Extending Landau-Ginzburg models to the point

Nils Carqueville; Flavio Montiel Montoya

doi:10.1007/s00220-020-03871-5

Extending Landau-Ginzburg models to the point

Carqueville, Nils;Montoya, Flavio Montiel 2018-09-28 00:00:00 Extending Landau-Ginzburg models to the point Nils Carqueville Flavio Montiel Montoya [email protected] [email protected] Fakult¨at fu¨r Mathematik, Universit¨at Wien, Austria We classify framed and oriented 2-1-0-extended TQFTs with val- ues in the bicategories of Landau-Ginzburg models, whose objects and 1-morphisms are isolated singularities and (either Z - or (Z × Q)- 2 2 graded) matrix factorisations, respectively. For this we present the relevant symmetric monoidal structures and ﬁnd that every object W ∈ k[x , . . . , x ] determines a framed extended TQFT. We then 1 n compute the Serre automorphisms S to show that W determines an oriented extended TQFT if the associated category of matrix factori- sations 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 construc- N+1 tion of the extended TQFT based on W = x given by Khovanov and Rozansky can be derived directly from the cobordism hypothesis. arXiv:1809.10965v2 [math.QA] 4 Dec 2020 Contents 1 Introduction 2 2 Bicategories of Landau-Ginzburg models 5 2.1 Deﬁnition of LG . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Monoidal structure for LG . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Symmetric monoidal structure for LG . . . . . . . . . . . . . . . . 11 2.4 Duality in LG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4.1 Adjoints for 1-morphisms . . . . . . . . . . . . . . . . . . . 12 2.4.2 Duals for objects . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Graded matrix factorisations . . . . . . . . . . . . . . . . . . . . . 15 gr 3 Extended TQFTs with values in LG and LG 16 3.1 Framed case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Oriented case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1 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 cer- tain 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 bor- disms of all dimensions from 0 to n is highly restrictive. Speciﬁcally, the cobor- dism 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 −→ B (1.1) 2,1,0 where σ = fr (or σ = or), without any mention of ∞-categories. The bicategories Bord of points, 1-manifolds with boundary and 2-manifolds with corners (all 2,1,0 2 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 (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 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) or TQFTs. Recall e. g. from [Ko] that such TQFTs Z : Bord → V are equiva- ne 2,1 lent to commutative Frobenius algebras in V, where V is a symmetric monoidal 1-category. Important examples are the categories of vector spaces, possibly with Z Z a Z - or Z-grading. In V = Vect or V = Vect , Dolbeault cohomologies of k k Calabi-Yau manifolds serve as examples of non-separable commutative Frobe- nius algebras (describing B-twisted sigma models). Another class of examples of generically non-separable Frobenius algebras (in Vect ) are the Jacobi algebras k[x , . . . , x ]/(∂ W, . . . , ∂ W ) of isolated singularities described by polynomi- 1 n x x 1 n als 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 : Bord → V can be extended to ne 2,1 the point if there is a symmetric monoidal bicategory B and an extended TQFT Z : Bord → B such that (with I ∈ B the unit object, and ∅ = I ) B Bord 2,1,0 2,1,0 ∼ ∼ V End (I ) and Z Z . (1.2) = = B B ne End σ (∅) Bord 2,1,0 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 is a restriction of an appropriate ne def defect TQFT Z , then Z can be extended to the point (at least as a framed ne ne def theory), with the bicategory B def associated to Z as target.” ne ne Let us unpack this statement and give concrete meaning to it. A 2-dimensional def defect TQFT is a symmetric monoidal functor Z on a category of stratiﬁed and ne decorated oriented 2-bordisms, see [DKR, CRS] or the review [Ca]. Restricting def Z to only trivially stratiﬁed bordisms (meaning that there are no 1- or 0- ne strata) 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 def ne 3 def from any defect TQFT Z (along the same lines as one constructs commutative ne Frobenius algebras from closed TQFTs). In the case of state sum models the 2-category is equivalent to the full subbicategory ssFrob ⊂ Alg of separable symmetric Frobenius algebras [DKR], and indeed End (k) = Vect where k ssFrob 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 . And in the case of Landau-Ginzburg models it should gr be the bicategory LG (or its Q-graded version LG ) 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 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. gr In Section 2 we review the bicategories LG and LG , and we present symmetric monoidal structures for them which on objects reduce to the sum of polynomi- als; the unit object is the zero polynomial, and its endomorphism categories are 2 Z equivalent to Vect and Vect , respectively. Moreover, we prove that every ob- k k gr ject in both LG and LG is fully dualisable (Corollaries 2.7 and 2.9). Careful and lengthy checks that the data we supply satisfy the coherence axioms of sym- metric monoidal bicategories are performed in the PhD thesis [MM] for the case gr LG, and we explain how they carry over to LG . It follows immediately from the cobordism hypothesis that every object in LG gr gr or LG determines an extended framed TQFT (with values in LG or LG ), while generically Landau-Ginzburg models cannot be extended to the point with target Alg . 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 (see (3.14)) is isomorphic to the unit 1-morphism I . In Section 3.2, we show that for a potential W ∈ k[x , . . . , x ] viewed as an 1 n object in LG we have S = I [n] where [n] is the n-fold shift functor which W W satisﬁes [2] = [0], cf. Section 2.1. Since I ≇ I [1] this implies that W determines W W an extended oriented TQFT (cf. Proposition 3.9) or or Z : Bord −→ LG (1.3) W 2,1,0 4 if and only if n is even, and we discuss the relation to Serre functors and Calabi- Yau categories in Remark 3.10. For a quasi-homogeneous potential W ∈ k[x , . . . , x ] viewed as an object in 1 n gr 1 LG we ﬁnd that S = I [n−2]{ c(W )}, where c(W ) is the central charge of W W W (see (2.40)) and {−} denotes the shift in Q-degree. Hence every potential W that satisﬁes the condition I = I [n − 2]{ c(W )} determines an extended oriented W W TQFT (cf. Proposition 3.14) or or gr Z : Bord −→ LG . (1.4) W,gr 2,1,0 If the hypersurface {W = 0} in weighted projective space is a Calabi-Yau variety (equivalently: if c(W ) = n − 2) then the trivialisability of S reduces to the n−2 (n − 2)-Calabi-Yau condition Σ = Id on the shift functor Σ = [1]{1} of the gr triangulated category LG (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 gr explicit control over the bicategories LG and LG 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 cobor- N+1 dism hypothesis as extended TQFTs that assign the potentials W = x to the point, for all N ∈ Z . >2 Acknowledgements We thank Ilka Brunner, Domenico Fiorenza, Jan Hesse, Daniel Murfet and Christoph Schweigert for helpful discussions. The work of N. C. is partially sup- ported by a grant from the Simons Foundation and by the stand-alone project P 27513-N27 of the Austrian Science Fund. The work of F. M. M. was supported by a fellowship from the Peters-Beer Foundation. 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 gr of graded Landau-Ginzburg models LG . 5 Our main reference for bicategories, pseudonatural transformations, modiﬁca- tions 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]. 2.1 Deﬁnition of LG Recall from [CM, Sect. 2.2] that for a ﬁxed ﬁeld k of characteristic zero, the bi- category of Landau-Ginzburg models LG is deﬁned as follows. An object is either the pair (k, 0) or a pair (k[x , . . ., x ], W ) where n ∈ Z and W ∈ k[x , . . . , x ] 1 n >0 1 n is a potential, i. e. the Jacobi algebra Jac = k[x , . . . , x ]/(∂ W, . . . , ∂ W ) (2.1) W 1 n x x 1 n is ﬁnite-dimensional over k. We often abbreviate lists of variables (x , . . . , x ) 1 n by x, and we often shorten (k[x], W ) to W . For two objects (k[x], W ) and (k[z], V ) we have LG (k[x], W ), (k[z], V ) = hmf k[x, z], V − W (2.2) 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 ) (or 0 1 simply by X for short), where X = X ⊕ X is a free Z -graded k[x, z]-module and d ∈ End (X) such that d = W · id . The twisted diﬀerentials d , d ′ X X X X k[x,z] X induce diﬀerentials |ζ| δ ′ : ζ 7−→ d ′ ◦ ζ − (−1) ζ ◦ d (2.3) X,X X X on the modules Hom (X, X ), and 2-morphisms in LG are even cohomology k[x,z] 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 ﬁnite- rank 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 0 1 1 0 [1]: X ⊕ X , d 7−→ X ⊕ X , −d (2.4) X X see e. g. [KST, Sect. 2.1]. It follows that [2] := [1] ◦ [1] = Id . (2.5) LG(W,V ) In fact we can allow any commutative unital ring k if we generalise the deﬁnition of potentials as in [CM, Def. 2.4]. 6 Horizontal composition in LG is given by functors ⊗: LG (k[y], W ), (k[z], W ) × LG (k[x], W ), (k[y], W ) 2 3 1 2 −→ LG (k[x], W ), (k[z], W ) (2.6) 1 3 which act on 1-morphisms as 0 0 1 1 0 1 1 0 (Y, X) 7−→ Y ⊗X ≡ (Y ⊗ X )⊕(Y ⊗ X ) ⊕ (Y ⊗ X )⊕(Y ⊗ X ) k[y] k[y] k[y] k[y] (2.7) with d = d ⊗ 1 + 1 ⊗ d , and analogously on 2-morphisms. It follows from Y ⊗X Y X [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 eﬀec- tive 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 ) ◦⊗ ◦((ι ) ×(ι ) ), where ι : k[x, z] ֒→ k[x, y, z] etc. x,z ∗ k[x,y,z] y,z x,y x,z 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 , . . . , x ], W ) ∈ LG, its unit 1-morphism is (I , d ) with 1 n W I ^ M I = k[x, x ] · θ (2.8) W i i=1 ′ ′ ′ ′ where x ≡ (x , . . . , x ) is another list of n variables, {θ } is a chosen k[x, x ]-basis 1 n ′ ⊕n of k[x, x ] , and x ,x ′ ∗ d = ∂ W · θ ∧ (−) + (x − x ) · θ (2.9) I i i i i W [i] i=1 where ′ ′ ′ ′ ′ W (x , . . . , x , x , . . . x ) − W (x , . . . , x , x , . . . x ) 1 i−1 1 i x ,x i n i+1 n ∂ W = (2.10) [i] ′ x − x ∗ ∗ and θ is deﬁned by linear extension of θ (θ ) = δ and to obey the Leibniz j i,j i i 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 . Finally, the left and right unitors λ : I ⊗ X −→ X , ρ : X ⊗ I −→ X (2.11) X V X W 7 for X ∈ LG(W, V ) are deﬁned as projection to θ-degree zero on the units I and I , respectively; their explicit inverses (in the homotopy category LG(W, V )) were worked out in [CM] to act as follows: n o X X X ′ ′ z ,z z ,z −1 λ (e ) = θ . . . θ ∂ d . . . ∂ d ⊗ e , i a a X X j X 1 l [a ] [a ] l 1 ji l>0 a <···<a j 1 l n o X X X ′ ′ −1 +l|e | x ,x x ,x ( ) i ρ (e ) = (−1) e ⊗ ∂ d . . . ∂ d θ . . . θ i j X X a a X 1 l [a ] [a ] 1 l ji l>0 a <···<a j 1 l (2.12) where {e } is a basis of the module X, and d is identiﬁed with the matrix i X representing it with respect to {e }. 