see e.g. [24]). The aim of this section is to explain a relationship between the Gauss-Manin system and the formal D-modules from Theorem 1.1, and thereby convince (hopefully) the reader that the aforementioned invariants can be recovered, up to minor details, starting from the category of matrix factorization.
Let us begin by recalling the definition of the Gauss-Manin system and the Brieskorn lattice.
The Gauss-Manin system can be defined as where Ω • (Y an ) (0) are the spaces of germs of holomorphic differential forms of appropriate degrees and the D-module structure is given by An important property of this D-module is that the operator ∂ t is invertible: The Brieskorn lattice is defined as follows: The canonical map Ω k (Y an ) (0) → G is known to induce an embedding G 0 ֒→ G whose image we will also denote by G 0 . G 0 is a free C{{u}}-submodule in G of rank µ and so G = G 0 [u −1 ] (this is one reason to call G 0 lattice).
Observe that [t, u] = u 2 , so it is natural to set ∂ u := u −2 t. Since the resulting C{{u}}[u −1 ][∂ u ]module structure on G encodes the original D-module structure (it is called the Fourier-Laplace transform of the latter), we will from now on forget the variable t and work exclusively with u.
We don't know whether or not the analytic versions of the Gauss-Manin system and the Brieskorn lattice we just described admit any categorical interpretation. Fortunately, due to the regularity of the D-module G it suffices for a number of questions (e.g. computing the aforementioned invariants of the singularity) to work with the following formal versions: These formal versions are precisely what we are able to reconstruct using the categorical approach. Namely, it was proved in [25, Section 2] that where Ω • (Y ) (0) are the spaces of formal differential forms. Moreover, the argument in [25] shows What about the D-module structure? It follows from the above definitions that the action of The connection on the twisted de Rham cohomology induced by (1.3), which we are able to produce starting from the category of matrix factorizations, differs from (5.1) by the term k 2 · 1 u , a kind of Tate twist (cf. the definition right after the equation (2.3)). Note that k, the number of variables w depends on, cannot be extracted from the category MF(w) due to the celebrated Knörrer periodicity: the categories MF(w) and MF(w + xy) are equivalent.
To summarize the previous discussion, Theorem 1.1 allows us to reproduce, up to the above Tate twist, the formal Gauss-Manin system and the formal Brieskorn lattice starting from the categorical data. As a consequence, the category of matrix factorizations does remember about the invariants that are encoded in the Gauss-Manin system, but only modulo information involving the number k. Let us explain this using two examples.
Our first example is the spectrum of w. It is defined as follows. The D-module G is regular and, thus, it carries the canonical decreasing filtration V α G (see section 3.5). This filtration induces a filtration on by the images of V α G ∩ G 0 . Then the spectrum is the function Its support is known to belong to the interval (0, k); moreover, d(α) = d(k −α) (cf. [21, Example

1.8]).
We can repeat this construction starting from the connection induced by (1.3) instead of (5.1).
The resulting V -filtration will differ from the previous one by a shift: V α new G = V α+ k 2 old G. Thus, we will obtain a shifted version of the spectrum d new (α) = d old (α + k 2 ), an even function with support in (− k 2 , k 2 ). Our second example is the Steenbrink Hodge filtration on the vanishing cohomology of w.
The latter is nothing but HP k (MF(w)) with its non-commutative Hodge filtration (see section 3.5), as one can infer by inspecting the formulas presented in [24]. Since k is not a categorical invariant, we recover the Steenbrink Hodge filtration only up to an undetermined shift.
Appendix A. Proofs for Section 2 To see that the latter equals 0, one can simplify the first part of the expression using the Leibniz rule as follows: A.2. Proof of Proposition 2.4. It suffices to show that any u-morphism, u-homotopic to 0, is a morphism of u-connections. Let Then, computing modulo terms u-homotopic to 0 (∼ above stands for 'u-homotopic').
A.3. Proof of Proposition 2.6. Due to Proposition 2.1, we only need to prove Lemma A.1. If one of the u-morphisms p(u), ι(u) is a morphism of u-connections then the other one is also a morphisms of u-connections.
Proof. Let us assume for instance that ι(u) is a morphism of u-connections: Thus, it suffices to show that the latter expression is u-homotopic to 0. By (2.2) and the Leibniz rule, it is u-homotopic to The latter coincides with Indeed, this follows from (2.2) and the equality which, in its turn, is due to (A.2) and (A.3).
A.4. Proof of Proposition 2.7. First of all, we need to show that (2.5) is a u-connection, i.e.
we need to verify the property (2.2): By Lemma A.1 to complete the proof it is enough to show that ι(u) is a morphism from ∇ to ∇ ′ , i.e.
This can be rewritten using (A.3) as follows: Using the Leibniz rule and (2.2), the left-hand side is u-homotopic to Repeating the argument at the end of the proof of Lemma A.1, the latter is u-homotopic to 0.

Appendix B. Useful formulas
In this appendix we list various "commutation relations" between the operators introduced in section 3.
To avoid duplicating results, we will view δ's, µ's, τ etc. as operators on C(A e ) where A e is the unitalization of A. Then most of the formulas below may be interpreted in two ways, namely, as equalities between operators whose domain is either C Using this notation, the operators b h (µ), b v (µ) and b(µ) are related as follows We will also omit subscripts in symbols like µ (i) n+1 , when it does not lead to confusion.

