A symplectic look at the Fargues-Fontaine curve
aa r X i v : . [ m a t h . S G ] J un A SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE
YANKI LEKILI AND DAVID TREUMANN Introduction
This paper discusses homological mirror symmetry for the Fargues-Fontaine curve.1.1.
The Fargues-Fontaine curve.
Let E be a local field and let C be a perfectoidfield of characteristic p . For each such pair ( E, C ), Fargues and Fontaine have defined an E -scheme that we will denote by FF E ( C ) — it is denoted by X C,E in [FF, Def. 6.5.1]. Itis in no sense a “curve over E ” or even a variety: it is not of finite type over E or overany other field. A scheme that is not of finite type over some base cannot be smooth orproper in the usual sense ([EGA, Vol 4, §
17; Vol 2, § FF E ( C ) resembles a closedRiemann surface is some peculiar ways: • It is noetherian of Krull dimension one. Moreover it is regular, so that the localring at each closed point of FF E ( C ) has a discrete valuation. • A nonzero rational function f (that is, a section of O FF over the generic point) has v ( f ) = 0 for at most finitely many of these valuations v , and P v v ( f ) = 0In fact FF E ( C ) even resembles the Riemann sphere: one hasPic( FF E ( C )) = Z and H ( O FF ) = 0There are some contrasts with the Riemann sphere: FF E ( C ) has indecomposable vectorbundles of higher rank, and its ´etale fundamental group is naturally isomorphic to theabsolute Galois group of E . Fargues has a program to apply these properties of FF E ( C )to the local Langlands correspondence [Fa].When E = Q p , FF E ( C ) is an important object in p -adic cohomology — it was introducedto organize some of the structures of p -adic Hodge theory. When E = F p (( z )), the analogousstructures are those of Hartl [H]. We have nothing to say about FF Q p ( C ) but we are ableto touch FF F p (( z )) ( C ) with mirror symmetry.1.2. Homological mirror symmetry.
Homological mirror symmetry (HMS) is a frame-work for relating Lagrangian Floer theory on a symplectic manifold to the homologicalalgebra of coherent sheaves on a scheme — often, a scheme that is seemingly unrelated tothe symplectic manifold.What symplectic structure could be mirror to FF F p (( z )) ( C )? We suggest the answer isa two-dimensional torus. There is already a very well-studied mirror relationship betweenthe symplectic torus and the Tate elliptic curve (over Z (( t ))), which we review in §
2. Toget the Fargues-Fontaine curve in place of the Tate curve, we introduce two changes: (1) We couple Lagrangian Floer theory to a locally constant sheaf of rings on the torus— the fiber of this sheaf of rings has characteristic p , and going around one of thecircles is the p th power map. (Going around the other circle is the identity map).(2) We set the Novikov parameter (this is the element t ∈ Z (( t )) in the ground ring ofthe Tate curve) to t = 1 — symplectically this is sort of like studying the limit asthe symplectic form goes to 0.Both of these maneuvers are unusual in symplectic geometry. The first turns out to bestraightforward, so that one obtains a Fukaya A ∞ -category with the usual properties. Thesecond is much more delicate and touches some folklore questions about “convergent powerseries Floer homology.”1.3. Lagrangian Floer theory on the torus.
Let T be a 2-dimensional torus, whichwe present as a quotient of R by Z and endow with the standard symplectic form dx dy .For each integer m , let L ( m ) ⊂ T denote the image of the line in R through the origin ofslope − m . Let L ( ∞ ) denote the image of the vertical line through the origin. We orient L ( m ) from left to right and L ( ∞ ) from top to bottom. The figure shows L (0) , L ( ∞ ) , and L (3) in a fundamental domain of T : L ( ∞ ) L (0) L (3) If m > m , then L ( m ) and L ( m ) meet transversely in ( m − m ) points. Lagrangian Floertheory gives algebraic structures to the free modulesCF( L ( m ) , L ( m ) ) := M x ∈ L ( m ∩ L ( m Λ , (1.3.1)where Λ is a suitable ring, about which more in § L ( m ) , L ( m ) endow (1.3.1) with a Z / Z -grading by making some additional topological choices), and CF ∗ ( L ( m ) , L ( m ) ) supportsa differential of degree +1 § § § L ( m ) , L ( m ) ) × CF( L ( m ) , L ( m ) ) → CF( L ( m ) , L ( m ) ) (1.3.2) SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE 3 whose value on ( x , x ) ∈ ( L ( m ) ∩ L ( m ) ) × ( L ( m ) ∩ L ( m ) ) is the summation X y ∈ L ( m ∩ L ( m y X u ∈M ( y,x ,x ) ± t area( u ) (1.3.3)The inner sum is infinite: it is indexed by the set of rigid pseudoholomorphic triangles u : ∆ → T (1.3.4)with vertices at x , x , y and one edge each along L ( m ) , L ( m ) , L ( m ) . The sign in ± t area( u ) depends on u , and on the choice of a spin structure on each of the oriented 1-manifolds L ( m i ) , see § Dehn twist.
There is a canonical identification of CF( L ( m ) , L ( n ) ) with CF( L (0) , L ( n − m ) ),induced by ( x, y ) ( x, y − mx )the m -fold Dehn twist around L ( ∞ ) . An old suggestion of Seidel’s [Z] is to use this identi-fication to package the triangle products as a graded ring structure on the sumΛ ⊕ ∞ M m =1 CF( L (0) , L ( m ) ) (1.4.1)The multiplication on (1.4.1) is associative and commutative for nontrivial reasons. Theassociativity is a consequence of a very general Floer-theoretic argument that studies 1-dimensional moduli spaces of pseudoholomorphic quadrilaterals § § The Floer cohomology of ( L (0) , L (0) ) . In (1.4.1), we have inserted the unit of thering by hand (the summand Λ, which we place in degree zero), but this can also bemotivated Floer-theoretically. The definition of CF in (1.3.1) is not the right one when L ( m ) = L ( m ) , or for any other pair that do not meet transversely. But if φ = { φ s } s ∈ R isa general Hamiltonian isotopy, thenCF( φ s L (0) , L (0) ) := M x ∈ φ s L (0) ∩ L (0) x · Λtogether with its differential, gives a cochain complex whose cohomology groups do notdepend on φ . These cohomology groups are Z / ( L (0) , L (0) ), cf. § Novikov ring Λ and Floer theory relative to a divisor. There is some flexibilityin choosing the ground ring Λ, but it should contain a ring of constants (let us use C forthis ring — later on it will be the same as the C of § t and all necessary YANKI LEKILI AND DAVID TREUMANN powers of it, and it should carry a topology in which all the sums (1.4.1) converge. Theconventional choice is the Novikov ring (2.7.3), which we will denote by Λ C :Λ C = ( ∞ X i =0 a i t λ i | a i ∈ C, λ i ∈ R and lim i →∞ λ i = ∞ ) (1.6.1)We can shrink those coefficients to C [[ t ]] by the following device of Seidel’s, called Floertheory “relative to a divisor.” Rather than computing the area of the triangles u , we fixa basepoint D ∈ T (in general, a symplectic divisor D ⊂ T ) and use the cardinality of u − ( D ) in place of symplectic area. If D is in the first quadrant and extremely close to(0 , D unlessthe triangle u is extremely acute — let us write area Z ( u ) for this discretized notion of area.With some additional care, by letting D → (0 ,
0) (see [LPe2, § C [[ t ]] · ⊕ ∞ M m =1 CF( L (0) , L ( m ) ) (1.6.2)1.7. Theorem [LPe2] . The C [[ t ]]-schemeProj C [[ t ]] · ⊕ ∞ M m =1 CF( L (0) , L ( m ) ) ! (1.7.1)is isomorphic to E Tate × Z [[ t ]] C [[ t ]]; the Tate elliptic curve over C [[ t ]] whose Weierstrass equa-tion is y + xy = x − b x − b where b , b ∈ Z [[ t ]] are the series b = ∞ X n =1 n t n − t n b = ∞ X n =1 (cid:18) n + 5 n (cid:19) t n − t n (1.7.2)1.8. Theta series.
The relationship between (1.6.2) and functions on the Tate ellipticcurve is more transparent when those functions are described in terms of θ -series. Set θ m,k := ∞ X i = −∞ t m i ( i − + ki z mi + k ; θ abs m,k := ∞ X i = −∞ t ( mi + k ) / (2 m ) z mi + k (1.8.1)The simplest of these series is the Jacobi function θ , = ∞ X i = −∞ t i ( i − z i = (1 + z ) ∞ Y i =1 (cid:2) (1 − t i )(1 + t i z )(1 + t i z − ) (cid:3) The others θ m,k are obtained by a change of variables from θ , . These series are doublyinfinite in z , but in formally expanding the product of two of them, the coefficient of z i t j has only finitely many nonzero contributions. The C [[ t ]]-linear span of the θ m,k (resp. theΛ C -linear span the θ abs m,k ) is closed under multiplication and graded by m , and is isomorphic SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE 5 as a graded ring to (1.6.2) (or (1.4.1) in the absolute case). The isomorphisms send( k/m, ∈ L (0) ∩ L ( m ) to θ m,k or to θ abs m,k .1.9. Fukaya category and homological mirror symmetry.
The triangle product(1.3.2) resembles a composition law in a category. It is part of a sequence of structures onthe CF(
L, L ′ ), µ n : CF( L n − , L n ) × · · · × CF( L , L ) → CF( L , L n ) (1.9.1)that are obtained by summing over the ( n + 1)-gons with sides along L , L , . . . , L n § § A ∞ -category. After passing to a triangulated envelope and splittingidempotents, we will call any of these A ∞ -structures a “Fukaya category” and denote it byFuk( T ) (in the absolute case) or Fuk( T, D ) (in the relative case).Kontsevich’s homological mirror symmetry conjecture, specialized to T , asks for a quasi-equivalence between Fuk( T ) and the derived category of coherent sheaves on an ellipticcurve. A version of this for complex elliptic curves was obtained in [PoZa]. When C = Z , Theorem 1.7, together with a generation result for Fuk( T, D ) [LPe2, § Z ”:Fuk( T, D ) ∼ = D b (Coh( E Tate ))The structure sheaf of E Tate is the image of L (0) under this equivalence.1.10. F -fields. Let Λ be a local system of rings on T , so that at each point x ∈ T we aregiven a ring Λ x , and along each path γ from x to y we are given a ring isomorphism ∇ γ : Λ x ∼ → Λ y (1.10.1)Suppose that each ring Λ x has the structures that we asked for in § C x ), distinguished elements of the form t a , and carriesa topology. The maps (1.10.1) should be continuous, carry each C x to C y , but leave theelements of the form t a alone ( ∇ γ ( t a ) = t a ).We will develop a version of Floer theory “with coefficients in Λ.” As in § C [[ t ]],where C is a locally constant sheaf of rings. In the absolute case, we would take Λ := Λ C (1.6.1).In the example of interest to us, the sheaf of rings is pulled back from S , along theprojection map f : T → S (1.10.2)Then Λ is determined by a ring C and an automorphism (the monodromy around the base S ) σ of C . It induces an automorphism of C [[ t ]] and of Λ C that fixes each t a . We areinterested in the case when C is perfect of characteristic p and σ is the p th root map.The map (1.10.2) (and the monodromy map σ ) is just for book-keeping, but it also hasan “occult” interpretation, in a way fitting in to the old analogy between number fieldsand three-manifolds, and between primes and knots. The generator in the fundamentalgroup of the base circle S and the Frobenius in the absolute Galois group of F p act on C YANKI LEKILI AND DAVID TREUMANN in the same way: by p th powers. See [T] for a little bit more about this. There is also anatural map from the set of closed points of FF E ( C ) (for E = F p (( z )) or any other localfield) to S , and in some sense this paper explores the idea that the SYZ mechanism formirror symmetry could apply § Lagrangian Floer theory — coupled to Λ . Let us putCF( L ( m ) , L ( m ) ; Λ) := M x ∈ L ( m ∩ L ( m Λ x (1.11.1)Lagrangian Floer theory coupled to Λ concerns algebraic structures on (1.11.1), for instancea triangle productCF( L ( m ) , L ( m ) ; Λ) × CF( L ( m ) , L ( m ) ; Λ) → CF( L ( m ) , L ( m ) ; Λ) (1.11.2)In some sense (1.11.1) is another free Λ-module on the intersection points L ( m ) ∩ L ( m ) ,but with many different Λ-module structures. The product (1.11.2) is not Λ-bilinear withrespect to any of them. To define it we give its value on a pair( x · b, x · a ) ∈ CF( L ( m ) , L ( m ) ; Λ) × CF( L ( m ) , L ( m ) ; Λ) (1.11.3)for any a ∈ Λ x and b ∈ Λ x , and extend bi-additively (or more precisely, Λ Z -bilinearly).The value on ( x · b, x · a ) is X y ∈ L ( m ∩ L ( m y X u ∈M ( y,x ,x ) ± t area( u ) or area Z ( u ) ∇ γ ( b ∇ γ ( a ∇ γ (1))) (1.11.4)where γ : y → x , γ : x → x , and γ : x → y are the three sides of the triangle u ,appearing in counterclockwise order. γ γ γ x · b x · ay The ± signs in the formula (1.11.4) are the same as they are in (1.3.3); in particular onecan arrange that they are identically +1.When the monodromy of Λ around L ( ∞ ) is trivial — equivalently, when Λ is pulledback along (1.10.2) — it is possible to package these triangle products into a graded ringstructure ∞ M m =1 CF( L (0) , L ( m ) ; Λ) (1.11.5) SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE 7
Theorem.
