Homological mirror symmetry for invertible polynomials in two variables

In this paper, we give a proof of homological mirror symmetry for two variable invertible polynomials, where the symmetry group on the $B$-side is taken to be maximal. The proof involves an explicit gluing construction of the Milnor fibres, and as an application, we prove derived equivalences between certain nodal stacky curves, some of whose irreducible components have non-trivial generic stabiliser.


Introduction
Consider an n × n matrix A with non-negative integer entries a ij . From this, we can define a polynomial w ∈ C[x 1 , . . . , x n ] given by In what follows w will always be quasi-homogeneous, and so we can associate to it a weight system (d 0 , d 1 , . . . , d n ; h), where w(t d 1 x 1 , . . . , t dn x n ) = t h w(x 1 , . . . , x n ), and d 0 := h − d 1 − · · · − d n . In [4], the authors define the transpose of w, denoted byw, to be the polynomial associated to A T ,w (x 1 , . . . ,x n ) = n i=1 n j=1x a ji j , and we call this the Berglund-Hübsch transpose. One can associate a weight system forw, denoted by (ď 0 ,ď 1 , . . . ,ď n ;ȟ), in the same way. We call a polynomial w invertible if the matrix A is invertible over Q, and if both w andw define isolated singularities at the origin (cf. Definition 2.1).
Since the t n+1 variable is uniquely determined by the other t i , we will think of Γ w as a subgroup of (C * ) n . It is a finite extension of C * , and is the group of diagonal transformations of A n which keep w semi-invariant with respect to the character (t 1 . . . , t n+1 ) → t n+1 . Homological Berglund-Hübsch mirror symmetry predicts: Conjecture 1. For any invertible polynomial, w, there is a quasi-equivalence mf(A n , Γ w , w) ≃ F(w) of pre-triangulated A ∞ -categories over C.
In the above, mf(A n , Γ w , w) is the category of Γ w -equivariant matrix factorisations of w, and F(w) is the Fukaya-Seidel category associated to a Morsification ofw, as defined in [37]. Conjecture 1 goes back to [40] and [41], and there have recently been many results in the direction of establishing it. It has been proven in several cases -in particular, for Brieskorn-Pham polynomials in any number of variables in [10], and for Thom-Sebastiani sums of polynomials of type A and D in [11]. Conjecture 1 is also established for all invertible polynomials in two variables in [13]. For each class of invertible polynomial, recent work of Kravets ([20]) establishes a full, strong, exceptional collection for mf(A n , Γ w , w) with n ≤ 3. In the case of chain polynomials in any number of variables, Hirano and Ouchi ( [15]) show that the category mf(A n , Γ w , w) has a tilting object, and a full, strong, exceptional collection whose length is the Milnor number ofw. For further discussion and background on Conjecture 1, see [7], and references therein.
There is also an extension of Conjecture 1 ( [8], [40], [21]), where rather than considering the maximal symmetry group, one considers certain subgroups of finite index. Correspondingly, one must then consider an 'orbifold Fukaya-Seidel' category, which incorporates a dual group in its data. Recently, this generalised conjecture which takes into account a symmetry group on the A-side was established in the Z/2-graded case for two variable invertible polynomials in [5].
The main focus of this paper is homological mirror symmetry where the (completion of the) Milnor fibre ofw,Vw is taken as the A-model. On the B-side, one extends the action of Γ w to A n+1 in a natural way, as described in Section 2 for the case of n = 2. The Lekili-Ueda conjecture predicts: of pre-triangulated A ∞ -categories over C.
In the above, W(Vw) is the wrapped Fukaya category of the Milnor fibre ofw. This category is completed with respect to cones and direct summands, as stated in Section 1.3. Recently, Conjecture 2 was established for simple singularities in any dimension in [30], and has subsequently been established in the Z/2-graded case in [12].
There is a trichotomy of cases depending on whether the weight d 0 is positive, negative, or zero. In the log general type case of d 0 > 0, there is a quasi-equivalence where Z w := Spec C[x 0 , x 1 , . . . , x n ]/(w + x 0 x 1 . . . x n ) \ (0) /Γ w .
This equivalence is a generalisation of ([31, Theorem 3.11]), where it was proven in the context of triangulated categories, and where C * ≃ Γ w . The generalisation to the case where Γ w is a finite extension of C * is straightforward, and the extension to the setting of dg-categories was studied in [38], [18], [6]. The main focus of this paper is the case of curves, for which the only invertible polynomial which is not of log general type is x 2 + y 2 . This, however, corresponds to the well-understood HMS statement for C * . We will therefore restrict ourselves to the log general type case for the remainder of the paper.
Recall that, since Z w is a proper stack, the subcategory perf Z w ⊆ coh Z w consists precisely of Ext-finite objects, meaning that X ∈ perf Z w if and only if i∈Z Ext i (X, Y ) is finite dimensional for every object Y ∈ coh Z w . On the symplectic side of the correspondence, it is clear that compact Lagrangians can have morphisms in only finitely many degrees with any other Lagrangian, but it is not known that this is necessarily not true for non-compact Lagrangians. This is reasonable to expect, however, and is certainly true in every known case. Therefore, in the log general type case, one expects that Conjecture 2 implies Establishing this quasi-equivalence in the case of curves is the main result of this paper.
Theorem 1.1. Let w be an invertible polynomial in two variables, andw its transpose. Then there is a quasi-equivalence F(Vw) ≃ perf Z w of Z-graded pre-triangulated A ∞ -categories over C, where F(Vw) and perf Z w are as in Section 1.3.
Remark 1.2. It should be reiterated that, although there is a trichotomy of cases depending on the weight d 0 , all but one invertible polynomials in two variables are of log general type, and this exception is well-understood. We are therefore free to state Theorem 1.1 in the context of invertible polynomials of log general type without making any assumptions on d 0 .
The first instances of the quasi-equivalence in (6) were established for the case of w = x 2 + y 3 in [23], and the cases of w = x 3 y + y 2 , w = x 3 + y 3 , and w = x 4 + y 2 in [27]. It was also established in [29] In [28], the authors use mirror symmetry arguments to deduce derived equivalences between rings of certain nodal stacky curves. We elaborate on these arguments in order to identify which Milnor fibres are graded symplectomorphic, and this enables us to deduce derived equivalences between nodal stacky curves with different numbers of irreducible components, some of which have nontrivial generic stabiliser. Corollary 1.3. For each n ≥ 1, q ≥ 2 let w loop = x n(q−1)+1 y + y q x, w chain = x nq+1 y + y q , each with maximal symmetry group. We then have quasi-equivalences perf Z w loop ≃ perf Z w chain of pre-triangulated A ∞ -categories over C. Similar, for n ≥ 1 and p ≥ 2 or n ≥ p = 2, let w ′ chain = x p y+y n(p−1) , w BP = x p +y np , each with maximal symmetry group. Then, we have quasi equivalences This is obtained by first proving that the Milnor fibres corresponding to the relevant Berglund-Hübsch transposes are graded symplectomorphic. This implies that their Fukaya categories are quasi-equivalent, which by Theorem 1.1 proves that the derived categories of perfect complexes of their mirrors are too. This corollary also appears as a special case of [9,Corollary 5.15], although was obtained there by a variation of GIT argument ( [3], [14]).
