Extending Landau-Ginzburg models to the point

We classify framed and oriented 2-1-0-extended TQFTs with values in the bicategories of Landau-Ginzburg models, whose objects and 1-morphisms are isolated singularities and (either $\mathbb{Z}_2$- or $(\mathbb{Z}_2 \times \mathbb{Q})$-graded) matrix factorisations, respectively. For this we present the relevant symmetric monoidal structures and find that every object $W \in \Bbbk[x_1,\dots,x_n]$ determines a framed extended TQFT. We then compute the Serre automorphisms $S_W$ to show that $W$ determines an oriented extended TQFT if the associated category of matrix factorisations is $(n-2)$-Calabi-Yau. The extended TQFTs we construct from $W$ assign the non-separable Jacobi algebra of $W$ to a circle. This illustrates how non-separable algebras can appear in 2-1-0-extended TQFTs, and more generally that the question of extendability depends on the choice of target category. As another application, we show how the construction of the extended TQFT based on $W=x^{N+1}$ given by Khovanov and Rozansky can be derived directly from the cobordism hypothesis.

We classify framed and oriented 2-1-0-extended TQFTs with values in the bicategories of Landau-Ginzburg models, whose objects and 1-morphisms are isolated singularities and (either Z 2 -or (Z 2 × Q)graded) matrix factorisations, respectively. For this we present the relevant symmetric monoidal structures and find that every object W ∈ k[x 1 , . . . , x n ] determines a framed extended TQFT. We then compute the Serre automorphisms S W to show that W determines an oriented extended TQFT if the associated category of matrix factorisations is (n − 2)-Calabi-Yau.
The extended TQFTs we construct from W assign the nonseparable Jacobi algebra of W to a circle. This illustrates how nonseparable algebras can appear in 2-1-0-extended TQFTs, and more generally that the question of extendability depends on the choice of target category. As another application, we show how the construction of the extended TQFT based on W = x N +1 given by Khovanov and Rozansky can be derived directly from the cobordism hypothesis.

Introduction
Fully extended topological quantum field theory is simultaneously an attempt to capture the quantum field theoretic notion of locality in a simplified rigorous setting, and a source of functorial topological invariants. In dimension n, such TQFTs have been formalised as symmetric monoidal (∞, n)-functors from certain categories of bordisms with extra geometric structure to some symmetric monoidal (∞, n)-category C.
The fact that such functors must respect structure and relations among bordisms of all dimensions from 0 to n is highly restrictive. Specifically, the cobordism hypothesis of [BD] as formalised in [Lu, AF] states that (in the case of bordisms with framings) a TQFT is already determined by what it assigns to the point, and that fully extended TQFTs with values in C are equivalent to fully dualisable objects in C. This is a strong finiteness condition. Similar relations hold for bordisms with other types of tangential structures; for example, fully extended TQFTs on oriented bordisms are argued to be described by homotopy fixed points of an induced SO(n)-action on fully dualisable objects in C.
In the present paper we are concerned with fully extended TQFTs in dimension n = 2. Following [SP, Ps] we take an extended framed (or oriented) 2-dimensional TQFT with values in a symmetric monoidal bicategory B (where B is called the target) to be a symmetric monoidal 2-functor Z : Bord σ 2,1,0 −→ B (1.1) with structure σ) are constructed in detail in [SP, Ps]. Moreover, these authors prove versions of the cobordism hypothesis (as we briefly review in Section 3), and the relevant SO(2)-homotopy fixed points were described in [HSV, HV, He]. The example for the target B that is dominant in the literature is the bicategory Alg k (or one of its variants, cf. [BD+, App. A]) of finite-dimensional k-algebras, finite-dimensional bimodules and bimodule maps, where k is some field. Using the cobordism hypothesis one finds that extended framed TQFTs with values in Alg k are classified by finite-dimensional separable k-algebras [Lu, SP], while in the oriented case the classification is in terms of separable symmetric Frobenius k-algebras [HSV].
On the other hand, non-separable algebras arise prominently in (non-extended) TQFTs. Recall e. g. from [Ko] that such TQFTs Z ne : Bord or 2,1 → V are equivalent to commutative Frobenius algebras in V, where V is a symmetric monoidal 1-category. Important examples are the categories of vector spaces, possibly with a Z 2 -or Z-grading. In V = Vect Z 2 k or V = Vect Z k , Dolbeault cohomologies of Calabi-Yau manifolds serve as examples of non-separable commutative Frobenius algebras (describing B-twisted sigma models). Another class of examples of generically non-separable Frobenius algebras (in Vect k ) are the Jacobi algebras k[x 1 , . . . , x n ]/(∂ x 1 W, . . . , ∂ xn W ) of isolated singularities described by polynomials W . The associated TQFTs are Landau-Ginzburg models with potential W .
Hence we are confronted with the following question: How do sigma models and Landau-Ginzburg models (and other non-extended TQFTs with non-separable Frobenius algebras) relate to fully extended TQFTs?
