# Integrable systems associated to open extensions of type A and D
Dubrovin–Frobenius manifolds

Alexey Basalaev  A. Basalaev:
Faculty of Mathematics, National Research University Higher School of
Economics, Usacheva str., 6, 119048 Moscow, Russian Federation, and
Skolkovo Institute of Science and Technology, Nobelya str., 3, 121205 Moscow,
Russian Federation<EMAIL_ADDRESS>
###### Abstract.

We investigate the solutions to open WDVV equation, associated to type A and D
Dubrovin–Frobenius manifolds. We show that these solutions satisfy some
stabilization condition and associate to both of them the systems of commuting
PDEs. In the type A we show that the system of PDEs constructed coincides with
the dispersionless modifiled KP hierarchy written in the Fay form.

##  1\. Introduction

Given a Dubrovin–Frobenius manifold, there are several constructions of an
integrable system associated to it (cf. [DZ], [GM] and [B2]). In particular,
it was proved that for type $A_{N}$ and $D_{N}$ Dubrovin–Frobenius manifolds
all these constructions provide the corresponding Drinfeld–Sokolov hierarchies
(cf. [LRZ]). The $A_{N}$ Drinfeld–Sokolov hierarchy can be defined as the
reductions of the KP and type $D_{N}$ Drinfeld–Sokolov hierarchy can be
defined as a 1–component reduction of the 2–component BKP hierarchies.

In [BDbN] the authors proposed a new construction of a system of commuting
PDEs associated to the family of $A$–type and $D$–type Dubrovin–Frobenius
manifolds. One of the main properties of this construction was the following:
consistency of the system of PDEs constructed was derived from WDVV equation
on the potential of a Dubrovin–Frobenius manifold. Comparing to the previous
approaches this construction associated dispersionless KP hierarchy and
dispersionless 1-component reduced 2–component BKP hierarchy in $A$ and $D$
types respectively.

###  1.1. Open WDVV equation

Motivated by studies of the open Gromov–Witten theories the new system of PDEs
called open WDVV was introduced in [HS12]. Given a solution
$F^{c}=F^{c}(t_{1},\dots,t_{N})$ to WDVV equation with the metric $\eta$, open
WDVV equation on a function $F^{o}=F^{o}(t_{0},t_{1},\dots,t_{N})$ reads

(1.1)
$\displaystyle\frac{{\partial}^{3}F^{c}}{{\partial}t_{\alpha}{\partial}t_{\beta}{\partial}t_{\mu}}\eta^{\mu\nu}\frac{{\partial}^{2}F^{o}}{{\partial}t_{\nu}{\partial}t_{\gamma}}+\frac{{\partial}^{2}F^{o}}{{\partial}t_{\alpha}{\partial}t_{\beta}}\frac{{\partial}^{2}F^{o}}{{\partial}t_{0}{\partial}t_{\gamma}}=\frac{{\partial}^{3}F^{c}}{{\partial}t_{\gamma}{\partial}t_{\beta}{\partial}t_{\mu}}\eta^{\mu\nu}\frac{{\partial}^{2}F^{o}}{{\partial}t_{\nu}{\partial}t_{\alpha}}+\frac{{\partial}^{2}F^{o}}{{\partial}t_{\gamma}{\partial}t_{\beta}}\frac{{\partial}^{2}F^{o}}{{\partial}t_{0}{\partial}t_{\alpha}},$

for any fixed $0\leq\alpha,\beta,\gamma\leq N$.

Similarly to “classical” WDVV equation, open WDVV equation is associativity
equation of the product defined by

$\frac{{\partial}}{{\partial}t_{\alpha}}\circ\frac{{\partial}}{{\partial}t_{\beta}}=\sum_{\gamma,\delta=1}^{N}\frac{{\partial}^{3}F^{c}}{{\partial}t_{\alpha}{\partial}t_{\beta}{\partial}t_{\gamma}}\eta^{\gamma\delta}\frac{{\partial}}{{\partial}t_{\delta}}+\frac{{\partial}^{2}F^{o}}{{\partial}t_{\alpha}{\partial}t_{\beta}}\frac{{\partial}}{{\partial}t_{0}},\quad
0\leq\alpha,\beta\leq N.$

From this point of view the function $F^{o}$, solving open WDVV equation,
defined an open extensions of a Dubrovin–Frobenius manifold given by $F^{c}$.

For $F^{c}$ being Dubrovin–Frobenius potential of type $A_{N}$ or $D_{N}$ the
solutions to open WDVV equation were investigated in [BB2]. In particular, for
the $A_{N}$ and $D_{N}$ Dubrovin–Frobenius potentials $F^{c}_{A_{N}}$ and
$F^{c}_{D_{N}}$ the authors constructed the functions $F^{o}_{A_{N}}$ and
$F^{o}_{D_{N}}$, being solutions to open WDVV equation. The function
$F^{o}_{A_{N}}$ appeared to be polynomial in $t_{0},t_{1},\dots,t_{N}$ and
$F^{o}_{D_{N}}$ polynomial in $t_{1},\dots,t_{N}$, but Laurent polynomial in
$t_{0}$.

The integrable hierarchies associated to the solutions of open WDVV equations
were investigated in [BCT1, BCT2] and [A]. In all these references only
$A_{N}$ type was assumed.

###  1.2. System of commuting PDEs

Extending the approach of [BDbN] we associate to the families
$\\{(F^{c}_{A_{N}},F^{o}_{A_{N}})\\}_{N\geq 1}$ and
$\\{(F^{c}_{D_{N}},F^{o}_{D_{N}})\\}_{N\geq 4}$ the systems of commuting PDEs.
Like in loc.cit. the first step on this way is the following stabilization
result.

In the A case for any $N<M$ and $1\leq\alpha\leq N$ we have

$\frac{{\partial}F_{A_{N}}^{o}}{{\partial}t_{\alpha}}\mid_{t_{N+1-\beta}=s_{\beta}}\
=\frac{{\partial}F_{A_{M}}^{o}}{{\partial}t_{\alpha}}\mid_{t_{M+1-\beta}=s_{\beta}},$

assumed as polynomials in $t_{0},s_{1},\dots,s_{M+1}$. In the D case for any
$4\leq N<M$ and $1\leq\alpha<N$ we have

$\displaystyle\frac{{\partial}F_{D_{N}}^{o}}{{\partial}t_{\alpha}}\mid_{t_{N-\beta}=s_{\beta},\
t_{N}=\bar{s}_{1}}\
=\frac{{\partial}F_{D_{M}}^{o}}{{\partial}t_{\alpha}}\mid_{t_{M-\beta}=s_{\beta},\
t_{M}=\bar{s}_{1}},$
$\displaystyle\frac{{\partial}F_{D_{N}}^{o}}{{\partial}t_{N}}\mid_{t_{N-\beta}=s_{\beta},\
t_{N}=\bar{s}_{1}}\
=\frac{{\partial}F_{D_{M}}^{o}}{{\partial}t_{M}}\mid_{t_{M-\beta}=s_{\beta},\
t_{M}=\bar{s}_{1}},$

assumed as polynomials in $\bar{s}_{1},s_{1},\dots,s_{M+1}$ and Laurent
polynomials in $t_{0}$.

Both systems of equations that we write express the higher derivatives
${\partial}_{\alpha}{\partial}_{\beta}f$ of a solution $f$ via
${\partial}_{1}{\partial}_{0}f,{\partial}_{1}{\partial}_{1}f,{\partial}_{1}{\partial}_{2}f,\dots$
that may be viewed as “initial data”.

For $A$ type this is the system of PDEs on a function
$f=f(t_{0},t_{1},t_{2},\dots)$

$\displaystyle{\partial}_{\alpha}{\partial}_{\beta}f$
$\displaystyle=\frac{{\partial}^{2}F^{c}_{A_{\kappa}}}{{\partial}t_{\alpha}{\partial}t_{\beta}}\mid_{t_{\kappa+1-\gamma}={\partial}_{1}{\partial}_{\gamma}f},\qquad\kappa=\alpha+\beta+1.$
$\displaystyle{\partial}_{0}{\partial}_{\alpha}f$
$\displaystyle=\frac{{\partial}F^{o}_{A_{\kappa}}}{{\partial}t_{\alpha}}\mid_{t_{\kappa+1-\gamma}={\partial}_{1}{\partial}_{\gamma}f,\
t_{0}={\partial}_{1}{\partial}_{0}f}$

where the abbreviate ${\partial}_{\alpha}={\partial}/{\partial}t_{\alpha}$. We
show in Section 3 that this system is well–defined and consistent. We also
prove in Theorem 3.5 that this system of equations coincides with the
dispersionless modified KP hierarchy written in the Fay form. We also provide
the construction extending new type A system to the full mKP hierarchy.