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]) End(I ) Jac . (2.13) W W 2.2 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, (M3) associator a: ◦( × Id ) → ◦ (Id × ), which is part of an adjoint LG LG equivalence, (M4) pentagonator π : (Id a)◦a◦(aId ) → a◦a (using shorthand notation LG LG explained below), (M5) left and right unitors l: ◦(I × Id ) → Id , r : ◦(Id × I) → Id , LG LG LG LG ′ ′ (M6) 2-unitors λ : 1 ◦ (l × 1) → (l ∗ 1) ◦ (a ∗ 1), ρ : r ◦ 1 → (1 ∗ (1 × r)) ◦ (a ∗ 1), and µ : 1 ◦ (r × 1) → (1 ∗ (1 × 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) 8 which is basically given by tensoring over k and taking sums of potentials. More precisely, according to [MM, Prop. 3.1.12], acts as (W, V ) ≡ (k[x], W ), (k[z], V ) 7−→ k[x, z], W + V ) ≡ W + V (2.15) on objects, while the functors on Hom catgories : LG×LG (V , V ), (W , W ) −→ LG V +V , W +W (2.16) (V ,V ),(W ,W ) 1 2 1 2 1 2 1 2 1 2 1 2 are given by ⊗ (up to a reordering of variables similar to the situation in Re- mark 2.1, see [MM, Def. 3.1.3]). Compatibility with horizontal composition is witnessed by the natural isomorphisms : ⊗◦(×) → ◦⊗ (U ,U ),(V ,V ),(W ,W ) 1 2 1 2 1 2 whose ((Y , Y ), (X , X ))-components are given by linearly extending 1 2 1 2 (Y Y ) ⊗ (X X ) −→ (Y ⊗ X ) (Y ⊗ X ) , 1 2 1 2 1 1 2 2 |f |·|e | 2 1 (f ⊗ f ) ⊗ (e ⊗ e ) 7−→ (−1) · (f ⊗ e ) ⊗ (f ⊗ e ) (2.17) 1 k 2 1 k 2 1 1 k 2 2 for Z -homogeneous module elements e , e , f , f , and isomorphisms on units 2 1 2 1 2 : I −→ I I (2.18) (W ,W ) W +W W W 1 2 1 2 1 2 are also standard, cf. [MM, Lem. 3.1.4]. (M2) The unit object in LG is I := (k, 0) . (2.19) Let 1 be the 2-category with a single object ∗ and only identity 1- and 2- morphisms. We deﬁne a strict 2-functor I : 1 → LG by setting I(∗) = I. (M3) The associator is a pseudonatural transformation a: ◦ ( × Id ) −→ ◦ (Id × ) ◦ A . (2.20) LG LG Here A is the rebracketing 2-functor (LG × LG) × LG → LG × (LG × LG), which we usually treat as an identity. The 1-morphism components a and 2- ((U,V ),W ) morphism components a of the associator are given by ((X,Y ),Z) −1 a = I , a = λ ◦ A ◦ ρ , (2.21) ((U,V ),W ) U+V +W ((X,Y ),Z) X,Y,Z X(Y Z) (XY )Z where A : X (Y Z) → (X Y ) Z is the rebracketing isomorphism for X,Y,Z LG, while λ and ρ are the 2-isomorphisms (2.11). The associator a and the pseudonatural transformation a : ◦ (Id × ) ◦ A −→ ◦ ( × Id ) (2.22) LG LG 9 − − −1 −1 with components a = I , a = λ ◦ A ◦ ρ U+V +W (XY )Z ((U,V ),W ) ((X,Y ),Z) X(Y Z) X,Y,Z are part of a biadjoint equivalence, see [MM, Lem. 3.2.5–3.2.6]. (M4) The pentagonator is an invertible modiﬁcation π : 1 ∗ (1 × a) ◦ a ∗ 1 ◦ 1 ∗ (a × 1 ) Id Id ××Id Id LG LG LG LG −→ a ∗ 1 ◦ a ∗ 1 (2.23) Id ×Id × ×Id ×Id LG LG LG LG where here and below we write vertical and horizontal composition of pseudonat- ural 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 π : (Id a) ◦ a ◦ (a Id ) −→ a ◦ a . (2.24) LG LG Its components are π = λ ◦ ⊗1 ⊗ . (((T,U),V ),W ) I ⊗I (T,U+V +W ) I (T +U+V,W ) T+U+V +W T+U+V +W T+U+V +W (2.25) (M5) The left and right (1-morphism) unitors are pseudonatural transformations l: ◦ (I × Id ) −→ Id , r : ◦ (Id × I) −→ Id (2.26) LG LG LG LG whose components are given by −1 l = I = r , l = λ ◦ ρ = r , (2.27) (∗,W ) W (W,∗) (1 ,X) X (X,1 ) ∗ X ∗ where we identify 1 × LG ≡ LG ≡ LG × 1 and I X ≡ X ≡ X I (see [MM, 0 0 Lem. 3.1.8 & 3.2.11 & 3.2.15] for details). The unitors l, r are part of biadjoint − − equivalences (l, l ), (r, r ) as explained in [MM, Lem. 3.2.13–3.2.15]. (M6) The 2-unitors are invertible modiﬁcations λ : 1 ◦ (l × 1) → (l ∗ 1) ◦ (a ∗ 1), ′ ′ ρ : r ◦ 1 → (1∗ (1×r))◦ (a∗ 1), and µ : 1◦ (r × 1) → (1∗ (1×l))◦ (a∗ 1), written here in the shorthand notation also employed in (M4) above, whose components are ′ −1 −1 ′ −1 λ = λ ◦ , ρ = ( ⊗ 1 ) ◦ λ , (V,W ) I ((∗,V ),W ) ((V,W ),∗) V +W I (V,W ) I V +W V +W ′ −1 µ = ρ . (2.28) ((V,∗),W ) I I V W Proposition 2.2. The data (M1)–(M6) endow LG with a monoidal structure. Proof. The straightforward but lengthy check of all coherence axioms is per- formed to prove Theorem 3.2.18 in [MM, Sect. 3.1–3.2]. 10 2.3 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) 7→ (ξ, ζ) on objects, 1- and 2-morphisms, (S2) syllepsis σ : 1 → b ◦ b, − − (S3) R: a ◦ b ◦ a → (Id b) ◦ a ◦ (b Id ) and S : a ◦ b ◦ a → (b Id ) ◦ LG LG LG a ◦ (Id b), LG 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 b: −→ ◦ τ (2.29) whose 1-morphism components b are given by I (up to a reordering of (V,W ) V +W variables, see [MM, Not. 3.1.2 & Lem. 3.3.5]), while the 2-morphism components b : (Y X) ⊗ b −→ b ⊗ (X Y ) (2.30) (X,Y ) (V ,V ) (W ,W ) 1 2 1 2 are deﬁned in [MM] as natural compositions of canonical module isomorphisms and structure maps of the bicategory LG. Explicitly, if {e } and {f } are bases a b of the underlying modules of X and Y , respectively, we have j j 1 m |e |·|f | −1 b : (f ⊗ e ) ⊗ θ . . . θ 7−→ (−1) δ . . . δ · λ (e ⊗ f ) . (2.31) (X,Y ) b a j ,0 j ,0 a b i i 1 m XY 1 m The braiding b and the pseudonatural transformation b : ◦ τ −→ (2.32) − − with components b = b and b = b are part of a biadjoint (W,V ) (Y,X) (V,W ) (X,Y ) equivalence, see [MM, Sect. 3.3.2]. N+1 Example 2.3. For a potential W = x , N ∈ Z , the matrix factorisation >2 b is precisely what is assigned to a “virtual crossing” in the construction (W,W ) of homological sl -tangle invariants of Khovanov and Rozansky [KR1] (see the second expression in [KR2, Eq. (A.9)]). 11 (S2) The syllepsis is an invertible modiﬁcation σ : 1 −→ b ◦ b (2.33) − −1 whose components σ : I → b ⊗ b are given by λ (up to (V,W ) V +W (V,W ) (V,W ) V +W a reordering of variables and a sign-less swapping of tensor factors, see [MM, Lem. 3.3.8]). (S3) The invertible modiﬁcations R: a ◦ b ◦ a −→ (Id b) ◦ a ◦ (b Id ) , LG LG − − − S : a ◦ b ◦ a −→ (b Id ) ◦ a ◦ (Id b) (2.34) LG LG have components R and S which act on basis elements, i. e. on ((U,V ),W ) ((U,V ),W ) tensor and wedge products of θ-variables, by a reordering with appropriate signs, see [MM, Lem. 3.3.11] for the lengthy explicit expressions. Theorem 2.4. The data (M1)–(M6) and (S1)–(S3) endow LG with a symmetric monoidal structure. Proof. It is shown in [MM, Sect. 3.1 & 3.3] that the data (S1)–(S3) are well-deﬁned and satisfy the coherence axioms for symmetric braidings. We note that instead of directly constructing the data (M1)–(M6) and (S1)–(S3) and verifying their coherence axioms, one could also employ Shul- man’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]. 2.4 Duality in LG 2.4.1 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 → I , coev : I → X ⊗ X, eev : X ⊗X → I X W X V X V and cg oev : I → X ⊗ X for every X ∈ LG(W, V ), X W subject to coherence axioms. In this section we recall the above data as con- structed in [CM] (this reference also spells out the coherence axioms). 12 ∨ (A1) Setting X = Hom (X, k[x, z]) and deﬁning the associated twisted dif- k[x,z] |φ|+1 ∨ ferential by d (φ) = (−1) φ◦d for homogeneous φ ∈ X , the left and right X X adjoints of X ∈ LG (k[x , . . . , x ], W ), (k[z , . . . , z ], V ) (2.35) 1 n 1 m are given by † ∨ † ∨ X = X [m] and X = X [n] , (2.36) respectively, where [m] is the m-th power of the shift functor [1] in (2.4) with itself. 0 D Hence if in a chosen basis d is represented by the block matrix ( ), then in D 0 T T 0 D 0 D 0 1 the dual basis d† is represented by ( ) if m is even, and by ( ) if m X T T −D 0 −D 0 1 0 † † is odd, and similarly for d †. It follows that X = X if m = n mod 2. (A2) To present the adjunction 2-morphisms † † ev : X ⊗ X −→ I , coev : I −→ X ⊗ X , X W X V † † eev : X ⊗ X −→ I , cg oev : I −→ X ⊗ X , (2.37) X V X W recall from [Li] the basic properties of residues (collected for our purposes in (x) n [CM, Sect. 2.4]), let {e } be a basis of X, and deﬁne Λ = (−1) ∂ d . . . ∂ d , i x X x X 1 n (z) Λ = ∂ d . . . ∂ d . In [CM] the theory of homological perturbation and z X z X 1 m associative Atiyah classes were used to obtain the following explicit expressions: X X ∗ +l|e | ( ) j ev (e ⊗ e ) = (−1) θ . . . θ X j a a i 1 l l>0 a <···<a 1 l " # ′ ′ x,x x,x (z) Λ ∂ d . . . ∂ d dz X X [a ] [a ] 1 ij · Res , ∂ V, . . . , ∂ V z z 1 m X X ∗ l+(n+1)|e | eev (e ⊗ e ) = (−1) θ . . . θ X j a a i 1 l l>0 a <···<a 1 l " # ′ ′ z,z z,z (x) ∂ d . . . ∂ d Λ dx X X [a ] [a ] l 1 ij · Res , ∂ W, . . . , ∂ W x x 1 n n o r+1 ′ ′ +mr+s z,z z,z ( ) m ∗ coev (γ) = (−1) ∂ d . . . ∂ d e ⊗ e , X X X i [b ] [b ] 1 r ij i,j n o ′ ′ x,x x,x (r¯+1)|e |+s ∗ j n cg oev (γ¯) = (−1) ∂ (d ) . . . ∂ (d ) e ⊗ e (2.38) X ¯ X ¯ X j [b ] [b ] r¯ 1 ji i,j where b , b and s , s ∈ Z are uniquely determined by requiring that b < i ¯ m n 2 1 ¯ ¯ n · · · < b , b < · · · < b , as well as γ¯θ¯ . . . θ¯ = (−1) θ . . . θ and γθ . . . θ = r 1 r¯ 1 n b b b b 1 r 1 r¯ (−1) θ . . . θ . 1 m Theorem 2.5. The data (A1)–(A2) endow the bicategory LG with left and right adjoints for every 1-morphism. Proof. This is [CM, Thm. 6.11]. (In fact LG even has a “graded pivotal” struc- ture, see [CM, Sect. 7].) 13 2.4.2 Duals for objects Endowing the symmetric monoidal bicategory LG with duals for objects amounts to specifying the following data: ∗ ∗ (D1) an object W ≡ (k[x], W ) ∈ LG for every W ≡ (k[x], W ) ∈ LG, ∗ ∗ (D2) 1-morphisms ev : W W → I and coev : I → W W such that there W W are 2-isomorphisms c : r ⊗ (I ev ) ⊗ a ⊗ (coev I ) ⊗ l −→ I , l (W,∗) W W ((W,W ),W ) W W W − − ∗ ∗ ∗ c : l ∗ ⊗ (ev I ) ⊗ a ⊗ (I coev ) ⊗ r −→ I . r (∗,W ) W W ∗ ∗ W W W ((W ,W ),W ) In this section we provide the above data; the explicit isomorphisms c , c are l r constructed in [MM, Ch. 4]. (D1) The dual of W ≡ (k[x], W ) is (k[x], −W ) ≡ W ≡ −W . (D2) The adjunction 1-morphisms exhibiting −W as the (left) dual are the matrix factorisations ev = I and coev = I (2.39) W W W W of W (x ) − W (x), viewed as 1-morphisms (−W ) W → I and I → W (−W ), respectively. Note that −W is also the right dual of W , with eev = I and cg oev = I W W W W viewed as 1-morphisms W (−W ) → I and I → (−W ) W . Proposition 2.6. The data (D1)–(D2) endow the monoidal bicategory LG with duals for every object. Proof. The cusp isomorphisms c , c are computed in terms of the unitors λ, ρ l r 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 , coev A A themselves have left and right adjoints, which in turn have left and right adjoints, and so on. Hence Proposition 2.6 together with Theorem 2.5 implies: Corollary 2.7. Every object of LG is fully dualisable. 14 2.5 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 gr section we recall the bicategory of graded Landau-Ginzburg models LG from [CM, CRCR, Mu] (see also [BFK]) and observe that it inherits the symmetric gr monoidal structure from LG. Moreover, every object in LG is fully dualisable. gr An object of LG is a pair (k[x , . . ., x ], W ) where now k[x , . . . , x ] is a 1 n 1 n graded ring by assigning degrees |x | ∈ Q to the variables x , and W ∈ i >0 i k[x , . . . , x ] is either zero or a potential of degree 2. The central charge of 1 n W ≡ (k[x , . . . , x ], W ) is the numerical invariant 1 n c(W ) = 3 1 − |x | . (2.40) i=1 gr A 1-morphism (k[x], W ) → (k[z], V ) in LG is a summand of a ﬁnite-rank matrix factorisation (X, d ) of V − W over k[x, z] such that the following four L L 0 0 1 1 conditions are satisﬁed: (i) the modules X = X and X = X q∈Q q q∈Q q are Q-graded, (ii) the action of x and z on X are respectively of Q-degree |x | i j i and |z |, (iii) the map d has Q-degree 1, and (iv) if we write {−} for the shift in j X i ⊕a ∼ i,q Q-degree and if X k[x, z]{q} for i ∈ {0, 1}, then {q ∈ Q | a 6= 0} i,q q∈Q must be a subset of i + G , where V −W G := |x |, . . . , |x |, |z |, . . . , |z | ⊂ Q and G := Z . (2.41) V −W 1 n 1 m 0 gr A 2-morphism in LG between two 1-morphisms (X, d ), (X , d ′) is a cohomol- X X ogy class of Z - and Q-degree 0 with respect to the diﬀerential δ in (2.3). 