A non-extended 2-dimensional TQFT Z ne : Bord σ 2,1 → V can be extended to the point if there is a symmetric monoidal bicategory B and an extended TQFT Z : Bord σ 2,1,0 → B such that (with I B ∈ B the unit object, and ∅ = I Bord σ 2,1,0 ) .
( 1.2) Clearly an extension, if it exists, is not unique, as it depends on the target B. We expect that the extendability of the known classes of non-separable TQFTs is captured by the following motto: "If a non-extended 2-dimensional TQFT Z ne is a restriction of an appropriate defect TQFT Z def ne , then Z ne can be extended to the point (at least as a framed theory), with the bicategory B Z def ne associated to Z def ne as target." Let us unpack this statement and give concrete meaning to it. A 2-dimensional defect TQFT is a symmetric monoidal functor Z def ne on a category of stratified and decorated oriented 2-bordisms, see [DKR, CRS] or the review [Ca]. Restricting Z def ne to only trivially stratified bordisms (meaning that there are no 1-or 0strata) which all carry the same decoration, one obtains a non-extended closed TQFT. As shown in [DKR, Ca] one can construct a pivotal 2-category B Z def ne from any defect TQFT Z def ne (along the same lines as one constructs commutative Frobenius algebras from closed TQFTs). In the case of state sum models the 2-category is equivalent to the full subbicategory ssFrob k ⊂ Alg k of separable symmetric Frobenius algebras [DKR], and indeed End ssFrob k (k) ∼ = Vect k where k is the unit object. For A-and B-twisted sigmal models, the bicategories are expected to be that of symplectic manifolds and Lagrangian correspondences [WW] and of Calabi-Yau varieties and Fourier-Mukai kernels [CW], respectively; in both cases the point serves as the unit object and its endomorphism category is equivalent to Vect Z C . And in the case of Landau-Ginzburg models it should be the bicategory LG (or its Q-graded version LG gr ) of isolated singularties and matrix factorisations [CM]. These are the "appropriate" bicategories we have in mind -if they admit a symmetric monoidal structure (as expected).
We stress that defect TQFT here only serves as a motivation to consider the bicategories above, and we will not mention defects again. A key point is that by choosing bicategories other than Alg k as targets for extended TQFTs Z, one can associate non-separable k-algebras to Z, namely what Z assigns to the circle and the pair-of-pants.
In the present paper we make the above precise for Landau-Ginzburg models. In Section 2 we review the bicategories LG and LG gr , and we present symmetric monoidal structures for them which on objects reduce to the sum of polynomials; the unit object is the zero polynomial, and its endomorphism categories are equivalent to Vect Z 2 k and Vect Z k , respectively. Moreover, we prove that every object in both LG and LG gr is fully dualisable (Corollaries 2.7 and 2.9). Careful and lengthy checks that the data we supply satisfy the coherence axioms of symmetric monoidal bicategories are performed in the PhD thesis [MM] for the case LG, and we explain how they carry over to LG gr .
It follows immediately from the cobordism hypothesis that every object in LG or LG gr determines an extended framed TQFT (with values in LG or LG gr ), while generically Landau-Ginzburg models cannot be extended to the point with target Alg k . Hence our results may be the first explicit demonstration of the general principle that the question of whether or not a given non-extended TQFT can be extended depends on the choice of the target for the extended theory.
To settle the question of extendability also in the oriented case, we use the results of [HSV, HV, He]: a fully dualisable object W determines an extended oriented TQFT if and only if the Serre automorphism S W : W → W (see (3.14)) is isomorphic to the unit 1-morphism I W .
In Section 3.2, we show that for a potential W ∈ If the hypersurface {W = 0} in weighted projective space is a Calabi-Yau variety (equivalently: if 1 3 c(W ) = n − 2) then the trivialisability of S W reduces to the (n − 2)-Calabi-Yau condition Σ n−2 ∼ = Id on the shift functor Σ = [1]{1} of the triangulated category LG gr (0, W ), as we show in Corollary 3.15. This is in line with the general discussion in [Lu,Sect. 4.2].
Finally, we illustrate the combined power of the cobordism hypothesis and the explicit control over the bicategories LG and LG gr by computing the actions of our extended TQFTs on various 2-bordisms: the saddle, the cap, the cup, and the pair-of-pants. This is done in terms of the explicit adjunction maps of [CM], for which we discuss two applications: • We explain (in Theorems 3.3 and 3.12, Remarks 3.6 and 3.16) how the non-separable Jacobi algebra and its residue pairing are recovered from the above extended TQFTs associated to a potential W .
• The "TQFTs with corners" constructed by Khovanov and Rozansky in [KR1] can be derived (as we do in Example 3.13) directly from the cobordism hypothesis as extended TQFTs that assign the potentials W = x N +1 to the point, for all N ∈ Z 2 .

Bicategories of Landau-Ginzburg models
In this section we collect the data that endows the bicategory of Landau-Ginzburg models LG with a symmetric monoidal structure in which every object has a dual and every 1-morphism has left and right adjoints. This is done in Sections 2.1-2.4. In Section 2.5 we explain how the analogous results hold for the bicategory of graded Landau-Ginzburg models LG gr .
