Abstract
This paper considers the (negative) cyclic open–closed map \({\mathcal{O}\mathcal{C}}^{-}\), 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 \({\mathcal{O}\mathcal{C}}^{-}\) 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 \({\mathcal{O}\mathcal{C}}^{-}\) in the semisimple case; we also consider the non-semisimple case.
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1 Introduction
Kontsevich conjectured [26] that enumerative mirror symmetry, an equality between Gromov–Witten invariants of a space X and period integrals on Y (see [4]) is a consequence of a homological mirror symmetry:
This paper focusses on the symplectic side of mirror symmetry. Barannikov [2] 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 \(A_\infty \)-category (see [16, 24], or [42]). Characterising the splitting has not been done in general, but results have been obtained in various settings.
Ganatra et al. [12] characterise the splitting when the VSHS is \(\mathbb {Z}\)-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 [1] 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.
1.1 Formal TEP-structures
Hertling [19, 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 \(\mathbb {K}\) be a field.
-
1.
A formal pre-T-structure over a \(\mathbb {K}\)-algebra R, is a pair \((\mathcal {E}, \nabla )\). Here \(\mathcal {E}\) is an R[[u]]-module and \(\nabla : Der_{\mathbb {K}}R \otimes \mathcal {E} \rightarrow u^{-1} \mathcal {E}\) a flat connection.
-
2.
If \(\mathcal {E}\) 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 \(\nabla : Der_{\mathbb {K}}(R[[u]]) \otimes \mathcal {E} \rightarrow u^{-2} \mathcal {E}\).
-
4.
A formal TEP-structure is a formal TE-structure equipped with polarisation, i.e. a symmetric, sesquilinear, covariantly constant pairing \(( \cdot , \cdot ): \mathcal {E} \otimes \mathcal {E} \rightarrow R[[u]]\), which restricts to a non-degenerate pairing \(( \cdot , \cdot ): \mathcal {E}/u\mathcal {E} \otimes \mathcal {E}/u\mathcal {E} \rightarrow R\).
Thus a VSHS in the sense of [2] 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 [36, §II.6]). The cyclic homology of an \(A_\infty \)-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 \(R = \Lambda [[H^*(X)]]\), where \(\Lambda \) is a Novikov ring. It is given by the \(S^1\)-equivariant quantum cohomology \(QH^*(X;R)[[u]]\). The connection is as defined in [6], or see Sect. 4.2. The pairing is given by the sesquilinear extension of the Poincaré pairing.
Definition 1.4
The TEP-structure \(HC_*^-(\mathcal {C})\) associated to an R-linear \(A_\infty \)-category is as defined in [24] or see Sect. 3.
Remark 1.5
In general, the TEP-structure associated to an \(A_\infty \)-category is only a pre-TEP-structure. If the non-commutative Hodge-de Rham spectral sequence degenerates, it is actually a TEP-structure. This is conjectured to hold for smooth and compact \(A_\infty \)-categories, see [25]. In the \({\mathbb {Z}}\)-graded setting, Kaledin [23] proves this conjectures holds. Our \(A_\infty \)-categories will only be \({\mathbb {Z}}/2\)-graded, and we will always assume the Hodge-de Rham spectral sequence degenerates, and can thus drop the prefix ‘pre’.
1.2 Cyclic open–closed map
An essential ingredient in proving that the enumerative invariants obtained from the Fukaya category agree with the Gromov–Witten invariants is an isomorphism of TEP-structures called the (negative) cyclic open–closed map. Let X be a closed symplectic manifold. Let Fuk(X) denote the Fukaya category of X, which is an \(A_\infty \)-category over the Novikov ring \(\Lambda \). Assume there exists a bulk-deformed Fukaya category \(Fuk^t(X)\). By this we mean a Fukaya category which is linear over the ring \(R = \Lambda [[H^*(X)]]\).
Conjecture 1.6
There exists a cyclic open-map \(\mathcal{O}\mathcal{C}^{-}: HC^-_*(Fuk^t(X)) \rightarrow QH^*(X;R)[[u]]\). This is a morphism of TEP-structures over \(\Lambda [[H^*(X)]]\).
Such a morphism has not been constructed in general. Partial results exist: [7, 10] construct cyclic open–closed maps in a wide range of settings. Ganatra et al. [12] have announced work proving this is an isomorphism of TP-structures when X is a projective Calabi–Yau manifold. In their case, \(R = \Lambda \), so they consider no bulk-deformations. Furthermore, Ohta–Sanda [31] show that both TE-structures considered come from a new algebraic structure they define, a ‘CH-structure’. They show that an isomorphism of the CH-structures associated to the Fukaya category and quantum cohomology would imply an isomorphism of associated TE-structures.
We prove a local version of this conjecture, focussing on the cyclic homology of the \(A_\infty \)-algebra associated to a single Lagrangian. We use the same technical assumptions as used by Solomon and Tukachinsky [47]: the moduli spaces of holomorphic disks need to be smooth orbifolds with corners, and the evaluation maps at boundary points are assumed to be submersions, see Assumptions 4.18. For us the main example satisfying these conditions is \(X = \mathbb{C}\mathbb{P}^n\) and L a Lagrangian torus fibre (see Lemma 4.19 for a proof of the assumptions). Another class of examples is given by flag varieties and their products, with the Lagrangian given by the real locus (see [47, Example 1.5]). The \(A_\infty \)-algebra \(CF^*(L,L)\) we use is equal up to sign to the \(A_\infty \)-algebra defined by [47], see Remark 4.41.
Theorem 1.7
(See Theorem 4.71) Let \(L\subset X\) be an oriented, relatively-spin Lagrangian submanifold equipped with a \(U(\Lambda )\)-local system. Suppose there exists a complex structure J such that (L, J) satisfy Assumptions 4.18. Then there exists a bulk-deformed Fukaya \(A_\infty \)-algebra \(CF^*(L,L)\). This is an \(R=\Lambda [[H^*(X)]]\)-linear, curved and filtered \(A_\infty \)-algebra. Furthermore, there exist a cyclic open–closed map
which is a morphism of pre-TE-structures over R.
The argument we use to show that the cyclic open–closed map is a morphism of T-structures is due to Ganatra et al. [11], as announced in [12]. The argument simplifies in our setting as the \(A_\infty \)-category we use is cyclic and strictly unital. This ensures that our construction of the cyclic open–closed map does not require higher order terms in u, as opposed to [10]. This comes at the cost of working over a field containing \({\mathbb {R}}\). To shows that the cyclic open–closed map respects the connection in the u-directions an extra ingredient is needed, which is that the cyclic open–closed map respects (Euler-)gradings.
As each component of the Fukaya category \(Fuk(\mathbb{C}\mathbb{P}^n)\) is generated by the Clifford torus (but with different local systems), we can thus construct a bulk-deformed Fukaya category \(Fuk(\mathbb{C}\mathbb{P}^n;R)\) over \(R = \mathbb {C}[[H^*(\mathbb{C}\mathbb{P}^n)]]\). We can thus define a global cyclic open–closed map using our setup:
Corollary 1.8
There exists a cyclic open–closed map
which is an isomorphism of TE-structures over R.
Remark 1.9
One reason we adopt the rather restrictive technical assumptions of [47] 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 [44]. Similar to [47, 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.
1.3 Image of the cyclic open–closed map for monotone symplectic manifolds
For the remainder of the introduction, let X be a monotone symplectic manifold. It is then possible to define the Fukaya category and quantum cohomology over \(\mathbb {C}\) (rather than over a Novikov ring). For ease of exposition in this introduction, we set all bulk-parameters equal to zero and the Novikov parameter to 1, so that \(R = {\mathbb {K}}= \mathbb {C}\). Because we then only have a connection in the u-direction, we call \(QH^*(X)[[u]]\) an E-structure (see Sect. 2.1).
By definition (see [41] for example), \(Fuk(X;\mathbb {C}) = \bigoplus _w Fuk(X)_w\). Here \(Fuk(X)_w\) is a \(\mathbb {C}\)-linear \(A_\infty \)-category with objects monotone Lagrangians with disk potential \(w \in \mathbb {C}\). We consider \(Fuk(X)_w\) as a weakly curved \(A_\infty \)-algebra with curvature \(w \cdot 1\). For a monotone symplectic manifold, quantum cohomology can also be defined over \(\mathbb {C}\) (see [29]). As a vector-space we have \(QH^*(X;\mathbb {C}) = H^*(X;\mathbb {C})\). The first Chern class defines a map:
Decompose quantum cohomology into generalised eigenspaces for this map:
The following was first proved by Ritter–Smith:
Theorem 1.10
([33, Theorem 9.5]) The open–closed map satisfies \(\mathcal{O}\mathcal{C}(HH_*(Fuk(X)_w)) \subset QH^*(X)_w\).
A natural question to ask is how this result extends to cyclic homology:
Question
What is the image of the cyclic open closed map \({\mathcal{O}\mathcal{C}}^{-}(HC^-_*(Fuk(X)_w)) \subset QH^*(X)[[u]]\)?
One might naively think we would have \({\mathcal{O}\mathcal{C}}^{-}(HC^-_*(Fuk(X)_w)) \subset QH^*(X)_w[[u]]\), but the latter is not necessarily invariant under the connection in the u-direction, so this is incompatible with Conjecture 1.6. Instead, taking inspiration from [24, Section 2.28], we apply the Hukuhara–Levelt–Turrittin theorem (see [21, 27, 49]) to decompose the quantum E-structure as a direct sum of \(\nabla _{\frac{d}{du}}\)-invariant submodules indexed by the eigenvalues of \(c_1 \star \):
Lemma 1.11
(Hukuhara–Levelt–Turrittin) There exists a unique decomposition of \(QH^*(X)[[u]]\) into \(\mathbb {C}[[u]]\) submodules:
such that each \(QH^*(X)[[u]]_w\) is invariant under \(u^2\nabla _{\frac{d}{du}}\), and \(QH^*(X)[[u]]_{w}/uQH^*(X)[[u]]_w = QH^*(X)_w\).
Conjecture 1.6, along with a slight extension of the results in [27], then shows that the cyclic open–closed map respects this decomposition:
Corollary 1.12
(See 6.5) \({\mathcal{O}\mathcal{C}}^{-}(HC^-_*(Fuk(X)_w)) \subset QH^*(X)[[u]]_w\).
Since we don’t actually prove the full conjecture, the corollary that follows from our Theorem 1.7 is:
Corollary 1.13
Let \(L \subset X\) be a monotone Lagrangian with disk potential w. Suppose (X, L) satisfies Assumptions 4.18, then:
Remark 1.14
The Hukuhara–Levelt–Turrittin decomposition has appeared before in the study of mirror symmetry. It was used first in [20] and later in [24] to introduce the notion of a Hodge structure of exponential type.
1.4 Semi-simple quantum cohomology
If we additionally assume that \(QH^*(X;{\mathbb {C}})\) is a semi-simple \({\mathbb {C}}\)-algebra (isomorphic as a ring to a direct sum of copies of \({\mathbb {C}}\)), we can completely determine the E-structure \(QH^*(X)[[u]]\). To this end, for \(\phi \in \mathbb {C}[u^{-1}]\), let \(\mathcal {E}^{\phi }:= (\mathbb {C}[[u]], \nabla _{\frac{d}{du}})\) denote the 1-dimensional TE-structure (over \(R = {\mathbb {C}}\)), with connection given by \(\nabla _{\frac{d}{du}} = \frac{d}{du} + \frac{d\phi }{du}\). We show the following, which was already obtained by [6], see also [9, 48]:
Lemma 1.15
(See Corollary 6.9) Assume \(QH^*(X)\) is semi-simple, then there exists a basis \(v_i \in QH^*(X)[[u]]\) such that \(u^2\nabla _{\frac{d}{du}}v_i = w_iv_i\), where the \(w_i\) are the eigenvalues of \(c_1 \star \). 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 \(\mu \) on quantum cohomology (see Lemma 6.8).
Definition 1.17
Given an E-structure \((\mathcal {E},\nabla )\) a splitting is a \({\mathbb {K}}\)-linear map \(s: \mathcal {E}/u\mathcal {E} \rightarrow \mathcal {E}\) splitting the natural projection \(\pi : \mathcal {E} \rightarrow \mathcal {E}/u\mathcal {E}\).
Example 1.18
The E-structure \(\mathcal {E}^{\phi }\) admits a splitting given by:
Remark 1.19
A choice of splitting is equivalent to a choice of opposite subspace as used by Barannikov [2] to obtain a Frobenius manifold from a VSHS. See also [18, 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 \(s^{ss}: QH^*(X) \rightarrow QH^*(X)[[u]]\) 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:
For a general E-structure, given two splittings \(s_1, s_2\), we obtain an element \(R \in Aut(\mathcal {E}/u\mathcal {E})[[u]]\) as \(R = \sum _{i \ge 0} u^iR_i\), with \(R_0 = Id\), and
Such R is called an R-matrix.
Remark 1.20
R-matrices were used by Givental [17] and Teleman [48] to classify semi-simple TFT’s. See also [32, 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 \(R \in Aut(QH^*(X))[[u]]\). A short computation shows that this is indeed the same R-matrix as defined by Teleman [48] 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 and Tu show the categorical version of Lemma 1.15:
Lemma 1.21
([1, Corollary 3.8]) Let \(\mathcal {C} = \bigoplus _w \mathcal {C}_w\) be a direct sum of strictly unital, smooth, finite-dimensional, cyclic and weakly curved \(A_\infty \)-categories of curvature \(w \cdot 1\). Assume \(HH^*(\mathcal {C})\) is semi-simple. Then there exists a splitting:
characterised by the equation \(u^2\nabla _{\frac{d}{du}}s(\alpha ) = ws(\alpha )\) for all \(\alpha \in HH_*(\mathcal {C}_w)\).
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 \(HH^*(Fuk(X))\) is semi-simple. Thus the previous lemma is indeed what was expected from Conjecture 1.6 and Lemma 1.15.
In Sect. 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.
Theorem 1.22
([1, Theorem 1.3]) Let X be a symplectic manifold with \(HH^*(Fuk(X))\) semi-simple. Then the category Fuk(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 \({\mathcal{C}\mathcal{O}}\) is a ring isomorphism, and use the Dubrovin-Teleman reconstruction theorem ([6, 48]) of semi-simple Frobenius manifolds. Our proof instead uses \({\mathcal{O}\mathcal{C}}^-\) and assumes Conjecture 1.6, which allows us to avoid appealing to the reconstruction theorem.
1.5 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 [50] for a proof in certain cases). Another example is the complete intersection of two quadric hypersurfaces in \(\mathbb{C}\mathbb{P}^5\) (see [43]). We conjecture:
Conjecture 1.23
When Fuk(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 \(\mathbb{C}\mathbb{P}^5\). The eigenvalue decomposition of the Fukaya category is as follows:
Smith proves an equivalence:
Theorem 1.24
([43, Theorem 1.1]) \(D^\pi Fuk(X)_0 \cong D^\pi Fuk(\Sigma _2)\), for \(\Sigma _2\) a genus 2 surface.
Assume that \(Fuk(X)_{\pm 8} \cong Fuk(pt)\), which [43, Section 1.6] expects. And note that the Fuk(pt) are considered here with curvature \(\pm 8\), so that \(HC^-_*(Fuk(X)_{\pm 8}) \cong \mathcal {E}^{\mp 8/u}\) (see Lemma 3.27). Also note that [42, Chapter 4] proves a natural isomorphism \(HC^{-}(D^\pi \mathcal {C}) \cong HC^{-}(\mathcal {C})\). We thus have an isomorphism of E-structures:
The cyclic open–closed map then carries this isomorphism to an isomorphism of E-structures:
An explicit computation shows that \(\Phi \) is unique up to rescaling the \(\Phi _i\) by constants \(\lambda _i \in {\mathbb {C}}\). Thus, the following splitting is well-defined:
Here \(s: {\mathbb {C}}\rightarrow \mathcal {E}^{\pm 8/u}\) is as defined in Example 1.18, \(s^{GW}\) denotes the canonical splitting on \(QH^*(\Sigma _2)[[u]]\), and \(\pi \) is the map given by setting \(u = 0\). Let \(s_2 = s^{GW}: QH^*(X) \rightarrow QH^*(X)[[u]]\) 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 \(\Sigma _2\).
1.6 Outline of the paper
Section 2 defines formal TEP-structures and related notations. In Sect. 2.2 we define semi-simple TEP-structures and interpret results of [1] using this language. Next in Sect. 3 we endow the cyclic homology of an \(A_\infty \)-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 Sect. 4.8. This relies on a structure equation for horocyclic operations, which we prove in Sect. 5. In Sect. 6 we study applications of Conjecture 1.6. In particular we show how Lemmas 1.11 and 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 \({\mathbb {Z}}\)-graded category (but at the cost of enlarging the coefficient ring). We also outline why we expect the properties of Sect. 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 \(\mathbb{C}\mathbb{P}^5\). Finally in Appendix 1 we prove a result which was missing in the literature about the orientation properties of gluing of holomorphic maps.
2 Formal TEP-structures
Let \(\mathbb {K}\) be a field of characteristic 0. Let R be a \({\mathbb {Z}}/2\)-graded commutative \({\mathbb {K}}\)-algebra.
Definition 2.1
(Formal T(EP)-structure)
-
1.
A formal pre-T-structure over \(\mathcal {M} = Spec(R)\), is a pair \((\mathcal {E}, \nabla )\). Here \(\mathcal {E}\) is a \({\mathbb {Z}}/2\)-graded R[[u]]-module, with u of even degree and \(\nabla : Der_{{\mathbb {K}}}R \otimes \mathcal {E} \rightarrow u^{-1} \mathcal {E}\) 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 \(\nabla : Der_{{\mathbb {K}}}(R[[u]]) \otimes \mathcal {E} \rightarrow u^{-2} \mathcal {E}\).
-
3.
A formal pre-TP-structure is a formal pre-T-structure equipped with a polarisation, i.e. a covariantly constant pairing
$$\begin{aligned} ( \cdot , \cdot )_{{\mathcal {E}}}: \mathcal {E} \otimes \mathcal {E} \rightarrow R[[u]], \end{aligned}$$which is R-linear, of even degree and u-sesquilinear. That is, for \(f(u) \in R[[u]]\), we have:
$$\begin{aligned} f(u)(a,b)_{\mathcal {E}}= (f(u)a,b)_{\mathcal {E}}= (-1)^{|f||a|}(a,f(-u)b)_{\mathcal {E}}. \end{aligned}$$ -
4.
For a formal pre-TEP-structure, we require that the pairing is also covariantly constant with respect to \(\nabla _{\partial _u}\). More precisely:
$$\begin{aligned} (\nabla _{u\partial _u}a,b)_{\mathcal {E}}+ (a,\nabla _{u\partial _u}b)_{\mathcal {E}}= u\partial _u(a,b)_{\mathcal {E}}. \end{aligned}$$ -
5.
If additionally \(\mathcal {E}\) is free and finitely-generated, we drop the prefix ‘pre’ from T- and TE-structures. Let \({\widetilde{{\mathcal {E}}}}= {\mathcal {E}}/u{\mathcal {E}}\). For a TP-structure, we additionally require that the restriction of the pairing \(( \cdot , \cdot )_{{\widetilde{{\mathcal {E}}}}}: {\widetilde{{\mathcal {E}}}}\otimes {\widetilde{{\mathcal {E}}}}\rightarrow R\) is non-degenerate.
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 \(R = {\mathbb {K}}\) a (pre-)E(P)-structure.
Definition 2.2
A morphism of pre-T(EP)-structures is an R[[u]]-module map \(F: {\mathcal {E}}_1 \rightarrow {\mathcal {E}}_2\) which respects connections and the pairing (if one exists). A morphism of T(EP)-structures is the same as a morphism of pre-T(EP)-structures.
Definition 2.3
Let \({\mathcal {E}}\) be a pre-T-structure over spec(R). An Euler-grading on \({\mathcal {E}}\) consists of an even degree \({\mathbb {K}}\)-linear map:
called the grading and a vector field \(E \in Der_{\mathbb {K}}R\) of even degree, called the Euler vector field, such that for \(f \in R\), \(a \in {\mathcal {E}}\) and \(X \in Der_{\mathbb {K}}R\):
If \({\mathcal {E}}\) is a pre-TP-structure, we additionally require that \((2u \partial _u + 2E)(a,b)_{\mathcal {E}}= (Gr(a),b)_{\mathcal {E}}+ (a,Gr(b))_{\mathcal {E}}\).