2 X,X We continue to write [−] for the Z -grading shift and {−} for the Q-grading gr shift. Translating [KST, Thm. 2.15] into our conventions we see that LG (W, V ) has the structure of a triangulated category with shift functor Σ := [1]{1} . (2.42) gr Since the categories LG (W, V ) are idempotent complete (cf. [KST, Lem. 2.11]) gr the construction of [DM] ensures that horizontal composition in LG can be deﬁned analogously to (2.6). Moreover, the units I of LG can naturally be endowed with an appropriate Q-grading (by setting |θ | = |x | − 1 and |θ | = i i gr 1 − |x |), and the associator α and unitors λ, ρ of LG are those of LG (as they gr manifestly have Q-degree 0). Hence LG is indeed a bicategory. 2 GR Without condition (iv) we still obtain a bicategory LG , with the same structures that we gr GR exhibit here for LG . As explained in [Mu, Lecture 3], the Hom categories LG (W, V ) are gr gr equivalent to inﬁnite direct sums of LG (W, V ) with itself, hence we can restrict to LG . 15 gr The bicategory LG 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 and from the structure maps α, λ, ρ, their inverses and (Q-degree 0) swapping maps, respectively. For a 1-morphism gr X ∈ LG (k[x , . . . , x ], W ), (k[z , . . ., z ], V ) (2.43) 1 n 1 m we deﬁne its left and right adjoint as † ∨ 1 † ∨ 1 X = X [m]{ c(V )} , X = X [n]{ c(W )} . (2.44) 3 3 The above shifts in Q-degree are necessary to render the adjunction maps ev , coev , eev , cg oev in (2.37) and (2.38) to be of Q-degree 0 so that they are X X X X gr gr indeed 2-morphisms in LG . Finally, the (left and right) dual of (k[x], W ) ∈ LG is (k[x], −W ) with the same grading, and the matrix factorisation underlying the adjunction 1-morphisms ev , coev , eev , cg oev is again I , but now viewed as W W W W W a Q-graded matrix factorisation. In summary, we have: gr Theorem 2.8. The bicategory LG inherits a symmetric monoidal structure gr from LG, every object of LG has a dual, and every 1-morphism has adjoints. gr Corollary 2.9. Every object of LG is fully dualisable. gr 3 Extended TQFTs with values in LG and LG gr In this section we study extended TQFTs with values in LG and LG . We brieﬂy review framed and oriented 2-1-0-extended TQFTs and their “classiﬁcation” in terms of fully dualisable objects and trivialisable Serre automorphisms, respec- gr tively. Then we observe that every object W ≡ (k[x , . . ., x ], W ) in LG or LG 1 n 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 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). 3.1 Framed case Recall from [SP, Sect. 3.2] and [Ps, Sect. 5] that there is a symmetric monoidal fr bicategory Bord of framed 2-bordisms. Its objects, 1- and 2-morphisms are, 2,1,0 roughly, disjoint unions of 2-framed points + and −, 2-framed 1-manifolds with 16 boundary and (equivalence classed of) 2-framed 2-manifolds with corners. For any symmetric monoidal bicategory B, the cobordism hypothesis, originally due fr to [BD], describes the 2-groupoid Fun (Bord , B) (of symmetric monoidal sym ⊗ 2,1,0 fr 2-functors Z : Bord → B, their symmetric monoidal pseudonatural trans- 2,1,0 formations and modiﬁcations) in terms of data internal to B that satisfy certain fr ﬁniteness conditions. Objects of Fun (Bord , B) are called extended framed sym ⊗ 2,1,0 TQFTs with values in B. fd 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 fd for the core of B , i. e. the subbicategory of B 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 fr fd Fun Bord , B −→ K (B ) , sym ⊗ 2,1,0 Z 7−→ Z(+) . (3.1) fr Note that thanks to the description of Bord as a symmetric monoidal bi- 2,1,0 category 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 ⊗ ev . Similarly, + + A A fr 2-morphisms in Bord can be decomposed into cylinders and adjunction 2- 2,1,0 morphisms 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 mod- els LG. As a direct consequence of the cobordism hypothesis and Corollary 2.7 we have: Proposition 3.2. Every object W ≡ (k[x , . . . , x ], W ) ∈ LG determines an 1 n extended framed TQFT fr fr fr Z : Bord −→ LG with Z (+) = W . (3.2) W 2,1,0 W 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 fr precise by relating Z to the (non-extended) closed oriented TQFT or Z : Bord −→ Vect (3.3) W k 2,1 which via the standard classiﬁcation in terms of commutative Frobenius algebras (see e. g. [Ko]) is described by the Jacobi algebra Jac with pairing h−, −i : Jac ⊗ Jac −→ k (3.4) W W k W 17 induced by the residue trace map φ dx Jac −→ k , φ 7−→ Res =: hφi , (3.5) W W ∂ W, . . . , ∂ W x x 1 n i. e. hφ, ψi = hφψi . W W To recover the k-algebra Jac with its multiplication µ : φ ⊗ ψ 7→ φψ, W Jac fr we want to show that Jac and µ are what Z assigns to “the” circle and W Jac W W “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 1 2 the equivalent description of 2-framed circles in terms of immersions ι: S → R 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 . In terms of the structure 1- fr morphisms of Bord (whose horizontal composition we write as #), we have 2,1,0 (see [DSPS, Sect. 1.2]) 1 † S = ev # ev . (3.6) + + This is the correct choice in the sense that for every integer k, there is a 2-framed 1 2 circle S , and for every pair (k, l) ∈ Z there is a pair-of-pants 2-morphism fr 1 1 1 S ⊔ S → S in Bord , and only for k = 0 = l do we get a multiplication. k l k+l 2,1,0 This is “the” 2-framed pair-of-pants for us. Theorem 3.3. For every (k[x , . . . , x ], W ) ∈ LG, we have that 1 n fr 1 fr Z (S ), Z (pair-of-pants) (3.7) W 0 W is isomorphic to (Jac , µ ) as a k-algebra. W Jac fr 1 † ∨ ∨ Proof. Note ﬁrst that Z (S ) ev ⊗ ev = ev ⊗(ev [0]) = I ⊗ ′ I W W W W k[x,x ] W 0 W W ∼ 2 is isomorphic in LG((k, 0), (k, 0)) = vect to the vector space Jac (viewed as a Z -graded vector space concentrated in even degree). One can check that an ∼ ∼ explicit isomorphism κ: I ⊗ I = End (I ) = Jac is given by linear W k[x,x ] LG W W ′ ∗ extension of p(x)q(x ) · e ⊗ e 7→ p(x)q(x) · δ , where p and q are polynomials i i,j and {e } is a basis of the k[x, x ]-module I . i W fr Next we prove that Z sends the pair-of-pants to the commutative multiplica- tion µ . For this we decompose the pair-of-pants into generators, namely into Jac cylinders over the left and right elbows ev : − ⊔ + → ∅ and ev : ∅ → − ⊔ +, + + respectively, and the “upside-down saddle” ev : ev # ev → 1 (which is ev + + −⊔+ called v in [DSPS, Ex. 1.1.7]). Then 1 1 1 pair-of-pants = 1 # ev # 1† : S ⊔ S −→ S . (3.8) ev ev ev + + 0 0 0 fr fr Hence if Z (ev ) = ev , then the functor Z sends this pair-of-pants to ev ev W W W 1 ⊗ ev ⊗1† , which by pre- and post-composition with the isomorphism ev ev ev W W 18 † κ: ev ⊗ ev Jac becomes a map µ : Jac ⊗ Jac → Jac . Noting that W W W W k W W both κ and ev act diagonally (with ev (e ⊗ e ) = δ since ev : (−W ) ev ev j i,j W W W i W → I has trivial target, cf. the explicit expression for ev in (2.38)), we ﬁnd ev that µ is indeed given by multiplication of polynomials, i. e. µ = µ . Jac fr To complete the proof we need to argue that Z assigns our choice of counit ev to the upside-down saddle ev , and not some other choice of adjunc- ev ev tion data. By [Ps, Thm. 3.17 & Thm. 8.1], extended framed TQFTs are equiva- lent to “coherent fully dual pairs” in their target bicategories, see [Ps, Def. 3.12] fr 1 for the details. For the algebra structure on Z (S ), we only need a “coher- W 0 ent dual pair” as deﬁned in [Ps, Def. 2.6]. One straightforwardly checks that (W, −W, ev , coev , c , c ) satisﬁes all the deﬁning properties of a coherent dual W W r l fr fr pair, ensuring that Z can indeed be chosen such that Z (ev ) = ev . (The ev ev W W + W key deﬁning properties of coherent dual pairs for us to check are the so-called swal- lowtail identities of [Ps, Def. 2.6], which can be viewed as consistency constraints on our cusp isomorphism c . But since c , c and all other 2-morphisms that ap- l l r pear in the swallowtail identities are structure maps of the underlying bicategory of LG, the coherence theorem for bicategories guarantees that the constraints are satisﬁed.) Remark 3.4. The ﬁnite-dimensional k-algebra Jac is typically not separable. N+1 For example, if W = x with N ∈ Z the algebra Jac has non-semisimple >2 W representations (as multiplication by x has non-trivial Jordan blocks) and hence cannot be separable. Thus Jac is not fully dualisable in the bicategory Alg of ﬁnite-dimensional k-algebras, bimodules and intertwiners [Lu, SP], so Jac cannot describe an extended TQFT with values in Alg . Proposition 3.2 and Theorem 3.3 explain how Jac does appear in an extended TQFT with values in LG, namely as the algebra assigned to the circle S and its pair-of-pants. For an algebra A ∈ Alg its Hochschild cohomology HH (A) is isomorphic to ev ⊗ ev , and for Hochschild homology one ﬁnds HH (A) ev ⊗ b ⊗ A A • A (A,A) coev . Similarly, for every object W ≡ (k[x , . . ., x ], W ) ∈ LG we may deﬁne A 1 n • † HH (W ) := ev ⊗ ev , HH (W ) := ev ⊗ b ⊗ coev . (3.9) W W • W (W,W ) W Thus by Theorem 3.3 we have HH (W ) Jac , and paralleling the ﬁrst part of the proof we ﬁnd HH (W ) = Jac [n] as Z -graded vector spaces (because • W 2 the matrix factorisations b and coev are I I and I I = I [n], (W,W ) W W W W respectively). Hence HH (W ) and HH (W ) precisely recover the Hochschild co- homology 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]: Corollary 3.5. For every W ≡ (k[x], W ) ∈ LG we have • • ∼ ∼ HH (W ) = HH MF(k[x], W ) , HH (W ) = HH MF(k[x], W ) . (3.10) • • 19 Remark 3.6. Proposition 3.2, Theorem 3.3 and Corollary 3.5 have direct ana- logues for the graded Landau-Ginzburg models of Section 2.5. Firstly, Theo- rem 3.1 and Corollary 2.9 immediately imply that every object (k[x , . . . , x ], W ) ∈ 1 n gr LG determines an extended TQFT fr fr gr Z : Bord −→ LG . (3.11) W,gr 2,1,0 Secondly, going through the proof of Theorem 3.3 we see that to the cir- 1 fr cle S and its pair-of-pants, Z assigns the Jacobi algebra Jac which is 0 W,gr now a Q-graded algebra with degree-preserving multiplication. We note that here it is important that the upside-down saddle ev : ev # ev → 1 in- ev + + −⊔+ † ∨ ∨ volves the left adjoint of ev : by (2.44) we have ev = ev [0]{0} = ev , so + W W W fr Z (pair-of-pants) really gives a map W,gr † † † ∼ ∼ Jac ⊗ Jac = ev ⊗ ev ev ⊗ ev −→ ev ⊗ ev = Jac . W k W W W W W W W W (3.12) (Incorrectly using the right adjoint ev = ev [2n]{ c(W )} would lead to un- W W wanted Q-degree shifts in the multiplication. In Remark 3.16 below however we are naturally led to use the right adjoint ev to obtain the correct graded trace map h−i on Jac .) W W gr Thirdly, for every (k[x , . . . , x ], W ) ∈ LG the matrix factorisation underlying 1 n 1 1 ∨ ∨ coev is I I = I [n]{ c(W )} = I [n − 2]{ c(W )}, and hence we have W W W W W 3 3 • 1 ∼ ∼ HH (W ) Jac , HH (W ) Jac [n − 2]{ c(W )} . (3.13) = = W • W 3.2 Oriented case An extended oriented TQFT with values in a symmetric monoidal bicategory B or or is a symmetric monoidal 2-functor Z : Bord → B. Here Bord is the bicat- 2,1,0 2,1,0 egory 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) or and their relations. Hence objects of Bord are disjoint unions of positively 2,1,0 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 fd points of the SO(2)-action induced on B by the SO(2)-action which rotates the fr framings in Bord . This was worked out in detail in [HSV, HV, He] as we 2,1,0 brieﬂy review next. fd An SO(2)-action on B is a monoidal 2-functor ̺ from the fundamental 2- fd groupoid Π (SO(2)) to the bicategory of autoequivalences of B . Since SO(2) is path-connected, Π (SO(2)) has essentially a single object ∗ which ̺ sends to fd the identity Id fd on B . Since π (SO(2)) Z the action of ̺ on 1-morphisms B 1 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 20 fd Z(+) =: A, the relevant choice for ̺(1 ) is the Serre automorphism S of A ∈ B . ∗ A By deﬁnition S is the 1-morphism S := r ⊗ 1 eev ⊗ b 1 ⊗ 1 eev ⊗ r : A −→ A . (3.14) A (A,∗) A A (A,A) A A A A Here we denote the braiding, horizontal composition and monoidal product in B gr by b, ⊗ and , respectively, as we do in LG and LG . fd SO(2) The bicategory of SO(2)-homotopy ﬁxed points K (B ) was deﬁned and endowed with a natural symmetric monoidal structure in [HV]. Objects of fd SO(2) fd K (B ) are pairs (A, σ ) where A ∈ B and σ is a trivialisation of the A A Serre automorphism S , i. e. a 2-isomorphism S → 1 in B. A 1-morphism A A A ′ fd SO(2) fd ′ (A, σ ) → (A , σ ′) in K (B ) is an equivalence F ∈ B (A, A ) such that A A λ ◦ (σ ′ ⊗ 1 ) ◦ S = ρ ◦ (1 ⊗ σ ) where S is the 2-isomorphism constructed F A F F F F A F ′ fd SO(2) in the proof of [HV, Prop. 2.8], and 2-morphisms F → F in K (B ) are 2-isomorphism F → F in B. Building on [Lu, SP, HSV, HV], extended ori- ented 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 or fd SO(2) Fun Bord , B −→ K (B ) , sym ⊗ 2,1,0 Z 7−→ Z(+) . (3.15) 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: Lemma 3.8. Let W ≡ (k[x , . . . , x ], W ) ∈ LG. Then S I [n]. 