The cyclic open-closed map, u-connections and R-matrices

This paper considers the (negative) cyclic open-closed map $\mathcal{OC}^{-}$, which maps the cyclic homology of the Fukaya category of a symplectic manifold to its $S^1$-equivariant quantum cohomology. We prove (under simplifying technical hypotheses) that this map respects the respective natural connections in the direction of the equivariant parameter. In the monotone setting this allows us to conclude that $\mathcal{OC}^{-}$ intertwines the decomposition of the Fukaya category by eigenvalues of quantum cup product with the first Chern class, with the Hukuhara-Levelt-Turrittin decomposition of the quantum cohomology. We also explain how our results relate to the Givental-Teleman classification of semisimple cohomological field theories: in particular, how the R-matrix is related to $\mathcal{OC}^{-}$ in the semisimple case; we also consider the non-semisimple case.


Introduction
Kontsevich conjectured [Kon95] that enumerative mirror symmetry, an equality between Gromov-Witten invariants of a space X and period integrals on Y (see [Can+92]) is a consequence of a homological mirror symmetry: This paper focusses on the symplectic side of mirror symmetry. [Bar01] shows that one can extract the Gromov-Witten invariants of X from a variation of semi-infinite Hodge structures (VSHS) associated to the quantum cohomology of X, together with a splitting of the Hodge filtration. This goes via the intermediary step of a Frobenius manifold. One approach to obtain enumerative invariants from the Fukaya category is thus to first associate a VSHS to it, and then to specify the correct splitting. It is by now well understood how to construct the structure of a VSHS on the cyclic homology of an A ∞ -category (see [Get93], [KKP08], or [She20]). Characterising the splitting has not been done in general, but results have been obtained in various settings. Ganatra-Perutz-Sheridan [GPS15] characterise the splitting when the VSHS is Z-graded and of Hodge-Tate type over a one-dimensional base. The geometric setting one should think of is the Fukaya category of a Calabi-Yau. In this case the splitting is determined by the VSHS itself. Secondly Amorim-Tu [AT19] show how the grading operator on quantum cohomology classifies the correct splitting when the Hochschild cohomology ring of the Fukaya category is semi-simple. The grading operator constitutes extra data, so the splitting is not necessarily determined intrinsically by the VSHS. The main examples are all Fano: complex projective space, or quadric hypersurfaces.

Formal TEP-structures
[Her03, Section 2.5] defines TERP-structures. We will only need TEP-structures. Furthermore, rather than working with holomorphic functions, we work with formal power series in the equivariant parameter. Hence we call them formal TEP-structures.
Definition 1.1 (see Definition 2.1). Let K be a field.
1. A formal pre-T-structure over a K-algebra R, is a pair (E, ∇). Here E is an R [[u]]-module and ∇ : Der K R ⊗ E → u −1 E a flat connection.
2. If E is free and finitely-generated, call this a formal T-structure.
3. A formal TE-structure is a formal T-structure together with an extension of the connection to a flat connection ∇ : 4. A formal TEP-structure is a formal TE-structure equipped with polarisation, i.e. a symmetric, sesquilinear, covariantly constant pairing (·, ·) : E ⊗ E → R [[u]], which restricts to a non-degenerate pairing (·, ·) : E/uE ⊗ E/uE → R.
Thus a VSHS in the sense of [Bar01] is a formal TP-structure.
Remark 1.2. A TEP-structure can be formalised to yield a formal TEP-structure, this process forgets information (the Stokes' data, see [Sab07,§II.6]). The cyclic homology of an A ∞ -category only yields a formal TEP-structure, which is why we will always be talking about formal TEP-structures. For ease of reading, we omit the word 'formal' from now on. We hope this doesn't cause any confusion.
Definition 1.3. The quantum TEP-structure is defined over R = Λ[[H * (X)]], where Λ is a Novikov ring. It is given by the S 1 -equivariant quantum cohomology QH * (X; R) [[u]]. The connection is as defined in [Dub99], or see Section 4.2. The pairing is given by the sesquilinear extension of the Poincaré pairing.
Definition 1.4. The TEP-structure HC − * (C) associated to an R-linear A ∞ -category is as defined in [KKP08] or see Section 3. Remark 1.9. One reason we adopt the rather restrictive technical assumptions of [ST16] is that we plan follow-up work in which we relate the results of this paper, which concern closed Gromov-Witten invariants, with the open Gromov-Witten invariants defined in [ST19]. Similar to [ST16,Remark 4.2] we expect that these restrictive technical assumptions can be removed, as their role is purely to simplify the analysis of moduli spaces of holomorphic disks.

Semi-simple quantum cohomology
If we additionally assume that QH * (X; C) is a semi-simple C-algebra (isomorphic as a ring to a direct sum of copies of C), we can completely determine the E-structure QH * (X) [[u]]. To this end, for φ ∈ C[u −1 ], let E φ := (C [[u]], ∇ d du ) denote the 1-dimensional TE-structure (over R = C), with connection given by We show the following, which was already obtained by [Dub99], see also [Tel12] and [GGI16]: Lemma 1.15 (see Corollary 6.9). Assume QH * (X) is semi-simple, then there exists a basis v i ∈ QH * (X) [[u]] such that u 2 ∇ d du v i = w i v i , where the w i are the eigenvalues of c 1 . We call the v i 'w i -flat sections'. Equivalently, there is an isomorphism of E-structures Remark 1.16. The semi-simplicity assumption is essential; diagonalisability of c 1 is insufficient. This is because we need a special property of the grading operator µ on quantum cohomology (see Lemma 6.8).
Definition 1.17. Given an E-structure (E, ∇) a splitting is a K-linear map s : E/uE → E splitting the natural projection π : E → E/uE.
Example 1.18. The E-structure E φ admits a splitting given by: Remark 1.19. A choice of splitting is equivalent to a choice of opposite subspace as used by Barannikov [Bar01] to obtain a Frobenius manifold from a VSHS. See also [Gro11, Section 2.1.7].
The quantum E-structure admits a canonical splitting. This splitting does not respect the decomposition of Lemma 1.11, but it is the one relevant for Gromov-Witten theory: When the quantum cohomology is semi-simple, the w i -flat sections define a second splitting s ss : QH * (X) → QH * (X) [[u]] given by: Note that whilst the v i are not unique, the associated splitting is uniquely determined, as any two choices of the v i are related by a constant matrix. This splitting preserves the decomposition of the quantum TE-structure: s ss (QH * (X) w ) ⊂ QH * (X) [[u]] w .
For a general E-structure, given two splittings s 1 , s 2 , we obtain an element R ∈ Aut(E/uE) [[u]] as R = i≥0 u i R i , with R 0 = Id, and s 1 (α) = i≥0 u i s 2 (R i (α)) for all α ∈ E/uE.
Such R is called an R-matrix.
Remark 1.20. R-matrices were used by Givental [Giv01] and [Tel12] to classify semi-simple TFT's. See also [PPZ15,chapter 2] for the definition of R-matrices and their action on cohomological field theories. Their definition of an R-matrix involves an additional 'symplectic' property, namely that R preserves the polarisation. The group of such symplectic R-matrices is called the Givental loop group. We do not consider this polarisation, so our R-matrices need not be elements of the Givental loop group.
The two splittings on the quantum E-structure are thus related by an R-matrix R ∈ Aut(QH * (X)) [[u]]. A short computation shows that this is indeed the same R-matrix as defined by Teleman [Tel12] to recover all (including higher genus) Gromov-Witten invariants of X from its genus 0, 3-point invariants.
By Corollary 1.12, we find the following: The R-matrix thus tells us how to change the naive/constant decomposition of quantum cohomology to be compatible with the cyclic open-closed map. Amorim  This lemma can be rephrased as the existence of an isomorphism of E-structures If QH * (X) is semi-simple, and the closed-open map is an isomorphism, then HH * (F uk(X)) is semi-simple. Thus the previous lemma is indeed what was expected from Conjecture 1.6 and Lemma 1.15. In Section 6.1 we explain how our Conjecture 1.6, if proved in appropriate generality, can be used to give an alternative proof of the following theorem of Amorim-Tu. . Let X be a symplectic manifold with HH * (F uk(X)) semi-simple. Then the category F uk(X) together with the closed-open map determine the big quantum cohomology as a Frobenius manifold.
Amorim and Tu prove their theorem under the assumption that CO is a ring isomorphism, and use the Dubrovin-Teleman reconstruction theorem ( [Dub99], [Tel12]) of semi-simple Frobenius manifolds. Our proof instead uses OC − and assumes Conjecture 1.6, which allows us to avoid appealing to the reconstruction theorem.