Proof. a) and b) follow from definitions; c) follows from a); d) and e) follow from b). Finally,
both hand sides of f) are equal to N µ (0) N . Part c) is proved similarly (the last case is just the Leibniz rule).
The following lemma is a corollary of the previous one: The remaining formulas involve the operator h e .
Proof. The equalities are straightforward except, perhaps, the very last one. We need to show From this lemma one easily deduces The formulas above hold true if we replace C(A e ) by C(A) and h e by h.
Appendix C. Proofs for Section 3 C.1. Proof of Proposition 3.1. We will start by proving Lemma C.1. ι(u) and p(u) are u-morphisms of mixed complexes.
Proof. Let us show that ι(u) is a u-morphism. The only part that is not immediate is the equality so we need to show that The first components of the two vectors coincide by Lemma B.5 b), so it remains to prove that Let us show now that p(u) is a u-morphism. The only non-trivial part is the equality

What remains is to use Lemmas B.4 b), B.5 c).
To complete the proof of Proposition 3.1, we will prove Proof.
The second equality is obvious. Let us prove the first one. By Lemmas B.1 c), B.
Again, the latter equality is obvious and we will prove only the former one. By Lemmas B.1 c), The first equality is due to Lemma B.
Proof. Part a): The first commutator equals 0 by Lemma B.3. Let us prove the second equality.
Writing both b e (µ) and U • (δ) in the matrix notation, the second equality is equivalent to where both hand sides are viewed as operators either on C(A) or on C + (A).

By Lemma B.2 c)
By Lemma B.2 b), the latter expression equals the right-hand side of (C.3). where To simplify * 22 , let us use the fact that h e : By Lemma B.5 a) the latter equals b(δ) n (h e n−1 (α)). Thus, * 22 = b(δ) n . It remains to show that * 21 = 0. By Lemma B.5 c) it is enough to prove that Thus, the coefficient "in front of" δ (i) n on the left-hand side of (C.4) equals if i = n − 1. We will prove that (C.5) vanishes and leave the second and the third cases to the reader: The interval for k in the third expression contains the interval for k in the second one, so Using the matrix expressions for b e and B e , the statement can be seen to follow from the following Since the last two equalities are quite straightforward, we will give proofs of the first two only.
The first equality follows from Lemma B.2 c): The proof of the second equality uses Lemmas B.1 f), B.2 c) and the computation we just did: Proof of Corollary 3.6. Let us show that where H(u) = H 0 + uH 1 : C e (A) → C e (A) and We will use the following property of N ′ : Finally, To explain the last transition above, we'll take α ∈ C(A) and notice that Let us simplify the remaining entry of the matrix (C.7): By reindexing and resumming one gets First, observe that where H(u) = u(1 − τ −1 )hhhN (see the proof of Part a) of Lemma C.2). Furthermore, by and similarly p(u)V(δ)ι(u) = V un (δ). It remains to compute p(u)Γι(u): To summarize, C.6. Proof of Proposition 3.8. Let us start by introducing the following operators on C e (A): They extend to C(A e ) and the extended operators satisfy the relations Obviously, To simplify * 22 , take α ∈ A ⊗ sA ⊗(n−2) and apply * 22 to h e n−1 (α): Observe that the extensions of a where a Then, clearly, ad(a) The following is easy to check using (3.2) In combination with Lemma B.1 b) this gives c) First, note that B e is the restriction of h e N onto C e (A). Then observe that Thus, The latter implies the statement.

Part c):
That str is a morphism of complexes ( is a special case of a more general fact (see, for instance, [27]) but it is also easy to prove in our special case. It is enough to show that str · µ (0) = µ (0) · str and str · τ = τ · str: It remains to verify that str commutes with b e (w) and B e : (in the above computation, we implicitly use the fact that |E ij | = |v i | + |v j |),

Part d):
The statement is an easy variation on the classical Hochschild-Kostant-Rosenberg map (cf. [4]).

Part e):
The quasi-isomorphism of complexes constructed in [26] is the composition of ǫ · str · exp(−b e (D(w))) with the embedding Since the latter embedding is a quasi-isomorphism (see Proposition 3.1), we conclude that ǫ · str · exp(−b e (D(w))) : (C e (A w ), b e ) → (Ω(Y ), −dw) is a quasi-isomorphism. It remains to apply Lemma 4.1.
D.3. Proof of Proposition 4.3. It will be convenient for us to work with ∇ instead of ∇ (see Corollary 3.6). The proof of Proposition 4.3 will be a combination of two Lemmas.
Let us extend exp(−b e (D(w))) to a map from C Π, e (A w ) to itself.
Proof. We will be using the following formulas obtained earlier (see the proof of Lemma D.1): In what follows, we represent b e (D(w)) by the matrix b(D(w)) 0 0 b(D(w)) as we agreed in section D.1. 11 Note the similarity between (3.11) and (D.4); the connections seem to be special cases of a more general formula which should work for arbitrary curved A∞ algebras.