1.1. Strategy of proof. Our strategy follows that of [29], where one reduces the proof of Theorem 1.1 to a deformation theory argument. For the case at hand, this approach is predicated on the proof of Conjecture 1 for curves given in [13].
On the A-side of the correspondence, we have that there is a restriction functor where we equip the vanishing cycle ∂S → with the induced (non-trivial) spin structure. Suppose that (S → i )μ i=1 is a collection of thimbles which generates F(w), whereμ is the Milnor number ofw, and that S → is the full subcategory of F(w) whose objects are (S → i )μ i=1 . Denote its A ∞ -endomorphism algebra by and its cohomology algebra A → := H * (A → ). Correspondingly, let S be the collection (S i )μ i=1 of vanishing cycles equipped with the non-trivial spin structure, considered as a full subcategory of the compact Fukaya category of the Milnor fibre, and A its A ∞ -endomorphism algebra. Poincaré duality tells us that we can identify H * (A) with as a vector space. In our case, we will deduce in Section 5 that the algebra structure on A is induced purely from the A → -bimodule structure of (A → ) ∨ [1 − n]. Namely, we have This is known as a trivial extension algebra of degree n − 1. By the argument of [35,Lemma 5.4], when the weightď 0 ̸ = 0, S split generates the compact Fukaya category of the Milnor fibre. Therefore, in order to characterise this category, it is sufficient to identify the A ∞ -structure on A which is given by A, up to gauge transformation (a.k.a. formal diffeomorphism).
On the algebro-geometric side of the correspondence, one can consider the Jacobi algebra, Since the singularity is isolated, this algebra has dimension µ < ∞, the Milnor number of w. Let J w be the set of exponents for a basis of this algebra, and consider the semi-universal unfoldings of w, Such unfoldings are universal in the sense that every other unfolding of w is induced from w by a change of coordinates; however, this change of coordinates is not unique. These semi-universal unfoldings are parametrised by µ complex parameters, and we set We can therefore consider w as a map and define To such a polynomial w u , we associate a stack V u , defined in the case of two variables in (26). In the case where the weight d 0 > 0, we want to compactify V u to a Calabi-Yau hypersurface in a quotient of weighted projective space by a finite group, although this is not possible for every u ∈ U . As previously mentioned, we extend the action of Γ w to A n+1 in a prescribed way, and define U + ⊆ U to be the subspace such that w u can be quasi-homogenised to W u ∈ C[x 0 , x 1 , . . . , x n ] with respect to this action. Following [29], one then defines for each u ∈ U + . It goes back to the work of Pinkham ([32]), that the fact that w is quasihomogeneous forces there to be a C * -action on U + . We therefore have that Y u ≃ Y v if and only if v = t · u for some t ∈ C * . By construction, the dualising sheaf of this stack is trivial and it is a compactification of V u .
For each u ∈ U + , there is a functor which is to be expounded upon in Section 2 for the case of curves. In any case where mf(A n , Γ w , w) has a tilting object, E, denote by S u the image of E by (17). It is then a theorem of Lekili and Ueda ([29,Theorem 4.1]) that S u split-generates perf Y u . Let A u be the minimal A ∞ -endomorphism algebra of S u . Then, by the work of Ueda in [42], we have that A u := H * (A u ) is also given by the degree n − 1 trivial extension algebra of the endomorphism algebra of E, and is, in particular, independent of u. In the case where Conjecture 1 is solved by exactly matching generators, as in [13], we have that, at the level of cohomology, the endomorphism algebra of the generators on both the A-, and B-sides are given by the same algebra, which we denote by A. In light of this, establishing the equivalence (6) boils down to identifying the A ∞ -structure given by the chain level endomorphism algebra on the B-side which matches with that of the A-side. With this perspective, homological mirror symmetry for invertible polynomials turns into a deformation theory problem.
Recall that for a graded algebra, A, the Hochschild cochain complex has a bigrading. Namely, we consider CC r+s (A, A) s to be the space of maps A ⊗r → A[s]. In general, if µ • is a minimal A ∞ -structure on A, then deformations which keep µ k for 1 ≤ k ≤ m fixed are controlled by i>m−2 HH 2 (A) −i (see, for example, [35,Section 3a]). In particular, the deformations of A to a minimal A ∞ -model with prescribed µ 2 are controlled by HH 2 (A) <0 = i>1 HH 2 (A) −i . Furthermore, note that HH 2 (A) 0 is the first order deformations of the algebra structure on A. It is natural to consider the functor which takes an algebra to the set of gauge equivalence classes of A ∞ -structures on that algebra. It is a theorem of Polishchuk ([33, Corollary 3.2.5]) that if HH 1 (A) <0 = 0, then this functor is represented by an affine scheme, U ∞ (A). Moreover, if dim HH 2 (A) <0 < ∞, then [33,Corollary 3.2.6] shows that this scheme is of finite type. This functor was first studied in the context of homological mirror symmetry in [25]. There is a natural C * -action on U ∞ (A) given by sending , and this is denoted by A → t * A. Note that the formal A ∞ -structure is the fixed point of this action. For each t ̸ = 0, we have that A and t * A are quasi-isomorphic, although not through a gauge transformation ([35, Section 3]). Now, for each u ∈ U + , we have that A u defines an A ∞ -structure on A with µ 2 given as in (10). Therefore, it defines a point in U ∞ (A), and so we get a map If we can show that (18) is an isomorphism, then we know that every A ∞ -structure on A is realised as the A ∞ -endomorphism algebra of S u for some u ∈ U + . In the case that the pair (w, Γ) is untwisted (see Definition 4.1), we have by a theorem of Lekili and Ueda ([29, Theorem 1.6]) that there is a C * -equivariant isomorphism of affine varieties U + ∼ − → U ∞ (A) which sends the origin to the formal A ∞ -structure. By removing the fixed point of the action on both sides, we have that this isomorphism descends to an isomorphism Therefore, in the case where w is untwisted, we have that, up to scaling, there is some u ∈ U + for which (6) holds.
We end this section by briefly remarking that the moduli of A ∞ -structures argument employed in this paper fits into a broader framework which has proven to be a fruitful approach to HMS, and whose scope is more wide-reaching than that of invertible polynomials. In [24] and [23], the authors establish HMS for the once punctured torus by studying the moduli space of A ∞ -structures on the degree one trivial extension algebra of the A 2 quiver. Interestingly, it was proven that M ∞ (A) ≃ M 1,1 , the moduli space of elliptic curves. Further connection was made to the moduli theory of curves in [26], where the authors show the moduli space of A ∞ -structures on a particular algebra coincides with the modular compactification of genus 1 curves with n marked points, as constructed in [39]. This then leads them to prove homological mirror symmetry for the n-punctured torus in [25].
1.2. Structure of paper. In Section 2, we recall some basic facts about invertible polynomials in two variables, as well as compute U + in the relevant cases. In Section 3, we study the symplectic topology of the Milnor fibre. In Section 4, we compute the relevant Hochschild cohomology for invertible polynomials in two variables. In Section 5, we recall some facts about generators and formality for Fukaya categories and the proper algebraic stacks under consideration. Section 6 is then a proof of Theorem 1.1 and Corollary 1.3.