For each a ∈ C , and each pair of integers m, k with m > k ≥
0, let θ m,k [ a ] denote the formal series θ m,k [ a ] := ∞ X i = −∞ t m i ( i − + ki z mi + k σ i ( a )Let θ abs m,k [ a ] denote the formal series θ abs m,k [ a ] := ∞ X i = −∞ t m ( mi + k ) z mi + k σ i ( a )Then the relative (resp. absolute) version of (1.11.5) is isomorphic, as a ring-without-unit,to the Z [[ t ]]-linear span of the θ m,k [ a ] (resp. to the Λ Z -linear span of the θ abs m,k [ a ]).1.13. Specializations of t . Fix a commutative ring C and an automorphism σ , cf. (1.10.2).The groups (1.11.1) are linear over Λ σC in the absolute case, and over C σ [[ t ]] in the relativecase. We will discuss the specializations t = 0 and t = 1. The case t = 0 we treat onlybriefly in § T , and its mirror relationship with the nodal cubic curve atthe “large complex structure limit.”The case t = 1 is more delicate. There is a class of symplectic manifolds and Lagrangiansubmanifolds (for instance, monotone Lagrangians in a Fano manifold, or in a genus twosurface) for which setting t = 1 is unproblematic, but the torus does not belong to thisclass. And indeed the series (1.3.3), (1.7.2) do not converge, in any archimedean or nonar-chimedean ring, when t = 1.An F -field can repair some (but only some) of the convergence. DefineCF( L ( m ) , L ( m ) ; C ) = M x ∈ L ( m ∩ L ( m C x (1.13.1)Setting t = 1 in (1.11.4) suggests, in a formal way, a mapCF( L ( m ) , L ( m ) ; C ) × CF( L ( m ) , L ( m ) ; C ) CF( L ( m ) , L ( m ) ; C )When C is complete with respect to a norm | · | , σ is the p th root map, and m < m < m ,this map has a nontrivial domain of convergence. In particular it defines a multiplicationon M m> CF( L (0) , L ( m ) ; m )where m = { x ∈ C : | x | < } .1.14. The Fargues-Fontaine graded ring.
A perfect field of characteristic p , completewith respect to a norm | · | , is since [Sc] known as a “perfectoid field of characteristic p .”Suppose ( C, | · | ) is such a field, and suppose furthermore that C is algebraically closed.Let E = F p (( z )), and let B ⊃ E be the set of bi-infinite formal series P i ∈ Z b i z i ∈ Q i ∈ Z Cz i with coefficients b i ∈ C , and which obey ∀ r ∈ (0 , | b i | r i → | i | → ∞ (1.14.1) YANKI LEKILI AND DAVID TREUMANN
This ring B coincides with what is called B (0 , in [FF, Ex. 1.6.5], and what is called O R ((0 , ϕ : B → B given by ϕ : (cid:16)X c i z i (cid:17) X c pi z i (i.e. ϕ ( f ( z )) = f ( z /p ) p ) (1.14.2)cuts B into “eigenspaces” B ϕ = z n := { f ∈ B | ϕ ( f ) = z n f } . The Fargues-Fontaine curveattached to ( E, C ) is FF E ( C ) := Proj ∞ M n =0 B ϕ = z n ! (1.14.3) Theorem. (4.3.2) is isomorphic as a graded-ring-without-unit to the irrelevant ideal of(1.14.3), i.e. ∞ M n =1 CF( L (0) , L ( n ) ; m ) ∼ = ∞ M n =1 B ϕ = z n (1.14.4)1.15. Annuli.
The degree zero piece B ϕ =1 of the Fargues-Fontaine graded ring (1.14.3)is isomorphic to E (= F p (( z )) in our case). The theorem (1.14.4) does not explain howthis part arises Floer-theoretically. As in § L (0) against itself — a version of Floer cohomology with the F -field turned on — butthe usual rules for making sense of the nontransverse intersection L (0) ∩ L (0) have to berevisited when t = 1.As we mentioned in § § { φ s } s ∈ R so that φ s L (0) and L (0) do meet transversely. The problem that weencounter is that the quasi-isomorphism type of CF( φ s L (0) , L (0) ; C ), with its bigon differen-tial, is no longer independent of φ . One still has natural maps between cochain complexesfor different φ , but they are not quasi-isomorphisms: the usual formula for the necessarycochain homotopies does not converge.This is a well-known problem with “convergent power series Floer cohomology.” It isdiscussed in print in [ChOh, p.3] and [Aur2, § φ ,fails because there are pseudoholomorphic annuli in T that have one sideon φ s L (0) and the other side on L (0) .For instance, Oszv´ath and Szab´o stick to “admissible” Heegaard diagrams to avoid prob-lems with annuli like these [OsSz, § Loud Floer cochains. If L and L ′ are in different homology classes there are noannuli between them. But there are infinitely many annuli between φ s L (0) and L (0) for any SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE 9 φ . If we fix an “autonomous” φ , then we can get these under control by considering largerand larger s : for s large, all of the annuli between φ s L (0) and L (0) have large area. Forinstance:Some of the constructions of [Lee] have inspired us, here. The complexes CF( φ s L (0) , L (0) ; C )for different s do not all have the same cohomology but there are natural cochain mapsCF( φ s L (0) , L (0) ; C ) → CF( φ s ′ L (0) , L (0) ; C )whenever s ′ > s . We will study the colimit of this filtered diagram. For large s the pictureof φ s L (0) is a sine wave with large amplitude (wrapped up around the torus), we callCF loud ( L (0) , L (0) ; C ) := lim −→ s CF( φ s L (0) , L (0) ; C ) (1.16.1)the loud Floer cochains on ( L (0) , L (0) ). The name was suggested to us by Johnson-Freyd.Now our point of view is the following:By shouting infinitely loud, all of the annuli break, along with whateverproblems they posed for noninvariance.We will not try to make this precise, but for a somewhat analogous precedent in the settingof periodic orbits, see what is called the “Latour trick” in [Hutc]. The Latour trick breaksup the periodic orbits of a closed 1-form by adding a large multiple of an exact 1-form.One could equivalently think of pushing the graph of the closed 1-form, for a long time,by the Hamiltonian flow of a primitive for the exact form. A large finite multiple of theexact form suffices to break up all the periodic orbits, while in (1.16.1) one has to pass tothe limit, but maybe “shouting loud” is not a worse metaphor for one process than for theother.We will show that the triangles with sides on L (0) , φ s L (0) , φ s + s ′ L (0) induce a multiplicationon CF loud ( L (0) , L (0) ; C ) and on HF loud ( L (0) , L (0) ; C ). Our construction of this multiplicationis quite crude: a better analysis would follow the construction of an A ∞ -structure onwrapped Floer cochains [AS], which we expect to apply here and give a richer structure.But our computations give an isomorphism of ringsHF ( L (0) , L (0) ; C ) ∼ = C σ [ z, z − ] (1.16.2)When σ is the p th root map, this is the Laurent polynomial ring F p [ z, z − ], a dense subringof F p (( z )). One can similarly define HF loud ( L ( m ) , L ( m ) ; C ), and a ring structure on it, for any m ∈ Q .One gets the same answer (1.16.2) when m is an integer. For m = d/r , we expect but willnot prove that HF is a dense subring of an r -dimensional division algebra over F p (( z ))whose invariant is m + Z ∈ Q / Z . The indecomposable vector bundles on the Fargues-Fontaine curve are classified by Q and those division algebras arise as their endomorphismrings [FF, Thm. 8.2.10].1.17. Acknowledgments.
We have benefitted from conversations, suggestions and corre-spondence from D. Auroux, B. Bhatt, A. Caraiani, K. Fukaya, M. Kim, K. Madapusi-Pera,B. Poonen and P. Seidel. YL is partially supported by the Royal Society URF \ R \ Some Floer-theoretic background
In this section we review some of Floer theory, making what simplifications are possiblewhen the target manifold is a 2d torus T .2.1. J -holomorphic polygons. Let D ⊂ C be the closed unit disk in the complex plane.We denote by D ◦ the open unit disk and ∂D = D − D ◦ the boundary of D . Let z =( z , . . . , z n ) denote an ordered ( n + 1)-tuple of points in ∂D . We require that the pointsof z are pairwise distinct and that the counterclockwise arc subtending z i − and z i (or z n and z ) does not contain any other point of z .Let L , L , . . . , L n be an ( n + 1)-tuple of one-dimensional submanifolds of T , and let( x , . . . , x n ) be a tuple of points in T with x ∈ L ∩ L n , x ∈ L ∩ L , . . . , x n ∈ L n ∩ L n − We write W ( x , . . . , x n ) for the set of pairs ( z , u ) where u : D → T is a continuous map,smooth away from z , that carries z i to x i and that maps the counterclockwise arc between z i and z i +1 into L i . It carries a topology in which a sequence ( z i , u i ) converges ( z , u ) if z i → z in ( ∂D ) × ( n +1) and u i → u in a suitable Sobolev space.Fix an almost complex structure J on T . A polygon ( z , u ) ∈ W ( x , . . . , x n ) is called J -holomorphic if the differential of u is C -linear on each tangent space of the interior D ◦ .Write f M ( x , . . . , x n ) ⊂ W ( x , . . . , x n ) for the subspace of J -holomorphic polygons. Thegroup PSL ( R ) of biholomorphisms of D ◦ acts on f M by reparametrization, and we denotethe quotient by M ( x , . . . , x n ).2.2. Transversely cut criteria.
Each connected component of M ( x , . . . , x n ) is labeledby a nonnegative integer called the analytic index of the component (or of any map in thecomponent). In case the conditions that cut f M out of W are transverse in a sense that wewill not review here, then each component is a topological manifold and the analytic indexcoincides with the dimension of this component. A formula for this dimension is givenbelow (2.3.1). When the “transversely cut” condition is satisfied, we call the componentsof dimension zero rigid polygons. SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE 11
The “transversely cut” condition is satisfied whenever the ( L , . . . , L n ) are in generalposition — that is, whenever the L i are pairwise transverse and L i ∩ L j ∩ L k is empty. On T or another surface, this triple intersection condition can be relaxed [Se2, Lem. 13.2],for instance M is transversely cut in a neighborhood of u as soon as u is not constant.Even some constant maps u are transversely cut, for instance: If ( L , L , L ) are pair-wise transverse, then at any triple intersection point x ∈ L ∩ L ∩ L , the tangent lines( T x L , T x L , T x L ) come in either clockwise or counterclockwise order. A constant map D → L ∩ L ∩ L contributes to M ( x, x, x ) if and only if they come in clockwise order L L L An example of a non-transversely cut quadrilateral in T (necessarily constant) is analyzedin [LPe1, Thm. 8]. We will encounter some non-transversely cut triangles in § Maslov index of an intersection point.
Let (
L, L ′ ) be an ordered pair of connectedone-dimensional submanifolds of T , and fix an orientation of both L and L ′ . Suppose that L and L ′ meet transversely at the point x , then we define mas( x ) ∈ Z / L ′ L mas = 0 LL ′ mas = 1If L and L ′ are not homologous to zero, one may lift the Maslov index to a Z -valuedinvariant by equipping L and L ′ (and T ) with gradings, see [LPe2, §
6] — let us denote this Z -valued Maslov index by mas Z ( x ). A formula for the dimension near u ∈ M ( x , . . . , x n )is mas Z ( x ) − mas Z ( x ) − · · · − mas Z ( x n ) (2.3.1)2.4. Sign of a rigid polygon.