For D type we introduce the system of PDEs on
$f=f(\bar{t}_{1},t_{0},t_{1},\dots)$

$\displaystyle{\partial}_{\alpha}{\partial}_{\beta}f$
$\displaystyle=\frac{{\partial}^{2}F^{c}_{D_{\kappa}}}{{\partial}t_{\alpha}{\partial}t_{\beta}}\mid_{t_{\kappa-\gamma}={\partial}_{1}{\partial}_{\gamma}f,\
t_{\kappa}={\partial}_{1}\bar{\partial}_{1}f},\qquad\kappa=\alpha+\beta-1.$
$\displaystyle{\partial}_{\alpha}\bar{\partial}_{1}f$
$\displaystyle=\frac{{\partial}^{2}F^{c}_{D_{\kappa}}}{{\partial}t_{\alpha}{\partial}t_{\kappa}}\mid_{t_{\kappa-\gamma}={\partial}_{1}{\partial}_{\gamma}f,\
t_{\kappa}={\partial}_{1}\bar{\partial}_{1}f},$
$\displaystyle{\partial}_{0}{\partial}_{\alpha}f$
$\displaystyle=\frac{{\partial}F^{o}_{D_{\kappa}}}{{\partial}t_{\alpha}}\mid_{t_{\kappa-\gamma}={\partial}_{1}{\partial}_{\gamma}f,\
t_{\kappa}={\partial}_{1}\bar{\partial}_{1}f,\
t_{0}={\partial}_{1}{\partial}_{0}f},\qquad\kappa=\alpha+1,$
$\displaystyle{\partial}_{0}\bar{\partial}_{1}f{\partial}_{0}{\partial}_{1}f$
$\displaystyle={\partial}_{1}\bar{\partial}_{1}f,$

where we abbreviate $\bar{\partial}_{1}={\partial}/{\partial}\bar{t}_{1}$. It
is proved in Theorem 4.2 that this system of equations is well-defined and
consistent.

Like in [BDbN] we derive consistency of both A and D type PDEs from open WDVV
equation above. Comparing to the previously cited references our work is the
first example of the system of PDEs constructed by the open WDVV solutions.

Following the relation of $D_{N}$ type Drinfeld–Sokolov hierarchy and also D
type hierarchy of [BDbN] to the 2–component BKP hierarchy, one would expect
that our D type system is connected to the modified BKP hierarchy. However the
modified version of BKP hierarchy is not yet fully settled.

Acknowledgements. The author acknowledges partial support by International
Laboratory of Cluster Geometry HSE University (RF Government grant, agreement
no. 075-15-2021-608 from 08.06.2021).

##  2\. Open potentials of type A and D

The Dubrovin–Frobenius potentials $F_{A_{N}}^{c}$ and $F^{c}_{D_{N}}$ were
first derived by B.Dubrovin via the geometry of the respective Coxeter groups
(cf. [D1]). Since then these potentials were constructed in the several
different ways by the other authors. We neither give the construction of these
Dubrovin–Frobenius manifolds nor provide the expansion of the respective
potentials referencing the reader to [BDbN].

###  2.1. $A_{N}$ open potential

The genus zero $A_{N}$ open potential $F^{o}_{A_{N}}$ was first found in
[BCT2]. It is a polynomial in $t_{1},\dots,t_{N}$ and $t_{0}$ defined by

$\left.\frac{{\partial}^{m+k}F^{o}_{A_{N}}}{{\partial}t_{\alpha_{1}}\ldots{\partial}t_{\alpha_{m}}({\partial}t_{0})^{k}}\right|_{t_{\ast}=0}=\begin{cases}(m+k-2)!,&\text{if
$\sum_{i=1}^{m}(N+2-\alpha_{i})+k=N+2$},\\\ 0,&\text{otherwise}.\end{cases}$

It is also connected to the $A_{N}$ unfolding coordinates by the following
identity (see [B3])

(2.1)
$\frac{{\partial}F^{o}_{A_{N}}}{{\partial}t_{0}}=\frac{t_{0}^{N+1}}{N+1}+\sum_{k=1}^{N}t_{0}^{k-1}v_{k}^{\mathrm{A}}(t_{1},\dots,t_{N})$

where $v_{\bullet}^{\mathrm{A}}$ are defined by

(2.2) $\displaystyle v_{\gamma}^{\mathrm{A}}$
$\displaystyle=\sum_{\begin{subarray}{c}\alpha_{1},\dots,\alpha_{N}\geq 0\\\
\sum_{k=1}^{N}(N+2-k)\alpha_{k}=N+2-\gamma\end{subarray}}\frac{(|\alpha|+\gamma-2)!}{(\gamma-1)!}\prod_{k=1}^{N}\frac{t_{k}^{\alpha_{k}}}{\alpha_{k}!}$

for $|\alpha|=\sum_{k=1}^{N}\alpha_{k}$.

It follows immediately from the definition above that $F^{o}_{A_{N}}$
satisfies the following quasihomogeneity condition

(2.3)
$\left(\sum_{k=1}^{N}(N+2-k)t_{k}\frac{{\partial}}{{\partial}t_{k}}+t_{0}\frac{{\partial}}{{\partial}t_{0}}\right)F^{o}_{A_{N}}=(N+2)F^{o}_{A_{N}}.$

It was shown in [BB2] that $F^{o}_{A_{N}}$ is the only polynomial satisfying
this quasihomogeneity condition, s.t. $(F^{c}_{A_{N}},F^{o}_{A_{N}})$ is a
solution to open WDVV equation and
${\partial}_{1}{\partial}_{0}F^{o}_{A_{N}}=1,{\partial}_{1}{\partial}_{\alpha}F^{o}_{A_{N}}=0$
for $1\leq\alpha\leq N$.

###  2.2. $D_{N}$ open potential

The genus zero $D_{N}$ open potential $F^{o}_{D_{N}}$ is a polynomial in
$t_{1},\dots,t_{N}$ and Laurent polynomial in $t_{0}$ defined by (cf. [BB2,
Section 5.2])

$\displaystyle
F^{o}_{D_{N}}:=\frac{t_{0}^{2N-1}}{2^{N-2}(2N-1)(2N-2)}+\frac{t_{N}^{2}}{2t_{0}}+\sum_{k=1}^{N-1}\frac{v_{k}^{\mathrm{D}}t_{0}^{2k-1}}{2^{k-1}(2k-1)}$

with the functions $v_{\bullet}^{\mathrm{D}}$ given by (see [BDbN, Corollary
4.6])

(2.4) $\displaystyle v_{b}^{\mathrm{D}}$
$\displaystyle=\sum_{\begin{subarray}{c}\alpha_{1},\dots,\alpha_{N-1}\geq 0\\\
\sum_{k=1}^{N-1}(N-k)\alpha_{k}=N-b\end{subarray}}\frac{(|\alpha|+2b-3)!}{(2b-2)!}\prod_{k=1}^{N-1}\frac{t_{k}^{\alpha_{k}}}{\alpha_{k}!}.$

The Laurent polynomial $F^{o}_{D_{N}}$ satisfies the following
quasihomogeneity condition

(2.5)
$\left(\sum_{k=1}^{N-1}2(N-k)t_{k}\frac{{\partial}}{{\partial}t_{k}}+Nt_{N}\frac{{\partial}}{{\partial}t_{N}}+t_{0}\frac{{\partial}}{{\partial}t_{0}}\right)F^{o}_{D_{N}}=(2N-1)F^{o}_{D_{N}}.$

It is immediate to see that the variable $t_{N}$ plays a special role in
$F^{o}_{D_{N}}$. In particular, the functions $v_{b}^{\mathrm{D}}$ do not
depend on $t_{N}$ and the only non–polynomial summand of $F^{o}_{D_{N}}$ is at
the same time the only appearance of $t_{N}$ in the open potential. This
variable is also special for $F^{c}_{D_{N}}$. In particular, we have (see
[BDbN, Section 3.3])

(2.6)
$F^{c}_{D_{N}}=\frac{1}{2}t_{1}^{2}t_{N-1}+\frac{1}{2}t_{1}t_{N}^{2}+\frac{1}{2}t_{1}\sum_{\alpha,\beta=2}^{N-1}t_{\alpha}t_{N-\alpha}+\phi(t_{2},\dots,t_{N-1})+v_{1}^{\mathrm{D}}\cdot\frac{t_{N}^{2}}{2},$

for some polynomial $\phi$ that does not depend on $t_{N}$ again. This
speciality of $t_{N}$ will result in the special form of the PDEs that we
obtain in the D case.