Remark 2.4
An Euler-grading differs from a more standard definition of graded in that \({\mathcal {E}}\) is not required to admit a direct sum decomposition into graded pieces.
Definition 2.5
For Euler-graded pre-T-structures \({\mathcal {E}}_1\), \({\mathcal {E}}_2\) 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 \(F \circ Gr_1 = Gr_2 \circ F\).
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 \(\nabla _{\partial _u} = \frac{1}{2u}Gr - \frac{1}{u}\nabla _E\).
2.1 E-structures
Throughout this subsection we take \({\mathbb {K}}= {\mathbb {C}}\).
Definition 2.9
(E(P)-structure) An E(P)-structure is a TE(P)-structure \({\mathcal {E}}\) over the \({\mathbb {K}}\)-algebra \(R = {\mathbb {K}}\), so that \({\mathcal {M}}= pt\). We thus only have a connection \(\nabla _{\frac{d}{du}}: {\mathcal {E}}\rightarrow u^{-2}{\mathcal {E}}\). For ease of notation we will often write \(\nabla \) for \(\nabla _{\frac{d}{du}}\) for an E(P)-structure.
Example 2.10
Let \(\mathcal {E}^{-w/u}\) be the 1-dimensional EP-structure \(\mathcal {E} = \mathbb {C}[[u]]\) with connection \(\nabla = \frac{d}{du} + \frac{w}{u^2}\) and pairing \((1,1) = 1\).
Definition 2.11
A splitting of an E-structure is a \({\mathbb {K}}\)-linear map \(s: {\widetilde{{\mathcal {E}}}}\rightarrow {\mathcal {E}}\) which splits the canonical map \(\pi : {\mathcal {E}}\rightarrow {\widetilde{{\mathcal {E}}}}\). If \({\mathcal {E}}\) is an EP-structure, we say s is P-compatible if \((s(a),s(b))_{\mathcal {E}}= (a,b)_{{\widetilde{{\mathcal {E}}}}}\) for all \(a,b \in {\widetilde{{\mathcal {E}}}}\).
As \({\mathcal {E}}\) 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 \({\mathcal {E}}\) is finitely generated. We can then write the connection on \({\mathcal {E}}\) as
Call A the connection matrix and \(A_0\) the residue. Given two splittings \(s_1\) and \(s_2\) we obtain an isomorphism
Writing \(R = \sum _{i \ge 0} u^i R_i\) for linear maps \(R_i: {\widetilde{{\mathcal {E}}}}\rightarrow {\widetilde{{\mathcal {E}}}}[[u]]\), 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 [27, Chapter 2] in our setup. This theorem is the first step in the Hukuhara–Levelt–Turrittin decomposition. See for example [28] for a modern statement.
Theorem 2.12
Given an E-structure \({\mathcal {E}}\) there exists a unique decomposition \({\mathcal {E}}= \bigoplus _w {\mathcal {E}}_w\) where the w are the eigenvalues of the residue \(A_0: {\widetilde{{\mathcal {E}}}}\rightarrow {\widetilde{{\mathcal {E}}}}\). This decomposition satisfies:
-
\(u^2\nabla ({\mathcal {E}}_w) \subset {\mathcal {E}}_w\) for all w,
-
\(\pi ({\mathcal {E}}_w) = {\widetilde{{\mathcal {E}}}}_w\), where \({\widetilde{{\mathcal {E}}}}_w\) denotes the w-generalised eigenspace of the residue.
The proof in [27, 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 \({\mathcal {E}}\) such that the vectors \(\pi (e_j) \in {\widetilde{{\mathcal {E}}}}\) are generalised eigenvectors for the residue \(A_0\). Write the connection as \(\nabla = \frac{d}{du} + u^{-2}\sum _{i\ge 0} A_iu^i\) in this basis. Then there exists another basis \(\{v_j\}\) for \({\mathcal {E}}\) such that the following hold:
-
\(\pi (v_j) = \pi (e_j)\).
-
Write the connection as \(\nabla = \frac{d}{du} + u^{-2}\sum _{i\ge 0} \widetilde{A}_iu^i\) in the basis \(\{v_i\}\). Then each \(\widetilde{A}_i\) respects the eigenvalue decomposition of \({\widetilde{{\mathcal {E}}}}\), that is \(\widetilde{A}_i|_{{\widetilde{{\mathcal {E}}}}_w}: {\widetilde{{\mathcal {E}}}}_w \rightarrow {\widetilde{{\mathcal {E}}}}_w\).
-
\(\pi _w \circ \widetilde{A}_1|_{{\widetilde{{\mathcal {E}}}}_w} = \pi _ w \circ A_1|_{{\widetilde{{\mathcal {E}}}}_w}: {\widetilde{{\mathcal {E}}}}_w \rightarrow {\widetilde{{\mathcal {E}}}}_w\) for all w. Here \(\pi _w: {\widetilde{{\mathcal {E}}}}\rightarrow {\widetilde{{\mathcal {E}}}}_w\).
Proof of Lemma 2.13
Consider a new basis \(\lbrace P(e_j) \rbrace \) for some invertible \(\mathbb {C}[[u]]\)-linear map P. The new connection matrix is \(\widetilde{A} = P^{-1}AP - P^{-1}\frac{dP}{du}\). If we set \(P = id + u^mT_m\), we find (see [36, Theorem 5.7]): \(\widetilde{A}_s = A_s\) for \(s < m\),
and a more complicated expression for \(\widetilde{A}_{>m}\). Let \({\widetilde{{\mathcal {E}}}}_w\) for \(w \in \mathbb {C}\) 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 \(ad_{A_0} = [A_0, \_]: \mathcal {W}_{A_0} \rightarrow \mathcal {W}_{A_0}\) is an isomorphism. Thus, there exists a \(T_m \in \mathcal {W}_{A_0}\) such that
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 \(\widetilde{A}_m({\widetilde{{\mathcal {E}}}}_w) \subset {\widetilde{{\mathcal {E}}}}_w\). We then find the \(T_m\) successively, starting with \(m=1\). Then set \(P = \prod _{m \ge 1} (id + u^mT_m)\), 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 \(\widetilde{A}_1 = A_1 + [A_0,T_1]\). As we need that \(\widetilde{A}_1|_{{\widetilde{{\mathcal {E}}}}_w}: {\widetilde{{\mathcal {E}}}}_w \rightarrow {\widetilde{{\mathcal {E}}}}_w\), we can choose \(T_1\) to only have entries in the off-diagonal blocks. That is, the restriction \(T_1: {\widetilde{{\mathcal {E}}}}_w \rightarrow {\widetilde{{\mathcal {E}}}}_w\) vanishes. But then the same holds for \([A_0,T_1]\). \(\square \)
Proof of Theorem 2.12
Let \({\mathcal {E}}_w = \langle v_j | \pi (v_j) \in {\widetilde{{\mathcal {E}}}}_w \rangle \). By construction, the \({\mathcal {E}}_w\) are \(u^2 \nabla \)-invariant. Uniqueness of the decomposition follows from the following lemma.
Lemma 2.14
Let \(f: {\mathcal {E}}\rightarrow {\mathcal {E}}'\) be a morphism of E-structures. Then for any choice of decomposition by eigenvalues \({\mathcal {E}}= \oplus _w {\mathcal {E}}_w\) by eigenvalues of \(A_0\), and any choice of decomposition of \({\mathcal {E}}'\) by the eigenvalues of \(A_0'\), we have that \(f({\mathcal {E}}_w) \subset {\mathcal {E}}'_w\).
Remark 2.15
Levelt [27, 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 \(f: {\mathcal {E}}\rightarrow {\mathcal {E}}'\) 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 \({\mathcal {E}}\) and \({\mathcal {E}}'\) as a matrix \(F = \sum _i u^iF_i\). Expand the connections \(\nabla , \nabla '\) 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 [14, Chapter 8.1]). Next, compare coefficients of \(u^{m+1}\). This yields:
where \(L(F_0, \dots , F_{m-g})\) denotes a linear combination of the \(F_{\le m}\) with vanishing constant term. By induction we can assume \(F_0, \dots , F_m\) vanish, which implies \(F_{m+1} = 0\). \(\square \)
2.2 Semi-simple TEP-structures
In this section we will interpret results from [1] in the language of TEP-structures. For simplicity, let \({\mathbb {K}}= {\mathbb {C}}\).
Definition 2.17
(semi-simple E(P)-structure) An E(P)-structure is semi-simple if there exists an isomorphism of E(P)-structures \({\mathcal {E}}\cong \bigoplus _w \mathcal {E}^{-w/u}\). Here the values \(w \in {\mathbb {C}}\) are allowed to occur with multiplicity. Let \(\xi = u^2\nabla _{\frac{d}{du}}: {\widetilde{{\mathcal {E}}}}\rightarrow {\widetilde{{\mathcal {E}}}}\) be the residue of the connection. Thus, \(\xi \) is given by multiplication by w on each summand \(\mathcal {E}^{-w/u}\).
The following two definitions are inspired by [1].
Definition 2.18
([1, Definition 3.9]) Let \({\mathcal {E}}\) be a semi-simple EP-structure with a specified element \(\omega \in {\widetilde{{\mathcal {E}}}}\). We call a \({\mathbb {C}}\)-linear map \(\mu : {\widetilde{{\mathcal {E}}}}\rightarrow {\widetilde{{\mathcal {E}}}}\) a grading operator and say it is:
-
1.
P-compatible: if \((\mu (x),y)_{{\widetilde{{\mathcal {E}}}}} +(x,\mu (y))_{{\widetilde{{\mathcal {E}}}}} = 0\) for all \(x,y \in {\widetilde{{\mathcal {E}}}}\),
-
2.
\(\xi \)-compatible: if the restriction \(\pi _w \circ \mu : {\widetilde{{\mathcal {E}}}}_{w} \rightarrow {\widetilde{{\mathcal {E}}}}_{w}\) vanishes. Here \(\pi _w: {\widetilde{{\mathcal {E}}}}\rightarrow {\widetilde{{\mathcal {E}}}}_w\) denotes the projection onto the w-eigenspace of \(\xi \),
-
3.
\(\omega \)-compatible: if \(\mu (\omega ) = r \omega \) for some \(r \in {\mathbb {C}}\) called the weight of \(\mu \).
Definition 2.19
([1, Definition 3.7]) Let \({\mathcal {E}}\) be an EP-structure with a specified element \(\omega \in {\widetilde{{\mathcal {E}}}}\), and \(s: {\widetilde{{\mathcal {E}}}}\rightarrow {\mathcal {E}}\) a splitting. We say the splitting is:
-
1.
P-compatible: if \((s(a),s(b))_{\mathcal {E}}= (a,b)_{{\widetilde{{\mathcal {E}}}}}\) for all \(a,b \in {\widetilde{{\mathcal {E}}}}\),
-
2.
Homogeneous: if \(\nabla _{\frac{d}{du}} s(a) \in u^{-1} Im(s) + Im(s)\) for all \(a \in {\widetilde{{\mathcal {E}}}}\),
-
3.
\(\omega \)-compatible: if \(\nabla _{u\frac{d}{du}} s(\omega ) \in r s(\omega ) + u^{-1}Im(s)\) for some \(r \in {\mathbb {C}}\).
Example 2.20
The EP-structure \(\mathcal {E}^{-w/u}\) admits a canonical homogeneous, P- and \(\omega \)-compatible splitting given by \(s^{can}(1) = 1 \in {\mathcal {E}}\). Here we have not specified the element \(\omega \in {\widetilde{{\mathcal {E}}}}\), as the splitting is \(\omega \)-compatible for any choice of \(\omega \). This is because, by definition, \(\nabla _{u\frac{d}{du}} s^{can}(a) = u^{-1}ws^{can}(a)\) for all \(a \in {\widetilde{{\mathcal {E}}}}\).
Example 2.21
A semi-simple E-structure \({\mathcal {E}}\) comes with a canonical splitting induced by the isomorphism \(\Phi : {\mathcal {E}}\cong \bigoplus _w \mathcal {E}^{-w/u}\) and the splitting \(s^{can}\) on each \({\mathcal {E}}^{-w/u}\). This splitting is independent of the choice of isomorphism \(\Phi \) as any two such isomorphisms are related by an isomorphism \(\Psi : \bigoplus _w \mathcal {E}^{-w/u} \rightarrow \bigoplus _w \mathcal {E}^{-w/u}\) and any such \(\Psi \) is necessarily independent of u. We denote this splitting by \(s^{ss}\) (the semi-simple splitting) and note that it is homogeneous and \(\omega \)-compatible, for any \(\omega \in {\widetilde{{\mathcal {E}}}}\). If \({\mathcal {E}}\) is a semi-simple EP-structure, \(s^{ss}\) is also P-compatible.
Remark 2.22
A splitting is homogeneous if and only if the associated connection matrix \(A = u^{-2}\sum _{i \ge 0} u^iA_i\) satisfies \(A_i = 0\) for \(i \ge 2\).
Amorim and Tu show the following for EP-structures coming from the cyclic homology of an \(A_\infty \)-category, see [1, Theorem 3.10]. We state their result in our more general setup. The proof is identical.
Theorem 2.23
Let \({\mathcal {E}}\) be a semi-simple EP-structure with a specified element \(\omega \in {\widetilde{{\mathcal {E}}}}\). Then there exists a bijection between the set of homogeneous, P- and \(\omega \)-compatible splittings \(s: {\widetilde{{\mathcal {E}}}}\rightarrow {\mathcal {E}}\) and the set of P-, \(\xi \)- and \(\omega \)-compatible grading operators \(\mu : {\widetilde{{\mathcal {E}}}}\rightarrow {\widetilde{{\mathcal {E}}}}\).
We refer the reader to [1, 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 \(R = \sum _{i\ge 0} u^iR_i\), where \(R_i: {\widetilde{{\mathcal {E}}}}\rightarrow {\widetilde{{\mathcal {E}}}}\) and \(R_0 = Id\) such that
The associated grading operator is then defined by \(\mu ^s = [\xi ,R_1]\), and one obtains the following relation on R:
One then checks all the required properties hold. Conversely, given a grading operator \(\mu : {\widetilde{{\mathcal {E}}}}\rightarrow {\widetilde{{\mathcal {E}}}}\), [1] show that there exists a unique R-matrix solving Eq. (39) and then define the splitting \(s^{\mu }\) by Eq. (38).
Now let \(\mathcal {H}\) be an Euler-graded TEP-structure over a \({\mathbb {C}}\)-algebra R. Let \(\widetilde{{\mathcal {H}}} = {\mathcal {H}}/u{\mathcal {H}}\). The following definition is originally due to Saito [37, Definition 3.1].
Definition 2.24
([1, Definition 4.1]) An element \(\zeta \in {\mathcal {H}}\) is called a primitive form if it satisfies:
-
(Primitivity) The map defined by
$$\begin{aligned} \rho ^{\zeta }: Der_{{\mathbb {C}}}(R) \rightarrow {\widetilde{{\mathcal {H}}}}, \; \rho ^{\zeta }(v) = [u \nabla _v \zeta ] \end{aligned}$$(40)is an isomorphism.
-
(Orthogonality) For any tangent vectors \(v_1, v_2 \in Der_{\mathbb {C}}(R)\), we have:
$$\begin{aligned} (u\nabla _{v_1} \zeta , u\nabla _{v_2} \zeta )_{{\mathcal {H}}} \in R. \end{aligned}$$(41) -
(Holonomicity) For any tangent vectors \(v_1, v_2, v_3 \in Der_{\mathbb {C}}(R)\), we have:
$$\begin{aligned} (u\nabla _{v_1} u\nabla _{v_2} \zeta , u\nabla _{v_3} \zeta )_{{\mathcal {H}}} \in R \oplus u\cdot R. \end{aligned}$$(42) -
(Homogeneity) There exists a constant \(r \in {\mathbb {C}}\) such that
$$\begin{aligned} Gr(\zeta ) = 2r \zeta . \end{aligned}$$(43)
If \(\zeta \) only satisfies the Primitivity property, we will call \(\zeta \) a primitive element, and call the TEP-structure \({\mathcal {H}}\) primitive if such \(\zeta \) exists.
Definition 2.25
Let \(\mathcal {H}\) be a primitive TEP-structure over \(R = {\mathbb {C}}[[t_1, \dots , t_n]]\). Let \({\mathcal {E}}\) be the EP-structure \({\mathcal {E}}:= {\mathcal {H}}\otimes _R {\mathbb {C}}\), where \({\mathbb {C}}\) is an R-module under the map \(t_i \mapsto 0\). For \(\omega \in {\widetilde{{\mathcal {E}}}}\) say \(\omega \) is primitive if there exists a primitive element \(\zeta \in {\mathcal {H}}\) such that \(\zeta |_{t= u = 0} = \omega \).
Amorim and Tu [1, Theorem 4.2] also prove a bijection between primitive forms and splittings, which is a bijection originally established in [38]. We rephrase their theorem to apply to our setup. As already observed by [1, 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 \(\mathcal {H}\) be a Euler-graded, primitive TEP-structure over \(R = {\mathbb {C}}[[t_1, \dots , t_n]]\) and let \(\omega \in {\widetilde{{\mathcal {E}}}}\) be primitive. Then there exists a natural bijection between the following two sets:
Definition 2.27
Let \(\mathcal {H}\) be a TEP-structure over \(R = {\mathbb {C}}[[t_1, \dots , t_n]]\). We say \({\mathcal {H}}\) is semi-simple if the associated EP-structure \({\mathcal {E}}:= {\mathcal {H}}\otimes _R {\mathbb {C}}\) is semi-simple.
For a semi-simple, Euler graded and primitive TEP-structure \({\mathcal {H}}\) as above, with a choice of primitive \(\omega \in {\widetilde{{\mathcal {E}}}}\), Theorems 2.23 and 2.26 thus combine to give a bijection between the set of P-, \(\xi \)- and \(\omega \)-compatible grading operators \(\mu : {\widetilde{{\mathcal {E}}}}\rightarrow {\widetilde{{\mathcal {E}}}}\) and the set of primitive forms \(\zeta \in {\mathcal {H}}\) with \(\zeta |_{t= 0, u=0} = \omega \).
Corollary 2.28
Let \(\mathcal {H}\) be a Euler-graded, primitive, semi-simple TEP-structure over \(R = {\mathbb {C}}[[t_1, \dots , t_n]]\) and let \(\omega \in {\widetilde{{\mathcal {E}}}}\) be primitive. Then there exists a natural bijection between the following two sets:
The relevance of this bijection is that given a primitive form \(\zeta \) as above, Saito and Takahashi [39] endow Spec(R) with the structure of a Frobenius manifold. A grading operator \(\mu \) on a semi-simple TEP-structure over a ring R thus gives rise to a Frobenius manifold \({\mathcal {M}}_{\mu }\). In chapter 6 we will come back to this construction.
3 TE-structure on the cyclic homology of an \(A_\infty \)-algebra
In this section we will define a TE-structure on the cyclic homology of an \(A_\infty \)-algebra. All of the definitions can easily be extended to \(A_\infty \)-categories. Let \(S_n[k]\) be the set of all partitions of \(\lbrace 1, \dots k\rbrace \) into n ordered sets of the form \((1, 2, \dots , k_1)\), \((k_1 + 1, \dots , k_1 + k_2), \dots ,(k_1 + \dots + k_{n-1} + 1, \dots , k_1 + \dots + k_n)\). Let (i : n) denote the ith set of the partition. The size of (i : n) is \(k_i\). We allow for the case \(k_i = 0\).
Let \({\mathbb {K}}\) be a field, and assume \({\mathbb {K}}\) is complete when equipped with the trivial valuation. Let R be a \(\mathbb {Z}/2\)-graded \({\mathbb {K}}\)-algebra with a complete valuation \(\zeta _R: R \rightarrow {\mathbb {R}}_{\ge 0} \cup \{ \infty \}\). Let A be a \(\mathbb {Z}/2\)-graded R-module with a complete valuation \(\zeta _A: A \rightarrow {\mathbb {R}}_{\ge 0}\cup \{ \infty \}\). Let \(|\alpha |\) denote the degree of \(\alpha \in A\), and \(|\alpha |':= |\alpha | - 1\) the shifted degree. For \(\alpha = (\alpha _1,\dots ,\alpha _k)\) let \(\epsilon (\alpha ) = \sum _{j=1}^k |\alpha _j|'\). For a partition \(P \in S_n[k]\) let \(\epsilon _i = \epsilon (\alpha ^{(i:n)})\).