By making some additional choices one may attach a signto each rigid polygon with boundary on ( L , . . . , L n ) — in other words one may define amap M ( x , . . . , x n ) → { , − } (2.4.1)We recall the recipe for (2.4.1) given in [Se1, §
7] — it depends on the choice of orientationfor each L i , and on the additional data of a basepoint ⋆ i ∈ L i in each L i . One requiresthat ⋆ i / ∈ L j for any j = i . The point ⋆ i endows L i with a nontrivial spin structure (alsoknown as bounding or Neveu-Schwarz spin structure) which is trivialized away from ⋆ i .If u | ∂D : ∂D → ∪ ni =0 L i preserves the counter-clockwise orientation of D , the sign is +1or − ∂D — i.e. it is ( − u − { ⋆ ,...,⋆ n } . Changing the orientation of L does not change this sign,changing the orientation of L n changes the signs by ( − mas( x )+mas( x n ) , while changing theorientation of any of the other L i changes the sign by ( − mas( x i ) .2 YANKI LEKILI AND DAVID TREUMANN
7] — it depends on the choice of orientationfor each L i , and on the additional data of a basepoint ⋆ i ∈ L i in each L i . One requiresthat ⋆ i / ∈ L j for any j = i . The point ⋆ i endows L i with a nontrivial spin structure (alsoknown as bounding or Neveu-Schwarz spin structure) which is trivialized away from ⋆ i .If u | ∂D : ∂D → ∪ ni =0 L i preserves the counter-clockwise orientation of D , the sign is +1or − ∂D — i.e. it is ( − u − { ⋆ ,...,⋆ n } . Changing the orientation of L does not change this sign,changing the orientation of L n changes the signs by ( − mas( x )+mas( x n ) , while changing theorientation of any of the other L i changes the sign by ( − mas( x i ) .2 YANKI LEKILI AND DAVID TREUMANN For short, we will sometimes write( − | x | := ( − mas( x ) (2.4.2)2.5. Floer cochain complexes.
Let t be a formal variable, and let Z [ t R ≥ ] denote thesemigroup algebra of R ≥ , i.e. the group of finite Z -linear combinations of symbols of theform t a , where a ∈ R ≥ , with the multiplication t a t b = t a + b . Let Λ be a Z [ t R ≥ ]-algebra.In a moment we will take Λ to be the Novikov completion of Z [ t R ≥ ] but the differential inCF( L, L ′ ) is given by a finite sum, so that in this section it might as well be Z [ t R ≥ ] itself.When L and L ′ intersect transversely, then we letCF( L, L ′ ) := M x ∈ L ∩ L ′ Λ (2.5.1)The choice of orientation for L and L ′ endows this group with a Z / ⊕ CF , where CF i ( L, L ′ ) := M x | mas( x )= i ΛFurther equipping L and L ′ with stars § µ : CF i ( L, L ′ ) → CF i +1 ( L, L ′ ) : x X y | mas( y )= i +1 y X u ∈M ( x,y ) ± t area( u ) (2.5.2)where the sign is given in § u ) := R D u ∗ ( dx dy ). The inner sum is finite. Ageneral argument using non-rigid bigons shows that µ µ = 0 — this is a case of the A ∞ -relations § T , this can be proved more simply by lifting the grading from Z / Z : the Z -grading is always concentrated in only two degrees.We have just described the “absolute” Floer cochain complex. After fixing a point D ∈ T , not on L or L ′ , we also have a “relative to D ” complex in which Λ in (2.5.1) canbe shrunk to Z [ t ] (or another Z [ t ]-algebra), and the expression t area( u ) in is replaced by t u − ( D ) .2.6. Example — special Lagrangians.
We will call a circle L ⊂ T a “special La-grangian” if it is the image under R → T of a straight line. If that straight line has theform y + mx = b then we will call m the slope of the special Lagrangian, otherwise wesay L has slope ∞ ; thus the possible slopes are m ∈ Q ∪ {∞} . If L is special with finiteslope, let us call the orientation under the parametrization x ( x, b − mx ) the “defaultorientation.”If L = L ′ are two special Lagrangians, of finite slopes m and m ′ , then they meet trans-versely in a set of cardinality | nd ′ − n ′ d | , if m = n/d and m ′ = n ′ /d ′ . All the intersectionpoints are in a single Maslov degree; with the default orientations, these degrees areCF( L, L ′ ) = ( CF ( L, L ′ ) if m ′ > m CF ( L, L ′ ) if m ′ < m There are no bigons and the differential (2.5.2) is zero.
SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE 13
Polygon maps.
Suppose that ( L , . . . , L n ) are in sufficiently general position thatall the M ( x , . . . , x n ) are transversely cut. The ( n + 1)-gon map is a multilinear map µ n : CF i n ( L n − , L n ) × · · · × CF i ( L , L ) → CF i + ··· + i n +2 − n ( L , L n ) (2.7.1)which carries ( x n , x n − , . . . , x ) to X y y X u ∈M ( y,x ,x ,...,x n ) ± t area( u ) (2.7.2)It is not defined until the L i are equipped with orientations and stars. Furthermore, theinner sum in (2.7.2) is usually infinite, so Λ should carry a topology in which it converges.The standard choice for Λ is one of Λ or Λ [ t − ], where Λ is the Novikov ringΛ = Λ Z = ( ∞ X i =0 a i t λ i | a i ∈ Z , λ i ∈ R ≥ and lim i →∞ λ i = ∞ ) (2.7.3)The fact that Λ can be taken to have Z -coefficients is a reflection of the fact that themoduli spaces M are not orbifolds — this holds for T and more generally for semipositivesymplectic manifolds.In the relative setting, as long as D does not lie on any L i , we replace t area ( u ) with t u − ( D ) , and (2.7.1) is multilinear over Z [[ t ]].2.8. Example — some triangle maps.
Suppose L , . . . , L k are special Lagrangians ofslopes m < m < · · · < m k < ∞ (2.8.1)If k = 2, then the set of rigid ( k + 1)-gons with boundary on L , . . . , L k is empty, and themaps µ k : CF( L k − , L k ) × · · · × CF( L , L ) → CF( L , L k )are zero — this is a consequence of § µ is the only interesting polygon map. In this section we explain how to compute µ indetail for Lagrangians of the form L ( m ) , L ( m ) , L ( m ) § L CF( L (0) , L ( m ) ) § µ k for k = 2 is that this product is strictlyassociative § µ k do not all vanish for k ≥ m < m , there are m − m intersection points between L ( m ) and L ( m ) , eachcontributing a basis element to CF( L ( m ) , L ( m ) ). Let us index those intersection points inthe following way. τ m ( x m − m ,κ ) := ( κ, − m κ ) (2.8.2) where κ ∈ { , m − m , . . . , m − m − m − m } and τ denotes the Dehn twist map τ : ( x, y ) ( x, y − x )Fix an irrational number ε and equip each L ( m ) with a star § ⋆ ( m ) = ⋆ ( m ) ,ε := ( ε, − mε ) (2.8.3)The Dehn twist carries L ( m ) to L ( m +1) and preserves the stars. As ε is irrational, the starsavoid the intersection points L ( m ) ∩ L ( m ) . We will compute the triangle mapsCF( L ( m ) , L ( m + m ) ) × CF( L (0) , L (( m )) ) → CF( L (0) , L ( m + m ) )by computing µ ( τ m x m ,κ , x m ,κ ). For example µ ( τ x , , x , ) = x , (cid:16) P i ∈ Z t i (cid:17) + x , (cid:16) P i ∈ Z t ( i + ) (cid:17) (absolute setting) x , (cid:16) P i ∈ Z t i (cid:17) + x , (cid:16) P i ∈ Z t i ( i +1) (cid:17) (relative setting) (2.8.4) Theorem.
Let E ( κ , κ ) = E m ,m ( κ , κ ) := m κ + m κ m + m (2.8.5)(which carries m Z × m Z into m + m Z ). Then in the absolute setting µ ( τ m x m ,κ , x m ,κ )is given by µ ( τ m x m ,κ , x m ,κ ) = X ℓ ∈ Z x m + m ,E ( κ ,κ + ℓ ) t ( ℓ + κ − κ ) / (2( m + m )) (2.8.6)Let the functions φ ( s ) and λ ( u, v ) = λ m ,m ( u, v ) be given by (cf. [LPe2, p. 83]) φ ( s ) := ⌊ s ⌋ s − ⌊ s ⌋⌊ s + 1 ⌋ ; λ ( u, v ) := m φ ( u ) + m φ ( v ) − ( m + m ) φ ( E ( u, v )) (2.8.7)Then in the relative setting µ ( τ m x m ,κ , x m ,κ ) is given by µ ( τ m x m ,.κ , x m ,κ ) = X ℓ ∈ Z x m + m ,E ( κ ,κ + ℓ ) t λ ( κ ,κ + ℓ ) (2.8.8)where we understand x m + m ,E ( κ ,κ + ℓ ) := x m + m ,κ if E ( κ , κ + ℓ ) = κ modulo 1. Proof.
The index ℓ in either sum (2.8.6), (2.8.8) determines a triangle, two of whose verticesare at x m ,κ and τ m x m ,κ . In the universal cover, the coordinates of all three vertices are( κ , , ( E ( κ , κ + ℓ ) , , ( κ + ℓ, − m ( κ + ℓ − κ ))as in the diagram SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE 15 L (0) L ( m + m ) L ( m ) x m ,κ = ( κ , x m ,κ = ( κ + ℓ, − m ( κ + ℓ − κ ))( E ( κ , κ + ℓ ) ,
0) = x m + m ,E ( κ ,κ + l ) On any of these triangles there are an even number of stars (2.8.3): for each i ∈ Z with κ < ε + i < κ + ℓ there are exactly two stars whose x -coordinate (in the universal cover)is ε + i , one along L ( m ) and the other along either L (0) or L ( m + m ) . Thus every summandin the triangle has sign +1.The exponent of t in (2.8.6) is the area of the ℓ th triangle, i.e.12 m ( κ + ℓ − κ )( E ( κ , κ + ℓ ) − κ ) = ( ℓ + κ − κ ) / (2( 1 m + 1 m )) . The more complicated exponent of t in (2.8.8) is the lattice area § u − ( D ) when D is in the first quadrant very closeto (0 ,
0) — see [LPe2]. (cid:3)
Theta functions.
Let θ m,k , θ abs m,k be as in (1.8.1) θ m,k := ∞ X i = −∞ t m i ( i − + ki z mi + k , θ abs m,k := ∞ X i = −∞ t ( mi + k ) / (2 m ) z mi + k The Jacobi theta function is θ , , and the others are obtained by a simple change of variables θ m,k ( t, z ) = z k θ , ( t m , t k z m ) , θ m,k ( t, t z ) = t k ( m − k ) / (2 m ) θ abs m,k ( t, z )Although these series are doubly infinite in z , when formally expanding the product of θ m,k and θ m ′ ,k ′ only finitely many terms contribute to the coefficient of any monomial z e t f — thesame goes for θ abs m,k and θ abs m ′ ,k ′ . This is a consequence of the convexity of the functions i m (cid:0) i (cid:1) + ki and i ( mi + k ) / (2 m ) in the exponent of t . That the resulting series θ m,k · θ m ′ ,k ′ or θ abs m,k · θ abs m ′ ,k ′ can be written as a linear combination of θ m + m ′ , , · · · , θ m + m ′ ,m + m ′ − is astandard but nontrivial fact about the theta functions, the formulas for these coefficientsis the same as (2.8.8) and (2.8.6): putting κ = k /m and κ = k /m , θ m ,k · θ m ,k = X ℓ ∈ Z θ m + m ,E ( κ ,κ + ℓ ) t λ ( κ ,κ + ℓ ) and θ abs m ,k · θ abs m ,k = X ℓ ∈ Z θ abs m + m ,E ( κ ,κ + ℓ ) t ( ℓ + κ − κ ) / (2( m + m )) In other words the map x m,k θ m,k or θ abs m,k is a ring homomorphism. One may verifythis directly (and we will do so in the next section when we turn on an F -field), but it isnatural to ask what is the Floer-theoretic origin of these series. Each summand of (1.8.1)is indexed by a right triangle, with one vertex at x m,k/m and sides along L (0) , L ( m ) , L ( ∞ ) .The exponent of t carries the area (or lattice area) of this right triangle and the exponentof z carries the number of times the vertical edge of the triangle wraps around L ( ∞ ) — this z can be interpreted as the monodromy of a rank one local system § θ , have the form (for i positive) x , / z mi + k = z i +1 The punctured torus, the large complex structure limit.
