Tropical Mirror Symmetry: Correlation functions

Andrey Losev

Wu Wen-Tsun Key Lab of Mathematics, Chinese Academy of Sciences, USTC, No.96,
JinZhai Road Baohe District, Hefei, Anhui, 230026, P.R.China
National Research University Higher School of Economics, Laboratory of Mirror
Symmetry, NRU HSE, 6 Usacheva str., Moscow, Russia, 119048

Vyacheslav Lysov

Okinawa Institute of Science and Technology,
1919-1 Tancha, Onna-son, Okinawa 904-0495, Japan

We formulate the mirror symmetry for correlation functions of tropical
observables. We prove the tropical mirror correspondence for correlation
functions of evaluation observables on toric space. The key point of the proof
is the localization of correlation functions for mirror states in type-B
higher topological quantum mechanics on trees. The correlation functions
localize to the correlation functions of holomorphic functions, defined
recursively in Landau-Ginzburg-Saito theory with exponential mirror
superpotential and tropical good section.

###### Contents

  1. 1 Introduction
  2. 2 Mirror Correspondence
    1. 2.1 Toric varieties
    2. 2.2 Gromov-Witten theory
    3. 2.3 Tropical GW invariants
    4. 2.4 Mirror relation
  3. 3 Mirror for correlation functions
    1. 3.1 Landau-Ginzburg-Saito theory
    2. 3.2 Correlation functions in Landau-Ginzburg-Saito theory
    3. 3.3 Cohomology and pairing
    4. 3.4 Contact terms from good section
  4. 4 HTQM on trees
    1. 4.1 Higher topological quantum mechanics
    2. 4.2 HTQM on trees
    3. 4.3 Amplitudes on trees
    4. 4.4 Amplitudes in homotopy notation
    5. 4.5 Deformation of HTQM by a state
    6. 4.6 Diagrammatic representation of deformed theory
  5. 5 Correlation functions in HTQM
    1. 5.1 3-point function
    2. 5.2 4-point function
    3. 5.3 Generating function
    4. 5.4 Invariance theorem
    5. 5.5 Recursion relation for correlation functions
  6. 6 Mirror for HTQM
    1. 6.1 A-model
    2. 6.2 Correlation functions in A-model
    3. 6.3 Dual variables
    4. 6.4 B-model
  7. 7 Localization of mirror states
    1. 7.1 Mirror states vs divisor states
    2. 7.2 Spectral sequence for $Q_{W}$-cohomology
    3. 7.3 Pairing and localization of states
    4. 7.4 Higher pairings in B-model
    5. 7.5 B-model deformation by a holomorphic function
    6. 7.6 Higher pairing for mirror states
  8. 8 Correlation functions for mirror states
    1. 8.1 4-point function invariance and holomorphic representatives
    2. 8.2 n-point function invariance and holomorphic representative
    3. 8.3 3-point functions
    4. 8.4 4-point functions
    5. 8.5 Contact terms
    6. 8.6 5-point function
    7. 8.7 Parallel transport of a good section
    8. 8.8 Localization of correlation functions

##  1 Introduction

The real tropical numbers is a set of real numbers, extended by
$\\{-\infty\\}$ with arithmetic operations: tropical addition
$x+_{T}y=\max(x,y)$ and tropical multiplication $x\ast_{T}y=x+y$. The set of
tropical numbers is a semigroup with respect to tropical addition. The
tropical numbers appeared at different times at several branches of
mathematics: Maslov introduced notion of tropical integration, while computer
scientist Imre Simon introduced and adjective tropical.

Mikhalkin [1], [2], [3] used tropical numbers to study geometric problems. The
collection of problems and methods become the Tropical Geometry research area.
In many cases the enumerative problems in algebraic geometry over complex
numbers have tropical counterparts. In particular, we can define the tropical
Gromov-Witten invariants by counting graphs passing through some cycles, see
Mikhalkin [2, 1].

Mikhalkin [2] observed that the number of tropical curves in $\mathbb{P}^{2}$
of degree 3, passing through 8 points in general position is 12. This number,
matches with counting of degree-3 algebraic curves passing through 8 points.
Gathmann and Markwig [4] showed that equality generalizes for all Gromov-
Witten invariants of $\mathbb{P}^{2}$. The matching of tropical and complex
invariants was formalized into:

Theorem (tropical correspondence): Gromov–Witten invariant coincides with its
tropical counterpart.

We propose to analyze the tropical correspondence theorem in context of mirror
symmetry.

Theorem (mirror symmetry): The Gromov–Witten invariants for $X$ equal to the
correlation functions in Landau-Ginzburg-Saito theory on mirror space
$X^{\vee}$.

The mirror symmetry for toric space $X$ relates [5] the GW invariants to
Landau-Ginzburg-Saito theory with exponential superpotentials, see [6] for
review.

Theorem (tropical mirror symmetry): The tropical Gromov–Witten invariants for
toric space $X$ of complex dimension $N$ equal to the correlation functions in
Landau-Ginzburg-Saito theory on mirror space
$X^{\vee}={\mathbb{C}^{\ast}}^{N}$ with certain exponential superpotential,
canonical holomorphic top form and mirror K. Saito’s good section.

Our main result is the proof of the tropical mirror symmetry theorem. There
are two possible application of our result: We can assume the tropical
correspondence theorem and use our proof as a new proof of mirror symmetry for
the toric spaces. Alternatively, we can assume the mirror symmetry theorem and
use our proof as a proof for the tropical correspondence for toric spaces.

In our proof we use several key ideas: tropical numbers as a scaling of the
exponential map, quantum mechanics representation of the tropical GW
invariants, mirror relation for quantum mechanics as summation over divisor
states and localization for HTQM correlation functions.

Real tropical numbers can be constructed from the field of real numbers with
usual addition and multiplication by performing the exponential map
$X=e^{\frac{x}{{\epsilon}}}$ followed by the limit ${\epsilon}\to 0$. The
exponential map turns multiplication into addition for any value of
${\epsilon}$, while the tropical addition requires the limit

$\lim_{{\epsilon}\to
0}\;\;{\epsilon}\ln(e^{\frac{x}{{\epsilon}}}+e^{\frac{y}{{\epsilon}}})=\max(x,y)=x+_{T}y.$

The scaling of exponential map generalizes to the complex geometry. The
scaling construction also provides an heuristic proof of the Tropical
Correspondence theorem.

The Gromov-Witten invariants over complex numbers can be described using
$A$-type twisted topological string theory [7]. In our work [8] we proved that
the tropical GW invariants can be described using the higher topological
quantum mechanics (HTQM) with circle action on graphs, introduced in [9].
Certainly, the HTQM can be constructed as the ${\epsilon}\to 0$ limit of
maximally degenerate complex structure (very long strings) in topological
string theory, but the fact that it captures all tree-level tropical GW
invariants is a novel result. Moreover in [10] (see also [9]) it was shown
that the quantum mechanics, similar to the one we described in previous paper,
provides a solution to the WDVV equations.

Authors of [11] used the 2d CFT to relate the sum over holomortex insertions
in A-type topological string with superpotential deformation of the 2d CFT. In
our paper [8] we showed that the sum over divisor states in A-type HTQM
amplitudes equals to the amplitudes in B-type HTQM. The B-type HTQM is the
deformation of the A-type HTQM by the boundary divisor states. In particular
we showed that for the case of toric spaces the B-type HTQM has exponential
superpotential.

The topological nature of the B-type topological string allows for the drastic
simplification for correlation functions of certain observables. In
particular, the correlation functions of the mirrored evaluation observables
can be written using the recursive construction, discussed in section 8 of
present work. The base of the recursion, the 3-points functions are evaluated
in terms of the residue formula, constructed from superpotential and
holomorphic top form. Such simplification is a reflection of a localization-
like phenomenon. Typically, localization in field theory and quantum mechanics
requires path integral formulation, while we showed that the localization
construction can be realized using the operator formalism in quantum
mechanics.

The recursive construction for implies that the superpotential alone is not
enough to evaluate the correlation functions for more than 3 observables! The
additional data is given in the form of good section, introduced and studied
by K. Saito [12]. The correlation functions in Landau-Ginzburg-Saito theory
have non-trivial dependence on the choice of good section. As a part of the
proof for the main theorem we derive the good section for mirror
superpotentials from the tropical mirror correspondence.

The structure of our paper is as follows: In section 2 we briefly review the
mirror symmetry and formulate it in the form of the equality for correlation
functions. In section 3.1 we review the recursive construction for correlation
functions in Landau-Ginzburg-Saito theory. In section 4 we define the Higher
Topological Quantum Mechanics on trees, describe the amplitudes and describe a
deformation of HTQM by special state. In section 5 we introduce the notion of
correlation functions for HTQM and discuss their properties. In section 6 we
briefly review the HTQM representation for tropical GW invariants and mirror
relation between A- and B- types HTQMs. In section 7 we will introduce a
notion of localization for mirror states, while in the last section we will
use the localized states to evaluate the correlation functions in mirror HTQM.

##  2 Mirror Correspondence

The mirror symmetry describes a relation between the A- and B-models. The
A-model of our interest is the theory of the Gromov-Witten invariants for a
Kahler manifold $X$ of dimension $N$. The B-model is the theory of complex
structure deformations on the dual complex manifold $X^{\vee}$ of same
dimension $N$. In well-known examples both $X$ and $X^{\vee}$ are compact
Calabi-Yau 3-folds.

We can generalize the correspondence away from three-dimensional Calabi-Yau
spaces by relaxing the compactness condition on $X^{\vee}$. The complex
structure deformations in that case are also parametrized by a holomorphic a
function $W$ and the corresponding B-model in commonly referred to as the
Landau-Ginzburg (LG) theory with holomorphic superpotential $W$.

In our paper we will discuss the A-model on a toric space $X$ of dimension
$N$. Furthermore, we will perform a certain scaling procedure to the geometry,
which transforms the usual GW invariants into the tropical ones. The
corresponding mirror B-model becomes the LG theory on
$X^{\vee}=\mathbb{C}^{\ast N}$ with exponential superpotential.

In the rest of this section we will briefly review the definition of toric
manifolds, tropical GW invariants and give a detailed formulation of the
mirror relation.

###  2.1 Toric varieties

Toric manifold $X$ is a compactification of ${\mathbb{C^{\ast}}}^{N}$. We can
represent $\mathbb{C}^{\ast N}=\mathbb{R}^{N}\times\mathbb{T}^{N}$ with the
radial part $\mathbb{R}^{N}$, equipped with standard coordinates
$r^{i},\;i=1,..,N$ and angular part, $N$-dimensional torus
$\mathbb{T}^{N}=(S^{1})^{N}$, with standard angular coordinates $\phi^{i}$.
Equivalently, we can say that the $\mathbb{C}^{\ast N}$ is a trivial
$N$-dimensional toric fibration over $\mathbb{R}^{N}$. We describe the
compactification of ${\mathbb{C^{\ast}}}^{N}$ using the fibraton data.

  * •

The radial part is compactified by the hyperplanes at infinity. Each
hyperplane is given in terms of the inside-pointing $N$-dimensional normal
vector $\vec{b}_{a}$ with components $b_{a}^{i}$. Each normal vector has
integer components i.e. $b^{i}_{a}\in\mathbb{Z}$. The two normal vectors with
proportional components describe the same hypersurface. We can fix this
ambiguity by choosing the primitive normal vector, such that
gcd$(b_{a}^{1},b_{a}^{2},...,b_{a}^{N})=1$. For toric space $X$ we will denote
the set of such normal vectors by $B_{X}=\\{\vec{b}_{a}\\}$.

  * •

In order to get a compactification of a complex manifold, we require that one
of the circles $S^{1}\subset\mathbb{T}^{N}$ inside the toric fibration shrinks
to zero when we approach each of the compactifying hypersurfaces. The choice
of a circle is given by a class in
$\pi_{1}(\mathbb{T}^{N})=H^{1}(\mathbb{T}^{N},\mathbb{Z})$. For the hyperplane
with normal vector $\vec{b}_{a}$ the corresponding class is the class of $\sum
b_{a}^{i}d\phi^{i}$.

We will refer to the compacting hyperplanes as compactifying divisors, and to
$B_{X}$ as the set of all compactifying divisors for toric space $X$.

###  2.2 Gromov-Witten theory

The Gromov-Witten invariant $N^{X}_{\beta}(C_{1},...,C_{n})$ counts the number
of algebraic curves of degree-$\beta$, genus-$0$ in complex space $X$, passing
through the cycles $C_{1},...,C_{n}$. The integral representation

$N^{X}_{\beta}(C_{1},...,C_{n})=\int_{\mathcal{M}_{0,n}(X,\beta)}\bigwedge_{\alpha=1}^{n}\;\;ev_{\alpha}^{\ast}{\gamma}_{\alpha},$
(2.1)

uses $\mathcal{M}_{0,n}(X,\beta)$, the moduli space of degree $\beta$ curves
in $X$ with $n$ marked points, compactified by quasi-maps, equipped with the
evaluation map

$ev_{\alpha}:\mathcal{M}_{0,n}(X,\beta)\to X:(\phi:\mathbb{P}^{1}\to
X,z_{1},...,z_{n})\mapsto\phi(z_{\alpha}).$ (2.2)

The ${\gamma}_{\alpha}$ are special representatives (smoothed out delta
functions on cycles) of the Poincare dual to the cycles $C_{\alpha}$. In
mathphysics literature the commonly used notation for the same GW invariant is
$\langle{\gamma}_{1},..,{\gamma}_{n}\rangle^{X}_{\beta,0}$ and we will use it
in our paper.

For a given cycles ${\gamma}_{k}\in H^{\ast}(X)$ we can organize the genus-0
GW invariants of different degrees into a single expression

$\langle{\gamma}_{1},..,{\gamma}_{n}\rangle^{X}=\sum_{\beta\in
H_{1,1}(X)}q^{\beta}\langle{\gamma}_{1},..,{\gamma}_{n}\rangle^{X}_{\beta,0}\;\;,$
(2.3)

where $q$ describes the Kahler moduli of $X$. For generic $X$ the GW
invariants are formal series in $q$, but for toric $X$ they simplify to the
polynomials in $q$.

Remark: In present paper we will restrict our attention to the GW invariants
for 3 or more observables i.e $n\geq 3$. There are interesting geometric
invariants of $X$, with natural description in the form of GW invariants for
two observables, but most of them can be reformulated as invariants with 3 or
more observables.

###  2.3 Tropical GW invariants

On toric manifold $X$ we can perform a tropical limit: a coordinate
transformation in the form of scaling
$(r^{k},\phi^{k})\to(r^{k}/{\epsilon},\phi^{k})$ followed by the
${\epsilon}\to 0$ limit. For more details see our previous paper [8] . The
limit of a smooth algebraic curve of genus zero in toric space $X$ is a circle
bundle over a tree. The tree is embedded by a piece-wise linear map into
radial part of $X$. The limiting curve is known as a tropical curve and was
extensively studied by Mikhalkin and collaborators [2] in context of Tropical
Geometry. The moduli of a tropical curve are position of a root vertex,
lengths and twists of internal edges of a tree. We discussed the moduli space
in our work [8], while for more detailed review see Mikhalkin’s book [3].

We can take the tropical limit for the differential forms ${\gamma}_{\alpha}$
to define the tropical A-model observables. Note that the tropical limit turns
smooth forms on toric space $X$ into singular forms on radial part of $X$.

We can define the tropical GW invariants as the integral over tropical moduli
space of tropical observables. In our work [8] we showed that the tropical GW
invariants can be written as a sum of amplitudes in Higher Topological Quantum
Mechanics (HTQM) on trees. We will provide the detailed definition of HTQM and
describe the amplitudes in section 4.

In many examples it was observed that the tropical GW invariants match with
the conventional GW invariants. This observation was formalized into:

Theorem (tropical correspondence): For the toric space $X$ and smooth
differential forms ${\gamma}_{k}\in H_{dR}^{\ast}(X)$ the Gromov-Witten
invariant $\langle{\gamma}_{1},..,{\gamma}_{n}\rangle^{X}_{\beta}$ matches
with tropical Gromov-Witten invariant
$\langle{\gamma}^{trop}_{1},..,{\gamma}^{trop}_{n}\rangle^{X}_{\beta}$ for the
tropical limit ${\gamma}_{k}^{trop}$ of the forms ${\gamma}_{k}$.

Evidence: There are several different types of evidence for the theorem:

  * •

Mikhalkin [2] counted the number degree-3 tropical curves in $\mathbb{P}^{2}$
passing through the $8$ points in general position, by presenting the
corresponding graphs, counted with proper multiplicities. The total number he
obtained was $12$, what matched with the well known $N_{3}=12$ answer for the
same problem for algebraic curves.

  * •

Gathmann and Markwig [13] derived the recursion formula for number of tropical
curves of degree-$d$, passing through the $3d-1$ points on $\mathbb{P}^{2}$.
Their result matches with the Kontsevich-Manin recursion formula [14] for
algebraic curves. Such matching essentially gives a prof of tropical
correspondence for $\mathbb{P}^{2}$.

  * •

There are several results on match for the descendant GW invariants: The case
of $\mathbb{P}^{1}$ was discussed by Böhm, Goldner and Markwig [15]. Markiw
and Rau [16] proved the equality for descendant invariants for
$\mathbb{P}^{2}$.

  * •

Our construction of tropical numbers and geometric objects as a scaling limit
${\epsilon}\to 0$ in cylindrical coordinates serves as an heuristic proof of
the theorem. The GW invariants do not depend on the choice of coordinates,
hence remain the same for any non-zero value of ${\epsilon}$.

  * •

In [8] we derived the tropical mirror superpotential for toric $X$ and it
matches with exponential mirror superpotential derived in [5], [17] for the
same toric space $X$. Given that the mirror symmetry relation holds for toric
spaces we can use our result as an evidence in favor of the tropical
correspondence theorem.

###  2.4 Mirror relation

The mirror of the $N$-dimensional toric manifold $X$ is a non-compact
$N$-dimensional Calabi-Yau $X^{\vee}=\mathbb{C}^{\ast N}$ with holomorphic
superpotential. We will use the toric representation $\mathbb{C}^{\ast
N}=\mathbb{R}^{N}\times\mathbb{T}^{N}$ with radial coordinates $r^{j}$ and
angular (holomorphic) coordinates $Y_{j}$. The holomorphic top form in these
coordinates is $\Omega=dY^{1}\wedge..\wedge dY^{N}$. Let us formulate several
relations, implied by the mirror correspondence theorem.

Definition: The Jacobi ring for superpotential $W$ is

$J_{W}=R_{\mathbb{C}^{\ast N}}/I_{W},$ (2.4)

where $R_{\mathbb{C}^{\ast N}}$ is the ring of holomorphic functions on
$\mathbb{C}^{\ast N}$. In our coordinates $R_{\mathbb{C}^{\ast N}}$ is the
ring of periodic functions of $Y$. The $I_{W}$ is the ideal generated by the
partial derivatives of $W$

$I_{W}=\left\\{\frac{\partial W}{\partial Y_{j}}\right\\}.$ (2.5)

Remark: If $W$ has isolated critical points then $J_{W}$ is finite-
dimensional.

Proposition (mirror for observables): The de Rahm cohomology of toric space
$X$ isomorphic (as a vector spaces !) to the Jacobi ring

$J:H_{dR}^{\ast}(X)\to J_{W_{X}}:\gamma\mapsto J_{\gamma}$ (2.6)

for mirror superpotential

$W_{X}=\sum_{\vec{b}\in
B_{X}}q_{\vec{b}}\;\;e^{i\langle\vec{b},\vec{Y}\rangle},$ (2.7)

where the sum is taken over the compactifying divisors $B_{X}$ of $X$.

We can refine the mirror relation for observables to the isomorphism of rings.

Proposition (mirror for Frobenius rings): The quantum cohomology of toric
space $X$ isomorphic (as graded-commutative Frobenius rings) to the Jacobi
ring for mirror superpotential.

The Frobenius ring structure $(C^{A},g^{A})$ on quantum cohomology ring of $X$
is determined by the 3-point GW invariants of $X$

$C^{A}_{{\alpha}{\beta}{\delta}}=\langle{\gamma}_{\alpha},{\gamma}_{\beta},{\gamma}_{\delta}\rangle^{X},\;\;\;g^{A}_{{\alpha}{\beta}}=\langle{\gamma}_{\alpha},{\gamma}_{\beta},1\rangle^{X}=\int_{X}{\gamma}_{\alpha}\wedge{\gamma}_{\beta},$
(2.8)

while the Frobenius ring structure $(C^{B},g^{B})$ for Jacobi ring can be
formulated using the residue formula. Hence the mirror relation for rings can
be formulated as equality

$C^{B}_{{\alpha}{\beta}{\delta}}=\sum_{dW_{X}=0}\frac{\Phi_{{\gamma}_{\alpha}}\;\Phi_{{\gamma}_{\beta}}\;\Phi_{{\gamma}_{\delta}}}{\det\partial_{k}\partial_{l}W_{X}},\;\;g^{B}_{{\alpha}{\beta}}=\sum_{dW_{X}=0}\frac{\Phi_{{\gamma}_{\alpha}}\;\Phi_{{\gamma}_{\beta}}}{\det\partial_{k}\partial_{l}W_{X}},$
(2.9)

where $\Phi_{\gamma}$ is a representative of a class $J_{\gamma}$. One can
show that (2.9) is independent on the choice of representative. There is a
further generalization of the mirror relation for rings, which includes all GW
invariants of $X$.

Let ${\gamma}_{1},..,{\gamma}_{n}$ be a basis in $H_{dR}^{\ast}(X)$ and
$T^{k}$ $-$ linear coordinates on this space in this basis. We can organize
all genus-0 GW invariants for $X$ into the generating function

$\mathcal{F}_{A}(T,q)=\sum_{j_{1},..,j_{n}=0}^{\infty}\langle\underbrace{{\gamma}_{1},..,{\gamma}_{1}}_{j_{1}},...,\underbrace{{\gamma}_{n},..,{\gamma}_{n}}_{j_{n}}\rangle^{X}\;\prod_{k=1}^{n}\frac{(T^{k})^{j_{k}}}{j_{k}!}.$
(2.10)

Parameters $q$ represent the Kahler moduli dependence for $X$ in A-model.