Let \(I \coprod J = [l]\) be a partition of [l] in the usual sense, not respecting the order of [l]. Equip the subsets I and J with the order induced from [l].
Definition 3.1
([47, Definition 1.1]) An n-dimensional, strictly unital, curved, cyclic, \(\mathbb {Z}/2\)-graded \(A_\infty \)-structure on A consists of:
-
R-multilinear maps \(\mathfrak {m}_k: A^{\otimes {k}} \rightarrow A[2-k]\). This means that for \(t \in R\) and \(\alpha _1, \dots , \alpha _k \in A\):
$$\begin{aligned} \mathfrak {m}_k(\alpha _1, \dots , t\alpha _i, \dots , \alpha _k) = (-1)^{|t|(|\alpha _1|' + \dots + |\alpha _{i-1}|' + 1)}t\mathfrak {m}_k(\alpha _1, \dots , \alpha _i, \dots , \alpha _k).\nonumber \\ \end{aligned}$$(48) -
A pairing \(\langle \;, \; \rangle : A \otimes A \rightarrow R[n]\).
-
An element \({\textbf{e}} \in A\) with \(|{\textbf{e}}| = 0\).
These satisfy the following relations:
-
1.
The \(A_\infty \) relations hold:
$$\begin{aligned} \sum _{P \in S_3[k]} (-1)^{\epsilon _1}\mathfrak {m}_{k_1 + 1 + k_3}(\alpha ^{(1:3)}, \mathfrak {m}_{k_2}(\alpha ^{(2:3)}),\alpha ^{(3:3)}) = 0.\end{aligned}$$(49) -
2.
The pairing \(\langle \;, \; \rangle \) is graded R-bilinear:
$$\begin{aligned} \langle a \alpha _1, \alpha _2 \rangle = a \langle \alpha _1, \alpha _2 \rangle = (-1)^{|a||\alpha _1|'}\langle \alpha _1, a \alpha _2 \rangle . \end{aligned}$$(50) -
3.
The pairing is graded anti-symmetric:
$$\begin{aligned} \langle \alpha _1, \alpha _2 \rangle = (-1)^{|\alpha _1|'|\alpha _2|'+1}\langle \alpha _2, \alpha _1 \rangle . \end{aligned}$$(51) -
4.
The pairing is cyclic: for \(\alpha = (\alpha _1, \dots , \alpha _k)\), we have
$$\begin{aligned} \langle \mathfrak {m}_k(\alpha _1, \dots , \alpha _k), \beta \rangle = (-1)^{|\beta |'\epsilon (\alpha )} \langle \mathfrak {m}_k(\beta , \alpha _1, \dots , \alpha _{k-1}), \alpha _k \rangle . \end{aligned}$$(52) -
5.
The unit is strict:
-
(a)
\(\mathfrak {m}_k(\alpha _1, \dots , \alpha _{i-1}, {\textbf{e}}, \alpha _{i+1}, \dots , \alpha _k) = 0, \; \forall k \ne 0,2,\)
-
(b)
\(\mathfrak {m}_2({\textbf{e}},\alpha ) = \alpha = (-1)^{|\alpha |}\mathfrak {m}_2(\alpha , {\textbf{e}})\).
-
(a)
-
6.
The \(A_\infty \)-operations respect the valuation:
-
(a)
\(\zeta _A({\mathfrak {m}}_k({\alpha })) \ge \zeta _A({\alpha })\),
-
(b)
\({\mathfrak {m}}_0 = w \cdot {\textbf{e}} + {\overline{{\mathfrak {m}}}}_0\), with \(\zeta _A({\overline{{\mathfrak {m}}}}_0) > 0\),
-
(c)
\(\zeta _R(\langle {\alpha }_1, {\alpha }_2 \rangle ) \ge \zeta _A({\alpha }_1) + \zeta _A({\alpha }_2)\).
-
(a)
Given an \(A_\infty \)-algebra A, recall the opposite \(A_\infty \)-algebra as defined by [42, definition 3.5]. As an R-module we have \(A^{op} = A\). But now:
Here, for \({\alpha }= ({\alpha }_1, \dots , {\alpha }_k)\), we set \(\dagger ({\alpha }) = \sum _{1 \le i <j \le k} |{\alpha }_i|'|{\alpha }_j|'\). If A is strictly unital with unit \({\textbf{e}} \in A\), then \(A^{op}\) is strictly unital with unit \({\textbf{e}}^{op}:= -{\textbf{e}}\).
We can also define:
Definition 3.2
(\(negative A_\infty \)-algebra) Let the negative \(A_\infty \)-algebra be given by \(A^{-}:= A\) as R-modules, but \(\mathfrak {m}_k^{-}({\alpha }_1, \dots , {\alpha }_k) = (-1)^{k-1} \mathfrak {m}_k({\alpha }_1, \dots , {\alpha }_k)\).
Note that there is an isomorphism of \(A_\infty \)-algebras \(A \cong A^{-}\) given by \({\alpha }\mapsto -{\alpha }\). Combining both definitions we have
Definition 3.3
(negative-\(opposite A_\infty \)-algebra) The negative-opposite algebra associated to A is given by \((A^{op})^{-} =: A^{-op}\). This is then an n-dimensional, strictly unital, cyclic, \(\mathbb {Z}/2\)-graded \(A_\infty \)-algebra with unit \({\textbf{e}} \in A\).
3.1 Hochschild invariants
Let \((A,\mathfrak {m})\) be an n-dimensional, cyclic, strictly unital, curved, \(\mathbb {Z}/2\)-graded \(A_{\infty }\)-algebra over R. Let \(\mathfrak {r} \subset R\) 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 [5, 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 \(A_\infty \)-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 \(\phi \in CC^*(A)\) and \(t\in R\) this means that:
Denote an element \(\alpha \in CC_*(A)\) by \(\alpha = \alpha _0[\alpha _1|\dots |\alpha _k]\), and for a subset \(I\subset \lbrace 1, \dots , k \rbrace \), write \(\alpha ^I\) for \(\bigotimes _{j \in I} \alpha _j\).
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: \([\phi , \psi ] = \phi \circ \psi - (-1)^{|\phi |'|\psi |'} \psi \circ \phi \). The \(A_\infty \)-structure maps \(\mathfrak {m}_k\) define a Hochschild chain \(\mathfrak {m} \in CC^2(A)\). The differential on \(CC^*(A)\) is then given by \([\mathfrak {m}, \_]\). Hochschild cohomology is defined as \(HH^*(A):= H^*(CC^*(A),[\mathfrak {m}, \_])\).
Finally we note that \(CC^*(A)\) admits an \(A_\infty \)-structure \(M^k\), defined by [16]. \(M^1\) is the differential \([\mathfrak {m}, \_]\). We will also need the \(M^2\) part of these operations.
Definition 3.9
The cup product on \(CC^*(A)\) is given by \(\psi \cup \phi := (-1)^{|\psi |}M^2(\psi ,\phi )\). Here
Since A is cyclic, we can consider the pairing:
Here, for \(\alpha \in CC_*(A)\) we use \(\widetilde{\alpha }\) to denote the tuple \((\alpha _1, \dots , \alpha _k)\).
Lemma 3.10
The pairing above descends to a pairing \(HH^*(A) \otimes HH_*(A) \rightarrow R\). Concretely, we have \(([\mathfrak {m},\phi ],\alpha ) + (-1)^{|\phi |}(\phi ,b(\alpha )) = 0\).
Proof
We first write out:
Next, we write out:
Where \(A_1 = (\epsilon _2+1)(|\alpha _0|'+\epsilon _1+\epsilon _3)\). The terms (66) and (70) agree up to a factor \((-1)^{|\phi |'}\), as required. We then use cyclic symmetry to make (69) agree with (65), up to the same factor. We compute
by 51, where \(A_2 = (\epsilon _2 + |\phi |')(|\alpha _0|'+\epsilon _1+\epsilon _3+1)\). And finally:
where \(A_3 = (|\alpha _0|'+\epsilon _3)(|\phi |'+\epsilon _2+\epsilon _1)\). Combining all the signs, we get the required equality. \(\square \)
We will also need the bilinear maps \(b^{1,1}, B^{1,1}: CC^*(A) \otimes CC_*(A) \rightarrow CC_*(A)\), defined for a Hochschild cochain \(\phi \in CC^*(A)\), and \(\alpha = \alpha _0[\alpha _1|\dots |\alpha _k] \in CC_*(A)\) by:
where \(\dagger = (\epsilon _3 + \epsilon _4 + \epsilon _5)(|\alpha _0|' + \epsilon _1 +\epsilon _2) +|\phi |'\epsilon _3\), and
There is a nice interplay between the cup product on \(CC^*(A)\), the \(A_\infty \)-module structure \(b^{1,1}\), and the pairing.
Lemma 3.11
Define the cap product \(\phi \cap \alpha = (-1)^{|\phi |}b^{1,1}(\phi ,\alpha )\). We then have:
Proof
This is an easy verification of signs using the cyclic symmetry of the pairing. \(\square \)
Definition 3.12
Set \(i\lbrace \phi \rbrace = b^{1,1}(\phi , \_ ) + u B^{1,1}(\phi , \_ ): CC^{-}_*(A) \rightarrow CC^-_*(A)\).
The curvature, or Lie derivative, \(\mathcal {L}: CC^*(A) \otimes CC_*(A) \rightarrow CC_*(A)\) defined in [16] can be written as:
Observe that \(\mathcal {L}_{\mathfrak {m}} = b\). An easy computation shows:
Lemma 3.13
For any \(\phi , \psi \in CC^*(A)\), we have: \([\mathcal {L}_\psi ,\mathcal {L}_\phi ] = \mathcal {L}_{[\psi ,\phi ]}\). In particular \([b,\mathcal {L}_\phi ] = \mathcal {L}_{[{\mathfrak {m}},\phi ]}\). Furthermore \([B,\mathcal {L}_\phi ] = 0\).
Remark 3.14
For any linear maps A and B of homogeneous degrees the commutator [A, B] is defined as the supercommutator: \([A,B] = AB - (-1)^{|A||B|}BA\). By extending linearly, this defines the commutator for all linear maps.
Getzler shows:
Proposition 3.15
([16, Theorem 2.2])
Getzler furthermore defines a connection in the base directions. We extend his definition to allow for the case when R is \({\mathbb {Z}}/2\)-graded.
Definition 3.16
The Getzler–Gauss–Manin connection is defined on the chain level by:
Here for a Hochschild cochain \(\phi \in CC^*(A)\) and a derivation \(v \in Der_{\mathbb {C}}R\), the Hochschild cochain \(v(\phi )\) is defined as
Getzler shows the connection descends to the level of cohomology and is flat. This endows \(HC^{-}_*(A)\) with a \({\mathbb {Z}}/2\)-graded pre-T-structure over Spec(R).
Sheridan proves the following holds over a field \({\mathbb {K}}\). Nothing in their proof breaks down if we work over a general ring. We thus have:
Theorem 3.17
([42, Theorem B.2]) Let \(F: \mathcal {C} \rightarrow \mathcal {D}\) be an R-linear \(A_\infty \)-morphism. Then
is a morphism of pre-T-structures.
3.2 The u-connection
The pre-T-structure \(HC^-_*(A)\) has been extended to a pre-TE-structure by [24]. We give another interpretation of this definition. First recall the notion of an Euler-grading on an \(A_\infty \)-algebra:
Definition 3.18
An Euler-grading on an n-dimensional, strictly unital, cyclic, \(\mathbb {Z}/2\)-graded \(A_\infty \)-algebra A consists of an Euler vector field \(E \in Der_{\mathbb {K}}R\) of even degree and an even degree map \(Gr: A \rightarrow A\) such that
and
Furthermore, we require that \(Gr({\textbf{e}}) = 0\). An R-linear \(A_\infty \)-morphism \(F: A \rightarrow B\) is said to be Euler-graded if \(E_A = E_B\) and \(F^k \circ Gr_A = Gr_B \circ F^k + (k-1)F^k\).
Now suppose that A is Euler-graded. Consider \(Gr: A \rightarrow A\) as a length-1 Hochschild cochain. Then define the operator \(Gr^{-}: CC^-_*(A) \rightarrow CC_*^-(A)\) by
where \(\Gamma (\alpha _0[\alpha _1|\dots |\alpha _k]) = -k \alpha _0[\alpha _1|\dots |\alpha _k]\) is the length operator on cyclic chains.
Lemma 3.19
The grading \(Gr^{-}\) descends to cyclic homology, and endows \(HC^-_*(A)\) with an Euler-graded T-structure.
Proof
Let \({\mathfrak {m}}' = \prod _k (2-k){\mathfrak {m}}_k \in CC^*(A)\). A short computation shows \([\Gamma ,b] = b - \mathcal {L}_{{\mathfrak {m}}'}\) and \([\Gamma ,B] = -B\). We thus have:
The second equality follows by Lemma 3.13. This shows \(Gr^-\) descends to cyclic homology. Next, observe that for \(f \in R[[u]]\) and \({\alpha }\in CC_*^-(A)\), we have \(\mathcal {L}_{Gr}(f\alpha ) = 2E(f){\alpha }+ fGr({\alpha })\). This shows that
holds on the chain level. Next, for \(v \in Der_\mathbb {C} R\), we want to compute \([Gr^-,\nabla _v]\). To this end, first observe that, after picking a basis for A, we have: \([\mathcal {L}_{Gr},v]({\alpha }) = [2E,v]({\alpha })\). Furthermore, a direct computation shows:
We thus have:
\(\square \)
Thus, as an Euler-graded pre-T-structure naturally admits an extension to a pre-TE-structure, any Euler-graded \(A_\infty \)-algebra naturally admits a pre-TE-structure on \(HC^-_*(A)\). For an arbitrary \(A_\infty \)-algebra \({\mathcal {C}}\), we will now define an Euler-graded deformation, and use this to define a u-connection on \(HC^-_*({\mathcal {C}})\).
Definition 3.20
Let \({\mathcal {C}}\) be any R-linear \(A_\infty \)-algebra. Define the \(R[s,s^{-1}]\)-linear \(A_\infty \)-algebra \({\mathcal {C}}^s:= {\mathcal {C}}\otimes _R R[s,s^{-1}]\), where s is of odd degree. The operations are defined by:
and extending s-linearly.
Lemma 3.21
Define \(Gr: hom({\mathcal {C}}^s, {\mathcal {C}}^s) \rightarrow hom({\mathcal {C}}^s, {\mathcal {C}}^s)\) by setting \(Gr(s^ka):= ks^ka\) for \(a \in hom({\mathcal {C}},{\mathcal {C}})\). This makes \({\mathcal {C}}^s\) a \({\mathbb {Z}}\)-graded algebra. In particular, by defining \(E = \frac{s}{2}\frac{d}{ds} \in Der_{{\mathbb {K}}} R[s,s^{-1}]\), \({\mathcal {C}}^s\) is an Euler-graded \(A_\infty \)-algebra.
Remark 3.22
The \(A_\infty \)-algebra \({\mathcal {C}}^s\) is also used in [5, section 3.1] to define the connection in the u direction.
Remark 3.23
The deformation \({\mathcal {C}}^s\) is canonical in the following sense: an \(A_\infty \)-morphism \(F: {\mathcal {C}}\rightarrow \mathcal {D}\) induces an \(A_\infty \)-morphism \(F^s: {\mathcal {C}}^s \rightarrow \mathcal {D}^s\) given by
This morphism is Euler-graded.
\({\mathcal {C}}^s\) is Euler-graded, so naturally comes with a u-connection \(\nabla ^s_{\frac{\partial }{\partial u}} = \frac{1}{2u}Gr - \frac{1}{u}\nabla ^{GGM}_E\) (see Definition 2.6). Define the u-connection on \(HC^-({\mathcal {C}})\) to be the restriction to \(s=1\) of \(\nabla ^s_{\frac{\partial }{\partial u}}\). One can check that indeed:
where \(\mathfrak {m}' = \sum _k (2-k)\mathfrak {m}_k\). Call this the canonical u-connection associated to an \(A_\infty \)-algebra. This makes \(HC^-_*({\mathcal {C}})\) into a pre-TE-structure.
Remark 3.24
In the deformation \({\mathcal {C}}^s\), s has odd degree. We can also define the \(R[e,e^{-1}]\)-linear \(A_\infty \)-algebra \(\mathcal {C}^e:= {\mathcal {C}}\otimes _R R[e,e^{-1}]\), 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 \({\mathfrak {m}}\) is \(\mathbb {Z}/2\)-graded. This is Euler-graded with \(E = e\partial _e\), grading operator \(Gr(e^ka) = (2k+|a|)e^ka\).
Lemma 3.25
Let \({\mathcal {C}}\) be an Euler-graded \(A_\infty \)-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 \(\nabla _{\frac{d}{du}}\) 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 3.15, we thus have:
\(\square \)
In particular, if we define a u-connection \(\widetilde{\nabla }_{\frac{d}{du}}\) on \(HC^-({\mathcal {C}})\) by restricting the connection \(\nabla ^e\) coming from the Euler-grading on \({\mathcal {C}}^e\) to \(e = 1\), then \(\widetilde{\nabla }_{\frac{d}{du}}\) agrees with the canonical u-connection.
Lemma 3.26
Let \(F: \mathcal {C} \rightarrow \mathcal {D}\) be an R-linear \(A_\infty \)-morphism. Then \(F_*: HC_*^-({\mathcal {C}}) \rightarrow HC^-_*(\mathcal {D})\) is a morphism of pre-TE-structures.
Proof
Let \(F^s: \mathcal {C}^s \rightarrow \mathcal {D}^s\) be the induced Euler-graded morphism. Now apply Theorem 3.17 to \(F^s\) to find that it respects \(\nabla ^{GGM}_E\) up to homotopy. As we also have that \([F^s,Gr] = 0\), we find that \(F^s\) respects \(\nabla _{\partial _u}\) up to homotopy. Restriction to \(s=1\) shows the result. \(\square \)
We finish this section with a comparison between the E-structures associated to a weakly curved \(A_\infty \)-algebra and its uncurved associated \(A_\infty \)-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 \(\mathbb {C}\)-linear \(A_\infty \)-algebra. Suppose that \((A,\mathfrak {m})\) is strictly unital and weakly curved, i.e. \(\mathfrak {m}_0 = w \cdot {\textbf{e}}\) for some \(w \in \mathbb {C}\). From \((A,\mathfrak {m})\) we can then obtain an uncurved \(A_\infty \)-algebra by setting \(\overline{\mathfrak {m}}_k = \mathfrak {m}_k\) for \(k\ge 1\), and \(\overline{\mathfrak {m}}_0 = 0\).
Lemma 3.27
\((A,\overline{\mathfrak {m}})\) is a unital, uncurved \(A_\infty \)-algebra with:
Furthermore, there exist an isomorphism of pre-E-structures:
Here on both sides the connection \(\nabla \) denotes the canonical connection defined above.
Proof
The Hochschild differentials satisfy \(b = {\overline{b}}\) as we are working with reduced chains. \(B = {\overline{B}}\) by definition. \(\mathfrak {m}' = \overline{\mathfrak {m}}' + 2w\cdot {\textbf{e}}\), and then from the fact that \({\textbf{e}}\) is a strict unit, we get that \(b^{1,1}(\mathfrak {m}',\_) = {\overline{b}}^{1,1}(\overline{\mathfrak {m}},\_) + 2w\cdot Id\). Furthermore \(B^{1,1}(\mathfrak {m}',\_) = {\overline{B}}^{1,1}(\overline{\mathfrak {m}}',\_)\) by definition of the reduced chains. The result then follows. \(\square \)
The residue of the connection \(\nabla _{\frac{d}{du}}\) is the map \(b^{1,1}(\mathfrak {m}', \_): HC^-(A)/uHC^-(A) = HH_*(A) \rightarrow HH_*(A)\). The following lemma shows that this decomposition is trivial, with the only eigenvalue being w. See also [34, Lemma 2.4] and [24, Section 2.2.7].
Lemma 3.28
Let \((A,\mathfrak {m})\) be a weakly curved \(A_\infty \)-algebra with finite dimensional Hochschild homology. Then the operator \(b^{1,1}(\mathfrak {m}',\_) - w\cdot Id: HH_*(A) \rightarrow HH_*(A)\) is nilpotent.