1 n W W Proof. According to Sections 2.3–2.4, the factors r , 1 , eev , b , 1 (W,∗) W W (W,W ) W and r in the deﬁning expression (3.14) are all given by the matrix factorisation underlying the unit I ∈ LG(W, W ), while the matrix factorisation underlying † † ∨ ∨ ∼ ∼ ∼ eev = eev [2n] = eev is I I [n] I [n]. This leads to S I [n]. (A = = = W W W W W W W W straighforward computation, taking into account subtleties of the kind mentioned in Remark 2.1, is carried out in the proof of [MM, Lem. 5.2.3] to construct an explicit isomorphism S → I only in terms of λ, ρ and standard swapping W W isomorphisms.) The general fact I ≇ I [1] (even Hom (I , I [1]) = 0 is true) together W W LG(W,W ) W W with Theorem 3.7 thus imply: Proposition 3.9. An object W ≡ (k[x , . . . , x ], W ) ∈ LG determines an ex- 1 n tended oriented TQFT or or or Z : Bord −→ LG with Z (+) = W (3.16) W 2,1,0 W if and only if n is even. 21 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 3 d a Serre functor S such that Σ S . The triangulated category LG(0, W ) is T T known to admit a Serre functor S = [n] = [n − 2]. Hence LG(0, W ) is LG(0,W ) weakly (n − 2)-Calabi-Yau, the Serre automorphism and Serre functor coincide in the sense that S ⊗ (−) = S , and the condition that S is trivialisable W LG(0,W ) W 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 Jac = k[x , . . . , x ]/(∂ W, . . . , ∂ W ) W 1 n x x 1 n 2 2 2 = k[x , . . . , x , y]/(∂ (W + y ), . . ., ∂ (W + y ), ∂ (W + y )) 1 n x x y 1 n = Jac 2 , (3.17) W +y every non-extended oriented Landau-Ginzburg model appears as part of an or or extended oriented TQFT Z or Z (depending on whether n is even W W +y or odd, respectively), namely as the commutative Frobenius algebra with or 1 or 1 underlying vector space Z (S ) or Z (S ). Note that for this argument W W +y 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. (ii) Instead of LG one can also consider the symmetric monoidal bicategory •/2 LG which is equal to LG except that the vector space of 2-morphisms • ′ (X, d ) → (X, d ) is deﬁned to be H (Hom (X, X ))/Z , i. e. X X k[x,z] 2 X,X both even and odd cohomology of the diﬀerential δ ′ in (2.3) are in- X,X cluded while ζ ∈ Hom (X, X ) and −ζ are identiﬁed after taking co- k[x,z] homology. Dividing out this Z -action circumvents the issue that with- out it the interchange law would only hold up to a sign, as we have |ζ |·|ξ | 2 1 (ζ ⊗ζ )◦(ξ ⊗ξ ) = (−1) (ζ ◦ξ )⊗(ζ ◦ξ ) for appropriately compos- 1 2 1 2 1 1 2 2 able homogeneous 2-morphisms. Such Z -quotients also appear in [KR1]; •/2 the bicategory LG is described in more detail in [MM, Sect. 5.3.1] (where it is denoted LG). •/2 In particular, for every (k[x , . . . , x ], W ) ∈ LG there is an even/odd 1 n isomorphism I I [n] for n even/odd. Hence by Lemma 3.8 every object W W •/2 •/2 of LG determines an extended oriented TQFT with values in LG . (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. A Serre functor of T is an additive equivalence S : T → T together with isomorphisms Hom (A, B) Hom (B, S (A)) that are natural in A, B ∈ T . T = T T 22 Part of this involves the natural diﬀerential Z -graded categories (with dif- ferential δ as above) studied in [Dy], whose even cohomologies are the X,X matrix factorisation categories of Section 2.1. Such bicategories of dif- ferential 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 in- volve 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 de- fect TQFT on super bordisms (which again has not been deﬁned in detail as far as we know), paralleling the non-super construction of [DKR] reviewed in [Ca]. Theorem 3.12. For every (k[x , . . . , x ], W ) ∈ LG with n even, we have that 1 n fr 1 fr fr fr Z (S ), Z (pair-of-pants), Z (cup)(1), Z (cap) (3.18) W 0 W W W is isomorphic to (recall (3.5) for the residue trace h−i ) Jac , µ , 1, hζ(−)i (3.19) W Jac W fr as a commutative Frobenius k-algebra, where the traces Z (cap) and h−i fr 1 induce the Frobenius pairings on Z (S ) and Jac , respectively, and ζ ∈ Jac W W W 0 is a uniquely determined invertible element. Proof. The isomorphism on the level of k-algebras was already established in fr Theorem 3.3, it remains to compute the action of Z on the cap and cup 2- morphisms. The cap is the bordism eev from the 2-framed circle ev # ev to 1 . We ﬁrst ev + ∅ + + fr † assume that Z sends it to the 2-morphism eev from ev ⊗ ev = ev ⊗ ev ev W W W W W W to I . Since ev : (k[x], −W )(k[x], W ) → (k, 0) has trivial target, only the sum- 0 W mand l = 0 contributes to the expression for eev in (2.38), and pre-composing ev with the isomorphism Jac ev ⊗ ev from the proof of Theorem 3.3 pro- W W W duces the residue trace h−i . 1 † Similarly, the cup: ∅ → S = ev # ev is equal to coev . Using the explicit + + ev 0 + fr expression for coev in (2.38) we see that post-composing Z (cup)(1) with the ev W W isomorphism ev ⊗ ev Jac is indeed the unit 1 ∈ Jac . W W W W To complete the proof we must investigate to what extent our choice of adjunc- tion data in LG gives rise to a “coherent fully dual pair” (where again we rely on 23 the result of [Ps] that extended framed TQFTs are equivalent to coherent fully dual pairs): if the coherent dual pair (W, −W, ev , coev , c , c ) can be lifted to W W r l fr fr a coherent fully dual pair then Z can be chosen such that Z (eev ) = eev . ev ev W W + W First we observe that by Lemma 3.8 there is a “fully dual pair” W, −W, ev , coev , I , I , c , c , µ , ǫ , µ , ǫ , ψ, φ (3.20) W W W W r l e e c c in the sense of [Ps, Def. 3.10], where φ := λ =: ψ and µ , ǫ , µ , ǫ are equal I e e c c to coev , ev , coev , ev up to appropriate composition with the iso- ev ev coev coev W W W W ±1 ±1 morphisms λ , ρ . As explained in the proof of [Ps, Thm. 3.16], every fully dual pair can be made coherent by changing only the counit 2-morphisms by composi- tion with an automorphism ζ of I (and possibly the cusp isomorphism c which W l however in our case is not necessary as observed in the proof of Theorem 3.3). Given a fully dual pair, the map ζ is uniquely determined by the cusp-counit equation of [Ps, Def. 3.12], which involves two adjunction maps on one side and none on the other. In our case one ﬁnds that the constraint reduces to the equal- ity of two linear maps Jac ⊗ Jac → k, one of which involves the residue trace W k W h−i pre-composed with ζ ∈ Aut(I ) ⊂ End(I ) = Jac , while the other is a W W W W composite of structure maps of the symmetric monoidal bicategory LG (without any adjunction maps). Paralleling the above proof we see that for even n, the extended oriented TQFT or Z also assigns the Frobenius algebra Jac to the oriented circle, pair-of-pants, cup and cap (up to an invertible element ζ ∈ Jac ). N+1 Example 3.13. For every N ∈ Z , the potential x determines an extended >2 •/2 oriented TQFT with values in the symmetric monoidal bicategory LG intro- duced in Remark 3.11(ii). We denote this TQFT by Z as it recovers – directly KR 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 or saddle bordisms in Bord . Except for the saddle we have already computed 2,1,0 all these assignments of Z for any potential W in Theorems 3.3 and 3.12, and KR N+1 for W = x they match the prescriptions of [KR1] (except for non-essential prefactors for the cap and cup morphisms). To establish that the TQFT Z indeed matches that of [KR1, Sect. 9] it re- KR mains to compute Z (saddle) = Z (cg oev ) and compare it to the explicit KR KR ev matrix expressions in [KR1, Page 81] (or Page 95 of arXiv:math/0401268v2 [math.QA]). Since Z (cg oev ) = cg oev this is another exercise in using the formu- KR ev ev + N+1 las (2.38) for adjunction 2-morphisms. This is carried out in [MM, Sect. 5.3.2], ﬁnding   e 1 0 0   −e 1 0 0   Z (saddle) = (3.21) KR   0 0 −1 1 0 0 −e −e 234 124 24 P a b c where the entries e := x x x ∈ k[x , x , x , x ] depend on four ijk 1 2 3 4 i j k a+b+c=N−1 N+1 variables as the source and target of cg oev involve four copies of x ∈ ev N+1 •/2 4 LG . Up to a minor normalisation issue the expression (3.21) agrees with that of [KR1]. In summary, we veriﬁed that the construction of [KR1, Sect. 9] can be under- N+1 stood as an application of the cobordism hypothesis to the potential W = x . gr We return to the bicategory LG of Section 2.5. All the above results in the gr present section have analogues or reﬁnements in LG . In particular: gr Proposition 3.14. An object W ≡ (k[x , . . . , x ], W ) ∈ LG determines an 1 n extended oriented TQFT or or gr or Z : Bord −→ LG with Z (+) = W (3.22) W,gr 2,1,0 W,gr gr if and only if [n − 2]{ c(W )} Id . LG (0,W ) Proof. By Theorem 3.7, W determines a TQFT as stated if and only if its Serre automorphism S is trivialisable. Paralleling the proof of Lemma 3.8 we see that, using (2.44), the matrix factorisation underlying S is eev = ∨ 2 2 1 eev [2n]{ c(W )}. Hence S is isomorphic to I [2n]{ c(W )} = I [n]{ c(W )} W W W 3 3 3 I [n − 2]{ c(W )}. gr/Z gr Let LG be the symmetric monoidal 2-category obtained from LG by re- gr gr placing the hom categories LG (W, V ) with the orbit categories LG (W, V )/Z obtained by dividing out the action of the shift functor Σ = [1]{1}, i. e. gr Hom (X, Y ) = Hom (X, Σ (Y )) (3.23) gr/Z LG LG k∈Z gr/Z gr for 1-morphism X, Y ∈ Ob(LG (W, V )) = Ob(LG (W, V )). It follows that in /Z gr k LG , we have X = Σ (X) for all 1-morphisms X and k ∈ Z (with 1 viewed as a 2-isomorphism of degree k). In the setting of orbit categories, Calabi-Yau varieties give rise to oriented extended TQFTs: gr Corollary 3.15. If for (k[x , . . . , x ], W ) ∈ LG the hypersurface {W = 0} in 1 n weighted projective space is a Calabi-Yau variety, then W determines an extended or gr/Z oriented TQFT Bord → LG . 2,1,0 More precisely, (3.21) agrees with the saddle morphism of [KR1] if the arbitrary polynomial r a b c d of degree N − 2 in loc. cit. is set to x x x x , and if non-scalar entries of the 1 2 3 4 a+b+c+d=N−2 matrix are multiplied by . 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 † . I I ,ev ⊗ ev W −W N+1 N+1 25 Proof. We write Y for the zero locus of W in weighted projective space. The variety Y is Calabi-Yau if and only if the condition c (Y ) = 0 is satisﬁed W 1 W by the ﬁrst Chern class, which in our normalisation convention is equivalent to P P n n |x | = |W| = 2. 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DOI: 10.1007/s00220-020-03871-5