The B-model generating function was defined in works of K.Saito [12], Blok-
Varchenko [18] and Dijkgraaf-Verlinde-Verlinde [19]. Namely

$\frac{\partial^{3}\mathcal{F}_{B}(T,q)}{\partial T^{\alpha}\;\partial
T^{\beta}\;\partial
T^{\delta}}=\sum_{dW=0}\frac{\Phi_{\alpha}(T)\cdot\Phi_{\beta}(T)\cdot\Phi_{\delta}(T)}{\det\partial_{k}\partial_{l}W(T)},$
(2.11)

where $W(T)$ is the deformation of mirror superpotential in special
coordinates, introduced by K. Saito, the holomorphic functions
$\Phi_{\alpha}(T)$ are partial derivatives of superpotential, i.e.

$\Phi_{\alpha}(T)=\frac{\partial W}{\partial T^{\alpha}}.$ (2.12)

The partial derivatives $\partial_{k}\partial_{l}W$ of $W$ are taken in
coordinates $Y$, where the holomorphic top form is $dY_{1}\wedge..\wedge
dY_{N}$.

Theorem (mirror correspondence): For toric space $X$ the generating function
of the GW invariants equals to the B-model generating function (2.11) for
deformations of mirror superpotential $W_{X}$ i.e.

$\mathcal{F}_{A}(T,q)=\mathcal{F}_{B}(T,q).$ (2.13)

Remark: The equality above is an equality for the formal series in $q$ and
$T$. For the toric GW invariant each coefficient in $T$-expansion is a
polynomial in $q$, rather than the formal series. Therefore for the case of
toric A-model the corresponding mirror B-model expression should also be a
polynomial. Hence we will focus on proving the equality of two polynomials for
the coefficients of formal series in $T$. In the next section we will give a
description for the B-model coefficients in the generating function expansion.

##  3 Mirror for correlation functions

For arbitrary three holomorphic functions we define the 3-point correlation
function

$\left\langle\Phi_{\alpha},\Phi_{\beta},\Phi_{\gamma}\right\rangle_{W}=\sum_{dW=0}\frac{\Phi_{\alpha}\cdot\Phi_{\beta}\cdot\Phi_{\gamma}}{\det\partial_{k}\partial_{l}W}.$
(3.1)

Then the generating function of B-model (2.11) can be written in the form

$\frac{\partial^{3}\mathcal{F}_{B}(T,q)}{\partial T^{\alpha}\;\partial
T^{\beta}\;\partial
T^{\gamma}}=\left\langle\Phi_{\alpha}(T),\Phi_{\beta}(T),\Phi_{\gamma}(T)\right\rangle_{W(T)}.$
(3.2)

In particular, the relation above implies that the cubical term in
$T$-expansion is given by

$\mathcal{F}_{B}(T,q)=\frac{1}{3!}T^{\alpha}T^{\beta}T^{\gamma}\left\langle\Phi_{\alpha},\Phi_{\beta},\Phi_{\gamma}\right\rangle_{W}+\mathcal{O}(T^{4}),$
(3.3)

where $\Phi_{\alpha}=\Phi_{\alpha}(0)$ are representatives for the classes in
Jacobi ring $J_{W}$ for $W=W(0)$. The quartic term in the expansion for
$\mathcal{F}_{B}$ can be expressed in the form

$\begin{split}&\frac{\partial^{4}\mathcal{F}_{B}}{\partial
T^{\alpha}\;\partial T^{\beta}\;\partial T^{\gamma}\;\partial
T^{\delta}}\Big{|}_{T=0}=\frac{\partial}{\partial
T^{\delta}}\Big{|}_{T^{\delta}=0}\left\langle\Phi_{{\alpha}},\Phi_{{\beta}},\Phi_{{\gamma}}\right\rangle_{W+T^{\delta}\Phi_{\delta}}+\left\langle\frac{\partial}{\partial
T^{\delta}}\Big{|}_{T^{\delta}=0}\Phi_{{\alpha}}(T),\Phi_{{\beta}},\Phi_{{\gamma}}\right\rangle_{W}\\\
&\qquad\qquad+\left\langle\Phi_{{\alpha}},\frac{\partial}{\partial
T^{\delta}}\Big{|}_{T^{\delta}=0}\Phi_{\beta}(T),\Phi_{{\gamma}}\right\rangle_{W}+\left\langle\Phi_{{\alpha}},\Phi_{{\beta}},\frac{\partial}{\partial
T^{\delta}}\Big{|}_{T^{\delta}=0}\Phi_{\gamma}(T)\right\rangle_{W}.\end{split}$
(3.4)

The expression above has the following interpretation: The first term is the
change in 3-point correlation function under the change of $W$ in the
direction of $\Phi_{\delta}$. The last three terms describe the change of the
functions $\Phi_{\alpha}(T)$ under the transport in direction of
$\Phi_{\delta}$. We can introduce connection $C_{W}$, so the change of a
function $\Phi_{\alpha}(T)$ in direction $\Phi_{\delta}$ is given by

$\frac{\partial}{\partial
T^{\delta}}\Big{|}_{T^{\delta}=0}\Phi_{{\alpha}}(T)=C_{W}(\Phi_{\delta},\Phi_{\alpha}).$
(3.5)

The combination of four terms in the formula (3.4) is known as the recursion
formula for the 4-point function in 2-dimensional topological theory. The
commonly used form for the recursion relation is (see also [20])

$\begin{split}\langle\Phi_{{\alpha}},\Phi_{\beta},\Phi_{{\gamma}},\Phi_{\delta}\rangle_{W}&=\frac{d}{d{\epsilon}}\Big{|}_{{\epsilon}=0}\langle\Phi_{{\alpha}},\Phi_{{\beta}},\Phi_{{\gamma}}\rangle_{W+{\epsilon}\Phi_{\delta}}+\langle
C_{W}(\Phi_{\delta},\Phi_{\alpha}),\Phi_{{\beta}},\Phi_{{\gamma}}\rangle_{W}\\\
&\qquad\qquad+\langle\Phi_{{\alpha}},C_{W}(\Phi_{\delta},\Phi_{\beta}),\Phi_{{\gamma}}\rangle_{W}+\langle\Phi_{{\alpha}},\Phi_{{\beta}},C_{W}(\Phi_{\delta},\Phi_{\gamma})\rangle_{W}.\end{split}$
(3.6)

Note that the 4-point function depends on a choice of holomorphic function
representatives for the classes in Jacobi ring.

The connection coefficients $C_{W}(\Phi_{\delta},\Phi_{\alpha})$ are referred
to in [20] as the contact terms. The fourth derivative of the generating
function can be written in terms of the 4-point function, defined by (3.6)

$\frac{\partial^{4}\mathcal{F}_{B}}{\partial T^{\alpha}\;\partial
T^{\beta}\;\partial T^{\gamma}\;\partial
T^{\delta}}\Big{|}_{T=0}=\langle\Phi_{{\alpha}},\Phi_{\beta},\Phi_{{\gamma}},\Phi_{\delta}\rangle_{W}.$
(3.7)

The recursion relation (3.6) can be further generalized to the case of
$n$-point functions. In physics literature the recursion relations of that
type are commonly discussed in context of the Landau-Ginzburg theory [19], so
we will adopt the same name for our review of it. We will give a detailed
version of the recursion formula and contact terms later in this section. The
expansion coefficients of the generating function become the n-point
functions, i.e.

$\frac{\partial^{n}\mathcal{F}_{B}(T,q)}{\partial T^{{\alpha}_{1}}...\partial
T^{{\alpha}_{n}}}\Big{|}_{T=0}=\langle\Phi_{{\alpha}_{1}},...,\Phi_{{\alpha}_{n}}\rangle_{W}.$
(3.8)

The mirror relation for generating functions implies the mirror relation for
the correlation functions:

Proposition (mirror for correlation functions): For toric space $X$ the GW
invariants fo cycles ${\gamma}_{\alpha}$ equal to the correlation function of
special representatives $SJ_{{\gamma}_{\alpha}}$ of the corresponding the
Jacobi ring classes $J_{{\gamma}_{\alpha}}$ for the mirror superpotential.

$\langle{\gamma}_{1},...,{\gamma}_{n}\rangle^{X}=\langle
SJ_{{\gamma}_{1}},...,SJ_{{\gamma}_{n}}\rangle_{W^{X}}.$ (3.9)

###  3.1 Landau-Ginzburg-Saito theory

The study of Landau-Ginzburg theory was motivated by the theory of critical
phenomena, which later grown into 2D CFT and eventially become part of the
topological string theory. For review see [17], namely the
$(2,2)$-supersymmetric sigma models on non-compact spaces in B-type twisting.
After the B-type twisting, the theory is not superconformal and further
requires setting the anti-holomorphic superpotential to zero, while keeping
the holomorphic superpotential $W$ fixed. B-twisting is anomalous that results
in appearence of the holomorphic top form.

Definition: The Landau-Ginzburg-Saito theory on complex space $X$ with
superpotential $W$ is a collection of the following data:

  1. 1.

non-compact complex manifold $X$ of dimension $N$;

  2. 2.

non-degenerate holomorphic top form

$\Omega\in\Omega^{N,0}(X);$ (3.10)

Note that the pair $(X,\Omega)$ can be constructed from non-compact Calabi-Yau
manifold.

  3. 3.

holomorphic function $W:X\to\mathbb{C}$, called superpotential;

  4. 4.

Good section $S:J_{W}\to R_{X}$.

The readers, familiar with mathphysics literature on the Landau-Ginzburg
models, might not be familiar with the notion of good section and its
significance for the LGS theory. Indeed all of the LGS correlation functions
can be obtained from the $T$-expansion of the generating function
$\mathcal{F}_{LG}(T)$ defined via the 3-point function

$\frac{\partial^{3}\mathcal{F}_{LG}(T)}{\partial T^{\alpha}\;\partial
T^{\beta}\;\partial T^{\gamma}}=\left\langle\frac{\partial W}{\partial
T^{\alpha}},\frac{\partial W}{\partial T^{\beta}},\frac{\partial W}{\partial
T^{\gamma}}\right\rangle_{W(T)}.$ (3.11)

However, in order to use the formula (3.11) we need to define the good times
$T$, which typically have complicated functional relation to the deformations
of $W$, linear on Jacobi ring. The good section data is equivalent to the
choice of good times $T$, but is has more straightforward meaning for the
recursive definition of correlation functions. In some simple cases
(polynomial superpotentials) the good section can be constructed from the
superpotential if we impose an extra requirement of homogeneity. For more
information about the good times and good section relation see [12] and [20].

The most studied LGS theories have complex manifold $\mathbb{C}^{N}$, with
coordinates $x_{j}$, canonical holomorphic top form
$\Omega=dx_{1}\wedge..\wedge dx_{N}$ and polynomial superpotential $W(x)$. The
simplest LGS model of this class has single variable ($N=1$) and polynomial
superpotential of degree $k$. The image of good section in this case is well
known and consists of monomials of degree up to $k-2$

$\hbox{Im}\;S\;=\mathbb{C}\langle 1,x,..,x^{k-2}\rangle.$ (3.12)

Note that in case of polynomial superpotential with two variables, the good
section is known only for limited classes of superpotentials.

Our main interest is the mirror of the A-model for toric manifold $X$ of
complex dimension $N$, given in terms of compactifying divisors $B_{X}$. The
corresponding LGS theory has complex manifold ${\mathbb{C}^{\ast}}^{N}$
equipped with cylindrical coordinates: the radial coordinates
$r^{j}\in\mathbb{R}$ and holomorphic angular coordinates $Y_{j}\in S^{1}$. The
holomorphic top form in this coordinates is

$\Omega=dY_{1}\wedge...\wedge dY_{N},$ (3.13)

while the superpotential is the exponential function, written using the
primitive normal vectors

$W_{X}=\sum_{\vec{b}\in B_{X}}q_{\vec{b}}\;e^{i\;b^{k}Y_{k}}.$ (3.14)

The good sections has not been constructed for all LGS theories with
exponential superpotentials. The simplest exponential superpotential is the
mirror superpotential for $X=\mathbb{P}^{1}$

$W_{\mathbb{P}^{1}}=e^{iY}+qe^{-iY}.$ (3.15)

The image of good section is

$\hbox{ Im}\;S_{\mathbb{P}^{1}}=\mathbb{C}\langle 1,e^{iY}\rangle.$ (3.16)

###  3.2 Correlation functions in Landau-Ginzburg-Saito theory

Definition: For holomorphic functions $\Phi_{\alpha},\;{\alpha}=1,..,n>2$ in
LGS theory on $\mathbb{C}^{\ast N}$ with superpotential $W$ the $n$-point
correlation function $\langle\Phi_{1},...,\Phi_{n}\rangle_{W}$ is defined
recursively:

  * •

the 3-point function is given by the residue formula

$\begin{split}\langle\Phi_{1},&\Phi_{2},\Phi_{3}\rangle_{W}=\sum_{dW=0}\frac{\Phi_{1}\Phi_{2}\Phi_{3}}{\det\partial_{j}\partial_{k}W}\end{split}$
(3.17)

  * •

The $(n+1)$-point function is defined recursively via $n$-point functions and
their derivatives according to formula below

$\begin{split}\langle&\Phi_{1},\Phi_{2},..,\Phi_{n},\Phi_{n+1}\rangle_{W}=\frac{d}{d{\epsilon}}\Big{|}_{{\epsilon}=0}\langle\Phi_{1},\Phi_{2},..,\Phi_{n}\rangle_{W+{\epsilon}\Phi_{n+1}}+\langle
C_{W}(\Phi_{1},\Phi_{n+1}),\Phi_{2},..,\Phi_{n}\rangle_{W}\\\
&\qquad+\langle\Phi_{1},C_{W}(\Phi_{2},\Phi_{n+1}),..,\Phi_{n}\rangle_{W}+..+\langle\Phi_{1},\Phi_{2},..,C_{W}(\Phi_{n},\Phi_{n+1})\rangle_{W}\end{split}$
(3.18)

Earlier we saw that the $n$-point correlation functions represent the
coefficient in the generating function (3.8) hence they are symmetric under
permutation of all arguments $\Phi_{1},\Phi_{2},...,\Phi_{n}$. The 3-point
function in our definition is manifestly symmetric. The symmetry of higher
point functions is rather obscure from the recursive definition and require
certain properties of the contact terms $C_{W}$. Following the literature [12]
and [20], we will formulate this properties in terms of the K. Saito’s
connection on Brieskorn cohomology.

Proposition: The correlation functions in (3.18) are symmetric if

  * •

connection is symmetric;

  * •

connection is flat;

  * •

connection preserves the metric.

In [20] was proposed a construction of $C_{W}$ in terms of the K. Saito’s good
section $S$. Such connection is manifestly symmetric, while the flatness and
metric preservation are derived from the properties of a good section.

###  3.3 Cohomology and pairing

In order to give a definition of good section and contact terms we will
introduce a cohomology theory, motivated by topological string theory of type
B.

Let us consider a graded vector space

$V_{LGS}=R_{\mathbb{C}^{\ast N}}\otimes\mathbb{C}[\psi_{\Phi}^{i}]$ (3.19)

for parity-odd variables $\psi_{\Phi}^{i}$. On $V_{LGS}$ there is a pair of
graded-commuting differentials

${\bf Q}_{W}=\frac{\partial W}{\partial
Y_{j}}\frac{\partial}{\partial\psi_{\Phi}^{j}},\;\;\;{\bf
G}_{-}=\frac{\partial}{\partial
Y_{j}}\frac{\partial}{\partial\psi_{\Phi}^{j}}.$ (3.20)

Remark: $V_{LGS}$ is isomorphic to the space of polyvector fields on
$\mathbb{C}^{\ast N}$, hence it is equipped with parity-odd symplectic
structure and holomorphic top form. The ${\bf G}_{-}$ is a Batalin–Vilkovisky
(BV) operator on $V_{LGS}$, which generalizes the divergence on vector fields
to polyvector fields.

Remark: The local holomorphic observables of dimension-0 in topological string
theory of type B can be identified with polyvector fields. The ${\bf G}_{-}$
is the action of the superpartner to certain $U(1)$-rotation, which preserves
insertion positions for these observables.

Definition: On $V_{LGS}$ there is $\mathbb{C}[[z]]$-valued Saito’s higher
residue pairing

${\bf K}(v_{1},v_{2})=\int_{{S^{1}}^{N}}d^{N}Y\int_{\mathbb{R}^{N}}d^{N}r\int
d^{N}\psi_{\Phi}d^{N}\psi_{R}\;\;v_{1}\wedge e^{-i\Lambda\\{{\bf Q}_{W}+z{\bf
G}_{-}+d_{R},L\\}}v_{2},$ (3.21)

where $\Lambda$ is a real parameter,

$L=\sum_{k=1}^{N}r^{k}\psi_{\Phi}^{k}$ (3.22)

is the localization function and

$d_{R}=\psi_{R}^{j}\frac{\partial}{\partial r^{j}}$ (3.23)

is the radial de Rham operator.

Remark: The integral form of the K. Saito’s pairing (3.21) was proposed in
[20].

Definition: The $\mathbb{C}$-valued higher pairings ${\bf K}^{(k)}$ are
defined as expansion coefficients in $z$-expansion of ${\bf K}$ i.e.

${\bf K}(v_{1},v_{2})=\sum_{k=0}^{\infty}z^{k}\;{\bf K}^{(k)}(v_{1},v_{2}).$
(3.24)

In section 7.4 we will discuss a similar pairing in details, so for now let us
list some properties of the pairing without the proof:

  * •

The operators ${\bf Q}_{W}-z{\bf G}_{-}$ and ${\bf Q}_{W}+z{\bf G}_{-}$ are
conjugated with respect to the pairing (3.21) i.e.

${\bf K}(({\bf Q}_{W}-z{\bf G}_{-})v_{1},v_{2})=-(-1)^{|v_{1}|}{\bf
K}(v_{1},({\bf Q}_{W}+z{\bf G}_{-})v_{2}).$ (3.25)

Hence we can descend the pairing to the pairing on cohomology $H^{\ast}({\bf
Q}_{W}-z{\bf G}_{-})\otimes H^{\ast}({\bf Q}_{W}+z{\bf G}_{-})$. One can show
that all cohomology for ${\bf Q}_{W}\pm z{\bf G}_{-}$ are holomorphic
functions.

  * •

The pairing on $H^{\ast}({\bf Q}_{W}-z{\bf G}_{-})\otimes H^{\ast}({\bf
Q}_{W}+z{\bf G}_{-})$ is independent of $\Lambda$. Hence we can choose
$\Lambda\to\infty$, what localizes the pairing on a sum over critical points
of $W$. In particular, the first two pairings on holomorphic functions

${\bf K}^{(0)}(\Phi_{1},\Phi_{2})=(2\pi
i)^{N}\sum_{dW=0}\frac{\Phi_{1}\Phi_{2}}{\det\partial_{i}\partial_{j}W}$
(3.26)

and

${\bf K}^{(1)}(\Phi_{1},\Phi_{2})=(2\pi
i)^{N}\frac{1}{2}\sum_{dW=0}(\partial_{k}\partial_{{l}}W)^{-1}\frac{\left(\Phi_{1}\;\partial_{k}\partial_{l}\Phi_{2}-\Phi_{2}\;\partial_{k}\partial_{l}\Phi_{1}\right)}{\det\partial_{m}\partial_{n}W}.$
(3.27)

  * •

We can use the pairing ${\bf K}^{(0)}$ to establish an isomorphism between the
Jacobi ring $J_{W}$ and $H^{\ast}({\bf Q}_{W})$.

Remark: The cohomology of ${\bf Q}_{W}+z{\bf G}_{-}$ were introduced by K.
Saito under the name of Brieskorn lattice.

###  3.4 Contact terms from good section

The construction of Jacobi ring comes with canonical projection
$\pi_{W}:R_{\mathbb{C}^{\ast N}}\to J_{W}$. Given a pair of homolorphic
functions $\Phi_{1}$ and $\Phi_{2}$ we can project their product
$\Phi_{1}\Phi_{2}$ to the class $\pi_{W}(\Phi_{1}\Phi_{2})$ in Jacobi ring
$J_{W}$. The section (which inverts $\pi_{W}$) $S_{W}:J_{W}\to
R_{\mathbb{C}^{\ast N}}$ turns this class into holomorphic function
$S_{W}\;\pi_{W}(\Phi_{1}\Phi_{2})$. The difference

$\Phi_{1}\Phi_{2}-S_{W}\;\pi_{W}(\Phi_{1}\Phi_{2})$ (3.28)

is trivial in Jacobi ring. An isomorphism between the $J_{W}$ and
$H^{\ast}({\bf Q}_{W})$ means that there exists a map
${\bf\Sigma}_{W}:R_{\mathbb{C}^{\ast N}}\to V_{LGS}$ such that

$\Phi_{1}\Phi_{2}-S_{W}\pi_{W}(\Phi_{1}\Phi_{2})={\bf
Q}_{W}{\bf\Sigma}_{W}(\Phi_{1}\Phi_{2}),$ (3.29)

and

${\bf\Sigma}_{W}S_{W}=0.$ (3.30)

The choice of such ${\bf\Sigma}_{W}$ is known as the choice of homotopy for
${\bf Q}_{W}$.

Definition: We define a contact term fo $\Phi_{1}$ and $\Phi_{2}$ in LGS
theory with section $S_{W}$

$C^{S}_{W}(\Phi_{1},\Phi_{2})=\pm{\bf
G}_{-}{\bf\Sigma}_{W}(\Phi_{1}\Phi_{2}).$ (3.31)

In other terms the product of two functions $\Phi_{1}\Phi_{2}$ can be
decomposed into the sum of the image of $S_{W}$ and a linear combination of
$\partial^{1}W,..,\partial^{N}W$, i.e.

$\Phi_{1}\Phi_{2}=S_{W}\pi_{W}(\Phi_{1}\Phi_{2})+\sigma_{k}\partial^{k}W$
(3.32)

The ${\bf\Sigma}_{W}(\Phi_{1}\Phi_{2})$ has the form
$\sigma_{k}(Y)\psi_{\Phi}^{k}$, so ${\bf G}_{-}$-action on it is

${\bf
G}_{-}{\bf\Sigma}_{W}(\Phi_{1}\Phi_{2})=\frac{\partial\sigma_{k}(Y)}{\partial
Y_{k}},$ (3.33)

i.e. just a divergence of the vector field $\sigma_{k}(Y)\partial_{Y_{k}}$.
Note that for a given $S_{W}$ the decomposition in (3.32) does not uniquely
fixes the $\sigma_{k}(Y)$. The freedom of choice $\sigma$ is fixed by the
choice of homotopy ${\bf\Sigma}_{W}$.