Part of the moti-vation for relative Floer theory is to make sense of the specialization t = 0 in the polygonsums. Setting t = 0 has the effect of discarding the polygons that contribute a positivepower of t , which in the relative case is the same as discarding the polygons that touch D . One gets the same effect by doing Floer theory for Lagrangians in the punctured torus T − D . When T − D is equipped with the right symplectic structure, one with infinite areain a neighborhood of D , this is called the “large volume limit” of Floer theory.We can treat the Lagrangians L ( m ) as boundary conditions for triangles u : ∆ → T − D ,and sum over them to obtain a map µ as before, this time defined on the free Z -modulesspanned by L ( m ) ∩ L ( m ) . For instance, µ ( τ x , , x , ) = x , + x , / · x , comes from a constant map and the two copies of x , / come from the twoshaded triangles in the figure L (2) L (1) L (0) D SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE 17
We also record µ ( τ x , , x , ) = x , + x , / + x , / µ ( τ x , , x , / ) = x , / + x , / µ ( τ x , / , x , / ) = x , / and µ ( τ x , / , µ ( τ x , / , x , / )) = x , / . If one sets z = x , , x = x , / , y = x , / , these equation imply in particular that y z + x = xyz This is the equation for a nodal plane cubic curve — the “large complex structure limit”that matches the large volume limit under mirror symmetry.2.11. A ∞ -relations. The moduli spaces M that parametrize non-rigid polygons are notusually compact. For example, suppose L , L , L , L are as in the diagram L L L L Denote the red-black, black-purple, and purple-blue intersection points by f , g , and h respectively, and the blue-red intersection point by hgf . Then M ( hgf, f, g, h ) includesthe following one-parameter family of quadrilaterals: ∗ ∗ (2.11.1)They all have the same image closure, but along the boundary may back-track along eitherthe blue or the red line. Near the ∗ s, the map u is biholomorphic to the map from the upperhalf-plane to C that sends z to z . At the extreme parameters, where the ∗ reaches all theway to the black or to the purple line, there is no such J -holomorphic quadrilateral — so M is not compact — but each of those extremes is occupied by a pair of J -holomorphictriangles.Since the quadrilaterals are not rigid, they do not contribute to µ ( h, g, f ). But thetriangles at the extremes are rigid, at one end they contribute to µ ( g, f ) and µ ( h, gf ),and at the other end to µ ( h, g ) and µ ( hg, f ). The interpolating family of quadrilateralsexhibits a relation between them. More generally there is a compactification (the Deligne-Mumford-Stasheff compactifica-tion) of M . When everything is transversely cut § L i with orientations and stars induces an orientation on M and its compactification —(2.4.1) is a special case of this orientation. The oriented compactification of M is used inthe proof (it essentially is the proof) of the following equations among the polygon maps µ n : X i + j = n +1 X ℓ
L, L ′ ) is like a morphism between objects. (2.11.2) expresses the fact that thesemorphism spaces are cochain complexes and that the composition law is associative upto chain homotopy in a strong sense. It falls short of being an A ∞ -category, for instancebecause CF( L, L ′ ) is defined only when L and L ′ meet transversely, so CF( L, L ) is undefinedand there is no “morphism” that could play the role of the identity map. A standard wayto address this problem is by analyzing the sense in which CF(
L, L ′ ) are invariant underHamiltonian isotopies — Floer’s theory of continuation.2.12. Example — identity maps.
There is a Floer cochain complex CF(
L, L ′ ), well-defined up to quasi-isomorphism, even if L and L ′ do not intersect transversely. In case L ′ meets both L and L transversely, then in the absolute setting (resp. relative setting)any Hamiltonian isotopy (resp. any Hamiltonian isotopy supported on the complement of D ) induces a quasi-isomorphism between CF( L , L ′ ) and CF( L , L ′ ) § L, L ′ ) by perturbing L .We illustrate this in an example that fills in the zeroth graded piece of (1.4.1). Let φ s denote the flow of H ( x, y ) = sin(2 πx ), i.e. φ s ( x, y ) = ( x, y − s cos(2 πx )) (2.12.1)Then for 0 < s < φ s L (0) meets L (0) transversely at the points x = ( . ,
0) and y = ( . , φ s L (0) , L (0) ) = Λ x ⊕ Λ y . In a suitable fundamental domain the picture is this: ⋆⋆x yφ s L (0) L (0) SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE 19
Equipping L (0) with its default orientation and φ s L (0) with the orientation induced by φ s : L (0) ∼ = φ s L (0) , the Maslov degrees areCF ( φ s L (0) , L (0) ) = x Λ CF ( φ s L (0) , L (0) ) = y Λ (2.12.2)There are two bigons contributing to the differential µ , of equal area A . Both bigons haveinput x and output y , and their signs § µ ( x ) = y ( t A − t A ) = 0Some variations of this computation are made in § § Continuation.
Let X H denote the Hamiltonian vector field of a function H : [0 , × T → R , and write φ s for its time s flow, φ s : T → T . Let us review how the quasi-isomorphism CF( L, L ′ ) → CF( φ s L, L ′ ) (2.13.1)works in the absolute setting. The map (2.13.1) goes back to [Fl, Thm. 4], our notation iscloser to the appendix of [Aur1]. It is defined in terms of a set M ( x, y, φ, β ) of maps u : [ −∞ , ∞ ] × [0 , → T that obey the boundary conditions u ([ −∞ , ∞ ] × { } ) ⊂ Lu ([ −∞ , ∞ ] × { } ) ⊂ L ′ u ( {∞} × [0 , { x } u ( −∞ , τ ) = φ s · τ ( y )and (with analytic index zero § X H -perturbed J -holomorphic curve equation: ∂u/∂σ + J ( ∂u/∂τ − β ( σ ) sX H ) = 0 (2.13.2)Here β (the “profile function”) is a monotone decreasing R -valued function on [ −∞ , ∞ ]with β ( σ ) = 1 for σ ≪ β ( σ ) = 0 for σ ≫
0. The formula for (2.13.1) is x X y ∈ φ s L ∩ L ′ y X u ∈M ( x,y,φ ) ± t topological energy of u (2.13.3)where the “topological energy” is (see Lemma 14.4.5 [Oh]). Z u ∗ ω + Z H ( τ, u ( ∞ , τ )) dτ − Z ∞−∞ β ′ ( σ ) s Z ( H τ ◦ u ) dτ dσ This is sometimes a negative quantity, so we must allow t − ∈ Λ.The homotopy inverse to (2.13.1) is just the continuation map for the reversed flow φ − s .To describe the chain homotopy between the compositeCF( φ s L, L ′ ) → CF( φ − s φ s L, L ′ ) = CF( L, L ′ ) → CF( φ s L, L ′ ) (2.13.4) and the identity, map, let B r ( σ ) (for each r >
0) be a function that vanishes on an intervalof length r and that agrees up to translation of σ with β ( σ ) when σ is to the left of thatinterval and with β ( − σ ) when σ is to the right of that interval. Then∆( y ) = X x x X u ± t topological energy of u (2.13.5)where the inner sum is over strips u : [ −∞ , ∞ ] × [0 , → T that solve (for some r , withindex − ∂u/∂σ + J ( ∂u/∂τ − B r ( σ ) sX H ) = 0 r ∈ R ≥ and that have u ( − , ⊂ L , u ( − , ⊂ L ′ , and u ( −∞ , τ ) = φ s · τ ( x ) and u ( ∞ , τ ) = φ s · τ ( y ).2.14. Example.
Suppose that L and L ′ are a pair of parallel, horizontal circles, at distance c apart. With φ s as in (2.12.1), φ s ( L ) ∩ L ′ is empty unless | s | > c . If | s | only slightly exceeds c , then φ s ( L ) ∩ L ′ has two intersection points, say x and y as in the diagram: ⋆⋆ LL ′ ⋆⋆x yφ s LL ′ Thus, CF(
L, L ′ ) = 0, while (noting that there are two bigons in the right picture, a smallone of area A and a large one of area A + c ) and CF( φ s L, L ′ ) = x Λ ⊕ y Λ, with differential µ ( x ) = y ( t A − t A + c )(see § L, L ′ ) = 0, the composite (2.13.4) is zero. By Floer’s theorem, the identitymap on CF( φ s L, L ′ ) is chain homotopic to zero, with (2.13.5) supplying the contractinghomotopy. Indeed the contracting homotopy is the geometric series∆( y ) = x · ( t − A + t − A + c + · · · + t − A + nc + · · · )The strip u n contributing the n th term in the series (of topological energy nc − A ) stretcheshorizontally across n + 2 fundamental domains, crossing the boundary of the fundamentaldomain exactly n + 1 times. Here is a picture of u | ∂ ([ −∞ , ∞ ] × [0 , and u | ∂ ([ −∞ , ∞ ] × [0 , SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE 21 ⋆x yφ s LL ′ Floer theory coupled to an F -field In this section we go over the constructions and calculations of §
2, coupling all the sumsover polygons to a sheaf of rings Λ of the kind discussed in § f : T → S (1.10.2), and set up somenotation for dealing with that case in § Cochain complex. If L and L ′ are two embedded circles in T , meeting transversely,let us write (just as in (1.11.1)) CF( L, L ′ ; Λ) := M x ∈ L ∩ L ′ Λ x (3.1.1)If a ∈ Λ x and we wish to regard it as an element of (3.1.1), we will write it as x · a . Weequip (3.1.1) with a Z / L and L ′ with orientations, just as in § L and L ′ with stars, we define a differential µ : CF i ( L, L ′ ; Λ) → CF i +1 ( L, L ′ ; Λ)by the following analog of (2.5.2): x · a X y | mas( y )= i +1 y X u ∈M ( y,x ) ± t area( u ) ∇ γ ′ (cid:16) a ∇ γ (1 Λ y ) (cid:17) (3.1.2)where • γ : I → L is the path along the L -side of the bigon u starting at y and ending at x , • γ ′ : I → L ′ is the path along the L ′ -side of u starting at x and ending at y .This differential does not obey anything like µ ( xa ) = µ ( x ) a — in fact µ ( x ) a is typicallyundefined and in general µ ( xa ) and µ ( x ) do not have any useful relationship with eachother.We obtain a Λ-version of the continuation map (2.13.1)CF( L, L ′ ; Λ) → CF( φ s L, L ′ ; Λ) (3.1.3)by multiplying each summand of (2.13.3) by ∇ γ ′ ( a ∇ γ ( ∇ ( φ − τs ( y ))(1 Λ y ))), where γ = u ( τ,
0) and γ ′ = u ( − τ,
1) are the paths along L and L ′ respectively, and φ − τs ( y ) is the2 YANKI LEKILI AND DAVID TREUMANN
1) are the paths along L and L ′ respectively, and φ − τs ( y ) is the2 YANKI LEKILI AND DAVID TREUMANN reverse of the trajectory from y to φ s y , i.e. X y ∈ φ s L ∩ L ′ y X u ∈M ( x,y,φ ) ± t topological energy of u ∇ γ ′ ( a ∇ γ ( ∇ ( φ − τs ( y ))(1 Λ y ))) (3.1.4)The same recipe as § a · y X x x X u ± t topological energy of u ∇ γ ′ (cid:0) a · ∇ ( φ τs ( y )) ∇ γ ∇ ( φ − τs ( x ))(1 Λ x ) (cid:1) (3.1.5)3.2. Polygon maps.