Speculations on the general case
When the quantum cohomology is not semi-simple, a basis of w-flat sections does not necessarily exist. However, sometimes it is still possible to construct a non-trivial R-matrix. Consider the case when the Fukaya category of X splits as follows: where the Y i are (not necessarily monotone) symplectic manifolds. This is expected to hold when X is a blow up (see [VWX20] for a proof in certain cases). Another example is the complete intersection of two quadric hypersurfaces in CP 5 (see [Smi12]). We conjecture: Conjecture 1.23. When F uk(X) splits up as above, then the Gromov-Witten invariants of X can be obtained from those of the Y i , together with the genus 0, 3 point invariants of X.
We will illustrate this conjecture when X is the complete intersection of two quadric hypersurfaces in CP 5 . The eigenvalue decomposition of the Fukaya category is as follows: F uk(X) = F uk(X) −8 ⊕ F uk(X) 0 ⊕ F uk(X) 8 .
An explicit computation shows that Φ is unique up to rescaling the Φ i by constants λ i ∈ C. Thus, the following splitting is well-defined: Here s : C → E ±8/u is as defined in Example 1.18, s GW denotes the canonical splitting on QH * (Σ 2 ) [[u]], and π is the map given by setting u = 0. Let s 2 = s GW : QH * (X) → QH * (X) [[u]] be the canonical splitting. These splittings s 1 and s 2 are related by an R-matrix. In Appendix B we show how to compute this R-matrix. We conjecture: Conjecture 1.25. This R-matrix recovers all (including higher genus) Gromov-Witten invariants of X from the genus 0, 3 point invariants of X and the all-genus Gromov-Witten invariants of Σ 2 .

Outline of the paper
Section 2 defines formal TEP-structures and related notations. In section 2.2 we define semi-simple TEPstructures and interpret results of [AT19] using this language. Next in section 3 we endow the cyclic homology of an A ∞ -algebra with a TE-structure. Section 4.3 outlines properties of the Fukaya category and the cyclic open-closed map which are sufficient to prove Conjecture 1.6 in a general setting. For a Fukaya category with a single Lagrangian we then construct a cyclic open-closed map satisfying these properties in section 4.8. This relies on a structure equation for horocyclic operations, which we prove in section 5. In section 6 we study applications of Conjecture 1.6. In particular we show how Lemma 1.11 and Lemma 1.15 follow from general considerations about TE-structures. We also explain an alternative proof of Theorem 1.22. In Appendix A we provide heuristics showing how a 'standard' definition of a Fukaya category (with multiple Lagrangians) can be modified to define a Z-graded category (but at the cost of enlarging the coefficient ring). We also outline why we expect the properties of section 4.3 (which are sufficient to prove that the cyclic open-closed map is a morphism of TE-structures) to hold for this Fukaya category. In Appendix B we show there exists a unique R-matrix for the intersection of quadrics in CP 5 . Finally in Appendix C we prove a result which was missing in the literature about the orientation properties of gluing of holomorphic maps.

Acknowledgements
I could not wish for a better supervisor than Nick Sheridan. His explanations, suggestions and comments have been invaluable. I would also like to thank Sara Tukachinsky for explaining a wide variety of ideas from her series of joint papers with Jake Solomon. I am also grateful to both Sara Tukachinsky and Jake Solomon for sharing an unpublished draft chapter proving the structure equations for geodesic operators [ST19, section 3.2]. My chapter 5 is an adaptation of their proof to the case of horocyclic constraints.

Formal TEP-structures
Let K be a field of characteristic 0. Let R be a Z/2-graded commutative K-algebra.
]-module, with u of even degree and ∇ : Der K R ⊗ E → u −1 E a flat connection of even degree.
2. A formal pre-TE-structure is a formal pre-T-structure together with an extension of the connection to a flat connection ∇ : 3. A formal pre-TP-structure is a formal pre-T-structure equipped with a polarisation, i.e. a covariantly constant pairing which is R-linear, of even degree and u-sesquilinear. That is, for we have: 4. For a formal pre-TEP-structure, we require that the pairing is also covariantly constant with respect to ∇ ∂u . More precisely:

5.
If additionally E is free and finitely-generated, we drop the prefix 'pre' from T-and TE-structures. Let E = E/uE. For a TP-structure, we additionally require that the restriction of the pairing (·, ·) E : As mentioned in the introduction, we will always be taking about formal T-structures, so we will forget about the 'formal'. Additionally, we call a (pre-)TE(P)-structure with R = K a (pre-)E(P)-structure. Definition 2.3. Let E be a pre-T-structure over spec(R). An Euler-grading on E consists of an even degree K-linear map: Gr called the grading and a vector field E ∈ Der K R of even degree, called the Euler vector field, such that for f ∈ R, a ∈ E and X ∈ Der K R: [Gr, If E is a pre-TP-structure, we additionally require that (2u∂ Remark 2.4. An Euler-grading differs from a more standard definition of graded in that E is not required to admit a direct sum decomposition into graded pieces.
Definition 2.5. For Euler-graded pre-T-structures E 1 , E 2 over R with grading operators Gr 1 and Gr 2 , and Euler-vector field E 1 = E 2 , a morphism of Euler-graded pre-T-structures is a morphism of pre-T-structures F which additionally satisfies F • Gr 1 = Gr 2 • F .
Definition 2.6. Given an Euler-graded pre-T-structure, we obtain an associated pre-TE-structure by setting: A short computation shows the total connection is flat, showing this is a valid definition. As a morphism of Euler-graded pre-T-structures respects the grading and the connection, we find: Lemma 2.7. A morphism of Euler-graded pre-T-structures is a morphism of associated pre-TE-structures.
Definition 2.8. An Euler-grading on a pre-TE-structure is an Euler-grading on the underlying T-structure, such that ∇ ∂u = 1 2u Gr − 1 u ∇ E .

E-structures
Definition 2.9 (E(P)-structure). An E(P)-structure is a TE(P)-structure E over the K-algebra R = K, so that M = pt. We thus only have a connection ∇ d du : E → u −2 E. For ease of notation we will often write ∇ for ∇ d du for an E(P)-structure.
Definition 2.11. A splitting of an E-structure is a K-linear map s : E → E which splits the canonical map As E is finitely generated and free, there always exists a splitting. A choice of splitting s defines an isomorphism: Note that the sum on the right side makes sense as E is finitely generated. We can then write the connection on E as ∇ =: Call A the connection matrix and A 0 the residue. Given two splittings s 1 and s 2 we obtain an isomorphism ], we find that R 0 = Id. The splittings s 1 and s 2 are then related via: The connection matrices are related via: which shows that the residue A 0 is independent of the choice of splitting. Such a matrix series R is called an R-matrix. Usually an extra condition, symplecticity, is imposed on R. This condition is satisfied when both splittings are P-compatible. We now rephrase a theorem by Levelt [Lev75, Chapter 2] in our setup. This theorem is the first step in the Hukuhara-Levelt-Turrittin decomposition. See for example [Mal83] for a modern statement.
Theorem 2.12. Given an E-structure E there exists a unique decomposition E = w E w where the w are the eigenvalues of the residue A 0 : E → E. This decomposition satisfies: The proof in [Lev75, Chapter 2] is easily seen to apply in our situation. As we will need a specific form of the next term in the connection matrix, A 1 , we will provide a proof. The main result we need is: Lemma 2.13. Let {e j } be a basis for E such that the vectors π(e j ) ∈ E are generalised eigenvectors for the residue A 0 . Write the connection as ∇ = d du + u −2 i≥0 A i u i in this basis. Then there exists another basis {v j } for E such that the following hold: -π(v j ) = π(e j ).
-Write the connection as Proof of Lemma 2.13. Consider a new basis {P (e j )} for some invertible C[[u]]-linear map P . The new connection matrix is A = P −1 AP − P −1 dP du . If we set P = id + u m T m , we find (see [Sab07,Theorem 5.7]): and a more complicated expression for A >m . Let E w for w ∈ C be the generalised eigenspaces for A 0 , then let These are the linear maps which vanish on the diagonal blocks of A 0 . A short computation shows that the restriction of the adjoint map That is, all entries of [A 0 , T m ] and −A m which are not in the diagonal blocks of A 0 agree. We thus have A m ( E w ) ⊂ E w . We then find the T m successively, starting with m = 1. Then set P = m≥1 (id + u m T m ), noting that this product is well-defined, as for each power of u, only finitely many terms in the product contribute. Then set v j = P (e j ). This shows the first two properties. For the final property, note that A 1 = A 1 + [A 0 , T 1 ]. As we need that A 1 | Ew : E w → E w , we can choose T 1 to only have entries in the off-diagonal blocks. That is, the restriction T 1 : E w → E w vanishes. But then the same holds for [A 0 , T 1 ].
Proof of Theorem 2.12. Let E w = v j |π(v j ) ∈ E w . By construction, the E w are u 2 ∇-invariant. Uniqueness of the decomposition follows from the following lemma.
Lemma 2.14. Let f : E → E be a morphism of E-structures. Then for any choice of decomposition by eigenvalues E = ⊕ w E w by eigenvalues of A 0 , and any choice of decomposition of E by the eigenvalues of A 0 , Remark 2.15. Levelt [Lev75, Chapter 2] proves this lemma when f is an isomorphism. Our proof of the general case is very similar.
Lemma 2.14 follows directly from: Lemma 2.16. Let f : E → E be a morphism of E-structures. Assume that the residues A 0 and A 0 have no eigenvalues in common, then f = 0.
Proof. Expand f in a basis for E and E as a matrix F = i u i F i . Expand the connections ∇, ∇ as usual with connection matrices A and A . As f respects connection, we obtain the equation: Expanding in powers of u, we find As A 0 and A 0 have no eigenvalues in common, this implies F 0 = 0 (see [Sab07,Lemma 2.16]). Next, compare coefficients of u m+1 . This yields: where L(F 0 , . . . , F m−g ) denotes a linear combination of the F ≤m with vanishing constant term. By induction we can assume F 0 , . . . , F m vanish, which implies F m+1 = 0.