##  3\. Dispersionless open–closed system of type A

The following stabilization condition was proved in [BDbN, Theorem 4.1]. For
any $M>N\geq 1$ and $\alpha,\beta$, s.t. $1\leq\alpha,\beta\leq N$,
$\alpha+\beta\leq N+1$ we have

$\left.\frac{{\partial}^{2}F^{c}_{A_{N}}}{{\partial}t_{\alpha}{\partial}t_{\beta}}\right|_{\forall\gamma\;t_{N+1-\gamma}=s_{\gamma}}=\left.\frac{{\partial}^{2}F^{c}_{A_{M}}}{{\partial}t_{\alpha}{\partial}t_{\beta}}\right|_{\forall\gamma\;t_{M+1-\gamma}=s_{\gamma}},$

understood as an equality of polynomials in $s_{\bullet}$. We show that it
also holds for $F^{o}_{A_{N}}$.

######  Proposition 3.1.

For any $N<M$ and $1\leq\alpha\leq N$ we have

$\frac{{\partial}F_{A_{N}}^{o}}{{\partial}t_{\alpha}}\mid_{t_{N+1-\beta}=s_{\beta}}=\frac{{\partial}F_{A_{M}}^{o}}{{\partial}t_{\alpha}}\mid_{t_{M+1-\beta}=s_{\beta}},$

assumed as polynomials in $t_{0},s_{1},\dots,s_{M+1}$.

###### Proof.

Note that the change of the variables above does not affect the variable
$t_{0}$ and we can consider $F^{o}_{A_{N}}$ and $F^{o}_{A_{M}}$ as polynomials
in $t_{0}$.

For the summands of $F^{o}_{A_{N}}$ and $F^{o}_{A_{M}}$ involving the variable
$t_{0}$ the statement follows from (2.1) and Lemma 4.2 of [BDbN].

Consider the free terms in $t_{0}$. We have for any $\kappa\geq 1$

$\left.\frac{{\partial}^{m+1}F^{o}_{A_{\kappa}}}{{\partial}t_{\alpha}t_{\kappa+1-\beta_{1}}\ldots{\partial}t_{\kappa+1-\beta_{m}}}\right|_{t_{\ast}=0}=\begin{cases}(m-1)!&\text{
if }\quad m+\sum_{i=1}^{m}\beta_{i}=\alpha,\\\
0,&\text{otherwise}.\end{cases}$

Both the value and the condition above do not depend on $\kappa$ what approves
the statement. ∎

For any $\alpha,\beta\in\mathbb{Z}_{\geq 0}$ and
$\gamma_{\bullet}\in\mathbb{Z}_{\geq 0}$ take $N=\alpha+\beta+1$ and denote
$\bar{x}:=N+1-x$. Set

(3.1) $\displaystyle
R^{\mathrm{A}}_{\alpha,\beta;\gamma_{1},\dots,\gamma_{m}}$
$\displaystyle=\frac{1}{m!}\left.\frac{{\partial}^{m+2}F_{A_{N}}^{c}}{{\partial}t_{\alpha}{\partial}t_{\beta}{\partial}t_{\bar{\gamma}_{1}}\cdots{\partial}t_{\bar{\gamma}_{m}}}\right|_{{\bf
t}=0},$ (3.2) $\displaystyle
R^{\mathrm{A},ext}_{\alpha;\gamma_{1},\dots,\gamma_{m}}$
$\displaystyle=\frac{1}{m!}\left.\frac{{\partial}^{m+1}F_{A_{N}}^{o}}{{\partial}t_{\alpha}{\partial}t_{\bar{\gamma}_{1}}\cdots{\partial}t_{\bar{\gamma}_{m}}}\right|_{{\bf
t}=0}.$

The proposition above and [BDbN, Theorem 4.1] assure that these coefficients
are well–defined for all natural $\alpha,\beta,\gamma_{\bullet}$. Note that
$R^{\mathrm{A}}_{0,\beta;\gamma_{1},\dots,\gamma_{m}}=0$ by construction.

Consider the system of PDEs

(3.3) $\displaystyle{\partial}_{\alpha}{\partial}_{\beta}f$
$\displaystyle=\sum_{m\geq
1}\sum_{\gamma_{\bullet}}R^{\mathrm{A}}_{\alpha,\beta;\gamma_{1},\dots,\gamma_{m}}\prod_{k=1}^{m}{\partial}_{1}{\partial}_{\gamma_{k}}f,$
(3.4) $\displaystyle{\partial}_{0}{\partial}_{\alpha}f$
$\displaystyle=\sum_{m\geq
1}\sum_{\gamma_{\bullet}}R^{\mathrm{A},ext}_{\alpha;\gamma_{1},\dots,\gamma_{m}}\prod_{k=1}^{m}{\partial}_{1}{\partial}_{\gamma_{k}}f.$

One notes immediately that these PDEs coincide with the PDEs presented in
Introduction.

######  Proposition 3.2.

The system (3.3),(3.4) is compatible. The function
$\tilde{f}=F_{A_{N}}^{c}+\int F^{o}_{A_{N}}dt_{0}$ satisfies this system for
$\alpha+\beta\leq N+1$.

###### Proof.

Compatibility of the system written follows from open WDVV equation in the
same way it was proved in [BDbN, Proposition 2.1]. The proof is parallel to
the one given in D case in the proof of Theorem 4.2.

Because of the special dependance of $F^{c}_{A_{N}}$ and $F^{o}_{A_{N}}$ on
the variable $t_{1}$ we have

${\partial}_{1}{\partial}_{\gamma}\tilde{f}=\begin{cases}{\partial}_{1}{\partial}_{\gamma}F^{c}_{A_{N}}=t_{\overline{\gamma}},\quad&\text{
if }\quad\gamma>0,\\\ {\partial}_{1}F^{o}_{A_{N}}=t_{0},\quad&\text{ if
}\quad\gamma=0.\end{cases}$

Let’s show that applied to $\tilde{f}$ assumed both (3.3) and (3.4) just
provide the series expansions of
${\partial}_{\alpha}{\partial}_{\beta}F^{c}_{A_{N}}$ and
${\partial}_{\alpha}F^{o}_{A_{N}}$.

It follows from (2.3) that
${\partial}_{\alpha}{\partial}_{\beta}F^{o}_{A_{N}}\equiv 0$ whenever
$\alpha+\beta\leq N+1$. The function $F^{c}_{A_{N}}$ does not depend on
$t_{0}$ and therefore (3.3) holds for $\tilde{f}$.

Similarly
${\partial}_{0}{\partial}_{\alpha}\tilde{f}={\partial}_{\alpha}F^{o}_{A_{N}}$
what approves (3.4) for $\tilde{f}$. ∎

###  3.1. The flows

For any $f=f(t_{0},t_{1},t_{2},\dots)$ denote
$p_{k}:={\partial}_{1}{\partial}_{k}f$. The first flows read

$\displaystyle{\partial}_{0}{\partial}_{1}f$ $\displaystyle=p_{0},$
$\displaystyle{\partial}_{0}{\partial}_{2}f$
$\displaystyle=\frac{p_{0}^{2}}{2}+p_{1},$
$\displaystyle{\partial}_{0}{\partial}_{3}f$
$\displaystyle=\frac{p_{0}^{3}}{3}+p_{1}p_{0}+p_{2},$
$\displaystyle{\partial}_{0}{\partial}_{4}f$
$\displaystyle=\frac{p_{0}^{4}}{4}+p_{1}p_{0}^{2}+p_{2}p_{0}+\frac{p_{1}^{2}}{2}+p_{3}.$

Denote by $P_{ij}(\gamma_{1},\dots,\gamma_{m})$ the number of all partitions
of $i_{1},\dots,i_{m}$ of $i$ and $j_{1},\dots,j_{m}$ of $j$, s.t.
$i_{k}+j_{k}=\gamma_{k}+1$ for all $k$. It was computed in [BDbN, Corollary
5.1] that the flows of (3.3) read:

(3.5) ${\partial}_{i}{\partial}_{j}f=\sum_{m\geq
1}\frac{(-1)^{m-1}}{m}\sum_{\gamma_{1}+\dots+\gamma_{m}=i+j-m}P_{ij}(\gamma_{1},\dots,\gamma_{m})\prod_{k=1}^{m}{\partial}_{1}{\partial}_{\gamma_{k}}f.$