Note that the dependence of contact term $C_{W}$ on the choice of homotopy
${\bf\Sigma}_{W}$ is $({\bf Q}_{W}+z{\bf G}_{-})$-exact. It was shown that the
correlation functions are well-defined in $H^{\ast}({\bf Q}_{W}+z{\bf
G}_{-})$, so the choice of homotopy does not affect the recursion formula.

We can define a natural projection $\pi:V_{LGS}\otimes\mathbb{C}[[z]]\to
V_{LGS}$, given by an evaluation at $z=0$. The projection $\pi$ is a chain
map, hence it induces projection on cohomology

$\pi:H^{\ast}({\bf Q}_{W}+z{\bf G}_{-})\to H^{\ast}({\bf Q}_{W}).$ (3.34)

The section $S_{W}$ induces a section $S_{Saito}:J_{W}=H^{\ast}({\bf
Q}_{W})\to H^{\ast}({\bf Q}_{W}+z{\bf G}_{-})$. Indeed, every holomorphic
function is both ${\bf Q}_{W}$\- and ${\bf G}_{-}$-closed, hence it describes
a class in $H^{\ast}({\bf Q}_{W}+z{\bf G}_{-})$, which we take as an image of
the $S_{Saito}$-map.

Definition: The good section $S_{Saito}:H^{\ast}({\bf Q}_{W})\to H^{\ast}({\bf
Q}_{W}+z{\bf G}_{-})$ is

  * •

a section for $\pi$ i.e.

$\pi\circ S_{Saito}=Id_{H^{\ast}({\bf Q}_{W})};$ (3.35)

  * •

the higher pairings (3.24) vanish for all pairs
$\Phi_{1},\Phi_{2}\in\hbox{Im}\;S_{Saito}$ i.e

${\bf K}^{(k)}(\Phi_{1},\;\Phi_{2})=0,\;\;\forall\;k>0;$ (3.36)

  * •

For a given section $S_{Saito}$ we can construct the corresponding contact
term and connection. The good section $S_{Saito}$ is preserved under the
parallel transport respect to this connection.

##  4 HTQM on trees

In previous work [8] we showed that the tropical Gromow-Witten invariants can
be described using the higher topological quantum mechanics (HTQM) on tree
graphs. In this section we will briefly review the definition on the HTQM on
trees, describe the amplitudes and construct a family of HTQMs as a
deformation of HTQM by a certain type of states.

###  4.1 Higher topological quantum mechanics

Definition: The higher topological quantum mechanics, HTQM (with the circle
action) $(V,Q,G_{\pm})$ is a collection of the following data:

  1. 1.

$\mathbb{Z}_{2}$-bi-complex $(V,Q,G_{-})$, namely:

     * •

$\mathbb{Z}_{2}$-graded vector space $V$, can be infinite-dimensional. There
is a decomposition of $V=V_{0}\oplus V_{1}$ into even $V_{0}$ and odd $V_{1}$
under the grading. We will use the notation $|v|\in\mathbb{Z}_{2}$ to describe
the grading of a vector $v\in V$;

     * •

pair of differentials $Q,G_{-}:V\to V$, such that

       * –

grading-odd operators: $|Qv|=|G_{-}v|=|v|+1$,

       * –

square to zero: $Q^{2}=G_{-}^{2}=0$;

     * •

two differentials graded-commute, i.e. $\\{Q,G_{-}\\}=0$.

  2. 2.

Unnormalized homotopy $G_{+}:V\to V$, such that

     * •

grading-odd operator: $|G_{+}v|=|v|+1$,

     * •

squares to zero $G_{+}^{2}=0$,

     * •

$\\{G_{+},G_{-}\\}=0$.

In case $V$ is infinite-dimensional we impose certain consistency conditions
on HTQM data $(V,Q,G_{\pm})$. We define the Hamiltonian operator
$H=\\{Q,G_{+}\\}:V\to V$. The consistency conditions are formulated in terms
of Hamiltonian:

  * •

The hamiltonian $H$ is such that the evolution operator $e^{-tH}$ is well
defined for $t\geq 0$ in the following sense:

    * –

it is a solution to the ODE

$(\partial_{t}+H)e^{-tH}=0,\;\;\;e^{-0\cdot
H}=1,\;\;\;t\in\mathbb{R}^{+}\cup\\{0\\};$ (4.1)

    * –

forms a 1-parameter semi-group with multiplication

$e^{-t_{1}H}e^{-t_{2}H}=e^{-(t_{1}+t_{2})H},\;\;\forall\;t_{1},t_{2}\in\mathbb{R}^{+}\cup\\{0\\}.$
(4.2)

  * •

We require that the $t\to\infty$ limit of the evolution operator exists and is
the projector on $\ker H$, i.e.

$\lim_{t\to+\infty}e^{-tH}=\Pi_{0}.$ (4.3)

  * •

The projector $\Pi_{0}$ obeys

$\Pi_{0}G_{\pm}=G_{\pm}\Pi_{0}=0.$ (4.4)

###  4.2 HTQM on trees

Definition: The HTQM $(V,Q,G_{\pm},\mu_{2},g)$ on a connected tree $\Gamma$
with distinct root vertex is the collection of the following data

  1. 1.

1-valent vertices are assigned the HTQM states i.e. $v_{a}\in
V,\;\;a=1,..,n_{1}=|V_{1}(\Gamma)|$.

  2. 2.

2-valent vertices are assigned observables $\mathcal{O}_{\alpha}\in V\otimes
V^{\ast}$ ${\alpha}=1,..,n_{2}=|V_{2}(\Gamma)|$.

  3. 3.

3-valent vertices are assigned the multiplication $\mu_{2}:V\otimes V\to V$
such that the triple $(Q,G_{-},\mu_{2})$ obeys

     * •

$\mu_{2}$ is graded commutative

$\mu_{2}(v,w)=(-1)^{|v||w|}\mu_{2}(w,v);$ (4.5)

     * •

$\mu_{2}$ is associative

$\mu_{2}(\mu_{2}(v,w),u)=\mu_{2}(v,\mu_{2}(w,u));$ (4.6)

     * •

Leibniz rule for $(\mu_{2},Q)$

$Q\mu_{2}(v,w)=\mu_{2}(Qv,w)+(-1)^{|v|}\mu_{2}(v,Qw);$ (4.7)

     * •

the pair $(G_{-},\mu_{2})$ obeys the 7-term relation for all $v,u,w\in V$

$\begin{split}G_{-}&\mu_{2}(\mu_{2}(v,w),u)=\mu_{2}(G_{-}\mu_{2}(v,w),u)+(-1)^{|w|(|v|-1)}\mu_{2}(w,G_{-}\mu_{2}(v,u))\\\
&\qquad+(-1)^{|v|}\mu_{2}(v,G_{-}\mu_{2}(w,u))-\mu_{2}(G_{-}v,\mu_{2}(w,u))-(-1)^{|v|}\mu_{2}(v,\mu_{2}(G_{-}w,u))\\\
&\qquad-(-1)^{|u|+|v|}\mu_{2}(v,\mu_{2}(w,G_{-}u)).\end{split}$ (4.8)

  4. 4.

the root 3-valent vertex assigned the multiplication

$\mu_{3}^{0}=g\circ\mu_{2}:V^{\otimes 3}\to\mathbb{R}$ (4.9)

constructed from Frobenius structure $(g,\mu_{2},Q)$, where the scalar
product, commonly referred to as the pairing, $g:V\otimes V\to\mathbb{R}$
obeys the following properties:

     * •

non-degeneracy on $V$;

     * •

the graded-symmetry

$g(v,w)=(-1)^{|v||w|}g(w,v);$ (4.10)

     * •

$Q$-invariance

$g(Qv,w)+(-1)^{|v|}g(v,Qw)=0;$ (4.11)

     * •

$G_{\pm}$-invariance

$g(G_{\pm}v,w)=(-1)^{|v|}g(v,G_{\pm}w);$ (4.12)

     * •

evolution invariance

$g(e^{-tH}v,w)=g(v,e^{-tH}w).$ (4.13)

###  4.3 Amplitudes on trees

Definition: The evolution operator in HTQM $(V,Q,G_{\pm})$ is

$U(t,dt,d{\varphi})=e^{-tH+G_{+}dt+G_{-}d{\varphi}}\in\Omega^{\ast}(\mathbb{R}^{+}\times
S^{1})\otimes End(V).$ (4.14)

Definition: For each tree $\Gamma$ we define pre-amplitude

$\mathcal{PA}_{\Gamma}:V^{\otimes n_{1}(\Gamma)}\otimes(V\otimes
V^{\ast})^{\otimes n_{2}(\Gamma)}\to\Omega^{\ast}(\mathcal{M}(\Gamma)),$
(4.15)

where $\mathcal{M}(\Gamma)$ is the moduli space of trees $\Gamma$, defined in
(4.19). Each connected tree $\Gamma$ defines a contraction in tensor algebra

$\langle\;\;\rangle_{\Gamma}:(V\otimes V^{\ast})^{\otimes E}\otimes V^{\otimes
n_{1}}\otimes(V^{\ast}\otimes V)^{\otimes n_{2}}\otimes(V^{\ast\otimes
2}\otimes V)^{\otimes(n_{3}-1)}\otimes V^{\ast\otimes 3}\to\mathbb{R},$ (4.16)

where $n_{3}$ is the number of 3-valent vertices in $\Gamma$. The pre-
amplitude on a tree $\Gamma$ with states $v_{a}$ and operators
$\mathcal{O}_{\alpha}$ is

$\mathcal{PA}_{\Gamma}(v_{a};\mathcal{O}_{\alpha})=\left\langle U^{\otimes
I(\Gamma)}\otimes 1^{\otimes
n_{1}}\bigotimes\limits_{a=1}^{n_{1}}v_{a}\bigotimes\limits_{{\alpha}=1}^{n_{2}}\mathcal{O}_{\alpha}\otimes\mu_{2}^{\otimes(n_{3}-1)}\otimes\mu_{3}^{0}\right\rangle_{\Gamma}\;.$
(4.17)

Note that the pre-amplitude on tree $\Gamma$ has no evolution operators on
external edges (edges attached to leaves of a tree $\Gamma$).

Definition: The amplitude on connected tree $\Gamma$ is an integral

$\mathcal{A}_{\Gamma}=\int_{\mathcal{M}^{0}(\Gamma)}\mathcal{PA}_{\Gamma},$
(4.18)

over moduli space of connected tree

$\mathcal{M}^{0}(\Gamma)=\left(\mathbb{R}^{+}\times S^{1}\right)^{I(\Gamma)}.$
(4.19)

###  4.4 Amplitudes in homotopy notation

Definition: The propagator $K:V\to V$ for HTQM $(V,Q,G_{\pm})$ is

$K=\lim_{T\to\infty}\int^{T}_{0}dt\;e^{-tH}\;G_{+}=\int^{\infty}_{0}dt\;e^{-tH}\;G_{+}.$
(4.20)

Note, that the integral has potential divergence, when the exponent vanishes
for states from $\ker H$. The $G_{+}$ in the expression (4.20) and the HTQM
property (4.4) evaluates $G_{+}v=0$ on all $v\in\ker H$, hence $Kv=0$ for such
states.

The propagator $K$ is a homotopy i.e.

$\\{Q,K\\}=\int^{\infty}_{0}dt\;e^{-tH}\;\\{Q,G_{+}\\}=-\int_{0}^{\infty}d\left(\;e^{-tH}\right)=e^{-tH}\Big{|}_{0}-e^{-tH}\Big{|}_{\infty}=1-\Pi_{0}.$
(4.21)

We can perform the moduli space integral in the amplitude definition (4.18)
and express the amplitude using propagators

$\mathcal{A}_{\Gamma}(v_{a};\mathcal{O}_{\alpha};K)=\left\langle(2\pi
KG_{-})^{\otimes
I(\Gamma)}\bigotimes\limits_{a=1}^{n_{1}(\Gamma)}v_{a}\bigotimes\limits_{{\alpha}=1}^{n_{2}(\Gamma)}\mathcal{O}_{\alpha}\otimes\mu_{2}^{\otimes(n_{3}(\Gamma)-1)}\otimes\mu_{3}^{0}\right\rangle_{\Gamma}\;.$
(4.22)

Note that the factors $2\pi G_{-}$ originate from the angular parts of the
moduli space integral.

It is very convenient to introduce a graphical representation for amplitudes:
we use solid lines for edges, equipped with propagator $2\pi KG_{-}$ and
dashed lines for edges without the propagator. We label 1-valent vertices by
the corresponding states $v_{a}$, 2-valent vertices by the corresponding
operators $\mathcal{O}_{\alpha}$. For each tree there is a single special
vertex, responsible for the pairing in HTQM, which can be either the 3-valent
special vertex equipped with $\mu_{3}^{0}$-multiplication, or the 2-valent
vertex, equipped with the pairing $g$. The graphical representation is not
unique. Below we present three graphical representations for the same
amplitude.

$\mu_{2}$$\mu^{0}_{3}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$\mathcal{O}_{1}$$\mathcal{O}_{2}$
$\mu_{2}$$\mu_{2}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$\mathcal{O}_{1}$$\mathcal{O}_{2}$$g$
$\mu_{3}^{0}$$\mu_{2}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$\mathcal{O}_{1}$$\mathcal{O}_{2}$
The amplitude, evaluated from the left representation is

$\mathcal{A}_{\Gamma}=(2\pi)^{3}\;\mu_{3}^{0}(v_{4},KG_{-}\mathcal{O}_{1}v_{3},KG_{-}\mu_{2}(v_{1},KG_{-}\mathcal{O}_{2}v_{2})).$
(4.23)

We can use the Frobenius structure for $(g,\mu_{2})$ to evaluate the same
amplitude, while moving the 3-valent special vertex with $\mu_{3}^{0}$ to the
edge with $v_{4}$-state and turning it into 2-valent vertex with pairing $g$,
i.e.

$\mathcal{A}_{\Gamma}=(2\pi)^{3}\;g(v_{4},\mu_{2}(KG_{-}\mathcal{O}_{1}v_{3},KG_{-}\mu_{2}(v_{1},KG_{-}\mathcal{O}_{2}v_{2}))).$
(4.24)

We can revaluate the same amplitude by moving the $\mu_{3}^{0}$ to the
different 3-vertex

$\mathcal{A}_{\Gamma}=(2\pi)^{3}\;\mu_{3}^{0}(v_{1},KG_{-}\mathcal{O}_{2}v_{2},KG_{-}\mu_{2}(v_{4},KG_{-}\mathcal{O}_{1}v_{3})).$
(4.25)

The two representation (4.23) and (4.25) are related by the $KG_{-}$-flip

$g(KG_{-}v,w)=g(v,KG_{-}w).$ (4.26)

We can derive the flip formula using the definition (4.20) of propagator

$\begin{split}g(KG_{-}v,w)&=\lim_{T\to\infty}\int^{T}_{0}dt\;g\left(e^{-tH}G_{+}G_{-}v,w\right)\\\
&=\lim_{T\to\infty}\int^{T}_{0}dt\;g\left(G_{+}G_{-}v,e^{-tH}w\right)=\lim_{T\to\infty}\int^{T}_{0}dt\;g\left(v,G_{+}G_{-}e^{-tH}w\right)\\\
&=g(v,KG_{-}w).\end{split}$ (4.27)

In the equality we used the integral representation (4.20) for the propagator,
the $G_{\pm}$-invariance of the pairing (4.12) and the evolution invariance of
the pairing (4.13).

###  4.5 Deformation of HTQM by a state

In our work [8] on tropical mirror we argued that the HTQM on trees admits a
“state-operator correspondence”-type relation. We can turn a HTQM state
$\Psi\in V$ into an operator

$\mathcal{O}_{\Psi}=\mu_{2}(\Psi,\cdot):V\to V$ (4.28)

acting as the $\mu_{2}$-multiplication by $\Psi$. Such relation allowed us to
indirectly study the HTQM deformation by a state, by the means of turning
state into an operator first. For the discussion in later parts of the paper
we introduce the notion of the HTQM, deformed by a state below.

Proposition (HTQM deformation by a state): Given HTQM
$(V,Q,G_{\pm},\mu_{2},g)$ on trees and an even state ${\epsilon}\Psi$ such
that

$Q\Psi=G_{-}\Psi=0$ (4.29)

there is an one-parameter family $(V,Q^{\epsilon},G_{\pm},\mu_{2},g)$ of HTQMs
on trees with differential

$Q^{\epsilon}=Q-[G_{-},\mu_{2}({\epsilon}\Psi,\cdot)]+\mathcal{O}({\epsilon}^{2}),$
(4.30)

i.e. the action on states is given by

$Q^{\epsilon}v=Qv-
G_{-}\mu_{2}({\epsilon}\Psi,v)+\mu_{2}({\epsilon}\Psi,G_{-}v)+\mathcal{O}({\epsilon}^{2}).$
(4.31)

Proof: For a proof we need to check that the family
$(V,Q^{\epsilon},G_{\pm},\mu_{2},g)$ satisfies the definitions from sections
4.1 and 4.2. The gradings of $G_{-}$ and $Q$ are odd, while the grading of
${\epsilon}\Psi$ is even hence the grading of $Q_{\epsilon}$ is odd i.e.

$|Q^{\epsilon}|=|[G_{-},\mu_{2}({\epsilon}\Psi,\cdot)]|=1+|{\epsilon}\Psi|=1.$
(4.32)

The $Q^{\epsilon}$ is a differential to the leading order in ${\epsilon}$,
what follows from the square-evaluation

$\begin{split}Q^{\epsilon}Q^{\epsilon}v&=Q^{2}v-QG_{-}\mu_{2}({\epsilon}\Psi,v)+Q\mu_{2}({\epsilon}\Psi,G_{-}v)-G_{-}\mu_{2}({\epsilon}\Psi,Qv)+\mu_{2}({\epsilon}\Psi,G_{-}Qv)\\\
&\quad\qquad+G_{-}\mu_{2}({\epsilon}\Psi,G_{-}\mu_{2}({\epsilon}\Psi,v))-G_{-}\mu_{2}({\epsilon}\Psi,\mu_{2}({\epsilon}\Psi,G_{-}v))-\mu_{2}({\epsilon}\Psi,G_{-}\mu_{2}({\epsilon}\Psi,G_{-}v))\\\
&=\frac{1}{2}\mu_{2}(G_{-}\mu_{2}({\epsilon}\Psi,{\epsilon}\Psi),G_{-}v)-\frac{1}{2}G_{-}\mu_{2}(G_{-}\mu_{2}({\epsilon}\Psi,{\epsilon}\Psi),v)=\mathcal{O}({\epsilon}^{2}).\end{split}$
(4.33)

In the equality we used Leibniz rule (4.7), associativity of the
multiplication $\mu_{2}$ and the 7-term relation (4.8).

Remark: The $Q^{\epsilon}Q^{\epsilon}=0$ holds for all orders in ${\epsilon}$
for the states ${\epsilon}\Psi$, such that

$G_{-}\mu_{2}({\epsilon}\Psi,{\epsilon}\Psi)=0.$ (4.34)

The condition above does not follow from the $G_{-}\Psi=0$, since the pair
$(G_{-},\mu_{2})$ does not obey the Leibniz rule.

The $Q^{\epsilon}$ and $G_{-}$ form a pair of graded-commuting differentials.
Indeed, we can simplify the graded commutator

$\begin{split}\\{Q^{\epsilon},G_{-}\\}v&=QG_{-}v-G_{-}\mu_{2}({\epsilon}\Psi,G_{-}v)+\mu_{2}({\epsilon}\Psi,G_{-}G_{-}v)+G_{-}Qv\\\
&\qquad-
G_{-}G_{-}\mu_{2}({\epsilon}\Psi,v)+G_{-}\mu_{2}({\epsilon}\Psi,G_{-}v)=0.\end{split}$
(4.35)

The pair $(Q^{\epsilon},\mu_{2})$ obey DGA: By construction $\mu_{2}$ is an
associative, graded commutative multiplication and $Q^{\epsilon}$ is the
differential, so we just need to check the Leibniz rule for
$(\mu_{2},Q^{\epsilon})$. Indeed, we can evaluate

$\begin{split}&\mu_{2}(Q^{\epsilon}v,w)+(-1)^{|v|}\mu_{2}(v,Q^{\epsilon}w)-Q^{\epsilon}\mu_{2}(v,w)\\\
&=-\mu_{2}(G_{-}\mu_{2}({\epsilon}\Psi,v),w)+\mu_{2}(\mu_{2}({\epsilon}\Psi,G_{-}v),w)-(-1)^{|v|}\mu_{2}(v,G_{-}\mu_{2}({\epsilon}\Psi,w))\\\
&\qquad\qquad+(-1)^{|v|}\mu_{2}(v,\mu_{2}({\epsilon}\Psi,G_{-}w))-\mu_{2}({\epsilon}\Psi,G_{-}\mu_{2}(v,w))+\mu_{2}(G_{-}\mu_{2}(v,w),{\epsilon}\Psi)\\\
&\qquad\qquad+(-1)^{|w|(|v|-1)}\mu_{2}(w,G_{-}\mu_{2}(v,{\epsilon}\Psi))+(-1)^{|v|}\mu_{2}(v,G_{-}\mu_{2}(w,{\epsilon}\Psi))\\\
&\qquad\qquad-\mu_{2}(G_{-}v,\mu_{2}(w,{\epsilon}\Psi))-(-1)^{|v|}\mu_{2}(v,\mu_{2}(G_{-}w,{\epsilon}\Psi))=0\end{split}$
(4.36)

In the equality we used the Leibniz rule (4.7) and 7-term relation (4.8).

The $Q^{\epsilon}$-invariance of the pairing follows from

$\begin{split}&g(Q^{\epsilon}v,w)+(-1)^{|v|}g(v,Q^{\epsilon}w)=-g(G_{-}\mu_{2}({\epsilon}\Psi,v),w)+g(\mu_{2}({\epsilon}\Psi,G_{-}v),w)\\\
&\qquad\qquad+(-1)^{|v|}g(v,\mu_{2}({\epsilon}\Psi,G_{-}w))-(-1)^{|v|}g(v,G_{-}\mu_{2}({\epsilon}\Psi,w))=0.\end{split}$
(4.37)

We used the $Q$-invariance of the pairing (4.11) and $G_{\pm}$ invariance of
the pairing (4.12) and Frobenius structure for $(g,\mu_{2})$.