Proof
Lemma 3.27 allows us to reduce this to the uncurved case. Then, on the chain level, \(b^{1,1}(\mathfrak {m}',\_): CC_*(A) \rightarrow CC_*(A)\) reduces the length of the chain by at least 1. Take a basis for \(HH_*(A)\) and pick representatives \(\alpha _i \in CC_*(A)\). Let \(N = \max _i length(\alpha _i)\), then \((b^{1,1}(\mathfrak {m}',\_))^{N+1} = 0 \) on \(HH_*(A)\). \(\square \)
Corollary 3.29
Let A be a \({\mathbb {C}}\)-linear, strictly unital and weakly curved \(A_\infty \)-algebra with curvature \(w\cdot {\textbf{e}}\). 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.
For future reference, we rephrase a result by Amorim and Tu [1, Corollary 3.8] in the language of TEP-structures:
Theorem 3.30
If A is an n-dimensional, strictly unital, cyclic, \(\mathbb {Z}/2\)-graded, smooth and finite dimensional \(A_\infty \)-algebra with \(HH^*(A)\) semi-simple, then \(HC_*^-(A)\) is a semi-simple TEP-structure.
4 Cyclic open–closed map respects connections
4.1 Coefficient rings
Consider the Novikov ring
Let Q have degree 0. For a \({\mathbb {Z}}\)-graded \({\mathbb {C}}\)-vector space U, let \({\mathbb {C}}[[U]]\) be the ring of formal functions on the completion of U at the origin. Explicitly, let \(\{v_i\}_{i \in I}\) be a homogeneous basis for U, and \(\{v_i^*\}_{i \in I}\) the dual basis for \(U^*\). Let \(\{t_i\}_{i \in I}\) be formal variables of degree \(-|v_i|\), then we have an isomorphism:
Each formal vector field \(v \in {\mathbb {C}}[[U]] \otimes U\) on U can be viewed as a derivation \(\partial _v: {\mathbb {C}}[[U]] \rightarrow {\mathbb {C}}[[U]]\). In coordinates, if \(v = \sum _i f_i v_i\), for some \(f_i \in {\mathbb {C}}[[U]]\), then \(\partial _v = \sum _i f_i \partial _{t_i}\). Define the vector fields
These are independent of the chosen basis.
For \(l \in \mathbb {Z}\), let U[l] denote the graded vector space with \(U[l]^i = U^{i+l}\). Then set:
Following [47], define the valuation \(\zeta _Q: Q_U \rightarrow {\mathbb {R}}_{\ge 0}\) by:
Let \(\mathcal {I}_U = \{ f \in Q_U | \zeta _Q(f) > 0 \} \subset Q_U\).
To account for gradings, we will also make use of the ‘universal Novikov ring’: \(\Lambda ^{e}:= \Lambda [e,e^{-1}]\), where e has degree 2. Let \(Q^{e}_U\) be defined using \(\Lambda ^e\) instead of \(\Lambda \).
Remark 4.1
A lot of our work is based on [47]. They use a different Novikov ring, more commonly used in Gromov–Witten theory. Instead of taking series in \(Q^{\mathbb {R}}\) they take series with terms \(T^{\beta }\) for \(\beta \in H_2(X,L)\). For them the monomial \(T^\beta \) has degree \(\mu (\beta )\), where \(\mu : H_2(X,L) \rightarrow \mathbb {Z}\) is the Maslov index. The graded map \(T^{\beta } \mapsto Q^{\omega (\beta )}e^{\mu (\beta )/2}\), allows us to compare their Novikov ring with the universal Novikov ring \(\Lambda ^e\). Note that \(\mu (\beta ) \in 2\mathbb {Z}\) as we assume our Lagrangian is orientable.
4.2 Quantum TE-structure
Let \((X,\omega )\) be a symplectic manifold and let \(U \subset H^*(X;\mathbb {C})\) be a graded \({\mathbb {C}}\)-vector subspace. For any ring R, let \(A^*(X;R)\) denote the space of differential forms on X with coefficients in R.
Definition 4.2
A bulk-deformation parameter over U is an element \(\gamma \in A^*(X;\mathcal {I}_U)\) with \(d\gamma = 0\), \(|\gamma | = 2\) and \([\gamma ] = \Gamma _U \in Q_U \otimes U\).
In Sect. 4.3 we will outline how a general proof that \({\mathcal{O}\mathcal{C}}^-\) respects T-structures can be upgraded to a result about TE-structures. In order for this to work, the T-structure needs to know about deformation in the direction of \(c_1(X)\), the first Chern class of X. To that end we make the following assumption:
Assumption 4.3
We assume there exists a \(Y\in Der_{\Lambda ^e} Q^e_U\) be such that \([Y(\gamma )] = c_1 \in H^*(X;Q_U^e)\). In this case, note that as \(|{\gamma }| = 2\), this implies \(|Y| = 0\).
Let \(\gamma \) 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 \(\eta _1 \star _\gamma \eta _2\). A general reference for the construction of the quantum cup product is [29], however, our coefficient ring includes the universal Novikov parameter e, so we sketch how to modify the definition. See also Definition 4.35 for a construction in our specific setup. Recall from [30] that the quantum cup product is defined as a sum over curve classes \(\beta \in H_2(X)\):
Here \((\eta _1 \star \eta _2)_{\beta }\) is defined by the equation
where \(GW^\beta _{0,3}\) denotes the genus 0, 3 point Gromov–Witten invariant in curve class \(\beta \). One can then extend this definition to take into account bulk deformations \({\gamma }\), to obtain the product \(\star _{\gamma }\) on quantum cohomology \(QH^*(X;Q_U)\). We then define (see also Definition 4.35) the product on \(QH^*(X;Q_U^e)\) by:
where \(c_1 = c_1(TX)\) is the first Chern class.
Definition 4.4
The quantum T-structure over \(Q^e_U \supset \Lambda ^e\) is given as a \(Q_U^e[[u]]\)-module by:
For \(v \in Der_{\Lambda }Q_U\) the quantum connection is defined by:
We now wish to extend the quantum T-structure to be defined over \(Q^e_U \supset \Lambda \). To this end observe that \(Der_\Lambda (Q_U^e) = Q_U^e \otimes _{Q_U} Der_\Lambda Q_U \oplus Q^e_U\langle \partial _e \rangle \). Extend the connection by setting:
Lemma 4.5
These above definitions make \(QH^*(X;Q_U^e)[[u]]\) into a T-structure over \(Q_U^e \supset \Lambda \).
Proof
The verification that \([\nabla _v,\nabla _w] = \nabla _{[v,w]}\) for \(v,w \in Der_{\Lambda } Q_U\) is standard, so we will not do it here. Instead we verify that \([\nabla _v,\nabla _{e\partial _e]} = 0\). The divisor equation for closed Gromov–Witten invariants shows that for \(v \in Der_\Lambda Q_U\) we have:
We also find that:
A direct verification then shows that \([\nabla _{e\partial _e},\nabla _v] = 0\) holds. \(\square \)
We will now define endow \(QH^*(X;Q_U^e)[[u]]\) with an Euler-graded T-structure.
Definition 4.6
Define the Euler vector field by \(E = e\partial _e + E_U\). 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 \(\eta \in H^*(X;{\mathbb {C}})\) and \(f \in Q_U^e[[u]]\), we have:
where \(\mu : H^p(X;Q_U^e) \rightarrow H^p(X;Q_U^e)\) is given by \(\mu (\eta ) = \frac{p-n}{2}\eta \).
A short computation then shows:
Lemma 4.7
The above definitions make \(QH^*(X;Q_U^e)[[u]]\) 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 \(QH^*(X;Q_U^e)[[u]]\) a TE-structure. Writing out the definitions of \(Gr^-\) and E yields the formula:
Remark 4.8
Defining \(\nabla _{Q\partial _Q}\eta := Q\partial _Q (\eta ) - u^{-1} [\omega ] \star _\gamma \eta \) extends the connection to include the Q-direction, to obtain a TE-structure over \(Q_U^e \supset {\mathbb {C}}\). We do not use this connection.
Remark 4.9
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 \(Q_U \supset \Lambda \):
Here \(\Lambda \) is considered as a \(\Lambda ^e\) module via the homomorphism \(\Lambda ^e \rightarrow \Lambda \) given by evaluation at \(e=1\).
There are alternative definitions of the quantum connections on \(QH^*(X;Q_U^e)[[u]]\) given by changing the signs:
Definition 4.10
The alternative connection in the u-direction is given by \(\nabla ^*_{\partial _u} = \frac{Gr^-}{2u} -u^{-1}\nabla ^*_E\). Writing the formulae out we find:
Define the Poincaré pairing
by \(\langle \eta _1, \eta _2 \rangle _X = \int _X \eta _1 \wedge \eta _2\). Now extend the Poincaré pairing u-linearly to a pairing
We then have:
for all \(v \in Der_\Lambda Q_U^e[[u]]\).
Remark 4.11
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.
4.3 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 \(\nabla _{\partial _u}\). We state these conditions in such a way that they should be easy to generalise to different geometric setups.
Let \(U \subset H^*(X;\mathbb {C})\) be a graded vector space, and \({\gamma }\) a bulk-deformation parameter over U satisfying Assumption 4.3. Let \(L \subset X\) be a Lagrangian submanifold. We define an Euler-graded \(A_\infty \)-algebra \(CF^*(L,L)[e]\) over \(Q^e:= Q_U[e,e^{-1}]\) in Sect. 4.7. The Euler vector field is given by \(E = e\partial _e + E_U\), where \(E_U\) is as in Sect. 4.1 and e is of degree 2. The Floer cochain complex \(CF^*(L,L)\) is then defined by restricting to \(e=1\): \(CF^*(L,L):= CF^*(L,L)[e] \otimes _{Q_U^e} Q_U\).
More generally, suppose there exists a bulk-deformed Fukaya category \(Fuk^t(X)\) 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 \(Fuk^t(X)[e]\) 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 Sect. 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:
This ensures the chain level pairing descends to a pairing:
We then apply Poincaré duality to the \(QH^*(X;Q^e)\) factor to obtain the map \({\mathcal{O}\mathcal{C}}_e^-\). It is uniquely determined by the property:
On the left, the pairing is the Poincaré pairing on X, and on the right the pairing is the open–closed pairing.
We expect more generally that given a construction of a chain level cyclic open–closed map, it can be extended to \(CC_*^-(Fuk^t(X)[e])\) by taking the Maslov index of the holomorphic disks into account.
Suppose the cyclic open–closed pairing satisfies the following properties.
Assumptions 4.12
-
1.
For \(v \in Der_{\Lambda ^e} Q_U^e\), there exists a pairing \(\langle \_, G_{v}(\_) \rangle : (C^*(X;Q_U^e)[[u]]) \otimes CC_*^-(Fuk^t(X)[e]) \rightarrow Q_U^e[[u]]\) such that for all \(\eta \) and \({\alpha }\) we have:
$$\begin{aligned}{} & {} \langle \nabla ^*_{v} \eta , {\mathcal{O}\mathcal{C}}_e^- (\alpha ) \rangle + (-1)^{|\eta ||v|}\langle \eta , {\mathcal{O}\mathcal{C}}_e^- (\nabla _{v} \alpha ) \rangle = v \left( \langle \eta , {\mathcal{O}\mathcal{C}}_e^-(\alpha )\rangle \right) \nonumber \\{} & {} \qquad +\, u^{-1}\left( \langle d\eta , G_{v} (\alpha ) \rangle + (-1)^{|\eta | + |v|}\langle \eta , G_{v}\left( (b+uB)(\alpha )\right) \rangle \right) . \end{aligned}$$(137)Here \(\nabla ^*\) is the sign-changed connection from (125). By (130), this implies that \({\mathcal{O}\mathcal{C}}_e^-\) is a morphism of T-structures over \(Q_U^e \supset \Lambda ^e\).
-
2.
There exists a \(\phi \in CC^1(Fuk^t(X)[e])\), such that \(Y(\mathfrak {m}^e) = e\partial _e({\mathfrak {m}}^e) + [{\mathfrak {m}}^e, \phi ]\).
-
3.
For any \(\eta \) and \({\alpha }\) we have:
$$\begin{aligned} \langle \eta , Y({\mathcal{O}\mathcal{C}}_e^-)(\alpha ) \rangle = \langle \eta , e\partial _e({\mathcal{O}\mathcal{C}}_e^-)({\alpha }) \rangle + \langle {\alpha }, {\mathcal{O}\mathcal{C}}_e^-(\mathcal {L}_\phi (\eta )) \rangle . \end{aligned}$$(138)Here, for any \(v \in Der_{\Lambda } Q_U^e\), we define:
$$\begin{aligned} \langle \eta , v({\mathcal{O}\mathcal{C}}_e^-)({\alpha }) \rangle := v(\langle \eta , {\mathcal{O}\mathcal{C}}_e^-(\alpha ) \rangle ) - \langle v(\eta ), {\mathcal{O}\mathcal{C}}_e^-(\alpha ) \rangle - (-1)^{|\eta ||v|} \langle \eta , {\mathcal{O}\mathcal{C}}_e^-(v(\alpha )) \rangle .\nonumber \\ \end{aligned}$$(139) -
4.
\({\mathcal{O}\mathcal{C}}_e^-\) respects the Euler-grading on cyclic invariants: \(Gr^{-} \circ {\mathcal{O}\mathcal{C}}_e^- = {\mathcal{O}\mathcal{C}}_e^- \circ Gr^{-}\).
Remark 4.13
Since we construct a cyclic open–closed map via a cyclic open–closed pairing, we have stated the required properties for the latter. An easy modification of the assumptions would allow them to be applied to an open–closed map defined directly as a chain map \(CC_*^-(Fuk^t(X)[e]) \rightarrow C^*(X;Q_U^e)[[u]]\).
Remark 4.14
To define the connections \(\nabla _v\) and \(\nabla _{e\partial _e}\) on cyclic homology, we need to choose a basis for all of the morphism spaces in \(Fuk^t(X)\). 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 \(H^*(X;{\mathbb {C}})\)).
Theorem 4.15
Suppose Assumptions 4.12 hold, then \(\mathcal{O}\mathcal{C}_e^-\), and hence \({\mathcal{O}\mathcal{C}}^-\), respects \(\nabla _{\partial _u}\) on homology.
First we will show that Assumptions 1, 2 and 3 show that \({\mathcal{O}\mathcal{C}}^e\) respects the connection \(\nabla _{e\partial _e}\). To this end define:
Definition 4.16
Define the pairing \(\langle \eta , G_{e\partial _e}({\alpha }) \rangle \) by
Proposition 4.17
The pairing \(\langle \eta , G_{e\partial _e}({\alpha }) \rangle \) satisfies
Proof
Applying Assumption 1, with \(v = Y\) yields:
using Assumption 2, this gives:
By the Cartan homotopy formula (Proposition 3.15) we can rewrite this as:
Then apply Assumption 3 to obtain:
This implies the result, by rewriting \(\langle \eta , e\partial _e({\mathcal{O}\mathcal{C}}_e^-)(\alpha ) \rangle \) using (139). \(\square \)
Proof of Theorem 4.15
The previous proposition implies \(\mathcal{O}\mathcal{C}^e_-\) respects \(\nabla _{e\partial _e}\) on homology. We will spell this out. By the properties of the Poincaré pairing (Eq. 130):
The above shows that on homology:
which shows that \({\mathcal{O}\mathcal{C}}_e^-(\nabla _{e\partial _e}({\alpha })) = \nabla _{e\partial _e}({\mathcal{O}\mathcal{C}}_e^-({\alpha }))\) on homology. As \(E = e\partial _e + E_U\), we combine this with Assumption 1, applied to \(v = E_U\) to find that the open–closed map respects \(\nabla _{E}\). Then, as \(\nabla _{\partial _u} = \frac{Gr^{-}}{2u} - u^{-1}\nabla _E\), and using the fact that \({\mathcal{O}\mathcal{C}}_e^-\) respects the Euler-grading (Assumption 4), we find that \({\mathcal{O}\mathcal{C}}_e^-\) respects the connection \(\nabla _{\partial _u}\). The statement about \({\mathcal{O}\mathcal{C}}^-\) follows by restriction to \(e = 1\).
4.4 Regularity assumptions
Let X be a 2n-dimensional symplectic manifold and J be an \(\omega \)-tame almost complex structure on X. Let \(L \subset X\) be an oriented Lagrangian equipped with a \(U(\Lambda )\)-local system and a relative spin structure \(\mathfrak {s}\). For us a relative spin structure comes with a choice of element \(w_{\mathfrak {s}} \in H^2(X;{\mathbb {Z}}/2)\) such that \(w_{\mathfrak {s}}|_L = w_2(TL) \in H^2(L;{\mathbb {Z}}/2)\).
For \(l \ge 0\), let \(\mathcal {M}_{l+1}(\beta )\) be the moduli space of stable J-holomorphic spheres with \(l+1\) marked points in homology class \(\beta \in H_2(X,\mathbb {Z})\). Let
be the evaluation map at the j’th marked point. For \(k\ge -1\), \(l\ge 0\), let \(\mathcal {M}_{k+1,l}(\beta )\) be the moduli space of J-holomorphic stable maps \((\mathbb {D},S^1) \rightarrow (X,L)\) in homology class \(\beta \in H_2(X,L)\) 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 \(\mathcal {M}_{k+1,l}(\beta )\), see [7, 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 [44, Chapter 3], where some of the marked points are constrained to lie on a geodesic in \(\mathbb {D}\). Our definition is entirely analogous, except that we replace ‘geodesic’ with ‘horocycle’. Let the smooth locus of \(\mathcal {M}_{k+1,l; \perp _i}(\beta ) \subset \mathcal {M}_{k+1,l}(\beta )\) 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 \(t \in (-1,1)\) 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 [13], 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 \(\mathcal {M}_{k+1,l; \perp _i}(\beta )\) as a fibre product of known spaces. Consider the forgetful map \(\mathcal {M}_{k+1,l}(\beta ) \rightarrow {\mathcal {M}}_{1,2} = D^2\), only remembering the i’th boundary marked point, and the first interior marked point. Here the identification \({\mathcal {M}}_{1,2} \cong D^2\) 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 \(I \hookrightarrow D^2\) given by the arc of the horocycle through \(-i\) and 0 with negative real part. This is a circle of radius \(\frac{1}{2}\) centred at \(-\frac{i}{2}\). 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 \(\partial I = \{1\} -\{ 0\}\). The orientation on \(\mathcal {M}_{k+1,l; \perp _i}(\beta )\) is then defined by the fibre-product orientation, as in [46, Section 2.2].
We assume the following:
Assumptions 4.18
-
1.
\(\mathcal {M}_{l+1}(\beta )\) is a smooth orbifold with corners.
-
2.
\(ev_0\) is a proper submersion.
-
3.
\(\mathcal {M}_{k+1,l}(\beta )\) is a smooth orbifold with corners.
-
4.
\(evb_0^\beta \) is a proper submersion.
-
5.
\(\mathcal {M}_{k+1,l; \perp _i}(X,\beta )\) is a smooth orbifold with corners.
-
6.
\(evb_0^\beta |_{\mathcal {M}_{k+1,l; \perp _i}(X,\beta )}\) is a proper submersion.
We will now show these assumptions hold in the following setup:
Lemma 4.19
The above assumptions hold for \(L \subset X\) 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 \(G_L \subset G_X\) whose action restricts to a transitive action on L.
Proof
This argument is the same as [44, Section 1.3.12], but for horocyclic rather than geodesic constraints. For Assumptions 4.18.1 and 4.18.2, [29, Proposition 7.4.3] show that if the above properties hold, all stable holomorphic maps in \(\mathcal {M}_{l+1}(\beta )\) are regular, it then follows from [35] that this space is a smooth orbifold with corners. As \(G_X\) acts on \(\mathcal {M}_{l+1}(\beta )\), \(G_X\) 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 4.18.3 and 4.18.4 hold in this situation by adapting the arguments for closed Riemann surfaces to Riemann surfaces with boundary (see [47, Remark 1.6]).
Furthermore, \(\mathcal {M}_{k+1,l; \perp _i}(\beta )\) is a smooth orbifold with corners as the maps \(I \hookrightarrow D^2\) and \(\mathcal {M}_{k+1,l}(\beta ) \rightarrow D^2\) are transverse (in the sense of [22, Definition 6.1]). This is clear for any curve mapping to the part of the horocycle not intersecting \(S^1\). The remaining case, for a point in the pre-image of \(-i \in D^2\) is more involved.