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Abstract

Extending Landau-Ginzburg models to the point Nils Carqueville Flavio Montiel Montoya [email protected] [email protected] Fakult¨at fu¨r Mathematik, Universit¨at Wien, Austria We classify framed and oriented 2-1-0-extended TQFTs with val- ues in the bicategories of Landau-Ginzburg models, whose objects and 1-morphisms are isolated singularities and (either Z - or (Z × Q)- 2 2 graded) matrix factorisations, respectively. For this we present the relevant symmetric monoidal structures and ﬁnd that every object W ∈ k[x , . . . , x ] determines a framed extended TQFT. We then 1 n compute the Serre automorphisms S to show that W determines an oriented extended TQFT if the associated category of matrix factori- sations 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 construc- N+1 tion of the extended TQFT based on W = x given by Khovanov and Rozansky can be derived directly from the cobordism hypothesis. arXiv:1809.10965v2 [math.QA] 4 Dec 2020 Contents 1 Introduction 2 2 Bicategories of Landau-Ginzburg models 5 2.1 Deﬁnition of LG . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Monoidal structure for LG . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Symmetric monoidal structure for LG . . . . . . . . . . . . . . . . 11 2.4 Duality in LG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4.1 Adjoints for 1-morphisms . . . . . . . . . . . . . . . . . . . 12 2.4.2 Duals for objects . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Graded matrix factorisations . . . . . . . . . . . . . . . . . . . . . 15 gr 3 Extended TQFTs with values in LG and LG 16 3.1 Framed case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Oriented case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1 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 cer- tain 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 bor- disms of all dimensions from 0 to n is highly restrictive. Speciﬁcally, the cobor- dism 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 −→ B (1.1) 2,1,0 where σ = fr (or σ = or), without any mention of ∞-categories. The bicategories Bord of points, 1-manifolds with boundary and 2-manifolds with corners (all 2,1,0 2 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 (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 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) or TQFTs. Recall e. g. from [Ko] that such TQFTs Z : Bord → V are equiva- ne 2,1 lent to commutative Frobenius algebras in V, where V is a symmetric monoidal 1-category. Important examples are the categories of vector spaces, possibly with Z Z a Z - or Z-grading. In V = Vect or V = Vect , Dolbeault cohomologies of k k Calabi-Yau manifolds serve as examples of non-separable commutative Frobe- nius algebras (describing B-twisted sigma models). Another class of examples of generically non-separable Frobenius algebras (in Vect ) are the Jacobi algebras k[x , . . . , x ]/(∂ W, . . . , ∂ W ) of isolated singularities described by polynomi- 1 n x x 1 n als 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 : Bord → V can be extended to ne 2,1 the point if there is a symmetric monoidal bicategory B and an extended TQFT Z : Bord → B such that (with I ∈ B the unit object, and ∅ = I ) B Bord 2,1,0 2,1,0 ∼ ∼ V End (I ) and Z Z . (1.2) = = B B ne End σ (∅) Bord 2,1,0 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 is a restriction of an appropriate ne def defect TQFT Z , then Z can be extended to the point (at least as a framed ne ne def theory), with the bicategory B def associated to Z as target.” ne ne Let us unpack this statement and give concrete meaning to it. A 2-dimensional def defect TQFT is a symmetric monoidal functor Z on a category of stratiﬁed and ne decorated oriented 2-bordisms, see [DKR, CRS] or the review [Ca]. Restricting def Z to only trivially stratiﬁed bordisms (meaning that there are no 1- or 0- ne strata) 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 def ne 3 def from any defect TQFT Z (along the same lines as one constructs commutative ne Frobenius algebras from closed TQFTs). In the case of state sum models the 2-category is equivalent to the full subbicategory ssFrob ⊂ Alg of separable symmetric Frobenius algebras [DKR], and indeed End (k) = Vect where k ssFrob 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 . And in the case of Landau-Ginzburg models it should gr be the bicategory LG (or its Q-graded version LG ) 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 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. gr In Section 2 we review the bicategories LG and LG , and we present symmetric monoidal structures for them which on objects reduce to the sum of polynomi- als; the unit object is the zero polynomial, and its endomorphism categories are 2 Z equivalent to Vect and Vect , respectively. Moreover, we prove that every ob- k k gr ject in both LG and LG is fully dualisable (Corollaries 2.7 and 2.9). Careful and lengthy checks that the data we supply satisfy the coherence axioms of sym- metric monoidal bicategories are performed in the PhD thesis [MM] for the case gr LG, and we explain how they carry over to LG . It follows immediately from the cobordism hypothesis that every object in LG gr gr or LG determines an extended framed TQFT (with values in LG or LG ), while generically Landau-Ginzburg models cannot be extended to the point with target Alg . 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 (see (3.14)) is isomorphic to the unit 1-morphism I . In Section 3.2, we show that for a potential W ∈ k[x , . . . , x ] viewed as an 1 n object in LG we have S = I [n] where [n] is the n-fold shift functor which W W satisﬁes [2] = [0], cf. Section 2.1. Since I ≇ I [1] this implies that W determines W W an extended oriented TQFT (cf. Proposition 3.9) or or Z : Bord −→ LG (1.3) W 2,1,0 4 if and only if n is even, and we discuss the relation to Serre functors and Calabi- Yau categories in Remark 3.10. For a quasi-homogeneous potential W ∈ k[x , . . . , x ] viewed as an object in 1 n gr 1 LG we ﬁnd that S = I [n−2]{ c(W )}, where c(W ) is the central charge of W W W (see (2.40)) and {−} denotes the shift in Q-degree. Hence every potential W that satisﬁes the condition I = I [n − 2]{ c(W )} determines an extended oriented W W TQFT (cf. Proposition 3.14) or or gr Z : Bord −→ LG . (1.4) W,gr 2,1,0 If the hypersurface {W = 0} in weighted projective space is a Calabi-Yau variety (equivalently: if c(W ) = n − 2) then the trivialisability of S reduces to the n−2 (n − 2)-Calabi-Yau condition Σ = Id on the shift functor Σ = [1]{1} of the gr triangulated category LG (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 gr explicit control over the bicategories LG and LG 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 cobor- N+1 dism hypothesis as extended TQFTs that assign the potentials W = x to the point, for all N ∈ Z . >2 Acknowledgements We thank Ilka Brunner, Domenico Fiorenza, Jan Hesse, Daniel Murfet and Christoph Schweigert for helpful discussions. The work of N. C. is partially sup- ported by a grant from the Simons Foundation and by the stand-alone project P 27513-N27 of the Austrian Science Fund. The work of F. M. M. was supported by a fellowship from the Peters-Beer Foundation. 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 gr of graded Landau-Ginzburg models LG . 5 Our main reference for bicategories, pseudonatural transformations, modiﬁca- tions 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]. 2.1 Deﬁnition of LG Recall from [CM, Sect. 2.2] that for a ﬁxed ﬁeld k of characteristic zero, the bi- category of Landau-Ginzburg models LG is deﬁned as follows. An object is either the pair (k, 0) or a pair (k[x , . . ., x ], W ) where n ∈ Z and W ∈ k[x , . . . , x ] 1 n >0 1 n is a potential, i. e. the Jacobi algebra Jac = k[x , . . . , x ]/(∂ W, . . . , ∂ W ) (2.1) W 1 n x x 1 n is ﬁnite-dimensional over k. We often abbreviate lists of variables (x , . . . , x ) 1 n by x, and we often shorten (k[x], W ) to W . For two objects (k[x], W ) and (k[z], V ) we have LG (k[x], W ), (k[z], V ) = hmf k[x, z], V − W (2.2) 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 ) (or 0 1 simply by X for short), where X = X ⊕ X is a free Z -graded k[x, z]-module and d ∈ End (X) such that d = W · id . The twisted diﬀerentials d , d ′ X X X X k[x,z] X induce diﬀerentials |ζ| δ ′ : ζ 7−→ d ′ ◦ ζ − (−1) ζ ◦ d (2.3) X,X X X on the modules Hom (X, X ), and 2-morphisms in LG are even cohomology k[x,z] 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 ﬁnite- rank 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 0 1 1 0 [1]: X ⊕ X , d 7−→ X ⊕ X , −d (2.4) X X see e. g. [KST, Sect. 2.1]. It follows that [2] := [1] ◦ [1] = Id . (2.5) LG(W,V ) In fact we can allow any commutative unital ring k if we generalise the deﬁnition of potentials as in [CM, Def. 2.4]. 6 Horizontal composition in LG is given by functors ⊗: LG (k[y], W ), (k[z], W ) × LG (k[x], W ), (k[y], W ) 2 3 1 2 −→ LG (k[x], W ), (k[z], W ) (2.6) 1 3 which act on 1-morphisms as 0 0 1 1 0 1 1 0 (Y, X) 7−→ Y ⊗X ≡ (Y ⊗ X )⊕(Y ⊗ X ) ⊕ (Y ⊗ X )⊕(Y ⊗ X ) k[y] k[y] k[y] k[y] (2.7) with d = d ⊗ 1 + 1 ⊗ d , and analogously on 2-morphisms. It follows from Y ⊗X Y X [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 eﬀec- tive 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 ) ◦⊗ ◦((ι ) ×(ι ) ), where ι : k[x, z] ֒→ k[x, y, z] etc. x,z ∗ k[x,y,z] y,z x,y x,z 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 , . . . , x ], W ) ∈ LG, its unit 1-morphism is (I , d ) with 1 n W I ^ M I = k[x, x ] · θ (2.8) W i i=1 ′ ′ ′ ′ where x ≡ (x , . . . , x ) is another list of n variables, {θ } is a chosen k[x, x ]-basis 1 n ′ ⊕n of k[x, x ] , and x ,x ′ ∗ d = ∂ W · θ ∧ (−) + (x − x ) · θ (2.9) I i i i i W [i] i=1 where ′ ′ ′ ′ ′ W (x , . . . , x , x , . . . x ) − W (x , . . . , x , x , . . . x ) 1 i−1 1 i x ,x i n i+1 n ∂ W = (2.10) [i] ′ x − x ∗ ∗ and θ is deﬁned by linear extension of θ (θ ) = δ and to obey the Leibniz j i,j i i 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 . Finally, the left and right unitors λ : I ⊗ X −→ X , ρ : X ⊗ I −→ X (2.11) X V X W 7 for X ∈ LG(W, V ) are deﬁned as projection to θ-degree zero on the units I and I , respectively; their explicit inverses (in the homotopy category LG(W, V )) were worked out in [CM] to act as follows: n o X X X ′ ′ z ,z z ,z −1 λ (e ) = θ . . . θ ∂ d . . . ∂ d ⊗ e , i a a X X j X 1 l [a ] [a ] l 1 ji l>0 a <···<a j 1 l n o X X X ′ ′ −1 +l|e | x ,x x ,x ( ) i ρ (e ) = (−1) e ⊗ ∂ d . . . ∂ d θ . . . θ i j X X a a X 1 l [a ] [a ] 1 l ji l>0 a <···<a j 1 l (2.12) where {e } is a basis of the module X, and d is identiﬁed with the matrix i X representing it with respect to {e }. 