######  Proposition 3.3.

Equation (3.4) is equivalent to the following equality of the formal power
series in $z$

$\sum_{\alpha\geq 1}{\partial}_{0}{\partial}_{\alpha}f\cdot
z^{\alpha}=-\log\left[1-\sum_{\alpha\geq
1}{\partial}_{1}{\partial}_{\alpha}f\cdot
z^{\alpha+1}-z\cdot{\partial}_{0}{\partial}_{1}f\right].$

###### Proof.

Introduce the notation $x_{\alpha}:={\partial}_{1}{\partial}_{\alpha-1}f$ and
$y_{\alpha}:={\partial}_{0}{\partial}_{\alpha}f$. Let also $\phi_{k,m}:=\sum
x_{\gamma_{1}}\dots x_{\gamma_{m}}$ where the summation is taken over all
$\gamma_{i}\geq 2$, s.t. $\gamma_{1}+\dots+\gamma_{m}=k$. Then (3.4) can be
written using (2.1) as

$y_{\alpha}=\sum_{m=1}^{\alpha}\frac{\phi_{\alpha,m}}{m}+\sum_{m=0}^{\alpha-1}\sum_{k=m+1}^{\alpha}\frac{y_{1}^{\alpha+1-k}}{(\alpha+1-k)!}\frac{(m+\alpha-k)!}{m!}\phi_{k-1,m}.$

For any polynomial $p=p(z)$ let $[z^{k}]p$ stand for the coefficient of
$z^{k}$ in the polynomial assumed. For $\Phi:=\sum_{k\geq 2}x_{k}z^{k}$ we
have $\phi_{k,m}=[z^{k}]\Phi^{m}$. Denote also $\Psi:=\sum_{k\geq
2}x_{k}z^{k}/y_{1}^{k}$. Note that this power series is obtained from $\Phi$
by a formal rescaling of $z$ variable.

Equation above is equivalent to

$\displaystyle
y_{\alpha}-\frac{y_{1}^{\alpha}}{\alpha}=y_{1}^{\alpha}[z^{\alpha}]\left(\sum_{k=2}^{\alpha+1}\sum_{m=1}^{k-1}\frac{(m+\alpha-k)!}{(\alpha+1-k)!m!}z^{\alpha+1-k}\Psi^{m}\right)$
$\displaystyle\Leftrightarrow\
y_{\alpha}-\frac{y_{1}^{\alpha}}{\alpha}=y_{1}^{\alpha}[z^{\alpha}]\left(\sum_{k=2}^{\alpha}\frac{z^{\alpha+1-k}}{(\alpha+1-k)}\frac{1}{(1-\Psi)^{\alpha+1-k}}+\sum_{m\geq
1}\frac{\Psi^{m}}{m}\right)$ $\displaystyle\Leftrightarrow\
y_{\alpha}=y_{1}^{\alpha}[z^{\alpha}]\left(-\log(1-\frac{z}{1-\Psi})-\log(1-\Psi)\right).$

Rescaling formally $z$ on the both sides and collecting $y_{\bullet}$ into a
power series the proposition follows. ∎

The following proposition shows that the dependance of
${\partial}_{1}{\partial}_{j}f$ on ${\partial}_{0}{\partial}_{\bullet}f$ is
given via the Schur polynomials.

######  Proposition 3.4.

Let $f$ satisfy (3.4) then we have

(3.6) ${\partial}_{1}{\partial}_{j}f=\sum_{m\geq
1}\frac{(-1)^{m-1}}{m!}\sum_{\gamma_{1}+\dots+\gamma_{m}=j+1}\prod_{k=1}^{m}{\partial}_{0}{\partial}_{\gamma_{k}}f.$

###### Proof.

Assume the notation introduced in the proof of Proposition 3.3. Exponentiating
the equality of Proposition 3.3 we get

$\displaystyle\Phi(z)=1-\exp(-\sum_{\alpha\geq
1}y_{\alpha}z^{\alpha})-y_{1}z.$

Expanding the exponent in a power series and combining the coefficients of
$z^{\alpha}$ on the both sides we get exactly the desired equation. ∎

Our goal now is to relate the system (3.3) and (3.4) to dispersionless mKP
hierarchy.

###  3.2. Dispersionless mKP hierarchy

Consider the notation

$D(z):=\sum_{k\geq 1}\frac{z^{-k}}{k}\frac{{\partial}}{{\partial}t_{k}}.$

For a function $f=f(t_{0},t_{1},t_{2},\dots)$, the dispersionless limit of mKP
hierarchy can be written in a Fay form as the following equality of the formal
power series in $z^{-1}$, $w^{-1}$.

(3.7) $e^{D(z)D(w)f}=\frac{z\cdot e^{-D(z){\partial}_{0}f}-w\cdot
e^{-D(w){\partial}_{0}f}}{z-w},$

The coefficient of $w^{-1}$ on the both sides gives the following equality

(3.8) $z-D(z){\partial}_{1}f=ze^{-D(z){\partial}_{0}f}.$

Substituting it back to (3.7) gives

(3.9) $e^{D(z)D(w)f}=1-\frac{D(z){\partial}_{1}f-D(w){\partial}_{1}f}{z-w}.$

This equation is exactly the Fay form of the dispersionless limit of KP
hierarchy of the function $f$ assumed as a function of $t_{1},t_{2},\dots$
with $t_{0}$ being fixed.

######  Theorem 3.5.

The system of equations (3.3) and (3.4) coincides with the dispersionless mKP
hierarchy after the change of the variables $t_{k}\to t_{k}/k$, $k\geq 1$.

###### Proof.

After the change of the variables given, the system (3.3) coincides with with
the Fay form of dispersionless KP by (3.5) (see also [BDbN]). It follows
immediately from Proposition 3.4 that (3.8) holds for a solution to (3.4).
Therefore substituting it to (3.3) on gets (3.4). ∎

###  3.3. h–deformation

Full mKP hierarchy can be obtained from its dispersionless limit by the
following procedure extending the approach of [NZ].

Let $\tau_{\hbar}(s,{\bf t}):=\tau(\hbar^{-1}s,\hbar^{-1}{\bf t})$. Assume it
to be expanded by $\log\tau_{\hbar}=\sum_{g\geq 0}\hbar^{g-2}F_{g}$. Denote
${f:=\hbar^{2}\log\tau_{\hbar}}$. Consider also the operators

$\Delta(z):=\hbar^{-1}\left(e^{\hbar
D(z)}-1\right),\quad\Delta^{o}:=\hbar^{-1}\left(e^{-\hbar{\partial}_{0}}-1\right).$

Note that $\Delta^{o}$ only involves the differentiations w.r.t. $t_{0}$ while
$\Delta(z)$ only the differentiations w.r.t. $t_{k}$, $k\geq 1$.

The $\hbar$–deformed version of (3.7) above is the following equation

(3.10) $e^{\Delta(z)\Delta(w)f}=\frac{z\cdot e^{-\Delta(z)\Delta^{0}f}-w\cdot
e^{-\Delta(w)\Delta^{0}f}}{z-w}.$

It coincides with the Fay form of mKP hierarchy (cf. [T, Section 3.2]).

Consider ${\partial}^{\hbar}_{k}$ defined by

${\partial}^{\hbar}_{0}:=\hbar^{-1}\left(e^{-\hbar{\partial}_{0}}-1\right),\quad\sum_{k\geq
1}\frac{z^{-k}}{k}{\partial}^{\hbar}_{k}=\Delta(z)$

Comparing the coefficients of $z^{-a}w^{-b}$ on the both sides of (3.10) one
gets exactly

$\displaystyle{\partial}_{\alpha}^{\hbar}{\partial}_{\beta}^{\hbar}f$
$\displaystyle=\sum_{m\geq
1}\sum_{\gamma_{\bullet}}R^{\mathrm{A}}_{\alpha,\beta;\gamma_{1},\dots,\gamma_{m}}\prod_{k=1}^{m}{\partial}_{1}{\partial}_{\gamma_{k}}^{\hbar}f,$
$\displaystyle{\partial}_{0}^{\hbar}{\partial}_{\alpha}^{\hbar}f$
$\displaystyle=\sum_{m\geq
1}\sum_{\gamma_{\bullet}}R^{\mathrm{A},ext}_{\alpha;\gamma_{1},\dots,\gamma_{m}}\prod_{k=1}^{m}{\partial}_{1}{\partial}_{\gamma_{k}}^{\hbar}f.$

This procedure recovers full mKP hierarchy from the dispersionless hierarchy
that we’ve constructed from the family of pairs
$\\{F^{c}_{A_{N}},F^{o}_{A_{N}}\\}_{N\geq 1}$.

