# Curved Fukaya algebras and the Dubrovin spectrum

Marco Castronovo  Department of Mathematics, Columbia University
<EMAIL_ADDRESS>
###### Abstract.

Under simplified axioms on moduli spaces of pseudo-holomorphic curves, we show
that weakly unobstructed Fukaya algebras of Floer-nontrivial Lagrangians in a
compact symplectic manifold must have curvature in the spectrum of an operator
introduced by Dubrovin, which acts on the big quantum cohomology. We use the
example of the complex Grassmannian $\operatorname{Gr}(2,4)$ to illustrate a
decoupling phenomenon, where the eigenvalues of finite energy truncations
become simple under explicit bulk-deformations.

Partially supported by an AMS-Simons Travel Grant.

##  1\. Introduction

###  1.1. Cohomological $q$-deformations

Let $X$ be a compact symplectic manifold, and denote $\Lambda$ the Novikov
field over $\mathbb{C}$ with formal parameter $q$. The cohomology
$\operatorname{H}(X;\mathbb{C})$ of $X$ has a $q$-deformation
$\operatorname{QH}(X)=\operatorname{H}(X;\mathbb{C})\otimes\Lambda$ known as
quantum cohomology, whose product is defined by counts of $J$-holomorphic
spheres in $X$. This is an associative $\Lambda$-algebra, with the classical
counterpart being recovered for $q=0$. See Abouzaid-McLean-Smith [2] for the
most recent approach to this construction, which also yields $q$-deformations
of many generalized cohomology theories.

###  1.2. Categorical $q$-deformations

Starting with Fukaya-Oh-Ohta-Ono [15] a relative analogue of the construction
above has been studied, whose output is a family of $A_{\infty}$ categories
$\mathcal{F}_{\lambda}(X)$ over $\Lambda$ known as Fukaya categories. General
Lagrangians $L\subset X$ have an obstructed $A_{\infty}$ algebra of Floer
cochains $\operatorname{CF}(L)$. To deal with this, one takes as objects of
$\mathcal{F}_{\lambda}(X)$ the pairs $(L,D)$ with $D\in\operatorname{MC}(L)$ a
solution of the projective Maurer-Cartan equation in $\operatorname{CF}(L)$
with curvature $\lambda\in\Lambda$. Morphisms are given by transverse
intersection points, with composition maps arising from counts of
$J$-holomorphic polygons in $X$ with Lagrangian boundary conditions, twisted
by $D$. It is expected that taking Hochschild cohomology recovers the
generalized eigenspace
$\operatorname{HH}(\mathcal{F}_{\lambda}(X))\cong\operatorname{QH}_{\lambda}(X)$
of eigenvalue $\lambda\in\Lambda$ for a suitable operator, whose action on
quantum cohomology is multiplication by the first Chern class $c_{1}$; see
Perutz-Sheridan [27] and Ganatra [18] for results in this direction.

###  1.3. The status of computations

While computations of quantum cohomology algebras are available in many
examples, this is not the case at the categorical level. One could hope to
begin the computation of a Fukaya category by finding finitely many
Lagrangians that generate the split-closed derived category of its
triangulated closure. General results of this kind were obtained for non-
compact targets: see Seidel [28] for Lefschetz fibrations; Chantraine-
Dimitroglou Rizell-Ghiggini-Golovko [8] and Ganatra-Pardon-Shende [19] for
Weinstein sectors. In the compact case, generators have been found in a
handful of examples using a variety of techniques: toric Fano manifolds
(Evans-Lekili [13]); Calabi-Yau and Fano hypersurfaces (Sheridan [30, 31]);
curves (Seidel [29] and Efimov [12]); some Grassmannians (Castronovo [5, 6]).

###  1.4. The multiplicity issue

Suppose that one has found a finite set of Lagrangians
$L_{i}\in\mathcal{F}_{\lambda}(X)$ which are nontrivial in the derived
category, i.e. with $\operatorname{HF}(L_{i})\neq 0$ for all $i$; this can
often be checked by examining the disk potential of $L_{i}$, see e.g. Sheridan
[31, Proposition 4.2] for monotone tori. A criterion is available for showing
that these objects generate, due to Abouzaid [1] for Weinstein domains and
announced in the compact case by Abouzaid-Fukaya-Oh-Ohta-Ono; see Sheridan
[31, Proposition 1.10] for a proof in the monotone setting. When
$\operatorname{dim}\operatorname{QH}_{\lambda}(X)=1$, this criterion implies
that any of the $L_{i}$ generates, so a key difficulty in computing Fukaya
categories is to address what happens if
$\operatorname{dim}\operatorname{QH}_{\lambda}(X)>1$. In the latter case one
could hope that if $\operatorname{HF}(L_{i},L_{j})=0$ for $i\neq j$ then
generation holds, but Floer cohomologies of pairs of Lagrangians are hard to
compute in practice, unless one knows in advance that the Lagrangians $L_{i}$
can be disjoined by Hamiltonian isotopies.

###  1.5. Generic simplicity

The starting point for this article is the work of Dubrovin [10]. Assume that
$\operatorname{H}(X;\mathbb{C})$ is concentrated in even degrees, and replace
$\operatorname{QH}(X)$ with the big quantum cohomology
$\operatorname{QH}^{B}(X)$. In many interesting examples this algebra is
semisimple for some values of the deformation parameter $B$, with a
decomposition into fields given by the generalized eigenspaces of the
$B$-dependent operator of multiplication by

$K^{B}=c_{1}-\sum_{j\neq 2}\frac{j-2}{2}B_{j}\quad.$

Here $c_{1}$ is corrected by a term depending on $B$ and the degrees of its
homogeneous components $B=\sum_{j}B_{j}$. When $X$ is a Fano variety, Dubrovin
conjectured [11, Conjecture 4.2.2] that $K^{B}$ has simple spectrum (i.e. all
eigenvalues have multiplicity one) for generic $B$ if and only if the derived
category of coherent sheaves on $X$ has a full exceptional collection; see
Halpern-Leistner [20] for some recent progress.

###  1.6. Curvatures are eigenvalues

The big quantum cohomology $\operatorname{QH}^{B}(X)$ is a generalization of
$\operatorname{QH}(X)$, where one includes in the $q$-deformed product
contributions coming from $J$-holomorphic spheres with extra marked points
constrained to map to a so-called bulk cycle in $X$ of homology class $B$
(Figure 2(A)). This construction has a relative analogue, yielding a family of
$A_{\infty}$ categories $\mathcal{F}^{B}_{\lambda}(X)$ whose objects are pairs
$(L,D)$ with $D\in\operatorname{MC}^{B}(L)$ a solution of the projective
Maurer-Cartan equation in the $A_{\infty}$ algebra of bulk-deformed Floer
cochains $\operatorname{CF}^{B}(L)$ with curvature $\lambda\in\Lambda$.
Composition maps in $\mathcal{F}^{B}_{\lambda}(X)$ arise from counts of
$J$-holomorphic polygons in $X$ with Lagrangian boundary conditions, with
extra interior and boundary marked points mapping to the bulk cycle and the
Maurer-Cartan chain respectively (Figure 2(B)). Even after fixing $L$, the
property $\operatorname{MC}^{B}(L)\neq\emptyset$ may depend on $B$, and one
should think of the curvature $\lambda\in\Lambda$ as a function depending on
the bulk-deformation $B$. We prove the following.

######  Theorem 1.1.

(Theorem 6.9) Assume that $D\in\operatorname{MC}^{B}(L)$ is a solution of the
projective Maurer-Cartan equation in the $A_{\infty}$ algebra of bulk-deformed
Floer cochains $\operatorname{CF}^{B}(L)$, and call $\lambda\in\Lambda$ the
curvature of the corresponding $A_{\infty}$ algebra
$\operatorname{CF}^{B}(L,D)$ twisted by $D$. Then
$\operatorname{HF}^{B}(L,D)\neq 0$ implies that $\lambda$ is an eigenvalue of
Dubrovin’s operator $K^{B}$.

This theorem generalizes a result of Auroux [3, Proposition 6.8], which dealt
with the case where $B=0$, $K^{0}=c_{1}$ and $D$ is a local system,
corresponding to the Maurer-Cartan solution $D$ having codegree one; see
Auroux [4, Lemma 4.1] for a comparison between local system and Maurer-Cartan
twists. Our proof follows Auroux’s approach closely. First, we choose as model
for Floer cochains $\operatorname{CF}(L)=C(L)\otimes\Lambda$ the normalized
smooth singular chains, and spell out a set of simplified axioms on moduli of
pseudo-holomorphic curves that allows us to use Fukaya’s general picture [14]
of correspondences and pull-push maps without appealing to virtual techniques;
see Section 2 for more on this approach. In Section 4, we construct a chain
level operation $\cap_{B,D}$ called BMC-deformed $q$-cap, which generalizes
Auroux’s $q$-deformation of the classical cap product [3, Definition 6.2] by
allowing a bulk-deformation parameter $B$ coupled with a Maurer-Cartan
deformation parameter $D$. When $D\in\operatorname{MC}^{B}(L)$, we prove that
this operation descends to a map
$\operatorname{QH}^{B}(X)\otimes\operatorname{HF}^{B}(L,D)\to\operatorname{HF}^{B}(L,D)$
(Proposition 4.5). In Section 5 we verify that $\cap_{B,D}$ endows the bulk-
deformed Floer cohomology $\operatorname{HF}^{B}(L,D)$ twisted by $D$ with the
structure of a $\operatorname{QH}^{B}(X)$-module (Proposition 5.6). Finally,
in Section 6 we reduce the main theorem to proving the identity
$K^{B}\cap_{B,D}[L]=\lambda[L]$, where $\lambda=W^{B}(D)$ is the curvature of
the $A_{\infty}$ algebra of bulk-deformed Floer cochains
$\operatorname{CF}^{B}(L,D)$ twisted by $D$. The notation $W^{B}(D)$ suggests
that $\lambda$ can also be thought of as the value at $D$ of the $B$-deformed
disk potential $W^{B}$ of $L$, for which we provide an explicit formula
(Corollary 6.4). The computation of this BMC-deformed $q$-cap product
(Proposition 6.7) is the hardest part of the proof, and the one that differs
the most from the case with no bulk or Maurer-Cartan deformations. Indeed, a
similar identity holds at the chain level in that case, while it is only true
in homology at this level of generality. The proof consists in checking that
$K^{B}\cap_{B,D}L-W^{B}(D)L$ is exact in $\operatorname{CF}^{B}(L,D)$, which
is done by combining degree arguments on the terms of the projective Maurer-
Cartan equation (Proposition 6.3) with Axioms (5) and (6) from Section 2.
These two axioms are identities relating chains obtained from correspondences
induced by moduli spaces of disks with geodesic constraints contributing to
$-\cap_{B,D}L$ to those induced by unconstrained disks, obtained by forgetting
the vacuous boundary constraint of mapping to $L$; see the discussion in
Section 2. One should expect, by analogy with the case of no bulk, that taking
Hochschild cohomology will recover the generalized eigenspace
$\operatorname{HH}(\mathcal{F}^{B}_{\lambda}(X))\cong\operatorname{QH}^{B}_{\lambda}(X)$
of eigenvalue $\lambda\in\Lambda$ for Dubrovin’s operator $K^{B}$. In light of
this, Theorem 6.9 gives evidence that the multiplicity issue arising in the
computation of $\mathcal{F}_{\lambda}(X)$ will disappear in the bulk-deformed
$\mathcal{F}^{B}_{\lambda}(X)$ whenever Dubrovin’s operator $K^{B}$ has
generically simple spectrum, thus simplifying the problem of describing
generators in such cases.

###  1.7. Finite energy decoupling

For explicit calculations, one might want to go beyond Dubrovin’s observation
that the spectrum of $K^{B}$ is often generically simple and ask the
following.

######  Question 1.2.

Given $X$, for which bulk parameters $B$ is the spectrum of $K^{B}$ simple?

For example, combining Dubrovin’s conjecture with results of Kapranov [24] and
Kuznetsov-Polishchuk [25], one can expect that $K^{B}$ will have generically
simple spectrum for homogeneous varieties $G/P$ at least in classical Lie
types; see Iritani-Koto [22, Proposition 5.11] for unconditional results in
this direction. In Section 7 we focus on what is perhaps the easiest example
of a new phenomenon called finite energy decoupling, that we believe will be
important to study further in order to address Question 1.2 in practice. After
fixing an energy level $\alpha\in\mathbb{R}_{>0}$, one can explicitly
determine those $B$ for which
$\operatorname{dim}\operatorname{QH}^{B}_{\lambda}(X)=1$ for all eigenvalues
$\lambda\in\Lambda$ of the truncation of $K^{B}$ modulo $q^{\alpha}$.
Truncation modulo a power of the formal parameter $q$ corresponds to
considering contributions of $J$-holomorphic spheres satisfying an energy
bound independent of the homotopy class. While finite energy decoupling is not
the same as decoupling the eigenvalues in the spectrum of $K^{B}$, the two
must be related: see Proposition 7.4 for some basic observations. An appealing
feature of finite energy truncations of $K^{B}$ is that they can be explicitly
computed by combining the WDVV equations with the knowledge of a few Gromov-
Witten numbers. This allows us to prove the following.

boxsize=0.3em

######  Theorem 1.3.

(Theorem 7.6) Consider the Grassmannian $X=\operatorname{Gr}(2,4)$ and the
energy level $\alpha=2$. The truncation of $K^{B}$ modulo $q^{2}$ has simple
spectrum when $B\neq 0$ is supported on the Schubert cycles
$X_{\ydiagram{2,0}},X_{\ydiagram{1,1}},X_{\ydiagram{2,2}}\subset\operatorname{Gr}(2,4)$,
while it doesn’t when it is supported on the Schubert cycle
$X_{\ydiagram{2,1}}\subset\operatorname{Gr}(2,4)$.

See Figure 1 for an illustration of this theorem, as well as the interactive
interface [7]. Theorem 7.6 comes close to being the first case where Question
1.2 is answered explicitly, and suggests a qualitative distinction between
bulk-deformations supported on different algebraic cycles. We hope to explore
this further in the future, by looking at energy truncation levels $\alpha>2$
as well as more intricate coalescing patterns for Dubrovin eigenvalues of
higher dimensional Grassmannians (see [5, Figure 1]).