Let L , L , . . . , L n be oriented, starred submanifolds of T as in § µ n : CF i n ( L n − , L n ; Λ) × · · · × CF i ( L , L ; Λ) → CF i + ··· + i n +2 − n ( L , L n ; Λ) (3.2.1)carrying ( x n · a n , x n − · a n − , . . . , x · a ) (with each a i ∈ Λ x i ) to X y y X u ± t area( u ) ∇ γ n ( a n ∇ γ n − ( a n − · · · ∇ γ ( a ∇ γ ( a ∇ ( γ (1)) · · · )))) ! (3.2.2)where u runs over the same set of rigid polygons as (2.7.1), the signs are just the same,and γ i is the path along the boundary of u going from x i to x i +1 , or from x n − to x .Each connected component of the Deligne-Mumford-Stasheff compactification of thespace of non-rigid polygons has the same vertices x , . . . , x n — that is, every u in thatcomponent has those same vertices. Moreover, the path from x i to x i +1 or from x n to x along u belongs to the same homotopy class, so that ∇ ( γ i ) : Λ x i → Λ x i +1 is locally constantin u . Thus the A ∞ -relations among the (3.2.1) hold for the usual reasons: X i + j = n +1 X ℓ
We can put the formulas in § L i . For each i let E i be a local system of Λ | L i -modules on L i . Then wedefine CF(( L i , E i ) , ( L i +1 , E i +1 ); Λ) = M x ∈ L i ∩ L i +1 Hom( E i,x , E i +1 ,x ) (3.3.1)The differential is modified by µ ( xf : E i,x → E i +1 ,x ) = X y X u ± t area( u ) ∇ γ ′ ◦ f ◦ ∇ γ (3.3.2)(where xf denotes f placed in the x th summand of (3.3.1)). In case E i = Λ | L i is the trivialsheaf of modules, then Hom( E i,x , E i +1 ,x ) = Hom(Λ x , Λ x ) is naturally identified with Λ x and(3.3.2) coincides with (3.1.2).The polygon maps µ n : CF(( L n − , E n − ) , ( L n , E n ); Λ) × · · · × CF(( L , E ) , ( L , E ); Λ) → CF(( L , E ) , ( L n , E n ); Λ) (3.3.3) SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE 23 are defined by sending ( f n , f n − , . . . , f ) to the formal expression X y y X u ∈M ( y,x ,x ,...,x n ) ± t area( u ) ∇ γ n ◦ f n ◦ ∇ γ n − ◦ f n − ◦ · · · ◦ ∇ γ ◦ f ◦ ∇ γ (3.3.4)We’ve left the degrees in (3.3.3) off for typesetting reasons; they are the same as in (2.7.1).The formula (3.3.4) specializes to (3.2.2) in case E i = Λ | L i . In general (3.3.4) can fail toconverge, unless the following “unitarity” condition is imposed on the E i :Each fiber of E i is locally free of finite rank over Λ | L i , and the monodromypreserves an Λ | L i -lattice.3.4. F -field. We would like to package the Λ-coupled triangle products among the L ( m ) into a graded ring, as in § L ( m ) , L ( n ) ; Λ)and CF( L (0) , L ( n − m ) ; Λ) exists only when Λ is pulled back along the projection map f : T → S : ( x, y ) + Z x + Z (3.4.1)To make this explicit, let σ : C → C be a ring automorphism, where C is commutative.We also let σ denote induced automorphism of C [[ t ]] or of Λ C , with σ ( t a ) = t a . We aremainly interested in the case that C perfect of characteristic p, σ ( c ) = c /p (3.4.2)in which case if f ( t ) = P c a t a belongs to C [[ t ]] or to Λ C then we can write σ ( f )( t ) = f ( t p ) /p .The quotient ( R × C ) / ∼ (3.4.3)of R × C by the equivalence relation ( x, c ) ∼ ( x + 1 , σ ( c )) is the ´etal´e space of a locallyconstant sheaf of rings on S — as are ( R × C [[ t ]]) / ∼ and ( R × Λ C ) / ∼ . We denote thepullback-to- T of these sheaves of rings by C , C [[ t ]], and Λ C .We will call (3.4.1) an “ F -field” on T . In diagrams, we keep track of it with a red line— the inverse image of a point close to the right edge of the fundamental domain [0 ,
1) of S , as in the figure on the left below. On the right we have drawn, in a different scale,the preimage of the red line in part of the universal cover of T , along with a triangle thatcontributes to µ ( x · b, x · a ). One understands that σ or σ − is to be applied every timeone crosses this “danger line” — σ if one crosses it from right to left, σ − if one crosses itfrom left to right. γ γ γ x · b x · ay Example.
Let Λ = Λ C be as in § L (0) and φ s L (0) be as in § ⋆⋆x yφ s L (0) L (0) We will compute the differential on CF( φ s L (0) , L (0) ; Λ) — this specializes to the exampleof § σ is trivial. As in that example we still have CF ( φ s L (0) , L (0) ; Λ) = x Λ andCF ( φ s L (0) , L (0) ; Λ) = y Λ, but the map µ is not Λ-linear so we must compute not just µ ( x ) but µ ( xa ) for all a ∈ Λ. The same two bigons in § A , contribute to µ ( xa ), but only of them crosses the “danger line”The left bigon contributes y · ( − t A a ) and the right bigon contributes y · ( t A σ ( a )), so thatthe differential is given by µ ( xa ) = yt A ( σ ( a ) − a )If we make the change of basis ( x, y ) → ( x, yt A ), thenHF ∼ = ker( a σ ( a ) − a ) HF ∼ = coker( a σ ( a ) − a )More suggestively, HF i ( φ s L (0) , L (0) ; Λ) is isomorphic to H i ( L (0) ; Λ | L (0) ), the cohomology ofthe circle L (0) with coefficients in Λ.3.6. Example.
With Λ and φ as in the previous example, let us compute CF( φ s L, L ′ ; Λ)when L and L ′ are two special Lagrangians that are parallel to L (0) and to each other, butthat do not intersect. Let c be the distance between (and therefore also the area between) L and L ′ . Suppose that | s | slightly exceed c , so that φ s ( L ) ∩ L ′ has two intersection pointsthat we again denote by x and y . ⋆⋆x yφ s LL ′ SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE 25
Then the differential on CF( φ s L, L ′ ; Λ) is µ ( xa ) = y ( t a σ ( a ) − t A + c a ). If c is not zero, thecomplex is acyclic, but it is interesting to note that the formal series x · X n ∈ Z σ − n ( a ) t nc a ∈ C x (3.6.1)are killed by µ . Since (3.6.1) has an infinite “tail”, it does not lie in Λ x and does notcontribute to CF( φ s L, L ′ ; Λ).Since L ∩ L ′ is empty the continuation maps associated to φ s and its reverse φ − s CF(
L, L ′ ; Λ) → CF( φ s L, L ′ ; Λ) and CF( φ s L, L ′ ; Λ) → CF(
L, L ′ ; Λ)both vanish. But the explicit contracting homotopy on CF ( φ s L, L ′ ; Λ) → CF ( φ s L, L ′ ; Λ)is interesting, it is given by the series∆( a · y ) = x ( t − A σ − ( a ) + t − A + c σ − ( a ) + · · · + t − A + nc σ − n − ( a ) + · · · ) (3.6.2)The strip u n contributing the t − A + nc term in this series is the same as in § ∇ γ ′ (cid:0) a · ∇ ( φ τs ( y )) ∇ γ ∇ ( φ − τs ( x ))(1 Λ x ) (cid:1) , which simplifiesto σ − n − ( a ).3.7. Computing the triangle maps.
Let L ( m ) (equipped with the same orientations andstars), x m,κ , and τ be as in § C be pulled back along f , with σ denoting thenontrivial monodromy, as in § L ( m ) , L ( m + m ) ; Λ) × CF( L (0) , L ( m ) ; Λ) → CF( L (0) , L ( m + m ) ; Λ)by computing µ ( τ m x m ,κ · b, x m ,κ · a ). Theorem.
Let E be as in (2.8.5). Then in the absolute case µ ( τ m x m ,κ , x m ,κ ) is givenby X ℓ ∈ Z x m + m ,E ( κ ,κ + ℓ ) t ( ℓ + κ − κ ) / (2( m + m )) σ ℓ −⌊ E ( κ ,κ + ℓ ) ⌋ ( b ) σ −⌊ E ( κ ,κ + ℓ ) ⌋ ( a ) (3.7.1)where we understand x m + m ,E ( κ ,κ + ℓ ) := x m + m ,κ if E ( κ , κ + ℓ ) = κ modulo 1. In therelative case, µ ( τ m x m ,κ , x m ,κ ) is given by X ℓ ∈ Z x m + m ,E ( κ ,κ + ℓ ) t λ ( κ ,κ + ℓ ) σ ℓ −⌊ E ( κ ,κ + ℓ ) ⌋ ( b ) σ −⌊ E ( κ ,κ + ℓ ) ⌋ ( a ) (3.7.2) Proof.
The triangles that contribute are exactly as in the proof in § ℓ ∈ Z . The ± sign and the exponent of t in (1.11.4) are the same as in § ∇ γ ( b ∇ γ ( a ∇ γ (1))). If I is an interval and γ : I → T is a path in T , write f ( γ ) for the number of times γ crosses the “danger line” § ∇ γ ( b ∇ γ ( a ∇ γ (1))) = σ f ( γ ) ( bσ f ( γ ) ( a )) (3.7.3)= σ f ( γ ) ( b ) σ f ( γ )+ f ( γ ) ( a ) (3.7.4)= σ f ( γ ) ( b ) σ − f ( γ ) ( a ) (3.7.5)The ℓ th triangle (pictured below) has f ( γ ) = ⌊ E ( κ , κ + ℓ ) ⌋ and f ( γ ) = ℓ −⌊ E ( κ , κ + ℓ ) ⌋ . L (0) L ( m + m ) L ( m ) x m ,κ = ( κ , x m ,κ = ( κ + ℓ, − m ( κ + ℓ − κ ))( E ( κ , κ + ℓ ) ,
0) = x m + m ,E ( κ ,κ + l ) (cid:3) Theta functions with F -field coupling. Keeping the notation of the previoussection, where Λ and C are pulled back along f , we may give ∞ M m =1 CF( L (0) , L ( m ) ; Λ) (3.8.1)the structure of a graded ring without unit. One may equip it with a unit by taking thedegree zero piece to be Λ C σ = HF ( L (0) , L (0) ; Λ) § § Theorem ( § . For each a ∈ C and each pair of integers m, k with m > k ≥
0, let θ m,k [ a ] denote the formal series θ m,k [ a ] := ∞ X i = −∞ t m i ( i − + ki z mi + k σ i ( a ) (3.8.2)Let θ abs m,k [ a ] denote the formal series θ abs m,k [ a ] := ∞ X i = −∞ t m ( mi + k ) z mi + k σ i ( a ) (3.8.3)Then the relative (resp. absolute) version of (3.8.1) is isomorphic (as a graded ring-without-unit) to the Z [[ t ]]-linear span of the θ m,k [ a ] (resp. Λ Z -linear span of the θ abs m,k [ a ]) viathe map x m,k/m · a θ m,k [ a ] (3.8.4)As in § θ [ a ] are indexed by right triangles. The factor of σ i ( a )plays the same role as the ∇ γ ( b ∇ γ ( a ∇ ( γ (1))) factor in (1.11.4). SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE 27 a · x , σ z · = z This triangle contributes t z σ ( a ) to the relative version of θ , [ a ] (3.8.2), and t . z σ ( a )to the absolute version (3.8.3). Presumably a “family Floer” argument along these lineswould prove the Theorem, but we will give a proof in terms of the explicit formulas. Proof.