Our main reference for bicategories, pseudonatural transformations, modifications etc. is [Be] (see [Le] for a quick reminder). Symmetric monoidal bicategories are reviewed in [Gu, SP] and [Sc,App. A.4]; duals for objects and adjoints for 1-morphisms are e. g. reviewed in [Ps, SP].

Definition of LG
Recall from [CM,Sect. 2.2] that for a fixed field k of characteristic zero, 1 the bicategory of Landau-Ginzburg models LG is defined as follows. An object is either the pair (k, 0) or a pair (k[x 1 , . . . , x n ], W ) where n ∈ Z 0 and W ∈ k[x 1 , . . . , x n ] is a potential, i. e. the Jacobi algebra LG for the Hom category. The right-hand side of (2.2) is the idempotent completion of the homotopy category of finite-rank matrix factorisations of the potential V − W over k [x, z]. We denote matrix factorisations of V − W by (X, d X ) (or simply by X for short), where X = X 0 ⊕ X 1 is a free Z 2 -graded k[x, z]-module and d X ∈ End 1 k[x,z] (X) such that d 2 on the modules Hom k[x,z] (X, X ′ ), and 2-morphisms in LG are even cohomology classes with respect to these differentials. Finally, the idempotent completion (−) ⊕ in (2.2) is obtained by considering only matrix factorisations which are direct summands (in the homotopy category of all matrix factorisations) of finiterank matrix factorisations. For more details, see [CM,Sect. 2.2].
In passing we note that the category LG(W, V ) has a triangulated structure with the shift functor [1] : LG(W, V ) → LG(W, V ) acting on objects as Horizontal composition in LG is given by functors which act on 1-morphisms as and analogously on 2-morphisms. It follows from [DM,Sect. 12] that the right-hand side of (2.7) is indeed a direct summand of a finite-rank matrix factorisation in the homotopy category over k[x, z], hence ⊗ is well-defined. Moreover, the associator in LG is induced from the standard associator for modules, and we will suppress it notationally.
Remark 2.1. One technical issue in rigorously exhibiting LG as a symmetric monoidal bicategory (as summarised in Sections 2.2-2.3) is to establish an effective bookkeeping device that keeps track of how to transform and interpret various mathematical entities. Exercising such care already for the functor ⊗ in (2.7) we can write it as (ι are the canonical inclusions, while (−) * and (−) * denote restriction and extension of scalars, respectively; [MM, Sect. 2.3-2.4] has more details.
For an object (k[x 1 , . . . , x n ], W ) ∈ LG, its unit 1-morphism is (I W , d I W ) with and θ * i is defined by linear extension of θ * i (θ j ) = δ i,j and to obey the Leibniz rule with Koszul signs, cf. [CM,Sect. 2.2]. In the following we will suppress the symbol ∧ when writing elements in or operators on I W .
Finally, the left and right unitors for X ∈ LG(W, V ) are defined as projection to θ-degree zero on the units I V and I W , respectively; their explicit inverses (in the homotopy category LG(W, V )) were worked out in [CM] to act as follows: where {e i } is a basis of the module X, and d X is identified with the matrix representing it with respect to {e i }.
In summary, the above structure makes LG into a bicategory, cf. [CM,Prop. 2.7]. Note that in LG it is straightforward to determine isomorphisms of commutative algebras (see e. g. [KR1]) (2.13)

Monoidal structure for LG
Endowing LG with a monoidal structure involves specifying the following data: (M1) monoidal product : LG × LG → LG, (M2) monoidal unit I ∈ LG, specified by a strict 2-functor I : 1 → LG, , which is part of an adjoint equivalence, (M4) pentagonator π : (Id LG a)•a•(a Id LG ) → a•a (using shorthand notation explained below), (M5) left and right unitors l : • (a * 1) (using shorthand notation), subject to the coherence axioms spelled out e. g. in [SP,Sect. 2.3]. In this section we provide the above data for LG, which come as no surprise to the expert. The coherence axioms are carefully checked in [MM, Ch. 3].
(M1) We start with the monoidal product. It is a 2-functor : LG × LG −→ LG (2.14) which is basically given by tensoring over k and taking sums of potentials. More precisely, according to [MM,Prop. 3.1.12], acts as on objects, while the functors on Hom catgories are given by ⊗ k (up to a reordering of variables similar to the situation in Re- , (X 1 , X 2 ))-components are given by linearly extending for Z 2 -homogeneous module elements e 1 , e 2 , f 1 , f 2 , and isomorphisms on units (M2) The unit object in LG is Let 1 be the 2-category with a single object * and only identity 1-and 2morphisms. We define a strict 2-functor I : 1 → LG by setting I( * ) = I.