To this end, let \(u \in {\mathcal {M}}_{k+1,l}(\beta )\) be a pre-image of the point \(-i \in D^2\). Let \(z_0\) denote the boundary marked point, and \(w_1, w_2\) the interior marked points partaking in the horocycle. The stable curve underlying u has at least 3 components, with one ‘middle’ disk component which has the marked point \(z_0\). There is then additionally a ‘right’ configuration which has the point \(w_1\), and a ‘left’ one which has \(w_2\). Both the left and the right configurations either are disks connected to the middle disks, or have a disk component which connects them to the middle disk. Let \(p_1\) and \(p_2\) denote the boundary points where the right and left configurations are connected to the middle disk. Now use a gluing argument to open up the point \(p_1\). By stability, the new middle disk (which might have \(w_1\) on it) must now have at least 4 boundary marked points, or 2 boundary and 1 interior marked point. A one parameter family can then be constructed by moving the point \(z_0\) towards \(p_2\). The original map u arises from the case where \(z_0\) merges with \(p_2\) and a new disk bubble is formed. The map u lies in the interior of this one parameter family (as we can move \(z_0\) to either side of \(p_2\)), so it lies in the same stratum as the family. The forgetful map to \(D^2\) is given by the location of \(p_2\) in this case, so we see that the map at the level of tangent spaces is indeed surjective onto \(T_{-i}S^1\) as required.
Finally note that \(G_L\) acts on \(\mathcal {M}_{k+1,l; \perp _i}(X,\beta )\), and the evaluations maps are equivariant. As \(G_L\) acts transitively on L, \(evb_0^\beta |_{\mathcal {M}_{k+1,l; \perp _i}(X,\beta )}\) is a proper submersion. \(\square \)
Example 4.20
The simplest example is \((\mathbb{C}\mathbb{P}^n, T_{cl})\), where \(T_{cl}\) denotes the Clifford torus. Here the groups acting are \(G_X = SU(n+1)\) and \(G_L = T^{n}\). Other examples (see [47, Example 1.5]) are \((\mathbb{C}\mathbb{P}^n, \mathbb{R}\mathbb{P}^n)\), or more generally flag varieties and Grassmannians with L being the real locus. In these cases \(G_X\) is a complex matrix group and \(G_L\) the real subgroup. Another class of examples are the quadric hypersurfaces with real locus \(S^n\):
In this case, \(G_X = SO(n+1, {\mathbb {C}})\) and \(G_L = SO(n+1, {\mathbb {R}})\), with the group acting in the obvious manner on the first \(n+1\) coordinates.
4.5 \(\mathfrak {q}\)-operations
This section follows [44, 47] closely. Let \(L \subset X\) be as in the previous section. Let
denote the monodromy representation of the local system on L.
Let \(A^*(L)\) denote differential forms on L with coefficients in \(\mathbb {C}\), and similarly for X. For \({\alpha }\in A^*(L)\), let \(|{\alpha }|\) denote its degree as a differential form, and similarly for differential forms on X. Also, let \(|{\alpha }|':= |{\alpha }| - 1\), and for an ordered set \({\alpha }= ({\alpha }_1, \dots , {\alpha }_k)\), write \(\epsilon ({\alpha }):= \sum _i |{\alpha }_i|' \in \mathbb {Z}\).
For \(k,l \ge 0\) and \(\beta \in H_2(X,L)\) with \((k,l,\beta ) \notin \lbrace (1,0, \beta _0), (0,0,\beta _0) \rbrace \), [47] define operations:
We extend their definition to take into account the local system and set:
Here \(\zeta (\alpha ) = 1 + \sum _{j=1}^{k} j|\alpha _j|'\). The special cases are as follows:
For the cleanest staments, we will use a sign convention differing from [47].
Definition 4.21
Let the operations \({\mathfrak {q}}_{k,l}^\beta : A^*(L)^{\otimes k} \otimes A^*(X)^{\otimes l} \rightarrow A^*(L)\) be defined by:
Here, for \({\alpha }= ({\alpha }_1, \dots , {\alpha }_k)\), we set \(\dagger ({\alpha }) = \sum _{1 \le i <j \le k} |{\alpha }_i|'|{\alpha }_j|'\). This is the sign coming from reversing the order of \({\alpha }\).
[44] also define closed operations:
by
with special cases:
We use these operations, without any sign change, so that \({\mathfrak {q}}_{\emptyset ,l} = {\mathfrak {q}}^{ST}_{\emptyset ,l}\). The quantum product \(\_ \star \_: A^*(X) \otimes A^*(X) \rightarrow A^*(X)\), is then given by
We also define new operations coming from the moduli spaces with horocyclic constraints \({\mathcal {M}}_{k+1,l,\perp _i}(\beta )\). We first define these using sign conventions similar to [47].
Definition 4.22
Let \(\mathfrak {q}^{ST,\beta }_{k,l;\perp _i}: A^*(L)^{\otimes k} \otimes A^*(X)^{\otimes l} \rightarrow A^*(L),\) be defined by
where \(\zeta _{\perp }(\alpha ;\gamma ) = |\alpha |' + |\gamma | + n\). Then, we modify the sign convention as before. We set
The sign \(\dagger ({\alpha })\) is as before. When \(i = 0\), \(\perp _{k+1}\) should be interpreted as \(\perp _0\).
For all of the above \(\mathfrak {q}^\beta \) operations, set
Here \(\mu : H_2(X,L) \rightarrow \mathbb {Z}\) is the Maslov-class, and e is of degree 2. We thus have operations
Let \(\langle \alpha _1, \alpha _2 \rangle _L = (-1)^{|\alpha _2|}\int _L \alpha _1 \wedge \alpha _2\) be the Poincaré pairing on L. [47] prove results about the operations \({\mathfrak {q}}^{ST}\), we state the analogous results for our operations \({\mathfrak {q}}\). These follows by a direct verification of signs from the results in [47].
Proposition 4.23
(Cyclic symmetry, see [47, Proposition 3.3]) For any \(\alpha = (\alpha _1, \dots , \alpha _{k+1})\) and \(\gamma = (\gamma _1, \dots , \gamma _l)\):
Proposition 4.24
(Degree property, see [47, Proposition 3.4]) For any \(\alpha = (\alpha _1, \dots , \alpha _{k})\) and \(\gamma = (\gamma _1, \dots , \gamma _l)\):
The last equality holds as L is orientable so the Maslov-index of any disk is even. By construction, we then have:
Proposition 4.25
(Unit property, see [47, Proposition 3.2]) For \(f \in A^0(L)\), \(\alpha _1, \dots , \alpha _k \in A^*(L;R)\) and \(\gamma \in A^*(X;Q)^{\otimes l}\), we have:
Proposition 4.26
(Top degree property, see [47, Proposition 3.12]) We have \((\mathfrak {q}^\beta _{k,l}(\alpha ;\gamma ))_n = 0\) for all lists \(\alpha \), \(\gamma \) for all \((k,l,\beta ) \notin \{ (1,0,\beta _0), (0,1,\beta _0), (2,0,\beta _0) \}\). Here \(\delta _n\) denotes the degree n part of a differential form \(\delta \in A^*(L;R)\).
Proposition 4.27
(Divisor property, see [47, Proposition 3.9]) Assume \(\gamma _1 \in A^2(X,L)\), \(d\gamma _1 = 0\), then
Proposition 4.28
(Energy-zero property, see [47, Proposition 3.8]) For \(k \ge 0\),
Proposition 4.29
(Fundamental class property, see [47, Proposition 3.7]) For \(k \ge 0\),
Let \(\gamma = (\gamma _1, \dots ,\gamma _l)\) be a list of differential forms on X. For subsets \(I \sqcup J = \{ 1, \dots , l\}\), define \(sign^\gamma (I,J)\) by the equation
or explicitly
Proposition 4.30
(Structure equation for \({\mathfrak {q}}_{k,l}\), see[47, Proposition 2.4]) For any \(\alpha = (\alpha _1, \dots , \alpha _{k})\) and \(\gamma = (\gamma _1, \dots , \gamma _l)\):
where
The new result we prove concerns the boundary of the moduli spaces \(\mathcal {M}_{k+1,l; \perp _0}(X,\beta )\). The proof is given in chapter 5.
Proposition 4.31
(structure equation for \({\mathfrak {q}}_{k,l,\perp _0}\))
Where
Additionally we need the following proposition, which is similar to [44, Lemma 3.10] and is proven in Sect. 5.4.
Proposition 4.32
(Unit on the horocycle) For \(\alpha = (\alpha _0, \dots , \alpha _k) \in A^*(L)^{\otimes {k+1}}\) and \(\gamma \in A^*(X)^{\otimes l}\) we have:
4.6 Bulk-deformed \(\mathfrak {q}\)-operations
Let \(U \subset H^*(X;{\mathbb {C}})\) be a graded vector subspace, and recall the definition of \(Q_U\) from Sect. 4.1. The same formulae as before then define \({\mathfrak {q}}\)-operations for differential forms with coefficients in \(Q_U^e\). Thus, for example we have:
We then have:
Proposition 4.33
(Linearity, see [47, Proposition 3.1]) The \({\mathfrak {q}}\)-operations are multilinear, in the sense that for \(f \in Q_U^e\), \({\alpha }= ({\alpha }_1, \dots , {\alpha }_k)\) and \({\gamma }= ({\gamma }_1, \dots , {\gamma }_l)\), we have:
and
Definition 4.34
A bulk-deformation pair over U is a pair \((b,{\gamma })\). Here \(\gamma \in \mathcal {I}_UA^*(X;Q_U)\) is a bulk-deformation parameter over U and \(b \in \mathcal {I}_U A^*(L,Q_U^e)\) with \(|b| = 1\).
For a bulk-deformation pair \((b,{\gamma })\) define the bulk-deformed operations:
Similarly define:
Finally we define the bulk-deformed horocyclic \(\mathfrak {q}\)-operations by:
We also define:
Definition 4.35
For a bulk-deformation parameter \({\gamma }_U\), the bulk-deformed quantum cup product is defined by:
by setting \(\eta _1 \star _{{\gamma }_U} \eta _2 = {\mathfrak {q}}^{{\gamma }_U}_{\emptyset ,2}(\eta _1,\eta _2)\).
Remark 4.36
When \(({\gamma },b)\) is a bulk-deformation pair, the degree assumptions on \(\gamma \) and b imply that the properties (4.23)–(4.32), with the exception of the energy-zero property (4.28), all hold for the operations \(\mathfrak {q}^{b,\gamma }\) with the same signs as before.
We will need the following lemma later on; it follows from an easy verification of signs.
Lemma 4.37
For \(v \in Der_{\Lambda ^e} Q^e\), \(\alpha \in A^*(L;Q^e)^{\otimes k}\) and \(\eta \in A^*(X;Q^e)^{\otimes l}\) all of homogeneous degrees, we have:
Here
A similar lemma holds for \(v = e \partial _e\), here one gets an additional term, as the \({\mathfrak {q}}\) operations depend on e. Also note that \(\partial _e({\gamma }) = 0\) by definition of a bulk-parameter. First, define operations weighted by the Maslov index \(\mu \): \(\widetilde{\mathfrak {q}}_{k,l}^{b,{\gamma }} = \sum _{\beta } \mu (\beta )\mathfrak {q}_{k,l}^{b,{\gamma },\beta }\). We then have:
Lemma 4.38
For \(\alpha \in A^*(L;Q^e)^{\otimes k}\) and \(\eta \in A^*(X;Q^e)^{\otimes l}\) all of homogeneous degrees, we have:
4.7 Fukaya \(A_\infty \)-algebra
Let \(U \subset H^*(X;{\mathbb {C}})\) and \(({\gamma },b)\) be a bulk-deformation pair over U. Assume \({\gamma }\) satisfies Assumption 4.3, fixing the derivation \(Y\in Der_{\Lambda ^e} Q^e_U\) such that \([Y({\gamma })] = c_1\). For ease of notation, write \(Q = Q_U\).
Solomon and Tukachinsky [47, Theorem 1] construct an \(A_\infty \)-algebra \(A^{ST}\) using the operations \({\mathfrak {q}}^{ST}_{k,0}\). We have different sign conventions for our operations \({\mathfrak {q}}\), but the following still holds.
Definition 4.39
Let \((A:= CF^*(L,L)[e]:= A^*(L;Q^e),\mathfrak {m}_k:= {\mathfrak {q}}^{b,{\gamma }}_{k,0},\langle \;, \; \rangle _L, 1)\). It follows directly from the properties of the \({\mathfrak {q}}\) operations that this forms an n-dimensional, strictly unital and cyclic \(A_\infty \)-algebra. It follows from the degree property 4.24, and the definition that \(|e| = 2\) that this \(A_\infty \)-algebra is Euler-graded with Euler vector field \(E = e\partial _e + E_U\). The grading operator is defined by \(Gr(f{\alpha }) = (|f| + |{\alpha }|)f{\alpha }\) for \(f \in Q^e\) and \({\alpha }\in A^*(L)\). Furthermore, this \(A_\infty \)-algebra is (possibly) curved. The valuation \(\zeta _A\) is induced by the valuation \(\zeta _Q\), defined in (111).
Definition 4.40
Let \(CF^*(L,L):= CF^*(L,L)[e] \otimes _{Q^e} Q\) be the \(A_\infty \)-algebra obtained by setting \(e=1\).
Remark 4.41
Recall the definition of the negative-opposite of an \(A_\infty \)-algebra (Definition 3.3). Then \(CF^*(L,L)\) is related to \(A^{ST}\) as \(CF^*(L,L)^{-op} = A^{ST}\).
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 \(A^*(L;\mathbb {C})\).
Recall the connection
We will now define a connection \(\widetilde{\nabla }\) which agrees with \(\nabla ^{GGM}\) up to homotopy. First, for \(v \in Der_\mathbb {C} Q^e\) define the length zero Hochschild cochain \(\phi _v:= v(b) \in A\). Also let \(\mathfrak {m}_{v}(\alpha ):= \mathfrak {q}^{b,{\gamma }}_{k,1}(\alpha ,v(\gamma ))\). Lemma 4.37 then shows:
Corollary 4.42
For \(v \in Der_{\Lambda ^e} Q^e\) we have:
Definition 4.43
For \(v \in Der_{\Lambda ^e Q^e}\), define \(J_v = i\{ \phi _v \}\).
The Cartan homotopy formula 3.15 then shows:
Corollary 4.44
The connection defined by:
satisfies:
We finish this section by showing that the property 4.12.2 holds. To this end, set \(\widetilde{\mathfrak {m}}_k = \widetilde{{\mathfrak {q}}}^{b,{\gamma }}_{k,0}\). Lemma 4.38 then shows:
Lemma 4.45
\(2e\partial _e(\mathfrak {m}^{b,{\gamma }}) = \widetilde{\mathfrak {m}}^{b,{\gamma }} - [\mathfrak {m}^{b,{\gamma }},\phi _{2e\partial _e}]\).
Applying Lemma 4.42 to \(v= Y\), and using the divisor equation (4.27) to rewrite \(2{\mathfrak {m}}^{b,{\gamma }}_{Y} = \widetilde{{\mathfrak {m}}}^{b,{\gamma }}\), we find:
Lemma 4.46
The Fukaya \(A_\infty \)-algebra thus defined satisfies property 4.12.2.
4.8 Closed–open and open–closed maps
In [7] an open–closed map is defined directly. In our setup one would want to define it using operations \(\mathfrak {p}_k: CC_*(A) \rightarrow A^*(X)\) given by:
where now the push forward is along the interior evaluation \(evi_1: \mathcal {M}_{k+1,1} \rightarrow X\). However, in the present setup the push-forward along interior evaluation is not well defined, as \(evi_1:\mathcal {M}_{k+1,l}(\beta ) \rightarrow X\) need not be a submersion.
The approach we thus take is to first construct a closed–open map, and then dualise to obtain an open–closed map. The fact that the closed–open map is an algebra homomorphism (a result first shown by [8, 13]), will then translate into the result that the open–closed map is a morphism of \(QH^*(X)\) modules (see also [13]). We will then construct a cyclic open–closed map, which was also done in [10]. We then upgrade the result that the open–closed map is a morphism of \(QH^*(X)\)-module to show that the cyclic open–closed map is a morphism of T-structures, using an argument due to [11]. We then show Assumptions 4.12 hold for the cyclic open–closed map in order to conclude that it respects the connection in the u-direction.
4.8.1 The closed–open map
We define the closed–open map \({\mathcal{C}\mathcal{O}}_e: A^*(X;Q^e) \rightarrow CC^*(A)\) on the chain level. Set
It follows from the unit property, Proposition 4.25, that the Hochschild cochain \({\mathcal{C}\mathcal{O}}_e(\eta )\) is reduced. Furthermore, it follows from the degree property, Proposition 4.24, that \(|{\mathcal{C}\mathcal{O}}_e(\eta )| = |\eta |\; (mod\;2)\), so that the closed–open map is a \(\mathbb {Z}/2\)-graded map.
Lemma 4.47
The closed–open map is a chain map. That is: \([\mathfrak {m},{\mathcal{C}\mathcal{O}}_e(\eta )] = {\mathcal{C}\mathcal{O}}_e(d\eta )\).
Proof
By definition:
which, by Proposition 4.30 equals \({\mathcal{C}\mathcal{O}}_e(d\eta )\). \(\square \)
Let \({\mathcal{C}\mathcal{O}}_e: H^*(X;Q^e) \rightarrow HH^*(A)\) be the induced map on cohomology. Next up we will prove the following, which is originally due to [8, 13] in different setups:
Proposition 4.48
The closed–open map induces a unital algebra homomorphism on cohomology.
To this end, we first define:
Definition 4.49
Let the homotopy operator \(H:A^*(X;Q^e)^{\otimes 2} \rightarrow CC^*(A)\) be given by
Unitality follows from the fundamental class property 4.29. The following lemma immediately implies that the closed–open map respects the product.
Lemma 4.50
The homotopy operator H satisfies:
Proof
We write down the terms one by one. Firstly:
Furthermore:
where \(\star = |\gamma _1| + (|\gamma _1|+1)\epsilon _1 + (|\gamma _2|+1)(\epsilon _1+\epsilon _2+\epsilon _3)\).
We then compute the homotopy terms:
Finally we find:
Lemma (4.50) then follows by applying the structure equation (4.31) with interior inputs \(\gamma _1 \otimes \gamma _2\). \(\square \)
We conclude this section with the following observations:
Lemma 4.51
\({\mathcal{C}\mathcal{O}}_e(c_1) = \widetilde{\mathfrak {m}}/2\).
Proof
Pick a representative \(\eta \in A^2(X,L)\) for the Maslov class \(\mu \). Then \(\eta \) also represents \(2c_1 \in H^*(X)\). The divisor property 4.27 shows that \({\mathcal{C}\mathcal{O}}_e({\gamma }_1) = \widetilde{\mathfrak {m}}\). \(\square \)
Furthermore, by definition of \({\mathfrak {m}}_v\) we have:
Lemma 4.52
For \(v \in Der_\mathbb {C} Q^e\) we have: \({\mathcal{C}\mathcal{O}}_e(v(\gamma )) = \mathfrak {m}_{v}\). It follows that \({\mathcal{C}\mathcal{O}}_e(E_U(\gamma )) = \mathfrak {m}_{E_U}\).
4.8.2 The open–closed map
The open–closed map will take the form \({\mathcal{O}\mathcal{C}}: HH_*(A) \rightarrow QH^*(X;Q^e)\). To this end, we first define the open–closed pairing \(A^*(X;Q^e) \otimes CC_*(A) \rightarrow Q^e\). We then show this descends to a pairing on (co)homology. Finally, by dualising the first factor, and using Poincaré duality, we obtain the open–closed map.
Definition 4.53
The open–closed pairing \(\langle \_,{\mathcal{O}\mathcal{C}}_e(\_) \rangle : A^*(X;Q^e) \otimes CC_*(A) \rightarrow Q^e\) is defined as
Here the pairing \((\cdot ,\cdot )\) is as in Eq. (63).
We will show (Lemma 4.55) that the open–closed pairing descends to homology, so that the following makes sense.
Definition 4.54
The open–closed map \({\mathcal{O}\mathcal{C}}: HH_*(A) \rightarrow QH^*(X;Q^e)\) is defined by requiring that
On the left, the pairing is the Poincaré pairing on X, and on the right the pairing is the open–closed pairing.
Since the closed–open map is a chain map (Lemma 4.47), and the pairing respects differentials (Lemma 3.10), the following is immediate.
Lemma 4.55
We have:
The open–closed pairing thus descends to (co)homology.