Semi-simple TEP-structures
In this section we will interpret results from [AT19] in the language of TEP-structures. For simplicity, let K = C.
Definition 2.17 (semi-simple E(P)-structure). An E(P)-structure is semi-simple if there exists an isomorphism of E(P)-structures E ∼ = w E −w/u . Here the values w ∈ C are allowed to occur with multiplicity. Let ξ = u 2 ∇ d du : E → E be the residue of the connection. Thus, ξ is given by multiplication by w on each The following two definitions are inspired by [AT19].
Definition 2.19 ([AT19, Definition 3.7]). Let E be an EP-structure with a specified element ω ∈ E, and s : E → E a splitting. We say the splitting is: Example 2.20. The EP-structure E −w/u admits a canonical homogeneous, P-and ω-compatible splitting given by s can (1) = 1 ∈ E. Here we have not specified the element ω ∈ E, as the splitting is ω-compatible for any choice of ω. This is because, by definition, ∇ u d du s can (a) = u −1 ws can (a) for all a ∈ E.
Example 2.21. A semi-simple E-structure E comes with a canonical splitting induced by the isomorphism Φ : E ∼ = w E −w/u and the splitting s can on each E −w/u . This splitting is independent of the choice of isomorphism Φ as any two such isomorphisms are related by an isomorphism Ψ : w E −w/u → w E −w/u and any such Ψ is necessarily independent of u. We denote this splitting by s ss (the semi-simple splitting) and note that it is homogeneous and ω-compatible, for any ω ∈ E. If E is a semi-simple EP-structure, s ss is also P-compatible.

Remark 2.22. A splitting s is homogeneous if and only if the associated connection matrix
Amorim and Tu show the following for EP-structures coming from the cyclic homology of an A ∞ -category, see [AT19, Theorem 3.10]. We state their result in our more general setup. The proof is identical.
Theorem 2.23. Let E be a semi-simple EP-structure with a specified element ω ∈ E. Then there exists a bijection between the set of homogeneous, P-and ω-compatible splittings s : E → E and the set of P-, ξ-and ω-compatible grading operators µ : E → E.
We refer the reader to [AT19, Theorem 3.10] for the details of the proof, but will say a few words about it. Given a splitting s as in the lemma, there exists a unique series The associated grading operator is then defined by µ s = [ξ, R 1 ], and one obtains the following relation on R: One then checks all the required properties hold. Conversely, given a grading operator µ : E → E, [AT19] show that there exists a unique R-matrix solving Equation (39) is an isomorphism.
-(Orthogonality) For any tangent vectors v 1 , v 2 ∈ Der C (R), we have: -(Holonomicity) For any tangent vectors v 1 , v 2 , v 3 ∈ Der C (R), we have: -(Homogeneity) There exists a constant r ∈ C such that If ζ only satisfies the Primitivity property, we will call ζ a primitive element, and call the TEP-structure H primitive if such ζ exists.
Definition 2.25. Let H be a primitive TEP-structure over R = C[[t 1 , . . . , t n ]]. Let E be the EP-structure E := H ⊗ R C, where C is an R-module under the map t i → 0. For ω ∈ E say ω is primitive if there exists a primitive element ζ ∈ H such that ζ| t=u=0 = ω.
Amorim and Tu [AT19, Theorem 4.2] also prove a bijection between primitive forms and splittings, which is a bijection originally established in [Sai83b]. We rephrase their theorem to apply to our setup. As already observed by [AT19, Remark 4.3], their proof applies to our more general setup (note that what they call a VSHS corresponds to what we call a TE-structure).
Theorem 2.26. Let H be a Euler-graded, primitive TEP-structure over R = C[[t 1 , . . . , t n ]] and let ω ∈ E be primitive. Then there exists a natural bijection between the following two sets: P := {ζ ∈ H|ζ is a primitive form with ζ| t=0,u=0 = ω}, For a semi-simple, Euler graded and primitive TEP-structure H as above, with a choice of primitive ω ∈ E, Theorems 2.23 and 2.26 thus combine to give a bijection between the set of P-, ξ-and ω-compatible grading operators µ : E → E and the set of primitive forms ζ ∈ H with ζ| t=0,u=0 = ω.
The relevance of this bijection is that given a primitive form ζ as above, Saito and Takahashi [ST08] endow Spec(R) with the structure of a Frobenius manifold. A grading operator µ on a semi-simple TEP-structure over a ring R thus gives rise to a Frobenius manifold M µ . In chapter 6 we will come back to this construction.

TE-structure on the cyclic homology of an A ∞ -algebra
In this section we will define a TE-structure on the cyclic homology of an A ∞ -algebra. All of the definitions can easily be extended to A ∞ -categories. Let S n [k] be the set of all partitions of {1, . . . k} into n ordered sets of the form (1, 2, . . . , k 1 ), (k 1 + 1, . . . , k 1 + k 2 ), . . . , (k 1 + · · · + k n−1 + 1, . . . , k 1 + · · · + k n ). Let (i : n) denote the ith set of the partition. The size of (i : n) is k i . We allow for the case k i = 0.

Hochschild invariants
Let (A, m) be an n-dimensional, cyclic, strictly unital, curved, Z/2-graded A ∞ -algebra over R. Let r ⊂ R be the maximal ideal of elements with positive valuation. Define the (reduced) Hochschild cochains of A: Also define the uncompleted (reduced) Hochschild chains of A to be: Following [CLT21, Section 3.5], we define: Definition 3.4. The completed reduced Hochschild chains and cochains are given by: Remark 3.5. In the remainder of this chapter, we will recall and define various operations on Hochschild (co)chains. For simplicity, we will often define these operations only for the uncompleted chains. They descend to operations on the completed Hochschild chains as the A ∞ -operations and the pairing are assumed to respect the valuation.
Remark 3.6. We need to be careful about the R-linearity of Hochschild cochains. For φ ∈ CC * (A) and t ∈ R this means that: Denote an element α ∈ CC * (A) by α = α 0 [α 1 | . . . |α k ], and for a subset I ⊂ {1, . . . , k}, write α I for j∈I α j . Definition 3.7. Hochschild homology is defined as: HH * (A) := H * (CC * (A), b). Here the differential b is given by: Note here that in the second sum, terms with k 2 = 0 are allowed.
Definition 3.8. The negative cyclic chain complex is given by Here the second differential B is defined by: The homology of the negative cyclic chain complex is called the negative cyclic homology, denoted HC − * (A). Hochschild cochains admit a differential too. First introduce the Gerstenhaber product, defined by: The Gerstenhaber bracket is then defined by: Finally we note that CC * (A) admits an A ∞ -structure M k , defined by [Get93]. M 1 is the differential [m, ]. We will also need the M 2 part of these operations.
. Finally the curvature, or Lie derivative, [Get93] can be written as: Observe that L m = b. An easy computation shows: Getzler furthermore defines a connection in the base directions. We extend his definition to allow for the case when R is Z/2-graded.
Definition 3.15. The Getzler-Gauss-Manin connection is defined on the chain level by: Here for a Hochschild cochain φ ∈ CC * (A) and a derivation v ∈ Der C R, the Hochschild cochain Getzler shows the connection descends to the level of cohomology and is flat. This endows HC − * (A) with a Z/2-graded pre-T-structure over Spec(R).
Sheridan proves the following holds over a field K. Nothing in their proof breaks down if we work over a general ring. We thus have: is a morphism of pre-T-structures.