The associator a and the pseudonatural transformation where here and below we write vertical and horizontal composition of pseudonatural transformations as • and * , respectively. We also typically use shorthand notation for the sources and targets of modifications obtained by whiskering; for example, the pentagonator is then written (2.24) Its components are The left and right (1-morphism) unitors are pseudonatural transformations l : whose components are given by , written here in the shorthand notation also employed in (M4) above, whose components are (2.28) LG with a monoidal structure. Proof.

Symmetric monoidal structure for LG
Endowing the monoidal bicategory LG with a symmetric braided structure amounts to specifying the following data: (S1) braiding b : → • τ as part of and adjoint equivalence (b, b − ), where τ : LG × LG → LG × LG is the strict 2-functor which acts as (ζ, ξ) → (ξ, ζ) on objects, 1-and 2-morphisms, subject to the coherence axioms spelled out e. g. in [SP,Sect. 2.3]. In this section we provide the above data which are discussed in detail in [MM,Sect. 3.3].
(S1) The braiding is a pseudonatural transformation The braiding b and the pseudonatural transformation is precisely what is assigned to a "virtual crossing" in the construction of homological sl N -tangle invariants of Khovanov and Rozansky [KR1] (see the second expression in [KR2,Eq. (A.9)]).
(S2) The syllepsis is an invertible modification W ) are given by λ −1 I V +W (up to a reordering of variables and a sign-less swapping of tensor factors, see [MM,Lem. 3.3.8]).
(S3) The invertible modifications have components R ((U,V ),W ) and S ((U,V ),W ) which act on basis elements, i. e. on tensor and wedge products of θ-variables, by a reordering with appropriate signs, see [MM,Lem. 3.3.11] for the lengthy explicit expressions.
We note that instead of directly constructing the data (M1)-(M6) and (S1)-(S3) and verifying their coherence axioms, one could also employ Shulman's method of constructing symmetric monoidal bicategories from symmetric monoidal double categories [Sh]. A double category of Landau-Ginzburg models was first studied in [MN].

Adjoints for 1-morphisms
Endowing LG with left and right adjoints for 1-morphisms amounts to specifying the following data: (A1) 1-morphisms † X, X † ∈ LG(V, W ) for every X ∈ LG(W, V ), (A2) 2-morphisms ev X : † X ⊗X → I W , coev X : I V → X ⊗ † X, ev X : X ⊗X † → I V and coev X : I W → X † ⊗ X for every X ∈ LG(W, V ), subject to coherence axioms. In this section we recall the above data as constructed in [CM] (this reference also spells out the coherence axioms).

Duals for objects
Endowing the symmetric monoidal bicategory LG with duals for objects amounts to specifying the following data: In this section we provide the above data; the explicit isomorphisms c l , c r are constructed in [MM,Ch. 4]. Proposition 2.6. The data (D1)-(D2) endow the monoidal bicategory LG with duals for every object.
Proof. The cusp isomorphisms c l , c r are computed in terms of the unitors λ, ρ and canonical swap maps in [MM,Lem. 4.6].
Recall that an object A of a symmetric monoidal bicategory B is fully dualisable if A has a dual and if the corresponding adjunction 1-morphisms ev A , coev A themselves have left and right adjoints. Hence Proposition 2.6 together with Theorem 2.5 implies: Corollary 2.7. Every object of LG is fully dualisable.

Graded matrix factorisations
Landau-Ginzburg models with an additional Q-or Z-grading appear naturally as (non-functorial) quantum field theories, in their relation to conformal field theories, as well as in representation theory and algebraic geometry. In this section we recall the bicategory of graded Landau-Ginzburg models LG gr from [CM, CRCR, Mu] (see also [BFK]) and observe that it inherits the symmetric monoidal structure from LG. Moreover, every object in LG gr is fully dualisable.
An object of LG gr is a pair (k[x 1 , . . . , x n ], W ) where now k[x 1 , . . . , x n ] is a graded ring by assigning degrees |x i | ∈ Q >0 to the variables x i , and W ∈ k[x 1 , . . . , x n ] is either zero or a potential of degree 2. The central charge of LG gr is a summand of a finite-rank matrix factorisation (X, d X ) of V − W over k[x, z] such that the following four conditions are satisfied: (i) the modules X 0 = g∈Q X 0 q and X 1 = g∈Q X 1 q are Q-graded, (ii) the action of x i and z j on X are respectively of Q-degree |x i | and |z j |, (iii) the map d X has Q-degree 1, and (iv) if we write {−} for the shift in Q-degree and if X i ∼ = q∈Q k[x, z]{q} ⊕a i,q for i ∈ {0, 1}, then {q ∈ Q | a i,q = 0} must 2 be a subset of i + G V −W , where G V −W := |x 1 |, . . . , |x n |, |z 1 |, . . . , |z m | ⊂ Q and G 0 := Z . (2.41) A 2-morphism in LG gr between two 1-morphisms (X, d X ), (X ′ , d X ′ ) is a cohomology class of Z 2 -and Q-degree 0 with respect to the differential δ X,X ′ in ( [DM] ensures that horizontal composition in LG gr can be defined analogously to (2.6). Moreover, the units I W of LG can naturally be endowed with an appropriate Q-grading (by setting |θ i | = |x i | − 1 and |θ * i | = 1 − |x i |), and the associator α and unitors λ, ρ of LG gr are those of LG (as they manifestly have Q-degree 0). Hence LG gr is indeed a bicategory.