Conventions.
Throughout this paper all Fukaya categories will be completed with respect to cones and direct summands. We will also denote the bounded derived category of coherent sheaves, its full subcategory consisting of perfect complexes, and the unbounded derived category of quasi-coherent sheaves on an algebraic stack X by coh X, perf X, and Qcoh X, respectively. For a dg-category A, we will also denote the unbounded derived category of right dg-modules as Mod A. All coefficient groups will be taken to be Z unless stated otherwise. By Z n we mean Z/nZ, and by Z (2) we mean the local ring of rational numbers with odd denominator.
1.4. Acknowledgements. The author would like to thank his Ph.D. supervisor Yankı Lekili for suggesting the project, his guidance throughout, and for careful reading of previous versions of this paper. He would also like to thank Jack Smith for his interest in the project, and his valuable feedback. The author is grateful to the anonymous referee for their helpful feedback and suggestions. This work was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], The EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London.

Invertible polynomials in two variables
In this paper, we will focus on invertible polynomials in two variables, as well as their unfoldings and quasi-homogenisations. As such, we will restrict ourselves to this case in the rest of the paper, and consider the variables x, y, z. The purpose of this section is to give a self-contained overview of the required background on invertible polynomials in two variables, and then to calculate the relevant spaces of semi-universal unfoldings.
Definition 2.1. Let A be a 2 × 2 matrix with non-negative integer coefficients. Let w andw be as above. We call w an invertible polynomial if A is invertible over Q, and w andw both have isolated singularities at the origin.
In what follows, we will always assume that p and q are always at least 2. For Brieskorn-Pham and chain polynomials, this is necessary for the origin to be a critical point of both w andw. In the loop case, if one of p or q is 1, then one can see that w andw are equivalent to x 2 + y 2 anď x 2 +y 2 by a change of variables.
The maximal symmetry group is defined as in (2), and to each t i we associate a character given by The group of characters for Γ w is given bŷ Let χ w := χ 3 , so that the elements of Γ w are the diagonal transformations of A 2 which keep w semi-invariant with respect to χ w , The subgroup ker χ w of Γ w are those elements which keep w invariant, and this is called the maximal diagonal symmetry group. There is an injective map and this fits into the short exact sequence where j w generates the cyclic group im(ϕ) ∩ ker χ w , and is called the grading element. Let Γ ⊆ Γ w be a subgroup of finite index containing ϕ(C * ), and for each Γ we denote χ := χ w | Γ . The statements of Theorem 1.1 and Corollary 1.3 require Γ = Γ w , since this avoids the problem of needing 'orbifold Fukaya(-Seidel) categories', as described in the introduction. Nevertheless, we will use Γ when what we say is valid for any Γ ⊆ Γ w , and Γ w when we specifically mean the maximal symmetry group.
The Jacobi algebra of w with Milnor number µ is given in (11). Let J w be as in the introduction, and semi-universal unfoldings of w be as in (12). Let U and w u be are as in (13) and (15), respectively. As already noted, Pinkham ([32]) observed that w being quasi-homogeneous means that the space U comes with a natural C * -action on it. Namely, the action on u ij is given by For a fixed u ∈ U , define R u := C[x, y]/(w u ), and observe that by scaling x, y, one can identify R u ≃ R t·u for t ∈ C * . The origin is the only fixed point of this action.
For a fixed Γ ⊆ Γ w , we would like to quasi-homogenise w u . In order to do this, however, we will need to extend the action of Γ to A 3 . The action on the z variable is chosen by setting This is done precisely so that x ∨ ∧ y ∨ ∧ z ∨ is isomorphic to χ as a Γ-module. With this weight, we want to restrict ourselves to the subspace U + ⊆ U for which w u is quasi-homogenisable, and has only positive powers of z. We define U + to be the subset of u ij in U which can be non-zero only if there exists a positive integer w ij such that and consider W u to be the quasi-homogenisation of w u for each u ∈ U + . Let J + ⊆ J w be the subset satisfying this condition.
For a fixed u ∈ U + , we set R u := C[x, y, z]/(W u ). By an abuse of notation, we will also denote the pullback of w to A 3 by w. We have that Y u is defined as in (16), and each Y u is the compactification of and the divisor at infinity The condition d 0 > 0 ensures that each Y u is a proper stack.
These W u fit together to form a family Following [29], we can then define and this gives us a family Note that since each fibre is the compactification of V u by X, and V u ≃ V t·u for t ∈ C * , we have that the fibres above points in the same C * -orbit of U + are isomorphic. Furthermore, the relative dualising sheaf of this family is Γ-equivariantly trivial, by construction, and since d 0 > 0, this trivialisation is unique up to scaling.
obtained by considering the 2-periodic free resolution of an R 0 -module, and replacing each free R 0 module with the R u -free resolution This is explained in detail, and in far greater generality, in [42,Section 3].
For the quotient stack Y u , since the dualising sheaf of Y u is trivial for each u ∈ U + , we have the Orlov equivalence The composition of (27) and Orlov equivalence gives the functor (17).

Unfoldings of loop polynomials.
In the case of a two variable loop polynomial w = x p y + y q x, we have µ = pq, and where d := gcd(p − 1, q − 1). Without loss of generality, we can assume that p ≥ q. One has that and where m, n is a fixed solution to The image of the injective homomorphism is an index d subgroup of Γ w ; however, we will only be interested in the maximal symmetry group, i.e. Γ = Γ w . A semi-universal unfolding is given by By definition, U + is the subspace of U containing elements such that there exists a positive integer w ij such that There are three possibilities for U + : Case I: For q > 2 the only solution to this is i = j = w ij = 1, and so U + = Spec C[u 11 ] = A 1 .
Case II: p > q = 2, we have i = j = w ij = 1, as well as j = 0, i = 1, and w ij = 2, and so Case III:

2.2.
Unfoldings of chain polynomials. In the case of a two variable chain polynomial w = x p y + y q , we have µ = pq − q + 1, and where d := gcd(p, q − 1).
Remark 2.1. It should be stressed that this is the Milnor number on the B-side. In the loop and Brieskorn-Pham cases the matrices defining the polynomials are symmetric, and the Milnor numbers of both sides will be the same, but this is not the case for chain polynomials.
One has that and where m, n is a fixed solution to The image of the injective homomorphism is an index d subgroup of Γ w , but again we will only be interested in the maximal symmetry group. A semi-universal unfolding is given by By definition, U + is the subspace of U containing elements such that there exists a positive integer w ij such that For chain polynomials, there are five different cases of U + to consider: Case I: When p, q > 2, the only solution is i = j = w ij = 1, and so U + = Spec C[u 1,1 ] = A 1 .

2.3.
Unfoldings of Brieskorn-Pham polynomials. In the case of a two variable Brieskorn- where d := gcd(p, q). One has that and where m, n is a fixed solution to The image of the injective homomorphism is an index d subgroup of Γ w , but as in the loop and chain cases, we are only interested in the maximal symmetry group. A semi-universal unfolding is given by By definition, U + is the subspace of U containing elements such that there exists a positive integer w ij such that For Brieskorn-Pham polynomials, we have the following five cases: Case II: In the case where p = 3 and q = 2, we have i = 1, j = 0 and w ij = 4, as well as i = j = 0 and w ij = 6, so Case III: In the case when p = q = 3, we have i = j = w ij = 1, as well as i = j = 0, w ij = 3, and so Case IV: In the case where p = 4, q = 2, we have j = 0, i = 2, w ij = 2, and i = j = 0, w ij = 4. Therefore Case V: In the case where p > 4 and q = 2, we have i = w ij = 2 and j = 0, so U + = Spec C[u 2,0 ] = A 1 .