(a) $X_{d}=X_{\ydiagram{2,0}}$
---
$t=0.5+i$

(b) $X_{d}=X_{\ydiagram{1,1}}$
---
$t=0.5+i$

(c) $X_{d}=X_{\ydiagram{2,1}}$
---
$t=0.5+i$

(d) $X_{d}=X_{\ydiagram{2,2}}$
---
$t=0.5+i$

(e) $X_{d}=X_{\ydiagram{2,0}}$
---
$t=1+2i$

(f) $X_{d}=X_{\ydiagram{1,1}}$
---
$t=1+2i$

(g) $X_{d}=X_{\ydiagram{2,1}}$
---
$t=1+2i$

(h) $X_{d}=X_{\ydiagram{2,1}}$
---
$t=1+2i$

(i) $X_{d}=X_{\ydiagram{2,0}}$
---
$t=1.5+3i$

(j) $X_{d}=X_{\ydiagram{1,1}}$
---
$t=1.5+3i$

(k) $X_{d}=X_{\ydiagram{2,1}}$
---
$t=1.5+3i$

(l) $X_{d}=X_{\ydiagram{2,2}}$
---
$t=1.5+3i$

Figure 1. Spectrum of the $\alpha=2$ truncation of $K^{B}$ with $B=0$ (red)
and $B=t[X_{d}]$ (blue) for different Schubert cycles
$X_{d}\subset\operatorname{Gr}(2,4)$ and $t\in\mathbb{C}^{\times}$, after
specializing to $q=1$. The zero eigenvalue has algebraic multiplicity two, and
it decouples whenever $X_{d}\neq X_{\ydiagram{2,1}}$.

###  1.8. Relation with closed-open maps

By analogy with the case $B=0$, one can expect the existence of algebra maps
$\mathcal{CO}^{B}_{\lambda}:\operatorname{QH}^{B}(X)\to\operatorname{HH}(\mathcal{F}^{B}_{\lambda}(X))$
obtained by bulk-deformation from ordinary closed-open maps, and whose length
$0$ part $(\mathcal{CO}^{B}_{\lambda})^{0}$ is related to our BMC-deformed
$q$-cap operation by
$m^{2}_{B,D}((\mathcal{CO}^{B}_{\lambda})^{0}(-),-)=-\cap_{B,D}-$. In this
language, our Theorem 6.9 should follow after verifying that: (1)
$(\mathcal{CO}^{B}_{\lambda})^{0}:\operatorname{QH}^{B}(X)\to\operatorname{HF}^{B}(L,D)$
is a unital algebra map; (2)
$(\mathcal{CO}^{B}_{\lambda})^{0}(K^{B})=\lambda[L]$, where $\lambda=W^{B}(D)$
is as in Corollary 6.4. See Sheridan [31, Corollary 2.10] for an alternative
proof (in the monotone setting) of Auroux’s result [3, Proposition 6.8] using
this approach. It is our understanding that Hugtenberg [21] carries out the
construction of length $0$ bulk-deformed closed-open maps in the de Rham model
for Floer cochains, under the same assumptions used by Solomon-Tukachinsky
[32]. Also, it is our understanding that Venugopalan-Woodward-Xu [33] carry
out the construction of full length bulk-deformed closed-open maps in the
pearly model for Floer cochains, under rationality assumptions that allow the
use of stabilizing divisors.

Acknowledgements I thank Mohammed Abouzaid for advertising bulk-deformations
to me in 2019. Denis Auroux provided crucial feedback at various points in the
project, especially suggesting initial versions of the forgetful axioms
(5)-(6). I thank Kenji Fukaya for useful comments regarding Section 2.
Finally, I thank Kai Hugtenberg and Chris Woodward for explaining their work
on bulk-deformed closed-open maps.

##  2\. Axiomatic framework

This section introduces some moduli spaces of $J$-holomorphic curves used
later in the article to define various types of operations. We assume that
such moduli spaces satisfy a set of simplified axioms, and discuss the
heuristic behind their choice. Stating and proving virtual analogues of these
axioms is an important and interesting problem, particularly open for the
forgetful axioms (5)-(6). See the monograph by Fukaya-Oh-Ohta-Ono [16] for the
most recent approach to this kind of questions, based on the notion of
Kuranishi structure.

###  2.1. Pull-push

Suppose that $M_{\otimes}$ and $M_{\odot}$ are closed manifolds of dimensions
$d_{\otimes}$ and $d_{\odot}$, and let $\mathcal{M}$ be a manifold with
corners of dimension $d$, endowed with maps
$\pi_{\otimes}:M_{\otimes}\leftarrow\mathcal{M}$ and
$\pi_{\odot}:\mathcal{M}\rightarrow M_{\odot}$.

######  Definition 2.1.

If $\pi_{\otimes}$ is a submersion, call $\kappa=(\pi_{\otimes},\pi_{\odot})$
a correspondence from $M_{\otimes}$ to $M_{\odot}$, also denoted
$\kappa:M_{\otimes}\rightsquigarrow M_{\odot}$.

Following Fukaya [14], a correspondence $\kappa:M_{\otimes}\rightsquigarrow
M_{\odot}$ induces a map $\kappa_{\\#}:C_{\bullet}(M_{\otimes})\to
C_{\bullet+d-d_{\otimes}}(M_{\odot})$, sometimes called pull-push, via the
following construction. A smooth singular chain on $M_{\otimes}$ is a linear
combination of smooth maps $\sigma:\Delta^{\bullet}\to M_{\otimes}$ from the
$\bullet$-dimensional simplex. In a well-behaved theory of manifolds with
corners (see e.g. Joyce [23]), the assumption that $\pi_{\otimes}$ is a
submersion guarantees the existence of fiber products
$\Delta^{\bullet}\prescript{}{\sigma}{\times}_{\pi_{\otimes}}M_{\otimes}$
which are manifolds with corners of dimension $\bullet+d-d_{\otimes}$. After
choosing a triangulation $\sigma_{i}$ for each of them, the fiber product
gives maps $\sigma_{i}:\Delta^{\bullet+d-d_{\otimes}}\to\mathcal{M}$. One can
define $\kappa_{\\#}(\sigma)=\sum_{i}\pi_{\odot}\circ\sigma_{i}$ and extend by
linearity to any smooth singular chain.

###  2.2. Correspondence axioms

The following axioms (1)-(4) guarantee that various moduli spaces of
$J$-holomorphic curves used to construct chain-level operations in the article
do give rise to correspondences in the sense of Definition 2.1. Fix an almost
complex structure $J$ compatible with the symplectic structure of $X$, and
suppress it from the notation. Denote $\mathcal{M}_{l}(X,\beta)$ the moduli
space of stable $J$-holomorphic spheres in $X$, with $l$ marked points and
class $\beta\in\pi_{2}(X)$. For each marked point one has a corresponding
evaluation map $\mathcal{M}_{l}(X,\beta)\to X$.

(1) Spheres: $\mathcal{M}_{l}(X,\beta)$ is a compact manifold of dimension
$2n+2c_{1}(\beta)+2l-6$, and the evaluation maps are submersions.

Denote $\mathcal{M}_{k,l}(L,\beta)$ the moduli space of stable $J$-holomorphic
disks in $X$ with boundary in $L$, with $k$ boundary marked points and $l$
interior marked points, of class $\beta\in\pi_{2}(X,L)$. For each interior
marked point one has a corresponding evaluation map
$\mathcal{M}_{k,l}(L,\beta)\to X$, and similarly for each boundary marked
point one has a corresponding evaluation map $\mathcal{M}_{k,l}(L,\beta)\to
L$.

(2) Disks: $\mathcal{M}_{k,l}(L,\beta)$ is a compact manifold with corners of
dimension $n+\mu(\beta)+k+2l-3$, and the evaluation maps are submersions.

Denote $\mathcal{G}^{\prime}_{2+k,1+l}(L,\beta)$ the moduli space of stable
$J$-holomorphic disks in $X$ with boundary in $L$, with $k$ boundary marked
points and $l$ interior marked points, of class $\beta\in\pi_{2}(X,L)$, with
one extra interior and one extra boundary marked point, both constrained to be
on a hyperbolic geodesic from the first boundary marked point. For each
interior marked point one has a corresponding evaluation map
$\mathcal{G}^{\prime}_{2+k,1+l}(L,\beta)\to X$, and similarly for each
boundary marked point one has a corresponding evaluation map
$\mathcal{G}^{\prime}_{2+k,1+l}(L,\beta)\to L$.

(3) Geodesic′: $\mathcal{G}^{\prime}_{2+k,1+l}(L,\beta)$ is a compact manifold
with corners of dimension $n+\mu(\beta)+k+2l$, and the evaluation maps are
submersions.

Denote $\mathcal{G}^{\prime\prime}_{2+k,2+l}(L,\beta)$ the moduli space of
stable $J$-holomorphic disks in $X$ with boundary in $L$, with $k$ boundary
marked points and $l$ interior marked points, of class $\beta\in\pi_{2}(X,L)$,
with two extra interior and one extra boundary marked point, all constrained
to be on a hyperbolic geodesic from the first boundary marked point. For each
interior marked point one has a corresponding evaluation map
$\mathcal{G}^{\prime\prime}_{2+k,2+l}(L,\beta)\to X$, and similarly for each
boundary marked point one has a corresponding evaluation map
$\mathcal{G}^{\prime\prime}_{2+k,2+l}(L,\beta)\to L$.

(4) Geodesic′′: $\mathcal{G}^{\prime\prime}_{2+k,2+l}(L,\beta)$ is a compact
manifold with corners of dimension $n+\mu(\beta)+k+2l+1$, and the evaluation
maps are submersions.

######  Notation 2.2.

Throughout the paper, correspondences $\kappa:M_{\otimes}\rightsquigarrow
M_{\odot}$ induced by moduli spaces $\mathcal{M}$ of stable $J$-holomorphic
spheres/disks will be described by designating some marked points as inputs,
denoted $\otimes$, and one marked point as output, denoted $\odot$. The
corresponding evaluations then specify maps
$\pi_{\otimes}:M_{\otimes}\leftarrow\mathcal{M}$ and
$\pi_{\odot}:\mathcal{M}\rightarrow M_{\odot}$ to the corresponding products
of $L$ and $X$.

###  2.3. Forgetful axioms

The correspondence axioms listed so far can be grouped in two classes: (1)-(2)
involve no geodesic constraint, while (3)-(4) do. The results of Section 6
rely on two additional axioms, which relate chains obtained from moduli of
disks with and without geodesic constraints as in (3) and (2) respectively.
Both axioms describe what happens when one forgets the constraint of having
the input boundary marked point on the hyperbolic geodesic mapping to
$L\subset X$. Since disks already have boundary on $L$, this is a vacuous
constraint and one expects to sweep out the same chain by evaluation from a
different moduli space where the constraint is removed. The result of this
operation depends on what kind of cycle the remaining interior marked point on
the hyperbolic geodesic was constrained to map to.

(5) Forget′: If $L\subset X\setminus Z$ with $|Z|^{\vee}=2$ and
$[Z]^{\vee}=c_{1}$ then
$\sum_{k_{0}+k_{1}=k}(Z\cap_{B,D}L)_{\beta}^{k_{0},k_{1};l}=\frac{\mu(\beta)}{2}q^{\beta}_{k,l}(D^{k},B^{l})\quad.$

(6) Forget′′: If $B=\sum_{j}B_{j}$ with $|B_{j}|^{\vee}=j\neq 2$ then
$\sum_{k_{0}+k_{1}=k}(B_{j}\cap_{B,D}L)_{\beta}^{k_{0},k_{1};l}=\frac{1}{l+1}\sum_{t=0}^{l}q^{\beta}_{k,l+1}(D^{k},B^{t},B_{j},B^{l-t})\quad.$

##  3\. Bulk and Maurer-Cartan deformations

This section reviews the definition of bulk-deformed quantum cup product
$\star_{B}$, as well as of the coupled bulk/Maurer-Cartan deformation
$m^{d}_{B,D}$ of the $A_{\infty}$ algebra of a Lagrangian $L\subset X$. These
constructions rely on the correspondence axioms (1)-(2) from Section 2.

boxsize=1em

(a) $\kappa:X^{2}\rightsquigarrow X$ induced by $\mathcal{M}_{3+l}(X,\beta)$
for $l\geq 0$

(b) $\kappa:L^{d}\rightsquigarrow L$ induced by
$\mathcal{M}_{d+k_{0}+\ldots+k_{d},l}(L,\beta)$ for $k_{0},\ldots,k_{d}\geq 0$
and $l\geq 0$

Figure 2. Some correspondences induced by moduli of stable $J$-holomorphic
spheres/disks. The interior/boundary marked points denoted $\blacksquare$ are
constrained to map to $B/D$.

###  3.1. B-deformations

Denote $C(X)$ the module of normalized smooth singular chains on $X$ with
$\mathbb{C}$ coefficients. Throughout the paper, fix a cycle $B\in C(X)$
concentrated in even degrees and call it bulk deformation parameter. Denote
$B=\sum_{j}B_{j}$ its decomposition into homogeneous components, indexed by
their codegrees $j=2n-|B_{j}|$.

######  Definition 3.1.

The B-deformed cup product on $C(X)\otimes\Lambda$ is

$Q_{1}\star_{B}Q_{2}=\sum_{l\geq
0}\sum_{\beta\in\pi_{2}(X)}q^{\omega(\beta)}\frac{1}{l!}(Q_{1}\star_{B}Q_{2})_{\beta}^{l}\quad,$

where the chain $(Q_{1}\star_{B}Q_{2})_{\beta}^{l}=\kappa_{\\#}(Q_{1}\times
Q_{2}\times B^{l})$ is obtained via pull-push from the correspondence $\kappa$
in Figure 2(A).

######  Proposition 3.2.

(see e.g. McDuff-Salamon [26]) The B-deformed cup product descends to
homology, and makes the corresponding $\Lambda$-module an algebra with unit
$[X]$ denoted $\operatorname{QH}^{B}(X)$.

###  3.2. BMC-deformations

Denote $C(L)$ the module of normalized smooth singular chains on $L$ with
$\mathbb{C}$ coefficients. Throughout the paper, fix a chain $D\in
C(L)\otimes\Lambda_{>0}$ concentrated in odd degrees and call it Maurer-Cartan
deformation parameter. Denote $D=\sum_{i}D_{i}$ its decomposition into
homogeneous components, indexed by their codegrees $i=n-|D_{i}|$.

######  Definition 3.3.

For $d\geq 0$ the BMC-deformed $A_{\infty}$ products on $C(L)\otimes\Lambda$
are

$m^{d}_{B,D}(C_{1},\ldots,C_{d})=\sum_{k_{0},\ldots,k_{d}\geq 0}\sum_{l\geq
0}\sum_{\beta\in\pi_{2}(X,L)}q^{\omega(\beta)}\frac{1}{l!}q^{\beta}_{d+k_{0}+\ldots+k_{d},l}(D^{k_{0}},C_{1},D^{k_{1}},\ldots,C_{d},D^{k_{d}},B^{l})\quad.$

When $\beta\neq 0$ the chain
$q^{\beta}_{d+k_{0}+\ldots+k_{d},l}(D^{k_{0}},C_{1},D^{k_{1}},\ldots,C_{d},D^{k_{d}},B^{l})=\kappa_{\\#}(D^{k_{0}}\times
C_{1}\times D^{k_{1}}\times\cdots\times C_{d}\times D^{k_{d}}\times B^{l})$ is
obtained via pull-push from the correspondence $\kappa$ in Figure 2(B). For
$\beta=0$ set $q^{0}_{\bullet,\bullet}=0$ except for $q^{0}_{1,0}=\partial$.