The Hamiltonian in deformed HTQM is given by

$H^{\epsilon}=\\{Q^{\epsilon},G_{+}\\}=H-\\{G_{+},[G_{-},\mu_{2}({\epsilon}\Psi,\cdot)]\\}=H-{\epsilon}\;V_{\Psi}.$
(4.38)

The last equality introduces $V_{\Psi}$, the linear in ${\epsilon}$ correction
to the Hamiltonian in deformed theory. The Cauchy problem for the evolution
operator

$(\partial_{t}+H-{\epsilon}V_{\Psi})e^{-tH^{\epsilon}}=0,\;\;\;e^{-0\cdot
H^{\epsilon}}=1,\;\;\;t\in\mathbb{R}^{+}$ (4.39)

can be solved in power series in ${\epsilon}$. The evolution operator for
deformed theory is

$\begin{split}e^{-tH^{\epsilon}}&=e^{-tH}+\int^{t}_{0}ds\;e^{(s-t)H}\;{\epsilon}V_{\Psi}\;e^{-sH}+\mathcal{O}({\epsilon}^{2})\\\
&=e^{-tH}+\int^{t}_{0}ds\;e^{(s-t)H}\\{G_{+},[G_{-},\mu_{2}({\epsilon}\Psi,\cdot)]\\}e^{-sH}+\mathcal{O}({\epsilon}^{2}).\end{split}$
(4.40)

We can rewrite the integral in more symmetric form

$\int^{t}_{0}ds\;e^{sH-
tH}\;{\epsilon}V_{\Psi}\;e^{-sH}=\int_{t_{1}>0,\;t_{2}>0,\;t_{1}+t_{2}=t}dt_{1}dt_{2}\;e^{-t_{2}H}\;{\epsilon}V_{\Psi}\;e^{-t_{1}H}.$
(4.41)

The symmetric formula is a common formula for perturbation theory in quantum
mechanics and has the following interpretation: The $t_{1}$ and $t_{2}$
describe the decomposition of the interval $[0,t]$ into two sub-intervals:
$[0,t_{1}]$ of length $t_{1}$ and $[t_{1},t]$ of length $t_{2}$. Each interval
is equipped with the evolution operator $e^{-t_{1}H}$ and $e^{-t_{2}H}$, while
the splitting point carries the insertion of the deformation
${\epsilon}V_{\Psi}$ of the Hamiltonian.

The evolution operators in deformed theory form a semi-group, what follows
from the analysis of the composition of two perturbative solutions (4.40) with
times $t$ and $s$

$\begin{split}e^{-sH^{\epsilon}}e^{-tH^{\epsilon}}&=e^{-(t+s)H}+\int^{t}_{0}dt_{1}\;e^{-(t+s-t_{1})H}\;{\epsilon}V_{\Psi}\;e^{-t_{1}H}\\\
&\qquad\qquad+\int^{s+t}_{t}dt_{1}\;e^{-(s+t-t_{1})H}\;{\epsilon}V_{\Psi}\;e^{-t_{1}H}+\mathcal{O}({\epsilon}^{2})\\\
&=e^{-(t+s)H^{\epsilon}}+\mathcal{O}({\epsilon}^{2}).\end{split}$

The evolution with respect to $H^{\epsilon}$ preserves the pairing. Indeed, we
can evaluate

$\begin{split}g(e^{-tH^{\epsilon}}v,w)&=g(v,e^{-tH}w)+\int_{t_{1}+t_{2}=t}dt_{1}dt_{2}\;g\left(e^{-t_{2}H}\;{\epsilon}V_{\Psi}\;e^{-t_{1}H}v,w\right)\\\
&=g(v,e^{-tH}w)+\int_{t_{1}+t_{2}=t}dt_{1}dt_{2}\;g\left(v,e^{-t_{1}H}\;{\epsilon}V_{\Psi}\;e^{-t_{2}H}w\right)\\\
&=g(v,e^{-tH^{\epsilon}}w).\end{split}$ (4.42)

We used the evolution invariance of the pairing (4.13) and the invariance of
the pairing with respect to the ${\epsilon}V_{\Psi}$-action, what follows from
the $Q^{\epsilon}$-invariance of the pairing (4.37) and $G_{\pm}$-invariance
of the pairing (4.12) and the following relation

$\begin{split}g({\epsilon}V_{\Psi}\;v,w)&=g(\\{G_{+},Q^{\epsilon}-Q\\}v,w)=g(G_{+}(Q^{\epsilon}-Q)v,w)+g((Q^{\epsilon}-Q)G_{+}v,w)\\\
&=(-1)^{2|v|+2}g(v,(Q^{\epsilon}-Q)G_{+}w)+(-1)^{2|v|+2}g(v,G_{+}(Q^{\epsilon}-Q)w)\\\
&=g(v,{\epsilon}V_{\Psi}\;w).\end{split}$ (4.43)

We use the semi-continuity of the kernel: The kernel of $H$ can only decrease
under the small deformation, hence $\dim\ker H^{\epsilon}\leq\dim\ker H$. The
possible decrease of the kernel is due to the obstruction for deformation of
$v^{0}\in\ker H$ into $v^{0{\epsilon}}\in\ker H^{\epsilon}$. Given a state
$v_{0}\in\ker H$ we can deform it by a $\mathcal{O}({\epsilon})$-term to
construct a state

$v^{0{\epsilon}}=v^{0}+{\epsilon}v^{1}+\mathcal{O}({\epsilon}^{2})\in\ker
H^{\epsilon},$ (4.44)

where the deformation ${\epsilon}v^{1}$ is a solution to

$H^{\epsilon}v^{0{\epsilon}}=-{\epsilon}V_{\Psi}v^{0}+{\epsilon}Hv^{1}+\mathcal{O}({\epsilon}^{2})=\mathcal{O}({\epsilon}^{2}).$
(4.45)

The solution for ${\epsilon}v^{1}$ can be written in the form

$\begin{split}{\epsilon}v^{1}&=\int^{\infty}_{0}dt\;e^{-tH}G_{+}G_{-}\mu_{2}({\epsilon}\Psi,v^{0})=KG_{-}\mu_{2}({\epsilon}\Psi,v^{0}).\end{split}$
(4.46)

Indeed, we can check

$\begin{split}H{\epsilon}v^{1}&=\int^{\infty}_{0}dt\;H\;e^{-tH}G_{+}G_{-}\mu_{2}({\epsilon}\Psi,v^{0})=-\int^{\infty}_{0}d\left(e^{-tH}G_{+}G_{-}\mu_{2}({\epsilon}\Psi,v^{0})\right)\\\
&=-e^{-tH}G_{+}G_{-}\mu_{2}({\epsilon}\Psi,v^{0})\Big{|}^{t=\infty}_{t=0}=G_{+}G_{-}\mu_{2}({\epsilon}\Psi,v^{0})-\Pi_{0}G_{+}G_{-}\mu_{2}({\epsilon}\Psi,v^{0})\\\
&=G_{+}G_{-}\mu_{2}({\epsilon}\Psi,v^{0})={\epsilon}\;V_{\Psi}v^{0},\end{split}$
(4.47)

where we replaced the $t\to\infty$-limit by a projector $\Pi_{0}$ and used
(4.4) to eliminate the term. The deformation (4.46) exists for all
$v^{0}\in\ker H$, hence $\dim\ker H^{\epsilon}=\dim\ker H$.

Remark: The integral in (4.46) acquires most of its value near $t=0$, since
for large $t$, the exponential operator $e^{-tH}$ is close to the projector
$\Pi_{0}$. Hence we can write an approximation for the integral in the form of
the finite region integral

${\epsilon}v_{1}=\int^{\infty}_{0}dt\;e^{-tH}G_{+}G_{-}\mu_{2}({\epsilon}\Psi,v_{0})\approx\int^{T}_{0}dt\;e^{-tH}G_{+}G_{-}\mu_{2}({\epsilon}\Psi,v_{0}).$
(4.48)

Let us choose a basis $v^{0}_{k}$ in $\ker H$. Since $\ker H$ and $\ker
H^{\epsilon}$ are of the same dimension then the corresponding deformed states
$v^{0{\epsilon}}_{k}$ form a basis in $\ker H^{\epsilon}$. We use the non-
degeneracy of the pairing $g$ to identify the vector space $V$ and its dual
$V^{\ast}$ to express the projector

$\Pi_{0}=\sum v^{0}_{k}{v_{k}^{0}}^{\ast}.$ (4.49)

The projector $\Pi_{0}^{\epsilon}$ on $\ker H^{\epsilon}$ is written using the
deformed states $v^{0{\epsilon}}_{k}$

$\begin{split}\Pi^{\epsilon}_{0}&=\sum
v^{0{\epsilon}}_{k}\;{v_{k}^{0{\epsilon}}}^{\ast}=\Pi_{0}+\sum{\epsilon}v^{1}_{k}\;{v_{k}^{0}}^{\ast}+{\epsilon}\sum
v^{0}_{k}\;{{\epsilon}v_{k}^{1}}^{\ast}+\mathcal{O}({\epsilon}^{2}).\end{split}$
(4.50)

The $t\to\infty$ limit of the deformed evolution operator (4.40) is

$\begin{split}\lim_{t\to\infty}&e^{-tH_{\epsilon}}=\Pi_{0}+\lim_{t\to\infty}\int^{t}_{0}ds\;e^{sH-
tH}\;{\epsilon}V_{\Psi}\;e^{-sH}+\mathcal{O}({\epsilon}^{2}).\end{split}$
(4.51)

We decompose the integration interval $[0,t]$ into three regions and evaluate
the limit for the integration over each region:

  * •

Right side of the interval: The $t_{2}$ is small, i.e $t_{2}\in[0,T]$ for some
finite $T$, while $t_{1}\approx t$ is very large and we can replace the
corresponding exponential factor by the projector $\Pi_{0}$. The integral
(4.41) in this region evaluates into

$\begin{split}\lim_{t\to\infty}&\int^{T}_{0}dt_{2}\;e^{-t_{2}H}\;{\epsilon}V_{\Psi}\;e^{-tH}e^{t_{2}H}=\int^{T}_{0}dt_{2}\;e^{-t_{2}H}\;{\epsilon}V_{\Psi}\cdot\Pi_{0}\\\
&=\int^{T}_{0}dt_{2}\;e^{-t_{2}H}{\epsilon}V_{\Psi}\sum
v^{0}_{k}{v_{k}^{0}}^{\ast}=\sum\int^{T}_{0}dt_{2}\;e^{-t_{2}H}G_{+}G_{-}\mu_{2}({\epsilon}\Psi,v^{0}_{k}){v_{k}^{0}}^{\ast}\\\
&\approx\sum\int^{\infty}_{0}dt_{2}\;e^{-t_{2}H}G_{+}G_{-}\mu_{2}({\epsilon}\Psi,v^{0}_{k}){v_{k}^{0}}^{\ast}=\sum
KG_{-}\mu_{2}({\epsilon}\Psi,v^{0}_{k}){v_{k}^{0}}^{\ast}\\\
&=\sum{\epsilon}v^{1}_{k}\;{v_{k}^{0}}^{\ast}.\end{split}$ (4.52)

  * •

Left side of the interval: The $t_{1}$ is such that $t_{1}<T$, while
$t_{2}\approx t$ is very large and we can replace the corresponding
exponential factor by the projector. The integral (4.41) in this region
evaluates into

$\begin{split}\lim_{t\to\infty}&\int^{T}_{0}dt_{1}\;e^{-tH}e^{t_{1}H}\;{\epsilon}V_{\Psi}\;e^{-t_{1}H}=\int^{T}_{0}dt_{1}\;\Pi_{0}\;{\epsilon}V_{\Psi}\;e^{-t_{1}H}\\\
&=\int^{T}_{0}dt_{1}\;\sum
v^{0}_{k}{v_{k}^{0}}^{\ast}\;{\epsilon}V_{\Psi}e^{-t_{1}H}=\sum
v^{0}_{k}\left[\int^{T}_{0}dt_{1}\;e^{-t_{1}H}G_{+}G_{-}\mu_{2}({\epsilon}\Psi,v^{0}_{k})\right]^{\ast}\\\
&\approx\sum
v^{0}_{k}\left[\int^{\infty}_{0}dt_{1}\;e^{-t_{1}H}G_{+}G_{-}\mu_{2}({\epsilon}\Psi,v^{0}_{k})\right]^{\ast}=\sum
v^{0}_{k}\left[KG_{-}\mu_{2}({\epsilon}\Psi,v^{0}_{k})\right]^{\ast}\\\ &=\sum
v^{0}_{k}\;{{\epsilon}v_{k}^{1}}^{\ast}.\end{split}$ (4.53)

  * •

Middle of the interval: For the middle region $t_{1}\in[t/2-T,t/2+T]$, i.e.
both $t_{1}$ and $t_{2}$ are large. The integral (4.41) in this region
evaluates into

$\begin{split}\lim_{t\to\infty}&\int^{t/2+T}_{t/2-T}dt_{2}\;e^{-t_{2}H}\;{\epsilon}V_{\Psi}\;e^{-t_{1}H}=\int^{T}_{-T}dt_{1}\;\Pi_{0}\;{\epsilon}V_{\Psi}\;\Pi_{0}\\\
&=\int^{T}_{-T}dt_{1}\;0=\mathcal{O}(T)\cdot 0=0.\end{split}$ (4.54)

The sum over three contributions

$\begin{split}\lim_{t\to\infty}e^{-tH_{\epsilon}}=\Pi_{0}+\sum{\epsilon}v^{1}_{k}\;{v_{k}^{0}}^{\ast}+\sum
v^{0}_{k}\;{{\epsilon}v_{k}^{1}}^{\ast}=\Pi_{0}^{\epsilon}\end{split}$ (4.55)

matches with our expression (4.50) for projector on $\ker H^{\epsilon}$.

We can use the (4.50)-representation to check the properties of projector in
deformed theory

$\begin{split}\Pi_{0}^{\epsilon}G_{\pm}v&=\Pi_{0}G_{\pm}v+\sum\left({\epsilon}v^{1}_{k}\;{v_{k}^{0}}^{\ast}+v^{0}_{k}\;{{\epsilon}v_{k}^{1}}^{\ast}\right)G_{\pm}v\\\
&=\sum{\epsilon}v^{1}_{k}\;g(v_{k}^{0},G_{\pm}v)+\sum
v^{0}_{k}\;g({\epsilon}v_{k}^{1},G_{\pm}v)=0\end{split}$ (4.56)

and

$\begin{split}G_{\pm}\Pi_{0}^{\epsilon}v&=G_{\pm}\Pi_{0}v+G_{\pm}\sum\left({\epsilon}v^{1}_{k}\;{v_{k}^{0}}^{\ast}+v^{0}_{k}\;{{\epsilon}v_{k}^{1}}^{\ast}\right)v\\\
&=\sum(G_{\pm}\;{\epsilon}v^{1}_{k})\;{v_{k}^{0}}^{\ast}+\sum(G_{\pm}v^{0}_{k})\;{{\epsilon}v_{k}^{1}}^{\ast}=0.\end{split}$
(4.57)

In equalities we used the $G_{\pm}$-invariance of the pairing,
$G_{\pm}v^{0}_{k}=0$ and expression (4.46) for $v^{1}_{k}$ to evaluate

$G_{\pm}\;{\epsilon}v^{1}_{k}=G_{\pm}\int^{\infty}_{0}dt\;e^{-tH}G_{+}G_{-}\mu_{2}({\epsilon}\Psi,v_{k}^{0})=0.$
(4.58)

$\hfill\blacksquare$

Definition: For any state $v$ in HTQM $(V,Q,G_{\pm})$ its leading order
deformation by $Q$-, $G_{-}$-closed state $\Psi$ is

$v^{{\epsilon}}=v+KG_{-}\mu_{2}({\epsilon}\Psi,v)+\mathcal{O}({\epsilon}^{2}).$
(4.59)

Proposition (preservation of closeness): If $v$ is $Q$\- and $G_{-}$-closed,
then the deformed state $v^{\epsilon}$ is $Q^{\epsilon}$\- and $G_{-}$\-
closed i.e.

$Q^{\epsilon}v^{{\epsilon}}=G_{-}v^{{\epsilon}}=\mathcal{O}({\epsilon}^{2}).$
(4.60)

Proof: The $G_{-}$-closeness if fairly straightforward

$G_{-}v^{{\epsilon}}=G_{-}v+G_{-}KG_{-}\mu_{2}({\epsilon}\Psi,v)=-G_{-}^{2}K\mu_{2}({\epsilon}\Psi,v)=0.$
(4.61)

The $Q^{\epsilon}$-action on the deformed state

$\begin{split}Q^{\epsilon}v^{\epsilon}&=Q^{\epsilon}v+QKG_{-}\mu_{2}({\epsilon}\Psi,v)+\mathcal{O}({\epsilon}^{2})\\\
&=Qv-
G_{-}\mu_{2}({\epsilon}\Psi,v)+QKG_{-}\mu_{2}({\epsilon}\Psi,v)+\mathcal{O}({\epsilon}^{2})\\\
&=-G_{-}\mu_{2}({\epsilon}\Psi,v)+(1-\Pi_{0})G_{-}\mu_{2}({\epsilon}\Psi,v)+\mathcal{O}({\epsilon}^{2})=\mathcal{O}({\epsilon}^{2}).\end{split}$
(4.62)

In the equality we used the homotopy formula (4.21), Leibniz rule (4.7) for
even state ${\epsilon}\Psi$ and the projector $\Pi_{0}$ property (4.4) from
the HTQM definition. $\hfill\blacksquare$

Remark: The formula (4.59) for deformation of a state $v$, describes a
connection on $Q+zG_{-}$-cohomology, fibered over the space of HTQM
deformations.

###  4.6 Diagrammatic representation of deformed theory

We can express the propagator for deformed theory as expansion in ${\epsilon}$
in terms of propagators in original theory

$\begin{split}K^{\epsilon}v&=\int^{\infty}_{0}dt\;e^{-tH^{\epsilon}}G_{+}v=\int^{\infty}_{0}dt\;e^{-tH}G_{+}v\\\
&\qquad\qquad+\int^{\infty}_{0}dt\;\int_{t_{1}+t_{2}=t,\;t_{1},\;t_{2}>0}dt_{1}dt_{2}\;e^{-t_{1}H}\\{G_{+},[G_{-},\mu_{2}({\epsilon}\Psi,\cdot)]\\}e^{-t_{2}H}G_{+}v+\mathcal{O}({\epsilon}^{2})\\\
&=Kv+\int_{(\mathbb{R}^{+})^{2}}dt_{1}dt_{2}\;e^{-t_{1}H}G_{+}[G_{-},\mu_{2}({\epsilon}\Psi,\cdot)]e^{-t_{2}H}G_{+}v+\mathcal{O}({\epsilon}^{2})\\\
&=Kv+K[G_{-},\mu_{2}({\epsilon}\Psi,\cdot)]Kv+\mathcal{O}({\epsilon}^{2}).\end{split}$

The propagator for deformed theory further simplifies if we use it in
$KG_{-}$-combination i.e.

$K^{\epsilon}G_{-}v=KG_{-}v+KG_{-}\mu_{2}({\epsilon}\Psi,KG_{-}v)+\mathcal{O}({\epsilon}^{2}).$
(4.63)

We can give a graphical representation for a propagator in deformed theory
(denoted as thick solid line) in terms of the diagrams in the original theory

$=$$+$${\epsilon}\Psi$

In case $G_{-}\mu_{2}({\epsilon}\Psi,{\epsilon}\Psi)=0$ the $KG_{-}$ in
deformed theory can be written to all orders in ${\epsilon}$ in the form

$K^{\epsilon}G_{-}v=KG_{-}+KG_{-}\mu_{2}({\epsilon}\Psi,KG_{-}v)+KG_{-}\mu_{2}({\epsilon}\Psi,KG_{-}\mu_{2}({\epsilon}\Psi,KG_{-}v))+\dots$
(4.64)

with graphical representation

$=$$+$${\epsilon}\Psi$$+$${\epsilon}\Psi$${\epsilon}\Psi$$+\dots$

The diagrammatic expression for the deformed state (4.59), denoted as the
thick dashed line, is the sum of two terms

$v^{\epsilon}$$=$$v$$+$${\epsilon}\Psi$$v$
In case $G_{-}\mu_{2}({\epsilon}\Psi,{\epsilon}\Psi)=0$ the higher order terms
of the state deformation take the form

$v^{\epsilon}$$=$$v$$+$${\epsilon}\Psi$$v$$+$${\epsilon}\Psi$${\epsilon}\Psi$$v$$+\dots$
The diagrammatic expression above describes the sum

$\begin{split}v^{\epsilon}&=v+KG_{-}\mu_{2}({\epsilon}\Psi,v)+KG_{-}\mu_{2}({\epsilon}\Psi,KG_{-}\mu_{2}({\epsilon}\Psi,v))+...\end{split}$
(4.65)

##  5 Correlation functions in HTQM

In this section we introduce the correlation functions for the states in HTQM.
The correlation functions obey certain nice properties such as symmetry in all
arguments, $Q$-invariance and recursion relation.

Definition: The total amplitude on spacial with 3 valent vertices
$\Psi_{1},...,\Psi_{n}$ in HTQM $(V,Q,G_{\pm},g,\mu_{2})$ is

$\langle\Psi_{1},...,\Psi_{n}\rangle_{Q}=\sum_{\Gamma,\sigma\in
S_{n}}\frac{\mathcal{A}_{\Gamma}(\Psi_{\sigma(1)},...,\Psi_{\sigma(n)};K)}{|Aut(\Gamma)|},$
(5.1)

where $|Aut(\Gamma)|$ is the symmetry factor for the tree $\Gamma$ and $K$ is
a homotopy (4.20). The summation is taken over all distinct 3-valent connected
trees $\Gamma$ with $n$ leaves and over possible assignment of states
$\Psi_{1},...,\Psi_{n}$ on leaves of $\Gamma$.

Remark: The sum over permutations in (5.1) makes the correlation function
$\langle\Psi_{1},...,\Psi_{n}\rangle_{Q}$ manifestly symmetric in all
arguments.