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]) End(I ) Jac . (2.13) W W 2.2 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, (M3) associator a: ◦( × Id ) → ◦ (Id × ), which is part of an adjoint LG LG equivalence, (M4) pentagonator π : (Id a)◦a◦(aId ) → a◦a (using shorthand notation LG LG explained below), (M5) left and right unitors l: ◦(I × Id ) → Id , r : ◦(Id × I) → Id , LG LG LG LG ′ ′ (M6) 2-unitors λ : 1 ◦ (l × 1) → (l ∗ 1) ◦ (a ∗ 1), ρ : r ◦ 1 → (1 ∗ (1 × r)) ◦ (a ∗ 1), and µ : 1 ◦ (r × 1) → (1 ∗ (1 × 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) 8 which is basically given by tensoring over k and taking sums of potentials. More precisely, according to [MM, Prop. 3.1.12], acts as (W, V ) ≡ (k[x], W ), (k[z], V ) 7−→ k[x, z], W + V ) ≡ W + V (2.15) on objects, while the functors on Hom catgories : LG×LG (V , V ), (W , W ) −→ LG V +V , W +W (2.16) (V ,V ),(W ,W ) 1 2 1 2 1 2 1 2 1 2 1 2 are given by ⊗ (up to a reordering of variables similar to the situation in Re- mark 2.1, see [MM, Def. 3.1.3]). Compatibility with horizontal composition is witnessed by the natural isomorphisms : ⊗◦(×) → ◦⊗ (U ,U ),(V ,V ),(W ,W ) 1 2 1 2 1 2 whose ((Y , Y ), (X , X ))-components are given by linearly extending 1 2 1 2 (Y Y ) ⊗ (X X ) −→ (Y ⊗ X ) (Y ⊗ X ) , 1 2 1 2 1 1 2 2 |f |·|e | 2 1 (f ⊗ f ) ⊗ (e ⊗ e ) 7−→ (−1) · (f ⊗ e ) ⊗ (f ⊗ e ) (2.17) 1 k 2 1 k 2 1 1 k 2 2 for Z -homogeneous module elements e , e , f , f , and isomorphisms on units 2 1 2 1 2 : I −→ I I (2.18) (W ,W ) W +W W W 1 2 1 2 1 2 are also standard, cf. [MM, Lem. 3.1.4]. (M2) The unit object in LG is I := (k, 0) . (2.19) Let 1 be the 2-category with a single object ∗ and only identity 1- and 2- morphisms. We deﬁne a strict 2-functor I : 1 → LG by setting I(∗) = I. (M3) The associator is a pseudonatural transformation a: ◦ ( × Id ) −→ ◦ (Id × ) ◦ A . (2.20) LG LG Here A is the rebracketing 2-functor (LG × LG) × LG → LG × (LG × LG), which we usually treat as an identity. The 1-morphism components a and 2- ((U,V ),W ) morphism components a of the associator are given by ((X,Y ),Z) −1 a = I , a = λ ◦ A ◦ ρ , (2.21) ((U,V ),W ) U+V +W ((X,Y ),Z) X,Y,Z X(Y Z) (XY )Z where A : X (Y Z) → (X Y ) Z is the rebracketing isomorphism for X,Y,Z LG, while λ and ρ are the 2-isomorphisms (2.11). The associator a and the pseudonatural transformation a : ◦ (Id × ) ◦ A −→ ◦ ( × Id ) (2.22) LG LG 9 − − −1 −1 with components a = I , a = λ ◦ A ◦ ρ U+V +W (XY )Z ((U,V ),W ) ((X,Y ),Z) X(Y Z) X,Y,Z are part of a biadjoint equivalence, see [MM, Lem. 3.2.5–3.2.6]. (M4) The pentagonator is an invertible modiﬁcation π : 1 ∗ (1 × a) ◦ a ∗ 1 ◦ 1 ∗ (a × 1 ) Id Id ××Id Id LG LG LG LG −→ a ∗ 1 ◦ a ∗ 1 (2.23) Id ×Id × ×Id ×Id LG LG LG LG where here and below we write vertical and horizontal composition of pseudonat- ural 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 π : (Id a) ◦ a ◦ (a Id ) −→ a ◦ a . (2.24) LG LG Its components are π = λ ◦ ⊗1 ⊗ . (((T,U),V ),W ) I ⊗I (T,U+V +W ) I (T +U+V,W ) T+U+V +W T+U+V +W T+U+V +W (2.25) (M5) The left and right (1-morphism) unitors are pseudonatural transformations l: ◦ (I × Id ) −→ Id , r : ◦ (Id × I) −→ Id (2.26) LG LG LG LG whose components are given by −1 l = I = r , l = λ ◦ ρ = r , (2.27) (∗,W ) W (W,∗) (1 ,X) X (X,1 ) ∗ X ∗ where we identify 1 × LG ≡ LG ≡ LG × 1 and I X ≡ X ≡ X I (see [MM, 0 0 Lem. 3.1.8 & 3.2.11 & 3.2.15] for details). The unitors l, r are part of biadjoint − − equivalences (l, l ), (r, r ) as explained in [MM, Lem. 3.2.13–3.2.15]. (M6) The 2-unitors are invertible modiﬁcations λ : 1 ◦ (l × 1) → (l ∗ 1) ◦ (a ∗ 1), ′ ′ ρ : r ◦ 1 → (1∗ (1×r))◦ (a∗ 1), and µ : 1◦ (r × 1) → (1∗ (1×l))◦ (a∗ 1), written here in the shorthand notation also employed in (M4) above, whose components are ′ −1 −1 ′ −1 λ = λ ◦ , ρ = ( ⊗ 1 ) ◦ λ , (V,W ) I ((∗,V ),W ) ((V,W ),∗) V +W I (V,W ) I V +W V +W ′ −1 µ = ρ . (2.28) ((V,∗),W ) I I V W Proposition 2.2. The data (M1)–(M6) endow LG with a monoidal structure. Proof. The straightforward but lengthy check of all coherence axioms is per- formed to prove Theorem 3.2.18 in [MM, Sect. 3.1–3.2]. 10 2.3 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) 7→ (ξ, ζ) on objects, 1- and 2-morphisms, (S2) syllepsis σ : 1 → b ◦ b, − − (S3) R: a ◦ b ◦ a → (Id b) ◦ a ◦ (b Id ) and S : a ◦ b ◦ a → (b Id ) ◦ LG LG LG a ◦ (Id b), LG 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 b: −→ ◦ τ (2.29) whose 1-morphism components b are given by I (up to a reordering of (V,W ) V +W variables, see [MM, Not. 3.1.2 & Lem. 3.3.5]), while the 2-morphism components b : (Y X) ⊗ b −→ b ⊗ (X Y ) (2.30) (X,Y ) (V ,V ) (W ,W ) 1 2 1 2 are deﬁned in [MM] as natural compositions of canonical module isomorphisms and structure maps of the bicategory LG. Explicitly, if {e } and {f } are bases a b of the underlying modules of X and Y , respectively, we have j j 1 m |e |·|f | −1 b : (f ⊗ e ) ⊗ θ . . . θ 7−→ (−1) δ . . . δ · λ (e ⊗ f ) . (2.31) (X,Y ) b a j ,0 j ,0 a b i i 1 m XY 1 m The braiding b and the pseudonatural transformation b : ◦ τ −→ (2.32) − − with components b = b and b = b are part of a biadjoint (W,V ) (Y,X) (V,W ) (X,Y ) equivalence, see [MM, Sect. 3.3.2]. N+1 Example 2.3. For a potential W = x , N ∈ Z , the matrix factorisation >2 b is precisely what is assigned to a “virtual crossing” in the construction (W,W ) of homological sl -tangle invariants of Khovanov and Rozansky [KR1] (see the second expression in [KR2, Eq. (A.9)]). 11 (S2) The syllepsis is an invertible modiﬁcation σ : 1 −→ b ◦ b (2.33) − −1 whose components σ : I → b ⊗ b are given by λ (up to (V,W ) V +W (V,W ) (V,W ) V +W a reordering of variables and a sign-less swapping of tensor factors, see [MM, Lem. 3.3.8]). (S3) The invertible modiﬁcations R: a ◦ b ◦ a −→ (Id b) ◦ a ◦ (b Id ) , LG LG − − − S : a ◦ b ◦ a −→ (b Id ) ◦ a ◦ (Id b) (2.34) LG LG have components R and S which act on basis elements, i. e. on ((U,V ),W ) ((U,V ),W ) tensor and wedge products of θ-variables, by a reordering with appropriate signs, see [MM, Lem. 3.3.11] for the lengthy explicit expressions. Theorem 2.4. The data (M1)–(M6) and (S1)–(S3) endow LG with a symmetric monoidal structure. Proof. It is shown in [MM, Sect. 3.1 & 3.3] that the data (S1)–(S3) are well-deﬁned and satisfy the coherence axioms for symmetric braidings. We note that instead of directly constructing the data (M1)–(M6) and (S1)–(S3) and verifying their coherence axioms, one could also employ Shul- man’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]. 2.4 Duality in LG 2.4.1 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 → I , coev : I → X ⊗ X, eev : X ⊗X → I X W X V X V and cg oev : I → X ⊗ X for every X ∈ LG(W, V ), X W subject to coherence axioms. In this section we recall the above data as con- structed in [CM] (this reference also spells out the coherence axioms). 12 ∨ (A1) Setting X = Hom (X, k[x, z]) and deﬁning the associated twisted dif- k[x,z] |φ|+1 ∨ ferential by d (φ) = (−1) φ◦d for homogeneous φ ∈ X , the left and right X X adjoints of X ∈ LG (k[x , . . . , x ], W ), (k[z , . . . , z ], V ) (2.35) 1 n 1 m are given by † ∨ † ∨ X = X [m] and X = X [n] , (2.36) respectively, where [m] is the m-th power of the shift functor [1] in (2.4) with itself. 0 D Hence if in a chosen basis d is represented by the block matrix ( ), then in D 0 T T 0 D 0 D 0 1 the dual basis d† is represented by ( ) if m is even, and by ( ) if m X T T −D 0 −D 0 1 0 † † is odd, and similarly for d †. It follows that X = X if m = n mod 2. (A2) To present the adjunction 2-morphisms † † ev : X ⊗ X −→ I , coev : I −→ X ⊗ X , X W X V † † eev : X ⊗ X −→ I , cg oev : I −→ X ⊗ X , (2.37) X V X W recall from [Li] the basic properties of residues (collected for our purposes in (x) n [CM, Sect. 2.4]), let {e } be a basis of X, and deﬁne Λ = (−1) ∂ d . . . ∂ d , i x X x X 1 n (z) Λ = ∂ d . . . ∂ d . In [CM] the theory of homological perturbation and z X z X 1 m associative Atiyah classes were used to obtain the following explicit expressions: X X ∗ +l|e | ( ) j ev (e ⊗ e ) = (−1) θ . . . θ X j a a i 1 l l>0 a <···<a 1 l " # ′ ′ x,x x,x (z) Λ ∂ d . . . ∂ d dz X X [a ] [a ] 1 ij · Res , ∂ V, . . . , ∂ V z z 1 m X X ∗ l+(n+1)|e | eev (e ⊗ e ) = (−1) θ . . . θ X j a a i 1 l l>0 a <···<a 1 l " # ′ ′ z,z z,z (x) ∂ d . . . ∂ d Λ dx X X [a ] [a ] l 1 ij · Res , ∂ W, . . . , ∂ W x x 1 n n o r+1 ′ ′ +mr+s z,z z,z ( ) m ∗ coev (γ) = (−1) ∂ d . . . ∂ d e ⊗ e , X X X i [b ] [b ] 1 r ij i,j n o ′ ′ x,x x,x (r¯+1)|e |+s ∗ j n cg oev (γ¯) = (−1) ∂ (d ) . . . ∂ (d ) e ⊗ e (2.38) X ¯ X ¯ X j [b ] [b ] r¯ 1 ji i,j where b , b and s , s ∈ Z are uniquely determined by requiring that b < i ¯ m n 2 1 ¯ ¯ n · · · < b , b < · · · < b , as well as γ¯θ¯ . . . θ¯ = (−1) θ . . . θ and γθ . . . θ = r 1 r¯ 1 n b b b b 1 r 1 r¯ (−1) θ . . . θ . 1 m Theorem 2.5. The data (A1)–(A2) endow the bicategory LG with left and right adjoints for every 1-morphism. Proof. This is [CM, Thm. 6.11]. (In fact LG even has a “graded pivotal” struc- ture, see [CM, Sect. 7].) 13 2.4.2 Duals for objects Endowing the symmetric monoidal bicategory LG with duals for objects amounts to specifying the following data: ∗ ∗ (D1) an object W ≡ (k[x], W ) ∈ LG for every W ≡ (k[x], W ) ∈ LG, ∗ ∗ (D2) 1-morphisms ev : W W → I and coev : I → W W such that there W W are 2-isomorphisms c : r ⊗ (I ev ) ⊗ a ⊗ (coev I ) ⊗ l −→ I , l (W,∗) W W ((W,W ),W ) W W W − − ∗ ∗ ∗ c : l ∗ ⊗ (ev I ) ⊗ a ⊗ (I coev ) ⊗ r −→ I . r (∗,W ) W W ∗ ∗ W W W ((W ,W ),W ) In this section we provide the above data; the explicit isomorphisms c , c are l r constructed in [MM, Ch. 4]. (D1) The dual of W ≡ (k[x], W ) is (k[x], −W ) ≡ W ≡ −W . (D2) The adjunction 1-morphisms exhibiting −W as the (left) dual are the matrix factorisations ev = I and coev = I (2.39) W W W W of W (x ) − W (x), viewed as 1-morphisms (−W ) W → I and I → W (−W ), respectively. Note that −W is also the right dual of W , with eev = I and cg oev = I W W W W viewed as 1-morphisms W (−W ) → I and I → (−W ) W . Proposition 2.6. The data (D1)–(D2) endow the monoidal bicategory LG with duals for every object. Proof. The cusp isomorphisms c , c are computed in terms of the unitors λ, ρ l r 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 , coev A A themselves have left and right adjoints, which in turn have left and right adjoints, and so on. Hence Proposition 2.6 together with Theorem 2.5 implies: Corollary 2.7. Every object of LG is fully dualisable. 14 2.5 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 gr section we recall the bicategory of graded Landau-Ginzburg models LG from [CM, CRCR, Mu] (see also [BFK]) and observe that it inherits the symmetric gr monoidal structure from LG. Moreover, every object in LG is fully dualisable. gr An object of LG is a pair (k[x , . . ., x ], W ) where now k[x , . . . , x ] is a 1 n 1 n graded ring by assigning degrees |x | ∈ Q to the variables x , and W ∈ i >0 i k[x , . . . , x ] is either zero or a potential of degree 2. The central charge of 1 n W ≡ (k[x , . . . , x ], W ) is the numerical invariant 1 n c(W ) = 3 1 − |x | . (2.40) i=1 gr A 1-morphism (k[x], W ) → (k[z], V ) in LG is a summand of a ﬁnite-rank matrix factorisation (X, d ) of V − W over k[x, z] such that the following four L L 0 0 1 1 conditions are satisﬁed: (i) the modules X = X and X = X q∈Q q q∈Q q are Q-graded, (ii) the action of x and z on X are respectively of Q-degree |x | i j i and |z |, (iii) the map d has Q-degree 1, and (iv) if we write {−} for the shift in j X i ⊕a ∼ i,q Q-degree and if X k[x, z]{q} for i ∈ {0, 1}, then {q ∈ Q | a 6= 0} i,q q∈Q must be a subset of i + G , where V −W G := |x |, . . . , |x |, |z |, . . . , |z | ⊂ Q and G := Z . (2.41) V −W 1 n 1 m 0 gr A 2-morphism in LG between two 1-morphisms (X, d ), (X , d ′) is a cohomol- X X ogy class of Z - and Q-degree 0 with respect to the diﬀerential δ in (2.3). 2 X,X We continue to write [−] for the Z -grading shift and {−} for the Q-grading gr shift. Translating [KST, Thm. 2.15] into our conventions we see that LG (W, V ) has the structure of a triangulated category with shift functor Σ := [1]{1} . (2.42) gr Since the categories LG (W, V ) are idempotent complete (cf. [KST, Lem. 2.11]) gr the construction of [DM] ensures that horizontal composition in LG can be deﬁned analogously to (2.6). Moreover, the units I of LG can naturally be endowed with an appropriate Q-grading (by setting |θ | = |x | − 1 and |θ | = i i gr 1 − |x |), and the associator α and unitors λ, ρ of LG are those of LG (as they gr manifestly have Q-degree 0). Hence LG is indeed a bicategory. 