The chain $D\in C(L)\otimes\Lambda_{>0}$ satisfies the projective Maurer-
Cartan equation if $m^{0}_{B,D}(1)=\lambda L$ for some $\lambda\in\Lambda$, in
which case $\lambda$ is called the curvature. Denote
$\operatorname{MC}^{B}(L)$ the set of all solutions $D$ of the projective
Maurer-Cartan equation.

######  Proposition 3.4.

(see Fukaya-Oh-Ohta-Ono [15]) The maps $m^{\bullet}_{B,D}$ are an $A_{\infty}$
algebra structure on the $\Lambda$-module $C(L)$. Moreover if
$D\in\operatorname{MC}^{B}(L)$ then $(m^{1}_{B,D})^{2}=0$, and the
BMC=deformed product $m^{2}_{B,D}$ descends to homology and makes the
corresponding $\Lambda$-module an algebra with unit $[L]$ denoted
$\operatorname{HF}^{B}(L,D)$.

##  4\. BMC deformation of cap product

This section introduces a new coupled bulk and Maurer-Cartan (BMC for short)
deformation $\cap_{B,D}$ of the quantum cap product operation. This
constructions relies on the correspondence axiom (3) from Section 2.

###  4.1. Chain-level definition

The definition of the BMC-deformed quantum cap product $\cap_{B,D}$ is given
at the chain-level as follows.

######  Definition 4.1.

The BMC-deformed cap product is

$Q\cap_{B,D}C=\sum_{k_{0},k_{1}\geq 0}\sum_{l\geq
0}\sum_{\beta\in\pi_{2}(X,L)}q^{\omega(\beta)}\frac{1}{l!}(Q\cap_{B,D}C)_{\beta}^{k_{0},k_{1};l}\quad.$

When $\beta\neq 0$ the chain
$(Q\cap_{B,D}C)_{\beta}^{k_{0},k_{1};l}=\kappa_{\\#}(D^{k_{0}}\times C\times
D^{k_{1}}\times Q\times B^{l})$ is obtained via pull-push from the
correspondence $\kappa$ in Figure 3 (A). For $\beta=0$ set
$(\cdot\cap_{B,D}\cdot)_{0}^{\bullet,\bullet;\bullet}=0$ except for
$(\cdot\cap_{B,D}\cdot)_{0}^{0,0;0}=\cap$.

boxsize=1em

(a) $\kappa:X\times L\rightsquigarrow L$ induced by
$\mathcal{G}^{\prime}_{2+k_{0}+k_{1},1+l}(L,\beta)$ for $k_{0},k_{1}\geq 0$
and $l\geq 0$

(b) $\kappa:X^{2}\times L\rightsquigarrow L$ induced by
$\mathcal{G}^{\prime\prime}_{2+k_{0}+k_{1},2+l}(L,\beta)$ for $k_{0},k_{1}\geq
0$ and $l\geq 0$

Figure 3. Some correspondences induced by moduli of stable $J$-holomorphic
spheres/disks with geodesic constraints. The interior/boundary marked points
denoted $\blacksquare$ are constrained to map to $B/D$.

Recorded below is a degree formula that will be used later for arguing that
certain chains are degenerate.

######  Lemma 4.2.

If $Q,C,B,D$ are homogeneous chains, then

$|(Q\cap_{B,D}C)_{\beta}^{k_{0},k_{1};l}|=|Q|+|C|-2n+\mu(\beta)+k_{0}+k_{1}+2l-(k_{0}+k_{1})(n-|D|)-l(2n-|B|)\quad.$

###  4.2. Relation with Maurer-Cartan equation

A priori, the BMC-deformed quantum cap product $\cap_{B,D}$ is only defined at
the chain level. The following discussion explains why it descends to homology
under the assumption that $D$ satisfies the projective Maurer-Cartan equation,
i.e. $D\in\operatorname{MC}^{B}(L)$.

######  Definition 4.3.

Define:

$h^{+}_{B,D}(C,E,Q)=\sum_{l\geq 0}\sum_{k_{0}\geq
0}\sum_{\begin{subarray}{c}k_{1}^{L}\geq 0\\\ k_{1}^{R}\geq
0\end{subarray}}\sum_{\beta\in\pi_{2}(X,L)}q^{\omega(\beta)}\frac{1}{l!}h^{+}_{B,D}(C,E,Q)_{\beta}^{k_{0};k_{1}^{L},k_{1}^{R};l}\quad\textrm{and}$
$h^{-}_{B,D}(C,E,Q)=\sum_{l\geq 0}\sum_{\begin{subarray}{c}k_{0}^{L}\geq 0\\\
k_{0}^{R}\geq 0\end{subarray}}\sum_{k_{1}\geq
0}\sum_{\beta\in\pi_{2}(X,L)}q^{\omega(\beta)}\frac{1}{l!}h^{-}_{B,D}(C,E,Q)_{\beta}^{k_{0}^{L},k_{0}^{R};k_{1};l}\quad,$

where the chain
$h^{+}_{B,D}(C,E,Q)_{\beta}^{k_{0};k_{1}^{L},k_{1}^{R};l}=\kappa_{\\#}(D^{k_{0}}\times
C\times D^{k_{1}^{R}}\times E\times D^{k_{1}^{L}}\times Q\times B^{l})$ is
obtained via pull-push from the correspondence $\kappa$ in Figure 4 (A), and
the chain
$h^{-}_{B,D}(C,E,Q)_{\beta}^{k_{0}^{L},k_{0}^{R};k_{1};l}=\kappa_{\\#}(D^{k_{0}^{L}}\times
E\times D^{k_{0}^{R}}\times C\times D^{k_{1}}\times Q\times B^{l})$ from the
one in Figure 4 (B).

boxsize=1em

(a) $\kappa:L^{2}\times X\rightsquigarrow L$ induced by
$\mathcal{G}^{\prime}_{3+k_{0}+k_{1}^{L}+k_{1}^{R},1+l}(L,\beta)$ for
$k_{0},k_{1}^{L},k_{1}^{R}\geq 0$ and $l\geq 0$

(b) $\kappa:L^{2}\times X\rightsquigarrow L$ induced by
$\mathcal{G}^{\prime}_{3+k_{0}^{L}+k_{0}^{R}+k_{1},1+l}(L,\beta)$ for
$k_{0}^{L},k_{0}^{R},k_{1}\geq 0$ and $l\geq 0$

Figure 4. Some correspondences induced by moduli of stable $J$-holomorphic
disks with single geodesic constraints. The interior/boundary marked points
denoted $\blacksquare$ are constrained to map to $B/D$.

######  Lemma 4.4.

If $|E|=n$ then the chains $h^{\pm}_{B,D}(C,E,Q)$ are degenerate.

###### Proof.

Assume that all chains are homogeneous. One can see that

$|h^{+}_{B,D}(C,E,Q)_{\beta}^{k_{0};k_{1}^{L},k_{1}^{R};l}|=n+\mu(\beta)-|C|^{\vee}-|E|^{\vee}-|Q|^{\vee}-(k_{0}+k_{1}^{L}+k_{1}^{R})(|D|^{\vee}-1)-l(|B|^{\vee}-2)+1\quad.$

The constraint of having an input boundary point mapping to $L$ is vacuous, so
one has

$h^{+}_{B,D}(C,E,Q)=(Q\cap_{B,D}C)_{\beta}^{k_{0},k_{1}^{L}+k_{1}^{R};l}\quad.$

Since the chain on the right has degree one less than what computed above when
$|E|=n$ (i.e. $|E|^{\vee}=0$) thanks to Lemma 4.2, the chain on the left is
degenerate. A similar argument applies to $h^{-}_{B,D}(C,E,Q)$. ∎

######  Proposition 4.5.

Up to signs one has

$m^{1}_{B,D}(Q\cap_{B,D}C)=(\partial
Q)\cap_{B,D}C+Q\cap_{B,D}m^{1}_{B,D}(C)+h^{\pm}_{B,D}(C,m^{0}_{B,D}(1),Q)\quad,$

and when $D\in\operatorname{MC}^{B}(L)$ the BMC-deformed cap product descends
to a map
$\operatorname{QH}^{B}(X)\otimes\operatorname{HF}^{B}(L,D)\to\operatorname{HF}^{B}(L,D)$.

###### Proof.

In the formula defining the BMC-deformed differential (Definition 3.3 with
$k=1$), separate the classical ($\beta=0$) and quantum ($\beta\neq 0$) parts:

$m^{1}_{B,D}(\cdot)=q^{0}_{1,0}(\cdot)+\sum_{k_{0},k_{1}\geq 0}\sum_{l\geq
0}\frac{1}{l!}q^{0}_{1+k_{0}+k_{1},l}(D^{k_{0}},\cdot,D^{k_{1}},B^{l})+$
$\underbracket{\sum_{\begin{subarray}{c}\beta\in\pi_{2}(X,L)\\\ \beta\neq
0\end{subarray}}\sum_{k_{0},k_{1}\geq 0}\sum_{l\geq
0}T^{\omega(\beta)}\frac{1}{l!}q^{\beta}_{1+k_{0}+k_{1},l}(D^{k_{0}},\cdot,D^{k_{1}},B^{l})}_{\tilde{m}^{1}_{B,D}(\cdot)}\quad.$

The classical part is $q^{0}_{1,0}=\partial$, and the claim to be proven can
be rewritten as

$\partial(Q\cap_{B,D}C)=\underbracket{\tilde{m}^{1}_{B,D}(Q\cap_{B,D}C)}_{(1)}+\underbracket{(\partial
Q)\cap_{B,D}C+Q\cap_{B,D}(\partial C)}_{(2)}+$
$\underbracket{Q\cap_{B,D}\tilde{m}^{1}_{B,D}(C)}_{(3)}+\underbracket{h^{+}(C,m^{0}_{B,D}(1),Q)}_{(4)}+\underbracket{h^{-}(C,m^{0}_{B,D}(1),Q)}_{(5)}\quad.$

The terms (1)-(5) arise from the following degenerations of disks contributing
to $Q\cap_{B,D}C$ (see Figure 5).

  1. (1)

Using Definition 4.1 and Definition 3.3, one can rewrite this term as

$\sum_{\begin{subarray}{c}\beta[1],\beta[2]\\\ \beta[1]\neq
0\end{subarray}}\sum_{\begin{subarray}{c}k_{0}[1]\\\
k_{1}[1]\end{subarray}}\sum_{l[1]}\sum_{\begin{subarray}{c}k_{0}[2]\\\
k_{1}[2]\end{subarray}}\sum_{l[2]}T^{\omega(\beta[1])}T^{\omega(\beta[2])}\frac{1}{l[1]!l[2]!}$
$q^{\beta[1]}_{1+k_{0}[1]+k_{1}[1],l[1]}(D^{k_{0}[1]},(Q\cap_{B,D}C)_{\beta[2]}^{k_{0}[2],k_{1}[2];l[2]},D^{k_{1}[1]},B^{l[1]})\quad.$

Each chain in the sum corresponds to disk bubbling as in Figure 5 (A), where
the interior marked point mapping to $Q$ moves along the geodesic constraint
all the way up to the input boundary point mapping to $C$ . If a disk
contributes $T^{\omega(\beta)}(Q\cap_{B,D}C)^{k_{0},k_{1},l}_{\beta}$ to
$Q\cap_{B,D}C$, then after this kind of degeneration its class breaks up
$\beta=\beta[1]+\beta[2]$, while the boundary marked points mapping to $D$
split into two groups $k_{0}=k_{0}[1]+k_{1}[2]$, with $k_{0}[\bullet]$ and
$k_{1}[\bullet]$ respectively on the lower and upper arcs of the two disks.
Similarly, the interior marked points mapping to $B$ split into two groups
$l=l[1]+l[2]$. Degenerations inducing different partitions of the $l$ interior
points into groups of size $l[1]$ and $l[2]$ contribute the same chain, and
since there are $l!(l[1]!l[2]!)^{-1}$ such partitions overall one gets the
coefficient

$T^{\omega(\beta)}\frac{1}{l!}\frac{l!}{l[1]!l[2]!}=T^{\omega(\beta[1])}T^{\omega(\beta[2])}\frac{1}{l[1]!l[2]!}\quad.$

  2. (2)

These terms correspond to degenerating the condition of mapping marked points
to $Q$ and $D$ to conditions of mapping them to $\partial Q$ and $\partial D$.
These are codimension one thanks to Lemma 4.2.

  3. (3)

Using Definition 4.1 and Definition 3.3, one can rewrite this term as

$\sum_{\begin{subarray}{c}\beta[1],\beta[2]\\\ \beta[1]\neq
0\end{subarray}}\sum_{\begin{subarray}{c}k_{0}[1]\\\
k_{1}[1]\end{subarray}}\sum_{l[1]}\sum_{\begin{subarray}{c}k_{0}[2]\\\
k_{1}[2]\end{subarray}}\sum_{l[2]}T^{\omega(\beta[1])}T^{\omega(\beta[2])}\frac{1}{l[1]!l[2]!}$
$(Q\cap_{B,D}q^{\beta[1]}_{1+k_{0}[1]+k_{1}[1],l[1]}(D^{k_{0}[1]},C,D^{k_{1}[1]},B^{l[1]}))_{\beta[2]}^{k_{0}[2],k_{1}[2];l[2]}\quad.$

Each chain in the sum corresponds to disk bubbling as in Figure 5 (B), where
the interior marked point mapping to $Q$ moves along the geodesic constraint
all the way up to the output boundary point. If a disk contributes
$T^{\omega(\beta)}(Q\cap_{B,D}C)^{k_{0},k_{1},l}_{\beta}$ to $Q\cap_{B,D}C$,
then after this kind of degeneration its class breaks up
$\beta=\beta[1]+\beta[2]$, while the boundary marked points mapping to $D$
split into two groups $k_{0}=k_{0}[1]+k_{1}[2]$, with $k_{0}[\bullet]$ and
$k_{1}[\bullet]$ respectively on the lower and upper arcs of the two disks.
Similarly, the interior marked points mapping to $B$ split into two groups
$l=l[1]+l[2]$. Degenerations inducing different partitions of the $l$ interior
points into groups of size $l[1]$ and $l[2]$ contribute the same chain, and
since there are $l!(l[1]!l[2]!)^{-1}$ such partitions overall one gets the
coefficient

$T^{\omega(\beta)}\frac{1}{l!}\frac{l!}{l[1]!l[2]!}=T^{\omega(\beta[1])}T^{\omega(\beta[2])}\frac{1}{l[1]!l[2]!}\quad.$

  4. (4)

Using Definition 4.3 and Definition 3.3, one can rewrite this term as

$\sum_{\begin{subarray}{c}\beta[1]\\\
\beta[2]\end{subarray}}\sum_{k[1]}\sum_{k_{0}[2]}\sum_{\begin{subarray}{c}k_{1}^{L}[2]\\\
k_{1}^{R}[2]\end{subarray}}\sum_{\begin{subarray}{c}l[1]\\\
l[2]\end{subarray}}T^{\omega(\beta[1])}T^{\omega(\beta[2])}\frac{1}{l[1]!l[2]!}$
$h^{+}_{B,D}(C,q^{\beta[1]}_{k[1],l[1]}(D^{k[1]},B^{l[1]}),Q)_{\beta[2]}^{k_{0}[2];k_{1}^{L}[2],k_{1}^{R}[2];l[2]}\quad.$