Remark: The sum over trees, weighted with the symmetry factors, in correlation
functions (5.1) is a signature of their relation to amplitudes in certain
Quantum Field Theory. We conjecture that the QFT is a BCOV-like theory [21],
see also [10]. We are working on further investigation of this conjecture.

Remark: The HTQM states $\Psi_{a}$, relevant for the mirror symmetry, are
even, i.e. $|\Psi_{a}|=0$. Hence, for simplicity, we will assume that
$|\Psi_{a}|=0$ for the rest of this section. Such assumption will drastically
reduce the complexity of the sign factors in various expressions.

###  5.1 3-point function

There is a single tree with 3-valent vertices and three leaves. Hence, the
3-point correlation function is just a sum over 3! possible permutations of
states $\Psi_{1},\Psi_{2},\Psi_{3}$ on the leaves of a tree below

$\Psi_{1}$$\Psi_{2}$$\Psi_{3}$
The amplitudes for each permutation are identical, hence we have a sum of
$3!=6$ identical terms. The symmetry factor of the graph above is
$|Aut\;\Gamma_{3}|=3!=6$, so the 3-point correlation function simplifies to

$\langle\Psi_{1},\Psi_{2},\Psi_{3}\rangle_{Q}=\frac{1}{3!}\cdot
3!\;\mu_{3}^{0}(\Psi_{1},\Psi_{2},\Psi_{3})=\mu_{3}^{0}(\Psi_{1},\Psi_{2},\Psi_{3}).$
(5.2)

The 3-point function (5.2) is invariant under the shift
$\Psi_{3}\to\Psi_{3}+Q\chi$, given that $Q\Psi_{1}=Q\Psi_{2}=0$. Indeed, we
can evaluate the difference

$\begin{split}\langle\Psi_{1},\Psi_{2},Q\chi\rangle_{Q}&=g(Q\chi,\mu_{2}(\Psi_{1},\Psi_{2}))=g(\chi,Q\mu_{2}(\Psi_{1},\Psi_{2}))\\\
&=g(\chi,\mu_{2}(Q\Psi_{1},\Psi_{2}))+g(\chi,\mu_{2}(\Psi_{1},Q\Psi_{2}))=0.\end{split}$
(5.3)

We used the $Q$-invariance of the pairing (4.11) and Leibniz rule (4.7) to
simplify the expression.

###  5.2 4-point function

There is a single tree with 3-valent vertices and four leaves. Hence the
4-point correlation function is just a sum over 4! possible distributions of
states $\Psi_{1},\Psi_{2},\Psi_{3},\Psi_{4}$ on the leaves of a tree. The 24
terms in a sum of three groups of $8$, with equal amplitudes in each group.
The 3 (independent) amplitudes are depicted below and are commonly referred to
as the $s-,t-,u-$diagrams

$\Psi_{4}$$\Psi_{1}$$\Psi_{3}$$\Psi_{2}$
$\Psi_{4}$$\Psi_{1}$$\Psi_{2}$$\Psi_{3}$
$\Psi_{3}$$\Psi_{1}$$\Psi_{2}$$\Psi_{4}$

The 4-point correlation function is the sum of three contributions

$\langle\Psi_{1},\Psi_{2},\Psi_{3},\Psi_{4}\rangle_{Q}=\mathcal{A}_{\Gamma_{4}}(\Psi_{1},\Psi_{2},\Psi_{3},\Psi_{4})+\mathcal{A}_{\Gamma_{4}}(\Psi_{1},\Psi_{3},\Psi_{2},\Psi_{4})+\mathcal{A}_{\Gamma_{4}}(\Psi_{1},\Psi_{4},\Psi_{2},\Psi_{3}).$
(5.4)

The number $8$ equals to the symmetry factor $|Aut\;\Gamma_{4}|=8$ of a tree
and is constructed as $2\cdot 2\cdot 2$. The amplitudes on graphs, related by
symmetry, are the same. Indeed, the amplitude for the first tree

$\begin{split}\mathcal{A}_{\Gamma_{4}}&(\Psi_{1},\Psi_{2},\Psi_{3},\Psi_{4})=g(\mu_{2}(\Psi_{3},\Psi_{4}),2\pi
KG_{-}\mu_{2}(\Psi_{1},\Psi_{2}))\\\ &=g(\mu_{2}(\Psi_{3},\Psi_{4}),2\pi
KG_{-}\mu_{2}(\Psi_{2},\Psi_{1}))=g(\mu_{2}(\Psi_{4},\Psi_{3}),2\pi
KG_{-}\mu_{2}(\Psi_{1},\Psi_{2})).\end{split}$ (5.5)

is invariant under the exchange of two pairs of states
$\Psi_{1}\leftrightarrow\Psi_{2}$ and $\Psi_{3}\leftrightarrow\Psi_{4}$.
Indeed, the $\Psi_{a}$ are even states and $\mu_{2}$ is graded-symmetric. The
last factor of 2 is related to the reflection of the tree, i.e. exchange of
two pairs $\Psi_{1},\Psi_{2}$ and $\Psi_{3},\Psi_{4}$ i.e.

$g(\mu_{2}(\Psi_{4},\Psi_{3}),KG_{-}\mu_{2}(\Psi_{1},\Psi_{2}))=g(KG_{-}\mu_{2}(\Psi_{4},\Psi_{3}),\mu_{2}(\Psi_{1},\Psi_{2})).$
(5.6)

The invariance of the amplitude follows from the $KG_{-}$-flip relation
(4.26).

The 4-point amplitude is invariant under the shift
$\Psi_{4}\to\Psi_{4}+Q\chi$, given that

$Q\Psi_{a}=G_{-}\Psi_{a}=G_{-}\chi=0,\;\;\;a=1,2,3.$ (5.7)

Indeed, we can evaluate

$\begin{split}\frac{2}{2\pi}&\langle\Psi_{1},\Psi_{2},\Psi_{3},Q\chi\rangle_{Q}=\sum_{\sigma\in
S_{3}}g(Q\chi,\mu_{2}(\Psi_{\sigma(3)},KG_{-}\mu_{2}(\Psi_{\sigma(1)},\Psi_{\sigma(2)})))\\\
&=\sum_{\sigma\in
S_{3}}g(\chi,Q\mu_{2}(\Psi_{\sigma(3)},KG_{-}\mu_{2}(\Psi_{\sigma(1)},\Psi_{\sigma(2)})))\\\
&=\sum_{\sigma\in
S_{3}}g(\chi,\mu_{2}(\Psi_{\sigma(3)},\\{Q,K\\}G_{-}\mu_{2}(\Psi_{\sigma(1)},\Psi_{\sigma(2)})))=\sum_{\sigma\in
S_{3}}g(\chi,\mu_{2}(\Psi_{\sigma(3)},G_{-}\mu_{2}(\Psi_{\sigma(1)},\Psi_{\sigma(2)})))\\\
&=2g(\chi,G_{-}\mu_{2}(\mu_{2}(\Psi_{1},\Psi_{2}),\Psi_{3}))=-2g(G_{-}\chi,\mu_{2}(\mu_{2}(\Psi_{1},\Psi_{2}),\Psi_{3}))=0.\end{split}$
(5.8)

In rewriting the equality we used the $Q$-invariance of the pairing (4.11),
Leibniz rule (4.7), the homotopy formula (4.21), the projector property (4.4),
the 7-term relation (4.8) for even $G_{-}$-closed states in the form

$\begin{split}G_{-}\mu_{2}(\mu_{2}(\Psi_{1},\Psi_{2}),\Psi_{3})&=\mu_{2}(G_{-}\mu_{2}(\Psi_{1},\Psi_{2}),\Psi_{3})+\mu_{2}(\Psi_{2},G_{-}\mu_{2}(\Psi_{1},\Psi_{3}))\\\
&\qquad+\mu_{2}(\Psi_{1},G_{-}\mu_{2}(\Psi_{2},\Psi_{3})).\end{split}$ (5.9)

The last equality in (5.8) uses the $G_{\pm}$-invariance of pairing (4.12) and
completes the proof of invariance.

###  5.3 Generating function

We introduce formal parameters $t_{k}$, such that $t_{k}^{2}=0$ and combine
$\Psi_{k}$ into even state

$\Psi=\Psi(t)=\sum t_{k}\Psi_{k}.$ (5.10)

The $n$-point correlation function of $\Psi$ has $t$-expansion with
coefficients being the $n$-point correlation functions i.e.

$\begin{split}\langle\underbrace{\Psi,...,\Psi}_{n}\rangle_{Q}=n!\cdot\langle
t_{1}\Psi_{1},t_{2}\Psi_{2},..,t_{n}\Psi_{n}\rangle_{Q}.\end{split}$ (5.11)

Definition: The n-point correlation function of $\Psi$ can be organized into
generating function

$\mathcal{F}_{0}(\Psi,K)=\sum_{k=3}^{\infty}\frac{1}{k!}\langle\underbrace{\Psi,...,\Psi}_{k}\rangle_{Q}.$
(5.12)

The generating function equals to the the sum over connected 3-valent trees

$\mathcal{F}_{0}(\Psi,K)=\sum_{\Gamma}\frac{\mathcal{A}_{\Gamma}(\Psi,...,\Psi;K)}{|Aut(\Gamma)|}.$
(5.13)

The diagrammatic expression for generating function is

$\Psi$$\Psi$$\Psi$$+\frac{1}{8}$$\frac{1}{6}$
$\Psi$$\Psi$$\Psi$$\Psi$$+\frac{1}{8}$
$\Psi$$\Psi$$\Psi$$\Psi$$\Psi$$+\frac{1}{8}$
$\Psi$$\Psi$$\Psi$$\Psi$$\Psi$$\Psi$$+\frac{1}{48}$
$\Psi$$\Psi$$\Psi$$\Psi$$\Psi$$\Psi$$+\dots$

###  5.4 Invariance theorem

In our analysis for the 3- and 4-point functions we observed the invariance
under the shift by a $Q$-exact term, given that the states were $Q$\- and
$G$-closed. The invariance generalizes to the n-point function.

Theorem (invariance): Given the states $\Psi_{1},\Psi_{2},...,\Psi_{n},\chi$
of HTQM on trees $(V,Q,G_{\pm},\mu_{2},g)$ such that

$Q\Psi_{\alpha}=G_{-}\Psi_{\alpha}=G_{-}\chi=0,\;\;{\alpha}=1..n$ (5.14)

the $(n+1)$-point correlation function vanishes i.e.

$\langle\Psi_{1},...,\Psi_{n},Q\chi\rangle_{Q}=0.$ (5.15)

Proof: We use the generating function (5.12) to prove the theorem. The change
in generating function can be expressed

$\delta\mathcal{F}_{0}(\Psi)=\mathcal{F}_{0}(\Psi+\delta\Psi)-\mathcal{F}_{0}(\Psi)=g(\delta\Psi,\tilde{{\gamma}})+\mathcal{O}(\delta\Psi)^{2}$
(5.16)

via the state $\tilde{{\gamma}}$, defined as a sum over 3-valent rooted trees,
weighted with symmetry factors. The first several trees of the sum
$\tilde{{\gamma}}$ are presented below

$\frac{1}{2}$ $\Psi$$\Psi$ $+\frac{1}{2}$ $\Psi$$\Psi$$\Psi$$+\frac{1}{2}$
$\Psi$$\Psi$$\Psi$$\Psi$$+\frac{1}{8}$ $\Psi$$\Psi$$\Psi$$\Psi$\+ $\dots$

The diagrammatic sum for $\tilde{{\gamma}}$ takes the form

$\begin{split}\tilde{{\gamma}}&=\frac{1}{2}\mu_{2}(\Psi,\Psi)+\frac{1}{2}\mu_{2}(\Psi,2\pi
KG_{-}\mu_{2}(\Psi,\Psi))+\frac{1}{2}\mu_{2}(\Psi,2\pi KG_{-}\mu_{2}(\Psi,2\pi
KG_{-}\mu_{2}(\Psi,\Psi)))\\\ &+\frac{1}{8}\mu_{2}(2\pi
KG_{-}\mu_{2}(\Psi,\Psi),2\pi KG_{-}\mu_{2}(\Psi,\Psi))+\dots\end{split}$
(5.17)

Note that $\tilde{{\gamma}}$ is an even state since $\Psi$ is even and
$KG_{-}$ is even operator. We also introduce a related even state

$\gamma=\Psi+2\pi KG_{-}\tilde{{\gamma}},$ (5.18)

used in [10], which is $G_{-}$-closed

$G_{-}{\gamma}=G_{-}\Psi-2\pi KG^{2}_{-}\tilde{{\gamma}}=0.$ (5.19)

The $\tilde{{\gamma}}$ and ${\gamma}$ obey the “root-cutting” relation

$\begin{split}\tilde{{\gamma}}=\frac{1}{2}\mu_{2}({\gamma},{\gamma}).\end{split}$
(5.20)

The representation (5.20) for $\tilde{{\gamma}}$ is very convenient for
deriving the recursive formula

$\begin{split}Q\tilde{{\gamma}}&=\frac{1}{2}Q\mu_{2}({\gamma},{\gamma})=\mu_{2}({\gamma},Q{\gamma})=\mu_{2}({\gamma},Q\Psi+2\pi
QKG_{-}\tilde{{\gamma}})\\\ &=\mu_{2}({\gamma},2\pi
G_{-}\tilde{{\gamma}})+\mu_{2}({\gamma},2\pi
KG_{-}Q\tilde{{\gamma}})=\pi\mu_{2}({\gamma},G_{-}\mu_{2}({\gamma},{\gamma}))+\mu_{2}({\gamma},2\pi
KG_{-}Q\tilde{{\gamma}})\\\
&=\frac{\pi}{3}G_{-}\mu_{2}({\gamma},\mu_{2}({\gamma},{\gamma}))+\mu_{2}({\gamma},2\pi
KG_{-}Q\tilde{{\gamma}}).\end{split}$ (5.21)

In our derivation we used Leibniz rule (4.7), homotopy formula (4.21), the
projector property (4.4) and the 7-term relation (4.8) for even,
$G_{-}$-closed state ${\gamma}$ in the form

$G_{-}\mu_{2}({\gamma},\mu_{2}({\gamma},{\gamma}))=3\mu_{2}({\gamma},G_{-}\mu_{2}({\gamma},{\gamma})).$
(5.22)

We can replace the $Q\tilde{{\gamma}}$ in the last expression and use
$G_{-}^{2}=0$ to get

$\begin{split}Q\tilde{{\gamma}}=\frac{\pi}{3}G_{-}\mu_{2}({\gamma},\mu_{2}({\gamma},{\gamma}))+\mu_{2}({\gamma},2\pi
KG_{-}\mu_{2}({\gamma},2\pi KG_{-}Q\tilde{{\gamma}})).\end{split}$ (5.23)

We can iterate the process for of the $Q\tilde{{\gamma}}$-replacement to
arrive into

$Q\tilde{{\gamma}}=\frac{\pi}{3}G_{-}\mu_{2}({\gamma},\mu_{2}({\gamma},{\gamma})).$
(5.24)

The invariance of the correlation functions follows from

$\begin{split}\mathcal{F}_{0}(\Psi+Q\chi)-\mathcal{F}_{0}(\Psi)&=g(Q\chi,\tilde{{\gamma}})=g(\chi,Q\tilde{{\gamma}})=\frac{\pi}{3}g(\chi,G_{-}\mu_{2}({\gamma},\mu_{2}({\gamma},{\gamma})))\\\
&=-\frac{\pi}{3}g(G_{-}\chi,\mu_{2}({\gamma},\mu_{2}({\gamma},{\gamma})))=0.\end{split}$
(5.25)

We used the $Q$-invariance of the pairing (4.11), the $G_{\pm}$-invariance of
the pairing (4.12) and the assumption of the theorem that $G_{-}\chi=0$.
$\hfill\blacksquare$

###  5.5 Recursion relation for correlation functions

Let us consider the HTQM $(V,Q,G_{\pm},\mu_{2},g)$ and four $Q$-,
$G_{-}$-closed states $\Psi_{a},a=1,..,4$. The $4$-point correlation function
(5.4) of such states can be written in the following form

$\begin{split}\langle\Psi_{1},\Psi_{2},&\Psi_{3},\Psi_{4}\rangle_{Q}=\mu_{3}^{0}\left(2\pi
KG_{-}\mu_{2}(\Psi_{4},\Psi_{1}),\Psi_{2},\Psi_{3}\right)+\mu_{3}^{0}\left(\Psi_{1},2\pi
KG_{-}\mu_{2}(\Psi_{4},\Psi_{2}),\Psi_{3}\right)\\\
&+\mu_{3}^{0}\left(\Psi_{1},\Psi_{2},2\pi
KG_{-}\mu_{2}(\Psi_{4},\Psi_{3})\right)\\\
&=\frac{d}{d{\epsilon}}\Big{|}_{{\epsilon}=0}\mu_{3}^{0}(\Psi^{\epsilon}_{1},\Psi^{\epsilon}_{2},\Psi^{\epsilon}_{3})=\frac{d}{d{\epsilon}}\Big{|}_{{\epsilon}=0}\langle\Psi^{\epsilon}_{1},\Psi^{\epsilon}_{2},\Psi^{\epsilon}_{{3}}\rangle_{Q_{\epsilon}},\end{split}$
(5.26)

where we used the leading order deformation (4.59) of states
$\Psi_{1},\Psi_{2},\Psi_{3}$ by the state $\Psi_{4}$ i.e.

$\Psi_{a}^{\epsilon}=\Psi_{a}+2\pi
KG_{-}\mu_{2}({\epsilon}\Psi_{4},\Psi_{a})+\mathcal{O}({\epsilon}^{2}),\;\;a=1,2,3.$
(5.27)

We can describe the relation (5.26) using diagrammatic representation

$\Psi_{1}^{\epsilon}$$\Psi_{2}^{\epsilon}$$\Psi_{3}^{\epsilon}$$\frac{d}{d{\epsilon}}|_{{\epsilon}=0}$$=$
$\Psi_{1}$$\Psi_{2}$$\Psi_{3}$$\Psi_{4}$$+$
$\Psi_{1}$$\Psi_{2}$$\Psi_{3}$$\Psi_{4}$$+$
$\Psi_{1}$$\Psi_{2}$$\Psi_{3}$$\Psi_{4}$

Theorem (recursion relation): The $(n+1)$-point correlation function for the
$Q$-, $G_{-}$-closed states $\Psi_{0},\Psi_{1},..,\Psi_{n}$ in HTQM on trees
$(V,Q,G_{\pm},\mu_{2},g)$ can be expressed as a derivative of $n$-point
correlation function in HTQM, deformed by the state $\Psi_{0}$, i.e.

$\langle\Psi_{1},...,\Psi_{n},\Psi_{0}\rangle_{Q}=\frac{d}{d{\epsilon}}\Big{|}_{{\epsilon}=0}\langle\Psi^{\epsilon}_{1},...,\Psi^{\epsilon}_{{n}}\rangle_{Q^{\epsilon}}.$
(5.28)

The deformed HTQM $(V,Q^{\epsilon},G_{\pm},\mu_{2},g)$ has differential

$Q^{\epsilon}=Q-2\pi[G_{-},\mu_{2}({\epsilon}\Psi_{0},\cdot)]+\mathcal{O}({\epsilon}^{2}),$
(5.29)

while deformed states are

$\Psi_{a}^{\epsilon}=\Psi_{a}+2\pi
KG_{-}\mu_{2}({\epsilon}\Psi_{0},\Psi_{a})+\mathcal{O}({\epsilon}^{2}).$
(5.30)

Proof: The key idea in our proof is to use the generating function for the
amplitudes (5.12). The generating function in deformed theory is

$\mathcal{F}_{0}(\Psi^{\epsilon},K^{\epsilon})=\sum_{k=3}^{\infty}\frac{1}{k!}\langle\underbrace{\Psi^{\epsilon},...,\Psi^{\epsilon}}_{k}\rangle_{Q^{\epsilon}}.$
(5.31)

Let us recall the formula for the change of generating function under
variation of the external state $\Psi$

$\mathcal{F}_{0}(\Psi+{\epsilon}\delta\Psi,K)=\mathcal{F}_{0}(\Psi,K)+g(\tilde{\gamma},{\epsilon}\delta\Psi)+\mathcal{O}({\epsilon}^{2}),$
(5.32)

where the state $\tilde{{\gamma}}$ is a sum over rooted trees (5.17).
Similarly we can derive the change of generating function under the change of
propagator $K$

$\mathcal{F}_{0}(\Psi,K+{\epsilon}\delta K)=\mathcal{F}_{0}(\Psi,K)+\pi
g(\tilde{\gamma},{\epsilon}\delta
K\tilde{\gamma})+\mathcal{O}({\epsilon}^{2}).$ (5.33)

The full change of the generating function is presented on the picture below

$\Psi$$\Psi$$\Psi$$...$$\delta$$=$ $\delta\Psi$$g$$\Psi$$\Psi$$...$$+$
$g$$\delta K$$\Psi$$\Psi$$...$$\Psi$$\Psi$$...$

We use the state deformation formula (4.59) for $\delta\Psi$ and the
propagator in deformed theory (4.63) for $\delta K$ in terms of the state
$\Psi_{0}$, i.e.