The bicategory LG gr also inherits a symmetric monoidal structure from LG. This is so because all 1-and 2-morphisms in the data (M1)-(M6), (S1)-(S3) are constructed from the units I W and from the structure maps α, λ, ρ, their inverses and (Q-degree 0) swapping maps, respectively.
For a 1-morphism we define its left and right adjoint as † (2.44) The above shifts in Q-degree are necessary to render the adjunction maps ev X , coev X , ev X , coev X in (2.37) and (2.38) to be of Q-degree 0 so that they are indeed 2-morphisms in LG gr . Finally, the (left and right) dual of (k[x], W ) ∈ LG gr is (k[x], −W ) with the same grading, and the matrix factorisation underlying the adjunction 1-morphisms ev W , coev W , ev W , coev W is again I W , but now viewed as a Q-graded matrix factorisation.
In summary, we have: Theorem 2.8. The bicategory LG gr inherits a symmetric monoidal structure from LG, every object of LG gr has a dual, and every 1-morphism has adjoints.
Corollary 2.9. Every object of LG gr is fully dualisable.

Extended TQFTs with values in LG and LG gr
In this section we study extended TQFTs with values in LG and LG gr . We briefly review framed and oriented 2-1-0-extended TQFTs and their "classification" in terms of fully dualisable objects and trivialisable Serre automorphisms, respectively. Then we observe that every object W ≡ (k[x 1 , . . . , x n ], W ) in LG or LG gr gives rise to an extended framed TQFT (Proposition 3.2 and Remark 3.6), and we show precisely when W determines an oriented theory (Propositions 3.9 and 3.14). We also show how the extended framed (or oriented) TQFTs recover the Jacobi algebras Jac W as commutative (Frobenius) k-algebras (Theorems 3.3 and 3.12, Remark 3.16), and we explain how a construction of Khovanov and Rozansky can be recovered as a special case of the cobordism hypothesis (Example 3.13).
To formulate the precise statement of the cobordism hypothesis, denote by B fd the full subbicategory of B whose objects are fully dualisable, and write K (B fd ) for the core of B fd , i. e. the subbicategory of B fd with the same objects and whose 1-and 2-morphisms are the equivalences and 2-isomorphisms of B, respectively. Then: Theorem 3.1 (Cobordism hypothesis for framed 2-bordisms, [Ps,Thm. 8.1]). Let B be a symmetric monoidal bicategory. There is an equivalence Note that thanks to the description of Bord fr 2,1,0 as a symmetric monoidal bicategory in terms of generators and relations given in [Ps], the action of Z is fully determined (up to coherent isomorphisms) by what it assigns to the point. For example, if Z(+) = A, then the 2-framed circle which is the horizontal composite of the two semicircles (or elbows) † ev + and ev + is sent to ev A ⊗ † ev A . Similarly, 2-morphisms in Bord fr 2,1,0 can be decomposed into cylinders and adjunction 2morphisms for ev + , coev + and their (multiple) adjoints; we will discuss several examples of such decompositions in the proofs of Theorems 3.3 and 3.12 below.
We now turn to the symmetric monoidal bicategory of Landau-Ginzburg models LG. As a direct consequence of the cobordism hypothesis and Corollary 2.7 we have: Proposition 3.2. Every object W ≡ (k[x 1 , . . . , x n ], W ) ∈ LG determines an extended framed TQFT This can be interpreted as "every Landau-Ginzburg model can be extended to the point as a framed TQFT". In the remainder of Section 3.1 we make this more precise by relating Z fr W to the (non-extended) closed oriented TQFT Z W : Bord or 2,1 −→ Vect k (3.3) which via the standard classification in terms of commutative Frobenius algebras (see e. g. [Ko]) is described by the Jacobi algebra Jac W with pairing −, − W : Jac W ⊗ k Jac W −→ k (3.4) induced by the residue trace map To recover the k-algebra Jac W with its multiplication µ Jac W : φ ⊗ ψ → φψ, we want to show that Jac W and µ Jac W are what Z fr W assigns to "the" circle and "the" pair-of-pants. However, there are infinitely many isomorphism classes of 2-framed circles (one for every integer), so we have to be more specific. Using the equivalent description of 2-framed circles in terms of immersions ι : S 1 → R 2 together with a normal framing [DSPS,Sect. 1.1], the correct choice is to take the standard circle embedding for ι together with outward pointing normals. We denote the corresponding 2-framed circle S 1 0 . In terms of the structure 1morphisms of Bord fr 2,1,0 (whose horizontal composition we write as #), we have (see [DSPS,Sect. 1 This is the correct choice in the sense that for every integer k, there is a 2-framed circle S 1 k , and for every pair (k, l) ∈ Z 2 there is a pair-of-pants 2-morphism S 1 k ⊔ S 1 l → S 1 k+l in Bord fr 2,1,0 , and only for k = 0 = l do we get a multiplication. This is "the" 2-framed pair-of-pants for us. is isomorphic to (Jac W , µ Jac W ) as a k-algebra.