##  4\. Dispersionless open–closed system of type D

The following stabilization condition was proved in [BDbN].

For any $4\leq N<N$ we have

$\displaystyle\left.\frac{{\partial}^{2}F^{c}_{D_{N}}({\bf
t})}{{\partial}t_{\alpha}{\partial}t_{\beta}}\right|_{\forall\gamma\;t_{\gamma}=s_{N-\gamma};\
t_{N}=s_{-1}}=\left.\frac{{\partial}^{2}F^{c}_{D_{M}}}{{\partial}t_{\alpha}{\partial}t_{\beta}}\right|_{\forall\gamma\;t_{\gamma}=s_{M-\gamma};\
t_{M}=s_{-1}},\qquad\forall\alpha+\beta<N,$
$\displaystyle\left.\frac{{\partial}^{2}F^{c}_{D_{N}}({\bf
t})}{{\partial}t_{N}{\partial}t_{\beta}}\right|_{\forall\gamma\;t_{\gamma}=s_{N-\gamma};\
t_{N}=s_{-1}}=\left.\frac{{\partial}^{2}F^{c}_{D_{M}}({\bf
t})}{{\partial}t_{M}{\partial}t_{\beta}}\right|_{\forall\gamma\;t_{\gamma}=s_{M-\gamma};\
t_{M}=s_{-1}},\qquad\forall\beta<N,$

understood as an equality of polynomials in $s_{\bullet}$.

######  Proposition 4.1.

For any $4\leq N<M$ and $1\leq\alpha<N$ we have

$\displaystyle\frac{{\partial}F_{D_{N}}^{o}}{{\partial}t_{\alpha}}\mid_{t_{N-\beta}=s_{\beta},\
t_{N}=s_{-1}}=\frac{{\partial}F_{D_{M}}^{o}}{{\partial}t_{\alpha}}\mid_{t_{M-\beta}=s_{\beta},\
t_{M}=s_{-1}},$
$\displaystyle\frac{{\partial}F_{D_{N}}^{o}}{{\partial}t_{N}}\mid_{t_{N-\beta}=s_{\beta},\
t_{N}=s_{-1}}=\frac{{\partial}F_{D_{M}}^{o}}{{\partial}t_{M}}\mid_{t_{M-\beta}=s_{\beta},\
t_{M}=s_{-1}},$

assumed as polynomials in $s_{-1},s_{1},\dots,s_{M}$ and Laurent polynomial in
$t_{0}$.

###### Proof.

Note that the change of the variables above does not affect the variable
$t_{0}$ and we can consider $F^{o}_{D_{N}}$ and $F^{o}_{D_{M}}$ as Laurent
polynomials in $t_{0}$.

The second equality is straightforward by the explicit form of
$F^{o}_{D_{N}}$. To show the first equality we only have to consider the
summands of $F^{o}_{D_{N}}$ and $F^{o}_{D_{M}}$ involving the variable
$t_{0}$. This follows from (2.4) and Theorem 4.9 of [BDbN]. ∎

For $\alpha,\beta<N$, and $\gamma_{k}<N$ denote $N=\alpha+\beta-1$ and
$\bar{N}:=-1$, $\bar{\gamma}:=N-\gamma$ for $1\leq\gamma<N$. Set

$\displaystyle
R^{(\mathrm{D},1)}_{\alpha,\beta;\bar{\gamma}_{1},\dots,\bar{\gamma}_{m}}=\left.\frac{1}{m!}\frac{{\partial}^{m+2}F^{c}_{D_{N}}}{{\partial}_{\alpha}{\partial}_{\beta}{\partial}t_{\gamma_{1}}\cdot\dots\cdot{\partial}t_{\gamma_{m}}}\right|_{{\bf
t}=0},$ $\displaystyle
R^{(\mathrm{D},2)}_{\alpha;\bar{\gamma}_{1},\dots,\bar{\gamma}_{m}}=\left.\frac{1}{m!}\frac{{\partial}^{m+3}F^{c}_{D_{N}}}{{\partial}t_{N}{\partial}t_{N}{\partial}t_{\alpha}{\partial}t_{\gamma_{1}}\cdot\dots\cdot{\partial}t_{\gamma_{m}}}\right|_{{\bf
t}=0},$ $\displaystyle
R^{(\mathrm{D},ext,1)}_{\alpha;\bar{\gamma}_{1},\dots,\bar{\gamma}_{m}}=\frac{1}{m!}\left.\frac{{\partial}^{m+1}F_{D_{N}}^{o}}{{\partial}t_{\alpha}{\partial}t_{\gamma_{1}}\cdots{\partial}t_{\gamma_{m}}}\right|_{{\bf
t}=0}.$

It was proved in [BDbN] that $R^{(\mathrm{D},1)}_{\bullet}$ and
$R^{(\mathrm{D},2)}_{\bullet}$ are well–defined. Note that despite
$F^{o}_{D_{N}}$ being Laurent polynomial, according to the proposition above
all coefficients $R^{\bullet}$ are well-defined.

The data above does not collect all the information of $F^{o}_{D_{N}}$ that is
stabilized as $N$ grows. In particular we have
${\partial}_{N}{\partial}_{N}F^{o}_{D_{N}}=1/t_{0}$, for which ${\bf t}=0$ is
not defined. However we still can add the corresponding flow (4.4) to the
system of PDEs we build.

For a function $f=f(t_{-1},t_{0},t_{1},t_{2},\dots)$ consider the system of
PDEs

(4.1) $\displaystyle{\partial}_{\alpha}{\partial}_{\beta}f$
$\displaystyle=\sum_{m\geq
1}\sum_{\gamma_{1},\dots,\gamma_{m}}R^{(\mathrm{D},1)}_{\alpha,\beta;\gamma_{1},\dots,\gamma_{m}}\prod_{k=1}^{m}{\partial}_{1}{\partial}_{\gamma_{a}}f$
(4.2) $\displaystyle{\partial}_{-1}{\partial}_{\alpha}f$
$\displaystyle={\partial}_{-1}{\partial}_{1}f\cdot\sum_{m\geq
1}\sum_{\gamma_{1},\dots,\gamma_{m}}R^{(\mathrm{D},2)}_{\alpha;\gamma_{1},\dots,\gamma_{m}}\prod_{k=1}^{m}{\partial}_{1}{\partial}_{\gamma_{a}}f,$
(4.3) $\displaystyle{\partial}_{0}{\partial}_{\alpha}f$
$\displaystyle=\sum_{m\geq
1}\sum_{\gamma_{1},\dots,\gamma_{m}}R^{(\mathrm{D},ext)}_{\alpha;\gamma_{1},\dots,\gamma_{m}}\prod_{k=1}^{m}{\partial}_{1}{\partial}_{\gamma_{a}}f,$
(4.4) $\displaystyle{\partial}_{0}{\partial}_{-1}f$
$\displaystyle\cdot{\partial}_{1}{\partial}_{0}f={\partial}_{1}{\partial}_{-1}f,$

for all $\alpha,\beta\geq 2$ and $\gamma_{\bullet}\geq-1$.

We show that this system of equations coincides with the system given in
Introduction after setting $t_{-1}=\bar{t}_{1}$.

######  Theorem 4.2.

  1. (1)

The numbers $R^{(\mathrm{D},1)}_{\alpha,\beta;\gamma_{1},\dots,\gamma_{m}}$
and $R^{(\mathrm{D},2)}_{\alpha,\beta;\gamma_{1},\dots,\gamma_{m}}$ are series
expansions of ${\partial}_{\alpha}{\partial}_{\beta}F_{D_{N}}^{c}$ and
${\partial}_{N}{\partial}_{\alpha}F_{D_{N}}^{c}$ respectively written in the
coordinates $t_{\bar{1}},\dots,t_{\bar{N}}$.

  2. (2)

The system (4.1),(4.2) and (4.3),(4.4) is compatible.

  3. (3)

The function

$\tilde{f}=F_{D_{N}}^{c}(t_{\overline{1}},\dots,t_{\overline{N}})+\int
F^{o}_{D_{N}}(t_{\overline{1}},\dots,t_{\overline{N}},t_{0})dt_{0}$

satisfies this system for $\alpha+\beta\leq N$.

###### Proof.

Part (1) follows immediately from (2.6). After this in order to show (2) we
may use the PDEs given in Introduction. Compatibility of (4.1) and (4.2) was
proved in [BDbN].