Each chain in the sum corresponds to disk bubbling as in Figure 5 (C), where a
group of boundary marked points mapping to $D$ on the upper arc of the disk
with geodesic constraint come together. If a disk contributes
$T^{\omega(\beta)}(Q\cap_{B,D}C)^{k_{0},k_{1},l}_{\beta}$ to $Q\cap_{B,D}C$,
then after this kind of degeneration its class breaks up
$\beta=\beta[1]+\beta[2]$, while the boundary marked points mapping to $D$
that are on the upper arc split into three groups
$k_{1}=k_{1}^{L}[2]+k[1]+k_{1}^{R}[2]$, with $k_{1}^{L}[2]$ and $k_{1}^{R}[2]$
respectively on the left and right of the attaching point of the disk bubble,
while $k[1]$ are on the new disk. The interior marked points mapping to $B$
split into two groups $l=l[1]+l[2]$. Degenerations inducing different
partitions of the $l$ interior points into groups of size $l[1]$ and $l[2]$
contribute the same chain, and since there are $l!(l[1]!l[2]!)^{-1}$ such
partitions overall one gets the coefficient

$T^{\omega(\beta)}\frac{1}{l!}\frac{l!}{l[1]!l[2]!}=T^{\omega(\beta[1])}T^{\omega(\beta[2])}\frac{1}{l[1]!l[2]!}\quad.$

  5. (5)

Using Definition 4.3 and Definition 3.3, one can rewrite this term as

$\sum_{\begin{subarray}{c}\beta[1]\\\
\beta[2]\end{subarray}}\sum_{k[1]}\sum_{\begin{subarray}{c}k_{0}^{L}[2]\\\
k_{0}^{R}[2]\end{subarray}}\sum_{k_{1}[2]}\sum_{\begin{subarray}{c}l[1]\\\
l[2]\end{subarray}}T^{\omega(\beta[1])}T^{\omega(\beta[2])}\frac{1}{l[1]!l[2]!}$
$h^{-}_{B,D}(C,q^{\beta[1]}_{k[1],l[1]}(D^{k[1]},B^{l[1]}),Q)_{\beta[2]}^{k_{0}^{L}[2],k_{0}^{R}[2];k_{1}[2];l[2]}\quad.$

Each chain in the sum corresponds to disk bubbling as in Figure 5 (D), where a
group of boundary marked points mapping to $D$ on the lower arc of the disk
with geodesic constraint come together. If a disk contributes
$T^{\omega(\beta)}(Q\cap_{B,D}C)^{k_{0},k_{1},l}_{\beta}$ to $Q\cap_{B,D}C$,
then after this kind of degeneration its class breaks up
$\beta=\beta[1]+\beta[2]$, while the boundary marked points mapping to $D$
that are on the upper arc split into three groups
$k_{0}=k_{0}^{L}[2]+k[1]+k_{0}^{R}[2]$, with $k_{0}^{L}[2]$ and $k_{0}^{R}[2]$
respectively on the left and right of the attaching point of the disk bubble,
while $k[1]$ are on the new disk. The interior marked points mapping to $B$
split into two groups $l=l[1]+l[2]$. Degenerations inducing different
partitions of the $l$ interior points into groups of size $l[1]$ and $l[2]$
contribute the same chain, and since there are $l!(l[1]!l[2]!)^{-1}$ such
partitions overall one gets the coefficient

$T^{\omega(\beta)}\frac{1}{l!}\frac{l!}{l[1]!l[2]!}=T^{\omega(\beta[1])}T^{\omega(\beta[2])}\frac{1}{l[1]!l[2]!}\quad.$

By Gromov compactness, the only other possible degenerations of a disk
contributing to $Q\cap_{B,D}C$ consist of sphere bubbling, or disk bubbling
arising from moving the interior marked point mapping to $Q$ all the way up to
the upper or lower arcs. None of these degenerations shows up in
$\partial(Q\cap_{B,D}C)$, because the first has codimension two while the
second violates the geodesic constraint. Finally, the BMC-defomred cap product
descends to homology whenever $D\in\operatorname{MC}^{B}(L)$ because in this
case $m^{0}_{B,D}(1)=W^{B}(D)L$, and hence the terms
$h^{\pm}_{B,D}(Q,m^{0}_{B,D}(1),C)$ are degenerate chains thanks to Lemma 5.4.

∎

boxsize=1em

(a) Term (1)

(b) Term (3)

(c) Term (4)

(d) Term (5)

Figure 5. Degenerations of disks contributing to $Q\cap_{B,D}C$.

##  5\. Module structure

This section explains why the quantum cup product $\cap_{B,D}$ of Section 4
makes $\operatorname{HF}^{B}(L,D)$ into a module over
$\operatorname{QH}^{B}(X)$. This relies on the correspondence axiom (4) from
Section 2.

###  5.1. Double geodesic constraints

The following chains will play an important role in checking that $\cap_{B,D}$
indeed gives a module structure (Proposition 5.6).

######  Definition 5.1.

The double BMC-deformed cap product is

$\Theta_{B,D}(Q_{1},Q_{2},C)=\sum_{k_{0},k_{1}\geq 0}\sum_{l\geq
0}\sum_{\beta\in\pi_{2}(X,L)}q^{\omega(\beta)}\frac{1}{l!}(\Theta_{B,D}(Q_{1},Q_{2},C))_{\beta}^{k_{0},k_{1};l}\quad,$

where the chain
$(Q\cap_{B,D}C)_{\beta}^{k_{0},k_{1};l}=\kappa_{\\#}(D^{k_{0}}\times C\times
D^{k_{1}}\times Q_{1}\times Q_{2}\times B^{l})$ is obtained via pull-push from
the correspondence $\kappa$ in Figure 3 (B).

boxsize=1em

(a) $\kappa:L^{2}\times X^{2}\rightsquigarrow L$ induced by
$\mathcal{G}^{\prime\prime}_{3+k_{0}+k_{1}^{L}+k_{1}^{R},2+l}(L,\beta)$ for
$k_{0},k_{1}^{L},k_{1}^{R}\geq 0$ and $l\geq 0$

(b) $\kappa:L^{2}\times X^{2}\rightsquigarrow L$ induced by
$\mathcal{G}^{\prime\prime}_{3+k_{0}^{L}+k_{0}^{R}+k_{1},2+l}(L,\beta)$ for
$k_{0}^{L},k_{0}^{R},k_{1}\geq 0$ and $l\geq 0$

Figure 6. Some correspondences induced by moduli of stable $J$-holomorphic
disks with double geodesic constraints. The interior/boundary marked points
denoted $\blacksquare$ are constrained to map to $B/D$.

######  Definition 5.2.

Define:

$H^{+}_{B,D}(C,E,Q_{1},Q_{2})=\sum_{l\geq 0}\sum_{k_{0}\geq
0}\sum_{\begin{subarray}{c}k_{1}^{L}\geq 0\\\ k_{1}^{R}\geq
0\end{subarray}}\sum_{\beta\in\pi_{2}(X,L)}q^{\omega(\beta)}\frac{1}{l!}H^{+}_{B,D}(C,E,Q_{1},Q_{2})_{\beta}^{k_{0};k_{1}^{L},k_{1}^{R};l}\quad\textrm{and}$
$H^{-}_{B,D}(C,E,Q_{1},Q_{2})=\sum_{l\geq
0}\sum_{\begin{subarray}{c}k_{0}^{L}\geq 0\\\ k_{0}^{R}\geq
0\end{subarray}}\sum_{k_{1}\geq
0}\sum_{\beta\in\pi_{2}(X,L)}q^{\omega(\beta)}\frac{1}{l!}H^{-}_{B,D}(C,E,Q_{1},Q_{2})_{\beta}^{k_{0}^{L},k_{0}^{R};k_{1};l}\quad,$

where the chain
$H^{+}_{B,D}(C,E,Q_{1},Q_{2})_{\beta}^{k_{0};k_{1}^{L},k_{1}^{R};l}=\kappa_{\\#}(D^{k_{0}}\times
C\times D^{k_{1}^{R}}\times E\times D^{k_{1}^{L}}\times Q_{1}\times
Q_{2}\times B^{l})$ is obtained via pull-push from the correspondence $\kappa$
in Figure 6 (A), and the chain
$H^{-}_{B,D}(C,E,Q_{1},Q_{2})_{\beta}^{k_{0}^{L},k_{0}^{R};k_{1};l}=\kappa_{\\#}(D^{k_{0}^{L}}\times
E\times D^{k_{0}^{R}}\times C\times D^{k_{1}}\times Q_{1}\times Q_{2}\times
B^{l})$ from the one in Figure 6 (B).

######  Lemma 5.3.

If $Q_{1},Q_{2},C,B,D$ are homogeneous chains, then

$|(\Theta_{B,D}(C,Q_{1},Q_{2}))_{\beta}^{k_{0},k_{1}^{L}+k_{1}^{R};l}|=n+\mu(\beta)-|C|^{\vee}-|Q_{1}|^{\vee}-|Q_{2}|^{\vee}-(k_{0}+k_{1}^{L}+k_{1}^{R})(|D|^{\vee}-1)-l(|B|^{\vee}-2)+1\quad.$

######  Lemma 5.4.

If $|E|=n$ then the chains $H^{\pm}_{B,D}(C,E,Q_{1},Q_{2})$ are degenerate.

###### Proof.

Assume that all chains are homogeneous. One can see that

$|H^{+}_{B,D}(C,E,Q_{1},Q_{2})_{\beta}^{k_{0};k_{1}^{L},k_{1}^{R};l}|=n+\mu(\beta)-|C|^{\vee}-|E|^{\vee}-|Q_{1}|^{\vee}-|Q_{2}|^{\vee}-(k_{0}+k_{1}^{L}+k_{1}^{R})(|D|^{\vee}-1)-l(|B|^{\vee}-2)+2\quad.$

The constraint of having an input boundary point mapping to $L$ is vacuous, so
one has

$H^{+}_{B,D}(C,E,Q_{1},Q_{2})=(\Theta_{B,D}(C,Q_{1},Q_{2}))_{\beta}^{k_{0},k_{1}^{L}+k_{1}^{R};l}\quad.$

Since the chain on the right has degree one less than what computed above when
$|E|=n$ (i.e. $|E|^{\vee}=0$) thanks to Lemma 5.3, the chain on the left is
degenerate. A similar argument applies to $H^{-}_{B,D}(C,E,Q_{1},Q_{2})$. ∎

###  5.2. Degeneration to single geodesic constraints

The following discussion explains how the double geodesic constraints
introduced earlier degenerate to single geodesic constraints entering the
definition of BMC-deformed quantum cap $\cap_{B,D}$, and the relation with the
module structure.

######  Lemma 5.5.

Up to signs and degenerate chains one has

$m^{1}_{B,D}(\Theta_{B,D}(Q_{1},Q_{2},C))=\Theta_{B,D}(Q_{1},Q_{2},m^{1}_{B,D}(C))+$
$Q_{1}\cap_{B,D}(Q_{2}\cap_{B,D}C)+(Q_{1}\star_{B}Q_{2})\cap_{B,D}C+H_{B,D}^{\pm}(C,m^{0}_{B,D}(1),Q_{1},Q_{2})\quad.$

###### Proof.

As in the proof of Proposition 4.5, separate the classical and quantum parts
$m^{1}_{B,D}=\partial+\tilde{m}^{1}_{B,D}$. The claim to be proven can then be
rewritten as

$\partial(\Theta_{B,D}(Q_{1},Q_{2},C))=\underbracket{\tilde{m}^{1}_{B,D}(\Theta_{B,D}(Q_{1},Q_{2},C))}_{(1)}+\underbracket{\Theta_{B,D}(Q_{1},Q_{2},\partial
C)}_{(2)}+$
$\underbracket{\Theta_{B,D}(Q_{1},Q_{2},\tilde{m}^{1}_{B,D}(C)}_{(3)}+\underbracket{Q_{1}\cap_{B,D}(Q_{2}\cap_{B,D}C)}_{(4)}+\underbracket{(Q_{1}\star_{B}Q_{2})\cap_{B,D}C}_{(5)}+$
$\underbracket{H^{+}_{B,D}(C,m^{0}_{B,D}(1),Q_{1},Q_{2})}_{(6)}+\underbracket{H^{-}_{B,D}(C,m^{0}_{B,D}(1),Q_{1},Q_{2})}_{(7)}\quad.$

The terms above arise from the following degenerations of disks contributing
to $\Theta_{B,D}(Q_{1},Q_{2},C)$ (see Figure 7).

  1. (1)

Using Definition 5.1 and Definition 3.3, one can rewrite this term as

$\sum_{\begin{subarray}{c}k_{0}[1]\\\
k_{1}[1]\end{subarray}}\sum_{l[1]}\sum_{\begin{subarray}{c}k_{0}[2]\\\
k_{1}[2]\end{subarray}}\sum_{l[2]}\sum_{\begin{subarray}{c}\beta[1]\neq 0\\\
\beta[2]\end{subarray}}T^{\omega(\beta[1])}T^{\omega(\beta[2])}\frac{1}{l[1]!l[2]!}$
$q^{\beta[1]}_{1+k_{0}[1]+k_{1}[1],l[1]}(D^{k_{0}[1]},(\Theta_{B,D}(Q_{1},Q_{2},C))_{\beta[2]}^{k_{0}[2],k_{1}[2];l[2]},D^{k_{1}[1]},B^{l[1]})\quad.$

Each chain in the sum corresponds to disk bubbling as in Figure 7 (A), where
the interior marked points mapping to $Q_{1},Q_{2}$ move along the geodesic
constraint all the way up to the input boundary point mapping to $C$ . If a
disk contributes
$T^{\omega(\beta)}(\Theta_{B,D}(Q_{1},Q_{2},C))^{k_{0},k_{1},l}_{\beta}$ to
$\Theta_{B,D}(Q_{1},Q_{2},C)$, then after this kind of degeneration its class
breaks up $\beta=\beta[1]+\beta[2]$, while the boundary marked points mapping
to $D$ split into two groups $k_{0}=k_{0}[1]+k_{1}[2]$, with $k_{0}[\bullet]$
and $k_{1}[\bullet]$ respectively on the lower and upper arcs of the two
disks. Similarly, the interior marked points mapping to $B$ split into two
groups $l=l[1]+l[2]$. Degenerations inducing different partitions of the $l$
interior points into groups of size $l[1]$ and $l[2]$ contribute the same
chain, and since there are $l!(l[1]!l[2]!)^{-1}$ such partitions overall one
gets the coefficient

$T^{\omega(\beta)}\frac{1}{l!}\frac{l!}{l[1]!l[2]!}=T^{\omega(\beta[1])}T^{\omega(\beta[2])}\frac{1}{l[1]!l[2]!}\quad.$

  2. (2)

This term corresponds to degenerating the condition of mapping a marked point
to $C$ to the condition of mapping it to $\partial C$. This is codimension one
thanks to Lemma 5.3.