Proof. Note first that Z fr is isomorphic in LG((k, 0), (k, 0)) ∼ = vect Z 2 to the vector space Jac W (viewed as a Z 2 -graded vector space concentrated in even degree). One can check that an explicit isomorphism κ : where p and q are polynomials and {e i } is a basis of the k[x, x ′ ]-module I W .
Next we prove that Z fr W sends the pair-of-pants to the commutative multiplication µ Jac W . For this we decompose the pair-of-pants into generators, namely into cylinders over the left and right elbows ev + : − ⊔ + → ∅ and † ev + : ∅ → − ⊔ +, respectively, and the "upside-down saddle" ev ev + : † ev + # ev + → 1 −⊔+ (which is called v 1 in [DSPS,Ex. 1.1.7]). Then pair-of-pants = 1 ev + # ev ev + # 1 † ev + : S 1 0 ⊔ S 1 0 −→ S 1 0 . (3.8) Hence Z fr W sends this pair-of-pants to 1 ev W ⊗ ev ev W ⊗1 † ev W , which by pre-and post-composition with the isomorphism κ : ev W ⊗ † ev W ∼ = Jac W becomes a map µ : Jac W ⊗ k Jac W → Jac W . Noting that both κ and ev ev W act diagonally (with ev ev W (e * i ⊗ e j ) = δ i,j since ev W : (−W ) W → I has trivial target, cf. the explicit expression for ev ev W in (2.38)), we find that µ is indeed given by multiplication of polynomials, i. e. µ = µ Jac W .
Remark 3.4. The finite-dimensional k-algebra Jac W is typically not separable. For example, if W = x N +1 with N ∈ Z 2 the algebra Jac W has non-semisimple representations (as multiplication by x has non-trivial Jordan blocks) and hence cannot be separable. Thus Jac W is not fully dualisable in the bicategory Alg k of finite-dimensional k-algebras, bimodules and intertwiners [Lu, SP], so Jac W cannot describe an extended TQFT with values in Alg k . Proposition 3.2 and Theorem 3.3 explain how Jac W does appear in an extended TQFT with values in LG, namely as the algebra assigned to the circle S 1 0 and its pair-of-pants. For an algebra A ∈ Alg k its Hochschild cohomology HH • (A) is isomorphic to ev A ⊗ † ev A , and for Hochschild homology one finds HH (3.9) Thus by Theorem 3.3 we have HH • (W ) ∼ = Jac W , and paralleling the first part of the proof we find HH • (W ) ∼ = Jac W [n] as Z 2 -graded vector spaces (because the matrix factorisations b (W,W ) and coev W are I W I W and I W ∼ = I † W = I ∨ W [n], respectively). Hence HH • (W ) and HH • (W ) precisely recover the Hochschild cohomology and homology of the 2-periodic differential graded category of matrix factorisations MF(k[x], W ) as computed in [Dy,Cor. 6.5 & Thm. 6.6]: (3.10) Remark 3.6. Proposition 3.2, Theorem 3.3 and Corollary 3.5 have direct analogues for the graded Landau-Ginzburg models of Section 2.5. Firstly, Theorem 3.1 and Corollary 2.9 immediately imply that every object (k[x 1 , . . . , x n ], W ) ∈ LG gr determines an extended TQFT Z fr W,gr : Bord fr 2,1,0 −→ LG gr . (3.11) Secondly, going through the proof of Theorem 3.3 we see that to the circle S 1 0 and its pair-of-pants, Z fr W,gr assigns the Jacobi algebra Jac W which is now a Q-graded algebra with degree-preserving multiplication. We note that here it is important that the upside-down saddle ev ev + : † ev + # ev + → 1 −⊔+ involves the left adjoint of ev + : by (2.44) we have † ev W = ev ∨ W [0]{0} = ev ∨ W , so Z fr W,gr (pair-of-pants) really gives a map (3.12) (Incorrectly using the right adjoint ev † W = ev ∨ W [2n]{ 2 3 c(W )} would lead to unwanted Q-degree shifts in the multiplication. In Remark 3.16 below however we are naturally led to use the right adjoint ev † W to obtain the correct graded trace map − W on Jac W .) Thirdly, for every (k[x 1 , . . . , x n ], W ) ∈ LG gr the matrix factorisation underlying coev W is (3.13)

Oriented case
An extended oriented TQFT with values in a symmetric monoidal bicategory B is a symmetric monoidal 2-functor Z : Bord or 2,1,0 → B. Here Bord or 2,1,0 is the bicategory of oriented 2-bordisms defined and explicitly constructed in [SP,Ch. 3]; see in particular Fig. 3.13 of loc. cit. for a list of the 2-morphism generators (to wit: the saddle, the upside-down saddle, the cap, the cup, and cusp isomorphisms) and their relations. Hence objects of Bord or 2,1,0 are disjoint unions fo positively and negatively oriented points, which we (also) denote + and −, respectively. It was argued in [Lu] that such 2-functors Z are classified by the homotopy fixed points of the SO(2)-action induced on B fd by the SO(2)-action which rotates the framings in Bord fr 2,1,0 . This was worked out in detail in [HSV, HV, He] as we briefly review next.