Ganatra [13] shows the closed–open map makes \(HH_*(A)\) into a \(QH^*(X)\)-module. We prove this in our setup.
Proposition 4.56
The open–closed map is a map of \(QH^*(X;Q^e)\)-modules.
To this end, first define:
Definition 4.57
Recall the map H from Definition 4.49. Then let
The following lemma follows directly from Lemma 4.50.
Lemma 4.58
The pairing G satisfies:
Proof of Proposition 4.56
Lemma 4.58 shows that on homology we have::
Thus \({\gamma }\star {\mathcal{O}\mathcal{C}}({\alpha }) = {\mathcal{O}\mathcal{C}}_e({\mathcal{C}\mathcal{O}}_e(\gamma ) \cap \alpha )\).
4.8.3 The cyclic open–closed map
In order to define the cyclic open–closed map, we first show the open–closed pairing descends to cyclic homology.
Definition 4.59
Extend the open–closed pairing (Definition 4.53) u-linearly to a map:
We then have:
Lemma 4.60
The open–closed pairing descends to a pairing \(QH_{-}^{*}(X;R^e) \otimes HC^{-}(A) \rightarrow Q^e[[u]]\).
Proof
We have already shown the open–closed pairing respects the first differential b. We are then done if \(\langle \eta ,\mathcal{O}\mathcal{C}(B(\alpha )) \rangle = 0\). Now
The first equality is by definition of B and \({\mathcal{O}\mathcal{C}}\). The second equality holds by definition of \({\mathcal{C}\mathcal{O}}_e\). The last equality follows by the top degree property 4.26. \(\square \)
Extend the Poincaré pairing on X u-linearly to a pairing \(QH^*(X;Q^e)[[u]] \otimes QH^*(X;Q^e)[[u]] \rightarrow Q^e[[u]]\).
Definition 4.61
The cyclic open–closed map \({\mathcal{O}\mathcal{C}}_e^-: HC^-_*(A) \rightarrow QH^*(X;Q^e)[[u]]\) is defined by requiring that
On the left, the pairing is the Poincaré pairing on X, and on the right the pairing is the open–closed pairing.
We now prove:
Theorem 4.62
The cyclic open–closed map \({\mathcal{O}\mathcal{C}}_e^-: HC^{-}(A) \rightarrow (QH^*(X;Q_U)[[u]],\nabla )\) is a morphism of T-structures over \(Q^e \supset \Lambda ^e\).
Our proof follows the same ideas as outlined by Ganatra-Perutz-Sheridan in talks. First we observe that the same reasoning as for Lemma 4.60 shows that:
Lemma 4.63
\(\langle \eta , \mathcal{O}\mathcal{C}(B^{1,1}(\phi , \alpha )) \rangle = 0\) for any \(\eta \in A^*(X;Q^e)\), \(\alpha \in CC_*(A)\) and \(\phi \in CC^*(A)\).
Next we define:
Definition 4.64
For \(v \in Der_{\Lambda ^e} Q^e\), set
where we have extended G from Definition 4.57u-linearly.
We then have:
Lemma 4.65
The pairing \(\widetilde{G}_v\) satisfies:
In order to prove this, we first show the following:
Lemma 4.66
We have:
Proof
Using Lemma 4.32, we find that:
We also write out the other terms.
By definition of the open–closed pairing, we also have:
Then, apply Lemma 4.37 to compute \(v({\mathfrak {q}}^{b,{\gamma }}_{k,1}(\widetilde{\alpha };\eta ))\). Keeping track of all the signs shows the result. \(\square \)
Proof of Lemma 4.65
As all the terms in the above equation are u-linear, we may assume \({\alpha }\) and \(\eta \) are independent of u, and then prove this order by order in u.
To verify the \(u^{-1}\) term, apply Lemma 4.58 with \(\gamma _1 = \eta \) and \(\gamma _2 = v(\gamma )\) and use Lemma 4.52 to compute \({\mathcal{C}\mathcal{O}}_e(v(\gamma )) = \mathfrak {m}_{v}\). Equality of the \(u^0\) terms is shown by Lemma 4.66.
Definition 4.67
Recall the definition of \(J_v\) from Definition 4.43. Then define the pairing:
Proposition 4.68
The pairing \(G_v\) satisfies:
Proof
This follows directly from Lemma 4.65, Corollary 4.44, and a verification of signs. \(\square \)
Proof of Theorem 4.62
By the properties of the Poincaré pairing 130:
Lemma 4.68 then shows that on homology:
which shows that \({\mathcal{O}\mathcal{C}}_e^-(\nabla _v({\alpha })) = \nabla _v({\mathcal{O}\mathcal{C}}_e^-({\alpha }))\).
We next show that the Assumptions 4.12 hold in our setup, so that the open–closed map respects u-connections. Assumption 4.12(1) is Lemma 4.68. Assumption 4.12(2) is Lemma 4.46 with \(\phi = \phi _{e\partial _e} - \phi _{Y}\).
The following lemma shows Assumption 4.12(3) holds in our setup.
Lemma 4.69
For any \(\eta \) and \({\alpha }\) we have:
Where \(\phi \) is as above.
Proof
First note that a computation similar to Lemma 4.66 shows that
We thus find that:
Similarly, we have:
The last equality follows from the divisor property 4.27. The result follows. \(\square \)
Finally we show that Assumption 4.12(4) holds:
Lemma 4.70
We have: \(Gr^- \circ {\mathcal{O}\mathcal{C}}_e^- = {\mathcal{O}\mathcal{C}}_e^- \circ Gr^-\).
Proof
First observe that for \({\alpha }= {\alpha }_0[{\alpha }_1, \dots , {\alpha }_k]\), with \({\alpha }_i \in A^*(L;{\mathbb {C}})\), we have:
We then compute
We thus have \(|{\mathcal{O}\mathcal{C}}_e^-({\alpha })| = 1 +\epsilon ({\alpha }) + n\). As the grading \(Gr^-\) on \(QH^*(X;Q^e_U)[[u]]\) is shifted down by n compared to the cohomological grading, we have
\(\square \)
Theorem 4.15 thus implies that the cyclic open–closed map \({\mathcal{O}\mathcal{C}}_e^-\) is a morphism of TE-structures. Now recall that \(CF^*(L,L) = A \otimes _{Q^e} Q\). The cyclic open–closed map then restricts at \(e=1\) to a map
We have thus proved Theorem 1.7.
Theorem 4.71
(Theorem 1.7) Under Assumptions 4.18 the cyclic open–closed map
is a morphism of TE-structures.
5 Analysis on the horocyclic moduli space
The main goal of this chapter is to prove Proposition 4.31. Finally, in the last section we proof Proposition 4.32.
The following sections use 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 \({\mathfrak {q}}^{ST}_{k,l,\perp }\), which were defined using the sign convention similar to [47].
Proposition 5.1
Here
Proposition 4.31 then follows from the above by a direct verification of signs. In the following, we prove Proposition 5.1 assuming that the \(U(\Lambda )\)-local system is trivial. The general result then follows easily.
Recall that the orientation on \(\mathcal {M}_{k+1,l; \perp _0}(\beta )\) is defined by the fibre-product orientation, as defined in [46, Section 2.2]. We take the orientation on I to be the positive orientation, so that \(\partial I = \{1\} -\{ 0\}\). 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).
5.1 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 \(I \times _{D^2} \partial \mathcal {M}_{k+1,l}(\beta )\). Let \(k = k_1 + k_2 + k_3\) and write \(\mathcal {M}_1:= \mathcal {M}_{k_1 + k_3+2,l}(\beta _1)\), \(\mathcal {M}_2:= \mathcal {M}_{k_2+1,l}(\beta _2)\). Finally write \(\mathcal {M}_{j,\perp } = I \times _{D^2} \mathcal {M}_{j}\). Let \(B_\perp \) 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 \(I \times _{D^2} \partial \mathcal {M}_{k+1,l}(\beta )\) can be decomposed into two components, \(B_{\perp ,1}\), where the bubbling is not at the zeroth marked point and \(B_{\perp ,2}\), where the bubbling is at the zeroth marked point.
Lemma 5.2
There exists diffeomorphisms:
The maps \(\phi _j\) change the orientation by \(sign(\phi _j)\), where:
The proof of this lemma uses:
Proposition 5.3
([47, Proposition 2.8]) Let \(k,l \in \mathbb {Z}_{\ge 0}\). Let \(P \in S_3[k]\) and \(\beta _1 + \beta _2 = \beta \in H_2(X,L)\). Let \(I \sqcup J = [l]\) be a partition. Let \(B \subset \partial \mathcal {M}_{k+1,l}(\beta )\) be the 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 lying on the bubble disk. Then the canonical diffeomorphism
changes orientation by the sign \((-1)^{\delta _1}\), with
Proof of Lemma 5.2
We can decompose \(\phi _1\) as
Here \(\hat{\theta }\) is the map induced by \(\theta \) from the Proposition 5.3. By [7, Lemma 8.2.3(4)], we have \(sign(\hat{\theta }) = sign(\theta )\). From Eq. (258) it is clear that \(sign(t) = -1\). Finally, from the associativity of the fibre product [7, Lemma 8.2.3(2)], \(sign(m_1) = 1\). Thus
Similarly, we decompose \(\phi _2\) as:
We can compute \(sign(m_2)\) as follows:
So that \(sign(m_2) = (-1)^{k_1+k_3+1}\), and thus:
Now we consider the boundary components coming from \(\partial I \times _{D^2} \mathcal {M}_{k+1,l}(\beta )\). First we set up some notation regarding the moduli spaces of holomorphic disks. We have
The superscript main here denotes that this is the top-dimensional stratum of the moduli space. The moduli spaces \(\widetilde{\mathcal {M}}^{main}(\beta )\) are oriented using the relative spin structure as in [7, Chapter 8]. Adding marked points and quotienting by \(Aut(D^2)\) gives the moduli spaces \(\mathcal {M}_{k+1,l}^{main}(\beta )\) as open subsets
Here we need to be careful about the ordering in \((S^1)^{k+1}\). We stick to the convention in [7], so that \((S^{1})^{k+1} = S^1_0 \times S^1_1 \times \dots \times S^1_{k}\). 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 [7]. This means that the orientation on \(\mathcal {M}_{k+1,l}(\beta )\) is such that there exists an orientation preserving local diffeomorphism:
Lemma 5.4
Let \(k\ge 1\) and \(l \ge 1\). 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 [7] the orientation on \(Aut(D^2)\) is given by considering the local diffeomorphism:
for three points \(z_0, z_1, z_2 \subset S^1\) 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 \(\widetilde{\mathcal {M}}^{main}(\beta ) \times (S^1)^k \times (D^2)^{l-1}\) 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 \((S^1)^k \times S^1_0 \cong (-1)^k (S^1)^{k+1}\). Finally, apply Eq. (274) and cancel the factor \(Aut(D^2)\) to obtain the result. \(\square \)
Similar considerations show:
Lemma 5.5
Let \(k = k_1 + k_2 + k_3\). Fixing the three boundary marked points with indices \(0, k_1+1\) and \(k_1 + k_2 +2\), we obtain a local diffeomorphism:
which changes orientation by \((-1)^{k + k_2} = (-1)^{k_1 + k_3}\).
Lemma 5.6
Fixing the first three marked points at \(0,1,\infty \) gives an orientation preserving local diffeomorphism:
We now want to study the boundary components \(\partial I \times _{D^2} \mathcal {M}_{k+1,l}(\beta )\). Observe that:
First we look at the case where the two interior marked points collide. This corresponds to \(\{ 0 \} \times _{D^2} \mathcal {M}_{k+1,l}(\beta )\). Let \(B_{\perp ,3}\) be a boundary component where the interior marked points labelled by I bubble off on a sphere. Note that \(1,2 \in I\). Together with the output marked point on the sphere, this gives at least 3 marked points.
For gluing the moduli spaces of holomorphic maps \(\widetilde{\mathcal {M}}^{main}_{\emptyset }(\beta _1)\) and \(\widetilde{\mathcal {M}}^{main}(\beta _2)\), we use the following:
Proposition 5.7
(Lemma C.1) The gluing map
is a local diffeomorphism which changes orientation by \((-1)^{w_{\mathfrak {s}}(\beta _1)}\).
Remark 5.8
This proposition is implicit in [44] and the statement was communicated to the author by Sara Tukachinsky. See also [15, 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 1.
We then prove:
Proposition 5.9
The canonical local diffeomorphism
changes orientation by \(sign(\phi _3) = (-1)^{1+w_{\mathfrak {s}}(\beta _1)}\). Here \(\beta = \beta _1 + \beta _2\).
Proof
Let (v, u) be a stable map, where \(v: S^2 \rightarrow X\), \(u: (D^2,S^1) \rightarrow (X,L)\) and \(evi_0(v) = evi_0(u) = x \in X\). 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 \(T_0\mathbb {D} \cong {\mathbb {C}}\), and cancelling the terms \(T_xX\), we obtain:
The extra change in sign then comes from Eq. (281). \(\square \)
Next we consider the case when one of the horocyclicly constrained points collides with the boundary marked point. This corresponds to \(\{1 \} \times _{D^2} \mathcal {M}_{k+1,l}(\beta )\). Here two disks bubble off on either side of the disk. Let \(B_{\perp ,4}\) be this boundary component. We show:
Proposition 5.10
The map
changes orientation with \(sign(\phi _4) = k_4(k_1 + k_2 + k_3) + k_2(k_3+k_5) +k_3\).
Applying [7, Lemma 8.3.5] twice shows:
Lemma 5.11
The gluing map:
is an orientation preserving local diffeomorphism. Here \(\beta = \beta _1 + \beta _2 + \beta _3\).
Proof of 5.10
Let \(u = (u_1,u_2,u_3) \in B_{\perp ,4}\) be a stable map. For simplicity, write \({\mathcal {M}}_1 = \mathcal {M}_{k_1+k_3+k_5+3,l_1}\), \({\mathcal {M}}_2 = \mathcal {M}_{k_2+1, l_2}(\beta _2)\) and \(\mathcal {M}_{k_4+1, l_3}(\beta _3)\). 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 \(\mathbb {C}\) is even dimensional, and \(\mathbb {R}\) 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 TL and noting that \(T_1D^2 \cong {\mathbb {C}}\). \(\square \)
5.2 Stokes’ theorem and push-forward
The next step is to apply Stokes’ theorem for the push-forward of differential forms.
Theorem 5.12
([46, Theorem 1]) Let M be a smooth orbifold with boundary. For a smooth submersion \(f: M\rightarrow N\) of relative dimension s, and \(\xi \in A^t(M)\). We have:
Recall the following facts about the push-forward of differential forms, see [46]:
Lemma 5.13
-
1.
Let \(f:M\rightarrow N\) be a proper submersion, \(\alpha \in A^*(N)\), \(\beta \in A^*(M)\). Then:
$$\begin{aligned} f_*(f^*\alpha \wedge \beta ) = \alpha \wedge f_*\beta . \end{aligned}$$(302) -
2.
Let
(303)be a pull-back diagram of smooth maps, where g and f are proper submersions. Let \(\alpha \in A^*(P)\). Then:
$$\begin{aligned} q_*p^*\alpha = f^*g_*\alpha . \end{aligned}$$(304)Similarly, if \(\beta \in A^*(M)\), then:
$$\begin{aligned} p_*q^*\beta = (-1)^{(dim(M)-dim(N))(dim(P)-dim(N))}g^*f_*\beta . \end{aligned}$$(305)
To obtain the structure equations for the \(\mathfrak {q}^{ST}_{\perp }\) operations, we will apply Stokes’ theorem with \(M = \mathcal {M}_{k+1,l, \perp _0}(\beta )\), \(N = L\), \(f = evb_0\) and \(\xi = \bigwedge _{i=1} evi^*\gamma _i \wedge \bigwedge _{j=1} evb_j^*\alpha _j\). When it is clear which evaluation maps are used, we will simply write \(evi^*\gamma \) for \(\bigwedge _{i=1} evi^*\gamma _i\) and similarly for the boundary evaluations.
The first term in Stokes’ theorem is:
by definition of the \(\mathfrak {q}\) operations, this equals:
by expanding the signs \(\zeta _{\perp }\), we find:
The second term in Stokes’ theorem reads:
The final term in Stokes’ theorem is given by restricting to the various boundary components \(B_{\perp ,i}\) for \(i = 1,2,3,4\). We first compute the overall sign \((-1)^{s+t}\). Note that \(|\xi | = |\gamma | + |\alpha |\), \(dim(M) \equiv k+1 \;(mod \; 2)\) and \(dim(L) = n\), so \(s \equiv k+1+n\). The overall sign is thus
5.2.1 \(B_{\perp ,1}\)
Here we will show:
Lemma 5.14
The terms in Stokes’ theorem coming from \((evb_0|_{B_{1,\perp }})_*\xi \) are
where \(C_1 = n+ |\gamma | + \zeta (\alpha ) + \epsilon (\alpha ) + sign^\gamma (J_1,J_2) + (|\gamma ^{J_2}|+1)\epsilon _1 + |\gamma ^{J_1}|\).
Recall that
changes orientation by \(\delta _1:= sign(\phi _1) = n + k_1 + k_2k_3\).
Denote everything associated with \(\mathcal {M}_{k_1 + k_3+2,l_1,\perp _0}(\beta _1)\) with a subscript 1 and everything associated with \(\mathcal {M}_{k_2+1,l_2}(\beta _2)\) with a subscript 2. Consider the commutative diagram:
Let
Define \(\delta _2\) by
We thus find:
Now compute
by using the relation (302), this equals:
using Eq. (304) we obtain:
which, by definition of \(\mathfrak {q}^{ST}\) equals:
where \(\delta _3 = \zeta (\alpha ^{(2:3)})\). Expanding \(\xi _1\) and rearranging gives:
where \(\delta _4 = (|\gamma ^{J_2}| + \epsilon _2)(\epsilon _3 + k_3)\). Finally by definition this equals:
where \(\delta _5 = \zeta (\alpha ^{(1:3)},\mathfrak {q}^{ST}(\alpha ^{(2:3)};\gamma ^{J_2}),\alpha ^{(3:3)}) + \zeta _{\perp }(\alpha ^{(1:3)},\mathfrak {q}^{ST}(\alpha ^{(2:3)};\gamma ^{J_2}),\alpha ^{(3:3)})\). Adding all signs together with the sign in Stokes’ theorem, we get an overall sign:
Here we have used [47, Lemma 2.9] to compute
5.2.2 \(B_{\perp ,2}\)
We repeat the above argument for the boundary component \(B_{\perp ,2}\). We show:
Lemma 5.15
The terms in Stokes’ theorem coming from \((evb_0|_{B_{2,\perp }})_*\xi \) are
where \( C_2 = n+ |\gamma | + \zeta (\alpha ) + \epsilon (\alpha ) + sign^\gamma (J_1,J_2) + |\gamma ^{J_2}|\epsilon _1 + 1\).
Recall that
changes orientation by \(\delta _1:= sign(\phi _1) = n + k_3 + k_2k_3 + 1\).
Denote everything associated with \(\mathcal {M}_{k_1 + k_3+2,l_1}(\beta _1)\) with a subscript 1, everything associated with \(\mathcal {M}_{k_2+1,l_2,\perp _0}(\beta _2)\) with a subscript 2. Consider the commutative diagram:
Let
\(\delta _2\) is as before:
Now compute
where \(\delta _3 = \zeta (\alpha ^{(2:3)}) + \zeta _{\perp }(\alpha ^{(2:3)}; \gamma ^{J_2})\). Expanding \(\xi _1\) and rearranging gives:
where \(\delta _4 = (|\gamma ^{J_2}| + \epsilon _2 + 1)(\epsilon _3 + k_3)\). By definition of the \(\mathfrak {q}^{ST}\) operations, this equals:
where \(\delta _5 = \zeta (\alpha ^{(1:3)},\mathfrak {q}^{ST}_{k_2,l_2,\perp _0}(\alpha ^{(2:3)};\gamma ^{J_2}),\alpha ^{(3:3)})\). Adding all signs together with the sign in Stokes’ theorem 5.12, we get an overall sign:
5.2.3 \(B_{\perp ,3}\)
We show:
Lemma 5.16
The terms in Stokes’ theorem coming from \((evb_0|_{B_{3,\perp }})_*\xi \) are
where \(C_3 = n + |\gamma | +\zeta (\alpha ) + \epsilon (\alpha ) + sign^\gamma (J_1,J_2)\).