u-connection
The pre-T-structure HC − * (A) has been extended to a pre-TE-structure by [KKP08]. We give another interpretation of this definition. First recall the notion of an Euler-grading on an A ∞ -algebra: Definition 3.17. An Euler-grading on an n-dimensional, strictly unital, cyclic, Z/2-graded A ∞ -algebra A consists of an Euler vector field E ∈ Der K R of even degree and an even degree map Gr : A → A such that and Furthermore, we require that Now suppose that A is Euler-graded. Consider Gr : A → A as a length-1 Hochschild cochain. Then define the operator Gr − : where is the length operator on cyclic chains.
Lemma 3.18. The grading Gr − descends to cyclic homology, and endows HC − * (A) with an Euler-graded T-structure.
We thus have: The second equality follows by Lemma 3.12. This shows Gr − descends to cyclic homology. Next, observe . This shows that holds on the chain level. Next, for v ∈ Der C R, we want to compute [Gr − , ∇ v ]. To this end, first observe that, after picking a basis for A, we have: Furthermore, a direct computation shows: We thus have: Thus, as an Euler-graded pre-T-structure naturally admits an extension to a pre-TE-structure, any Euler-graded A ∞ -algebra naturally admits a pre-TE-structure on HC − * (A). For an arbitrary A ∞ -algebra C, we will now define an Euler-graded deformation, and use this to define a u-connection on HC − * (C).
and extending s-linearly.
Lemma 3.20. Define Gr : hom(C s , C s ) by setting Gr(s k a) := ks k a for a ∈ hom(C, C). This makes C s a Z-graded algebra. In particular, by defining E = s 2 d ds ∈ Der K R[s, s −1 ], C s is an Euler-graded A ∞ -algebra. Remark 3.21. The A ∞ -algebra C s is also used in [CLT21, section 3.1] to define the connection in the u direction.
Remark 3.22. The deformation C s is canonical in the following sense: an A ∞ -morphism F : C → B induces an A ∞ -morphism F s : C s → D s given by This morphism is Euler-graded.
Define the u-connection on HC − (C) to be the restriction to s = 1 of ∇ s ∂ ∂u . One can check that indeed: where m = k (2 − k)m k . Call this the canonical u-connection associated to an A ∞ -algebra. This makes HC − * (C) into a pre-TE-structure.
Remark 3.23. In the deformation C s , s has odd degree. We can also define the R[e, e −1 ]-linear A ∞ -algebra , where e is of even degree. The operations are defined by: Here |a| is 0 if a has even degree or 1 if a has odd degree. Note that we can divide by 2 because m is Z/2-graded. This is Euler-graded with E = e∂ e , grading operator Gr(e k a) = (2k + |a|)e k a.
Lemma 3.24. Let C be an Euler-graded A ∞ -algebra over R with grading Gr and Euler-vector field E. Then the canonical u-connection agrees up to homotopy with the u-connection coming from the Euler-grading.
Proof. Choose an R-basis for all morphism spaces. This defines an operator Let ∇ d du denote the canonical u-connection. The u-connection defined using the Euler grading is given by Using the definition of deg we can rewrite this as: The properties of E and Gr show: By the Cartan homotopy formula 3.14, we thus have: In particular, if we define a u-connection ∇ d du on HC − (C) by restricting the connection ∇ e coming from the Euler-grading on C e to e = 1, then ∇ d du agrees with the canonical u-connection.
Proof. Let F s : C s → D s be the induced Euler-graded morphism. Now apply Theorem 3.16 to F s to find that it respects ∇ GGM E up to homotopy. As we also have that [F s , Gr] = 0, we find that F s respects ∇ ∂u up to homotopy. Restriction to s = 1 shows the result.
The following is a rephrasing of a result by Amorim and Tu, [AT19, Corollary 3.8].
Theorem 3.26. If A is an n-dimensional, strictly unital, cyclic, Z/2-graded, smooth and finite dimensional A ∞ -algebra with HH * (A) semi-simple, then HC − * (A) is a semi-simple TEP-structure.
We finish this section with a comparison between the E-structures associated to a weakly curved A ∞algebra and its uncurved associated A ∞ -algebra. We use this to conclude that the eigenvalue decomposition of the negative cyclic homology is trivial. For simplicity, here we assume A is a C-linear A ∞ -algebra. Suppose that (A, m) is strictly unital and weakly curved, i.e. m 0 = w · e for some w ∈ C. From (A, m) we can then obtain an uncurved A ∞ -algebra by setting m k = m k for k ≥ 1, and m 0 = 0.
Furthermore, there exist an isomorphism of pre-E-structures: Here on both sides the connection ∇ denotes the canonical connection defined above.
Proof. The Hochschild differentials satisfy b = b as we are working with reduced chains. B = B by definition. m = m + 2w · e, and then from the fact that e is a strict unit, we get that b 1,1 (m , ) = b 1,1 (m, ) + 2w · Id. Furthermore Corollary 3.29. Let A be a C-linear, strictly unital and weakly curved A ∞ -algebra with curvature w · e. Assume the Hodge-de Rham spectral sequence of A degenerates. Then, in the eigenvalue decomposition of Theorem 2.12, the E-structure HC − * (A) has just a single summand associated to the eigenvalue w.

Cyclic open-closed map respects connections 4.1 Coefficient rings
Consider the Novikov ring ] be the ring of formal functions on the completion of U at the origin. Explicitly, let {v i } i∈I be a homogeneous basis for U , and {v * i } i∈I the dual basis for U * . Let {t i } i∈I be formal variables of degree −|v i |, then we have an isomorphism: These are independent of the chosen basis.
Then set: Following [ST16], define the valuation ζ Q : Let To account for gradings, we will also make use of the 'universal Novikov ring': Λ e := Λ[e, e −1 ], where e has degree 2. Let Q e U be defined using Λ e instead of Λ.
Remark 4.1. A lot of our work is based on [ST16]. They use a different Novikov ring, more commonly used in Gromov-Witten theory. Instead of taking series in Q R they take series with terms T β for β ∈ H 2 (X, L). For them the monomial T β has degree µ(β), where µ : H 2 (X, L) → Z is the Maslov index. The graded map T β → Q ω(β) e µ(β)/2 , allows us to compare their Novikov ring with the universal Novikov ring Λ e . Note that µ(β) ∈ 2Z as we assume our Lagrangian is orientable.

Quantum TE-structure
Let (X, ω) be a symplectic manifold and let U ⊂ H * (X; C) be a graded C-vector subspace. For any ring R, let A * (X; R) denote the space of differential forms on X with coefficients in R.
Let γ be a bulk-deformation parameter over U . We now consider the quantum cohomology QH * (X; Q e U ). As a vector space this is just H * (X; Q e U ), but the product is given by the bulk-deformed quantum cup product η 1 γ η 2 . A general reference for the construction of the quantum cup product is [MS04], however, our coefficient ring includes the universal Novikov parameter e, so we sketch how to modify the definition. See also Definition 4.32 for a construction in our specific setup. Recall from [MS17] that the quantum cup product is defined as a sum over curve classes β ∈ H 2 (X): Here (η 1 η 2 ) β is defined by the equation where GW β 0,3 denotes the genus 0, 3 point Gromov-Witten invariant in curve class β. One can then extend this definition to take into account bulk deformations γ, to obtain the product γ on quantum cohomology QH * (X; Q U ). We then define the product on QH * (X; Q e U ) by: where c 1 = c 1 (T X) is the first Chern class.
For v ∈ Der Λ Q U the quantum connection is defined by: We now wish to extend the quantum T-structure to be defined over Q e U ⊃ Λ. To this end observe that Extend the connection by setting: Lemma 4.5. These above definitions make QH * (X; w] for v, w ∈ Der Λ Q U is standard, so we will not do it here. Instead we verify that [∇ v , ∇ e∂e] = 0. The divisor equation for closed Gromov-Witten invariants shows that for v ∈ Der Λ Q U we have: We also find that: A direct verification then shows that [∇ e∂e , ∇ v ] = 0 holds.
Define the Euler vector field by E = e∂ e + E U . Define the grading operator by taking into account the cohomological degrees, the grading on the coefficient rings and the degree of u, but with the grading shifted down by n so that, for η ∈ H * (X; C) and f ∈ Q e U [[u]], we have: where µ : H p (X; Q e U ) → H p (X; Q e U ) is given by µ(η) = p−n 2 η. A short computation then shows: Lemma 4.6. The above definitions make QH * (X; Q e U ) [[u]] into an Euler-graded T-structure. As the quantum T-structure is Euler-graded, Definition 2.6 endows it with a connection in the u-direction: This makes QH * (X; Q e U )[[u]] a TE-structure. Writing out the definitions of Gr − and E yields the formula: to obtain a TE-structure over Q e U ⊃ C. We do not use this connection. Remark 4.8. We can use the same formula 123 to define a TE-structure on QH * (X; Q U ) [[u]]. There then is a natural isomorphism of TE-structures over Q U ⊃ Λ: Here Λ is considered as a Λ e module via the homomorphism Λ e → Λ given by evaluation at e = 1.
There are alternative definitions of the quantum connections on QH * (X; Q e U ) [[u]] given by changing the signs: The alternative connection in the u-direction is given by Writing the formulae out we find: Define the Poincaré pairing by η 1 , η 2 X = X η 1 ∧ η 2 . Now extend the Poincaré pairing u-linearly to a pairing We then have: ]. Remark 4.9. It is customary to extend the Poincaré pairing sesquilinearly to the quantum TE-structure, to obtain a TEP-structure where the polarisation can be matched up with the higher residue pairing on cyclic homology. However, since we don't mention the polarisation in this paper, we use the u-linear extension as it simplifies the proof that the cyclic open-closed map is a morphism of TE-structures.