An SO(2)-action on B fd is a monoidal 2-functor ̺ from the fundamental 2groupoid Π 2 (SO(2)) to the bicategory of autoequivalences of B fd . Since SO(2) is path-connected, Π 2 (SO(2)) has essentially a single object * which ̺ sends to the identity Id B fd on B fd . Since π 1 (SO(2)) ∼ = Z the action of ̺ on 1-morphisms is essentially determined by its value on the identity 1 * corresponding to 1 ∈ Z. It was argued in [Lu,Rem. 4.2.5] that for an oriented TQFT Z as above with Z(+) =: A, the relevant choice for ̺(1 * ) is the Serre automorphism S A of A ∈ B fd . By definition S A is the 1-morphism (3.14) Here we denote the braiding, horizontal composition and monoidal product in B by b, ⊗ and , respectively, as we do in LG and LG gr .
The bicategory of SO(2)-homotopy fixed points K (B fd ) SO(2) was defined and endowed with a natural symmetric monoidal structure in [HV]. Objects of Building on [Lu, SP, HSV, HV], extended oriented TQFTs with values in B were classified by fully dualisable objects with trivialisable Serre automorphisms in [He]: Theorem 3.7 (Cobordism hypothesis for oriented 2-bordisms, [He,Cor. 5.9]). Let B be a symmetric monoidal bicategory. There is an equivalence We return to the symmetric monoidal bicategory LG. To determine extended oriented TQFTs with values in LG we have to compute Serre automorphisms for all objects: Proof. According to Sections 2.3-2.4, the factors r (W, * ) , 1 W , ev W , b (W,W ) , 1 W * and r − W in the defining expression (3.14) are all given by the matrix factorisation underlying the unit I W ∈ LG(W, W ), while the matrix factorisation underlying ev if and only if n is even.
Remark 3.10. Let d ∈ Z. Following [Ke], we say that a k-linear, Hom-finite triangulated category T with shift functor Σ is weakly d-Calabi-Yau if T admits a Serre functor 3 S T such that Σ d ∼ = S T . The triangulated category LG(0, W ) is known to admit a Serre functor S LG(0,W ) ∼ = [n] = [n − 2]. Hence LG(0, W ) is weakly (n − 2)-Calabi-Yau, the Serre automorphism and Serre functor coincide in the sense that S W ⊗ (−) ∼ = S LG(0,W ) , and the condition that S W is trivialisable is equivalent to the condition that the Serre functor is isomorphic to the identity.
Remark 3.11. (i) Proposition 3.9 can be interpreted as "every Landau-Ginzburg model with an even number of variables can be extended to the point as an oriented TQFT". However, since for odd (and even) n there is an isomorphism of Frobenius algebras . . . , x n , y]/(∂ x 1 (W + y 2 ), . . . , ∂ xn (W + y 2 ), ∂ y (W + y 2 )) = Jac W +y 2 , (3.17) every non-extended oriented Landau-Ginzburg model appears as part of an extended oriented TQFT Z or W or Z or W +y 2 (depending on whether n is even or odd, respectively), namely as the commutative Frobenius algebra with underlying vector space Z or W (S 1 ) or Z or W +y 2 (S 1 ). Note that for this argument to work we need to ensure that this Frobenius algebra is really isomorphic to the associated Jacobi algebra, as we do with Theorems 3.3 and 3.12.
In particular, for every (k[x 1 , . . . , x n ], W ) ∈ LG •/2 there is an even/odd isomorphism I W ∼ = I W [n] for n even/odd. Hence by Lemma 3.8 every object of LG •/2 determines an extended oriented TQFT with values in LG •/2 .
(iii) A better way to deal with the signs in the interchange law mentioned in part (ii) above is to incorporate them into a richer conceptual structure. Part of this involves the natural differential Z 2 -graded categories (with differential δ X,X ′ as above) studied in [Dy], whose even cohomologies are the matrix factorisation categories of Section 2.1. Such bicategories of differential graded matrix factorisation categories are studied in [BFK], and demanding their monoidal product to be made up of differential graded functors produces Koszul signs in the interchange law.
A wider perspective on Koszul signs and parity issues in Landau-Ginzburg models as discussed here is that they are thought to be the topological twists of supersymmetric quantum field theories, see e. g. [HK+, LL, HL]. Formalising this construction in a functorial field theory setting would involve symmetric monoidal super 2-functors on super bicategories of super bordisms, which is a theory whose details to our knowledge have not been worked out. Relatedly, we expect the graded pivotal bicategory LG of [CM] to arise as the bicategory associated to a non-extended oriented defect TQFT on super bordisms (which again has not been defined in detail as far as we know), paralleling the non-super construction of Proof. The isomorphism on the level of k-algebras was already established in Theorem 3.3, it remains to compute the action of Z fr W on the cap and cup 2morphisms.