  3. (3)

Using Definition 5.1 and Definition 3.3, one can rewrite this term as

$\sum_{\begin{subarray}{c}k_{0}[1]\\\
k_{1}[1]\end{subarray}}\sum_{l[1]}\sum_{\begin{subarray}{c}k_{0}[2]\\\
k_{1}[2]\end{subarray}}\sum_{l[2]}\sum_{\begin{subarray}{c}\beta[1]\\\
\beta[2]\neq
0\end{subarray}}T^{\omega(\beta[1])}T^{\omega(\beta[2])}\frac{1}{l[1]!l[2]!}$
$\left(\Theta_{B,D}(Q_{1},Q_{2},q^{\beta[2]}_{1+k_{0}[2]+k_{1}[2],l[2]}(D^{k_{0}[2]},C,D^{k_{1}[2]},B^{l[2]})\right)_{\beta[1]}^{k_{0}[1],k_{1}[1];l[1]}\quad.$

Each chain in the sum corresponds to disk bubbling as in Figure 7 (B), where
the interior marked points mapping to $Q_{1},Q_{2}$ move along the geodesic
constraint all the way up to the output boundary point. If a disk contributes
$T^{\omega(\beta)}(\Theta_{B,D}(Q_{1},Q_{2},C))^{k_{0},k_{1},l}_{\beta}$ to
$\Theta_{B,D}(Q_{1},Q_{2},C)$, then after this kind of degeneration its class
breaks up $\beta=\beta[1]+\beta[2]$, while the boundary marked points mapping
to $D$ split into two groups $k_{0}=k_{0}[1]+k_{1}[2]$, with $k_{0}[\bullet]$
and $k_{1}[\bullet]$ respectively on the lower and upper arcs of the two
disks. Similarly, the interior marked points mapping to $B$ split into two
groups $l=l[1]+l[2]$. Degenerations inducing different partitions of the $l$
interior points into groups of size $l[1]$ and $l[2]$ contribute the same
chain, and since there are $l!(l[1]!l[2]!)^{-1}$ such partitions overall one
gets the coefficient

$T^{\omega(\beta)}\frac{1}{l!}\frac{l!}{l[1]!l[2]!}=T^{\omega(\beta[1])}T^{\omega(\beta[2])}\frac{1}{l[1]!l[2]!}\quad.$

  4. (4)

Using Definition 4.1, one can rewrite this term as

$\sum_{\begin{subarray}{c}k_{0}[1]\\\
k_{1}[1]\end{subarray}}\sum_{l[1]}\sum_{\begin{subarray}{c}k_{0}[2]\\\
k_{1}[2]\end{subarray}}\sum_{l[2]}\sum_{\begin{subarray}{c}\beta[1]\\\
\beta[2]\end{subarray}}T^{\omega(\beta[1])}T^{\omega(\beta[2])}\frac{1}{l[1]!l[2]!}$
$\left(Q_{1}\cap_{B,D}(Q_{2}\cap_{B,D}C)_{\beta[2]}^{k_{0}[2],k_{1}[2];l[2]}\right)_{\beta[1]}^{k_{0}[1],k_{1}[1];l[1]}\quad.$

Each chain in the sum corresponds to disk bubbling as in Figure 7 (C), where
one of the interior marked points mapping to $Q_{1},Q_{2}$ move along the
geodesic constraint all the way up to the nearest boundary point. If a disk
contributes
$T^{\omega(\beta)}(\Theta_{B,D}(Q_{1},Q_{2},C))^{k_{0},k_{1},l}_{\beta}$ to
$\Theta_{B,D}(Q_{1},Q_{2},C)$, then after this kind of degeneration its class
breaks up $\beta=\beta[1]+\beta[2]$, while the boundary marked points mapping
to $D$ split into two groups $k_{0}=k_{0}[1]+k_{1}[2]$, with $k_{0}[\bullet]$
and $k_{1}[\bullet]$ respectively on the lower and upper arcs of the two
disks. Similarly, the interior marked points mapping to $B$ split into two
groups $l=l[1]+l[2]$. Degenerations inducing different partitions of the $l$
interior points into groups of size $l[1]$ and $l[2]$ contribute the same
chain, and since there are $l!(l[1]!l[2]!)^{-1}$ such partitions overall one
gets the coefficient

$T^{\omega(\beta)}\frac{1}{l!}\frac{l!}{l[1]!l[2]!}=T^{\omega(\beta[1])}T^{\omega(\beta[2])}\frac{1}{l[1]!l[2]!}\quad.$

  5. (5)

Using Definition 3.1 and Definition 4.1, one can rewrite this term as

$\sum_{\begin{subarray}{c}k_{0}[2]\\\
k_{1}[2]\end{subarray}}\sum_{\begin{subarray}{c}l[1]\\\
l[2]\end{subarray}}\sum_{\begin{subarray}{c}\beta[1]\\\
\beta[2]\end{subarray}}T^{\omega(\beta[1])}T^{\omega(\beta[2])}\frac{1}{l[1]!l[2]!}\left((Q_{1}\star_{B}Q_{2})_{\beta[1]}^{l[1]}\cap_{B,D}C\right)_{\beta[2]}^{k_{0}[2],k_{1}[2];l[2]}\quad.$

Each chain in the sum corresponds to sphere bubbling as in Figure 7 (D), where
the two interior marked points mapping to $Q_{1},Q_{2}$ come together along
the geodesic constraint. If a disk contributes
$T^{\omega(\beta)}(\Theta_{B,D}(Q_{1},Q_{2},C))^{k_{0}[2],k_{1}[2],l}_{\beta}$
to $\Theta_{B,D}(Q_{1},Q_{2},C)$, then after this kind of degeneration its
class breaks up $\beta=\beta[1]+\beta[2]$ with $\beta[1]\in\pi_{2}(X)$ and
$\beta[2]\in\pi_{2}(X,L)$, while the interior marked points mapping to $B$
split into two groups $l=l[1]+l[2]$. Degenerations inducing different
partitions of the $l$ interior points into groups of size $l[1]$ and $l[2]$
contribute the same chain, and since there are $l!(l[1]!l[2]!)^{-1}$ such
partitions overall one gets the coefficient

$T^{\omega(\beta)}\frac{1}{l!}\frac{l!}{l[1]!l[2]!}=T^{\omega(\beta[1])}T^{\omega(\beta[2])}\frac{1}{l[1]!l[2]!}\quad.$

  6. (6)

Using Definition 3.3 with $k=0$ and Definition 5.2, one can rewrite this term
as

$\sum_{\begin{subarray}{c}l[1]\\\
l[2]\end{subarray}}\sum_{\begin{subarray}{c}k_{0}[1]\\\
k_{0}[2]\end{subarray}}\sum_{\begin{subarray}{c}k_{1}^{L}[1]\\\
k_{1}^{R}[1]\end{subarray}}\sum_{\begin{subarray}{c}\beta[1]\\\
\beta[2]\end{subarray}}T^{\omega(\beta[1])}T^{\omega(\beta[2])}\frac{1}{l[1]!l[2]!}H^{+}_{B,D}(C,q^{\beta[2]}_{k_{0}[2],l[2]}(D^{k_{0}[2]},B^{l[2]}),Q_{1},Q_{2})_{\beta[1]}^{k_{0}[1];k_{1}^{L}[1],k_{1}^{R}[1];l[1]}\quad.$

Each chain in the sum corresponds to disk bubbling as in Figure 7 (E), where a
group of boundary marked points mapping to $D$ on the upper arc of the disk
with geodesic constraint come together. If a disk contributes
$T^{\omega(\beta)}(\Theta_{B,D}(Q_{1},Q_{2},C))^{k_{0}[2],k_{1}[2],l}_{\beta}$
to $\Theta_{B,D}(Q_{1},Q_{2},C)$, then after this kind of degeneration its
class breaks up $\beta=\beta[1]+\beta[2]$, while the boundary marked points
mapping to $D$ that are on the upper arc split into three groups
$k_{1}=k_{1}^{L}[2]+k[1]+k_{1}^{R}[2]$, with $k_{1}^{L}[2]$ and $k_{1}^{R}[2]$
respectively on the left and right of the attaching point of the disk bubble,
while $k[1]$ are on the new disk. The interior marked points mapping to $B$
split into two groups $l=l[1]+l[2]$. Degenerations inducing different
partitions of the $l$ interior points into groups of size $l[1]$ and $l[2]$
contribute the same chain, and since there are $l!(l[1]!l[2]!)^{-1}$ such
partitions overall one gets the coefficient

$T^{\omega(\beta)}\frac{1}{l!}\frac{l!}{l[1]!l[2]!}=T^{\omega(\beta[1])}T^{\omega(\beta[2])}\frac{1}{l[1]!l[2]!}\quad.$

  7. (7)

Using Definition 3.3 with $k=0$ and Definition 5.2, one can rewrite this term
as

$\sum_{\begin{subarray}{c}l[1]\\\
l[2]\end{subarray}}\sum_{\begin{subarray}{c}k_{0}^{L}[1]\\\
k_{0}^{R}[1]\end{subarray}}\sum_{\begin{subarray}{c}k_{1}[1]]\\\
k_{0}[2]\end{subarray}}\sum_{\begin{subarray}{c}\beta[1]\\\
\beta[2]\end{subarray}}T^{\omega(\beta[1])}T^{\omega(\beta[2])}\frac{1}{l[1]!l[2]!}H^{-}_{B,D}(C,q^{\beta[2]}_{k_{0}[2],l[2]}(D^{k_{0}[2]},B^{l[2]}),Q_{1},Q_{2})_{\beta[1]}^{k_{0}^{L}[1],k_{0}^{R}[1];k_{1}[1];l[1]}\quad.$

Each chain in the sum corresponds to disk bubbling as in Figure 7 (F), where a
group of boundary marked points mapping to $D$ on the lower arc of the disk
with geodesic constraint come together. If a disk contributes
$T^{\omega(\beta)}(\Theta_{B,D}(Q_{1},Q_{2},C))^{k_{0}[2],k_{1}[2],l}_{\beta}$
to $\Theta_{B,D}(Q_{1},Q_{2},C)$, then after this kind of degeneration its
class breaks up $\beta=\beta[1]+\beta[2]$, while the boundary marked points
mapping to $D$ that are on the lower arc split into three groups
$k_{0}=k_{0}^{L}[1]+k[2]+k_{0}^{R}[2]$, with $k_{0}^{L}[1]$ and $k_{0}^{R}[1]$
respectively on the left and right of the attaching point of the disk bubble,
while $k[2]$ are on the new disk. The interior marked points mapping to $B$
split into two groups $l=l[1]+l[2]$. Degenerations inducing different
partitions of the $l$ interior points into groups of size $l[1]$ and $l[2]$
contribute the same chain, and since there are $l!(l[1]!l[2]!)^{-1}$ such
partitions overall one gets the coefficient

$T^{\omega(\beta)}\frac{1}{l!}\frac{l!}{l[1]!l[2]!}=T^{\omega(\beta[1])}T^{\omega(\beta[2])}\frac{1}{l[1]!l[2]!}\quad.$

By Gromov compactness, the only other possible degenerations of a disk
contributing to $\Theta_{B,D}(Q_{1},Q_{2},C)$ consist of sphere bubbling, or
disk bubbling arising from moving the interior marked points mapping to
$Q_{1},Q_{2}$ all the way up to the upper or lower arcs. None of these
degenerations shows up in $\partial(\Theta_{B,D}(Q_{1},Q_{2},C))$, because the
first has codimension two while the second violates the geodesic constraint.

∎

######  Proposition 5.6.

When $D\in\operatorname{MC}^{B}(L)$ the following properties hold:

  1. (1)

$[X]\cap_{B,D}[C]=[C]$ ;

  2. (2)

$[Q_{1}]\cap_{B,D}([Q_{2}]\cap_{B,D}[C])=([Q_{1}]\star_{B}[Q_{2}])\cap_{B,D}[C]$
.

###### Proof.

For part (1), separating the classical ($\beta=0$) and quantum ($\beta\neq 0$)
parts of the BMC-deformed cap product in Definition 4.1 one gets:

$X\cap_{B,D}C=\sum_{k_{0},k_{1}\geq 0}\sum_{l\geq
0}T^{\omega(\beta)}\frac{1}{l!}(X\cap_{B,D}C)_{0}^{k_{0},k_{1};l}+\sum_{k_{0},k_{1}\geq
0}\sum_{l\geq 0}\sum_{\begin{subarray}{c}\beta\in\pi_{2}(X,L)\\\ \beta\neq
0\end{subarray}}T^{\omega(\beta)}\frac{1}{l!}(X\cap_{B,D}C)_{\beta}^{k_{0},k_{1};l}\quad.$

In the classical part, $(X\cap_{B,D}C)_{0}^{k_{0},k_{1};l}=0$ for
$(k_{0},k_{1},l)\neq(0,0,0)$ and $(X\cap_{B,D}C)_{0}^{0,0;0}=X\cap C=C$.
Regarding the quantum part, observe that the constraint of mapping an interior
point to $X$ is vacuous, hence removing it one gets

$(X\cap_{B,D}C)_{\beta}^{k_{0},k_{1};l}=q^{\beta}_{1+k_{0}+k_{1},l}(D^{k_{0}},C,D^{k_{1}},B^{l})\quad\textrm{for
any}\;\beta\neq 0\quad.$

When all chains are homogeneous, from Lemma 4.2 one gets

$|(X\cap_{B,D}C)_{\beta}^{k_{0},k_{1};l}|=|q^{\beta}_{1+k_{0}+k_{1},l}(D^{k_{0}},C,D^{k_{1}},B^{l})|+1\quad,$

hence the chain on the left is degenerate for all $\beta\neq 0$. Part (2)
follows from the identity

$m^{1}_{B,D}(\Theta_{B,D}(Q_{1},Q_{2},C))=\Theta_{B,D}(Q_{1},Q_{2},m^{1}_{B,D}(C))+$
$Q_{1}\cap_{B,D}(Q_{2}\cap_{B,D}C)+(Q_{1}\star_{B}Q_{2})\cap_{B,D}C+H_{B,D}^{\pm}(C,m^{0}_{B,D}(1),Q_{1},Q_{2})\quad.$

proved in Lemma 5.5 and Lemma 5.4, which can be used because
$m^{0}_{B,D}(1)=W^{B}(D)L$ thanks to the assumption
$D\in\operatorname{MC}^{B}(L)$. ∎

boxsize=1em

(a) Term (1)

(b) Term (3)

(c) Term (4)

(d) Term (5)

(e) Term (6)

(f) Term (7)

Figure 7. Degenerations of disks contributing to
$\Theta_{B,D}(Q_{1},Q_{2},C)$.

##  6\. Curvatures are in the Dubrovin spectrum

This section contains the main result of the article, showing that the
curvature of the $A_{\infty}$ algebra $\operatorname{CF}^{B}(L,D)$ is in the
spectrum of Dubrovin’s operator $K^{B}$. The proof makes use of the forgetful
axioms (5)-(6) from Section 2.