Outline of proof of Theorem 1.7
In this section we give an outline of the proof of Theorem 1.7. We will state sufficient conditions which imply that the cyclic open-closed map respects the connection ∇ ∂u . We state these conditions in such a way that they should be easy to generalise to different geometric setups.
Let U ⊂ H * (X; C) be a graded vector space, and γ a bulk-deformation parameter over U satisfying Assumption 4.3. Let L ⊂ X be a Lagrangian submanifold. We define an Euler-graded A ∞ -algebra CF * (L, L)[e] over Q e := Q U [e, e −1 ] in Section 4.7. The Euler vector field is given by E = e∂ e + E U , where E U is as in Section 4.1 and e is of degree 2. The Floer cochain complex CF * (L, L) is then defined by restricting to e = 1: CF * (L, L) := CF * (L, L)[e] ⊗ Q e U Q U . More generally, suppose there exists a bulk-deformed Fukaya category F uk t (X) defined over Q U . By using e to take into account the Maslov index of holomorphic disks, it should be possible to construct an Euler-graded Fukaya category F uk t (X)[e] over Q e . In Appendix A, we construct such an Euler-graded Fukaya category geometrically. In the appendix, U will be the 1-dimensional vector-space spanned by the first Chern class.
Assume there exists a cyclic open-closed map which is the restriction to e = 1 of a map In Section 4.8.3, we will construct a cyclic open-closed map by defining a chain level pairing (which we call the cyclic open-closed pairing) We show that it satisfies so that it descends to a pairing We then apply Poincaré duality to the QH * (X; Q e ) factor to obtain the map OC − e . It is uniquely determined by the property: such that for all η and α we have: 3. For any η and α we have: Here, for any v ∈ Der Λ Q e U , we define: 4. OC − e respects the Euler-grading on cyclic invariants: ]. Remark 4.12. To define the connections ∇ v and ∇ e∂e on cyclic homology, we need to choose a basis for all of the morphism spaces in F uk t (X). Assumption 2 is required to hold with respect to the same bases as used to define the connections. The same holds for assumption 3. On quantum cohomology, we take the derivatives with respect to the standard constant basis (i.e. one in H * (X; C)).
Theorem 4.13. Suppose assumptions 4.10 hold, then OC − e , and hence OC − , respects ∇ ∂u on homology. Proof. First we will show that Assumptions 1, 2 and 3 show that OC e respects the connection ∇ e∂e . Applying assumption 1, with v = Y yields: using Assumption 2, this gives: By the Cartan homotopy formula (Proposition 3.14) we can rewrite this as: where η, G e∂e (α) = η, G Y (α) + (−1) |η| η, OC − e (i{φ}(α)) . Then apply Assumption 3 to obtain: This shows that: The last equation implies OC e − respects ∇ e∂e on homology. We will spell this out. By the properties of the Poincaré pairing (Equation (130)): The above shows that on homology: which shows that OC − e (∇ e∂e (α)) = ∇ e∂e (OC − e (α)) on homology. As E = e∂ e + E U , we combine this with Assumption 1, applied to v = E U to find that the open-closed map respects ∇ E . Then, as ∇ ∂u = Gr cyc 2u − u −1 ∇ E , and using the fact that OC − e respects the Euler-grading (Assumption 4), we find that OC − e respects the connection ∇ ∂u . The statement about OC − follows by restriction to e = 1.

Regularity assumptions
Let X be a 2n-dimensional symplectic manifold and J be an ω-tame almost complex structure on X. Let L ⊂ X be an oriented Lagrangian equipped with a Λ * -local system and a relative spin structure s. For us a relative spin structure comes with a choice of element w s ∈ H 2 (X; Z/2) such that w s | L = w 2 (T L) ∈ H 2 (L; Z/2).
For l ≥ 0, let M l+1 (β) be the moduli space of stable J-holomorphic spheres with l + 1 marked points in homology class β ∈ H 2 (X, Z). Let ev β j : M l+1 (β) → X (147) be the evaluation map at the j'th marked point. For k ≥ −1, l ≥ 0, let M k+1,l (β) be the moduli space of J-holomorphic stable maps (D, S 1 ) → (X, L) in homology class β ∈ H 2 (X, L) with one boundary component, k + 1 anti-clockwise ordered boundary marked points, and l interior marked points. Let be the evaluation maps at the i'th boundary and j'th interior marked points respectively. The relative spin structure determines an orientation on the moduli spaces M k+1,l (β), see [Fuk+09,Chapter 8].
We will also need a moduli space of disks with a horocyclic constraint. Recall that a horocycle in a disk is given by a circle tangent to the boundary. These moduli spaces are similar to the ones used in [ST19, Chapter 3], where some of the marked points are constrained to lie on a geodesic in D. Our definition is entirely analogous, except that we replace 'geodesic' with 'horocycle'. Let the smooth locus of M k+1,l;⊥i (β) ⊂ M k+1,l (β) be the subset defined by requiring the first and second interior marked points w 1 and w 2 to lie at −t and t respectively for t ∈ (−1, 1) and fixing the i'th boundary point z i at −i. Equivalently, we require that z i , w 1 , w 2 lie on a horocycle in anti-clockwise ordering. This moduli space also appeared in [Gan12], where it was used to show that the closed-open map is an algebra homomorphism.
We now give a more formal definition of the moduli space M k+1,l;⊥0 (β) as a fibre product of known spaces. Consider the forgetful map M k+1,l (β) → M 1,1 (β) = D 2 , only remembering the zeroth boundary marked point, and the first interior marked point. Here the identification M 1,1 (β) ∼ = D 2 is achieved by using an automorphism of the disk to map the boundary marked point to −i, and the interior marked point to 0. Consider the inclusion I → D 2 given by the arc of the horocycle through −i and 0 with negative real part. This is a circle of radius 1 2 centred at − i 2 . The condition on the order of the marked points means that second interior lies on the semi-circle with negative real part. We then define: Take the orientation on I to be the positive orientation, so that ∂I = {1} − {0}. The orientation on M k+1,l;⊥0 (β) is then defined by the fibre-product orientation, as in [ST20, Section 2.2].
We assume the following: Assumptions 4.14.
We will now show these assumptions hold in the following setup: Lemma 4.15. The above assumptions hold for L ⊂ X a Lagrangian and a complex structure J with the following properties: -J is integrable.
-There exists a Lie group G X acting J-holomorphically and transitively on X.
-There exist a Lie subgroup G L ⊂ G X whose action restricts to a transitive action on L.
Proof. This argument is the same as [ST19, Section 1.3.12], but for horocyclic rather than geodesic constraints. For assumptions 4.14.1 and 4.14.2, [MS04,Proposition 7.4.3] show that if the above properties hold, all stable holomorphic maps in M l+1 (β) are regular, it then follows from [RRS08] that this space is a smooth orbifold with corners. As G X acts on M l+1 (β), G X acts transitively on X, and ev 0 is equivariant with respect to this action, ev 0 is a proper submersion. Solomon and Tukachinsky show assumptions 4.14.3 and 4.14.4 hold in this situation by adapting the arguments for closed Riemann surfaces to Riemann surfaces with boundary (see [ST16, Remark 1.6]).

q-operations
This section follows [ST16] and [ST19] closely. Let L ⊂ X be as in the previous section. Let denote the monodromy representation of the local system on L.
[ST19] also define closed operations: by with special cases: q ST,β0 ∅,1 We use these operations, without any sign change, so that q ∅,l = q ST ∅,l . The quantum product : A * (X) ⊗ A * (X) → A * (X), is then given by We also define new operations coming from the moduli spaces with horocyclic constraints M k+1,l,⊥i (β). We first define these using sign conventions similar to [ST16].
For all of the above q β operations, set Here µ : H 2 (X, L) → Z is the Maslov-class, and e is of degree 2. We thus have operations The following lemma then follows directly from the degree property.
The new result we prove concerns the boundary of the moduli spaces M k+1,l;⊥0 (X, β). The proof is given in chapter 5.
Proof. Let p i : M k+1,l;⊥i (β) → M k,l (β) be the map given by forgetting the boundary marked point z i and the horocyclic constraint. p i is injective, as the location of the boundary marked point z i can be reconstructed from the interior marked points w 1 , w 2 . Fixing the interior marked point w 1 and the boundary marked points z j for j = i, we see that as z i moves between the two adjacent boundary marked points z j and z j+1 , the point w 2 sweeps out a lunar arc between the two horocycles through w 1 and z j , and through w 1 and z j+1 (see Figure  1). As these lunar arcs cover the entire unit disk we see that the image of k i=0 p i : M k+1,l;⊥i (β) → M k,l (β) is an open dense subset. The sign follows from a computation entirely analogous to [ST19, Lemma 3.10]. As the sign in our horocyclic operations q ST k,l,⊥i differ from the one [ST19] use for their geodesic operations q ST k,l;a,e , we need to modify the signs in their results to account for this difference. This is an easy verification. The second equality follows from a cyclic symmetry property of the q ⊥ operations, which is the direct analogue of Proposition 4.20.
Remark 4.33. When (γ, b) is a bulk-deformation pair, the degree assumptions on γ and b imply that the properties 4.20 -4.29, with the exception of the energy-zero property 4.25, all hold for the operations q b,γ with the same signs as before.
We will need the following lemma later on; it follows from an easy verification of signs.
Lemma 4.34. For v ∈ Der Λ e Q e , α ∈ A * (L; Q e ) ⊗k and η ∈ A * (X; Q e ) ⊗l all of homogeneous degrees, we have: Here A similar lemma holds for v = e∂ e , here one gets an additional term, as the q operations depend on e. Also note that ∂ e (γ) = 0 by definition of a bulk-parameter. First, define operations weighted by the Maslov index µ: q b,γ k,l = β µ(β)q b,γ,β k,l . We then have:

Fukaya A ∞ -algebra
Let U ⊂ H * (X; C) and (γ, b) be a bulk-deformation pair over U . Assume γ satisfies Assumption 4.3, fixing the derivation Y ∈ Der Λ e Q e U such that [Y (γ)] = c 1 . For ease of notation, write Q = Q U . Solomon and Tukachinsky [ST16, Theorem 1] construct an A ∞ -algebra A ST using the operations q ST k,0 . We have different sign conventions for our operations q, but the following still holds. k,0 , , L , 1). It follows directly from the properties of the q operations that this forms an n-dimensional, strictly unital and cyclic A ∞ -algebra. It follows from the degree property 4.21, and the definition that |e| = 2 that this A ∞ -algebra is Euler-graded with Euler vector field E = e∂ e + E U . The grading operator is defined by Gr(f α) = (|f | + |α|)f α for f ∈ Q e and α ∈ A * (L). Furthermore, this A ∞ -algebra is (possibly) curved. The valuation ζ A is induced by the valuation ζ Q , defined in (111).  Whenever we have to pick a basis for A in order to compute derivatives, we will always pick a constant basis, i.e. one in A * (L; C).
Recall the connection We will now define a connection ∇ which agrees with ∇ GGM up to homotopy. First, for v ∈ Der C Q e define the length zero Hochschild cochain φ v := v(b) ∈ A b,γ . Also let m v (α) := q b,γ k,1 (α, v(γ)). Lemma 4.34 then shows: We then define the modified connection.
Corollary 4.40. The connection defined by: is equal to ∇ on homology.
Proof. Let the homotopy be given by H = i{φv} u . The Cartan homotopy formula 3.14 then shows: Also, set m k = q b,γ k,0 . Lemma 4.35 then shows:

Closed-open and open-closed maps
In this section we will construct a closed-open map and prove it is an algebra homomorphism (a result first shown by [Fuk+10] and [Gan12]). We will then construct an open-closed map, and show it is a morphism of QH * (X) modules (see also [Gan12]). We will then construct a cyclic open-closed map, which was also done in [Gan19]. Finally we show that the cyclic open-closed map is a morphism of T-structures, using an argument due to [GPS]. We then show Assumptions 4.10 hold for the cyclic open-closed map in order to conclude that it respects the connection in the u-direction.
Let CO e : H * (X; Q e ) → HH * (A) be the induced map on cohomology. Next up we will prove the following, which is originally due to [Fuk+10] and [Gan12] in different setups: To this end, we first define: Definition 4.45. Let the homotopy operator H : A * (X; Q e ) ⊗2 → CC * (A) be given by Unitality follows from the fundamental class property 4.26. The following lemma immediately implies that the closed-open map respects the product.
We conclude this section with the following observations: Lemma 4.47. CO e (c 1 ) = m/2.
Furthermore, by definition of m v we have: Lemma 4.48. For v ∈ Der C Q e we have: CO e (v(γ)) = m v . It follows that CO e (E W (γ)) = m E W .

The open-closed map
The open-closed map will take the form Here the pairing (·, ·) is as in Equation (63).
We will show (Lemma 4.52) that the open-closed pairing descends to homology, so that the following makes sense. Remark 4.51. It might be more aesthetic to define the open-closed map directly. Suppose we were able to define operations p k : CC * (A) → A * (X) given by: where now the push forward is along the interior evaluation evi 1 : M k+1,1 → X. This approach is taken by [Fuk+09]. In the present setup the push-forward along interior evaluation is not well defined, as evi 1 : M k+1,l (β) → X need not be a submersion. Instead, we effectively define p k as a chain map to distributions on X.
Since the closed-open map is a chain map (Lemma 4.43), the following is immediate.
The open-closed pairing thus descends to (co)homology.
Ganatra [Gan12] shows the closed-open map makes HH * (A) into a QH * (X)-module. We prove this in our setup.
The following lemma follows directly from Lemma 4.46.
where G is given by (the u-linear extension of) Definition 222. We then have: Lemma 4.60. The pairing G v satisfies: In order to prove this, we first show the following: Proof. Using Lemma 4.29, we find that: We also write out the other terms.
(238) Then, apply Lemma 4.34 to compute v(q b,γ k,1 ( α; η)). Keeping track of all the signs shows the result. Proof of Lemma 4.60. As all the terms in the above equation are u-linear, we may assume α and η are independent of u, and then prove this order by order in u.
To verify the u −1 term, apply Lemma 4.55 with γ 1 = η and γ 2 = v(γ) and use Lemma 4.48 to compute CO e (v(γ)) = m v . Equality of the u 0 terms is shown by Lemma 4.61.
We next show that the Assumptions 4.10 hold in our setup, so that the open-closed map respects uconnections. Assumption 4.10(1) is Lemma 4.60. Assumption 4.10(2) is lemma 4.42 with φ = φ e∂e − φ Y . Assumption 4.10(3) holds in our setup: Lemma 4.62. For any η and α we have: Where φ is as above.
Proof. First note that a computation similar to Lemma 4.61 shows that (242) We thus find that: Similarly, we have: The last equality follows from the divisor property 4.24. The result follows.
Finally we show that assumption 4.10(4) holds: Lemma 4.63. We have: We then compute We thus have |OC − e (α)| = 1 + (α) + n. As the grading Gr − on QH * (X; Q e U ) [[u]] is shifted down by n compared to the cohomological grading, we have

Analysis of boundary of horocyclic moduli space
The goal of this section is to prove Proposition 4.28. This chapter follows the method of proof explained to the author in an unpublished draft by Jake Solomon and Sara Tukachinsky. We prove the following result for the operations q ST k,l,⊥ , which were defined using the sign convention similar to [ST16].
Proposition 5.1. Here Proposition 4.28 then follows from the above by a direct verification of signs. In the following, we prove Proposition 5.1 assuming that the Λ * -local system is trivial. The general result then follows easily.
Recall that the orientation on M k+1,l;⊥0 (β) is defined by the fibre-product orientation, as defined in [ST20, Section 2.2]. We take the orientation on I to be the positive orientation, so that ∂I = {1} − {0}. The boundary is then identified as: We now further decompose each of the terms in the boundary. For each, we identity them with a fibre product of other moduli spaces (both with and without horocyclic constraints).

Signs of boundary components
In this section we identify boundary components of the moduli spaces with fibre products of different moduli spaces. We compute the difference in orientation between the induced orientation on the boundary components, and the fibre product orientation. First we consider the boundary components coming from I × D 2 ∂M k+1,l (β). Let k = k 1 + k 2 + k 3 and write M 1 := M k1+k3+2,l (β 1 ), M 2 := M k2+1,l (β 2 ). Finally write M j,⊥ = I × D 2 M j . Let B ⊥ be a boundary component where a disk bubbles off at the (k 1 + 1)-th boundary point, with k 2 of the boundary marked points and the interior marked points labelled by J. The boundary I × D 2 ∂M k+1,l (β) can be decomposed into two components, B ⊥,1 , where the bubbling is not at the zeroth marked point and B ⊥,2 , where the bubbling is at the zeroth marked point.
Lemma 5.2. There exists diffeomorphisms: The maps φ j change the orientation by sign(φ j ), where: The proof of this lemma uses: changes orientation by the sign (−1) δ1 , with Proof of Lemma 5.2. We can decompose φ 1 as Hereθ is the map induced by θ from the Proposition 5.
The superscript main here denotes that this is the top-dimensional stratum of the moduli space. The moduli spaces M main (β) are oriented using the relative spin structure as in [Fuk+09,Chapter 8]. Adding marked points and quotienting by Aut(D 2 ) gives the moduli spaces M main k+1,l (β) as open subsets Here we need to be careful about the ordering in (S 1 ) k+1 . We stick to the convention in [Fuk+09], so that (S 1 ) k+1 = S 1 0 × S 1 1 × · · · × S 1 k . Here S 1 i is the circle corresponding to the i'th boundary marked point. The orientation of a quotient by a Lie group is defined as in [Fuk+09]. This means that the orientation on M k+1,l (β) is such that there exists an orientation preserving local diffeomorphism: Lemma 5.4. Let k ≥ 1 and l ≥ 1. Fixing the 0th boundary marked point at −i, and the first interior marked point at 0 defines a local diffeomorphism: which changes orientation by (−1) k .
Proof. Recall that in [Fuk+09] the orientation on Aut(D 2 ) is given by considering the local diffeomorphism: for three points z 0 , z 1 , z 2 ⊂ S 1 in counter-clockwise ordering. By definition, this map is orientation preserving. One can check that the map is also orientation preserving. Now multiply both sides by M main (β) × (S 1 ) k × (D 2 ) l−1 on the left, and commute the various terms through to obtain a local diffeomorphism: The sign (−1) k here comes from the change in ordering from (S 1 ) k × S 1 0 ∼ = (−1) k (S 1 ) k+1 . Finally, apply Equation (275) and cancel the factor Aut(D 2 ) to obtain the result.
Lemma 5.6. Fixing the first three marked points at 0, 1, ∞ gives an orientation preserving local diffeomorphism: We now want to study the boundary components ∂I × D 2 M k+1,l (β). Observe that: First we look at the case where the two interior marked points collide. This corresponds to {0} × D 2 M k+1,l (β). Let B ⊥,3 be a boundary component where the interior marked points labelled by I bubble off on a sphere. Note that 1, 2 ∈ I. Together with the output marked point on the sphere, this gives at least 3 marked points. For gluing the moduli spaces of holomorphic maps M main ∅ (β 1 ) and M main (β 2 ), we use the following: Proposition 5.7 (Lemma C.1). The gluing map is a local diffeomorphism which changes orientation by (−1) ws(β1) .
Remark 5.8. This proposition is implicit in [ST19] and the statement was communicated to the author by Sara Tukachinsky. See also [GZ17, Remark 2.7]. As far as the author is aware, the proof of this statement has not appeared in any literature before. We thus prove it in Appendix C.
We will compute the change in orientation locally at (v, u). By definition of the fibre product orientation, we have: by Lemma 5.4 we have: Now use Proposition 5.7 to rewrite this as: By definition of the fibre product orientation, this is isomorphic to Next, we rearrange the terms, to obtain: By Lemmas 5.4 and 5.6, this is isomorphic to: Again, by definition of the fibre product orientation, this is isomorphic to: Thus, as T 0 D ∼ = C, and cancelling the terms T x X, we obtain: The extra change in sign then comes from Equation (282).
Next we consider the case when one of the horocyclicly constrained points collides with the boundary marked point. This corresponds to {1} × D 2 M k+1,l (β). Here two disks bubble off on either side of the disk. Let B ⊥,4 be this boundary component. We show: Proposition 5.10. The map changes orientation with sign(φ 4 ) = k 4 (k 1 + k 2 + k 3 ) + k 2 (k 3 + k 5 ) + k 3 .
Proof of 5.10. Let u = (u 1 , u 2 , u 3 ) ∈ B ⊥,4 be a stable map. For simplicity, write M 1 = M k1+k3+k5+3,l1 , M 2 = M k2+1,l2 (β 2 ) and M k4+1,l3 (β 3 ). We first note that by definition of the fibre product orientation: We then use Lemma 5.4 to write: which, by Proposition 5.11 is isomorphic to: By applying the definition of the fibre product orientation twice, this is isomorphic to: Commuting the various terms through, noting that C is even dimensional, and R is odd dimensional, this gives: where A = k 4 (k 1 + k 2 + k 3 ) + n(k 1 + k 3 + k 5 ) + k 2 (k 3 + k 5 ). Then apply Lemmas 5.4 and 5.5 to find: where B = k 1 + k 2 + k 4 + k 5 . Finally, apply the definition of the fibre product orientation twice to obtain: The result then follows by cancelling the factors T L and noting that T 1 D 2 ∼ = C.