$\begin{split}{\epsilon}\delta KG_{-}&=K^{\epsilon}G_{-}-KG_{-}=2\pi
KG_{-}\mu_{2}({\epsilon}\Psi_{0},KG_{-}\cdot),\\\
{\epsilon}\delta\Psi&=\Psi^{\epsilon}-\Psi=2\pi
KG_{-}\mu_{2}({\epsilon}\Psi_{0},\Psi).\end{split}$ (5.34)

Using the $KG_{-}$-flip (4.26) and relations for the sum over rooted trees
(5.18) we can rewrite the generating function in deformed theory

$\begin{split}\mathcal{F}_{0}&(\Psi^{\epsilon},K^{\epsilon})-\mathcal{F}_{0}(\Psi,K)=g(\tilde{\gamma},{\epsilon}\delta\Psi)+\pi
g(\tilde{\gamma},{\epsilon}\delta
K\tilde{\gamma})+\mathcal{O}({\epsilon}^{2})\\\
&=g({\epsilon}\Psi_{0},\mu_{2}(2\pi KG_{-}\tilde{\gamma},\Psi))+\pi
g({\epsilon}\Psi_{0},\mu_{2}(2\pi
KG_{-}\tilde{\gamma},KG_{-}\tilde{\gamma}))+\mathcal{O}({\epsilon}^{2})\\\
&=g({\epsilon}\Psi_{0},\tilde{{\gamma}})-\frac{1}{2}g({\epsilon}\Psi_{0},\mu_{2}(\Psi,\Psi))+\mathcal{O}({\epsilon}^{2})\\\
&=\mathcal{F}_{0}(\Psi+{\epsilon}\Psi_{0},K)-\mathcal{F}_{0}(\Psi,K)-\frac{1}{2}g({\epsilon}\Psi_{0},\mu_{2}(\Psi,\Psi))+\mathcal{O}({\epsilon}^{2}).\end{split}$
(5.35)

The derivative of the relation becomes

$\begin{split}\sum_{k=3}^{\infty}&\frac{d}{d{\epsilon}}\Big{|}_{{\epsilon}=0}\frac{1}{k!}\langle\underbrace{\Psi^{\epsilon},...,\Psi^{\epsilon}}_{k}\rangle_{Q_{\epsilon}}=\frac{d}{d{\epsilon}}\Big{|}_{{\epsilon}=0}\mathcal{F}_{0}(\Psi+{\epsilon}\Psi_{0},K)-\frac{1}{2}\;g(\Psi_{0},\mu_{2}(\Psi,\Psi))\\\
&=\frac{3}{3!}\langle\Psi,\Psi,\Psi_{0}\rangle_{Q}+\sum_{k=4}^{\infty}\frac{1}{k!}k\langle\underbrace{\Psi,...,\Psi}_{k-1},\Psi_{0}\rangle_{Q}-\frac{1}{2}\langle\Psi,\Psi,\Psi_{0}\rangle_{Q}\\\
&=\sum_{k=3}^{\infty}\frac{1}{k!}\langle\underbrace{\Psi,...,\Psi}_{k},\Psi_{0}\rangle_{Q}.\end{split}$
(5.36)

The equality holds at each order in $\Psi$ so

$\frac{d}{d{\epsilon}}\Big{|}_{{\epsilon}=0}\langle\underbrace{\Psi^{\epsilon},...,\Psi^{\epsilon}}_{k}\rangle_{Q_{\epsilon}}=\langle\underbrace{\Psi,...,\Psi}_{k},\Psi_{0}\rangle_{Q}$
(5.37)

what completes the proof of the theorem. $\hfill\blacksquare$

##  6 Mirror for HTQM

###  6.1 A-model

Tropical Gromov-Witten theory on toric manifold $X$ of complex dimension $N$
defines the A-type HTQM on trees, also denoted as the A-model in this section.
In this section we will describe the HTQM data $(V,Q,G_{\pm},g,\mu_{2})$ for
A-model.

Let us consider pair $|\omega,\vec{m}\rangle$, of tropical form $\omega$ on
$\mathbb{C}^{\ast N}$, and $N$-dimensional integer-valued vector $\vec{m}$.
The pairing on $|\omega,\vec{m}\rangle$ is the integration of corresponding
form forms

$g(|\omega_{1},\vec{m}_{1}\rangle,|\omega_{2},\vec{m}_{2}\rangle)=\delta_{\vec{m}_{1}+\vec{m}_{2},\vec{0}}\int_{\mathbb{C}^{\ast
N}}\omega_{1}\wedge\omega_{2}.$ (6.1)

The vector space $V_{A}$ is the space of tropical differential forms on
$\mathbb{C}^{\ast N}$, equipped with the integer vector, i.e.

$V_{A}=\Omega^{\ast}_{trop}(\mathbb{C}^{\ast
N})\otimes\mathbb{R}\langle\vec{m}\;|\;\vec{m}\in\mathbb{Z}^{N}\rangle.$ (6.2)

The $\mathbb{Z}_{2}$-grading of the state $|\omega,\vec{m}\rangle$ is the
grading of the form $\omega$, a degree of the differential form mod $2$. The
differential $Q$ is the de Rham operator on $\mathbb{C}^{\ast N}$

$Q|\omega,\vec{m}\rangle=|d\omega,\vec{m}\rangle.$ (6.3)

The $G_{\pm}$ on states is a contraction with the constant radial (angular)
vector field $\vec{m}$

$\begin{split}G_{+}|\omega,\vec{m}\rangle&=|\iota^{R}_{\vec{m}}\omega,\vec{m}\rangle=|\iota_{m^{i}\partial_{r^{i}}}\omega,\vec{m}\rangle,\\\
G_{-}|\omega,\vec{m}\rangle&=|\iota^{\Phi}_{\vec{m}}\omega,\vec{m}\rangle.\end{split}$
(6.4)

The multiplication $\mu_{2}:V\otimes V\to V$ is the wedge product on
differential forms supplemented with addition of corresponding vectors

$\mu_{2}(|\omega_{1},\vec{m}_{1}\rangle,|\omega_{2},\vec{m}_{2}\rangle)=|\omega_{1}\wedge\omega_{2},\vec{m}_{1}+\vec{m}_{2}\rangle.$
(6.5)

The pair $(Q,\mu_{2})$ is essentially an external derivative and the wedge
product, hence obeys the DGA properties. The pair $(G_{-},\mu_{2})$ obeys the
7-term relation.

The Hamiltonian

$H=\\{Q,G_{+}\\}$ (6.6)

is the Lie derivative along the constant radial vector field $\vec{m}$

$H|\omega,\vec{m}\rangle=\\{Q,G_{+}\\}|\omega,\vec{m}\rangle=|\\{d,\iota^{R}_{\vec{m}}\\}\omega,\vec{m}\rangle=|{\cal
L}^{R}_{\vec{m}}\omega,\vec{m}\rangle.$ (6.7)

The evolution operator $e^{-tH}$, defined as solution to

$(\partial_{t}+H)e^{-tH}=(\partial_{t}+{\cal L}^{R}_{\vec{m}})e^{-tH}=0$ (6.8)

is a 1-parameter family of diffeomorphisms $\Phi^{t}_{\vec{m}}:r^{i}\mapsto
r^{i}-m^{i}t$

$e^{-tH}|\omega,\vec{m}\rangle=|(\Phi^{t}_{\vec{m}})^{\ast}\omega,\vec{m}\rangle.$
(6.9)

The composition property (4.2) naturally holds for diffeomorphisms. Since the
vector fields are constant vector fields the corresponding flows do not
develop any singularities, hence composition is valid for all values of $t$.

###  6.2 Correlation functions in A-model

In our paper [8] we showed that the tropical GW invariant of genus-0 and
degree-$\beta$ on toric space $X$ is the sum of the A-type HTQM amplitudes

$\langle{\gamma}_{1},...,{\gamma}_{n}\rangle^{X}_{\beta}=\sum_{\Gamma}\mathcal{A}_{\Gamma}(\Psi_{{\gamma}_{1}},..,\Psi_{{\gamma}_{n}},\Psi_{b_{1}},..,\Psi_{b_{B}})$
(6.10)

with two types of states:

  * •

Evaluation states $\Psi_{{\gamma}_{k}}=|{\gamma}_{k},\vec{0}\rangle$,
constructed from the tropical forms ${\gamma}_{k}$ on X. The space
$\Omega^{trop}(X)$ of tropical forms on $X$ is a subspace of tropical forms on
$\mathbb{C}^{\ast N}$ with good behaviour at compactifying divisors.

  * •

Divisor states $\Psi_{\vec{b}_{a}}=|1,\vec{b}_{a}\rangle$, where $\vec{b}_{a}$
are primitive normal vectors for compactifying divisors of $X$.

Using the definition (5.1) we can replace the sum over the amplitudes by the
correlation function and formulate the HTQM representation for the tropical GW
invariants in the form:

Theorem (HTQM representation of tropical GW): For toric space $X$, given in
terms of boundary divisors $B_{X}$, and tropical cycles ${\gamma}_{k}$, the
tropical GW invariant matches with the HTQM correlation function i.e.

$\langle{\gamma}_{1},...,{\gamma}_{n}\rangle^{X}_{\beta}=\frac{1}{d_{1}!\cdot..\cdot
d_{B}!}\langle\Psi_{{\gamma}_{1}},..,\Psi_{{\gamma}_{n}},\underbrace{\Psi_{\vec{b}_{1}},..,\Psi_{\vec{b}_{1}}}_{d_{1}},..,\underbrace{\Psi_{\vec{b}_{B}},..,\Psi_{\vec{b}_{B}}}_{d_{B}}\rangle_{Q}.$
(6.11)

Here $\Psi_{{\gamma}_{a}}$ are the evaluation states for tropical cycles
${\gamma}_{a}$, $\Psi_{\vec{b}}$ are divisor states for boundary divisors from
$B_{X}$ of cardinality $B=|B_{X}|$. The degree of the map $\beta\in
H_{1,1}(X)$ determines the number $d_{a}$ of divisor states
$\Psi_{\vec{b}_{a}}$ of a given type via the corresponding tropical
intersection number of $\beta$ and boundary divisor with normal vector
$\vec{b}_{a}$.

The GW theory is defined on classes of cycles $C_{\alpha}$. The change of a
cycle within the same class leads to the shift of an evaluation observable
(Poincare-dual form) ${\gamma}$ by an exact form

${\gamma}\to{\gamma}+d\lambda.$ (6.12)

The shift of evaluation observable ${\gamma}$ changes the corresponding
A-model state $\Psi_{\gamma}$ by a $Q$-exact term

$\Psi_{\gamma}\to\Psi_{{\gamma}+d\lambda}=\Psi_{\gamma}+Q\Psi_{\lambda}.$
(6.13)

Furthermore, since the states $\Psi_{\gamma}$ and $\Psi_{\lambda}$ carry
trivial integer vector, they are $G_{-}$-closed, i.e.

$G_{-}\Psi_{\lambda}=G_{-}|\lambda,\vec{0}\rangle=0,\;\;G_{-}\Psi_{\gamma}=G_{-}|{\gamma},\vec{0}\rangle=0$
(6.14)

The forms ${\gamma}_{a}$ are closed forms so is the corresponding HTQM states
$\Psi_{{\gamma}_{1}}$. Hence we can use the invariance theorem (5.4) to verify
that the tropical GW invariants in HTQM representation (6.11) are defined on
cohomology classes of ${\gamma}_{a}$.

###  6.3 Dual variables

It is convenient to introduce angular variables $Y_{j}\in S^{1}$, dual to the
integer vector components $m^{i}\in\mathbb{Z}$. We introduce a Fourier
transform of a state

$\sum_{\vec{m}\in\mathbb{Z}^{N}}c_{\vec{m}}|\omega_{\vec{m}},\vec{m}\rangle\mapsto\Psi=\sum_{\vec{m}\in\mathbb{Z}^{N}}e^{i\langle\vec{m},\vec{Y}\rangle}c_{\vec{m}}|\omega_{\vec{m}},\vec{m}\rangle\in
V_{B}=\Omega^{trop}(\mathbb{C}^{\ast N})\otimes C^{\infty}(\mathbb{T}^{N}).$
(6.15)

For convenience we describe the differential forms on $X$ using Grassmann
variables:

$dr^{j}=\psi_{R}^{j},\;\;\;d\phi^{j}=\psi_{\Phi}^{j}.$ (6.16)

The differentials on mirror states in new notations become first and second
order differential operators

$\begin{split}Q\Psi&=\sum_{\vec{m}\in\mathbb{Z}^{N}}e^{i\langle\vec{m},\vec{Y}\rangle}c_{\vec{m}}|d\omega_{\vec{m}},\vec{m}\rangle=d_{R}\Psi=\psi_{R}^{k}\frac{\partial}{\partial
r^{k}}\Psi,\\\
G_{-}\Psi&=\sum_{\vec{m}\in\mathbb{Z}^{N}}e^{i\langle\vec{m},\vec{Y}\rangle}c_{\vec{m}}|\iota^{\Phi}_{\vec{m}}\omega,\vec{m}\rangle=-i\frac{\partial}{\partial
Y_{k}}\frac{\partial}{\partial\psi_{\Phi}^{k}}\Psi,\\\
G_{+}\Psi&=\sum_{\vec{m}\in\mathbb{Z}^{N}}e^{i\langle\vec{m},\vec{Y}\rangle}c_{\vec{m}}|\iota^{R}_{\vec{m}}\omega,\vec{m}\rangle=-i\frac{\partial}{\partial
Y_{k}}\frac{\partial}{\partial\psi_{R}^{k}}\Psi.\end{split}$ (6.17)

The multiplication $\mu_{2}$, on forms becomes multiplication of functions on
superspace with coordinates $r,Y,\psi_{R},\psi_{\Phi}$

$\mu_{2}(\Psi_{1},\Psi_{2})=\Psi_{1}\cdot\Psi_{2},$ (6.18)

while the pairing is the integration over superspace

$g(\Psi_{1},\Psi_{2})=\int d\mu\;\;\Psi_{1}\Psi_{2},$ (6.19)

where Berezin integration measure for $\dim_{\mathbb{C}}X=N$ is

$d\mu=d^{N}r\;d^{N}Y\;d^{N}\psi_{\Phi}d^{N}\psi_{R}.$ (6.20)

The integrating region (for Grassmann-even variables) is the $N$-dimensional
torus $(S^{1})^{N}$ for $Y$-variables and Euclidean space $\mathbb{R}^{N}$ for
$r$-variables.

Remark: The differential operator representation (6.17) of the HTQM data
$(V,Q,G_{\pm},\mu_{2},g)$ allows for an easy check of HTQM definitions from
sections 4.1 and 4.2. In particular the 7-term relation (4.8) for $G_{-}$ is a
property of the second order differential operator in representation (6.17).

The divisor states $|1,\vec{b}\rangle$ become exponential functions of $Y$

$\Psi_{\vec{b}}(Y)=e^{i\langle\vec{b},\vec{Y}\rangle}=e^{ib^{k}Y_{k}}.$ (6.21)

For the tropical form

${\gamma}={\gamma}_{i_{1}..i_{k}j_{1}..j_{l}}(r)\;d\phi^{i_{1}}\wedge..\wedge
d\phi^{i_{k}}\wedge dr^{j_{1}}\wedge..\wedge dr^{j_{l}}$ (6.22)

the corresponding evaluation state is

$\Psi_{\gamma}={\gamma}_{i_{1}..i_{k}j_{1}..j_{l}}(r)\;\psi_{\Phi}^{i_{1}}\cdot..\cdot\psi_{\Phi}^{i_{k}}\cdot\psi_{R}^{j_{1}}\cdot..\cdot\psi_{R}^{j_{l}}.$
(6.23)

Example: The evaluation state for $U(1)$-invariant Poincare dual of the point
on $\mathbb{P}^{1}$ is

${\gamma}=\frac{1}{2\pi}\delta(r-r_{0})d\phi
dr\mapsto\Psi_{\gamma}=|{\gamma},0\rangle=\frac{1}{2\pi}\delta(r-r_{0})\psi_{\Phi}\psi_{R}.$
(6.24)

###  6.4 B-model

In our work [8] we showed that the deformation of A-model, the HTQM
$(V_{A},Q,G_{\pm},\mu_{2},g)$, by divisor states $\Psi_{\vec{b}}$ for toric
$X$ is also an HTQM $(V_{B},Q_{X},G_{\pm},\mu_{2},g)$, which we will denote as
the B-type HTQM or B-model for short.

Remark: The two-dimensional version of the deformation by compactifying
divisors of toric space was discussed in A-I-B mirror paper [11]. However, the
deformation of observables was not discussed there.

Divisor states obey

$G_{-}\mu_{2}(\Psi_{\vec{b}_{1}},\Psi_{\vec{b}_{2}})=0,$ (6.25)

hence the deformation of the differential $Q$, which we discussed in section
4.5 holds beyond the linearized level. The deformation of the A-model
differential by all divisors $B_{X}$ of $X$ is

$Q_{X}=Q-2\pi\sum_{\vec{b}\in
B_{X}}[G_{-},\mu_{2}(q_{\vec{b}}\Psi_{\vec{b}},\cdot)]=\psi_{R}^{j}\frac{\partial}{\partial
r^{j}}+2\pi i\sum_{\vec{b}\in
B_{X}}q_{\vec{b}}\frac{\partial\Psi_{\vec{b}}}{\partial
Y_{j}}\frac{\partial}{\partial\psi_{\Phi}^{j}}.$ (6.26)

The same differential can be written as

$Q_{X}=Q+2\pi i\frac{\partial W_{X}(Y)}{\partial
Y_{j}}\frac{\partial}{\partial\psi_{\Phi}^{j}}=Q_{W_{X}},$ (6.27)

using the mirror superpotential

$W_{X}(Y)=\sum_{\vec{b}\in B_{X}}q_{\vec{b}}\;\Psi_{\vec{b}}=\sum_{\vec{b}\in
B_{X}}q_{\vec{b}}\;e^{i\langle\vec{b},\vec{Y}\rangle}.$ (6.28)

We can absorb some $q_{\vec{b}}$ by the redefinition of $Y_{j}$ to obtain more
familiar (at least for the $\mathbb{P}^{N}$ case) form of the superpotential
with fewer parameters $q_{\vec{b}}$.

Remark: The exponential mirror superpotentials for toric spaces were derived
by Givental [5] and by Hori and Vafa [17] using different methods.

Definition: For toric space $X$ the deformation (4.59) of an A-model
evaluation state $\Psi_{\gamma}$ by the divisor states $\Psi_{\vec{b}}$ is a
mirror state $\Psi_{\gamma}^{X}$. We can write the mirror state using the
A-model notations in the form

$\begin{split}\Psi_{\gamma}^{X}&=\Psi_{\gamma}+2\pi
KG_{-}\mu_{2}(W_{X},\Psi_{\gamma})+(2\pi)^{2}KG_{-}\mu_{2}(W_{X},KG_{-}\mu_{2}(W_{X},\Psi_{\gamma}))+...\end{split}$
(6.29)

For an A-model state $\Psi_{\gamma}$ corresponding to the tropical form
${\gamma}\in\Omega^{k,l}(X)$ the sum (6.29) terminates after min$(k,l)+1$
terms. Indeed, the action of $KG_{-}$ lowers the degree of the form by
$(1,1)$, hence it can be applied to $(k,l)$-form $\Psi_{\gamma}$ at most
min$(k,l)$ times.

By preservation of closeness proposition (4.60), the mirror state
$\Psi_{\gamma}^{X}$ is $Q_{X}$\- and $G_{-}$-closed, i.e.

$Q_{X}\Psi_{\gamma}^{X}=G_{-}\Psi_{\gamma}^{X}=0.$ (6.30)

Example: For $X=\mathbb{P}^{1}$ the space of boundary normal vectors is
$B_{\mathbb{P}^{1}}=\\{+1,-1\\}$, hence the mirror superpotential equals

$W_{\mathbb{P}^{1}}(Y)=\sum_{b\in
B_{\mathbb{P}^{1}}}q_{b}\;e^{ibY}=q_{+}e^{iY}+q_{-}e^{-iY}.$ (6.31)

We can absorb the $q_{+}$ by constant shift of $Y$ to arrive into more
familiar form of the superpotential

$W_{\mathbb{P}^{1}}=e^{iY}+qe^{-iY}$ (6.32)

with $q=q_{+}q_{-}$. The B-model differential is

$Q_{\mathbb{P}^{1}}=\psi_{R}\frac{\partial}{\partial r}+2\pi i\frac{\partial
W_{\mathbb{P}^{1}}}{\partial
Y}\frac{\partial}{\partial\psi_{\Phi}}=\psi_{R}\frac{\partial}{\partial
r}-2\pi(q_{+}e^{iY}-q_{-}e^{-iY})\frac{\partial}{\partial\psi_{\Phi}}.$ (6.33)

The mirror state $\Psi^{\mathbb{P}^{1}}_{\gamma}$ for the A-model evaluation
observable ${\gamma}\in H^{\ast}_{dR}(\mathbb{P}^{1})$ contains three terms

$\begin{split}\Psi^{\mathbb{P}^{1}}_{\gamma}&=\Psi_{\gamma}+2\pi
KG_{-}\mu_{2}(W_{\mathbb{P}^{1}},\Psi_{\gamma})\\\ &=\Psi_{\gamma}+2\pi
q_{+}\int^{\infty}_{0}dt\;e^{-tH}\;G_{+}G_{-}(e^{iY}\Psi_{\gamma})+2\pi
q_{-}\int^{\infty}_{0}dt\;e^{-tH}\;G_{+}G_{-}(e^{-iY}\Psi_{\gamma}).\end{split}$
(6.34)

Indeed, the top form on $\mathbb{P}^{1}$ has degree $(1,1)$, hence we can
apply the deformation only once for each of two boundary divisors. The mirror
state for the constant form ${\gamma}=1\in H_{dR}^{0}(\mathbb{P}^{1})$ has a
trivial deformation

$\Psi^{\mathbb{P}^{1}}_{1}=\Psi_{1}=1.$ (6.35)

Consider the Poincare dual of the point $\phi=\phi_{0}$ and $r=r_{0}$ in
tropical coordinates on $\mathbb{P}^{1}$

$P_{0}=\delta(\phi-\phi_{0})\delta(r-r_{0})\;d\phi dr.$ (6.36)

The $U(1)$-averaging over $\phi_{0}$ of the form $P_{0}$ is

$P=\frac{1}{2\pi}\delta(r-r_{0})\;d\phi dr\in H_{dR}^{1,1}(\mathbb{P})$ (6.37)

The corresponding A-model state equals

$\Psi_{P}=\frac{1}{2\pi}\delta(r-r_{0})\psi_{\Phi}\psi_{R}.$ (6.38)

The corresponding mirror state is

$\begin{split}\Psi^{\mathbb{P}^{1}}_{P}&=\frac{1}{2\pi}\delta(r-r_{0})\psi_{\Phi}\psi_{R}+q_{+}e^{iY}\int\limits^{\infty}_{0}dt\;\delta(r-r_{0}-t)+q_{-}e^{-iY}\int\limits^{\infty}_{0}dt\;\delta(r-r_{0}+t)\\\
&=\frac{1}{2\pi}\delta(r-r_{0})\psi_{\Phi}\psi_{R}+q_{+}e^{iY}\Theta(r-r_{0})+q_{-}e^{-iY}\Theta(r_{0}-r).\end{split}$
(6.39)

In our work [8] we proved the mirror symmetry for the A-type HTQM
$(V_{A},Q,G_{\pm},\mu_{2},g)$ and the B-type HTQM
$(V_{B},Q_{X},G_{\pm},\mu_{2},g)$. The proof was essentially a summation over
the divisor states $\Psi_{\vec{b}}$ for all divisors of the toric space $X$.
The HTQM mirror implies the following statement for the correlation functions.