Symplectic topology of the Milnor fibre
Let Σ be a smooth, compact, orientated surface of genus g > 0 with b > 0 connected boundary The surface to have in mind is the Milnor fibre of an invertible polynomial,Vw. Note that by an abuse of notation, we will not distinguish between the Milnor fibre and its completion, since what we mean will be clear from context. 3.1. Graded symplectomorphisms. In this subsection, we recall some facts about graded symplectic surfaces with the goal of providing a self-contained summary of Lemma 3.1. This provides criteria to ascertain when two graded symplectic surfaces are graded symplectomorphic, and is the key step in establishing Corollary 1.3.
For a 2n-dimensional symplectic manifold, (X, ω), there is a natural Lagrangian Grassmannian bundle LGr(T X) → X, whose fibre at x ∈ X is the Grassmannian of Lagrangian n−planes in T x X. Recall ( [37], [34]) that we say (X, ω) is Z-gradeable if it admits a lift to LGr(T X), the fibrewise universal cover of the Lagrangian Grassmannian bundle. This is possible if and only if 2c 1 (X) = 0 in H 2 (X), and this implies that K ⊗2 X , the square of the canonical bundle, is trivial. If X is gradeable, then a grading is given by a choice of homotopy class of trivialisation of K ⊗2 X . For a trivialising section Θ ∈ Γ(X, K ⊗2 X ), one has a map α X : LGr Given a compact, exact Lagrangian submanifold, L, this defines a section of LGr(T X) by considering the tangent space to L at each point. We say that L is gradeable with respect to a grading on . This is possible if and only if the Maslov class of L vanishes, where the Maslov class is defined by the homotopy class of the map L → LGr(T X) As explained in [37, Section 13(c)], on a (real) 2-dimensional surface, Σ, gradings correspond to trivialisations of the real projectivised tangent bundle, P R (T Σ) ≃ LGr(T Σ). Recall that a line field is a section of P R (T Σ). Supposing that a grading of Σ is chosen such that α Σ is as above, then one can define a line field on the surface given by η = α −1 Σ (1). Conversely, a nowhere vanishing line field gives rise to a map α Σ by recording the anticlockwise angle between the line field and any other line in the tangent plane. In this way, line fields correspond naturally to gradings on a surface, Σ.
Given a line field, η, which grades Σ, and a Lagrangian, L, represented by an embedded curve γ : S 1 → Σ, the map which corresponds to the Maslov class is given by recording the anticlockwise angle from η x to T x L at each point x ∈ L. The Maslov class vanishes, and hence L is gradeable with respect to η, if and only if the sections γ * η and γ * T L are homotopic in γ * P R (T Σ). A grading of L is a choice of homotopy between them.
We denote the space of line fields by G(Σ) := π 0 (Γ(Σ, P R (T Σ))), and this has the natural structure of a torsor over the group of homotopy classes of maps Σ → S 1 , which we identify with H 1 (Σ). With this in mind, consider the trivial circle fibration which induces the exact sequence Note that the orientation of Σ induces an orientation on each tangent fibre, and so the map ι is unique up to homotopy. For each line field, we can associate an element [η] ∈ H 1 (P R (T Σ)) by considering the Poincaré-Lefschetz dual of [η(Σ)] ∈ H 2 (P R (T Σ), ∂P R (T Σ)). These are precisely the elements such that ι * ([η])([S 1 ]) = 1, and this is the content of [28, Lemma 1.1.2].
As already mentioned, for an embedded curve γ : S 1 → Σ, there is a corresponding section of the Lagrangian Grassmannian,γ : S 1 → P R (T Σ). This is given by (γ, [T γ]), where [T γ] is the projectivisation of the tangent space to the curve γ.
Definition 3.1. Given a line field, η, on Σ, and an immersed curve γ : S 1 → Σ, we define the winding number of γ with respect to η as where ⟨·, ·⟩ : This pairing only depends on the homotopy class of η, as well as the regular homotopy class of γ. Recall that, for the case of surfaces, the Maslov number of a Lagrangian is precisely its winding number with respect to the line field used to grade the surface. Therefore, a Lagrangian is gradeable with respect to a line field if and only if its winding number with respect to this line field vanishes. Since we will be considering the Milnor fibre of a Lefschetz fibration, we must consider the grading on the Milnor fibre which is induced by the restriction of the unique grading of C 2 to Σ. This is crucial so that the functor (7) is graded, and therefore that (9) holds. The Lagrangian thimbles are contractible, and therefore gradeable, so each vanishing cycle is also gradeable with respect to the grading on the Milnor fibre induced from the restriction of the grading of C 2 . With this, we have that the grading on the Milnor fibre is given by a line field ℓ such that w ℓ (γ i ) = 0 for each vanishing cycle γ i : S 1 → Σ. Since the vanishing cycles form a basis of H 1 (Σ), the fact that the winding number around each Lagrangian is zero implies that the homotopy class of ℓ is unique.
For any symplectomorphism ϕ : Σ 1 → Σ 2 and η 2 ∈ G(Σ 2 ), one can consider the line field on Σ 1 given by If one has (Σ 1 ; η 1 ) and (Σ 2 ; η 2 ), where η 1 and η 2 are line fields used to grade the surfaces Σ 1 and Σ 2 , respectively, we say that a symplectomorphism ϕ : Σ 1 → Σ 2 is graded if ϕ * η 2 is homotopic to η 1 . If one takes Σ 1 = Σ 2 , then we define Symp(Σ; ∂Σ) to be the space of symplectomorphisms of Σ which fix ∂Σ pointwise. One can then define the pure symplectic mapping class group of Σ as M(Σ; ∂Σ) := π 0 (Symp(Σ; ∂Σ)), and observe that this group acts on G(Σ) as in (47) For a given line field η, consider the winding numbers around the boundary components. For two line fields to be homotopic, it is necessary for the winding numbers around each boundary component to agree, although this is definitely not sufficient. In particular, one can have two line fields which agree on the boundary, but which differ along interior non-separating curves.
Recall that for a closed, orientated Riemann surface, Σ, a theorem of Atiyah in [1] proves the existence of a quadratic form φ : S(Σ) → Z 2 , where S(Σ) is the space of spin structures on Σ, φ does not depend on the complex structure of Σ, and the associated bilinear form on H 1 (Σ; Z 2 ) is the cup product. Note that S(Σ) is a torsor over H 1 (Σ; Z 2 ), and φ being a quadratic form on S(Σ) means that it is a quadratic form on H 1 (Σ; Z 2 ) for any choice of basepoint. Moreover, the associated bilinear form doesn't depend on the basepoint. He also proves that there are precisely two orbits of the mapping class group of Σ on S(Σ), and these are distinguished by the invariant φ, which is known as the Atiyah invariant. In [19], Johnson gives a topological interpretation of the Atiyah invariant by proving that it is the Arf invariant of the corresponding quadratic form on H 1 (Σ, Z 2 ).
The Arf invariant is well studied in topology, and we briefly recount some basic facts about it, as well as some computation techniques. Let (V , (− · −)) be a vector space over Z 2 with a non-degenerate bilinear form, and q : V → Z 2 a quadratic form satisfying It is well-known that the Gauß sum and the sign is the Arf invariant of the quadratic form. I.e.
To compute the Arf invariant, one can just compute the Gauß sum, although, except in particularly nice circumstances, this can become computationally intractable quite quickly. One can also find a base change to a symplectic basis where the formula simplifies, although we will not do this. Instead, consider the basis {e 1 , . . . , e 2n } of V , and the matrix defined by where i ̸ = j. Such a matrix defines an even quadratic form on a Z (2) module, V , whose mod 2 reduction gives the bilinear pairing on V . The precise module structure of V is not important, since det f is well defined mod 8, and this value only depends on q. One then has The standard reference for further discussion of these facts is [16,Chapter 9].
Returning to the case at hand, recall that a non-vanishing vector field induces a spin structure on any compact Riemann surface with boundary. If the winding number around each boundary component with respect to this vector field is 2 mod 4, then this spin structure extends to the closed Riemann surface obtained by capping off the boundary components with discs, Σ. Any vector field also yields a line field by considering the projectivisation, and each embedded curve has an even winding number with respect to this line field. Conversely, it is shown in [28, Lemma 1.1.4] that if each embedded curve has even winding number with respect to a line field, then this line field arises as the projectivisation of a vector field. In light of this, in the case when two line fields have matching winding numbers around boundary components, arise from the projectivisation of vector fields, and where these vector fields define spin structures which extend to Σ, one must check that the corresponding Atiyah invariants of these spin structures agree.
A useful fact is that, by the Poincaré-Hopf index theorem, (see, for example, [17,Chapter 3]) for any compact S ⊆ Σ, we have where χ(S) is the Euler characteristic. It is therefore clear that the winding number does not descend to a homomorphism from H 1 (Σ). What is true, however, is that one can consider for each line field η the following homomorphism, given by the mod 2 reduction of the winding number: From this, we can define the following invariant. otherwise.
In the case when σ(η) = 0, and so η is the projectivisation of a vector field, v, we need to check when the spin structure on Σ defined by v extends to a spin structure on Σ, and if it does, calculate the corresponding Atiyah invariant.
For a line field (not necessarily coming from the projectivisation of a vector field), η, the existence of a quadratic form where α i are simple closed curves, and whose associated bilinear form is twice the intersection pairing on H 1 (Σ; Z 4 ) is established in [28,Proposition 1.2.2]. It is proven in [28, Lemma 1.2.3] that for g(Σ) ≥ 2, two line fields, η, θ, lie in the same M(Σ; ∂Σ)-orbit if the winding numbers agree on each boundary component, and q η = q θ . In the case when η and θ come from the projectivisation of vector fields, but the corresponding spin structures do not extend to Σ, or when the two line fields do not arise as the projectivisation of vector fields, it is enough to show that σ(η) = σ(θ), and that the winding numbers on the boundary components agree. In the case where η and θ are line fields such that σ(η) = σ(θ) = 0, and w η (∂ i (Σ)) = w θ (∂ i (Σ)) ∈ 2 + 4Z for each i ∈ {1, . . . , b}, we must compare the corresponding Atiyah invariants.

Recall that the inclusion ∂Σ
The kernel of the intersection pairing on H 1 (Σ; Z 2 ) is spanned by the image of i * , and the cokernel is naturally identified with H 1 (Σ; Z 2 ), where Σ is as above. The intersection form on H 1 (Σ; Z 2 ) descends to a non-degenerate intersection form on H 1 (Σ; Z 2 ).
By the fact that σ(η) = σ(θ) = 0, we have that the function is well defined, where q is either q η or q θ . By (53), we have that q/2(∂ i Σ) ≡ 0 mod 2 for each i ∈ {1, . . . , b}. Since the kernel of the intersection pairing on H 1 (Σ; Z 2 ) is spanned by the boundary curves, q/2 descends to a non-singular quadratic form and Arf(q) gives the last invariant required to ascertain whether two line fields are in the same M(Σ; ∂Σ)-orbit in the case where g(Σ) ≥ 2. In the case when g = 1, we definẽ where α and β are non-separating curves which project to a basis of H 1 (Σ; Z 2 )/im(i * ).