######  Notation 6.1.

Decompose the chains $D=\sum_{i}D_{i}$ and $B=\sum_{j}B_{j}$ into homogeneous
components, with indices corresponding to cohomological degrees: $i=n-|D_{i}|$
and $j=2n-|B_{j}|$. Denote $\textbf{i}(\bullet)=(i_{1},\ldots,i_{\bullet})$
and $\textbf{j}(\bullet)=(j_{1},\ldots,j_{\bullet})$ unspecified sequences of
length $\bullet$ in the set of indices appearing in the sums above. For a
fixed length $\bullet$, sums over all such sequences are denoted
$\sum_{\textbf{i}(\bullet)}\quad\textrm{and}\quad\sum_{\textbf{j}(\bullet)}$.
Write $D_{\textbf{i}(\bullet)}=(D_{i_{1}},\ldots,D_{i_{\bullet}})$ and
$B_{\textbf{j}(\bullet)}=(B_{j_{1}},\ldots,B_{j_{\bullet}})$ for the
corresponding sequences of homogeneous components. Also define the following
quantities, called total shifted degrees:
$|\textbf{i}(\bullet)|=(i_{1}-1)+\ldots+(i_{\bullet}-1)$ and
$|\textbf{j}(\bullet)|=(j_{1}-2)+\ldots+(j_{\bullet}-2)$.

######  Lemma 6.2.

$|q^{\beta}_{l_{0},l}(D_{\textbf{i}(l_{0}},B_{\textbf{j}(l)})|=n+\mu(\beta)-2-|\textbf{i}(l_{0})|-|\textbf{j}(l)|$

######  Proposition 6.3.

Suppose that $D\in\operatorname{MC}^{B}(L)$. Then for any $d\in\mathbb{Z}$
with $d\neq n$ one has

$\sum_{l_{0},l\geq
0}\sum_{\textbf{i}(l_{0}),\textbf{j}(l)}\sum_{\begin{subarray}{c}\beta\in\pi_{2}(X,L)\\\
\mu(\beta)=2+d-n\\\
+|\textbf{i}(l_{0})|+|\textbf{j}(l)|\end{subarray}}T^{\omega(\beta)}\frac{1}{l!}q^{\beta}_{l_{0},l}(D_{\textbf{i}(l_{0})},B_{\textbf{j}(l)})=0\quad.$

###### Proof.

The assumption $D\in\operatorname{MC}^{B}(L)$ means that $D$ solves the
$B$-deformed weak Maurer-Cartan equation, i.e. $m^{0}_{B,D}(1)=W^{B}(D)L$ for
some $W^{B}(D)\in\Lambda$; see Section 3. By Definition 3.3 with $k=0$, this
is the same as

$\sum_{l_{0}\geq 0}\sum_{l\geq
0}\sum_{\beta\in\pi_{2}(X,L)}T^{\omega(\beta)}\frac{1}{l!}q^{\beta}_{l_{0},l}(D^{l_{0}},B^{l})=W^{B}(D)L\quad.$

Breaking up $D$ and $B$ into homogeneous components, this becomes

$\sum_{l_{0}\geq 0}\sum_{l\geq
0}\sum_{\beta\in\pi_{2}(X,L)}\sum_{\textbf{i}(l_{0}),\textbf{j}(l)}T^{\omega(\beta)}\frac{1}{l!}q^{\beta}_{l_{0},l}(D_{\textbf{i}(l_{0})},B_{\textbf{j}(l)})=W^{B}(D)L\quad.$

The degree of the right hand side is $n$, while the left hand side is a sum of
terms of different degrees. Using Lemma 6.2 to group terms with the same
degree $d\in\mathbb{Z}$ one gets:

$\sum_{d\in\mathbb{Z}}\sum_{l_{0}\geq 0}\sum_{l\geq
0}\sum_{\textbf{i}(l_{0}),\textbf{j}(l)}\sum_{\begin{subarray}{c}\beta\in\pi_{2}(X,L)\\\
\mu(\beta)=2+d-n\\\
+|\textbf{i}(l_{0})|+|\textbf{j}(l)|\end{subarray}}T^{\omega(\beta)}\frac{1}{l!}q^{\beta}_{l_{0},l}(D_{\textbf{i}(l_{0})},B_{\textbf{j}(l)})=W^{B}(D)L\quad,$

and the claim follows by degree comparison with the right hand side. ∎

When $\mu(\beta)=2+|\textbf{i}(l_{0})|+|\textbf{j}(l)|$, writing
$q^{\beta}_{l_{0},l}(D_{\textbf{i}(l_{0})},B_{\textbf{j}(l)})=n^{\beta}_{l_{0},l}(D_{\textbf{i}(l_{0})},B_{\textbf{j}(l)})L$
one can interpret
$n^{\beta}_{l_{0},l}(D_{\textbf{i}(l_{0})},B_{\textbf{j}(l)})\in\mathbb{Q}$ as
the number of $J$-holomorphic disks of class $\beta$ through a given basepoint
of $L$, with $l_{0}$ boundary points mapping to $D$ and $l$ interior points
mapping to $B$. Examining the $d=n$ case in the proof of Proposition 6.3, one
gets a formula for $W^{B}(D)$ in terms of such numbers.

######  Corollary 6.4.

$W^{B}(D)=\sum_{l_{0},l\geq
0}\sum_{\textbf{i}(l_{0}),\textbf{j}(l)}\sum_{\begin{subarray}{c}\beta\in\pi_{2}(X,L)\\\
\mu(\beta)=2\\\
+|\textbf{i}(l_{0})|+|\textbf{j}(l)|\end{subarray}}\frac{1}{l!}n^{\beta}_{l_{0},l}(D_{\textbf{i}(l_{0})},B_{\textbf{j}(l)})T^{\omega(\beta)}$

The following rescaling of the Maurer-Cartan deformation chain $D$ will play a
crucial role in the proof of Theorem 6.9.

######  Definition 6.5.

For any $D=\sum_{i}D_{i}$, its rescaling is
$\widetilde{D}=\sum_{i}\frac{i-1}{2}D_{i}$.

######  Lemma 6.6.

$m^{1}_{B,D}(\widetilde{D})=\sum_{l\geq 0}\sum_{k\geq
0}\sum_{\textbf{i}(k)}\sum_{\beta\in\pi_{2}(X,L)}T^{\omega(\beta)}\frac{|\textbf{i}(k)|}{2}\frac{1}{l!}q^{\beta}_{k,l}(D_{\textbf{i}(k)},B^{l})$

###### Proof.

Using Definition 3.3 with $k=1$ and Definition 6.5

$m^{1}_{B,D}(\widetilde{D})=\sum_{i}\sum_{l_{0},l_{1}}\sum_{l\geq
0}\sum_{\beta\in\pi_{2}(X,L)}T^{\omega(\beta)}\frac{i-1}{2}\frac{1}{l!}q^{\beta}_{1+l_{0}+l_{1},l}(D^{l_{0}},D_{i},D^{l_{1}},B^{l})=$
$\sum_{i}\sum_{k\geq 0}\sum_{l_{0}+l_{1}=k}\sum_{l\geq
0}\sum_{\beta\in\pi_{2}(X,L)}T^{\omega(\beta)}\frac{i-1}{2}\frac{1}{l!}q^{\beta}_{1+k,l}(D^{l_{0}},D_{i},D^{l_{1}},B^{l})\quad.$

Breaking up $D$ into homogeneous components and rearranging the sum one gets

$m^{1}_{B,D}(\widetilde{D})=\sum_{\beta\in\pi_{2}(X,L)}T^{\omega(\beta)}\sum_{l\geq
0}\frac{1}{l!}\sum_{k\geq
0}\sum_{\textbf{i}(k)}\sum_{i}\sum_{l_{0}+l_{1}=k}\frac{i-1}{2}q^{\beta}_{1+k,l}(D_{i_{1}},\ldots,D_{i_{l_{0}}},D_{i},D_{i_{l_{0}+1}},\ldots,D_{i_{k}},B^{l})\quad.$

Observe that for any $k\geq 0$ one can rewrite

$\sum_{\textbf{i}(k)}\sum_{i}\sum_{l_{0}+l_{1}=k}\frac{i-1}{2}q^{\beta}_{1+k,l}(D_{i_{1}},\ldots,D_{i_{l_{0}}},D_{i},D_{i_{l_{0}+1}},\ldots,D_{i_{k}},B^{l})=\sum_{\textbf{i}(1+k)}\frac{|\textbf{i}(1+k)|}{2}q^{\beta}_{1+k,l}(D_{\textbf{i}(1+k)},B^{l})\quad,$

thus obtaining

$m^{1}_{B,D}(\widetilde{D})=\sum_{\beta\in\pi_{2}(X,L)}T^{\omega(\beta)}\sum_{l\geq
0}\frac{1}{l!}\sum_{k\geq
0}\sum_{\textbf{i}(1+k)}\frac{|\textbf{i}(1+k)|}{2}q^{\beta}_{1+k,l}(D_{\textbf{i}(1+k)},B^{l})\quad.$

The last expression is the one claimed after shifting the index $k$, because
$|\textbf{i}(0)|=0$. ∎

######  Proposition 6.7.

For any $D\in\operatorname{MC}^{B}(L)$, one has
$K^{B}\cap_{B,D}[L]=W^{B}(D)[L]$ .

###### Proof.

Assume that $L\subset X\setminus Z$ for some divisor $Z$ Poincaré dual to a
lift of $c_{1}$ in $H^{2}(X,L)$, so that $\beta\cdot[Z]=\mu(\beta)/2$ for
every $\beta\in\pi_{2}(X,L)$. Decomposing the bulk-deformation cycle
$B=\sum_{j}B_{j}$ into homogeneous components

$K^{B}\cap_{B,D}[L]=[Z]\cap_{B,D}[L]-\sum_{j}\frac{j-2}{2}[B_{j}]\cap_{B,D}[L]\quad.$

Recall that $m^{0}_{B,D}(1)=W^{B}(D)L$ thanks to the assumption
$D\in\operatorname{MC}^{B}(L)$, so it suffices to check that
$K^{B}\cap_{B,D}L-m^{0}_{B,D}(1)$ is $m^{1}_{B,D}$-exact at the chain level.
We verify that the rescaled chain $\widetilde{D}$ (Definition 6.5) is a
primitive. In the definition of BMC-deformed cap product (Definition 4.1), one
can group together contributions with the same number $l_{0}+l_{1}=k$ of
boundary points mapping to $D$ and get

$Z\cap_{B,D}L=\sum_{l_{0},l_{1}\geq 0}\sum_{l\geq
0}\sum_{\beta\in\pi_{2}(X,L)}T^{\omega(\beta)}\frac{1}{l!}(Z\cap_{B,D}L)^{l_{0},l_{1};l}_{\beta}$
$=\sum_{\beta\in\pi_{2}(X,L)}T^{\omega(\beta)}\sum_{l\geq
0}\frac{1}{l!}\sum_{k\geq
0}\sum_{l_{0}+l_{1}=k}(Z\cap_{B,D}L)^{l_{0},l_{1};l}_{\beta}\quad.$

From the axioms of Section 2, removing the vacuous boundary constraint of an
input marked point mapping to $L$ yields an equality of chains

$\sum_{l_{0}+l_{1}=k}(Z\cap_{B,D}L)_{\beta}^{l_{0},l_{1};l}=\frac{\mu(\beta)}{2}q^{\beta}_{k,l}(D^{k},B^{l})\quad.$

Using this one can rewrite

$Z\cap_{B,D}L=\sum_{k\geq 0}\sum_{l\geq
0}\sum_{\beta\in\pi_{2}(X,L)}T^{\omega(\beta)}\frac{\mu(\beta)}{2}\frac{1}{l!}q^{\beta}_{k,l}(D^{k},B^{l})\quad.$

Definition 3.3 with $k=0$ then gives

$K^{B}\cap_{B,D}L-m^{0}_{B,D}(1)=\sum_{k\geq 0}\sum_{l\geq
0}\sum_{\beta\in\pi_{2}(X,L)}T^{\omega(\beta)}\left(\frac{\mu(\beta)}{2}-1\right)\frac{1}{l!}q^{\beta}_{k,l}(D^{k},B^{l})-\sum_{j}\frac{j-2}{2}B_{j}\cap_{B,D}L\quad.$

For each $k\geq 0$ and $l\geq 0$, break up $D=\sum_{i}D_{i}$ into homogenous
components and isolate the contributions of disk classes
$\beta\in\pi_{2}(X,L)$ with Maslov index
$\mu(\beta)=2+|\textbf{i}(k)|+|\textbf{j}(l)|$, which are the ones
contributing to $W^{B}(D)$ by Corollary 6.4:

$K^{B}\cap_{B,D}L-m^{0}_{B,D}(1)=$ $\underbracket{\sum_{k\geq 0}\sum_{l\geq
0}\sum_{\textbf{i}(k)}\sum_{\textbf{j}(l)}\sum_{\begin{subarray}{c}\beta\in\pi_{2}(X,L)\\\
\mu(\beta)=2+|\textbf{i}(k)|+|\textbf{j}(l)|\end{subarray}}T^{\omega(\beta)}\frac{|\textbf{i}(k)|}{2}\frac{1}{l!}q^{\beta}_{k,l}(D_{\textbf{i}(k)},B_{\textbf{j}(l)})}_{\textrm{(1)}}$
$+\underbracket{\sum_{k\geq 0}\sum_{l\geq
0}\sum_{\textbf{i}(k)}\sum_{\textbf{j}(l)}\sum_{\begin{subarray}{c}\beta\in\pi_{2}(X,L)\\\
\mu(\beta)=2+|\textbf{i}(k)|+|\textbf{j}(l)|\end{subarray}}T^{\omega(\beta)}\frac{|\textbf{j}(k)|}{2}\frac{1}{l!}q^{\beta}_{k,l}(D_{\textbf{i}(k)},B_{\textbf{j}(l)})}_{\textrm{(2)}}$
$+\underbracket{\sum_{k\geq 0}\sum_{l\geq
0}\sum_{\textbf{i}(k)}\sum_{\textbf{j}(l)}\sum_{\begin{subarray}{c}\beta\in\pi_{2}(X,L)\\\
\mu(\beta)\neq
2+|\textbf{i}(k)|+|\textbf{j}(l)|\end{subarray}}T^{\omega(\beta)}\left(\frac{\mu(\beta)}{2}-1\right)\frac{1}{l!}q^{\beta}_{k,l}(D_{\textbf{i}(k)},B_{\textbf{j}(l)})}_{\textrm{(3)}}$
$-\underbracket{\sum_{j}\frac{j-2}{2}B_{j}\cap_{B,D}L}_{\textrm{(4)}}\quad.$

Term (1) resembles $m^{1}_{B,D}(\widetilde{D})$ from Lemma 6.6, but only
includes the contributions of disk classes $\beta\in\pi_{2}(X,L)$ with Maslov
index $\mu(\beta)=2+|\textbf{i}(k)|+|\textbf{j}(l)|$. Term (3) can be
rewritten as