Let us give the proof first in the relative case. Fix m , m ∈ Z ≥ , k ∈ { , . . . , m − } , k ∈ { , . . . , m − } , and a, b ∈ C [[ t ]]. The product of θ m ,k [ b ] and θ m ,k [ a ] is bydefinition X i ,i ∈ Z × Z σ i ( b ) σ i ( a ) t m ( i ) + m ( i ) + k i + k i z m i + m i + k + k (3.8.5)We may also index the sum by triples ( r, c, d ) where ( c, d ) ∈ Z × Z and r ∈ n , , . . . , m + m gcd( m ,m ) − o .First, for ℓ ∈ Z , we define d and r via ℓ = m + m gcd( m , m ) d + r It then follows that if we set e ( r ) = e m ,m ,k ,k ( r ) := (cid:22) m r + k + k m + m (cid:23) ∈ (cid:26) , , . . . , m gcd( m , m ) − (cid:27) and κ = k /m , κ = k /m , we have ⌊ E ( κ , κ + ℓ ) ⌋ = m d gcd( m , m ) + e ( r ) (3.8.6)The triple ( r, c, d ) (and the integer ℓ ) is determined as the unique solution to (cid:18) i i (cid:19) = (cid:18) − m / gcd( m , m )1 m / gcd( m , m ) (cid:19) (cid:18) c − e ( r ) d (cid:19) + (cid:18) r (cid:19) To save space in the exponents, let us write g := gcd( m , m ). After reindexing the sum(3.8.5) is X r X c X d σ c − e ( r )+ m d/g + r ( b ) σ c − e ( r ) − dm /g ( a ) t (cid:3) z (cid:3) (3.8.7)where (cid:3) = m (cid:0) c − e ( r ) − dm /g (cid:1) + k ( c − e ( r ) − dm /g )+ m (cid:0) c − e ( r )+ dm /g + r (cid:1) + k ( c − e ( r ) + dm /g + r ) (3.8.8)and (cid:3) = m ( c − e ( r ) − dm /g ) + m ( c − e ( r ) + dm /g + r ) + k + k (3.8.9) We note that (cid:3) = ( m + m ) c + ( m r + k + k ) − ( m + m ) e ( r ) does not depend on d , and furthermore that k ( r ) = k m ,m ,k ,k ( r ) := m r + k + k − ( m + m ) e ( r )belongs to { , . . . , m + m − } . (Note that k ( r ) / ( m + m ) is the fractional part of E ( κ , κ + ℓ ).) Thus the sum (3.8.7) is the same as X r X c X d σ c − e ( r )+ dm /g + r ( b ) σ c − e ( r ) − dm /g ( a ) t (cid:3) ! z ( m + m ) c + k ( r ) Since σ acts trivially on t , this is the same as X r X c σ c X d σ − e ( r )+ dm /g + r ( b ) σ − e ( r ) − dm /g ( a ) t (cid:3) ! z ( m + m ) c + k ( r ) Now we claim (cid:3) = λ ( κ , κ + ℓ ) + ( m + m ) (cid:18) c (cid:19) + k ( r ) c (3.8.10)where (cid:3) is as in (3.8.8) and λ is as in (2.8.7) and ℓ = m + m g d + r .Taking (3.8.10) for granted, we obtain that θ m ,k [ b ] · θ m ,k [ a ] is equal to m m g − X r =0 X c σ c X d σ − e ( r ) − dm /g ( a ) σ − e ( r )+ dm /g + r ( b ) t λ ( κ ,κ + ℓ ) ! t ( m + m ) ( c ) + k ( r ) c z ( m + m ) c + k ( r ) which is equal to X r θ m + m ,k ( r ) "X d σ − e ( r )+ dm /g + r ( b ) σ − e ( r ) − dm /g ( a ) t λ ( κ ,κ + ℓ ) (3.8.11)We now compare (3.8.11) to ( b · x m ,k /m )( a · x m ,k /m ) which is given by (3.7.2) X ℓ ∈ Z (cid:0) σ ℓ −⌊ E ( κ ,κ + ℓ ) ⌋ ( b ) σ −⌊ E ( κ ,κ + ℓ ) ⌋ ( a ) t λ ( κ ,κ + ℓ ) (cid:1) x m + m ,E ( κ ,κ + ℓ ) Writing ℓ = m + m gcd( m ,m ) d + r and using the formula (3.8.6), we can rewrite this as X r X d σ − e ( r )+ dm /g + r ( b ) σ − e ( r ) − dm /g ( a ) t λ ( κ ,κ + ℓ ) ! x m + m ,k ( r ) (3.8.12)It is now evident that under our map (3.8.4), the two expressions (3.8.11) and (3.8.12)agree. SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE 29
It remains to verify the claim in (3.8.10). This will be a direct computation. We have λ ( κ , κ + ℓ ) = m φ ( κ ) + m φ ( κ + ℓ ) − ( m + m ) φ ( m dg + e ( r ) + k ( r ) m + m )= m ( ℓ ( ℓ + κ ) − ℓ ( ℓ + 1)2 ) − ( m + m ) ( dm g + e ( r ))( dm g + e ( r ) + k ( r ) m + m ) − ( dm g + e ( r ))( dm g + e ( r ) + 1)2 ) ! = m ( ℓκ + ℓ ( ℓ − − ( dm g + e ( r )) k ( r ) − ( m + m ) ( dm g + e ( r ))( dm g + e ( r ) − m κ ( d ( m + m ) g + r ) + m ( d ( m + m ) g + r )( d ( m + m ) g + r − − ( dm g + e ( r )) k ( r ) − ( m + m ) ( dm g + e ( r ))( dm g + e ( r ) − k ( r ) = m r + k + k − ( m + m ) e ( r ), we get λ ( κ , κ + ℓ ) = m κ ( d ( m + m ) g + r ) + m ( d ( m + m ) g + r )( d ( m + m ) g + r − − ( dm g + e ( r ))( m κ + m κ + m r − e ( r )( m + m )) − ( m + m ) ( dm g + e ( r ))( dm g + e ( r ) − m + m ) m m g d + m m g ( r + κ − κ ) d + m κ r + m r ( r − m + m ) e ( r )( e ( r ) + 1)2 − ( m κ + m κ + m r ) e ( r )We can rewrite this in a symmetric form as follows: λ ( κ , κ + ℓ ) = ( m + m ) m m g d + m m g ( r + κ − κ ) d + m ( e ( r ) − r )( e ( r ) − r + 1)2 + m e ( r )( e ( r ) + 1)2 − k ( e ( r ) − r ) − k e ( r )and this in turn can be seen to be equal to λ ( κ , κ + ℓ ) = m (cid:18) − e ( r ) − m d/g (cid:19) + k ( − e ( r ) − m d/g )+ m (cid:18) − e ( r ) + m d/g + r (cid:19) + k ( − e ( r ) + m d/g + r ) This completes the proof of the claim (3.8.10) and hence the proof of the theorem in therelative case.In the absolute case, the product of θ abs m ,k [ b ] and θ abs m ,k [ a ] is given by X i ,i ∈ Z × Z σ i ( b ) σ i ( a ) t ( m i k m + ( m i k m z m i + m i + k + k (3.8.13)Performing the same re-indexing using (3.8), we arrive at X r X c X d σ c − e ( r )+ m d/g + r ( b ) σ c − e ( r ) − dm /g ( a ) t (cid:3) abs z (cid:3) abs (3.8.14)where (cid:3) abs = ( m ( c − e ( r ) − dm /g ) + k ) m + ( m ( c − e ( r ) + dm /g + r ) + k ) m (3.8.15)and (cid:3) abs = (cid:3) given as before by (3.8.9). Following the same steps, the only difference inthe calculation is the verification of the analogue of equation (3.8.10) which now takes theform: (cid:3) abs = ( ℓ + κ − κ ) m m m + m ) + (( m + m ) c + k ( r )) m + m ) (3.8.16)Recalling that ℓ = ( m + m ) d/g + r , k ( r ) = m r + k + k − ( m + m ) e ( r ) and κ i = k i /m i for i = 1 ,
2, we can compare the equations (3.8.15) and (3.8.16) directly to verify the claim.This completes the proof in the absolute case. (cid:3) Specializing the Novikov parameter At t = 0 . The specialization t = 0 renders uninteresting the absolute version of themaps µ n , at least if we also set t a = 0 for every a >
0. But it is a standard part of relativeFloer theory. In fact it is part of the motivation for relative Floer theory — in any sumover triangles (say), the contribution from triangles which are not disjoint from D vanishes,so that working with t = 0 is closely related to replacing the closed symplectic manifold T with the open T − D . See [LPe2, § S n for the n th graded piece of (3.8.1). Let us also put S := C σ [[ t ]]— here C σ denotes the σ -fixed subring of C . Then S • is a graded C σ [[ t ]]-algebra — it isassociative and commutative by Theorem 1.12, § C is a perfect field and σ is the p th root map, then C σ = F p . In any case there is an isomorphism in the category of C σ -schemes Proj( S × C σ [[ t ]] C σ ) = colim (cid:20) Spec( C ) P /Ci ◦ σi ∞ (cid:21) where i and i ∞ are the inclusions of C -schemes Spec( C ) → P /C with coordinates 0 and ∞ , respectively.If C is a field then Proj( S × S C σ ) is a one-dimensional scheme, which can be coveredby two affine charts. It fails to be regular at a unique point and the complement of thispoint is isomorphic to Spec( C [ x, x − ]). For the other chart take the complement of any SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE 31 other point — one obtains an affine Zariski neighborhood of the non-regular point that isisomorphic to the spectrum of a subring of C [ y ], namely { f ∈ C [ y ] : σ ( f (0)) = f (1) } This ring is in some sense an order in a Dedekind domain but if C has infinite degree over C σ then it is not of finite type.4.2. Floer cochains at t = 1 . Let C and σ be as in § C pulled back along f fromthe sheaf whose ´etal´e space is (3.4.3). If L and L ′ are one-dimensional submanifolds thatintersect transversely, we will write (similar to (2.5.1))CF( L, L ′ ; C ) = M x ∈ L ∩ L ′ C x (4.2.1)This supports a Z / µ ( x · a ) = X y | mas( y )=mas( x )+1 y X u ∈M ( y,x ) ±∇ γ ′ (cid:16) a ∇ γ (1 Λ y ) (cid:17) (4.2.2)with γ and γ ′ as in (3.1.2). (4.2.2) is a finite sumIf we further endow C with a topology, for which σ is continuous, we can investigate thealgebraic structures on (4.2.1) induced by (3.2.1). That is, we study the sums X y y X u ±∇ γ n ( a n ∇ γ n − ( a n − · · · ∇ γ ( a ∇ ( γ ( a ∇ γ (1)) · · · )))) ! (4.2.3)(4.2.2) and (4.2.3) are simply the specializations one obtains by setting t and every power t a to 1 in the formulas from §
3. The sums P u in (4.2.3) might diverge or converge in thetopological ring C , so that at best the mapCF( L n − , L n ; C ) × · · · × CF( L , L ; C ) CF( L , L n ; C ) (4.2.4)is only partially defined. In many cases, the domain of convergence is reduced to a point,but we will see that the triangle maps are not trivial.4.3. The triangle products at t = 1 . Suppose that C is complete with respect to anonarchimedean norm | · | , and that σ ( c ) = | c | /p for some p >
1. With p prime and C, σ as in (3.4.2), the pair ( C, | · | ) is a perfectoidfield of characteristic p [Sc, § ∇ γ for the sheaf of rings C are continuous but(crucially) they d not preserve the norms.Write O C := { c ∈ C : | c | ≤ } m C := { c ∈ C : | c | < } ;then O C is the ring of integers in C and m is the unique maximal ideal of O C . They areboth stable by the σ -action so that they determine locally constant subsheaves of C that2 YANKI LEKILI AND DAVID TREUMANN
1. With p prime and C, σ as in (3.4.2), the pair ( C, | · | ) is a perfectoidfield of characteristic p [Sc, § ∇ γ for the sheaf of rings C are continuous but(crucially) they d not preserve the norms.Write O C := { c ∈ C : | c | ≤ } m C := { c ∈ C : | c | < } ;then O C is the ring of integers in C and m is the unique maximal ideal of O C . They areboth stable by the σ -action so that they determine locally constant subsheaves of C that2 YANKI LEKILI AND DAVID TREUMANN we denote by O C and m C . The fiber of m at x is the set of topologically nilpotent elementsin C x .Following the notation of (4.2.1) setCF( L, L ′ ; m ) := M x ∈ L ∩ L ′ m x (4.3.1)It is an open subgroup of CF( L, L ′ ; C ).Suppose L , L , and L are special of finite slopes m , m , and m , in the sense of § x ∈ L ∩ L x ∈ L ∩ L y ∈ L ∩ L to µ ( x · b, x · a ) has the form (3.7.5) σ f ( γ ) ( b ) σ − f ( γ ) ( a ), where γ and γ are the two edgesof u incident with the output vertex y . If (and only if) m < m < m , then f ( γ ) and f ( γ ) all have the same sign — with perhaps finitely many exceptions where one of f ( γ )and f ( γ ) are zero — so that when | a | < | b | < | σ f ( γ ) ( b ) σ − f ( γ ) ( a ) | = | b | p − f ( γ | a | p f ( γ is very rapidly decreasing as the side lengths of the triangles go to infinity. The triangleproduct µ : CF( L , L ; m ) × CF( L , L ; m ) → CF( L , L ; m )is therefore convergent when m < m < m . In particular we have a graded ring (for now,without unit) ∞ M m =1 CF( L (0) , L ( m ) ; m ) (4.3.2)4.4. The irrelevant ideal in the Fargues-Fontaine graded ring.