Gluing cylinders.
In this subsection we describe a general construction of graded surfaces by gluing cylinders. This allows us to reduce the computation of topological invariants of these surfaces to the combinatorics of how they are glued. We then provide explicit descriptions of the Milnor fibres of invertible polynomials in two variables, as well as the corresponding computations of the topological invariants.
Let A(ℓ, r; m) denote m disjoint cylinders placed in a column, each with r marked points on the right boundary component, and ℓ marked points on the left. Considering each cylinder as a rectangle with top and bottom identified, for each k ∈ {1, . . . , m}, counting top-to-bottom in the column, we label the marked points on the right (resp. left) boundary component of the k th cylinder as p + r(k−1) , . . . , p + rk−1 (resp. p − ℓ(k−1) , . . . , p − ℓk−1 ). The reasoning for the labelling is that we would like to keep track of where the marked points are on each individual cylinder, as well as where each marked point is on the right (resp. left) side of the column of cylinders with respect to the total ordering p + 0 , . . . , p + m i r i −1 (resp. p − 0 , . . . , p − m i ℓ i −1 ).

Given a collection of cylinders
A(ℓ 1 , r 1 ; m 1 ), A(ℓ 2 , r 2 ; m 2 ), . . . , A(ℓ n , r n ; m n ), such that r i m i = ℓ i+1 m i+1 , where i is counted mod n, and corresponding permutations σ i ∈ S m i r i , we can glue these cylinders together in the following way. For each i ∈ {1, . . . n} and j ∈ {0, . . . , m i r i − 1}, we glue a small segment of the boundary component p + by attaching a strip. See Figure 1 for an example.
For each i ∈ {1, . . . , n}, the number of boundary components arising from gluing the i th and (i + 1) st columns can be computed as follows. Consider the permutations The number of boundary components between the i th and (i + 1) st columns will then be given by the number of cycles in the decomposition of σ −1 i τ ℓ i+1 σ i τ r i ∈ S m i r i . Note that if m i = m i+1 then we simply get the commutator. To compute the homology groups of Σ, one can construct a ribbon graph Γ(ℓ 1 , . . . , ℓ n ; r 1 , . . . , r n ; m 1 , . . . , m n ; σ 1 , . . . , σ n ) ⊆ Σ, onto which the surface deformation retracts. To do this, let there be a topological disc D 2 for each of the cylinders. For each disc, attach a strip which has one end on the top, and the other end on the bottom. Then, attach a strip which connects two discs if there is a strip which connects the corresponding cylinders. These strips must be attached in such a way as to respect the cyclic ordering given by the gluing permutation. One can then deformation retract this onto a ribbon graph, whose cyclic ordering at the nodes is induced from the ordering of the strips on each cylinder. If there is no ambiguity, we will refer to this graph as Γ(Σ).
Since the embedding of Γ(Σ) into Σ induces an isomorphism on homology, the homology groups of Σ can be easily computed. Namely, since the graph is connected, we have H 0 (Σ) = Z. Since . A basis for the first homology of the graph is given by an integral cycle basis, and so the basis of the first homology for Σ is given by loops which retract onto these cycles.
Although there is no natural choice of grading on a surface glued in this way, in what follows we will only consider the case where the line field used to grade the surface is horizontal on each cylinder and parallel to the boundary components on attaching strips.

Loop Polynomials.
In the case of loop polynomialsw =x py +y qx , we have that n = 3 in the above construction, and we glue the cylinders where σ 1 and σ 2 are the identity elements in S q−1 and S p−1 , respectively, and σ 3 ∈ S (p−1)(q−1) is given by where in this case i ∈ {0, . . . , q − 2} and k 3 ∈ {1, . . . , p − 1}. Call the resulting surface Σ loop (p, q).
For the basis of homology, we begin by considering the compact curves in each cylinder, γ i . Together with these curves, we construct the basis for the first homology of the surface as follows. On each of the cylinders in the left and right columns, we take the curves to be approximately horizontal. We must therefore only describe the behaviour of the curves in the middle cylinder. Consider the curve which goes from the ((p − 1)(k 1 − 1) + j) th position on the left hand boundary to the ((q − 1)(k 3 − 1) + i) th position on the right hand boundary. In accordance with the construction of [13, Section 3], this curve must wind 2π degrees in the cylinder. This winding goes in the downwards direction, since we are thinking of the argument of thex coordinate increasing in this direction. These curves form a basis of the first homology, since they retract onto a basis for the graph, Γ(Σ loop (p, q)). The line field, ℓ, used to grade the surface is approximately horizontal on each cylinder, and approximately parallel to the boundary on the connecting strips. By construction, we have σ(ℓ) = 0. See Figure 2 for the case ofw =x 4y +xy 3 . There is only one boundary component between the first and second columns, as well as the second and third. With the line field ℓ given above, these components have winding numbers −2(q − 1) and −2(p − 1), respectively. To calculate the number of boundary components arising from gluing the third and first columns, note that in this case τ r 3 can be written as and τ ℓ 1 can be written as With this description, one can see that σ −1 As such, the length of a cycle is the least common multiple of (p−1) and (q−1), which is (p−1)(q−1) gcd(p−1,q−1) . There are therefore gcd(p − 1, q − 1) boundary components coming from gluing the third column to the first, each of winding number −2 (p−1)(q−1) gcd(p−1,q−1) . We can then compute the genus from (52), which yields and so the genus is By construction, the surface Σ loop (p, q) is graded symplectomorphic to the Milnor fibre of the polynomialw =x py +xy q . To see this, consider the ribbon graph which corresponds to the orientable surfaceVw. To construct this graph, first consider a disc D 2 for each of the neck regions of the construction of the Milnor fibre in [13, Section 3.1]. Then, attach a thin strip which connects two discs if there is at least one vanishing cycle which goes between them. The cyclic ordering of the strips at each disc is determined by the ordering of the vanishing cycles passing through a corresponding neck region. This graph can then be embedded intoVw in such a way that all intersections occur on the interior of the discs, and away from the discs, the vanishing cycles are on the interior of the attaching strips. One can deformation retract this onto a graph with the induced cyclic ordering at the vertices. Call this graph Γ(Vw), and observe that it is on-the-nose the same as Γ(Σ loop (p, q)), and so the corresponding surfaces with boundary are symplectomorphic. See Figure  3 for an example of p = 4, q = 3. To see that the two surfaces are graded symplectomorphic, consider the corresponding fat graphs in both cases. In this situation one can see that the description of the line field used to grade Σ loop (p, q) agrees with the description of the line field used to gradeVw, as in [13,Section 3.7], and this shows that the surfaces are graded symplectomorphic.

Chain polynomials.
In the case of chain polynomials, we havew =x p +xy q , and we will show that the Milnor fibre can be constructed by gluing where σ 1 is the identity element in S q−1 , and σ 2 ∈ S (p−1)(q−1) is given by where in this case i ∈ {0, . . . , (p − 1)(q − 1) − 1}. Call the resulting surface Σ chain (p, q).
For the basis of homology, we begin by including the compact curves in each cylinder, γ i . Together with these curves, we construct a basis for homology as follows. On the cylinders in the first column, we take the curves to be approximately horizontal. In the cylinder in the second column, the curve going from the (p − 1)(k 2 − 1) + j) th position on the left hand side to the i th position on the right hand side winds 2π (1−k 1 ) mod p−1 p−1 + i (p−1)(q−1) degrees, again in the downwards direction. This is in accordance with the description of the curves as in [13,Section 5.2]. Together, these curves form a basis for the first homology, since they retract onto a basis of the corresponding ribbon graph, Γ(Σ chain (p, q)). As in the loop case, the line field, ℓ, used to grade the surface is approximately horizontal on each cylinder, and approximately parallel to the boundary on the connecting strips. By construction, we have σ(ℓ) = 0.
There is only one boundary component which arises from gluing the first and second columns, and the winding number around this boundary component is −2(q − 1). To compute the number of boundary components, and their winding numbers, arising from gluing the second column to the first, observe that in this case, τ r 2 is just the permutation j → j − 1, and τ ℓ 1 is of the same form as (61). The permutation σ −1 2 τ ℓ 1 σ 2 τ r 2 ∈ S (p−1)(q−1) is given by Therefore the length of a cycle in the above permutation is (p−1)(q−1) gcd(p−1,q) . From this, we see that there are gcd(p − 1, q) boundary components arising from this gluing, and each boundary component has winding number −2 (p−1)(q−1) gcd(p−1,q) . Therefore there are 1 + gcd(p − 1, q) boundary components in total, and we conclude from (52) that As in the loop case, we claim that the surface constructed above is graded symplectomorphic toVw. To see this, we can construct a ribbon graph corresponding toVw as in the case of loop polynomials. This graph also matches Γ(Σ chain (p, q)) on-the-nose, and this establishes that Σ chain (p, q) andVw are symplectomorphic. To see that they are graded symplectomorphic, observe that in the corresponding fat graphs, the description of the line field above agrees with the description as in [13,Section 5.3], and this shows that the surfaces are graded symplectomorphic.