$(3)=\sum_{\begin{subarray}{c}d\in\mathbb{Z}\\\ d\neq
n\end{subarray}}\sum_{k\geq 0}\sum_{l\geq
0}\sum_{\textbf{i}(k)}\sum_{\textbf{j}(l)}\smashoperator[r]{\sum_{\begin{subarray}{c}\beta\in\pi_{2}(X,L)\\\
\mu(\beta)=2+d-n\\\
+|\textbf{i}(k)|+|\textbf{j}(l)|\end{subarray}}^{}}T^{\omega(\beta)}\left(\frac{d-n}{2}+\frac{|\textbf{i}(k)|}{2}+\frac{|\textbf{j}(l)|}{2}\right)\frac{1}{l!}q^{\beta}_{k,l}(D_{\textbf{i}(k)},B_{\textbf{j}(l)})=$
$\sum_{\begin{subarray}{c}d\in\mathbb{Z}\\\ d\neq
n\end{subarray}}\frac{d-n}{2}\underbracket{\left(\sum_{k\geq 0}\sum_{l\geq
0}\sum_{\textbf{i}(k)}\sum_{\textbf{j}(l)}\smashoperator[r]{\sum_{\begin{subarray}{c}\beta\in\pi_{2}(X,L)\\\
\mu(\beta)=2+d-n\\\
+|\textbf{i}(k)|+|\textbf{j}(l)|\end{subarray}}^{}}T^{\omega(\beta)}\frac{1}{l!}q^{\beta}_{k,l}(D_{\textbf{i}(k)},B_{\textbf{j}(l)})\right)}_{\textrm{(3a)}}$
$+\underbracket{\sum_{\begin{subarray}{c}d\in\mathbb{Z}\\\ d\neq
n\end{subarray}}\left(\sum_{k\geq 0}\sum_{l\geq
0}\sum_{\textbf{i}(k)}\sum_{\textbf{j}(l)}\smashoperator[r]{\sum_{\begin{subarray}{c}\beta\in\pi_{2}(X,L)\\\
\mu(\beta)=2+d-n\\\
+|\textbf{i}(k)|+|\textbf{j}(l)|\end{subarray}}^{}}T^{\omega(\beta)}\frac{|\textbf{i}(k)|}{2}\frac{1}{l!}q^{\beta}_{k,l}(D_{\textbf{i}(k)},B_{\textbf{j}(l)})\right)}_{\textrm{(3b)}}$
$+\underbracket{\sum_{\begin{subarray}{c}d\in\mathbb{Z}\\\ d\neq
n\end{subarray}}\left(\sum_{k\geq 0}\sum_{l\geq
0}\sum_{\textbf{i}(k)}\sum_{\textbf{j}(l)}\smashoperator[r]{\sum_{\begin{subarray}{c}\beta\in\pi_{2}(X,L)\\\
\mu(\beta)=2+d-n\\\
+|\textbf{i}(k)|+|\textbf{j}(l)|\end{subarray}}^{}}T^{\omega(\beta)}\frac{|\textbf{j}(l)|}{2}\frac{1}{l!}q^{\beta}_{k,l}(D_{\textbf{i}(k)},B_{\textbf{j}(l)})\right)}_{\textrm{(3c)}}\quad.$

Observe that $(3a)=0$ thanks to the assumption $D\in\operatorname{MC}^{B}(L)$
and Proposition 6.3, and that $(1)+(3b)=m^{1}_{B,D}(\widetilde{D})$, so that
overall

$K^{B}\cap_{B,D}L-m^{0}_{B,D}(1)=(1)+(2)+(3a)+(3b)+(3c)+(4)=$
$m^{1}_{B,D}(\widetilde{D})+\underbracket{\sum_{k\geq 0}\sum_{l\geq
0}\sum_{\textbf{i}(k)}\sum_{\textbf{j}(l)}\sum_{\beta\in\pi_{2}(X,L)}T^{\omega(\beta)}\frac{|\textbf{j}(l)|}{2}\frac{1}{l!}q^{\beta}_{k,l}(D_{\textbf{i}(k)},B_{\textbf{j}(l)})}_{\textrm{(2)+(3c)}}-\underbracket{\sum_{j}\frac{j-2}{2}B_{j}\cap_{B,D}L}_{\textrm{(4)}}\quad.$

To conclude, it suffices to check that $(2)+(3c)=(4)$. Using the definition of
BMC-deformed cap product (Definition 4.1) and grouping together the terms with
the same number $l_{0}+l_{1}=k$ of input boundary marked points mapping to $D$
one gets

$(4)=\sum_{\beta\in\pi_{2}(X,L)}T^{\omega(\beta)}\sum_{j}\frac{j-2}{2}\sum_{l\geq
0}\frac{1}{l!}\sum_{k\geq
0}\sum_{l_{0}+l_{1}=k}(B_{j}\cap_{B,D}L)^{l_{0},l_{1};l}_{\beta}\quad.$

From the axioms of Section 2, removing the vacuous boundary constraint of an
input boundary marked point mapping to $L$ yields an equality of chains

$\sum_{l_{0}+l_{1}=k}(B_{j}\cap_{B,D}L)^{l_{0},l_{1};l}_{\beta}=\frac{1}{l+1}\sum_{t=0}^{l}q^{\beta}_{k,l+1}(D^{k},B^{t},B_{j},B^{l-t})\quad,$

and substitution in the previous equation gives

$(4)=\sum_{\beta\in\pi_{2}(X,L)}T^{\omega(\beta)}\sum_{k\geq 0}\sum_{l\geq
0}\frac{1}{(l+1)!}\sum_{j}\frac{j-2}{2}\sum_{t=0}^{l}q^{\beta}_{k,l+1}(D^{k},B^{t},B_{j},B^{l-t})\quad.$

Breaking up $B$ into homogeneous components yields

$(4)=\sum_{\beta\in\pi_{2}(X,L)}T^{\omega(\beta)}\sum_{k\geq 0}\sum_{l\geq
0}\frac{1}{(l+1)!}\sum_{\textbf{j}(l)}\sum_{j}\frac{j-2}{2}\sum_{t=0}^{l}q^{\beta}_{k,l+1}(D^{k},B_{\textbf{j}(t)},B_{j},B_{\textbf{j}(l-t)})\quad,$

and rewriting the sum

$\sum_{\textbf{j}(l)}\sum_{j}\sum_{t=0}^{l}\frac{j-2}{2}q^{\beta}_{k,l+1}(D^{k},B_{\textbf{j}(t)},B_{j},B_{\textbf{j}(l-t)})=\sum_{\textbf{j}(l+1)}\frac{|\textbf{j}(l+1)|}{2}q^{\beta}_{k,l+1}(D^{k},B_{\textbf{j}(l+1)})$

one gets

$(4)=\sum_{\beta\in\pi_{2}(X,L)}T^{\omega(\beta)}\sum_{k\geq 0}\sum_{l\geq
0}\frac{1}{(l+1)!}\sum_{\textbf{j}(l+1)}\frac{|\textbf{j}(l+1)|}{2}q^{\beta}_{k,l+1}(D^{k},B_{\textbf{j}(l+1)})\quad,$

which is precisely $(2)+(3c)$ after a shift of the index $l$, using the fact
that $\textbf{j}(0)=0$.

∎

######  Remark 6.8.

The fact that $Z\cap_{B,D}L-m^{0}_{B,D}(1)$ is only exact, as opposed to zero,
is a major difference between Maurer-Cartan deformations where $D$ has
cohomological degree 1 and general ones.

######  Theorem 6.9.

For any $D\in\operatorname{MC}^{B}(L)$, $\operatorname{HF}^{B}(L,D)\neq 0$
implies that $W^{B}_{L}(D)$ is an eigenvalue of the Dubrovin operator
$K^{B}\star_{B}-:\operatorname{QH}^{B}(X)\to\operatorname{QH}^{B}(X)$ .

###### Proof.

The assumption $D\in\operatorname{MC}^{B}(L)$ guarantees that the curvature
$W^{B}(D)\in\Lambda$ is defined, and the BMC-deformed cap product of chains
descends to a map on homology
$\cap_{B,D}:\operatorname{QH}^{B}(X)\otimes\operatorname{HF}^{B}(L,D)\to\operatorname{HF}^{B}(L,D)$
(Proposition 4.5). The curvature $W^{B}(D)$ is an eigenvalue of
$K^{B}\star_{B}-:\operatorname{QH}^{B}(X)\to\operatorname{QH}^{B}(X)$ if and
only if
$(K^{B}-W^{B}(D)[X])\star_{B}-:\operatorname{QH}^{B}(X)\to\operatorname{QH}^{B}(X)$
is not invertible. Assume by contradiction that this is not the case. Then
such operator is surjective, and there is some
$[Q]\in\operatorname{QH}^{B}(X)$ such that
$(K^{B}-W^{B}(D)[X])\star_{B}[Q]=[X]$. Observe that
$(K^{B}-W^{B}(D)[X])\cap_{B,D}[L]=0$ (Proposition 5.6 and 6.7). Capping this
equation by $[Q]$ and using again the module structure of
$\operatorname{HF}^{B}(L,D)$ over $\operatorname{QH}^{B}(X)$ (Proposition 5.6)
one gets

$0=[Q]\cap_{B,D}((K^{B}-W^{B}(D)[X])\cap_{B,D}[L])=$
$=([Q]\star_{B}(K^{B}-W^{B}(D)[X]))\cap_{B,D}[L]=[X]\cap_{B,D}[L]=[L]\quad.$

This is against the fact $\operatorname{HF}^{B}(L,D)\neq 0$ by assumption,
because $[L]$ is the unit of this algebra (Proposition 3.4). ∎

##  7\. Decoupling phenomenon

This section discusses an example where the spectrum of some finite energy
truncation of the Dubrovin operator $K^{B}$ decouples for explicit bulk
parameters $B$, and explains how this is related to decoupling of the full
Dubrovin operator.

###  7.1. Classical cohomology

Let $X=\operatorname{Gr}(2,4)$ be the Grassmannians of complex planes in
$\mathbb{C}^{4}$. This is a $4$-dimensional smooth Fano variety, that can be
realized as a quadric hypersurface in $\mathbb{P}^{5}$ under the Plücker
embedding. As symplectic structure $\omega$ take the Fubini-Study form on
$\mathbb{P}^{5}$ restricted to $X$. The cohomology
$\operatorname{H}(X;\mathbb{C})$ has dimension $6$ and is concentrated in even
degrees, with a basis given by classes $\sigma_{d}=\operatorname{PD}([X_{d}])$
Poincaré dual to Schubert cycles $X_{d}\subset\operatorname{Gr}(2,4)$. The
labels $d\subset 2\times 2$ are Young diagrams in a grid with two rows and two
columns. The codegree $|X_{d}|^{\vee}=|\sigma_{d}|$ is twice the number of
boxes in the corresponding Young diagram $d$, and we order the basis elements
as follows: boxsize=0.3em

degree | 0 | 2 | 4 | 4 | 6 | 8
---|---|---|---|---|---|---
class | $\sigma_{0}=\sigma_{\emptyset}$ | $\sigma_{1}=\sigma_{\ydiagram{1,0}}$ | $\sigma_{2}=\sigma_{\ydiagram{2,0}}$ | $\sigma_{3}=\sigma_{\ydiagram{1,1}}$ | $\sigma_{4}=\sigma_{\ydiagram{2,1}}$ | $\sigma_{5}=\sigma_{\ydiagram{2,2}}$

The first Chern class of $X$ is $c_{1}=4\sigma_{\ydiagram{1,0}}$. Cup products
of Schubert classes can be computed combinatorially from their Young diagrams,
using the Littlewood-Richardson rule. We record for future reference the
intersection matrix $g$ of $X$ and note that $g=g^{-1}$:

$\begin{pmatrix}0&0&0&0&0&1\\\ 0&0&0&0&1&0\\\ 0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\
0&1&0&0&0&0\\\ 1&0&0&0&0&0\end{pmatrix}$

###  7.2. Big $q$-deformation and Dubrovin operator

Denote $t=(t_{0},t_{1},t_{2},t_{3},t_{4},t_{5})\in\mathbb{C}^{6}$ the
coordinates on the complex cohomology $\operatorname{H}(X;\mathbb{C})$ induced
by the basis of Schubert classes. Following Fulton-Pandharipande [17], the
genus $0$ Gromov-Witten potential of $X$ is a formal power series
$\Phi\in\mathbb{Q}[[t]]$ given by the sum $\Phi=\Phi_{c}+\hat{\Phi}$ of a
classical part and a quantum part. Denoting
$n=(n_{0},n_{1},n_{2},n_{3},n_{4},n_{5})\in\mathbb{N}^{5}$ and
$|n|=n_{0}+n_{1}+n_{2}+n_{3}+n_{4}+n_{5}$, the classical part is:

$\Phi_{c}=\sum_{|n|=3}\langle\sigma_{\emptyset}^{n_{0}}\sigma_{\ydiagram{1,0}}^{n_{1}}\sigma_{\ydiagram{2,0}}^{n_{2}}\sigma_{\ydiagram{1,1}}^{n_{3}}\sigma_{\ydiagram{2,1}}^{n_{4}}\sigma_{\ydiagram{2,2}}^{n_{5}},[X]\rangle\frac{t_{0}^{n_{0}}t_{1}^{n_{1}}t_{2}^{n_{2}}t_{3}^{n_{3}}t_{4}^{n_{4}}t_{5}^{n_{5}}}{n_{0}!n_{1}!n_{2}!n_{3}!n_{4}!n_{5}!}\quad.$
$=\frac{1}{2}t_{1}^{2}t_{3}+\frac{1}{2}t_{1}^{2}t_{2}+\frac{1}{2}t_{0}t_{3}^{2}+\frac{1}{2}t_{0}t_{2}^{2}+t_{0}t_{1}t_{4}+\frac{1}{2}t_{0}^{2}t_{5}\quad,$

where we used the Littlewood-Richardson rule to simplify the expression.
Denoting $\hat{n}=(n_{2},n_{3},n_{4},n_{5})\in\mathbb{N}^{4}$ and
$|\hat{n}|=n_{2}+n_{3}+2n_{4}+3n_{5}$, the quantum part is:

$\hat{\Phi}=\sum_{d=1}^{\infty}\left(\sum_{|\hat{n}|=4d+1}N(\hat{n})\frac{t_{2}^{n_{2}}t_{3}^{n_{3}}t_{4}^{n_{4}}t_{5}^{n_{5}}}{n_{2}!n_{3}!n_{4}!n_{5}!}\right)e^{dt_{1}}\quad.$

Here $N(\hat{n})\in\mathbb{Q}$ is a Gromov-Witten number, counting the stable
$J$-holomorphic spheres of class
$\beta=d\operatorname{PD}(\sigma_{\ydiagram{1,0}})$ with $n_{2}$ points
mapping to $\operatorname{PD}(\sigma_{\ydiagram{2,0}})$, $n_{3}$ points
mapping to $\operatorname{PD}(\sigma_{\ydiagram{1,1}})$, $n_{4}$ points
mapping to $\operatorname{PD}(\sigma_{\ydiagram{2,1}})$, $n_{5}$ points
mapping to $\operatorname{PD}(\sigma_{\ydiagram{2,2}})$. These curves are
rigid thanks to the dimension formula for the moduli space of $J$-holomorphic
spheres of class $\beta$ and

$|\hat{n}|=n_{2}\left(\frac{1}{2}|\sigma_{\ydiagram{2,0}}|-1\right)+n_{3}\left(\frac{1}{2}|\sigma_{\ydiagram{1,1}}|-1\right)+n_{4}\left(\frac{1}{2}|\sigma_{\ydiagram{2,1}}|-1\right)+n_{5}\left(\frac{1}{2}|\sigma_{\ydiagram{2,2}}|-1\right)$
$=\operatorname{dim}_{\mathbb{C}}(X)+\langle c_{1},\beta\rangle-3=4d+1\quad.$

The product structure on the big quantum cohomology $\operatorname{QH}^{t}(X)$
is given by

$\sigma_{i}\star_{t}\sigma_{j}=\sum_{e,f}\Phi_{ije}g^{ef}\sigma_{f}\quad,$

where $\Phi_{ije}$ denotes the third-order partial derivative of $\Phi$ with
respect to $t_{i},t_{j},t_{e}$, and $g^{ef}$ are the entries of the inverse
intersection matrix $g^{-1}$. Dubrovin’s operator acts on
$\operatorname{QH}^{t}(X)$ multiplying by

$K^{t}=4\sigma_{\ydiagram{1,0}}+t_{0}\sigma_{\emptyset}-t_{2}\sigma_{\ydiagram{2,0}}-t_{3}\sigma_{\ydiagram{1,1}}-2t_{4}\sigma_{\ydiagram{2,1}}-3t_{5}\sigma_{\ydiagram{2,2}}\quad.$

Setting $e^{t_{1}}=q$, we can think of this as $t$-dependent family of
operators with spectra in the Novikov field $\Lambda$. For the purposes of
studying the simplicity of such spectra, we can set $t_{0}=0$ thanks to the
following.