Theorem (tropical mirror for HTQMs): On toric space $X$, with boundary
divisors $\vec{b}_{1},...,\vec{b}_{B}$ the sum over divisor states in the
A-model’s correlation function of evaluation states $\Psi_{{\gamma}_{a}}$
equals to the correlation function of the corresponding mirror states
$\Psi^{X}_{{\gamma}_{a}}$ in B-model with mirror superpotential $W_{X}$ i.e.

$\sum_{k=0}^{\infty}\frac{1}{k!}\;\langle\Psi_{{\gamma}_{1}},...,\Psi_{{\gamma}_{n}},\underbrace{W_{X},..,W_{X}}_{k}\rangle_{Q}=\langle\Psi^{X}_{{\gamma}_{1}},...,\Psi^{X}_{{\gamma}_{n}}\rangle_{Q_{W_{X}}}.$
(6.40)

Note that the $q$-dependence in the A-model is due to the divisor states in
$W_{X}$, while in B-model both evaluation states and differential $Q_{X}$ have
nontrivial $q$-dependence.

##  7 Localization of mirror states

In the A-model formulation evaluation states and divisor states look very
different and does not have any simple relation. In case $X=\mathbb{P}^{1}$
the evaluation state for the point observable at $r=r_{0}$ and divisor states
are

$\Psi_{P}=\frac{1}{2\pi}\delta(r-r_{0})\psi_{\Phi}\psi_{R},\;\;\;\Psi_{+}=e^{iY},\;\;\Psi_{-}=e^{-iY}.$
(7.1)

In section 6.4 we constructed the $\mathbb{P}^{1}$-mirror state for the
evaluation state of a point observable

$\Psi^{\mathbb{P}^{1}}_{P}=\frac{1}{2\pi}\delta(r-r_{0})\psi_{\Phi}\psi_{R}+q_{+}e^{iY}\Theta(r-r_{0})+q_{-}e^{-iY}\Theta(r_{0}-r).$
(7.2)

The expression (7.2) for mirror state turns into a pure divisor state
$\Psi_{\pm}$ in the limit $r_{0}\to\pm\infty$, i.e.

$\begin{split}\lim_{r_{0}\to-\infty}&\Psi^{\mathbb{P}^{1}}_{P}=q_{+}e^{iY}=q_{+}\Psi_{+}\\\
\lim_{r_{0}\to+\infty}&\Psi^{\mathbb{P}^{1}}_{P}=q_{-}e^{-iY}=q_{-}\Psi_{-}.\end{split}$
(7.3)

There is a natural geometric interpretation of this relation: The point at
finite position $r=r_{0}$ becomes a compactifying divisor (a point at infinity
in case of $\mathbb{P}^{1}$) when we move $r_{0}$ to infinity.

The relation between mirror states and divisor states plays a key role for the
localization of the B-model correlation functions. Note, that the limit of the
state, often referred to as the point-wise limit, in general, may not commute
with the amplitude/correlation function evaluation for the same state. We will
discuss this potential problem carefully in the next section.

###  7.1 Mirror states vs divisor states

The pair of A-model states, corresponding to the Poincare duals of points
$r_{0}$ and $r_{1}$ are related by a $Q$-exact term

$\begin{split}\Psi_{1}-\Psi_{0}&=\frac{1}{2\pi}\delta(r-r_{1})\psi_{\Phi}\psi_{R}-\frac{1}{2\pi}\delta(r-r_{0})\psi_{\Phi}\psi_{R}\\\
&=Q\left(-\frac{1}{2\pi}\Theta(r-r_{1})\psi_{\Phi}+\frac{1}{2\pi}\Theta(r-r_{0})\psi_{\Phi}\right)=Q\chi_{01}.\end{split}$
(7.4)

The tropical form

$\begin{split}\chi_{01}&=\frac{1}{2\pi}\left[\Theta(r-r_{0})-\Theta(r-r_{1})\right]\psi_{\Phi}.\end{split}$
(7.5)

is a tropical form on $\mathbb{P}^{1}$ since the difference of two
$\Theta$-functions has finite support. We can take a limit $r_{1}\to-\infty$
for $\chi_{01}$ to define

$\begin{split}\chi_{+}=\lim_{r_{1}\to-\infty}\chi_{01}&=\frac{1}{2\pi}(\Theta(r-r_{0})-1)\psi_{\Phi}=-\frac{1}{2\pi}\Theta(r_{0}-r)\psi_{\Phi}.\end{split}$
(7.6)

The tropical form $\chi_{+}$ can be used to turn the mirror state into the
boundary divisor state, i.e.

$\begin{split}q_{+}\Psi_{+}=\Psi^{\mathbb{P}^{1}}_{P}+Q_{\mathbb{P}^{1}}\chi_{+}.\end{split}$
(7.7)

The relation (7.7) is equivalent to the statement that the mirror state
$\Psi^{\mathbb{P}^{1}}_{P}$ and the holomorphic function $\Psi_{+}=e^{iY}$
represent the same cohomology as a forms in $V_{B}$, i.e.

$\Psi^{\mathbb{P}^{1}}_{P}=q_{+}\Psi_{+}\in
H^{\ast}(Q_{\mathbb{P}^{1}},V_{B}).$ (7.8)

Note that the tropical form $\chi_{+}$ is $G_{-}$-closed, i.e.

$G_{-}\chi_{+}=G_{-}\left(-\frac{1}{2\pi}\Theta(r_{0}-r)\psi_{\Phi}\right)=0,$
(7.9)

hence we can formulate a stronger equality in cohomology

$\Psi^{\mathbb{P}^{1}}_{P}=q_{+}\Psi_{+}\in
H^{\ast}(Q_{\mathbb{P}^{1}}+zG_{-},V_{B}\otimes\mathbb{R}[[z]]).$ (7.10)

We can take $r_{1}\to+\infty$ limit for (7.5) to define a different tropical
form

$\chi_{-}=\frac{1}{2\pi}\Theta(r-r_{0})\psi_{\Phi}$ (7.11)

such that

$q_{-}\Psi_{-}=\Psi^{\mathbb{P}^{1}}_{P}+Q_{\mathbb{P}^{1}}\chi_{-}$ (7.12)

Similarly to the previous case

$G_{-}\chi_{-}=G_{-}\left(\frac{1}{2\pi}\Theta(r-r_{0})\psi_{\Phi}\right)=0,$
(7.13)

hence

$\Psi^{\mathbb{P}^{1}}_{P}=q_{-}\Psi_{-}\in
H^{\ast}(Q_{\mathbb{P}^{1}}+zG_{-},V_{B}\otimes\mathbb{R}[[z]]).$ (7.14)

We can choose more general $Q_{\mathbb{P}^{1}}$-exact term, so that the mirror
state will become a holomorphic function, which is not one of the
compactifying divisor states for $\mathbb{P}^{1}$

$\begin{split}\hat{\Psi}_{P}&=\Psi^{\mathbb{P}^{1}}_{P}+Q_{\mathbb{P}^{1}}\hat{\chi}=\Psi^{\mathbb{P}^{1}}_{P}+Q_{\mathbb{P}^{1}}\left(\frac{1}{2\pi}\Theta(r-r_{0})\psi_{\Phi}+\frac{1}{2\pi}qe^{-2iY}\psi_{\Phi}\right)\\\
&=q^{2}e^{-3iY}.\end{split}$ (7.15)

The key difference from the $\chi_{\pm}$ is that the tropical form
$\hat{\chi}$ is not $G_{-}$-closed, i.e.

$G_{-}\hat{\chi}=\left(\frac{1}{2\pi}\Theta(r-r_{0})\psi_{\Phi}+\frac{q}{2\pi}\;e^{-2iY}\psi_{\Phi}\right)=-\frac{q}{\pi}\;e^{-2iY}\neq
0.$ (7.16)

hence $\hat{\Psi}_{P}$ and $\Psi_{P}$ belong to different classes in
$(Q_{\mathbb{P}^{1}}+zG_{-})$-cohomology. Indeed, a simple computation shows
that

$q^{2}e^{-3iY}=e^{iY}+\frac{qz}{\pi}\;e^{-2iY}\in
H^{\ast}(Q_{\mathbb{P}^{1}}+zG_{-},V_{B}\otimes\mathbb{R}[[z]]).$ (7.17)

In the remaining part of this section we will show that all mirror states
admit holomorphic function representatives in $H^{\ast}(Q_{W}+zG_{-})$ and
discuss some properties of these representatives.

###  7.2 Spectral sequence for $Q_{W}$-cohomology

A mirror state describes a class in $H^{\ast}(Q_{W},V_{B})$. The
$Q_{W}$-differential

$Q_{W}=\psi^{j}_{R}\frac{\partial}{\partial r^{j}}+2\pi i\frac{\partial
W}{\partial Y_{j}}\frac{\partial}{\partial\psi_{\Phi}^{j}}=d_{R}+2\pi i\;{\bf
Q}_{W}$ (7.18)

is a sum of two (graded-) commuting differentials, the radial de Rham
differential

$d_{R}=\psi^{j}_{R}\frac{\partial}{\partial r^{j}},\;\;\;d_{R}^{2}=0$ (7.19)

and the LGS differential (3.20)

${\bf Q}_{W}=\frac{\partial W}{\partial
Y_{k}}\frac{\partial}{\partial\psi_{\Phi}^{k}},\;\;\;{\bf Q}_{W}^{2}=0.$
(7.20)

Hence we can use a spectral sequence to evaluate the cohomology of the
$Q_{W}$. The spectral sequence converges at the second step

$H^{\ast}(Q_{W},V_{B})=H^{\ast}({\bf Q}_{W},H^{\ast}({d_{R}},\textlnot
V_{B})),$ (7.21)

since the only cohomology of the radial de Rham operator on $\mathbb{R}^{N}$
are constant ($r$-independent) forms of degree 0 in $\psi_{R}$ and is
isomorphic to the LSG vector space (3.19). The ${\bf Q}_{W}$ is differential
in LGS theory, hence we can express the cohomology via the Jacobi ring (2.4)
for superpotential W

$H^{\ast}(Q_{W},V_{B})=J_{W}.$ (7.22)

An isomorphism (7.22) means that for every $Q_{W}$-closed B-model state we can
find a holomorphic function from the same $Q_{W}$-cohomology class on $V_{B}$.

###  7.3 Pairing and localization of states

Definition: The holomorphic germ for a B-model state $\Psi$ is an evaluation
of the state $\Psi$ at $r=\psi=0$, i.e.

$\Phi(Y)=\Psi(r,Y,\psi_{\Phi},\psi_{R})\Big{|}_{\psi=r=0}.$ (7.23)

Definition: For states $\Psi_{1},\Psi_{2}$ in B-model with superpotential $W$
we introduce a pairing

$g^{\Lambda}_{W}(\Psi_{1},\Psi_{2})=g\left(\Psi_{1},e^{\Lambda
Q_{W}(L)}\Psi_{2}\right),$ (7.24)

where $g$ is the B-model pairing, $\Lambda$ is a real parameter and $L$ is a
localization function

$L=\sum_{k=1}^{N}r^{k}\psi^{k}_{\Phi}.$ (7.25)

Note: The exponent in the pairing evaluates into

$Q_{W}(L)=\sum_{k=1}^{N}\psi^{k}_{\Phi}\psi^{k}_{R}+2\pi
i\sum_{k=1}^{N}r^{k}\frac{\partial W}{\partial Y_{k}}.$ (7.26)

Hence the $Q_{W}(L)$ in (7.26) is an oscillating function in parity-even
variables $r$ and $Y$ what makes the radial integral converging.

The pairing $g^{\Lambda}_{W}$ matches with the B-model pairing (6.19) for
$\Lambda=0$. The pairing $g^{\Lambda}_{W}$ is $Q_{W}$-invariant, indeed

$\begin{split}g^{\Lambda}_{W}(Q_{W}\Psi_{1},\Psi_{2})&=g\left(Q_{W}\Psi_{1},e^{\Lambda
Q_{W}(L)}\Psi_{2}\right)=-(-1)^{|\Psi_{1}|}g\left(\Psi_{1},Q_{W}e^{\Lambda
Q_{W}(L)}\Psi_{2}\right)\\\
&=-(-1)^{|\Psi_{1}|}g^{\Lambda}_{W}(\Psi_{1},Q_{W}\Psi_{2}).\end{split}$
(7.27)

We used the $Q_{W}$-invariance of the B-model pairing and an identity

$Q_{W}e^{Q_{W}(L)}=Q_{W}e^{Q_{W}L+LQ_{W}}=e^{Q_{W}(L)}Q_{W}.$ (7.28)

The $Q_{W}$-invariance of the pairing implies that it is well-defined on
$H^{\ast}(Q_{W},V_{B})$.

Lemma: On $Q_{W}$-closed states the pairing $g^{\Lambda}_{W}$ is independent
of $\Lambda$.

Proof: The derivative of the pairing evaluates into

$\begin{split}\frac{d}{d\Lambda}&g^{\Lambda}_{W}(\Psi_{1},\Psi_{2})=\frac{d}{d\Lambda}g\left(\Psi_{1},e^{\Lambda
Q_{W}(L)}\Psi_{2}\right)=g\left(\Psi_{1},(Q_{W}L+LQ_{W})e^{\Lambda
Q_{W}(L)}\Psi_{2}\right)\\\ &=g\left(\Psi_{1},Le^{\Lambda
Q_{W}(L)}Q_{W}\Psi_{2}\right)-(-1)^{|\Psi_{1}|}g\left(Q_{W}\Psi_{1},e^{\Lambda
Q_{W}(L)}\Psi_{2}\right)=0\end{split}$ (7.29)

We used the $Q_{W}$ invariance of the B-model pairing and $Q_{W}$-closeness of
both $\Psi_{1}$ and $\Psi_{2}$ and an identity (7.28). $\hfill\blacksquare$

The $\Lambda$-independence means that the pairing $g^{\Lambda}_{W}$ matches
with the B-model pairing $g$ on $H^{\ast}(Q_{W},V_{B})$. The B-model pairing
$g$ is non-degenerate on $V_{B}$ and $Q_{W}$-invariant, hence $g$ is non-
degenerate on $Q_{W}$-cohomology, so is the $g^{\Lambda}_{W}$.

Proposition: The $Q_{W}$-closed state $\Psi$ and its holomorphic germ $\Phi$
are in the same $Q_{W}$-cohomology class i.e.

$\Psi=\Phi\in H^{\ast}(Q_{W},V_{B}).$ (7.30)

Proof: Since $Q_{W}(L)$ in (7.26) is an oscillating function in parity-even
variables $r$ and $Y$, then the integral will localize near critical points of
$Q_{W}(L)$. Moreover, the localization exponent is scaled by $\Lambda$, so we
can choose large $\Lambda$ to eliminate the subleading corrections to the
saddle point formula. The critical points of $Q_{W}(L)$ are determined from

$\begin{split}\frac{\partial}{\partial r^{k}}Q_{W}(L)&=2\pi i\frac{\partial
W}{\partial Y_{k}}=0,\;\;\;\frac{\partial}{\partial Y_{m}}Q_{W}(L)=2\pi
i\sum_{k=1}^{N}r^{k}\frac{\partial^{2}W}{\partial Y_{k}\partial Y_{m}}=0,\\\
\frac{\partial}{\partial\psi^{k}_{\Phi}}Q_{W}(L)&=\psi^{k}_{R}=0,\;\;\;\frac{\partial}{\partial\psi^{k}_{R}}Q_{W}(L)=-\psi^{k}_{\Phi}=0.\end{split}$
(7.31)

Under the assumption that the critical points of $W$ are isolated, the rank of
$\frac{\partial^{2}W}{\partial Y_{l}\partial Y_{m}}$ is maximal, hence the
integral in the limit $\Lambda\to\infty$ localizes to the critical points of
$W$ and origin in all other variables

$r^{k}=\psi^{k}_{R}=\psi^{k}_{\Phi}=0,\;\;Y=Y_{0},\;\;dW(Y_{0})=0.$ (7.32)

The ratio of determinants for the integration around the critical point is

$\det\left(\frac{\partial^{2}Q_{W}(L)}{\partial\psi^{k}_{\Phi}\partial\psi^{l}_{R}}\right)\cdot\frac{(2\pi)^{N}}{\det\left(\frac{\partial^{2}Q_{W}(L)}{\partial
Y_{k}\partial
r^{l}}\right)}=\frac{(2\pi)^{N}\Lambda^{N}}{(2\pi\Lambda)^{N}\det\left(\frac{\partial^{2}W}{\partial
Y_{k}\partial Y_{l}}\right)}.$ (7.33)

The leading order saddle point formula for the pairing integral localizes the
$g^{\Lambda}_{W}$-pairing to

$g^{\Lambda}_{W}(\Psi_{1},\Psi_{2})=\sum_{dW=0}\frac{1}{\det\left(\frac{\partial^{2}W}{\partial
Y_{k}\partial Y_{l}}\right)}\;\;\Psi_{1}\cdot\Psi_{2}\Big{|}_{r=\psi=0}.$
(7.34)

Note that the $\Lambda$-dependence drops out from the leading saddle point
formula. We observe that the pairing $g^{\Lambda}_{W}$ for B-model states
localizes to the corresponding holomorphic germs. In particular, we have an
equality

$0=g^{\Lambda}_{W}(\Psi,\Psi^{\prime})-g^{\Lambda}_{W}(\Phi,\Psi^{\prime})=g^{\Lambda}_{W}(\Psi-\Phi,\Psi^{\prime}).$
(7.35)

The equality holds for all $Q_{W}$-closed states $\Psi^{\prime}$ and pairing
$g_{W}^{\Lambda}$ is non-degenerate on $H^{\ast}(Q_{W},V_{B})$, hence $\Psi$
and $\Phi$ represent the same class in $Q_{W}$-cohomology, i.e.

$\Psi=\Phi\in H^{\ast}(Q_{W},V_{B}),$ (7.36)

what completes the proof of the proposition. $\hfill\blacksquare$

###  7.4 Higher pairings in B-model

Definition: For a B-model with superpotential $W$ we can introduce a
$\mathbb{C}[[z]]$-valued pairing on B-model states

$\mathcal{K}_{W}(\Psi_{1},\Psi_{2})=\int d\mu\;\Psi_{1}\cdot
e^{\Lambda\\{Q_{W}+zG_{-},L\\}}\Psi_{2}=g\left(\Psi_{1},e^{\Lambda\\{Q_{W}+zG_{-},L\\}}\Psi_{2}\right),$
(7.37)

with the integration measure (7.24), real parameter $\Lambda$ and localization
function

$L=\sum_{k=1}^{N}r^{k}\psi^{k}_{\Phi}.$ (7.38)

The pairing $\mathcal{K}_{W}$ matches with the B-model pairing (6.19) for
$\Lambda=0$. The pairing $\mathcal{K}_{W}$ is a pairing between
$H^{\ast}(Q_{W}-zG_{-})$ and $H^{\ast}(Q_{W}+zG_{-})$. Indeed, the conjugation
formula

$\begin{split}\mathcal{K}_{W}((Q_{W}-zG_{-})\Psi_{1},\Psi_{2})&=g\left((Q_{W}-zG_{-})\Psi_{1},e^{\Lambda\\{Q_{W}+zG_{-},L\\}}\Psi_{2}\right)\\\
&=-(-1)^{|\Psi_{1}|}g\left(\Psi_{1},(Q_{W}+zG_{-})e^{\Lambda\\{Q_{W}+zG_{-},L\\}}\Psi_{2}\right)\\\
&=-(-1)^{|\Psi_{1}|}g\left(\Psi_{1},e^{\Lambda\\{Q_{W}+zG_{-},L\\}}(Q_{W}+zG_{-})\Psi_{2}\right)\\\
&=-(-1)^{|\Psi_{1}|}\mathcal{K}_{W}(\Psi_{1},(Q_{W}+zG_{-})\Psi_{2}).\end{split}$
(7.39)

Lemma: On $H^{\ast}(Q_{W}-zG_{-})\otimes H^{\ast}(Q_{W}+zG_{-})$ the pairing
(7.37) is independent on $\Lambda$.

Proof: The derivative of the pairing evaluates into

$\begin{split}\frac{d}{d\Lambda}&\mathcal{K}_{W}(\Psi_{1},\Psi_{2})=\frac{d}{d\Lambda}g(\Psi_{1},e^{\Lambda\\{Q_{W}+zG_{-},L\\}}\Psi_{2})=g(\Psi_{1},\\{Q_{W}+zG_{-},L\\}e^{\Lambda\\{Q_{W}+zG_{-},L\\}}\Psi_{2})\\\
&=g(\Psi_{1},(Q_{W}+zG_{-})Le^{\Lambda\\{Q_{W}+zG_{-},L\\}}\Psi_{2})+g(\Psi_{1},L(Q_{W}+zG_{-})e^{\Lambda\\{Q_{W}+zG_{-},L\\}}\Psi_{2})\\\
&=-(-1)^{|\Psi_{1}|}g((Q_{W}-zG_{-})\Psi_{1},Le^{\Lambda\\{Q_{W}+zG_{-},L\\}}\Psi_{2})+g(\Psi_{1},Le^{\Lambda\\{Q_{W}+zG_{-},L\\}}(Q_{W}+zG_{-})\Psi_{2})\\\
&=0.\end{split}$

In the relation we used $Q_{W}$-invariance of the B-model pairing (4.11),
$G_{-}$-invariance of the B-model pairing (4.12) an equality

$(Q_{W}+zG_{-})e^{\Lambda\\{Q_{W}+zG_{-},L\\}}=e^{\Lambda\\{Q_{W}+zG_{-},L\\}}(Q_{W}+zG_{-}).$
(7.40)

to complete the proof. $\hfill\blacksquare$

The $\Lambda$-independence means that the pairing $\mathcal{K}_{W}$ matches
with the B-model pairing $g$ on $H^{\ast}(Q_{W},\ker G_{-})$. The B-model
pairing $g$ is non-degenerate on $V_{B}$ and $Q_{W}$-, $G_{-}$-invariant,
hence $g$ is non-degenerate on $H^{\ast}(Q_{W},\ker G_{-})$, so is the
$\mathcal{K}_{W}$.