Let C be an alge-braically closed field of characteristic p that is complete with respect to a norm | · | . Let B and ϕ be as in § B = (X i ∈ Z b i z i | ∀ r ∈ (0 , , | b i | r i → | i | → ∞ ) (4.4.1)This appears in [KS, Def. 21] and in [FF, Ex. 1.6.5]. Fargues and Fontaine define a versionof B for every local field E , (4.4.1) is the case when E = F p (( z )). Below, we are takingadvantage of the fact that when E has equal characteristic each element of B has a uniqueseries expansion, something that is not clear when E has mixed characteristic [FF, Rem.1.6.7].Let ϕ be as in (1.14.2), i.e. the automorphism of B given by ϕ ( P c i z i ) = P c pi z i . Thehomogeneous coordinate ring of FF E ( C ) is F p (( z )) ⊕ B ϕ = z ⊕ B ϕ = z ⊕ · · · (4.4.2)We will prove the theorem of § SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE 33
Proof of (1.14.4) . Suppose that a ∈ C has | a | <
1. Then the sequence | σ i ( a ) | = | a | p − i of real numbers is bounded as i → ∞ and very rapidly decreasing as i → −∞ , and r mi + k | a | p − i → | i | → ∞ for any r ∈ (0 , m and k the expression X i ∈ Z z mi + k σ i ( a ) (4.4.3)belongs to B (1.14.1). Applying ϕ to (4.4.3) gives P i ∈ Z z mi + k σ i − ( a ) — re-indexing thisseries gives X i ∈ Z z m ( i +1)+ k σ i ( a ) = z m X i ∈ Z z mi + k σ i ( a )so that (4.4.3) belongs to B ϕ = z m . But (4.4.3) is θ m,k [ a ] | t =1 , so that by § x m,k/m · a θ m,k [ a ] | t =1 (4.4.4)intertwines µ with the ring structure on B .If f = P b i z i belongs to B ϕ = z m , then b mi + k = σ m ( b k ), so f is determined by b , . . . , b m − .To obey (1.14.1), the elements b , . . . , b m − must all belong to m . The map f m − X k =0 x m,k/m · b k gives the inverse isomorphism to CF( L (0) , L ( m ) ; m ) ∼ = B ϕ = z m . (cid:3) SYZ duality.
The degree one part of (1.14.4) is an isomorphismCF( L (0) , L (1) ; m ) ∼ = Hom FF ( O , O (1)) (4.5.1)where O (1) is the Serre line bundle on (1.14.3). In general it seems that CF( L, L ′ ; m )captures the set of homomorphisms between two vector bundles on FF whenever L and L ′ are (or are just isotopic to, if we replace CF by HF) special Lagrangians § m and m ′ with m strictly less than m ′ . But for other kinds of homomorphismsor Ext groups in Coh( FF ), another construction must be necessary — one that we onlypartially understand. In the next two sections § § FF .The closed points of FF E ( C ) are naturally parametrized by the Z -orbits of E -untilts ofthe perfectoid field C . When E = F p (( z )), an “ E -untilt” is just a continuous homomorphism i : E → C — such a homomorphism must carry z to a nonzero element of m and converselyevery nonzero element of m extends to a map from F p (( z )), the Z -action is generated by i σ ◦ i . There is a map (closed points of FF ( E, C )) → R / Z (4.5.2)It is defined for any E . When E = F p (( z )) it carries the Z -orbit of ι : F p (( z )) → C to the Z -coset of log p (log( | i ( z ) | − )). We expect that (4.5.2) is the SYZ dual to (3.4.1), and thatthe skyscraper sheaves have something to do with fibers of (3.4.1). Skyscraper sheaves and L ( ∞ ) . If ζ ∈ C is invertible, let us denote by L ζ ( ∞ ) thespecial Lagrangian L ( ∞ ) equipped with the rank one local system of C | L ( ∞ ) -modules (i.e.,a local system of C -modules) whose fiber at (0 ,
0) is C (0 , = C and whose monodromy(in the direction of the default orientation, top to bottom) is multiplication by ζ . Let e m denote (0 ,
0) regarded as the unique intersection point of L ( m ) and L ( ∞ ) , so thatCF( L ( m ) , L ζ ( ∞ ) ; C ) = CF ( L ( m ) , L ζ ( ∞ ) ; C ) = e m · C. (4.6.1)If C is algebraically closed then one also has Hom FF ( O ( m ) , δ ) ∼ = C for any skyscrapersheaf δ . If | ζ | < δ is the skyscraper sheaf supported at the Z -orbit of the untilt F p (( z )) → C , then we expect that for any surjection q : O (1) → δ , there is an isomorphismmaking the diagram CF( L (0) , L (1) ; m ) (4.5.1) (cid:15) (cid:15) µ ( e · , − ) / / CF( L (0) , L ζ ( ∞ ) ; C ) (cid:15) (cid:15) ✤✤✤ Hom( O , O (1)) q ◦ / / Hom( O , δ ) (4.6.2)commute; instead of constructing this isomorphism here let us verify that the two rows of(4.6.2) have the same kernel. We may find i : O → O (1) such that0 → O i −→ O (1) q −→ δ → O , O ) = F p (( z )), the ground field of (1.14.3). We will show that the kernel of µ ( e · , − ) has thestructure of a one-dimensional F p (( z ))-module.In general the triangle map µ ( e · b, x , · a ) (4.2.4) is given by e · (cid:0)P i ∈ Z ( − i bζ i σ i ( a ) (cid:1) x , · ae · b σ ζ (4.6.3)with the figure at the right illustrating the triangle that contributes the i = 5 term (forthe sign, see § b · θ , [ a ] at t = 1 and z = − ζ , it converges whenever | ζ | and | a | are both less than one.Thus the top row of (4.6.2) is isomorphic to the map m → C sending a to ϑ ( a ) := P n ∈ Z ( − ζ ) n a p − n . This map obeys ϑ ( a p ) = ( − ζ ) ϑ ( a )i.e. it intertwines the F p (( z ))-module structure on C given by the homomorphism z
7→ − ζ with the F p (( z ))-module structure on m given by ( z, a ) a p . The kernel is therefore an F p (( z ))-module. The image of this kernel under the isomorphism m ∼ = B ϕ = z (given by SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE 35 a P a p i z − i (1.14.4)) is the set of b ∈ B ϕ = z whose set of zeroes is exactly { ( − ζ ) p n } n ∈ Z .The function h ( z ) = X i ∈ Z a p i z − i ! ∞ Y n =0 (1 + ζ p n /z ) ! − (the meromorphic part of the Weierstrass factorization [FF, Ch. 2]) belongs to C (( z )) andobeys the functional equation h ( z /p ) p = ( ζ + z ) h ( z ), i.e. its coefficients obey the recursion h pn − ζ h n = h n − (4.6.4)—for each h n − there are exactly p solutions in h n to (4.6.4), so the set of such h ( z ) is aone-dimensional F p (( z ))-submodule of C (( z )).4.7. Ore adjoint.
Let L ζ ( ∞ ) be as in § L ( ∞ ) and L ( m ) in (4.6.1),the Maslov index of the intersection point is 1, so thatCF( L ζ ( ∞ ) , L ( m ) ; C ) = CF ( L ζ ( ∞ ) , L ( m ) ; C ) = e m · C The triangle sumCF ( L ζ ( ∞ ) , L (0) ; C ) × CF ( L (1) , L ζ ( ∞ ) ; C ) CF ( L (1) , L (0) ; C )is formally given by X n ∈ Z ( − n σ − n ( bζ n a ) = X n ∈ Z ( − ζ ) np n ( ab ) p n (4.7.1)It is the same triangles as (4.6.3) that contribute to (4.7.1), but they are decorated differ-ently. For instance the triangle contributing the n = 5 summand is e · ae · b σ − ζ Even if | ζ | <
1, the n → −∞ tail of (4.7.1) does not converge unless ab = 0. Even so, itis interesting in a formal way. In [Poon], Poonen following [Ore] attaches to each series ofthe form f ( a ) = P u n a p n an “adjoint” series f † ( a ) := P u p n − n a p n — let us call it the Oreadjoint. Evidently (4.7.1) is exactly ϑ † [ ba ].Under some hypotheses on f Poonen shows that the kernels of f and f † are Pontrjagindual to each other in a canonical fashion. These hypotheses are not satisfied by ϑ ( a ),but as ker( ϑ ) (being the additive group of a local field) is Pontrjagin self-dual, and as ϑ † does not converge in any case, we are perhaps free to speculate that “ker( ϑ † )” is somehowmorally isomorphic to F p (( z )). This speculation is consistent with mirror symmetry: onthe Fargues-Fontaine curve there indeed is a short exact sequence0 → Hom( O (1) , O (1)) → Hom( O (1) , δ ) → Ext ( O (1) , O ) → coming from the resolution O → O (1) of the skyscraper sheaf δ , and the vanishing ofExt ( O (1) , O (1)) = H ( FF ; O ). The kernel of (4.7.2) is naturally isomorphic to H ( FF ; O ),i.e. to F p (( z )). The middle group is isomorphic to C and to HF ( L (1) , L ζ ( ∞ ) ; C ). But weemphasize that Ext ( O (1) , O ) is not isomorphic to HF( L (1) , L (0) ; C ), nor to any open sub-group of it.4.8. Loud Floer cochains on L (0) . Let { φ s } s ∈ R be as in (2.12.1): φ s ( x, y ) = ( x, y − s cos(2 πx )) (4.8.1)We can try to compare CF( L, L ′ ; C ) and CF( φ s L, L ′ ; C ) by specializing to t = 1 in (3.1.4).In some cases, for instance if L and L ′ are parallel to L (0) as in § L, L ′ ; C ) → CF( φ s L, L ′ ; C )without any problems. But this map is not always a quasi-isomorphism. The series definingthe homotopy (3.1.5) may not converge at t = 1 — (3.6.2) is a vivid example of this.We will analyze the continuation maps between the groups CF( φ s L (0) , L (0) ; C ). If n isan integer and n < s < n + 1, then L (0) meets φ s L (0) in 2 n + 2 points. In the fundamentaldomain [0 , × [0 , x -coordinate < . x -coordinate > .