Let $\kappa=\alpha+\beta-1$. In the following formulae we use
${\partial}_{\bullet}{\partial}_{1}{\partial}_{\alpha}f={\partial}_{1}{\partial}_{\bullet}{\partial}_{\alpha}f$.
By the chain rule we have

$\displaystyle{\partial}_{0}$
$\displaystyle({\partial}_{\alpha}{\partial}_{\beta}f)=\sum_{\delta=1}^{\kappa}{\partial}_{0}{\partial}_{1}{\partial}_{\bar{\delta}}f\cdot\frac{{\partial}^{3}F^{c}_{D_{\kappa}}}{{\partial}t_{\delta}{\partial}t_{\alpha}{\partial}t_{\beta}}\mid_{t_{\kappa-\gamma}={\partial}_{1}{\partial}_{\gamma}f,\
t_{\kappa}={\partial}_{1}\bar{\partial}_{1}f},$
$\displaystyle={\partial}_{1}\frac{{\partial}F^{o}_{D_{\kappa}}}{{\partial}t_{\bar{\delta}}}\cdot\frac{{\partial}^{3}F^{c}_{D_{\kappa}}}{{\partial}t_{\delta}{\partial}t_{\alpha}{\partial}t_{\beta}}={\partial}_{1}{\partial}_{1}{\partial}_{0}f\cdot\frac{{\partial}F^{o}_{D_{\kappa}}}{{\partial}t_{\bar{\delta}}{\partial}t_{0}}\frac{{\partial}^{3}F^{c}_{D_{\kappa}}}{{\partial}t_{\delta}{\partial}t_{\alpha}{\partial}t_{\beta}}+\sum_{\sigma=1}^{\kappa}{\partial}_{1}{\partial}_{1}{\partial}_{\sigma}f\cdot\frac{{\partial}F^{o}_{D_{\kappa}}}{{\partial}t_{\bar{\delta}}{\partial}t_{\bar{\sigma}}}\frac{{\partial}^{3}F^{c}_{D_{\kappa}}}{{\partial}t_{\delta}{\partial}t_{\alpha}{\partial}t_{\beta}},$

where we skip the variable substitution on the second line in order to
simplify the formulae. Similarly we have

$\displaystyle{\partial}_{\beta}{\partial}_{0}$
$\displaystyle{\partial}_{\alpha}f=\sum_{\delta=1}^{\kappa}{\partial}_{\beta}{\partial}_{1}{\partial}_{\bar{\delta}}f\cdot\frac{{\partial}^{2}F^{o}_{D_{\kappa}}}{{\partial}t_{\delta}{\partial}t_{\alpha}}\mid_{t_{\kappa-\gamma}={\partial}_{1}{\partial}_{\gamma}f,\
t_{\kappa}={\partial}_{1}\bar{\partial}_{1}f,\
t_{0}={\partial}_{1}{\partial}_{0}f},$
$\displaystyle\qquad\qquad+{\partial}_{\beta}{\partial}_{1}{\partial}_{0}f\cdot\frac{{\partial}^{2}F^{o}_{D_{\kappa}}}{{\partial}t_{0}{\partial}t_{\alpha}}\mid_{t_{\kappa-\gamma}={\partial}_{1}{\partial}_{\gamma}f,\
t_{\kappa}={\partial}_{1}\bar{\partial}_{1}f,\
t_{0}={\partial}_{1}{\partial}_{0}f},$
$\displaystyle=\sum_{\delta=1}^{\kappa}{\partial}_{1}\frac{{\partial}^{2}F^{c}_{D_{\kappa}}}{{\partial}t_{\beta}{\partial}t_{\bar{\delta}}}\cdot\frac{{\partial}^{2}F^{o}_{D_{\kappa}}}{{\partial}t_{\delta}{\partial}t_{\alpha}}+{\partial}_{1}\frac{{\partial}F^{o}_{D_{\kappa}}}{{\partial}t_{\beta}}\cdot\frac{{\partial}^{2}F^{o}_{D_{\kappa}}}{{\partial}t_{0}{\partial}t_{\alpha}}$
$\displaystyle=\sum_{\sigma=1}^{\kappa}{\partial}_{1}{\partial}_{1}{\partial}_{\sigma}f\cdot\sum_{\delta=1}^{\kappa}\left(\frac{{\partial}^{3}F^{c}_{D_{\kappa}}}{{\partial}t_{\beta}{\partial}t_{\bar{\delta}}{\partial}_{\bar{\sigma}}}\frac{{\partial}^{2}F^{o}_{D_{\kappa}}}{{\partial}t_{\delta}{\partial}t_{\alpha}}+\frac{{\partial}^{2}F^{o}_{D_{\kappa}}}{{\partial}t_{\beta}{\partial}t_{\bar{\sigma}}}\frac{{\partial}^{2}F^{o}_{D_{\kappa}}}{{\partial}t_{0}{\partial}t_{\alpha}}\right)+{\partial}_{1}{\partial}_{1}{\partial}_{0}f\cdot\frac{{\partial}^{2}F^{o}_{D_{\kappa}}}{{\partial}t_{\beta}{\partial}t_{0}}\cdot\frac{{\partial}^{2}F^{o}_{D_{\kappa}}}{{\partial}t_{0}{\partial}t_{\alpha}}.$

The coefficient of ${\partial}_{1}{\partial}_{1}{\partial}_{0}f$ on the both
sides is the same if and only if

$\displaystyle\sum_{\delta=1}^{\kappa}\frac{{\partial}F^{o}_{D_{\kappa}}}{{\partial}t_{0}{\partial}t_{\bar{\delta}}}\frac{{\partial}^{3}F^{c}_{D_{\kappa}}}{{\partial}t_{\delta}{\partial}t_{\alpha}{\partial}t_{\beta}}=\frac{{\partial}^{2}F^{o}_{D_{\kappa}}}{{\partial}t_{\beta}{\partial}t_{0}}\frac{{\partial}^{2}F^{o}_{D_{\kappa}}}{{\partial}t_{0}{\partial}t_{\alpha}}.$

The metric $\eta$ defined by $F^{c}_{D_{\kappa}}$ has only the following
non–zero entries $\eta^{\kappa\kappa}=1$ and
$\eta^{\alpha\beta}=\delta^{\alpha+\beta,N}$. With our choice of $\bar{\beta}$
this is equivalent to $\eta^{\alpha\beta}=\delta^{\alpha,\bar{\beta}}$.

By open WDVV equation the difference of LHS and RHS is equal to
$\frac{{\partial}^{2}F^{o}_{D_{\kappa}}}{{\partial}t_{\alpha}{\partial}t_{\beta}}\frac{{\partial}^{2}F^{o}_{D_{\kappa}}}{{\partial}t_{0}{\partial}t_{0}}$
that vanishes for $\alpha+\beta\leq\kappa$ and $\alpha,\beta\neq\kappa$ due to
(2.5). For any $\sigma\neq 0$ the coefficients of
${\partial}_{1}{\partial}_{1}{\partial}_{\sigma}f$ of the expressions above
coincide by the same reasoning.

Consider now part (3). Because of the special dependance of $F^{c}_{N}$ and
$F^{o}_{N}$ on the variable $t_{1}$ we have

${\partial}_{1}{\partial}_{\overline{\gamma}}\tilde{f}=\begin{cases}{\partial}_{1}{\partial}_{\overline{\gamma}}F^{c}_{N}=t_{\gamma},\quad&\text{
if }\quad\overline{\gamma}>0,\\\
{\partial}_{1}{\partial}_{\overline{\gamma}}F^{c}_{N}=t_{N},\quad&\text{ if
}\quad\overline{\gamma}=-1,\\\ {\partial}_{1}F^{o}_{N}=t_{0},\quad&\text{ if
}\quad\overline{\gamma}=0.\end{cases}$

To show (3) let’s show that applied to $\tilde{f}$ assumed both (3.3) and
(3.4) just provide the series expansions of
${\partial}_{\alpha}{\partial}_{\beta}F^{c}_{N}$ and
${\partial}_{\alpha}F^{o}_{N}$.

It follows from (2.3) that
${\partial}_{\alpha}{\partial}_{\beta}F^{o}_{A_{N}}\equiv 0$ whenever
$\alpha+\beta\leq N+1$. The function $F^{c}_{N}$ does not depend on $t_{0}$
and therefore (3.3) holds for $\tilde{f}$.