Recall that
changes orientation by \(\delta _1:= sign(\phi _3) = w_{\mathfrak {s}}(\beta _1)+1\).
Denote everything associated with \(\mathcal {M}_{\emptyset ,l_1+1}(\beta _1)\) with a subscript 1, and those for \(\mathcal {M}_{k+1,l_2+1}(\beta _2)\) with a subscript 2. Consider the commutative diagram:
Let
\(\delta _2\) is defined by:
so that:
Now compute
where \(\delta _3 = reldim(p_2)|\xi _1| \equiv 0\; (mod \; 2)\). Using the commutative diagram, we find:
where \(\delta _4 = w_{\mathfrak {s}}(\beta _1)\). By definition this equals:
where \(\delta _5 = \zeta (\alpha )\). Adding all signs together with the sign in Stokes’ theorem 5.12, we get an overall sign:
5.2.4 \(B_{\perp ,4}\)
Lemma 5.17
The terms in Stokes’ theorem coming from \((evb_0|_{B_{4,\perp }})_*\xi \) are
where \(C_4 = n+ |\gamma | + \zeta (\alpha ) + \epsilon (\alpha ) + sign^\gamma (J_1,J_2,J_3) + (|\gamma ^{J_2}|+1)\epsilon _1 + (|\gamma ^{J_3}|+1)(\epsilon _1 + \epsilon _2 + \epsilon _3) + |\gamma ^{J_2}|\).
Recall that
changes orientation by \(\delta _1:= sign(\phi _4) = k_4(k_1 + k_2 + k_3) + k_2(k_3+k_5) + k_3\).
Denote everything associated with \(\mathcal {M}_{k_1+k_3+k_5+3,l_1}(\beta _1)\) with a subscript 1, everything associated with \(\mathcal {M}_{k_2+1,l_2}(\beta _2)\) with a subscript 2 and subscript 3 for \(\mathcal {M}_{k_4+1, l_3}(\beta _3)\). Consider the commutative diagram:
Let
\(\delta _2\) is defined by:
so that:
Now compute
where \(\delta _3 = k_4(|\gamma ^{J_1}| + |\gamma ^{J_2}|+k_5 +\epsilon _5 +k_3+\epsilon _3 + k_1 + \epsilon _1+ \epsilon _2)\).
where \(\delta _4 = k_4(k_1+k_3+k_5)\)
where \(\delta _5 = \zeta (\alpha ^{(2:5)})\)
where \(\delta _6 = \zeta (\alpha ^{(4:5)})\)
where \(\delta _7 = (|\gamma ^{J_1}|+\epsilon _1+k_1+\epsilon _3+k_3)(\epsilon _4 + |\gamma ^{J_3}|) + (|\gamma ^{J_2}|+\epsilon _2)(k_5 +\epsilon _5 +k_3+\epsilon _3+\epsilon _4 +|\gamma ^{J_3}|)\)
where \(\delta _8 = \zeta (\alpha ^{(1:5)}, \mathfrak {q}^{ST}_{k_2,l_2}(\alpha ^{(2:5)};\gamma ^{J_2}), \alpha ^{(3:5)}, \mathfrak {q}^{ST}_{k_4,l_4}(\alpha ^{(4:5)};\gamma ^{J_3}), \alpha ^{(5:5)})\). Adding all signs together with the sign in Stokes’ theorem 5.12, we get an overall sign:
The above computation relies on the following lemma, which is similar to [47, Lemma 2.9]:
Lemma 5.18
Let \(P \in S_5[k]\), then
Proof
All our computations here will be done modulo two. From the definition of \(\zeta \) we have:
Similarly we have:
Finally:
Adding up these three terms gives the result. \(\square \)
5.3 Concluding the proof of the structure equation
The last step of the proof of Proposition 5.1 is to combine the terms coming from \(B_{\perp ,1}\) with term (311) coming from \((evb_0)_*(d\xi )\). Because the disk bubbles in \(B_{\perp ,1}\) must be stable, there are no disks contributing with \(\beta _2 =\beta _0\). These contributions are exactly provided by the term \((evb_0)_*(d\xi )\) in Stokes’ theorem. The same holds for \(B_{\perp ,2}\) and the terms coming from \(d(evb_0)_*\xi \). We then sum all the terms together with those from \(B_{\perp ,3}\) and \(B_{\perp ,4}\). Finally multiply by \((-1)^{n+|\gamma | + \zeta (\alpha ) + \epsilon (\alpha )}\) to get Proposition 5.1.
To prove Proposition 4.31, one easily adapts the signs. The main computation one needs is the following, which can be directly verified:
Lemma 5.19
We have:
5.4 Unit on the horocycle
This proof follows that of [44, Lemma 3.10] very closely. Let \(p_i: \mathcal {M}_{k+2,l; \perp _i}(\beta ) \rightarrow \mathcal {M}_{k+1,l}(\beta )\) be the map given by forgetting the boundary marked point \(z_i\) and the horocyclic constraint. This map is injective, as there is a unique horocycle through any two interior points \(w_1\) and \(w_2\). Fixing the interior marked point \(w_1\) and the boundary marked points \(z_j\) for \(j \ne i\), 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 Fig. 1). As these lunar arcs cover the entire unit disk we see that the image of
is an open dense subset.
the coloured regions show the areas swept out by \(w_2\)
We next compute the sign of \(p_i\):
Lemma 5.20
The map \(p_i: \mathcal {M}_{k+2,l; \perp _i}(\beta ) \rightarrow \mathcal {M}_{k+1,l}(\beta )\) is a local diffeomorphism with \(sign(p_i) = n + i\).
Proof
The orientation on \({\mathcal {M}}_{k+2,l,\perp _i}(\beta ) = I \times _{D^2} {\mathcal {M}}_{k+2,l}(\beta )\) is defined by the fibre-product orientation (see [46, Section 2.2]). At the level of tangent spaces, this means that we have a natural identification
To calculate the sign of \(p_i\), we multiply both sides by a factor of \(D^2\) and \(Aut(D^2)\). This does not change the sign. Let \({\mathbb {R}}_i\) denote the tangent space to the factor \(S^1_i\) in the definition of the orientation on \({\mathcal {M}}_{k+2,l}(\beta )\) (see Eq. 273). Thus, \(sign(p_i)\) is equal to the sign of the map:
Bringing the summand \({\mathbb {R}}_i\) to the second position, which gives a sign of \((-1)^{i+n}\), and cancelling the obvious summands, we find that \(sign(p_i)\) is given by the sign of:
By drawing a local picture, it is clear that the natural map \(T_t I \oplus {\mathbb {R}}_i \rightarrow T_t D^2\) is orientation preserving. We have thus shown that \(sign(p_i) = i + n\). \(\square \)
Proof of Proposition 4.32
Let \(evi_j\) and \(evb_j\) denote the evaluation maps for the space \({\mathcal {M}}_{k+1,l}(\beta )\). Use \(evi^i_j\) and \(evb^i_j\) for those of \({\mathcal {M}}_{k+2,l,\perp _i}\). The map \(p_i\) then satisfies \(evi_j \circ p_i = evi^i_i\) and:
Let \(\xi = \bigwedge _{j=1}^l (evi_j)^*\gamma _j \wedge \bigwedge _{j=0}^{k} (evb_j)^* {\alpha }_j\) and \(\xi _i = p^*_i \xi \). Then as the images of \(p_i\) for \(1\le i \le k+1\) form an open dense set in \({\mathcal {M}}_{k+1,l}(\beta )\), we have:
A short sign computation shows that
Take the partition with \((1:2) = (1, \dots , i-1)\). We then compute:
Combining this, we have shown:
Taking into account the sign changes between the operations \({\mathfrak {q}}^ST\) and \({\mathfrak {q}}\), the first result of Proposition 4.32 follows. The second equality follows from a cyclic symmetry property of the \(\mathfrak {q}_\perp \) operations, which is the direct analogue of Proposition 4.23.
6 Applications of Conjecture 1.6 in the monotone setting
Let \((X,\omega )\) be a 2n dimensional monotone symplectic manifold, i.e \(c_1 = c_1(TX) = \tau [\omega ] \in H^2(X;\mathbb {R})\) for some \(\tau \in \mathbb {R}_{>0}\). We can then consider quantum cohomology \(QH^*(X) = H^*(X;\mathbb {C})\) with coefficients in \(\mathbb {C}\), as for example in [41].
Definition 6.1
(Quantum E-structure) The quantum E-structure is the \(\mathbb {C}[[u]]\)-module \(QH^*(X)[[u]]\) with connection:
Here \(\star \) is the quantum cup product, and \(\mu : QH^*(X) \rightarrow QH^*(X)\) is the grading operator with \(\mu (\alpha ) = \frac{p-n}{2}\alpha \) for \(\alpha \in QH^p(X)\). The residue of the connection is given by \(c_1 \star \). The canonical map \(\pi : QH^*(X)[[u]]\rightarrow QH^*(X)\) is given by evaluation at \(u=0\). We will often write \(\nabla \) for \(\nabla _{\frac{d}{du}}\).
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. [18]), we call this the Gromov–Witten splitting.
Decompose quantum cohomology as a direct sum of generalised eigenspaces of \(c_1 \star \)
Proposition 2.12 shows that we can extend this decomposition by eigenvalues of \(c_1 \star \) to the E-structure \(QH^*(X)[[u]]\).
Lemma 6.3
There exists a unique decomposition of \(QH^*(X)[[u]]\):
which is compatible with the connection:
and respects the eigenvalue decomposition of \(QH^*(X)\):
Remark 6.4
Note that in general \(QH^*(X)[[u]]_w \ne QH^*(X)_w[[u]]\). This is because the map \(u^2\nabla \) does not map \(QH^*(X)_w[[u]]\) to itself, as in general \(\mu \) will not, see Example 6.10. Similarly, the Gromov–Witten splitting does not respect the eigenvalue decomposition. So in general \(s^{GW}(QH^*(X)_w) \not \subset QH^*(X)[[u]]_w\). See Example 6.10.
Lemma 2.14 together with Corollary 3.29 then shows:
Corollary 6.5
Let X be a monotone symplectic manifold. Assume Conjecture 1.6 holds. Then the cyclic open–closed map respects the decompositions, that is:
6.1 Semi-simple quantum cohomology
Now assume additionally that the quantum cohomology ring \(QH^*(X)\) is semi-simple. This is the case for example for \(X = \mathbb{C}\mathbb{P}^n\). The main result is:
Theorem 6.6
Let X be a monotone symplectic manifold with semi-simple quantum cohomology. Then the EP-structure \(QH^*(X)[[u]]\) is semi-simple in the sense of Definition 2.17.
Remark 6.7
This theorem is a rephrasing of results in [6, 9].
In order to prove this theorem, we need the following lemma. This was first observed by Dubrovin [6], but see also [48, Remark 8.2 (iii)] or [9, Lemma 2.4.4].
Lemma 6.8
([6, Lemma 3.2]) When \(QH^*(X)\) is semi-simple, \(0 = \pi _w \circ \mu |_{QH_w^*(X)}: QH_w^*(X) \rightarrow QH_w^*(X)\). I.e. the diagonal blocks for a matrix representation of \(\mu \) vanish, when we choose a basis of eigenvectors for \(c_1 \star \).
Proof of theorem 6.6
By Proposition 2.12, after changing to a basis for the \({\mathbb {C}}[[u]]\)-modules \(QH^*(X)[[u]]_w\) the connection is given by
We also have that \(B_{-1}|_{QH^*(X)_w} = \pi _w \circ \mu |_{QH^*(X)_w} = 0\) by Lemma 6.8. As \(B_{-1}\) also respects the decomposition of \(QH^*(X)\) into eigenspaces for \(c_1 \star \), we find that \(B_{-1} = 0\). The connection on \(QH^*(X)[[u]]_w\) is thus given by:
The connection \(\widetilde{\nabla }^w\) on \(QH^*(X)[[u]]_w \otimes \mathcal {E}^{\frac{w}{u}}\) is then given by:
As in [49, Chapter 2, case 1], there now exists a further change of basis so that the connection matrix on \(QH^*(X)[[u]]_w \otimes \mathcal {E}^{\frac{w}{u}}\) vanishes. Equivalently, there exists an isomorphism \(QH^*(X)[[u]]_w \cong \mathcal {E}^{-\frac{w}{u}}\). \(\square \)
We thus obtain the following result, which is a part of [9, Proposition 2.5.1].
Corollary 6.9
When \(QH^*(X)\) is semi-simple, there exists a basis \(v_i \in QH^*(X)[[u]]\) such that \(u^2\nabla _{\frac{d}{du}}v_i = w_iv_i\), where the \(w_i\) are the eigenvalues of \(c_1 \star \).
We call the basis \(\{v_{i}\}\) a w-flat basis. As in Example 2.21, we obtain a splitting:
There then exists a unique transformation \(R = \sum _{i \ge 0} u^iR_i\), where \(R_i: QH^*(X) \rightarrow QH^*(X)\) such that
We will now show R agrees with the R-matrix as defined by Teleman [48]. Start with a basis for \(QH^*(X)\) consisting of eigenvectors \(\{e_{i}\}\) for \(c_1 \star \). Thus in this basis for \(QH^*(X)[[u]]\) the connection reads:
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:
which agrees with the relation in [48, Proposition 8.5].
Example 6.10
Consider a basis 1, H for \(S^2\), where H is the point class. Then \(c_1 = 2 H\). The quantum multiplication table reads
Thus \(QH^*(S^2) \cong \frac{\mathbb {C}[H]}{\langle H^2 - 1 \rangle }\), which is semi-simple. The decomposition into eigenspaces for \(c_1\) is:
Now \(\mu (v) = \frac{w}{2}\) and \(\mu (w) = \frac{v}{2}\). Thus the \(\mathbb {C}[[u]]\)-modules \(\langle 1+H \rangle _{\mathbb {C}[[u]]}\) and \(\langle 1-H \rangle _{\mathbb {C}[[u]]}\) are not invariant under \(u^2\nabla \). In fact, we can solve the differential equation \(u^2 \nabla _{\frac{d}{du}} \widetilde{w} = 2 \widetilde{w}\) directly, and find
where
Similarly we obtain a solution to \(u^2 \nabla _{\frac{d}{du}} \widetilde{v} = 2 \widetilde{v}\):
where
The R-matrix in the basis (v, w) is thus:
which indeed agrees with the R-matrix computed in [1, Example 5.4] for the cyclic homology of \(Fuk(S^2)\).
We now rephrase [1, Theorem 5.9] and provide an alternative proof. The proof in [1] uses the closed–open map and the Dubrovin–Teleman reconstruction theorem [48]. 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 \({\mathcal{O}\mathcal{C}}: HH_*(Fuk(X)) \rightarrow QH^*(X)\) is an isomorphism and \(HH^*(Fuk(X))\) is semi-simple. Let \(\mu ^{{\mathcal{O}\mathcal{C}}} = {\mathcal{O}\mathcal{C}}^{-1} \circ \mu \circ {\mathcal{O}\mathcal{C}}: HH_*(Fuk(X)) \rightarrow HH_*(Fuk(X))\) be the pull-back of the grading operator \(\mu \) on \(QH^*(X)\). Then the Frobenius manifold \({\mathcal {M}}_{\mu ^{{\mathcal{O}\mathcal{C}}}}\) associated to \(\mu ^{{\mathcal{O}\mathcal{C}}}\) (see Sect. 2.2) is isomorphic to the big quantum cohomology of X. Here Fuk(X) denotes a non-bulk deformed Fukaya category defined over \(\Lambda \) (or \({\mathbb {C}}\) in the monotone case).
Proof
As the open–closed map is an isomorphism, so is the closed–open map \({\mathcal{C}\mathcal{O}}: QH^*(X) \rightarrow HH^*(Fuk(X))\). Thus, \(QH^*(X)\) is also semi-simple. Lemma 6.6 then shows \(QH^*(X)[[u]]\) is a semi-simple EP-structure. Let \(R = {\mathbb {C}}[[H^*(X)]]\) parametrise bulk-deformations. Let \(QH^*(X;R)[[u]]\) denote the quantum TEP-structure over R.
Let \(Fuk^t(X)\) denote the bulk-deformed Fukaya category. As \({\mathcal{C}\mathcal{O}}\) is an isomorphism, this is a versal deformation of Fuk(X), and can thus be extracted from the categorical data of Fuk(X).
Now apply the bijection between grading operators and primitive forms (Corollary 2.28) to the TEP-structure \(QH^*(X;R)[[u]]\). It follows directly from the definition of the product and metric in [39], that the Frobenius manifold associated to the grading operator \(\mu \) and the primitive \(\omega = 1 \in QH^*(X)\) is indeed the big quantum cohomology ring \(QH^*(X;R)\).
Amorim and Tu [1, Corollary 3.8] show that as \(HH^*(Fuk(X))\) is semi-simple, \(HC^-(Fuk^t(X))\) is a semi-simple TEP-structure. The grading operator \(\mu ^{{\mathcal{O}\mathcal{C}}}\) on \(HH_*(Fuk(X))\) is pulled back from the grading operator on \(QH^*(X)\). The primitive element \(\omega \in HH_*(Fuk(X))\) is defined as \({\mathcal{O}\mathcal{C}}^{-1}(1)\).
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:
respects the grading operator and the primitive element \(\omega \). Thus, Corollary 2.28 equips the TEP-structures \(HC^-(Fuk^t(X))\) and \(QH^*(X;R)[[u]]\) with the same primitive form under the cyclic open–closed map. Hence, the associated Frobenius manifolds \({\mathcal {M}}_{\mu ^{{\mathcal{O}\mathcal{C}}}}\) and \(QH^*(X;R)\) are isomorphic. \(\square \)
6.2 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 \(\mathbb{C}\mathbb{P}^5\), which is a monotone symplectic manifold. The eigenvalue decomposition of the Fukuya category is as follows:
Smith proves an equivalence:
Theorem 6.12
([43, Theorem 1.1]) \(D^\pi Fuk(X)_0 \cong D^\pi Fuk(\Sigma _2)\), for \(\Sigma _2\) a genus 2 surface.
Assume that \(Fuk(X)_{\pm 8} \cong Fuk(pt)\), which [43, Section 1.6] expects. And note that the Fuk(pt) are considered here with curvature \(\pm 8\), so that \(HC^-_*(Fuk(X)_{\pm 4}) \cong \mathcal {E}^{\mp 8/u}\). Also note that [42, Chapter 4] proves a natural isomorphism \(HC_*^{-}(D^\pi \mathcal {C}) \cong HC_*^{-}(\mathcal {C})\). We thus have an isomorphism of TE-structures:
As the TEP-structure associated to an \(A_\infty \)-category is canonical, Conjecture 1.6 implies:
Lemma 6.13
There exists an isomorphism of TEP-structures:
We prove Lemma 6.13 in Appendix B by a direct computation, providing evidence for the conjecture. Moreover, we show that this isomorphism is unique.
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Acknowledgements
This paper was mostly written during my Ph.D at the University of Edinburgh. I could not have wished 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 [44, section 3.2]. My chapter 5 is an adaptation of their proof to the case of horocyclic constraints. I was partly supported by the Royal Society Research Grant for Research Fellows: RGFnR1n181009ERC, the ERC Starting Grant 850713—HMS and EPSRC Grant EP/W015749/1.
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Appendices
Appendix A: Euler-grading on Fukaya category
In this appendix we will explain how a ‘standard’ definition of the Fukaya category (see for example [40]), can be adapted to define an Euler-graded \(A_\infty \)-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 Sect. 4.3. We will show why Assumption 4.12(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.
1.1 A.1: Euler-grading
Let X be a symplectic manifold. Let \(\mathcal {L}X \rightarrow X\) denote the Grassmanian bundle of oriented Lagrangian subspaces of TX. Note that an oriented Lagrangian L comes with a canonical section \(s_L: L \rightarrow \mathcal {L}X\).
Let e have degree 2. We then define a Fukaya category over \(\Lambda [e,e^{-1}]\), denoted by Fuk(X)[e], as follows. Objects in Fuk(X)[e] are given by oriented, relatively spin Lagrangian submanifolds \(L \subset X\). For any two (transverse) oriented Lagrangians \(L_1, L_2\) and \(p \in L_1 \cap L_2\), denote by \(\mathcal {L}(L_1,L_2,p)\) the set of homotopy classes of paths \(\widetilde{p}: [0,1] \rightarrow \mathcal {L}_pX\) with \(\widetilde{p}(0) = T_pL_1\) and \(\widetilde{p}(1) = T_p L_2\). Then define:
As in [40, Section 11 g], there is a grading on \(CF^*(L_1,L_2)\) given by the Maslov-index for the pair of paths \((\widetilde{p},\widetilde{p}(1))\). The \(\Lambda [e,e^{-1}]\)-module structure is given by defining \(e \cdot {\widetilde{p}}\) to be the homotopy class of paths which have Maslov-index \(Ind(e \cdot {\widetilde{p}}) = Ind({\widetilde{p}}) + 2\).