5. They are linearly ordered by the x -coordinate and after listing them inthat order we will name them z ( s ) n , . . . , z ( s ) − n , ξ ( s ) − n , . . . , ξ ( s ) n More explicitly, z ( s ) i and ξ ( s ) i are the two solutions to i − s cos(2 πx ) = 0, i.e. for a suitablebranch of the inverse cosine function: z ( s ) i = 12 π arccos( i/s ) ξ ( s ) i = 1 − π arccos( i/s )The case 1 < s < ⋆⋆ z ( s )1 z ( s )0 z ( s ) − ξ ( s )1 ξ ( s )0 ξ ( s ) − The rules of § z ( s ) i the Maslov index 0 and each ξ ( s ) i the Maslov index 1,so that CF ( φ s L (0) , L (0) ; C ) = n M i = − n z ( s ) i · C CF ( φ s L (0) , L (0) ; C ) = n M i = − n ξ ( s ) i · C SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE 37
Each ξ ( s ) i is the output vertex of exactly two bigons, and the other vertex of both bigonsin z ( s ) i . There is an “upward” bigon whose boundary passes through (0 . , s ) + Z anda “downward” on whose boundary passes through (0 , − s ) + Z . If one places a star at(or close to, as in the figure above) (0 . , s ) + Z and (0 , − s ) + Z , then the sign of everydownward bigon is 1 and the sign of every upward bigon is −
1. The downward bigonscross the danger line exactly once, and the upward bigons exactly never, so that µ ( z ( s ) i · a ) = ξ ( s ) i · ( − a + σ ( a ))with the upward bigon contributing − a and the downward bigon contributing σ ( a ).If s ′ > s then for a suitable choice of profile function β (on that is very close to the“linear cascades” limit considered in [Aur1]), the continuation mapCF( φ s L (0) , L (0) ; C ) → CF( φ s ′ L (0) , L (0) ; C ) (4.8.2)simply sends z ( s ) i · a to z ( s ′ ) i · a and ξ ( s ) i · a to ξ ( s ′ ) i · a . In particular it defines a filtereddiagram of cochain complexes (indexed by s > s / ∈ Z , with respect to the usual orderingof real numbers s . Let CF loud ( L (0) , L (0) ) denote the direct limit of this diagramCF loud ( L (0) , L (0) ; C ) := lim −→ s> | s / ∈ Z CF( φ s L (0) , L (0) ; C )Since each map (4.8.2) is the inclusion of a direct summand of cochain complexes, CF loud is a model for the homotopy colimit of cochain complexes as well. Explicitly,CF = M i ∈ Z z i · C CF = M i ∈ Z ξ i · C µ ( z i · a ) = ξ i · ( − a + σ ( a )) (4.8.3)4.9. Triangles between the φ s L (0) . Continuing with the notation of § s , s ′ , and s + s ′ are in Z , and describe the triangles between L (0) , φ s L (0) , and φ s + s ′ L (0) . The “output” corners of these triangles are z ( s + s ′ ) i and ξ ( s + s ′ ) i on L (0) ∩ φ s + s ′ L (0) , and the other two corners in counterclockwise order are φ s z ( s ′ ) i , φ s ξ ( s ′ ) i ∈ φ s L (0) ∩ φ s + s ′ L (0) , z ( s ) i , ξ ( s ) i ∈ L (0) ∩ φ s L (0) . For each such triangle there is a unique pair of integers i and j so that the triangle liftsto R with boundary on the x -axis, the graph of y = i − s cos(2 πx ), and the graph of y = i + j − ( s + s ′ ) cos(2 πx ). The only non-empty moduli spaces of triangles are M (cid:16) z ( s + s ′ ) i + j , φ s z ( s ′ ) j , z ( s ) i (cid:17) M (cid:16) ξ ( s + s ′ ) i + j , φ s z ( s ′ ) j , ξ ( s ) i (cid:17) M (cid:16) ξ ( s + s ′ ) i + j , φ s ξ ( s ′ ) j , z ( s ) i (cid:17) (4.9.1)with − s < i < s and − s ′ < j < s ′ . When i/s = j/s ′ , there is something tricky aboutthe latter two moduli spaces (they are not transversely cut § i/s = j/s ′ . Then each space (4.9.1) contains exactly one triangle. The natureof this triangle depends on which of i/s or j/s ′ is larger. The triangle of M (cid:16) z ( s + s ′ ) i + j , φ s z ( s ′ ) j , z ( s ) i (cid:17) is the one bounded bymax(0 , ( i + j ) − ( s + s ′ ) cos(2 πx )) ≤ y ≤ i − s cos(2 πx ) if j/s ′ < i/s (4.9.2)min(0 , ( i + j ) − ( s + s ′ ) cos(2 πx )) ≥ y ≥ i − s cos(2 πx ) if j/s ′ > i/s (4.9.3)For legibility, in the following illustration of these triangles the curves φ s L (0) and φ s + s ′ L (0) are drawn in a different aspect ratio than in the diagram of § ∼ s + s ′ fundamental domains, stacked on top of eachother.In this and the following diagrams, φ s + s ′ L (0) is purple, φ s L (0) is blue, and L (0) is black.The left side shows the typical case where i/s > j/s ′ , and the right side shows the typicalcase when i/s < j/s ′ .In M (cid:16) ξ ( s + s ′ ) i + j , φ s z ( s ′ ) j , ξ ( s ) i (cid:17) we have the triangle(4 . . ∪ { min(0 , i − s cos(2 πx )) ≥ y ≥ ( i + j ) − ( s + s ′ ) cos(2 πx ) } if js ′ < is (4 . . ∪ { max(0 , i − s cos(2 πx )) ≤ y ≤ ( i + j ) − ( s + s ′ ) cos(2 πx ) } if js ′ > is (4.9.4)In M (cid:16) ξ ( s + s ′ ) i + j , φ s ξ ( s ′ ) j , z ( s ) i (cid:17) we have the triangle(4 . . ∪ { min( i − s cos(2 πx ) , i + j − ( s + s ′ ) cos(2 πx ))) ≥ y ≥ } if js ′ < is (4 . . ∪ { max( i − s cos(2 πx ) , i + j − ( s + s ′ ) cos(2 πx )) ≤ y ≤ } if js ′ > is (4.9.5) SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE 39
Now we discuss the triangles with i/s = j/s ′ . For generic s and s ′ , it is only possiblethat i/s = j/s ′ when i = j = 0. In that case M (cid:16) z ( s + s ′ )0 , φ s z ( s ′ )0 , z ( s )0 (cid:17) again contains asingle point (the constant map with value z = ( . , M (cid:16) ξ ( s + s ′ )0 , φ s z ( s ′ )0 , ξ ( s )0 (cid:17) or for M (cid:16) ξ ( s + s ′ )0 , φ s ξ ( s ′ )0 , z ( s )0 (cid:17) .These spaces each contain two points, which are degenerate triangles (they are at theboundary of the Deligne-Mumford-Stasheff compactification) which are not maps out of atriangle but out of a wedge sum of triangle and a bigon:The degenerate maps in M (cid:16) ξ ( s + s ′ )0 , φ s z ( s ′ )0 , ξ ( s )0 (cid:17) and M (cid:16) ξ ( s + s ′ )0 , φ s ξ ( s ′ )0 , z ( s )0 (cid:17) collapse thetriangle part to a point (to ξ = ( . , N NN N (4.9.6)The top two figures indicate the two points of M (cid:16) ξ ( s + s ′ )0 , φ s z ( s ′ )0 , ξ ( s )0 (cid:17) , the bottom two arethe two points of M (cid:16) ξ ( s + s ′ )0 , φ s ξ ( s ′ )0 , z ( s )0 (cid:17) . Though they are not transversely cut they haveanalytic index zero — more precisely they have index +1 along the constant triangle andindex − Triangle products on CF loud . For short, let us put A ( s ) := CF( φ s L (0) , L (0) ; C ).The triangles in the previous section, together with the identification CF( φ s + s ′ , φ s L (0) ; C )of and CF( φ s ′ L (0) , L (0) ; C ), give a multiplication A ( s ) × A ( s ′ ) → A ( s + s ′ ) specifically • (Coming from (4.9.2) and (4.9.3))( z ( s ) i · a, z ( s ′ ) j · b ) z ( s + s ′ ) i + j · ab • (Coming from (4.9.4))( ξ ( s ) i · a, z ( s ′ ) j · b ) ξ ( s + s ′ ) i + j · ( aσ ( b ) if j/s ′ < i/sab if j/s ′ > i/s (4.10.1) • (Coming from (4.9.5))( z ( s ) i · a, ξ ( s ′ ) j · b ) ξ ( s + s ′ ) i + j · ( ab if j/s ′ < i/sσ ( a ) b if j/s ′ > i/s (4.10.2)Since M (cid:16) ξ ( s + s ′ )0 , φ s z ( s ′ )0 , ξ ( s )0 (cid:17) and M (cid:16) ξ ( s + s ′ )0 , φ s ξ ( s ′ )0 , z ( s )0 (cid:17) are not transversely cut, theycarry a virtual fundamental class rather than an orientation. We will simply put( ξ ( s )0 · a, z ( s ′ ) · b ) ξ ( s + s ′ )0 · ab ( z ( s )0 a, ξ ( s ′ )0 b ) ξ ( s + s ′ )0 · σ ( a ) b (4.10.3)as though the left two degenerate triangles displayed in (4.9.6) contributed nothing. Thesame issue can also be addressed by introducing a Hamiltonian perturbation ψ of L (0) (butnot φ s L (0) or φ s + s ′ L (0) ) supported in a very small neighborhood of z and ξ . In that caseall moduli spaces are transversely cut and the triangle products µ ( ψz ( s )0 · a, φ s ξ ( s ′ )0 · b ) and µ ( ψξ ( s )0 · a, φ s z ( s ′ )0 · b ) are well-defined, though the specific formula will depend on ψ —(4.10.3) is consistent with some of these ψ .Now we use the products A ( s ) × A ( s ′ ) → A ( s + s ′ ) to define a multiplication on lim s A ( s ) ,i.e. on CF loud ( L (0) , L (0) ; C ). It is not quite straightforward, because the products (4.10.1)and (4.10.2) are not eventually constant as s and s ′ grow — it depends on which of i/s and j/s ′ are larger. The square A ( s ) × A ( s ′ ) (cid:15) (cid:15) / / A ( s + s ′ ) (cid:15) (cid:15) A ( S ) × A ( S ′ ) / / A ( S + S ′ ) (4.10.4)does not commute for all s, s ′ , S, S ′ with s < S and s ′ < S ′ . We address this in the followingcrude way: we choose an irrational number e >
0, and note that since i/s − j/ ( es ) hasconstant sign for s >
0, (4.10.4) does commute when s ′ = es and S ′ = eS . The inducedmultiplication on the colimit is explicitly ( z i · a, z j · b ) z i + j · ab and( ξ i · a, z j · b ) ξ i + j · ( aσ ( b ) if j/e < iab if j/e ≥ i ( z i · a, ξ j · b ) ξ i + j · ( ab if j/e < iσ ( a ) b if j/e ≥ i Thus we get one binary operation on CF loud ( L (0) , L (0) ; C ) for every irrational e >
0. Thesemultiplications are genuinely different for different e . Moreover, they are not associative;they do however, obey the Leibniz rule µ ( ww ′ ) = µ ( w ) w ′ + wµ ( w ′ ), with µ as in SYMPLECTIC LOOK AT THE FARGUES-FONTAINE CURVE 41 (4.8.3). It is likely that they can be extended to an A ∞ -structure on CF loud ( L (0) , L (0) ; C )(and even more likely that there is such an A ∞ -structure on a complex quasi-isomorphicto it, defined along the lines of [AS]), but we will not construct it. Instead we simplynote that the induced multiplication on HF := ker( µ ) (and even on HF ⊕ HF ,though the degree 1 part vanishes if C is algebraically closed and σ is the p th root map), isassociative, and independent of e . Indeed it is simply the Laurent polynomial ring C σ [ z ± ]under the assignment P c i z i P z i · c i . References [AS] M. Abouzaid, P. Seidel, An open string analogue of Viterbo functoriality Geom. Topol. 14 (2010),no. 2, 627–718.[Aur1] D. Auroux, Fukaya categories of symmetric products and bordered Heegaard-Floer homology. J.G¨okova Geom. Topol. GGT 4 (2010), 1–54.[Aur2] D. Auroux, Special Lagrangian fibrations, wall-crossing, and mirror symmetry. Surv. Differ. Geom.,13, Int. Press, Somerville, MA, 2009, 1–47.[ChOh] C. Cho and Y. Oh, Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toricmanifolds. Asian J. Math. 10 (2006), no. 4, 773–814.[EGA] J. Dieudonne and A. Grothendieck, ´El´ements de g´eom´etrie alg´ebrique, Pub. Math. IHES, 1960–1967[Fa] L. Fargues, Geometrization of the local Langlands correspondence, an overview, arXiv:1602.00999,preprint.[FF] L. Fargues and J. Fontaine, ‘Courbes et fibr´es vectoriels en th´eorie de Hodge p-adique. Ast´erisque2018, no. 406, xiii+382 pp.[Fl] A. Floer, Symplectic fixed points and holomorphic spheres. Comm. Math. Phys. 120 (1989), no. 4,575–611.[H] U. Hartl, Period spaces for Hodge structures in equal characteristic. Ann. of Math. (2) 173 (2011),no. 3, 1241–1358[Hutc] M. Hutchings, Reidemeister torsion in generalized Morse theory. Forum Math. 14 (2002), no. 2,209–244.[HuLe] M. Hutchings, Y-J Lee, Circle-valued Morse theory and Reidemeister torsion. Geom. Topol. 3(1999), 369–396.[KS] M. Kontsevich and Y. Soibelman, Homological mirror symmetry and torus fibrations. Symplecticgeometry and mirror symmetry (Seoul, 2000), 203–263, World Sci. Publ., River Edge, NJ, 2001.[Lee] H. Lee, Homological mirror symmetry for open Riemann surfaces from pair-of-pants decompositions,arXiv:1608.04473, preprint.[LPe1] Y. Lekili and T. Perutz, Fukaya categories of the torus and Dehn surgery. Proc. Natl. Acad. Sci.USA 108 (2011), no. 20, 8106–8113.[LPe2] Y. Lekili, and T. Perutz, Arithmetic mirror symmetry for the 2-torus arXiv:1211.4632, unpublished.[LPo] Y. Lekili and A. Polishchuk, Arithmetic mirror symmetry for genus 1 curves with n marked points.Selecta Math. (N.S.) 23 (2017), no. 3, 1851–1907.[Oh] Y. Oh, Symplectic topology and Floer homology. Vol. 2. Floer homology and its applications. NewMathematical Monographs, 29. Cambridge University Press, Cambridge, 2015. xxiii+446 pp.[Ore] O. Ore, On a special class of polynomials Trans. Amer. Math. Soc. 35 (1933), no. 3, 559–584.[OsSz] P. Oszv´ath and Z. Szab´o, Holomorphic disks and topological invariants for closed three-manifolds.Ann. of Math. (2) 159 (2004), no. 3, 1027–1158.[Poli] A. Polishchuk, Massey and Fukaya products on elliptic curves. Adv. Theor. Math. Phys. 4 (2000),no. 6, 1187–1207.2 YANKI LEKILI AND DAVID TREUMANN