Similarly
${\partial}_{0}{\partial}_{\alpha}\tilde{f}={\partial}_{\alpha}F^{o}_{N}$ what
approves (3.4) for $\tilde{f}$. ∎

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# Singular symplectic spaces and holomorphic membranes

Sergey Galkin and Grigory Mikhalkin  PUC-Rio, Matemática, rua Marquês de São
Vicente 225, Gávea, Rio de Janeiro, Brazil  (on leave from HSE University,
Russian Federation<EMAIL_ADDRESS>Université de Genève,
Mathématiques, rue du Conseil-Général 7-9, 1205 Genève, Suisse
<EMAIL_ADDRESS>
###### Abstract.

We set up a topological framework for degenerations of symplectic manifolds
into singular spaces paying a special attention to the behavior of Lagrangian
manifolds and their (holomorphic) membranes. We show that degenerations into
singular toric varieties provide a source of exotic Lagrangian tori.

Research is supported in part by the SNSF-grants 178828, 204125 and CNPq-grant
PQ 315747

##  1\. Degenerations to singular symplectic spaces

###  1.1. Symplectic singular spaces

Kähler varieties are complex analytic manifolds with a compatible symplectic
structure. Any smooth complex projective variety $X$ or, more generally,
complex submanifold of a Kähler variety $Y$ is a Kähler manifold itself.
Singular projective varieties are more delicate objects. At the same time they
are very useful as they might come (somewhat paradoxically) as degenerations
of rigid (undeformable) Kähler manifolds such as
${\mathbb{C}}{\mathbb{P}}^{2}$.

For the purposes of this paper it is convenient to keep the topological set-up
as much as possible and the algebraic set-up as little as possible. With this
in mind, we substitute singular projective varieties with the following rather
general class of spaces generalizing complex algebraic varieties: singular
symplectic spaces in ${\mathbb{C}}{\mathbb{P}}^{N}$. Furthermore, we may
replace ${\mathbb{C}}{\mathbb{P}}^{N}$ by another Kähler variety of
sufficiently large dimension as follows.

Let $Y$ be a (smooth, but possibly non-compact) Kähler manifold of dimension
$N$ with the symplectic form $\omega$. Let $\Sigma\subset X\subset Y$ be
closed subspaces such that

$X^{\circ}=X\smallsetminus\Sigma$

is a smooth connected complex submanifold of $Y$ of dimension $n$. The form
$\omega$ is a symplectic form on $X^{\circ}$. Its top exterior power
$\omega^{n}$ is a volume form on $X^{\circ}$ and is defined everywhere on $Y$.
Its cohomology class $[\omega^{n}]$ can be evaluated on any $2n$-cycle in $Y$.

######  Definition 1.1.

We say that $(X,\Sigma)$ is a singular symplectic space of (complex) dimension
$n$ in the ambient Kähler manifold $Y$ if $\Sigma$ and $X$ are closed
subspaces of $Y$ with $\Sigma\subset X$ such that the following conditions
hold.

  1. (1)

The complement $X^{\circ}=X\smallsetminus\Sigma$ is a smooth connected
$n$-dimensional complex submanifold of $Y$.

  2. (2)

For any $x\in X^{\circ}$ the inclusion homomorphism
$\phi_{x}:H_{2n}(X;{\mathbb{R}})\to
H_{2n}(X,X\smallsetminus\\{x\\};{\mathbb{R}})=H_{2n}({\mathbb{C}}^{n},{\mathbb{C}}^{n}\smallsetminus\\{0\\};{\mathbb{R}})={\mathbb{R}}$
is an isomorphism.

In particular, the fundamental class $[X]\in H_{2n}(Y;{\mathbb{R}})$ is well-
defined as the image of the fundamental class of
$({\mathbb{C}}^{n},{\mathbb{C}}^{n}\smallsetminus\\{0\\})$ under the
composition of $\phi^{-1}$ and the homomorphism induced by the inclusion
$X\subset Y$.

  3. (3)

We have

$\int\limits_{X^{\circ}}\omega^{n}=[\omega^{n}][X],$

where the right-hand side is defined by pairing of $[X]\in
H_{2n}(Y;{\mathbb{R}})$ and $\omega^{n}\in H^{2n}(Y;{\mathbb{R}})$.

In particular, $X^{\circ}$ has a finite volume with respect to $\omega$.

We denote the restriction of the symplectic form from $Y$ to $X^{\circ}$ also
with $\omega$. It turns $X^{\circ}$ into a Kähler manifold.

######  Remark 1.2.

Below by default we use Čech cohomology $H^{k}$ for our topological spaces in
consideration. The corresponding homology groups (over fields ${\mathbb{Q}}$
and ${\mathbb{R}}$) are obtained as dual vector spaces of the corresponding
cohomology groups. Note, however, that in most applications our main examples
are CW-complexes, so in this case we may also equivalently choose singular or
cellular cohomology as the principal homology theory.

######  Definition 1.3.

A singular symplectic space $(X,\Sigma)$ is called a
${\mathbb{Q}}\Gamma$-space if the following conditions hold.

  1. (1)

The inclusion homomorphism

(1) $\iota:H^{2}(X;{\mathbb{R}})\to H^{2}(X^{\circ};{\mathbb{R}})$

is injective.

  2. (2)

The canonical class $K\in H^{2}(X^{\circ};{\mathbb{R}})$ is in the image of
(1).

Recall that $\bar{a}\in H^{2}(X;{\mathbb{R}})$ is called a cohomological
extension of $a\in H^{2}(X^{\circ};{\mathbb{R}})$ if $a$ is the image of
$\bar{a}$ under (1). The following proposition is simply a translation of
Definition 1.3 in the language of cohomological extension.

######  Proposition 1.4.

Suppose that $(X,\Sigma)$ is a ${\mathbb{Q}}\Gamma$-space in $Y$.

  1. (1)

A cohomological extension of any class in $H^{2}(X^{\circ};{\mathbb{R}})$ is
unique.

  2. (2)

A cohomological extension of the canonical class $K\in
H^{2}(X^{\circ};{\mathbb{R}})$ exists.

For brevity, we denote the unique extension of the canonical class of a
${\mathbb{Q}}\Gamma$-space $X$ also by $K\in H^{2}(X;{\mathbb{R}})$.

######  Remark 1.5.

The second condition in Definition 1.3 mimics (in the context of symplectic
singular spaces) the key property of ${\mathbb{Q}}$-Gorenstein algebraic
varieties.

The dense smooth part $X^{\circ}$ is naturally a symplectic $(2n)$-manifold
with the symplectic structure $\omega$ induced by the symplectic form on $Y$.
The cohomological extension in $H^{2}(X;{\mathbb{R}})$ of $[\omega]\in
H^{2}(Y;{\mathbb{R}})$ is given by the cohomology class of the symplectic form
in $Y$. Where confusion is not possible we denote it still with $[\omega]\in
H^{2}(X;{\mathbb{R}})$.

######  Definition 1.6.

We say that a ${\mathbb{Q}}\Gamma$-space $X\subset Y$ is monotone if the
classes $K$ and $[\omega]$ are proportional, i.e. $K=\alpha[\omega]\in
H^{2}(X;{\mathbb{R}})$ for some constant $\alpha\in{\mathbb{R}}$.

###  1.2. Lagrangian submanifolds of ${\mathbb{Q}}\Gamma$-spaces

Recall that an orientable Lagrangian (or just only totally real) submanifold
$L$ of a symplectic manifold $X^{\circ}$ admits a canonical trivialization of
the top exterior power $\Lambda^{n}(T^{*}X^{\circ})$ of the cotangent bundle
$T^{*}X^{\circ}$ (computed with respect to any symplectically compatible
almost complex structure) given by a real volume form on $L$. Even if $L$ is
non-orientable, we get a trivialization of $\Lambda^{n}(T^{*}X^{\circ})$ over
any orientation preserving loop on $L$.

For a smooth compact surface $B\subset X^{\circ}$ with boundary such that
$\partial B\subset L$ is a collection of orientation-preserving loops in $L$,
we define $K(B)\in{\mathbb{Z}}$ as the obstruction to extending this
trivialization from $\partial B$ to $B$. Since twice $\partial B$ is always
orientation-preserving, for a general smooth closed surface $B\subset
X^{\circ}$ with $\partial B\subset L$ we may consistently define
$K(B)\in\frac{1}{2}{\mathbb{Z}}$ as one half of the obstruction to extending
the corresponding trivialization on $2\partial B$. Clearly, the value $K(B)$
depends only on $[B]\in H_{2}(X^{\circ},L)$. The homomorphism

(2) $\mu:H_{2}(X^{\circ},L;{\mathbb{Q}})\to{\mathbb{Q}}$

defined by $\mu([B])=-2K(B)$ is called the Maslov index of $[B]$.