######  Lemma 7.1.

The spectra of $K^{t}$ and $K^{\hat{t}}$ differ by a translation of
$t_{0}\in\mathbb{C}$. In particular, one is simple if and only if the other
is.

###### Proof.

Since
$K^{t}=K^{\hat{t}}+t_{0}\sigma_{\emptyset}=K^{\hat{t}}+t_{0}\operatorname{Id}$,
the characteristic polynomials $p_{K^{t}}(\lambda)$ of $K^{t}$ and
$p_{K^{\hat{t}}}(\lambda)$ of $K^{\hat{t}}$ are related by
$p_{K^{t}}(\lambda)=p_{K^{\hat{t}}}(\lambda-t_{0})$. ∎

From now on, denote $\hat{t}=(t_{2},t_{3},t_{4},t_{5})\in\mathbb{C}^{4}$, and
think of $K^{\hat{t}}$ as a family of operators with spectra in the Novikov
field $\Lambda$.

###  7.3. Finite energy truncations

After fixing a basis, represent the Dubrovin operator with a matrix
$M^{\hat{t}}$ with entries in $\Lambda$. Decompose $M^{\hat{t}}=\sum_{d\geq
0}M^{\hat{t}}_{d}q^{d}$, where $M^{\hat{t}}_{d}$ are matrices with entries in
$\mathbb{C}$ containing the coefficients of $q^{d}$. The sum runs over Novikov
exponents $d\geq 0$ since the Gromov-Witten potential $\Phi$ and its third-
order derivatives do not contain negative powers of the formal parameter $q$.

######  Definition 7.2.

The truncation of $K^{\hat{t}}$ at energy $\alpha\in\mathbb{R}_{\geq 0}$ is
the $\Lambda$-linear operator $K^{\hat{t}}_{<\alpha}$ associated to the matrix

$M^{\hat{t}}_{<\alpha}=\sum_{0\leq d<\alpha}M^{\hat{t}}_{d}q^{d}\quad.$

######  Remark 7.3.

While suppressed in the notation, the finite energy truncations
$K^{\hat{t}}_{<\alpha}$ do depend on the basis chosen to represent
$K^{\hat{t}}$ with a matrix. To see this, observe for example that the
following matrices are similar over $\Lambda$ but have different finite energy
truncations:

$\begin{pmatrix}q^{-1}&0\\\ 0&q^{-2}\end{pmatrix}\begin{pmatrix}0&q\\\
q&0\end{pmatrix}\begin{pmatrix}q&0\\\
0&q^{2}\end{pmatrix}=\begin{pmatrix}0&q^{2}\\\ 1&0\end{pmatrix}$

Simplicity of the spectrum of $K^{\hat{t}}$ and its truncations are can be
related as follows. Denote by
$\Delta^{\hat{t}}=\operatorname{Res}(p_{K^{\hat{t}}},p_{K^{\hat{t}}}^{\prime})\in\Lambda$
the discriminant of $K^{\hat{t}}$, i.e. the resultant of the characteristic
polynomial of $K^{\hat{t}}$ and its derivative. Also denote
$\Delta^{\hat{t}}_{<\alpha}\in\Lambda$ the discriminant of the truncation
$K^{\hat{t}}_{<\alpha}$.

######  Proposition 7.4.

The following implications hold:

  1. (1)

if $K^{\hat{t}}$ has simple spectrum, then there is some
$\alpha>\operatorname{val}(\Delta^{\hat{t}})$ such that
$K^{\hat{t}}_{<\alpha}$ has simple spectrum ;

  2. (2)

if $\Delta^{\hat{t}}_{<\alpha}\neq 0$ modulo $q^{\alpha}$, then $K^{\hat{t}}$
has simple spectrum .

###### Proof.

(1) If $K^{\hat{t}}$ has simple spectrum then $\Delta^{\hat{t}}\neq 0$, and so
it has finite valuation $\operatorname{val}(\Delta^{\hat{t}})<\infty$. Now
observe that $\Delta^{\hat{t}}=\Delta^{\hat{t}}_{<\alpha}$ modulo $q^{\alpha}$
because the discriminant is a polynomial function of the matrix entries. Since
the left hand side is nonzero modulo $q^{\alpha}$ for
$\alpha>\operatorname{val}(\Delta^{\hat{t}})$, it must be the case that
$\Delta^{\hat{t}}_{<\alpha}\neq 0$, hence $K^{\hat{t}}_{<\alpha}$ has simple
spectrum.
(2) Using again $\Delta^{\hat{t}}=\Delta^{\hat{t}}_{<\alpha}$ modulo
$q^{\alpha}$ and the assumption that the right hand side is nonzero, it
follows that $\Delta^{\hat{t}}\neq 0$, hence $K^{\hat{t}}$ has simple
spectrum. ∎

Crucially, finite energy truncations are determined by finitely many Gromov-
Witten numbers, and these are typically computable from a small set using the
recursion relations given by the WDVV equations. In our case the Gromov-Witten
numbers $N(\hat{n})\in\mathbb{Q}$ with $|\hat{n}|=1$ are written below (see
e.g. Di Francesco-Itzykson [9, Proposition 9]), and can be obtained by
recursion from $N(0,0,1,1)=1$:

$N(5,0,0,0)=0$ | $N(4,1,0,0)=0$ | $N(3,2,0,0)=1$ | $N(3,0,1,0)=0$
---|---|---|---
$N(2,1,1,0)=1$ | $N(2,0,0,1)=0$ | $N(1,1,0,1)=1$ | $N(1,0,2,0)=1$
$N(0,0,1,1)=1$ |  |  |

Note that these numbers are non-negative integers, because the target is a
homogeneous variety; see [17, Lemma 14]. Below is the matrix of the truncation
$(K^{\hat{t}})_{<2}$ with respect to the basis of Schubert classes. Its
calculation only involves the Gromov-Witten numbers $N(\hat{n})\in\mathbb{Q}$
with $|\hat{n}|=1$.

$0$ |  $(-2t_{2}t_{3}+2t_{4}+4t_{4}t_{5})q$ | $3t_{3}q$ |  $3t_{2}q$ | $4q$ |  $0$
---|---|---|---|---|---
$4$ |  $(1/2t_{2}^{2}t_{3}+t_{2}t_{4}+t_{5})q$ | $(2t_{2}t_{3}+2t_{4})q$ |  $t_{2}^{2}q$ | $3t_{2}q$ |  $4q$
$-t_{2}$ |  $4$ | $(1/2t_{2}t_{3}^{2}+t_{3}t_{4})q$ |  $(1/2t_{2}^{2}t_{3}+t_{2}t_{4}+t_{5})q$ | $(2t_{2}t_{3}+2t_{4})q$ |  $3t_{3}q$
$-t_{3}$ |  $4$ | $(1/2t_{2}^{2}t_{3}+t_{2}t_{4}+t_{5})q$ |  $1/6t_{2}^{3}q$ | $t_{2}^{2}q$ |  $3t_{2}q$
$-2t_{4}$ |  $-t_{2}-t_{3}+(1/12t_{2}^{3}t_{3}^{2}-1/2t_{2}^{2}t_{3}t_{4}+2t_{2}t_{3}t_{5}-1/2t_{2}t_{4}^{2}-t_{4}t_{5}-3t_{4}t_{5}^{2})q$ | $4$ |  $4$ | $(1/2t_{2}^{2}t_{3}+t_{2}t_{4}+t_{5})q$ |  $(-2t_{2}t_{3}+2t_{4}+4t_{4}t_{5})q$
$-3t_{5}$ |  $-2t_{4}$ | $-t_{2}$ |  $-t_{3}$ | $4$ |  $0$

###  7.4. Decoupling

The Grassmannian $\operatorname{Gr}(2,4)$ is perhaps one of the simplest
examples where the the spectrum of the Dubrovin operator is not simple for all
bulk-parameters.

######  Lemma 7.5.

(see for example [5, Proposition 1.12]) The eigenvalues of $K^{0}=c_{1}$ are
$\pm 4\sqrt{2}$, $\pm 4\sqrt{2}i$, $0$. Moreover $0$ has algebraic
multiplicity $2$.

This section analyzes the spectrum of the finite energy truncation
$K^{\hat{t}}_{<2}$ computed earlier one bulk-parameter at the time, i.e.
allowing only one bulk-parameter to be non-vanishing. This corresponds to
analyzing the quantum contributions of $J$-holomorphic spheres with incidence
condition on a single Schubert class.

  * •

Schubert class $\sigma_{2}=\sigma_{\ydiagram{2,0}}$: $t_{3}=t_{4}=t_{5}=0$.
The truncation $K^{\hat{t}}_{<2}$ specializes to the $t_{2}$-dependent family
of operators

$\begin{pmatrix}0&0&0&3t_{2}q&4q&0\\\ 4&0&0&t_{2}^{2}q&3t_{2}q&4q\\\
-t_{2}&4&0&0&0&0\\\ 0&4&0&1/6t_{2}^{3}q&t_{2}^{2}q&3t_{2}q\\\
0&-t_{2}&4&4&0&0\\\ 0&0&-t_{2}&0&4&0\\\ \end{pmatrix}$

whose discriminant is
$\Delta^{t_{2}}_{<2}=(-1208925819614629174706176t_{2}^{2})q^{8}+o(q^{8})$. In
particular, these have simple spectrum for $t_{2}\neq 0$.

  * •

Schubert class $\sigma_{3}=\sigma_{\ydiagram{1,1}}$, $t_{2}=t_{4}=t_{5}=0$.
The truncation $K^{\hat{t}}_{<2}$ specializes to the $t_{3}$-dependent family
of operators

$\begin{pmatrix}0&0&3t_{3}q&0&4q&0\\\ 4&0&0&0&0&4q\\\ 0&4&0&0&0&3t_{3}q\\\
-t_{3}&4&0&0&0&0\\\ 0&-t_{3}&4&4&0&0\\\ 0&0&0&-t_{3}&4&0\\\ \end{pmatrix}$

whose discriminant is
$\Delta^{t_{3}}_{<2}=(-1208925819614629174706176t_{3}^{2})q^{8}+o(q^{8})$. In
particular, these have simple spectrum for $t_{3}\neq 0$.

  * •

Schubert class $\sigma_{4}=\sigma_{\ydiagram{2,1}}$, $t_{2}=t_{3}=t_{5}=0$.
The truncation $K^{\hat{t}}_{<2}$ specializes to the $t_{4}$-dependent family
of operators

$\begin{pmatrix}0&2t_{4}q&0&0&4q&0\\\ 4&0&2t_{4}q&0&0&4q\\\
0&4&0&0&2t_{4}q&0\\\ 0&4&0&0&0&0\\\ -2t_{4}&0&4&4&0&2t_{4}q\\\
0&-2t_{4}&0&0&4&0\\\ \end{pmatrix}$

whose discriminant is $\Delta^{t_{4}}_{<2}=0$. In particular, these have non-
simple spectrum for all $t_{4}\in\mathbb{C}$.

  * •

Schubert class $\sigma_{5}=\sigma_{\ydiagram{2,2}}$, $t_{2}=t_{3}=t_{4}=0$.
The truncation $K^{\hat{t}}_{<2}$ specializes to the $t_{5}$-dependent family
of operators

$\begin{pmatrix}0&0&0&0&4q&0\\\ 4&t_{5}q&0&0&0&4q\\\ 0&4&0&t_{5}q&0&0\\\
0&4&t_{5}q&0&0&0\\\ 0&0&4&4&t_{5}q&0\\\ -3t_{5}&0&0&0&4&0\\\ \end{pmatrix}$

whose discriminant is
$\Delta^{t_{5}}_{<2}=(1888946593147858085478400t_{5}^{2})q^{9}+o(q^{9})$. In
particular, these have simple spectrum for $t_{5}\neq 0$.

The calculations above have been used to produce Figure 1, and can be
summarized in the following theorem.

######  Theorem 7.6.

Consider the Grassmannian $X=\operatorname{Gr}(2,4)$ and the energy
$\alpha=2$. The truncation of $(K^{B})_{<2}$ has simple spectrum when $B\neq
0$ is supported on the Schubert cycles
$X_{\ydiagram{2,0}},X_{\ydiagram{1,1}},X_{\ydiagram{2,2}}\subset\operatorname{Gr}(2,4)$,
while it doesn’t when it is supported on the Schubert cycle
$X_{\ydiagram{2,1}}\subset\operatorname{Gr}(2,4)$.

In view of Proposition 7.4, this theorem suggests that $K^{B}$ could have
simple spectrum whenever $B$ is supported on the Schubert cycles
$X_{\ydiagram{2,0}},X_{\ydiagram{1,1}},X_{\ydiagram{2,2}}\subset\operatorname{Gr}(2,4)$.
However, one cannot conclude this with the given information, since
$(\Delta^{B})_{<2}=0$ modulo $q^{2}$ in each of these cases. Instead, the
calculations with $B$ supported on the Schubert cycle
$X_{\ydiagram{2,1}}\subset\operatorname{Gr}(2,4)$ say that in this case either
$K^{B}$ has non-simple spectrum, or $\operatorname{val}(\Delta^{B})\geq 2$.

## References

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