The cap is the bordism ev ev + from the 2-framed circle ev + # ev † + to 1 ∅ , so Z fr W sends it to the 2-morphism ev ev W from ev W ⊗ ev † W = ev W ⊗ † ev W to I 0 . Since ev W : (k[x], −W ) (k[x], W ) → (k, 0) has trivial target, only the summand l = 0 contributes to the expression for ev ev W in (2.38), and pre-composing with the isomorphism Jac W ∼ = ev W ⊗ † ev W from the proof of Theorem 3.3 produces the residue trace − W .
Similarly, the cup : ∅ → S 1 0 = ev + # † ev + is equal to coev ev + . Using the explicit expression for coev ev W in (2.38) we see that post-composing Z fr W (cup)(1) with the isomorphism ev W ⊗ † ev W ∼ = Jac W is indeed the unit 1 ∈ Jac W .
Paralleling the above proof we see that for even n, the extended oriented TQFT Z or W also assigns the Frobenius algebra Jac W to the oriented circle, pair-of-pants, cup and cap.
Example 3.13. For every N ∈ Z 2 , the potential x N +1 determines an extended oriented TQFT with values in the symmetric monoidal bicategory LG •/2 introduced in Remark 3.11(ii). We denote this TQFT by Z KR as it recovers -directly from the cobordism hypothesis -the explicit construction that Khovanov and Rozansky gave in [KR1,Sect. 9]. In loc. cit. the authors determine their TQFT by describing what it assigns to the point +, the circle, the cap, the cup and the saddle bordisms in Bord or 2,1,0 . Except for the saddle we have already computed all these assignments of Z KR for any potential W in Theorems 3.3 and 3.12, and for W = x N +1 they match the prescriptions of [KR1] (except for non-essential prefactors for the cap and cup morphisms).
To establish that the TQFT Z KR indeed matches that of [KR1,Sect. 9] it remains to compute Z KR (saddle) = Z KR ( coev ev + ) and compare it to the explicit matrix expressions in [KR1,Page 81] (or Page 95 of arXiv:math/0401268v2 [math.QA]). Since Z KR ( coev ev + ) = coev ev x N+1 this is another exercise in using the formulas (2.38) for adjunction 2-morphisms. This is carried out in [MM,Sect. 5 where the entries e ijk := a+b+c=N −1 x a i x b j x c k ∈ k[x 1 , x 2 , x 3 , x 4 ] depend on four variables as the source and target of coev ev x N+1 involve four copies of x N +1 ∈ LG •/2 . Up to a minor normalisation issue 4 the expression (3.20) agrees with that of [KR1].
In summary, we verified that the construction of [KR1,Sect. 9] can be understood as an application of the cobordism hypothesis to the potential W = x N +1 .
We return to the bicategory LG gr of Section 2.5. All the above results in the present section have analogues or refinements in LG gr . In particular: Proposition 3.14. An object W ≡ (k[x 1 , . . . , x n ], W ) ∈ LG gr determines an extended oriented TQFT Proof. We write Y W for the zero locus of W in weighted projective space. The variety Y W is Calabi-Yau if and only if the condition c 1 (Y W ) = 0 is satisfied 4 More precisely, (3.20) agrees with the saddle morphism of [KR1] if the arbitrary polynomial r of degree N − 2 in loc. cit. is set to a+b+c+d=N −2 x a 1 x b 2 x c 3 x d 4 , and if non-scalar entries of the matrix are multiplied by 1 2 . The latter seems to be a typo in [KR1] as without these factors the expression would not be closed with respect to the differential δ IW I−W ,ev † x N +1 ⊗ ev x N +1 .
by the first Chern class, which in our normalisation convention is equivalent to n i=1 |x i | = |W | = 2. This implies 1 3 c(W ) = n i=1 (1 − |x i |) = n − 2, and hence according to the proof of Proposition 3.14 we have that is the (n − 2)-fold product of the shift functor Σ = [1]{1} of LG gr (0, W ) with itself. One way to see that the condition Σ n−2 ∼ = Id LG gr (0,W ) is satisfied is to note that since the Calabi-Yau variety Y W is (n − 2)-dimensional, the triangulated category D b (coh(Y W )) is (n − 2)-Calabi-Yau. But by [Or,Thm. 3.11 & Rem. 3.12] this latter category is triangle equivalent to LG gr (0, W ).
Remark 3.16. There is also an analogue of Theorem 3.12 for Z fr W,gr : We already saw in Remark 3.6 that Z fr W,gr sends the circle and pair-of-pants to Jac W as a graded algebra. As in the proof of Theorem 3.12 we find that Z fr W,gr (cup)(1) gives the unit 1 ∈ (Jac W ) 0 of degree 0 (because coev ev W is of Q-degree 0).