2 GR Without condition (iv) we still obtain a bicategory LG , with the same structures that we gr GR exhibit here for LG . As explained in [Mu, Lecture 3], the Hom categories LG (W, V ) are gr gr equivalent to inﬁnite direct sums of LG (W, V ) with itself, hence we can restrict to LG . 15 gr The bicategory LG 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 and from the structure maps α, λ, ρ, their inverses and (Q-degree 0) swapping maps, respectively. For a 1-morphism gr X ∈ LG (k[x , . . . , x ], W ), (k[z , . . ., z ], V ) (2.43) 1 n 1 m we deﬁne its left and right adjoint as † ∨ 1 † ∨ 1 X = X [m]{ c(V )} , X = X [n]{ c(W )} . (2.44) 3 3 The above shifts in Q-degree are necessary to render the adjunction maps ev , coev , eev , cg oev in (2.37) and (2.38) to be of Q-degree 0 so that they are X X X X gr gr indeed 2-morphisms in LG . Finally, the (left and right) dual of (k[x], W ) ∈ LG is (k[x], −W ) with the same grading, and the matrix factorisation underlying the adjunction 1-morphisms ev , coev , eev , cg oev is again I , but now viewed as W W W W W a Q-graded matrix factorisation. In summary, we have: gr Theorem 2.8. The bicategory LG inherits a symmetric monoidal structure gr from LG, every object of LG has a dual, and every 1-morphism has adjoints. gr Corollary 2.9. Every object of LG is fully dualisable. gr 3 Extended TQFTs with values in LG and LG gr In this section we study extended TQFTs with values in LG and LG . We brieﬂy review framed and oriented 2-1-0-extended TQFTs and their “classiﬁcation” in terms of fully dualisable objects and trivialisable Serre automorphisms, respec- gr tively. Then we observe that every object W ≡ (k[x , . . ., x ], W ) in LG or LG 1 n 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 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). 3.1 Framed case Recall from [SP, Sect. 3.2] and [Ps, Sect. 5] that there is a symmetric monoidal fr bicategory Bord of framed 2-bordisms. Its objects, 1- and 2-morphisms are, 2,1,0 roughly, disjoint unions of 2-framed points + and −, 2-framed 1-manifolds with 16 boundary and (equivalence classed of) 2-framed 2-manifolds with corners. For any symmetric monoidal bicategory B, the cobordism hypothesis, originally due fr to [BD], describes the 2-groupoid Fun (Bord , B) (of symmetric monoidal sym ⊗ 2,1,0 fr 2-functors Z : Bord → B, their symmetric monoidal pseudonatural trans- 2,1,0 formations and modiﬁcations) in terms of data internal to B that satisfy certain fr ﬁniteness conditions. Objects of Fun (Bord , B) are called extended framed sym ⊗ 2,1,0 TQFTs with values in B. fd 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 fd for the core of B , i. e. the subbicategory of B 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 fr fd Fun Bord , B −→ K (B ) , sym ⊗ 2,1,0 Z 7−→ Z(+) . (3.1) fr Note that thanks to the description of Bord as a symmetric monoidal bi- 2,1,0 category 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 ⊗ ev . Similarly, + + A A fr 2-morphisms in Bord can be decomposed into cylinders and adjunction 2- 2,1,0 morphisms 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 mod- els LG. As a direct consequence of the cobordism hypothesis and Corollary 2.7 we have: Proposition 3.2. Every object W ≡ (k[x , . . . , x ], W ) ∈ LG determines an 1 n extended framed TQFT fr fr fr Z : Bord −→ LG with Z (+) = W . (3.2) W 2,1,0 W 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 fr precise by relating Z to the (non-extended) closed oriented TQFT or Z : Bord −→ Vect (3.3) W k 2,1 which via the standard classiﬁcation in terms of commutative Frobenius algebras (see e. g. [Ko]) is described by the Jacobi algebra Jac with pairing h−, −i : Jac ⊗ Jac −→ k (3.4) W W k W 17 induced by the residue trace map φ dx Jac −→ k , φ 7−→ Res =: hφi , (3.5) W W ∂ W, . . . , ∂ W x x 1 n i. e. hφ, ψi = hφψi . W W To recover the k-algebra Jac with its multiplication µ : φ ⊗ ψ 7→ φψ, W Jac fr we want to show that Jac and µ are what Z assigns to “the” circle and W Jac W W “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 1 2 the equivalent description of 2-framed circles in terms of immersions ι: S → R 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 . In terms of the structure 1- fr morphisms of Bord (whose horizontal composition we write as #), we have 2,1,0 (see [DSPS, Sect. 1.2]) 1 † S = ev # ev . (3.6) + + This is the correct choice in the sense that for every integer k, there is a 2-framed 1 2 circle S , and for every pair (k, l) ∈ Z there is a pair-of-pants 2-morphism fr 1 1 1 S ⊔ S → S in Bord , and only for k = 0 = l do we get a multiplication. k l k+l 2,1,0 This is “the” 2-framed pair-of-pants for us. Theorem 3.3. For every (k[x , . . . , x ], W ) ∈ LG, we have that 1 n fr 1 fr Z (S ), Z (pair-of-pants) (3.7) W 0 W is isomorphic to (Jac , µ ) as a k-algebra. W Jac fr 1 † ∨ ∨ Proof. Note ﬁrst that Z (S ) ev ⊗ ev = ev ⊗(ev [0]) = I ⊗ ′ I W W W W k[x,x ] W 0 W W ∼ 2 is isomorphic in LG((k, 0), (k, 0)) = vect to the vector space Jac (viewed as a Z -graded vector space concentrated in even degree). One can check that an ∼ ∼ explicit isomorphism κ: I ⊗ I = End (I ) = Jac is given by linear W k[x,x ] LG W W ′ ∗ extension of p(x)q(x ) · e ⊗ e 7→ p(x)q(x) · δ , where p and q are polynomials i i,j and {e } is a basis of the k[x, x ]-module I . i W fr Next we prove that Z sends the pair-of-pants to the commutative multiplica- tion µ . For this we decompose the pair-of-pants into generators, namely into Jac cylinders over the left and right elbows ev : − ⊔ + → ∅ and ev : ∅ → − ⊔ +, + + respectively, and the “upside-down saddle” ev : ev # ev → 1 (which is ev + + −⊔+ called v in [DSPS, Ex. 1.1.7]). Then 1 1 1 pair-of-pants = 1 # ev # 1† : S ⊔ S −→ S . (3.8) ev ev ev + + 0 0 0 fr fr Hence if Z (ev ) = ev , then the functor Z sends this pair-of-pants to ev ev W W W 1 ⊗ ev ⊗1† , which by pre- and post-composition with the isomorphism ev ev ev W W 18 † κ: ev ⊗ ev Jac becomes a map µ : Jac ⊗ Jac → Jac . Noting that W W W W k W W both κ and ev act diagonally (with ev (e ⊗ e ) = δ since ev : (−W ) ev ev j i,j W W W i W → I has trivial target, cf. the explicit expression for ev in (2.38)), we ﬁnd ev that µ is indeed given by multiplication of polynomials, i. e. µ = µ . Jac fr To complete the proof we need to argue that Z assigns our choice of counit ev to the upside-down saddle ev , and not some other choice of adjunc- ev ev tion data. By [Ps, Thm. 3.17 & Thm. 8.1], extended framed TQFTs are equiva- lent to “coherent fully dual pairs” in their target bicategories, see [Ps, Def. 3.12] fr 1 for the details. For the algebra structure on Z (S ), we only need a “coher- W 0 ent dual pair” as deﬁned in [Ps, Def. 2.6]. One straightforwardly checks that (W, −W, ev , coev , c , c ) satisﬁes all the deﬁning properties of a coherent dual W W r l fr fr pair, ensuring that Z can indeed be chosen such that Z (ev ) = ev . (The ev ev W W + W key deﬁning properties of coherent dual pairs for us to check are the so-called swal- lowtail identities of [Ps, Def. 2.6], which can be viewed as consistency constraints on our cusp isomorphism c . But since c , c and all other 2-morphisms that ap- l l r pear in the swallowtail identities are structure maps of the underlying bicategory of LG, the coherence theorem for bicategories guarantees that the constraints are satisﬁed.) Remark 3.4. The ﬁnite-dimensional k-algebra Jac is typically not separable. N+1 For example, if W = x with N ∈ Z the algebra Jac has non-semisimple >2 W representations (as multiplication by x has non-trivial Jordan blocks) and hence cannot be separable. Thus Jac is not fully dualisable in the bicategory Alg of ﬁnite-dimensional k-algebras, bimodules and intertwiners [Lu, SP], so Jac cannot describe an extended TQFT with values in Alg . Proposition 3.2 and Theorem 3.3 explain how Jac does appear in an extended TQFT with values in LG, namely as the algebra assigned to the circle S and its pair-of-pants. For an algebra A ∈ Alg its Hochschild cohomology HH (A) is isomorphic to ev ⊗ ev , and for Hochschild homology one ﬁnds HH (A) ev ⊗ b ⊗ A A • A (A,A) coev . Similarly, for every object W ≡ (k[x , . . ., x ], W ) ∈ LG we may deﬁne A 1 n • † HH (W ) := ev ⊗ ev , HH (W ) := ev ⊗ b ⊗ coev . (3.9) W W • W (W,W ) W Thus by Theorem 3.3 we have HH (W ) Jac , and paralleling the ﬁrst part of the proof we ﬁnd HH (W ) = Jac [n] as Z -graded vector spaces (because • W 2 the matrix factorisations b and coev are I I and I I = I [n], (W,W ) W W W W respectively). Hence HH (W ) and HH (W ) precisely recover the Hochschild co- homology 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]: Corollary 3.5. For every W ≡ (k[x], W ) ∈ LG we have • • ∼ ∼ HH (W ) = HH MF(k[x], W ) , HH (W ) = HH MF(k[x], W ) . (3.10) • • 19 Remark 3.6. Proposition 3.2, Theorem 3.3 and Corollary 3.5 have direct ana- logues for the graded Landau-Ginzburg models of Section 2.5. Firstly, Theo- rem 3.1 and Corollary 2.9 immediately imply that every object (k[x , . . . , x ], W ) ∈ 1 n gr LG determines an extended TQFT fr fr gr Z : Bord −→ LG . (3.11) W,gr 2,1,0 Secondly, going through the proof of Theorem 3.3 we see that to the cir- 1 fr cle S and its pair-of-pants, Z assigns the Jacobi algebra Jac which is 0 W,gr now a Q-graded algebra with degree-preserving multiplication. We note that here it is important that the upside-down saddle ev : ev # ev → 1 in- ev + + −⊔+ † ∨ ∨ volves the left adjoint of ev : by (2.44) we have ev = ev [0]{0} = ev , so + W W W fr Z (pair-of-pants) really gives a map W,gr † † † ∼ ∼ Jac ⊗ Jac = ev ⊗ ev ev ⊗ ev −→ ev ⊗ ev = Jac . W k W W W W W W W W (3.12) (Incorrectly using the right adjoint ev = ev [2n]{ c(W )} would lead to un- W W wanted Q-degree shifts in the multiplication. In Remark 3.16 below however we are naturally led to use the right adjoint ev to obtain the correct graded trace map h−i on Jac .) W W gr Thirdly, for every (k[x , . . . , x ], W ) ∈ LG the matrix factorisation underlying 1 n 1 1 ∨ ∨ coev is I I = I [n]{ c(W )} = I [n − 2]{ c(W )}, and hence we have W W W W W 3 3 • 1 ∼ ∼ HH (W ) Jac , HH (W ) Jac [n − 2]{ c(W )} . (3.13) = = W • W 3.2 Oriented case An extended oriented TQFT with values in a symmetric monoidal bicategory B or or is a symmetric monoidal 2-functor Z : Bord → B. Here Bord is the bicat- 2,1,0 2,1,0 egory 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) or and their relations. Hence objects of Bord are disjoint unions of positively 2,1,0 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 fd points of the SO(2)-action induced on B by the SO(2)-action which rotates the fr framings in Bord . This was worked out in detail in [HSV, HV, He] as we 2,1,0 brieﬂy review next. fd An SO(2)-action on B is a monoidal 2-functor ̺ from the fundamental 2- fd groupoid Π (SO(2)) to the bicategory of autoequivalences of B . Since SO(2) is path-connected, Π (SO(2)) has essentially a single object ∗ which ̺ sends to fd the identity Id fd on B . Since π (SO(2)) Z the action of ̺ on 1-morphisms B 1 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 20 fd Z(+) =: A, the relevant choice for ̺(1 ) is the Serre automorphism S of A ∈ B . ∗ A By deﬁnition S is the 1-morphism S := r ⊗ 1 eev ⊗ b 1 ⊗ 1 eev ⊗ r : A −→ A . (3.14) A (A,∗) A A (A,A) A A A A Here we denote the braiding, horizontal composition and monoidal product in B gr by b, ⊗ and , respectively, as we do in LG and LG . fd SO(2) The bicategory of SO(2)-homotopy ﬁxed points K (B ) was deﬁned and endowed with a natural symmetric monoidal structure in [HV]. Objects of fd SO(2) fd K (B ) are pairs (A, σ ) where A ∈ B and σ is a trivialisation of the A A Serre automorphism S , i. e. a 2-isomorphism S → 1 in B. A 1-morphism A A A ′ fd SO(2) fd ′ (A, σ ) → (A , σ ′) in K (B ) is an equivalence F ∈ B (A, A ) such that A A λ ◦ (σ ′ ⊗ 1 ) ◦ S = ρ ◦ (1 ⊗ σ ) where S is the 2-isomorphism constructed F A F F F F A F ′ fd SO(2) in the proof of [HV, Prop. 2.8], and 2-morphisms F → F in K (B ) are 2-isomorphism F → F in B. Building on [Lu, SP, HSV, HV], extended ori- ented 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 or fd SO(2) Fun Bord , B −→ K (B ) , sym ⊗ 2,1,0 Z 7−→ Z(+) . (3.15) 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: Lemma 3.8. Let W ≡ (k[x , . . . , x ], W ) ∈ LG. Then S I [n]. 