Stokes' theorem and push-forward
The next step is to apply Stokes' theorem for the push-forward of differential forms.
Recall the following facts about the push-forward of differential forms, see [ST20]: Lemma 5.13.
1. Let f : M → N be a proper submersion, α ∈ A * (N ), β ∈ A * (M ). Then: be a pull-back diagram of smooth maps, where g and f are proper submersions. Let α ∈ A * (P ). Then: Similarly, if β ∈ A * (M ), then: To obtain the structure equations for the q ST ⊥ operations, we will apply Stokes' theorem with M = M k+1,l,⊥0 (β), N = L, f = evb 0 and ξ = i=1 evi * γ i ∧ j=1 evb * j α j . When it is clear which evaluation maps are used, we will simply write evi * γ for i=1 evi * γ i and similarly for the boundary evaluations.
The first term in Stokes' theorem is: by definition of the q operations, this equals: by expanding the signs ζ ⊥ , we find: The second term in Stokes' theorem reads: 1,0 (q ST,β k,l,⊥0 (α, γ)).

Proof of Proposition 5.1
The last step of the proof is to combine the terms coming from B ⊥,1 with term 312 coming from (evb 0 ) * (dξ). Because the disk bubbles in B ⊥,1 must be stable, there are no disks contributing with β 2 = β 0 . These contributions are exactly provided by (evb 0 ) * (dξ). The same holds for B ⊥,2 and the terms coming from d(evb 0 ) * ξ. We then sum all the terms together with those from B ⊥,3 and B ⊥,4 . Finally multiply by (−1) n+|γ|+ζ(α)+ (α) to get Proposition 5.1.
Here is the quantum cup product, and µ : QH * (X) → QH * (X) is the grading operator with µ(α) = p−n 2 α for α ∈ QH p (X). The residue of the connection is given by c 1 . The canonical map π : QH * (X) [[u]] → QH * (X) is given by evaluation at u = 0. We will often write ∇ for ∇ d du .
Definition 6.2. Quantum cohomology admits a canonically defined splitting, given by: As this is the splitting relevant for Gromov-Witten theory (see e.g. [Gro11]), we call this the Gromov-Witten splitting.
Decompose quantum cohomology as a direct sum of generalised eigenspaces of c 1 Proposition 2.12 shows that we can extend this decomposition by eigenvalues of c 1 to the E-structure QH * (X) [[u]].
which is compatible with the connection: and respects the eigenvalue decomposition of QH * (X): The R-matrix then changes bases to v i = R(e i ), in which the connection can be expanded as We thus obtain the recursive relation for R: Thus QH * (S 2 ) ∼ =
The R-matrix in the basis (v, w) is thus: which indeed agrees with the R-matrix computed in [AT19, Example 5.4] for the cyclic homology of F uk(S 2 ).
We now rephrase [AT19, Theorem 5.9] and provide an alternative proof. The proof in [AT19] uses the closed-open map and the Dubrovin-Teleman reconstruction theorem [Tel12]. Our proof instead uses the cyclic open-closed map and assumes Conjecture 1.6. In particular, it does not rely on the Dubrovin-Teleman reconstruction theorem.
Theorem 6.11. Let X be a symplectic manifold such that OC : HH * (F uk(X)) → QH * (X) is an isomorphism and HH * (F uk(X)) is semi-simple. Let µ OC = OC −1 • µ • OC : HH * (F uk(X)) → HH * (F uk(X)) be the pull-back of the grading operator µ on QH * (X). Then the Frobenius manifold M µ OC associated to µ OC (see Section 2.2) is isomorphic to the big quantum cohomology of X. Here F uk(X) denotes a non-bulk deformed Fukaya category defined over Λ (or C in the monotone case).
Let F uk t (X) denote the bulk-deformed Fukaya category. As CO is an isomorphism, this is a versal deformation of F uk(X), and can thus be extracted from the categorical data of F uk(X). Now apply the bijection between grading operators and primitive forms (Corollary 2.28) to the TEPstructure QH * (X; R) [[u]]. A short check shows that the Frobenius manifold associated to the grading operator µ and the primitive ω = 1 ∈ QH * (X) is indeed the big quantum cohomology ring QH * (X; R).
Amorim and Tu [AT19, Corollary 3.8] show that as HH * (F uk(X)) is semi-simple, HC − (F uk t (X)) is a semi-simple TEP-structure. The grading operator µ OC on HH * (F uk(X)) is pulled back from the grading operator on QH * (X). The primitive element ω ∈ HH * (F uk(X)) is defined as OC −1 (1). Now consider the bulk deformed cyclic open-closed map By conjecture 1.6, this is an isomorphism of TEP-structures. Furthermore, the cyclic open-closed map:

Example: intersection of quadrics
We will now give an example where, even though the quantum cohomology is not semi-simple, it is still possible to construct an R-matrix. Let X be a complete intersection of two quadric hypersurfaces in CP 5 , which is a monotone symplectic manifold. The eigenvalue decomposition of the Fukuya category is as follows: Smith proves an equivalence: Theorem 6.12 ([Smi12, Theorem 1.1]). D π F uk(X) 0 ∼ = D π F uk(Σ 2 ), for Σ 2 a genus 2 surface.

A Euler-grading on Fukaya category
In this appendix we will explain how a 'standard' definition of the Fukaya category (see for example [Sei08]), can be adapted to define an Euler-graded A ∞ -category. We will also show that bulk-deformations by c 1 are unobstructed. We will thus construct a Fukaya category satisfying the properties required in section 4.3. We will show why Assumption 4.10(2) holds in this setup. We already verified these assumptions for the case of a Fukaya category with a single Lagrangian, but here we work with a Fukaya category with multiple objects.