In the ‘standard’ Fukaya category, the product operations are defined by counting holomorphic disks as follows. Let \(L_0, \dots , L_k\) be pairwise transversely intersecting Lagrangians. Choose points \(p_0, \dots , p_k\), with \(p_i \in L_{i-1} \cap L_i\), where \(L_{-1}\) denotes \(L_k\). Let \(z_0, \dots , z_k \in \partial \mathbb {D}\) be marked points. Then the coefficient of \(p_0\) in \(m_k(p_1, \dots , p_k)\) is given by counting the number of holomorphic maps \(u: \mathbb {D}{\setminus } \{z_0, \dots z_k\} \rightarrow X\), which extend to continuous maps \(u: \mathbb {D} \rightarrow X\) with marked points \(u(z_i) = p_i\). Furthermore the boundary in between \(z_i\) and \(z_{i+1}\), \(\partial _i \mathbb {D}\) is required to satisfy \(u(\partial _i \mathbb {D}) \subset L_i\). Let \({\mathcal {M}}^0(p_0, \dots , p_k; [u])\) denote the zero-dimensional component of the moduli space of such disks in the homotopy class [u]. Then,
Given lifts \({\widetilde{p}}_1, \dots , {\widetilde{p}}_k\), where \({\widetilde{p}}_i \in \mathcal {L}(L_{i-1}, L_i, p_i)\), and a holomorphic disk u as above. We explain how to determine a lift \({\widetilde{p}}_0 \in \mathcal {L}(L_0, L_k, p_0)\). To this end, concatenate the paths \({\widetilde{p}}_i\) together with the paths \(s_{L_i}(u|_{\partial _i \mathbb {D}})\), to obtain a path \(\gamma \in \mathcal {L}X\). This is a path starting at \(T_{p_0}L_0\) and finishing at \(T_{p_0}L_k\). As the bundle \(u^*TX \rightarrow \mathbb {D}\) is trivial, we obtain a projection \(\mathcal {L}|_{im(u)}X \rightarrow \mathcal {L}_{p_0}X\). The projection of \(\gamma \) defines the lift \({\widetilde{p}}_0 \in \mathcal {L}(L_0,L_k,p_0)\). In other words, we have \([\gamma \circ ({\widetilde{p}}_0)^{-1}] = 0 \in \pi _1(\mathcal {L}X, T_{p_0}L_0)\). So the Maslov index of the loop \(\gamma \circ ({\widetilde{p}}_0)^{-1}\) vanishes. Alternatively, there exists a lift \(\widetilde{u}: \mathbb {D}{\setminus }\{ z_0, \dots , z_k\} \rightarrow \mathcal {L}X\) covering u, which by including small extra chords \(\gamma _i\) at the punctures \(z_i\) extends to a continuous map \(\widetilde{u}: \mathbb {D} \rightarrow \mathcal {L}\) with \(\widetilde{u} \circ \gamma _i = {\widetilde{p}}_i\) for \(i = 0,\dots , k\).
The above procedure determines the lift \({\widetilde{p}}_0\) as a function of \({\widetilde{p}}_1, \dots , {\widetilde{p}}_k\) and the class of the disk [u]. We then define:
Lemma A.1
The above definition endows Fuk(X)[e] with the structure of a \(\mathbb {Z}\)-graded \(A_\infty \)-category. The grading operator is given by \(Gr({\widetilde{p}}):= Ind({\widetilde{p}}){\widetilde{p}}\).
Proof
The fact that the product operations define an \(A_\infty \)-structure follows directly from the verification for Fuk(X). By (one of the many) definition(s) of the Maslov index, we have \(Ind(u) = Ind({\widetilde{p}}_0) - \sum _{i = 1} Ind({\widetilde{p}}_i)\) so that the product satisfies \([Gr, {\mathfrak {m}}^e_k] = (2-k) {\mathfrak {m}}^e_k\) as required. \(\square \)
1.2 A.2: Bulk-deforming by \(c_1\)
Let U be a 1-dimensional vector space, with grading 2. We thus identify \(Q_U:= \Lambda [[t]]\) for a formal parameter t of degree 0. We will now construct a \(Q_U^e\)-linear category \(Fuk^t[e]\). One should think of this as bulk-deforming Fuk(X)[e] by \({\gamma }= tc_1\).
Let \(\mathcal {S}X:= S^1(\Lambda ^n_{\mathbb {C}}T)X \xrightarrow []{\pi } X\) be the circle bundle associated to the top (complex) exterior power of the tangent bundle of X. By [3, Chapter 11] there exist a global angular form \(\theta \in \Omega ^1(\mathcal {S}X)\) such that \(d\theta = \pi ^*c_1\), where we have picked a representative \(c_1 \in \Omega ^2(X)\) for the first Chern class of TX.
Consider the map \(\Phi : {\mathcal {L}X}\rightarrow {\mathcal {S}X}\) defined by sending an oriented Lagrangian subspace to its orientation class. For an oriented Lagrangian \(l:L \hookrightarrow X\) we thus obtain a map \(\widetilde{l} = \Phi \circ s_L: L \rightarrow {\mathcal {S}X}\). Define the element \(\alpha _L = \widetilde{l}^*\theta \in \Omega ^1(L)\). This satisfies \(d {\alpha }_L = l^*c_1\). Thus, the element \((c_1,{\alpha }_L) \in H^2(X,L)\) represents the Maslov class of L. Note that here we are using a de Rham model for relative cohomology as in [3, Chapter 6].
For \({\alpha }= ({\alpha }_{L_0}, \dots , {\alpha }_{L_k})\) with \({\alpha }_{L_i} \in \Omega ^*(L_i)\) as defined above, set:
Note that this only depends on the homotopy class of u. We then define:
Lemma A.2
The product \({\mathfrak {m}}^{t,e}\) makes \(Fuk^t(X)[e]\) into an Euler-graded \(A_\infty \)-category with \(E = e\partial _e\).
We will now show that Assumption 4.12(2) holds for \(Fuk^t(X)[e]\). To this end, we need to pick a basis for each morphism space \(CF^*(L, L')\). Here, a basis is determined by a choice of designated paths \({\widetilde{p}}\in \mathcal {L}(L, L', p)\) for each \(p \in L \cup L'\). Given such a choice of paths, define the length-1 Hochschild cochain \(\phi \in CC^*\left( Fuk^t(X)[e]\right) \) by:
for the designated path \({\widetilde{p}}\). Then extend \(\phi \) linearly in t and e. Note that this integral is well-defined. For if \([{\widetilde{p}}] = [{\widetilde{p}}'] \in \mathcal {L}(L,L',p)\), then there exists a homotopy \(h: [0,1]^2 \rightarrow \mathcal {L}_pX\). We then have:
But \(\pi \circ \Phi \circ h\) is the constant map \([0,1]^2 \rightarrow \{p\}\). Thus the latter integral is zero.
Lemma A.3
We have: \(\partial _{t}({\mathfrak {m}}^{t,e}) = e \partial _e(\mathfrak {m}^{t,e}) + [{\mathfrak {m}}^{t,e},\phi ]\).
Proof
Fix Lagrangians \(L_0, \dots , L_k\) and intersection points \(p_i \in L_{i-1} \cap L_i\). Let \({\widetilde{p}}_i \in \mathcal {L}(L_{i-1},L_i,p_i)\) be the designated paths defining the bases. Let \(u \in {\mathcal {M}}^0(p_0, \dots , p_k;[u])\) be a holomorphic disk contributing to the product \(\mathfrak {m}^{t,e}({\widetilde{p}}_1, \dots , {\widetilde{p}}_k)\). Let \({\widetilde{p}}_0' \in \mathcal {L}(L_0, L_k, p_0)\) be the output path defined by u. We can then write \({\widetilde{p}}_0' = e^{s}{\widetilde{p}}_0\) for some \(s \in \mathbb {Z}\).
The contribution to \(e\partial _e(\mathfrak {m}^{t,e})({\widetilde{p}}_1, \dots , {\widetilde{p}}_k)\) from this disk is thus given by:
The contribution to \(\partial _t(\mathfrak {m}^{t,e})({\widetilde{p}}_1, \dots , {\widetilde{p}}_k)\) from u is given by:
Finally, the contribution of this disk to \([\mathfrak {m}^{t,e},\phi ]({\widetilde{p}}_1, \dots , {\widetilde{p}}_k)\) is given by:
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 \(\gamma \subset {\mathcal {L}X}\) used to define the lift \({\widetilde{p}}_0'\), and consider the loop \(\gamma \circ ({\widetilde{p}}'_0)^{-1} \subset {\mathcal {L}X}\). By construction \([\gamma \circ ({\widetilde{p}}'_0)^{-1}] = 0 \in \pi _1(\mathcal {L}X, T_{p_0}L_0)\). The loop \(\Phi \circ (\gamma \circ ({\widetilde{p}}'_0)^{-1}): S^1 \rightarrow {\mathcal {S}X}\) is then also null-homologous. As before, this means that we can construct a lift \(\widetilde{u}: \mathbb {D}{\setminus }\{ z_0, \dots , z_k\} \rightarrow {\mathcal {S}X}\) covering u. By including small extra chords \(\gamma _i\) at the punctures \(z_i\), this extends to a continuous map \(\widetilde{u}: \mathbb {D} \rightarrow {\mathcal {S}X}\) with \(\widetilde{u}(\gamma _i) = {\widetilde{p}}_i\) for \(i = 1,\dots , k\) and \(\widetilde{u}(\gamma _0) = {\widetilde{p}}_0'\). We then use Stokes’ theorem to obtain:
There are two different kinds of boundary to \(\widetilde{u}(\partial \mathbb {D})\). The chords \(\gamma _i\) get mapped to the paths \(\phi ({\widetilde{p}}_i)\) and the segments \(\partial _i \mathbb {D}\) to the image of the Lagrangians \(L_{i}\) under \(\Phi \circ s_{L_i}\). We thus have:
For the second equality, we have used that \(\widetilde{u}|_{\partial _i \mathbb {D}} = \Phi \circ s_{L_i} \circ u|_{\partial _i \mathbb {D}}\). The last equality here follows as \(\theta \) is the global angular form on \({\mathcal {S}X}\) and the fact that \(Ind({\widetilde{p}}_0) + 2s = Ind({\widetilde{p}}_0')\). The path \(\Phi \circ ({\widetilde{p}}_0' \circ ({\widetilde{p}}_0)^{-1}): S^1 \rightarrow SX\) then has winding number s, as \({\mathcal {L}X}\) consists of oriented Lagrangian subspaces. We have thus proved Eq. (430). \(\square \)
For an open–closed map defined using holomorphic disks, exactly the same reasoning would show Assumption 4.12(3) holds.
Appendix B: Example: intersection of quadrics
Let X be a complete intersection of two quadric hypersurfaces in \(\mathbb{C}\mathbb{P}^5\). In this section we will show:
Lemma B.1
There exists an isomorphism of TE-structures over \(\mathbb {C} \supset \mathbb {C}\):
First, observe that for both \(QH^*(X)[[u]]\) and \(QH^*(\Sigma _2)[[u]]\), the TEP-structure associated to the odd-degree cohomology is trivial, as here both \(c_1 \star \) and \(\mu \) vanish on the odd-degree cohomology.
Take the basis \(\langle -2H, 1 \rangle \) for \(QH^*(\Sigma _2)[[u]]\), and the standard basis \(\langle 1 \rangle \) for both factors \(\mathcal {E}^{\pm \frac{8}{u}}\). The connection \(\nabla _{\frac{d}{du}}\) for \(\mathcal {E}^{\frac{-8}{u}} \oplus QH^*(\Sigma _2)[[u]]\oplus \mathcal {E}^{\frac{8}{u}}\) is then given by:
Start out with the basis \(\langle 1,H, H^{\cup 2},H^{\cup 3}\rangle \) for \(QH^*(X)[[u]]\). The connection is then given by:
We will show there exists a basis \(\langle v_0, v_1, v_2, v_3\rangle \) for \(QH^*(X)[[u]]\) such that the connection for \(QH^*(X)[[u]]\) agrees with the connection in Eq. (436). To this end, on \(QH^*(X)[[u]]\), first change to a basis \(\langle {\widetilde{v}}_0, {\widetilde{v}}_1, {\widetilde{v}}_2, {\widetilde{v}}_3\rangle \) consisting of generalised eigenvectors for \(c_1 \star \). The change of basis matrix is given by
Thus, for example, \({\widetilde{v}}_1 = \frac{2}{3}H - \frac{1}{6}H^{\cup 3}\). In this new basis, the connection is given by:
Thus, in the basis \(\langle {\widetilde{v}}_0, {\widetilde{v}}_1, {\widetilde{v}}_2, {\widetilde{v}}_3\rangle \) for \(QH^*(X)[[u]]\), the \(u^{-2}\) term of the connection agrees with the one in Eq. (436). In Sect. B.1, we show there exists a unique R-matrix \(R = \sum _{i=0}^{\infty } u^iR_i\) such that in the basis \(\langle v_0, v_1, v_2, v_3\rangle \), where \(v_i = R({\widetilde{v}}_i)\), the connection on \(QH^*(X)[[u]]\) is also given by Eq. (436). We thus define the isomorphism \(\Phi : QH^*(X)[[u]]\rightarrow \mathcal {E}^{\frac{8}{u}} \oplus QH^*(\Sigma _2)[[u]]\oplus \mathcal {E}^{\frac{-8}{u}}\) by
1.1 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
We then find:
Thus, Eq. (445) 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
Similarly we need to have
These conditions hold for \(i=0\), so assume they hold for i. We will now write out all the entries of \(R_{i+1}\) we determined, and the conditions on \(R_i\). Conditions:
We also found the solutions:
Finally, we also find a relation between the entries \(x^{1}_{11}\) and \(x^{1}_{22}\):
The entries \(x^1_{00}\), \(x^1_{12}\) and \(x^1_{33}\) are as of yet undetermined. Now observe that if we are to solve the next recurrence for \(R_{i+2}\) the conditions obtained above should hold with i replaced by \(i+1\). Equation (450) thus determines \(x^1_{00}\) in terms of the entries of \(R_{i+1}\) we already found. We obtain:
Similarly, Eq. (451) determines \(x^1_{33}\):
Next, observe that the coefficients in front of \(x^1_{11}\) and \(x^1_{22}\) in Eqs. 453 and 465 are linearly independent. Thus these two equations can be solved to determine \(x^1_{11}\) and \(x^1_{22}\). This yields:
We are thus left with a single entry, \(x^1_{21}\), to be determined, and a single condition, Eq. (452) to be satisfied. We will use this condition to determine \(x^1_{21}\). For \(R_{i+1}\) this condition reads:
We now substitute in the solutions we found. This gives a new relation for the entries \(x^{0}\) of \(R_i\):
Again, assume that this condition is satisfies. We want to construct \(R_{i+1}\) so that this condition holds with \(i+1\) instead of i. The relation is:
Again, all of these entries have already been determined in terms of the entries \(x^0\) of \(R_i\). So substitute these in. This gives a new relation for the variables \(x^0\) which includes a term \(x^0_{12}\) with a coefficient which is non-zero for all \(i > 0\). We have not written out the equation, as it is rather large. Again, we want this relation to hold for the variables \(x^1\) with \(i+1\) instead of i. This equation then determines \(x^{1}_{21}\) in terms of the other entries \(x^{1}\) which have already been determined. Substituting in the solutions for these entries gives:
By construction, the matrix \(R_{i+1}\) satisfies the conditions we assumed held for \(R_i\). Thus, this recursion uniquely determines R.
Appendix 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 \((-1)^{w_{\mathfrak {s}}(\beta _1)}\).
Here we will use notation and definitions from [45]. Instead of using Pin\(^\pm \) structures, we will use spin structures.
Proof
It suffices to prove this locally. To this end, let \(v: S^2 \rightarrow X\) and \(u: (\mathbb {D},S^1) \rightarrow (X,L)\) be holomorphic maps of degree \(\beta _1\) and \(\beta _2\) respectively. Assume \(u(0) = v(0)\). For a fixed gluing parameter, consider the glued map \(u \# v: (\mathbb {D},S^1) \rightarrow (X,L)\). 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 \(i: L \rightarrow X\) consists of a triangulation of X and L and a vector bundle \(V \rightarrow X^3\) such that \(i^*w_2(V) = w_2(TL)\), and a spin structure \(\mathfrak {p}\) on \(TL \oplus V|_L\). Here \(w_2\) denotes the second Stiefel-Whitney class.
We will now explain briefly, following [45], how to orient \(Det(D_u)\). First, up to homotopy, we may assume that \(u: (\mathbb {D},S^1) \rightarrow (X^3,L^2)\). We then consider the Fredholm problem \(D_u \oplus D^0_u\) on \(u^*(TX \oplus V)\) for some choice of Fredholm operator \(D^0_u\) on the bundle \(u^*(V\otimes {\mathbb {C}}) \rightarrow \mathbb {D}\). The spin structure \(\mathfrak {p}\) on \(TL \oplus V|_L\) defines a canonical orientation of \(Det(D_u \oplus D^0_u)\) (see [45, Proposition 2.8]). Furthermore, as \(u^*(V\otimes {\mathbb {C}}) \rightarrow \mathbb {D}\), and \(\mathbb {D}\) is contractible, there exists a spin structure \(\widetilde{\mathfrak {p}}_0^u\) on \(u^*(V\otimes {\mathbb {C}})\). By restricting to \((u|_L)^*V \rightarrow S^1\), this defines a spin structure \(\mathfrak {p}_0^u\), thus equipping \(Det(D_0^u)\) with an orientation. Solomon shows ([45, Lemma 2.11]) this orientation does not depend on the choice of spin structure \(\widetilde{\mathfrak {p}}_0^u\). As
we obtain a canonical orientation on \(Det(D_u)\).
The procedure for \(D_{u \# v}\) is similar. We pick a Fredholm operator \(D_0^{u \# v}\) and a spin structure \(\widetilde{\mathfrak {p}}_0^{u \# v}\) and obtain an isomorphism:
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 475 as
Now let \(D_0^v\) be an arbitrary (complex) Cauchy-Riemann operator on \(v^*(V \otimes \mathbb {C}) \rightarrow S^2\). We can then glue (see [51, Section 2.4]) the bundle \(v^*(TX \oplus (V \otimes {\mathbb {C}})) \rightarrow S^2\) with \(u^*(TX \oplus (V \otimes {\mathbb {C}})) \rightarrow \mathbb {D}\) 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 \(u|_{\partial \mathbb {D}}^*\mathfrak {p}\). This spin structure was also used for the canonical orientation of \(Det(D_{u \# v} \oplus D_0^{u \# v})\). The result is that the induced map on determinant line bundles:
respects orientations (see [51, Theorem 4.3.3]). Substituting this into Eq. (478), and noting that \(Det(D_v \oplus D_0^v) \cong Det(D_v) \otimes Det(D_0^v)\), we can rewrite the isomorphism as:
Here we have included in the notation the spin structures that we consider on the boundary. Finally, gluing of the operators \(D_0^u\) and \(D_0^v\) allows us to compare them with \(D_0^{u \# v}\). 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:
The spin structure \( \mathfrak {p}_0^{u \# v}\) extends over all of \(\mathbb {D}\), but \(\mathfrak {p}_0^u\) might not. As \(\mathfrak {p}_0^u\) extends over \(u^*V \rightarrow \mathbb {D}\), general obstruction theory tells us that the obstruction to extending \(\mathfrak {p}_0^u\) over \((u \# v)^*V \rightarrow \mathbb {D}\) is given by \(w_2(V)(v_*([S^2])\).
Now, for any bundle \(F \rightarrow \mathbb {D}\), the spin structures on \(F|_{S^1} \rightarrow S^1\) are classified by whether or not they extend over the entirety of \(\mathbb {D}\). Combining this with [45, Lemma 2.10], we find that the orientations in Eq. (483) agree if \(w_2(V)(v_*([S^2]) = 0\), and are opposed otherwise. \(\square \)
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