1 n W W Proof. According to Sections 2.3–2.4, the factors r , 1 , eev , b , 1 (W,∗) W W (W,W ) W and r in the deﬁning expression (3.14) are all given by the matrix factorisation underlying the unit I ∈ LG(W, W ), while the matrix factorisation underlying † † ∨ ∨ ∼ ∼ ∼ eev = eev [2n] = eev is I I [n] I [n]. This leads to S I [n]. (A = = = W W W W W W W W straighforward computation, taking into account subtleties of the kind mentioned in Remark 2.1, is carried out in the proof of [MM, Lem. 5.2.3] to construct an explicit isomorphism S → I only in terms of λ, ρ and standard swapping W W isomorphisms.) The general fact I ≇ I [1] (even Hom (I , I [1]) = 0 is true) together W W LG(W,W ) W W with Theorem 3.7 thus imply: Proposition 3.9. An object W ≡ (k[x , . . . , x ], W ) ∈ LG determines an ex- 1 n tended oriented TQFT or or or Z : Bord −→ LG with Z (+) = W (3.16) W 2,1,0 W if and only if n is even. 21 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 3 d a Serre functor S such that Σ S . The triangulated category LG(0, W ) is T T known to admit a Serre functor S = [n] = [n − 2]. Hence LG(0, W ) is LG(0,W ) weakly (n − 2)-Calabi-Yau, the Serre automorphism and Serre functor coincide in the sense that S ⊗ (−) = S , and the condition that S is trivialisable W LG(0,W ) W 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 Jac = k[x , . . . , x ]/(∂ W, . . . , ∂ W ) W 1 n x x 1 n 2 2 2 = k[x , . . . , x , y]/(∂ (W + y ), . . ., ∂ (W + y ), ∂ (W + y )) 1 n x x y 1 n = Jac 2 , (3.17) W +y every non-extended oriented Landau-Ginzburg model appears as part of an or or extended oriented TQFT Z or Z (depending on whether n is even W W +y or odd, respectively), namely as the commutative Frobenius algebra with or 1 or 1 underlying vector space Z (S ) or Z (S ). Note that for this argument W W +y 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. (ii) Instead of LG one can also consider the symmetric monoidal bicategory •/2 LG which is equal to LG except that the vector space of 2-morphisms • ′ (X, d ) → (X, d ) is deﬁned to be H (Hom (X, X ))/Z , i. e. X X k[x,z] 2 X,X both even and odd cohomology of the diﬀerential δ ′ in (2.3) are in- X,X cluded while ζ ∈ Hom (X, X ) and −ζ are identiﬁed after taking co- k[x,z] homology. Dividing out this Z -action circumvents the issue that with- out it the interchange law would only hold up to a sign, as we have |ζ |·|ξ | 2 1 (ζ ⊗ζ )◦(ξ ⊗ξ ) = (−1) (ζ ◦ξ )⊗(ζ ◦ξ ) for appropriately compos- 1 2 1 2 1 1 2 2 able homogeneous 2-morphisms. Such Z -quotients also appear in [KR1]; •/2 the bicategory LG is described in more detail in [MM, Sect. 5.3.1] (where it is denoted LG). •/2 In particular, for every (k[x , . . . , x ], W ) ∈ LG there is an even/odd 1 n isomorphism I I [n] for n even/odd. Hence by Lemma 3.8 every object W W •/2 •/2 of LG determines an extended oriented TQFT with values in LG . (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. A Serre functor of T is an additive equivalence S : T → T together with isomorphisms Hom (A, B) Hom (B, S (A)) that are natural in A, B ∈ T . T = T T 22 Part of this involves the natural diﬀerential Z -graded categories (with dif- ferential δ as above) studied in [Dy], whose even cohomologies are the X,X matrix factorisation categories of Section 2.1. Such bicategories of dif- ferential 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 in- volve 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 de- fect TQFT on super bordisms (which again has not been deﬁned in detail as far as we know), paralleling the non-super construction of [DKR] reviewed in [Ca]. Theorem 3.12. For every (k[x , . . . , x ], W ) ∈ LG with n even, we have that 1 n fr 1 fr fr fr Z (S ), Z (pair-of-pants), Z (cup)(1), Z (cap) (3.18) W 0 W W W is isomorphic to (recall (3.5) for the residue trace h−i ) Jac , µ , 1, hζ(−)i (3.19) W Jac W fr as a commutative Frobenius k-algebra, where the traces Z (cap) and h−i fr 1 induce the Frobenius pairings on Z (S ) and Jac , respectively, and ζ ∈ Jac W W W 0 is a uniquely determined invertible element. Proof. The isomorphism on the level of k-algebras was already established in fr Theorem 3.3, it remains to compute the action of Z on the cap and cup 2- morphisms. The cap is the bordism eev from the 2-framed circle ev # ev to 1 . We ﬁrst ev + ∅ + + fr † assume that Z sends it to the 2-morphism eev from ev ⊗ ev = ev ⊗ ev ev W W W W W W to I . Since ev : (k[x], −W )(k[x], W ) → (k, 0) has trivial target, only the sum- 0 W mand l = 0 contributes to the expression for eev in (2.38), and pre-composing ev with the isomorphism Jac ev ⊗ ev from the proof of Theorem 3.3 pro- W W W duces the residue trace h−i . 1 † Similarly, the cup: ∅ → S = ev # ev is equal to coev . Using the explicit + + ev 0 + fr expression for coev in (2.38) we see that post-composing Z (cup)(1) with the ev W W isomorphism ev ⊗ ev Jac is indeed the unit 1 ∈ Jac . W W W W To complete the proof we must investigate to what extent our choice of adjunc- tion data in LG gives rise to a “coherent fully dual pair” (where again we rely on 23 the result of [Ps] that extended framed TQFTs are equivalent to coherent fully dual pairs): if the coherent dual pair (W, −W, ev , coev , c , c ) can be lifted to W W r l fr fr a coherent fully dual pair then Z can be chosen such that Z (eev ) = eev . ev ev W W + W First we observe that by Lemma 3.8 there is a “fully dual pair” W, −W, ev , coev , I , I , c , c , µ , ǫ , µ , ǫ , ψ, φ (3.20) W W W W r l e e c c in the sense of [Ps, Def. 3.10], where φ := λ =: ψ and µ , ǫ , µ , ǫ are equal I e e c c to coev , ev , coev , ev up to appropriate composition with the iso- ev ev coev coev W W W W ±1 ±1 morphisms λ , ρ . As explained in the proof of [Ps, Thm. 3.16], every fully dual pair can be made coherent by changing only the counit 2-morphisms by composi- tion with an automorphism ζ of I (and possibly the cusp isomorphism c which W l however in our case is not necessary as observed in the proof of Theorem 3.3). Given a fully dual pair, the map ζ is uniquely determined by the cusp-counit equation of [Ps, Def. 3.12], which involves two adjunction maps on one side and none on the other. In our case one ﬁnds that the constraint reduces to the equal- ity of two linear maps Jac ⊗ Jac → k, one of which involves the residue trace W k W h−i pre-composed with ζ ∈ Aut(I ) ⊂ End(I ) = Jac , while the other is a W W W W composite of structure maps of the symmetric monoidal bicategory LG (without any adjunction maps). Paralleling the above proof we see that for even n, the extended oriented TQFT or Z also assigns the Frobenius algebra Jac to the oriented circle, pair-of-pants, cup and cap (up to an invertible element ζ ∈ Jac ). N+1 Example 3.13. For every N ∈ Z , the potential x determines an extended >2 •/2 oriented TQFT with values in the symmetric monoidal bicategory LG intro- duced in Remark 3.11(ii). We denote this TQFT by Z as it recovers – directly KR 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 or saddle bordisms in Bord . Except for the saddle we have already computed 2,1,0 all these assignments of Z for any potential W in Theorems 3.3 and 3.12, and KR N+1 for W = x they match the prescriptions of [KR1] (except for non-essential prefactors for the cap and cup morphisms). To establish that the TQFT Z indeed matches that of [KR1, Sect. 9] it re- KR mains to compute Z (saddle) = Z (cg oev ) and compare it to the explicit KR KR ev matrix expressions in [KR1, Page 81] (or Page 95 of arXiv:math/0401268v2 [math.QA]). Since Z (cg oev ) = cg oev this is another exercise in using the formu- KR ev ev + N+1 las (2.38) for adjunction 2-morphisms. This is carried out in [MM, Sect. 5.3.2], ﬁnding   e 1 0 0   −e 1 0 0   Z (saddle) = (3.21) KR   0 0 −1 1 0 0 −e −e 234 124 24 P a b c where the entries e := x x x ∈ k[x , x , x , x ] depend on four ijk 1 2 3 4 i j k a+b+c=N−1 N+1 variables as the source and target of cg oev involve four copies of x ∈ ev N+1 •/2 4 LG . Up to a minor normalisation issue the expression (3.21) agrees with that of [KR1]. In summary, we veriﬁed that the construction of [KR1, Sect. 9] can be under- N+1 stood as an application of the cobordism hypothesis to the potential W = x . gr We return to the bicategory LG of Section 2.5. All the above results in the gr present section have analogues or reﬁnements in LG . In particular: gr Proposition 3.14. An object W ≡ (k[x , . . . , x ], W ) ∈ LG determines an 1 n extended oriented TQFT or or gr or Z : Bord −→ LG with Z (+) = W (3.22) W,gr 2,1,0 W,gr gr if and only if [n − 2]{ c(W )} Id . LG (0,W ) Proof. By Theorem 3.7, W determines a TQFT as stated if and only if its Serre automorphism S is trivialisable. Paralleling the proof of Lemma 3.8 we see that, using (2.44), the matrix factorisation underlying S is eev = ∨ 2 2 1 eev [2n]{ c(W )}. Hence S is isomorphic to I [2n]{ c(W )} = I [n]{ c(W )} W W W 3 3 3 I [n − 2]{ c(W )}. gr/Z gr Let LG be the symmetric monoidal 2-category obtained from LG by re- gr gr placing the hom categories LG (W, V ) with the orbit categories LG (W, V )/Z obtained by dividing out the action of the shift functor Σ = [1]{1}, i. e. gr Hom (X, Y ) = Hom (X, Σ (Y )) (3.23) gr/Z LG LG k∈Z gr/Z gr for 1-morphism X, Y ∈ Ob(LG (W, V )) = Ob(LG (W, V )). It follows that in /Z gr k LG , we have X = Σ (X) for all 1-morphisms X and k ∈ Z (with 1 viewed as a 2-isomorphism of degree k). In the setting of orbit categories, Calabi-Yau varieties give rise to oriented extended TQFTs: gr Corollary 3.15. If for (k[x , . . . , x ], W ) ∈ LG the hypersurface {W = 0} in 1 n weighted projective space is a Calabi-Yau variety, then W determines an extended or gr/Z oriented TQFT Bord → LG . 2,1,0 More precisely, (3.21) agrees with the saddle morphism of [KR1] if the arbitrary polynomial r a b c d of degree N − 2 in loc. cit. is set to x x x x , and if non-scalar entries of the 1 2 3 4 a+b+c+d=N−2 matrix are multiplied by . 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 † . I I ,ev ⊗ ev W −W N+1 N+1 25 Proof. We write Y for the zero locus of W in weighted projective space. The variety Y is Calabi-Yau if and only if the condition c (Y ) = 0 is satisﬁed W 1 W by the ﬁrst Chern class, which in our normalisation convention is equivalent to P P n n |x | = |W| = 2. 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Extending Landau-Ginzburg models to the point

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