A.1 Euler-grading
Let X be a symplectic manifold. Let LX → X denote the Grassmanian bundle of oriented Lagrangian subspaces of T X. Note that an oriented Lagrangian L comes with a canonical section s L : L → LX.
Let e have degree 2. We then define a Fukaya category over Λ[e, e −1 ], denoted by F uk(X)[e], as follows. Objects in F uk(X)[e] are given by oriented, relatively spin Lagrangian submanifolds L ⊂ X. For any two (transverse) oriented Lagrangians L 1 , L 2 and p ∈ L 1 ∩ L 2 , denote by L(L 1 , L 2 , p) the set of homotopy classes of paths p : [0, 1] → L p X with p(0) = T p L 1 and p(1) = T p L 2 . Then define: As in [Sei08, Section 11g], there is a grading on CF * (L 1 , L 2 ) given by the Maslov-index for the pair of paths ( p, p(1)). The Λ[e, e −1 ]-module structure is given by defining e · p to be the homotopy class of paths which have Maslov-index Ind(e · p) = Ind( p) + 2.
In the 'standard' Fukaya category, the product operations are defined by counting holomorphic disks as follows. Let L 0 , . . . , L k be pairwise transversely intersecting Lagrangians. Choose points p 0 , . . . , p k , with p i ∈ L i−1 ∩ L i , where L −1 denotes L k . Let z 0 , . . . , z k ∈ ∂D be marked points. Then the coefficient of p 0 in m k (p 1 , . . . , p k ) is given by counting the number of holomorphic maps u : D \ {z 0 , . . . z k } → X, which extend to continuous maps u : D → X with marked points u(z i ) = p i . Furthermore the boundary in between z i and z i+1 , ∂ i D is required to satisfy u(∂ i D) ⊂ L i . Let Given lifts p 1 , . . . , p k , where p i ∈ L(L i−1 , L i , p i ), and a holomorphic disk u as above. We explain how to determine a lift p 0 ∈ L(L 0 , L k , p 0 ). To this end, concatenate the paths p i together with the paths s Li (u| ∂iD ), to obtain a path γ ∈ LX. This is a path starting at T p0 L 0 and finishing at T p0 L k . As the bundle u * T X → D is trivial, we obtain a projection L| im(u) X → L p0 X. The projection of γ defines the lift p 0 ∈ L(L 0 , L k , p 0 ). In other words, we have [γ • ( p 0 ) −1 ] = 0 ∈ π 1 (LX, T p0 L 0 ). So the Maslov index of the loop γ • ( p 0 ) −1 vanishes. Alternatively, there exists a lift u : D \ {z 0 , . . . , z k } → LX covering u, which by including small extra chords γ i at the punctures z i extends to a continuous map u : D → L with u • γ i = p i for i = 0, . . . , k.
The above procedure determines the lift p 0 as a function of p 1 , . . . , p k and the class of the disk [u]. We then define: Lemma A.1. The above definition endows F uk(X)[e] with the structure of a Z-graded A ∞ -category. The grading operator is given by Gr( p) := Ind( p) p.
Proof. The fact that the product operations define an A ∞ -structure follows directly from the verification for F uk(X). By (one of the many) definition(s) of the Maslov index, we have Ind(u) = Ind( p 0 ) − i=1 Ind( p i ) so that the product satisfies [Gr, m e k ] = (2 − k)m e k as required.
A.2 Bulk-deforming by c 1 Let U be a 1-dimensional vector space, with grading 2. We thus identify Q U := Λ[[t]] for a formal parameter t of degree 0. We will now construct a Q e U -linear category F uk t [e]. One should think of this as bulk-deforming F uk(X)[e] by γ = tc 1 .
Let SX := S 1 (Λ n C T )X π − → X be the circle bundle associated to the top (complex) exterior power of the tangent bundle of X. By [BT82, Chapter 11] there exist a global angular form θ ∈ Ω 1 (SX) such that dθ = π * c 1 , where we have picked a representative c 1 ∈ Ω 2 (X) for the first Chern class of T X.
Consider the map Φ : LX → SX defined by sending an oriented Lagrangian subspace to its orientation class. For an oriented Lagrangian l : L → X we thus obtain a map l = Φ • s L : L → SX. Define the element α L = l * θ ∈ Ω 1 (L). This satisfies dα L = l * c 1 . Thus, the element (c 1 , α L ) ∈ H 2 (X, L) represents the Maslov class of L. Note that here we are using a de Rham model for relative cohomology as in [BT82,Chapter 6].
For α = (α L0 , . . . , α L k ) with α Li ∈ Ω * (L i ) as defined above, set: Note that this only depends on the homotopy class of u. We then define: It suffices to prove the statement in the lemma for the contribution of each disk u separately. It thus suffices to show: for every disk u as above. To this end, recall the path γ ⊂ LX used to define the lift p 0 , and consider the loop γ • ( p 0 ) −1 ⊂ LX. By construction [γ • ( p 0 ) −1 ] = 0 ∈ π 1 (LX, T p0 L 0 ). The loop Φ • (γ • ( p 0 ) −1 ) : S 1 → SX is then also null-homologous. As before, this means that we can construct a lift u : D \ {z 0 , . . . , z k } → SX covering u. By including small extra chords γ i at the punctures z i , this extends to a continuous map u : D → SX with u(γ i ) = p i for i = 1, . . . , k and u(γ 0 ) = p 0 . We then use Stokes' theorem to obtain: There are two different kinds of boundary to u(∂D). The chords γ i get mapped to the paths φ( p i ) and the segments ∂ i D to the image of the Lagrangians L i under Φ • s Li . We thus have: For the second equality, we have used that u| ∂iD = Φ • s Li • u| ∂iD . The last equality here follows as θ is the global angular form on SX and the fact that Ind( p 0 ) + 2s = Ind( p 0 ). The path Φ • ( p 0 • ( p 0 ) −1 ) : S 1 → SX then has winding number s, as LX consists of oriented Lagrangian subspaces. We have thus proved Equation (421).
For an open-closed map defined using holomorphic disks, exactly the same reasoning would show Assumption 4.10(3) holds.

B Example: intersection of quadrics
Let X be a complete intersection of two quadric hypersurfaces in CP 5 . In this section we will show: Lemma B.1. There exists an isomorphism of TE-structures over C ⊃ C: First, observe that for both QH * (X) [[u]] and QH * (Σ 2 )[[u]], the TEP-structure associated to the odd-degree cohomology is trivial, as here both c 1 and µ vanish on the odd-degree cohomology. Take

B.1 Construction of the R-matrix
We need R to satisfy: Equating powers of u yields the relation: We will show this equation has a unique solution R with R 0 = Id. Suppose we have solved this equation up to R i . Let Thus, Equation (436) uniquely determines the entries x 1 01 , x 1 02 , x 1 03 , x 1 13 , x 1 23 ,x 1 10 , x 1 20 , x 1 30 , x 1 31 and x 1 32 . These are the entries of R i+1 which are in the off-diagonal blocks with respect to the Jordan-decomposition of J. Furthermore, the entry x 1 21 is also determined, but a solution only exists provided that We also found the solutions:

C Orientation properties of gluing at interior points
In this section we will prove: Lemma C.1. The gluing map is a local diffeomorphism which changes orientation by (−1) ws(β1) .
Here we will use notation and definitions from [Sol06]. Instead of using Pin ± structures, we will use spin structures.
Proof. It suffices to prove this locally. To this end, let v : S 2 → X and u : (D, S 1 ) → (X, L) be holomorphic maps of degree β 1 and β 2 respectively. Assume u(0) = v(0). For a fixed gluing parameter, consider the glued map u#v : (D, S 1 ) → (X, L). The gluing map gives an isomorphism of determinant lines We need to compute the change in orientation of this map. Recall first that the relative spin structure on i : L → X consists of a triangulation of X and L and a vector bundle V → X 3 such that i * w 2 (V ) = w 2 (T L), and a spin structure p on T L ⊕ V | L . Here w 2 denotes the second Stiefel-Whitney class.
We will now explain briefly, following [Sol06], how to orient Det(D u ). First, up to homotopy, we may assume that u : (D, S 1 ) → (X 3 , L 2 ). We then consider the Fredholm problem D u ⊕ D 0 u on u * (T X ⊕ V ) for some choice of Fredholm operator D 0 u on the bundle u * (V ⊗ C) → D. The spin structure p on T L ⊕ V | L defines a canonical orientation of Det(D u ⊕ D 0 u ) (see [Sol06, Proposition 2.8]). Furthermore, as u * (V ⊗ C) → D, and D is contractible, there exists a spin structure p u 0 on u * (V ⊗ C). By restricting to (u| L ) * V → S 1 , this defines a spin structure p u 0 , thus equipping Det(D u 0 ) with an orientation. Solomon shows ([Sol06, Lemma 2.11]) this orientation does not depend on the choice of spin structure p u 0 . As we obtain a canonical orientation on Det(D u ). The procedure for D u#v is similar. We pick a Fredholm operator D u#v To orient Det(D v ), we note that D v is a (complex) Cauchy Riemann problem, and thus obtains a canonical orientation from the complex structure on ker(D v ) and coker(D v ).
We can thus rewrite the isomorphism 466 as Now let D v 0 be an arbitrary (complex) Cauchy-Riemann operator on v * (V ⊗ C) → S 2 . We can then glue (see [WW15, Section 2.4]) the bundle v * (T X ⊕ (V ⊗ C)) → S 2 with u * (T X ⊕ (V ⊗ C)) → D and obtain a Cauchy-Riemann operator Here we have to be careful about spin structures. The gluing map equips the right-hand side with the same spin structure on the boundary as on the left-hand side. Thus, on both sides the spin structure is given by u| * ∂D p. This spin structure was also used for the canonical orientation of Det(D u#v ⊕ D u#v 0 ). The result is that the induced map on determinant line bundles: Here we have included in the notation the spin structures that we consider on the boundary. Finally, gluing of the operators D u 0 and D v 0 allows us to compare them with D u#v 0 . Again, the gluing map equips the bundles both before and after with the same spin structure. The result is an orientation preserving isomorphism: We have thus reduced our main problem to figuring out the change in orientation between: Det(D u#v The spin structure p u#v 0 extends over all of D, but p u 0 might not. As p u 0 extends over u * V → D, general obstruction theory tells us that the obstruction to extending p u 0 over (u#v) * V → D is given by w 2 (V )(v * ([S 2 ]). Now, for any bundle F → D, the spin structures on F | S 1 → S 1 are classified by whether or not they extend over the entirety of D. Combining this with [Sol06, Lemma 2.10], we find that the orientations in Equation (474) agree if w 2 (V )(v * ([S 2 ]) = 0, and are opposed otherwise.