Brieskorn-Pham polynomials.
In the case of Brieskorn-Pham polynomials, we havě w =x p +y q , where (p, q) ̸ = (2, 2). Consider the surface obtained by gluing one cylinder to itself with the permutation σ ∈ S (p−1)(q−1)−1 , which is given by where in this case i is a point on the right boundary, and is considered as an element of {0, . . . , (p − 1)(q − 1) − 2}. Call this surface Σ BP (p, q).
For the basis of homology, we take a compact vertical curve in the cylinder, γ 1 , as well as one curve which is approximately parallel to the boundary along each of the connecting strips. On the interior of the cylinder, we have that the curve beginning in the j th position on the left hand side and ending at the i th position on the right hand side must wind 2π i+(−j) mod [(p−1)(q−1)−1] (p−1)(q−1)−1 degrees in the downwards direction, in accordance with the description of the curves in [13, Section 6.2]. Together, these curves form a basis for the first homology of Σ BP (p, q), since they retract onto a basis of the corresponding ribbon graph, Γ(Σ BP (p, q)). As in the previous two cases, the line field, ℓ, used to grade the surface is approximately horizontal on the cylinder, and approximately parallel to the boundary on the connecting strips. Again, by construction, we have σ(ℓ) = 0.
Let τ be the permutation i → i − 1, so the number of boundary components is given by the number of cycles in the decomposition [σ, τ ] ∈ S (p−1)(q−1)−1 . The commutator is given by and so the length of a cycle will be given by (p−1)(q−1)−1 gcd(p,q) . There are therefore gcd(p, q) boundary components arising from this gluing, and each has winding number −2 (p−1)(q−1)−1 gcd(p,q) . Therefore, we have As in the previous cases, we deduce that that Σ BP (p, q) is graded symplectomorphic to the Milnor fibre.
3.3. Symplectic cohomology of the Milnor fibre. In this subsection we utilise the explicit descriptions of the Milnor fibres of invertible polynomials given above to calculate the module structure of symplectic cohomology of these surfaces. By combining this with Theorem 3.2 below, we will be able to deduce the correct mirror curves in the proof of Theorem 1.1.
The symplectic cohomology of surfaces admits a particularly simple description -namely, for any Riemann surface, Σ g,b , of genus g > 0 with b > 0 boundary components, we have where w η (∂ i Σ g,b ) is the winding number of the line field η about the boundary component ∂ i Σ g,b .
This was first described in the case of one puncture in [36,Example 3.3], and the generalisation to more than one puncture follows by the same argument. Note that the grading convention in [36] is shifted by one from ours.
In the case of loop polynomials,w =x py +y qx , we saw in Section 3.2.1 that the Milnor fibre is a 2 + gcd(p − 1, q − 1)-times punctured surface of genus g loop = 1 2 (pq − 1 − gcd(p − 1, q − 1)). Consider Σ g,b =Vw, and let ℓ be the line field used to grade the surface as in Section 3.2.1. We then have by (66) and the analysis in Section 3.2.1, that In the case of chain polynomials,w =x p +xy q , we have that the Milnor fibre is a (1 + gcd(p − 1, q))−times punctured surface of genus g chain = 1 2 (pq − p + 1 − gcd(p − 1, q)). Let ℓ be the line field used to grade the surface, as in Section 3.2.2. We then have by (66) and the analysis in Section 3.2.2 that In the case of Brieskorn-Pham polynomials, we have that the Milnor fibre is a gcd(p, q)-times punctured surface of genus g BP = 1 2 ((p − 1)(q − 1) + 1 − gcd(p, q)). Let ℓ be the line field used to grade the surface, as in Section 3.2.3. Then, by (66) and the analysis in Section 3.2.3, we have As previously mentioned, the comparison of the symplectic cohomology of the Milnor fibre and the Hochschild cohomology of the Fukaya category of the Milnor fibre will be crucial in our mirror symmetry argument. To this end, we have the following theorem of Lekili and Ueda: Note that assumingď 0 > 0 is crucial, as can be seen if one considersw =x 2 +y 2 .

Graded symplectomorphisms between Milnor fibres.
It is a natural question to ask which Milnor fibres are graded symplectomorphic, and in this subsection we utilise Lemma 3.1 to determine this. Since the genera, number of boundary components, and winding numbers around boundary components of the Milnor fibres were calculated above, it is easy to check when these match. This gives the potential graded symplectomorphisms, although one must also check that the corresponding Arf invariants agree whenever they are defined. We use the method described in Section 3.1 to compute the Arf invariant when necessary. By establishing which Milnor fibres are graded symplectomorphic, Corollary 1.3 follows immediately once Theorem 1.1 is proven.
Observe that for each q ≥ 2 and n ≥ 1, we have thatw loop =x (q−1)n+1y +y qx , andw chain = x qn+1 +y qx have the same genus, number of boundary components, and winding numbers along each boundary component. In the case of q odd, this is enough to give a graded symplectomorphism by Lemma 3.1, since σ = 0 in both cases, and −2(q − 1) ≡ 0 mod 4. In the case where q and n are both even, we again have that the Milnor fibres are graded symplectomorphic. In the case where q is even and n is odd, it remains to check that the relevant Arf invariants agree.
For a graded symplectomorphism between the Milnor fibres of a chain and Brieskorn-Pham polynomial, we have thatw ′ chain =x p +y n(p−1)x andw BP =x p +y np for each p ≥ 2 and n ≥ 1 have the same genus, number of boundary components, and winding numbers along each boundary component. In the case where n is even and p is odd, we have that −2(n(p − 1) − 1) ≡ 0 mod 4, and so Lemma 3.1 gives us a graded symplectomorphism between the Milnor fibres. Similarly, for p = 2 and n odd, Lemma 3.1 yields a graded symplectomorphism between Milnor fibres. In all other cases, we must check the relevant Arf invariants.
The only possibility for a graded symplectomorphism between the Milnor fibres of a loop and Brieskorn-Pham polynomial is that both are symplectomorphic to a Milnor fibre of a chain polynomial. For such a graded symplectomorphism to exist, we requirew loop =x qy +y qx ,w chain = x q+1 +y qx , andw BP =x q+1 +y q+1 . It should be noted that the potential graded symplectomorphisms discussed above are the only such possibilities.
3.4.1. Graded symplectomorphisms between the Milnor fibres of loop and chain polynomials. In the case of loop polynomials of the formw loop =x (q−1)n+1y +y qx , we have that there are q +1 boundary components. Recall the basis of the first homology of the Milnor fibre given in [13,Section 3]. An elementary calculation shows that if we remove the Lagrangian V loop xy , as well as the Lagrangians ..,q−2} , then the restriction of the intersection form is non-degenerate.
In the case of chain polynomials of the formw chain =x qn+1 +y qx , we consider the basis of Lagrangians for the first homology group of the Milnor fibre as given in [13,Section 5]. By removing the Lagrangian V chaiň xy , as well as the Lagrangians { i V chaiň xw } i∈{0,...,q−2} , the restriction of the intersection form to the remaining Lagrangians is non-degenerate.
Let U n be the n × n matrix given by (U n ) i,j = 1 if i ≥ j 0 otherwise . Then we have that f chain = U q−1 ⊗ U qn + (U q−1 ⊗ U qn ) T . On the other hand, f loop is the block matrix given by In both cases, one can explicitly compute that the determinant is nq + 1, and so, in particular, we have Arf(q chain ) = Arf(q loop ). Therefore, by Lemma 3.1, the surfaces are graded symplectomorphic, and their respective compact Fukaya categories are quasi-equivalent.