Remark: We can introduce an expansion for the pairing in $z$ to define higher
pairings

$\mathcal{K}_{W}(\Psi_{1},\Psi_{2})=\sum_{k=0}^{\infty}z^{k}\;\mathcal{K}_{W}^{(k)}(\Psi_{1},\Psi_{2}),$
(7.41)

such that the $\mathcal{K}_{W}^{(0)}$ is identical to the $g^{\Lambda}_{W}$
from the section 7.3.

The argument of exponent in (7.37) is a sum of two terms: a function

$\\{Q_{W},L\\}=Q_{W}(L)=\sum_{k=1}^{N}\psi^{k}_{\Phi}\psi^{k}_{R}+2\pi
i\sum_{k=1}^{N}r^{k}\frac{\partial W}{\partial Y_{k}}$ (7.42)

and first-order differential operator

$\\{G_{-},L\\}=-i\sum_{k=1}^{N}r^{k}\frac{\partial}{\partial Y_{k}}.$ (7.43)

We can use the Zassenhaus formula

$e^{A+B}=e^{A}\;e^{B}\;e^{-\frac{1}{2}[A,B]}\;e^{\frac{1}{6}(2[B,[A,B]]+[A,[A,B]])}\dots$
(7.44)

to express the localization exponent as product of the $Q_{W}(L)$-localizaton
exponent and differential operator with coefficients in $z$, i.e.

$\begin{split}e^{\Lambda\\{Q_{W}+zG_{-},L\\}}=e^{\Lambda
Q_{W}(L)}\cdot\mathcal{D}(z,\Lambda,r,Y,\partial_{Y}).\end{split}$ (7.45)

The representation (7.45) allows us to conclude that the higher pairings also
localize to $\psi=r=0$, i.e. the value of the pairing $\mathcal{K}_{W}$ is the
same for $Q_{W}$-, $G_{-}$-closed states $\Psi,\Psi^{\prime}$ and their
holomorphic germs $\Phi,\Phi^{\prime}$, i.e.

$\mathcal{K}_{W}(\Psi,\Psi^{\prime})=\mathcal{K}_{W}(\Phi,\Phi^{\prime}).$
(7.46)

The localization exponent to first order in $z$

$\begin{split}e^{\Lambda\\{Q_{W}+zG_{-},L\\}}=e^{\Lambda Q_{W}(L)}\left(1+\pi
z\Lambda^{2}\;\sum_{k,l=1}^{N}r^{k}r^{l}\frac{\partial^{2}W}{\partial
Y_{k}\partial Y_{l}}-iz\Lambda\sum_{k=1}^{N}r^{k}\frac{\partial}{\partial
Y_{k}}+\mathcal{O}(z^{2})\right)\end{split}$ (7.47)

allows us to evaluate the $\mathcal{K}_{W}^{(1)}$-pairing

$\begin{split}\mathcal{K}_{W}^{(1)}(\Psi_{1},\Psi_{2})&=\int
d\mu\;\;\Psi_{1}\cdot\;e^{\Lambda
Q_{W}(L)}\left(\pi\Lambda^{2}\;r^{k}r^{l}\frac{\partial^{2}W}{\partial
Y_{k}\partial Y_{l}}-i\Lambda r^{k}\frac{\partial}{\partial
Y_{k}}\right)\Psi_{2}\\\
&=-\pi\sum_{dW=0}\;\;(\partial_{Y_{k}}\partial_{Y_{l}}W)^{-1}\frac{\left(\Psi_{1}\partial_{Y_{k}}\partial_{Y_{l}}\Psi_{2}-\Psi_{2}\partial_{Y_{k}}\partial_{Y_{l}}\Psi_{1}\right)}{\det\partial_{Y_{k}}\partial_{Y_{l}}W}\Big{|}_{r=\psi=0}.\end{split}$
(7.48)

Remark: In case of single $Y$-variable the pairing simplifies into

$\begin{split}\mathcal{K}_{W}^{(1)}(\Psi_{1},\Psi_{2})&=-2\pi\sum_{W^{\prime}=0}\;\;\frac{1}{2}\frac{\Psi_{1}\Psi_{2}^{\prime\prime}-\Psi_{1}^{\prime\prime}\Psi_{2}}{(W^{\prime\prime})^{2}}\Big{|}_{r=\psi=0}.\end{split}$
(7.49)

Remark: The pairings $\mathcal{K}^{(0)}_{W}$ and $\mathcal{K}^{(1)}_{W}$ on
holomorphic functions match with the corresponding higher pairing components
(3.26) and (3.27) for the LGS theory.

In proposition (7.30) we showed that $Q_{W}$-closed B-model state $\Psi$ and
its holomorphic germ represent the same class in $H^{\ast}(Q_{W},V_{B})$. The
pairing $\mathcal{K}_{W}$ allows us to make a stronger statement:

Proposition: The $Q_{W}$-, $G_{-}$-closed state $\Psi$ and its holomorphic
germ $\Phi$ are in the same class of $H^{\ast}(Q_{W}+zG_{-},V_{B})$ and there
exists a tropical form ($z$-independent) $\chi$ such that

$\Psi=\Phi+Q_{W}\chi,\;\;\;G_{-}\chi=0.$ (7.50)

Proof: The pairing of both representatives with arbitrary $\Psi^{\prime}\in
H^{\ast}(Q_{W}-zG_{-})$ are identical i.e.

$\mathcal{K}_{W}(\Psi^{\prime},\Psi)=\mathcal{K}_{W}(\Psi^{\prime},\Phi)\;\;\Longrightarrow\mathcal{K}_{W}(\Psi^{\prime},\Psi-\Phi)=0.$
(7.51)

The non-degeneracy of the pairing leads to

$\Psi-\Phi=0\in H^{\ast}(Q_{W}+zG_{-}),$ (7.52)

what implies the the existence of the tropical form $\chi$ such that

$\Psi-\Phi=(Q_{W}+zG_{-})\chi.$ (7.53)

By construction $\Psi$ and $\Phi$ are independent of $z$ and we can choose
$\chi$, which is $z$-independent, hence $G_{-}\chi=0$, what concludes the
proof. $\hfill\blacksquare$

Example: The holomorphic germ for the $\mathbb{P}^{1}$-mirror state (6.39) is

$\Phi_{\gamma}=\Psi^{\mathbb{P}^{1}}_{\gamma}\Big{|}_{r=\psi=0}=q_{+}e^{iY}\Theta(-r_{0})+q_{-}e^{-iY}\Theta(r_{0}).$
(7.54)

For $r_{0}<0$ it coincides with the $\Psi_{+}$-divisor, while for $r_{0}>0$ it
coincides with $\Psi_{-}$-divisor. For both cases we showed that the
corresponding tropical forms $\chi_{\pm}$ are $G_{-}$-closed.

###  7.5 B-model deformation by a holomorphic function

In this section we will describe a one-parameter family of deformations for
the B-type HTQM $(V_{B},Q_{W},G_{\pm},\mu_{2},g)$ by a holomorphic function
$\Phi$. The holomorphic function belongs to the B-model space of states, hence
we can use construction from section 4.5 for the leading order deformation

$\begin{split}Q^{\epsilon}_{W}&=Q_{W}-2\pi[G_{-},\mu_{2}({\epsilon}\Phi,\cdot)]=Q_{W}+{\epsilon}\;2\pi
i\frac{\partial\Phi}{\partial
Y_{k}}\frac{\partial}{\partial\psi^{k}_{\Phi}}\\\ &=Q+2\pi i\frac{\partial
W}{\partial Y_{k}}\frac{\partial}{\partial\psi^{k}_{\Phi}}+2\pi
i{\epsilon}\;\frac{\partial\Phi}{\partial
Y_{k}}\frac{\partial}{\partial\psi^{k}_{\Phi}}=Q_{W^{\epsilon}}.\end{split}$
(7.55)

The last equality allows us to describe the deformed differential
$Q^{\epsilon}_{W}$ in the form of differential in B-model with deformed
superpotential

$W^{\epsilon}=W+{\epsilon}\;\Phi.$ (7.56)

The deformation of a B-model state is defined as (a finite) expansion in $W$
and A-model propagators $K$

$\begin{split}\Psi^{\epsilon}&=\Psi+2\pi
K_{W}G_{-}\mu_{2}({\epsilon}\Phi,\Psi)+\mathcal{O}({\epsilon}^{2})\\\
&=\Psi+2\pi KG_{-}\mu_{2}({\epsilon}\Phi,\Psi)+2\pi KG_{-}\mu_{2}(W,2\pi
KG_{-}\mu_{2}({\epsilon}\Phi,\Psi))...+\mathcal{O}({\epsilon}^{2}).\end{split}$
(7.57)

The deformation of a mirror state $\Psi=\Psi_{\gamma}^{W}$ simplifies into

$\begin{split}[\Psi_{\gamma}^{W}]^{\epsilon}&=\Psi_{\gamma}^{W}+2\pi
K_{W}G_{-}\mu_{2}({\epsilon}\Phi,\Psi_{\gamma}^{W})+\mathcal{O}({\epsilon}^{2})\\\
&=\Psi_{\gamma}+2\pi KG_{-}\mu_{2}(W,\Psi_{\gamma})+2\pi
KG_{-}\mu_{2}({\epsilon}\Phi,\Psi_{\gamma})\\\ &\;\;\;\;\;\;\;\;\;+2\pi
KG_{-}\mu_{2}(W,2\pi KG_{-}\mu_{2}(W,\Psi_{\gamma}))+2\pi
KG_{-}\mu_{2}({\epsilon}\Phi,2\pi KG_{-}\mu_{2}(W,\Psi_{\gamma}))\\\
&\;\;\;\;\;\;\;\;\;+2\pi KG_{-}\mu_{2}(W,2\pi
KG_{-}\mu_{2}({\epsilon}\Phi,\Psi_{\gamma}))+...+\mathcal{O}({\epsilon}^{2})\\\
&=\Psi_{\gamma}+2\pi KG_{-}\mu_{2}(W+{\epsilon}\Phi,\Psi^{W}_{\gamma})\\\
&\;\;\;\;\;\;\;\;\;+2\pi KG_{-}(W+{\epsilon}\Phi,2\pi
KG_{-}\mu_{2}(W+{\epsilon}\Phi,\Psi^{W}_{\gamma}))+...+\mathcal{O}({\epsilon}^{2})\\\
&=\Psi_{\gamma}^{W_{\epsilon}}\end{split}$

Hence we conclude that the deformation $[\Psi_{\gamma}^{W}]^{\epsilon}$ of the
mirror state $\Psi_{\gamma}^{W}$ by a holomorphic function in B-model with
superpotential $W$ is a mirror state $\Psi_{\gamma}^{W_{\epsilon}}$ for the
same ${\gamma}$ but in B-model with deformed superpotential $W^{\epsilon}$.

###  7.6 Higher pairing for mirror states

The holomorphic germs for mirror states in B-model can be used to construct a
good section in the corresponding LGS theory.

Definition: The tropical good section in mirror LGS theory for toric space $X$
with superpotential $W$ is a linear span of holomorphic germs for mirror
states

$\hbox{Im}\;S^{trop}_{W}=\mathbb{C}\langle\Phi^{W}_{\gamma}\;|\;{\gamma}\in
H_{dR}^{\ast}(X)\rangle.$ (7.58)

Remark: A similar construction of the good section from holomorphic germs of
harmonic states was described in section 7.2 and 7.3 of [20].

The definition (3.36) of a good section in LGS requires vanishing of higher
pairings.

Proposition (higher pairing for tropical good section): Higher LGS pairings
(3.21) for tropical good section vanish. Moreover

${\bf
K}_{W}(\Phi^{W}_{{\gamma}_{1}},\Phi^{W}_{{\gamma}_{2}})=\int_{X}\;{\gamma}_{1}\wedge{\gamma}_{2},\;\;\;\;\;\;\forall{\gamma}_{1},{\gamma}_{2}\in
H_{dR}^{\ast}(X).$ (7.59)

Proof: By construction in (6.29) the mirror states $\Psi_{\gamma}^{W}$ are
$G_{-}$\- and $Q_{W}$-closed, hence $\Psi_{\gamma}^{W}\in H^{\ast}(Q_{W}\pm
zG_{-})$ and the higher B-model pairing (7.37) for such states is independent
of $\Lambda$ so we can relate it at different values of $\Lambda$. In
particular, we use

  * •

$\Lambda\to\infty$ limit gives us the LG pairing for holomorphic functions
i.e.

$\mathcal{K}_{W}(\Psi^{W}_{{\gamma}_{1}},\Psi^{W}_{{\gamma}_{2}})=\mathcal{K}_{W}(\Phi^{W}_{{\gamma}_{1}},\Phi^{W}_{{\gamma}_{2}})={\bf
K}_{W}(\Phi^{W}_{{\gamma}_{1}},\Phi^{W}_{{\gamma}_{2}});$ (7.60)

  * •

$\Lambda=0$ gives us the B-model pairing $g$ i.e.

$\mathcal{K}_{W}(\Psi^{W}_{{\gamma}_{1}},\Psi^{W}_{{\gamma}_{2}})=g(\Psi^{W}_{{\gamma}_{1}},\Psi^{W}_{{\gamma}_{2}}).$
(7.61)

The mirror states $\Psi^{W}_{{\gamma}_{k}}$ can be written in the following
schematic form

$\Psi^{W}_{{\gamma}}=\Psi_{{\gamma}}+G_{-}\chi^{W}_{{\gamma}}$ (7.62)

for the A-model state

$\chi^{W}_{{\gamma}}=-2\pi K\mu_{2}(W,\Psi_{\gamma})-2\pi K\mu_{2}(W,2\pi
KG_{-}\mu_{2}(W,\Psi_{\gamma}))+\dots$ (7.63)

The B-model pairing $g$ on representation (7.62) simplifies into

$\begin{split}g(\Psi^{W}_{{\gamma}_{1}},\Psi^{W}_{{\gamma}_{2}})&=g(\Psi_{{\gamma}_{1}},\Psi_{{\gamma}_{2}})+g(\Psi_{{\gamma}_{1}},G_{-}\chi^{W}_{{\gamma}_{2}})+g(G_{-}\chi_{{\gamma}_{1}}^{W},\Psi_{{\gamma}_{2}})+g(G_{-}\chi_{{\gamma}_{1}}^{W},G_{-}\chi_{{\gamma}_{2}}^{W})\\\
&=g(\Psi_{{\gamma}_{1}},\Psi_{{\gamma}_{2}})+g(G_{-}\Psi_{{\gamma}_{1}},\chi^{W}_{{\gamma}_{2}})-g(\chi_{{\gamma}_{1}}^{W},G_{-}\Psi_{{\gamma}_{2}})-g(\chi_{{\gamma}_{1}}^{W},G^{2}_{-}\chi_{{\gamma}_{2}}^{W})\\\
&=g(\Psi_{{\gamma}_{1}},\Psi_{{\gamma}_{2}}).\end{split}$ (7.64)

We used the $G_{\pm}$-invariance of the pairing (4.12) to simplify the
expression

The A-model pairing on evaluation states $\Psi_{\gamma}$ is the intersection
of the cohomology classes on $X$ i.e.

$g(\Psi_{{\gamma}_{1}},\Psi_{{\gamma}_{2}})=\int_{X}\;{\gamma}_{1}\wedge{\gamma}_{2}.$
(7.65)

The proof of the proposition is the following: We use (7.60) to turn LGS
higher pairing into the B-model higher pairing on mirror states. We use (7.61)
to turn the B-model higher pairing into ordinary pairing which according to
(7.64) simplifies to the intersection of cohomology classes. The higher
parings are $z$-expansion of ${\bf K}_{W}$ hence

${\bf
K}^{(k)}_{W}(\Phi_{{\gamma}_{1}}^{W},\Phi^{W}_{{\gamma}_{2}})=0,\;\;\forall\;k>0.$
(7.66)

what completes the proof of the proposition. $\hfill\blacksquare$

Example: The toric description of $\mathbb{P}^{2}$ consists of three
compactifying divisors with primitive normal vectors: $(1,0)$, $(0,1)$ and
$(-1,-1)$ so the mirror superpotential

$W_{\mathbb{P}^{2}}=e^{iY_{1}}+e^{iY_{2}}+qe^{-iY_{1}-iY_{2}}$ (7.67)

The de Rham cohomology of $\mathbb{P}^{2}$ are one-dimensional in degrees 0,
2, and 4. The image of tropical good section is

$\hbox{Im}\;S^{trop}_{\mathbb{P}^{2}}=\mathbb{C}\langle
1,qe^{-iY_{1}-iY_{2}},e^{iY_{1}+iY_{2}}\rangle.$ (7.68)

##  8 Correlation functions for mirror states

The correlation function invariance theorem from section 5.4 tells us that

$\langle\Psi_{1},...,\Psi_{n},Q_{X}\chi\rangle_{Q_{X}}=0$ (8.1)

for $Q_{X}$\- $G_{-}$-closed states $\Psi_{a}$ and a $G_{-}$-closed state
$\chi$. In previous section we showed that there exists a tropical form $\chi$
such that it can turn a mirror state into a holomorphic germ. Moreover, in
section 7.5 we saw that the deformation of B-model by a holomorphic function
leaves it in the same class but with different superpotential. Hence, our
strategy for evaluating correlation functions of mirror states in B-model will
be the following:

  1. 1.

Replace one mirror state by the corresponding holomorphic germ.

  2. 2.

Deform the superpotential and mirror states by the holomorphic germ.

  3. 3.

Apply the recursion formula for correlation functions to reduce the number of
arguments by one.

  4. 4.

Repeat steps 1-3 till the number of arguments reaches 3.

The simplification of the theory deformation while using the holomorphic germ
rather than the mirror state itself suggest an alternative strategy for the
correlation function evaluation: We replace all mirror states by the
corresponding holomorphic germs and evaluate the correlation function. The
application of this strategy to the 4pt functions immediately gives zero
answer. The reason for that is our definition of the B-model amplitudes via an
expansion in A-model amplitudes. In A-model the propagator acts trivially on
the holomorphic functions what is the source of zero answers.

To resolve the puzzle we recall that the A-model amplitudes for tropical GW
invariants have a particular number of divisor states, determined by the
degree. The mirror states can provide additional divisor states, but if we
already have too many (from holomorphic germs) the amplitude vanishes due to
the degree selection for the tropical GW invariants. Since, all non-trivial GW
invariants have degree bigger than zero, the corresponding A-model amplitudes
have at least one divisor state, hence we can always replace a single mirror
state by the holomorphic germ. Later we will see that a single replacement is
just enough to prove our main theorem.

###  8.1 4-point function invariance and holomorphic representatives

Using the symmetries of the amplitudes we can rewrite the 4pt function in the
form

$\begin{split}2&\langle\Psi_{1},\Psi_{2},\Psi_{3},\Psi_{4}\rangle_{Q_{W}}=g(\Psi_{4},\sum_{\sigma\in
S_{3}}\mu_{2}(\Psi_{\sigma(1)},K_{W}G_{-}\mu_{2}(\Psi_{\sigma(2)},\Psi_{\sigma(3)})))\\\
&=g(\Psi_{4},\Psi_{123})=g(\Phi_{4}+Q_{W}\chi_{+},\Psi_{123})=g(\Phi_{4},\Psi_{123})=2\langle\Psi_{1},\Psi_{2},\Psi_{3},\Phi_{4}\rangle_{Q_{W}}\end{split}$
(8.2)

The equality on the second line requires

$g(Q_{W}\chi_{+},\Psi_{123})=0.$ (8.3)

In (7.7) we showed that $\chi$ for $\mathbb{P}^{1}$-model does not belong to
tropical forms on $\mathbb{P}^{1}$, hence the difference

$g(Q_{W}\chi_{+},\Psi_{123})-g(\chi_{+},Q_{W}\Psi_{123})$ (8.4)

might not be zero. The difference is controlled by the boundary term of the
$r$-integration

$\begin{split}g&(Q_{W}\chi_{+},\Psi_{123})-g(\chi_{+},Q_{W}\Psi_{123})=\int
d\mu\;Q(\chi_{+}\Psi_{123})=\int d\mu\;d_{R}(\chi_{+}\Psi_{123})\\\
&=\lim_{r\to-\infty}\int_{S^{1}}dY\;\Theta(r_{4}-r)\
\Psi_{123}\Big{|}_{\psi=0}=\lim_{r\to-\infty}\int_{S^{1}}dY\;\Psi_{123}\Big{|}_{\psi=0}.\end{split}$
(8.5)

We can repeat the analysis for the case different holomorphic representative
$\Phi_{4}^{\prime}=qe^{-iY}$ and $\chi_{-}$

$\begin{split}g&(Q_{W}\chi_{-},\Psi_{123})-g(\chi_{-},Q_{W}\Psi_{123})=\lim_{r\to+\infty}\int_{S^{1}}dY\;\Psi_{123}\Big{|}_{\psi=0}.\end{split}$
(8.6)

Both boundary terms vanish if $Y$-independent part of
$\Psi_{123}\Big{|}_{\psi=0}$ has finite support. We can evaluate the
$\Psi_{123}$ for $\mathbb{P}^{1}$-mirror states

$\begin{split}\int_{S^{1}}&dY\;\Psi_{123}\Big{|}_{\psi=0}=\frac{1}{2}\sum_{\sigma\in
S_{3}}\int_{S^{1}}dY\;\mu_{2}(\Psi_{\sigma(3)},2\pi
K_{W}G_{-}\mu_{2}(\Psi_{\sigma(1)},\Psi_{\sigma(2)}))\Big{|}_{\psi=0}\\\ &=\pi
q\sum_{\sigma\in
S_{3}}\left[\Theta(\min(r_{\sigma(1)},r_{\sigma(2)})-r))\Theta(r-r_{\sigma(3)})+\Theta(r-\max(r_{\sigma(1)},r_{\sigma(2)}))\Theta(r_{\sigma(3)}-r)\right]\end{split}$

and observe that the products of $\Theta$-functions indeed have the finite
support.

We can perform the same analysis for
$\langle\Psi_{1},\Psi_{2},\Phi_{3},\Psi_{4}\rangle_{Q_{W}}$, where we replaced