3.4.2.
Graded symplectomorphisms between the Milnor fibres of chain and Brieskorn-Pham polynomials. In the case of chain polynomials of the formw ′ chain =x p +y n(p−1)x , and Brieskorn-Pham polynomials of the formw BP =x p +y np , we have that the there are p boundary components. In the chain case, we remove V chaiň xy , as well as the Lagrangians { i V chaiň xw } i∈{0,...,p−3} from the collection of Lagrangians which form a basis of the first homology of the Milnor fibre, and the restriction of the intersection form to the remaining Lagrangians is non-degenerate. In the Brieskorn-Pham case, if we remove the Lagrangians { l,np−2 V BP 0 } l∈{0,...,p−2} from the collection of Lagrangians which form a basis of the first homology group of the Milnor fibre, as described in [13,Section 6], then the restriction of the intersection form to the remaining Lagrangians is likewise non-degenerate.
In the case of chain polynomials, we have that f chain ′ is given by removing the top and left p − 2 rows and columns from In the case of Brieskorn-Pham polynomials, we have that In both cases, we have that We therefore have by Lemma 3.1 that the Milnor fibres are graded symplectomorphic.

Hochschild cohomology via matrix factorisations
In this section, we make the necessary Hochschild cohomology computations which will later enable us to deduce the existence of an affine scheme of finite type which represents the moduli functor of A ∞ -structures on the graded algebras we are interested in. This is the main computational component of the paper, and we include the entire calculation for completeness, although a computation of HH n (Y ) for n ≤ 2 would have sufficed.
Suppose once more that we are in the setting of Section 2, and we have that w is an invertible polynomial in two variables such that d 0 > 0, Γ is a subgroup of Γ w of finite index containing ϕ(C * ), and W u the quasi-homogenisation of a semi-universal unfolding corresponding to u ∈ U + . Denote V = {x, y, z}, S := Sym V = C[x, y, z], and so R u = S/(W u ), and W u ∈ (S ⊗ χ) Γ (recall χ = χ w | Γ ). Equation (28) implies that This vastly simplifies the calculation at hand, since a theorem of Ballard, Favero, and Katzarkov ([2, Theorem 1.2]) reduces the computation of the Hochschild cohomology of the category of Γequivariant matrix factorisations of W u to studying the cohomology of certain Koszul complexes, which in nice cases reduces to studying the Jacobi algebra of W u . To this end, consider an element γ ∈ ker χ, and V γ the subspace of V of γ-invariant elements. Let S γ := Sym V γ , and N γ the complement of V γ in V , so that V ≃ V γ ⊕ N γ as a Γ-module. Denote by W γ the restriction of W u to Spec S γ , and consider the Koszul complex where S γ sits in cohomological degree 0, and the differential is the contraction with Denote by H i (dW γ ) the i th cohomology group of the Koszul complex. The zeroth cohomology of (68) is isomorphic to the Jacobi algebra of W γ , and if W γ has an isolated critical point at the origin, then C * (dW γ ) is a resolution. Our main tool for computing Hochschild cohomology is the following theorem: ). Let w be an invertible polynomial in two variables, Γ be a subgroup of Γ w of finite index containing ϕ(C * ) acting on A 3 = Spec S, and W u ∈ S be a non-zero element of degree χ. Assume that the singular locus of the zero set Z (−Wu)⊞Wu of the Thom-Sebastiani sum −W u ⊞ W u is contained in the product of the zero sets Z Wu × Z Wu . Then HH t (A 3 , Γ, W u ) is isomorphic to In the case where the Γ-action on V satisfies dim(S ⊗ ρ) Γ < ∞ for any ρ ∈Γ, one then has for every t ∈ Z. To see this, note that the complex C * (dW γ ) is always bounded, and the group ker χ is finite. Therefore, each direct summand of (70) is finite dimensional, and there are only finitely many u contributing to a fixed t.
Theorem 4.1 is a minor modification of [2,Theorem 1.2], where the difference is in the convention for the Koszul complex. In our case, when there is an additional C * -action on V , then (70) is equivariant with respect to it. In particular, in the case of u = 0 ∈ U + , we have that there is an additional C * -action on V given by t · (x, y, z) = (x, y, tz), and this induces an additional C * -action on HH * (Y 0 ). Denote by HH * (Y 0 ) <0 the negative weight part of this action. We refer the reader to [2] for a proof of Theorem 4.1.
It should be emphasised that being (un)twisted is a property of a pair (w, Γ), rather than its category of matrix factorisations. Indeed, we will see below that the polynomial w = x 3 y + y 2 is twisted and w = x 2 y + y 2 x is not, although Corollary 1.3 shows that the Hochschild cohomology of their respective categories of matrix factorisations are isomorphic. A pair (w, Γ) being untwisted ensures that all of the deformations corresponding to HH 2 (Y 0 ) <0 come from semi-universal unfoldings of the polynomial w. This is a key step in the proof of [29, Theorem 1.6], a special case of which appears as Theorem 6.1. By an abuse of notation, we will refer to a polynomial w as being (un)twisted to mean that the pair (w, Γ w ) is (un)twisted. 4.1. Loop polynomials. Consider W 0 = x p y+y q x with the only restriction that p, q ≥ 2. Without loss of generality, we can consider p ≥ q. This has weights as in (29), where we again set d := gcd(p − 1, q − 1). As explained in Section 2, we extend the action of Γ w ≃ C * × µ d to A 3 as in (24) so that we now have The group of characters is given bŷ and we take m, n to be the same fixed solution to (32) as in Section 2.1. We write each charac- d ,m , span{y ∨ } ≃ ρ (p−1) d ,−n , χ ≃ ρ pq−1 d ,m−n , and ker χ ≃ µ pq−1 . We have that Jac w is given as in (30). Since we are in the situation of an affine cone over an isolated hypersurface singularity, [29, Section 3.1] shows that we must have l = 0 in (70). Furthermore, there are no contributions when u < −1, and the only possible contribution for u = −1 comes from when N γ = span{x, y}, or z / ∈ V γ . When γ ∈ ker χ is the identity element, we have V γ = V , N γ = 0, and for s > 0, and that for 0 ≤ n ≤ 2 (p−1)(q−1) In the case where u ≡ 0 mod (p−1)(q−1) gcd(p−1,q) , we also have where k = upq (p−1)(q−1) − 1. These contribute to C(k) to HH 2u (Y 0 ) and HH 2u+1 (Y 0 ), respectively.
In the case where V γ = span{y}, there cannot be a contribution. There are p − 1 such elements of ker χ which fix y and nothing else.
In the case where V γ = span{z}, N γ = span{x, y}, we have for each n ∈ Z ≥0 , there are contributions from the summands and these contribute C( (n+1)pq gcd(p−1,q) −1) to HH for s > 0, and that for 0 ≤ n ≤ 2(p−1)(q−1)−1 d + 1, we have that HH n (Y 0 ) is given by and k the unique solution to (80) Note that this is twisted in the case p = q = 3, and p = 4, q = 2, but is otherwise untwisted.

4.4.
Unfoldings of invertible polynomials. Of course, Theorem 4.1 can also be used to compute the Hochschild cohomology of the category of matrix factorisations of an unfolded polynomial. For the polynomials where dim U + > 1, we will need Hochschild cohomology calculations of unfolded polynomials in order to be able to isolate the correct mirror. Towards this end we will need to (at least partially) calculate HH 2 (Y u ) in these cases.
Proof. In each of the cases we consider, the sequence (∂ x W u , ∂ y W u ) is a regular sequence in S. Therefore, the cohomology of the Koszul complex, (68), will be concentrated in degrees 0 and −1, and the only contributions to HH 2 (Y u ) can come from (Jac Wu ⊗ χ) Γ and (Jac Wu ⊗ x ∨ ∧ y ∨ ) Γ . Note that if the latter term contributes to HH 2 (Y u ), then the polynomial is twisted, and we will not consider it.
The two loop polynomials we must consider are w = x p y + y 2 x for p > 2 and w = x 2 y + y 2 x. In the former case, the unfolding is given by W u = x p y + y 2 x + u 1,1 xyz + u 1,0 xz 2 . For a contribution to HH 2 (Y u ), there must be an element of Jac Wu which is proportional to χ. Note that if u 1,1 = 0 then dim(Jac Wu ⊗χ) Γ = 0. On the other hand, we have that dim(Jac Wu ⊗χ) Γ = 0 if u 1,0 ̸ = 0. In the case w = x 2 y+y 2 x, we have that dim(Jac Wu ⊗χ) Γ < 3 unless u 1,1 ̸ = 0, and the other coefficients are zero.
The only chain polynomials which need to be considered are w = x p y + y 2 for p > 3 and w = x 2 y + y 2 . In the former case, note that if u 1,1 = 0, or u 1,1 , u 2,0 ̸ = 0, then HH 2 (Y u ) = 0. In the latter case, note that dim HH 2 (Y u ) < 2 unless u 0,1 ̸ = 0 and the other coefficients are zero. □

Generators and formality
In this section we recall and implement the results of various authors to establish the required generation statements for the compact Fukaya category of the Milnor fibre, and also the category of perfect complexes on Y u for any u ∈ U + , as outlined in Section 1.1.
As in the previous sections, letVw be the Milnor fibre of the transpose of an invertible polynomial in two variables such that d 0 > 0. Let {S i }μ i=1 be a distinguished basis of vanishing cycles, and let S be the full subcategory of F(Vw) whose objects are {S i }μ i=1 . As in Section 1.1, denote by A the total A ∞ -endomorphism algebra of S, Sinceď 0 > 0, the argument of [35,Lemma 5.4] then shows that S split-generates F(Vw), and so On the B-side, let w : A 2 → A be an invertible polynomial in two variables such that d 0 > 0. In each case, we aim to associate U + to the moduli space of A ∞ -structures on a fixed quiver algebra.
In order to do this, for each u ∈ U + we must find generators S u of perf Y u such that (i) the isomorphism class of the cohomology level endomorphism algebra End(S u ) does not depend on u ∈ U + , and (ii) the generator S 0 at 0 ∈ U + admits a C * -equivariant structure such that the cohomological grading on End(S u ) is proportional to the weight of the C * -action. If we find generators which satisfy condition (i), then we can think of deformations of Y in terms of deformations of the A ∞ -structures on the cohomology level endomorphism algebra. Condition (ii) will be necessary to deduce that end(S 0 ) is formal.
Recall ([13, Theorem 2]) that mf(A 2 , Γ w , w) has a tilting object, E, for any two variable invertible polynomial w. For each u ∈ U + , let S u be the image of E under the pushforward functor mf(A 2 , Γ, w) → mf(A 3 , Γ, W u ) ≃ coh Y u .
It is then a consequence of [29, Theorem 4.1] that S u split-generates perf Y u .
Let A u be the minimal model of the dg-endomorphism algebra of S u , end(S u ). As discussed in Section 4, one has a quasi-equivalence and therefore, by the Morita invariance of Hochschild cohomology, an isomorphism HH * (Y u ) ≃ HH * (A u ).
The cohomology algebra A u := H * (A u ) is independent of u, and by [42, Theorem 1.1], is isomorphic as a vector space to (9). On both the A-, and B-sides, the algebra structure is given as in (10), since A → is the quiver algebra of a quiver with no cycles, and so HH 2 (A → , (A → ) ∨ [−1]) = (HH 1 (A → )) ∨ = 0.
By exploiting the additional C * -action, one can prove a general statement for the formality of A 0 . This is done by first showing that the cohomological grading on End * (S 0 ) is proportional (equal in the case of curves) to the weight of the C * -action. This follows from the fact that the dualising sheaf of Y 0 is trivial as an O Y 0 -module, but has weight one with respect to the additional C * -action. Since C * is reductive, the chain homotopy to take end(S 0 ) to a minimal A ∞ -structure can be made C * -equivariant. Since µ d lowers the cohomological degree by 2, the only map which can be non-zero is µ 2 . In particular, this means that HH * (Y 0 ) ≃ HH * (A), and so the computations in Section 4 imply that the moduli space of A ∞ -structures on A is represented by an affine scheme of finite type. Furthermore, combining equation (87) with Theorem 3.2, and the calculations in Section 3.3 gives us that the A ∞ -structure on A, the endomorphism algebra of the generators of F(Vw), is not formal.

Homological mirror symmetry for invertible polynomials in two variables
In this section, we bring together the previous sections of the paper to establish Theorem 1.1 and Corollary 1.3. As noted above, the computations of Section 4 together with (87) mean that the moduli space of A ∞ -structures on A is represented by an affine scheme of finite type, U ∞ (A), for any untwisted invertible polynomial w. As explained in Section 1.1, we would like to identify U ∞ (A) with the space U + corresponding to w by showing that the map (18) is an isomorphism. To this end, we utilise the following special case of [29, Theorem 1.6]: Theorem 6.1. Let w be an untwisted invertible polynomial in two variables such that d 0 > 0, and Γ be a subgroup of Γ w containing ϕ(C * ) as a subgroup of finite index. Let A → be the endomorphism algebra of a tilting object in mf(A 2 , Γ, w), and let A be the degree 1 trivial extension algebra of A → . Then there is a C * -equivariant isomorphism U + ∼ − → U ∞ (A) which sends 0 ∈ U + to the formal A ∞ -structure on A.
This isomorphism descends to the quotient by the C * -action, and so we get an isomorphism U + \ (0) /C * ∼ − → M ∞ (A). It should be reiterated that the polynomial being untwisted is a crucial assumption, as can be seen by considering, for example, w = x 3 y + y 2 . In this case, we have that HH 2 (Y 0 ) <0 = C(3) ⊕ C(2) ⊕2 ⊕ C(1), but U + = A 3 .
Proof of Theorem 1.1. In each case, we know that the A ∞ -structure on F(Vw) is not formal, and so is represented by a point in M ∞ (A). By Theorem 6.1, this, in turn, represents the A ∞ -structure corresponding to the dg-enhancement of the derived category of perfect sheaves on a semi-universal unfolding of w. In the cases where dim U + = 1, we have that M ∞ (A) is a single point, and so the semi-universal unfolding (up to scaling) corresponding to this point must be the mirror. Note that in the cases w = x 2 y + y q for q > 2 and w = x p + y 2 for p > 4, we have C[x, y, z]/(x 2 y + y q + yz 2 ) ≃ C[x, y, z]/(x 2 y + y q + xyz), C[x, y, z]/(x p + y 2 + x 2 z 2 ) ≃ C[x, y, z]/(x p + y 2 + xzy) by completing the square.
In the case where dim U + > 1, we must exclude the points in M ∞ (A) other than the claimed mirror. In the case w = x p y+y 2 for p > 3, we have by Lemma 4.2 that dim HH 2 (Y u ) = 0 < dim SH 2 (Vw) unless u = (0, 1). By Theorem 3.2, we must therefore have that the mirror is identified with Y u for u = (0, 1) ∈ U + . A similar argument in the cases w = x 2 y + y 2 x and w = x p y + xy 2 for p > 2 leads to identifying the mirrors as Y u for u = (0, 0, 0, 1) and u = (0, 1), respectively.
In the case of w = x 3 + y 2 , we follow the same argument as in [24]. Namely, we have that if Y u is an elliptic curve, then HH * (Y u ) exists in only finitely many degrees by the Hochschild-Kostant-Rosenberg theorem. Since the symplectic cohomology of the Milnor fibre is non-trivial in arbitrarily large degree, by Theorem 3.2, we have that the mirror cannot be smooth. We therefore have that the mirror must be the nodal cubic W u = x 3 + y 2 + xz 4 + In the cases where the polynomial is twisted, this result has already been established in [27] by different means. Our construction of the Milnor fibres agrees with the surfaces constructed in [27], and the mirrors established there are precisely the mirrors we claim.
The only invertible polynomial where d 0 ̸ > 0 is w = x 2 + y 2 , for which d 0 = 0. This, however, corresponds to the mirror symmetry statement for C * , which is already well established. Therefore, Theorem 1.1 is true in this case, too. □ Proof of Corollary 1.3. By observing that the results of Section 3.4 show that the relevant compact Fukaya categories are quasi-equivalent, Theorem 1.1 establishes that the derived categories of perfect complexes of their mirrors are, too. □