######  Proposition 1.7.

If $(X,\Sigma)$ is a ${\mathbb{Q}}\Gamma$-space and $L\subset X^{\circ}$ is a
Lagrangian then the homomorphism (2) factors through the inclusion
homomorphism

(3) $H_{2}(X^{\circ},L;{\mathbb{Q}})\to H_{2}(X,L;{\mathbb{Q}}).$

###### Proof.

The difference of two classes in $H_{2}(X^{\circ},L;{\mathbb{Q}})$ with the
same image under (3) comes from an absolute ${\mathbb{Q}}$-cycle in
$X^{\circ}$ that is homologous in $X$ to a class supported by $L$. By the
second condition in Definition 1.3 the canonical class vanishes on this
difference. ∎

######  Corollary 1.8.

If the homomorphism (3) is surjective then there exists a unique homomorphism

(4) $\mu:H_{2}(X,L;{\mathbb{Q}})\to{\mathbb{Q}}$

extending (2).

This homomorphism is still called the Maslov index.

Since $X\subset Y$ and $Y$ are Kähler, we may compute the symplectic area of a
surface $B\subset X$ with boundary on a Lagrangian submanifold $L$. Since $L$
is Lagrangian, the result $\omega(B)$ only depends on its homology class, thus
producing the area homomorphism

(5) $\omega:H_{2}(X,L;{\mathbb{R}})\to{\mathbb{R}}.$

######  Definition 1.9.

Let $(X,\Sigma)$ be a ${\mathbb{Q}}\Gamma$-space. A Lagrangian submanifold
$L\subset X^{\circ}$ is called monotone in $X$ if (3) is an epimorphism and
the homomorphisms (5) and (4) are proportional.

Clearly, if $L\subset X$ is monotone then the ${\mathbb{Q}}\Gamma$-space $X$
is also monotone, and the proportionality coefficients in Definitions 1.6 and
1.9 differ by multiplication by $-2$.

###  1.3. Symplectic degenerations

Let $0\in T\subset{\mathbb{C}}$ be the open unit complex disk (which we view
as a symplectic surface of finite area), $Y$ be a Kähler manifold (e.g.
${\mathbb{C}}{\mathbb{P}}^{N}$), and $(\mathcal{X},\Sigma)$ be a singular
symplectic space in $Y\times T$. Denote by

(6) $p:\mathcal{X}\to T$

the restriction of the projection $Y\times T\to T$ to $\mathcal{X}$, and by
$X_{t}=p^{-1}(t)$ the fiber over $t\in T$.

######  Definition 1.10.

Let $(\mathcal{X},\Sigma)$ be a singular symplectic space in $Y\times T$ with
$\Sigma\subset Y\times\\{0\\}$, such that $X=\mathcal{X}\cap(Y\times\\{0\\})$
is compact and $(X,\Sigma)$ is a singular space in $Y=Y\times\\{0\\}$. We say
that $X\subset Y$ is a closed degeneration of $X_{t}\subset Y=Y\times\\{t\\}$,
$t\neq 0$, if the following conditions hold.

  1. (1)

The map (6) is proper (in particular, each $X_{t}$ is compact).

  2. (2)

The restriction $p|_{\mathcal{X}^{\circ}}$ is a submersion.

In this case we say that $X_{t}$, $t\in T\smallsetminus\\{0\\}$, is a
symplectic smoothing of $(X,\Sigma)$ and that $(\mathcal{X},p)$ is a
symplectic deformation of $(X,\Sigma)$.

We have $X^{\circ}=X\smallsetminus\Sigma=\mathcal{X}^{\circ}\cap X$. The fiber
$X=X_{0}$ is called the central fiber. Fibers $X_{t}$, $t\neq 0$, are called
general fibers.

######  Proposition 1.11.

If $(\mathcal{X},p)$ is a symplectic deformation then the inclusion of the
central fiber $X\subset\mathcal{X}$ induces an isomorphism in all cohomology
groups $H^{k}(\mathcal{X};{\mathbb{R}})\to H^{k}(X;{\mathbb{R}})$ (and thus
also in all homology groups $H_{k}(X;{\mathbb{R}})\to
H_{k}(\mathcal{X};{\mathbb{R}})$).

###### Proof.

Recall that our basic homology theory is Čech cohomology and that $Y$ is a
Kähler manifold. For homology computations, we may use coverings by small open
disks in $Y$ and by products of small disks in $Y\times T$. Since $X$ is
compact, any such open covering of $X$ is induced by an open covering of
$p^{-1}(U)$ with the same nerve, where $0\in U\subset T$ is a subdisk of $T$.
But $p^{-1}(U)$ is a deformational retract of $\mathcal{X}$ and thus the
corresponding inclusion induces isomorphisms in homology. ∎

######  Proposition 1.12.

A smooth embedded path $\Gamma\subset T$ connecting $0$ and $t$ defines a
closed set $\Sigma_{\gamma}\subset X_{t}$ of zero $\omega^{n}$-volume, and a
symplectomorphism

$\Phi_{\gamma}:X^{\circ}\to X_{t}\smallsetminus\Sigma_{\gamma}.$

###### Proof.

The complement $\tilde{\Gamma}=p^{-1}(\Gamma)\smallsetminus\Sigma$ is a smooth
$(2n+1)$-submanifold of $Y\times T$ with boundary composed of $X^{\circ}$ and
$X_{t}$. The form $\omega$ is symplectic on each $X_{s}$, $s\in T$. The
radical of the restriction of $\omega$ to the interior of $\tilde{\Gamma}$
gives the characteristic foliation ${\mathcal{F}}$ defining a connection on

$p|_{\tilde{\Gamma}}:\tilde{\Gamma}\to\Gamma$

that preserves the symplectic form. The leaf of ${\mathcal{F}}$ passing
through a point $x_{0}\in X^{\circ}$ is a closed interval whose other end is a
point $x_{t}\in X_{t}$. We define $\Phi_{\gamma}(x_{0})=x_{t}$ and
$\Sigma_{\gamma}=X_{t}\smallsetminus\Phi_{\gamma}(X^{\circ})$. The locus
$\Sigma_{\gamma}$ is clearly closed. It remains to prove that
$\int\limits_{X_{t}\smallsetminus\Sigma_{\gamma}}\omega^{n}=\int\limits_{X_{t}}\omega^{n}$.
But the latter integral coincides with the evaluation of cohomological class
$[\omega^{n}]$ (which can be represented in Čech cohomology with the help of
De Rham isomorphism) at $[X_{t}]$. By Proposition 1.11 $[X_{t}]\in
H_{2n}(\mathcal{X};{\mathbb{R}})$ is in the image of the inclusion
homomorphism from $H_{2n}(X;{\mathbb{R}})$. With the help of a
($\mathcal{X}^{\circ}$-valued) section of $p$ we see that $[X_{t}]$ has to be
the image of $[X]$. Conservation of symplectic form by $\Phi_{\gamma}$ implies
that

$\int\limits_{X_{t}\smallsetminus\Sigma_{\gamma}}\omega^{n}=\int\limits_{X\smallsetminus\Sigma}\omega^{n}=[\omega^{n}][X]=[\omega^{n}][X_{t}].$

∎

The following statement can be formally obtained from Proposition 1.12 by
restricting to a subdisk $s,t\in T^{\prime}\subset T\smallsetminus\\{0\\}$ and
translating it by $-s$, so that the smooth fiber $X_{s}$ becomes the new
central fiber with $X^{\circ}_{s}=X_{s}$.

######  Corollary 1.13.

Any two general fibers $X_{t},X_{s}$, $s,t\in T$ are symplectomorphic.

######  Definition 1.14.

A symplectic deformation $(\mathcal{X},p)$ in $Y\times T$ is called a
${\mathbb{Q}}\Gamma$-deformation if both $(\mathcal{X},\Sigma)$ and
$(X,\Sigma)$ are ${\mathbb{Q}}\Gamma$-spaces. Accordingly, we say that $X_{t}$
is a ${\mathbb{Q}}\Gamma$-smoothing of $(X,\Sigma)$.

Main algebro-geometric source of ${\mathbb{Q}}\Gamma$-deformations are
${\mathbb{Q}}$-Gorenstein deformations in the sense of [9].

Suppose that $A\subset X^{\circ}$ is a subset.

######  Definition 1.15.

We say that $(X_{t},A_{t})$ is a ${\mathbb{Q}}\Gamma$-smoothing of $(X,A)$ if
$X_{t}$ is a ${\mathbb{Q}}\Gamma$-smoothing of $(X,\Sigma)$, and there exists
a path $\Gamma\subset T$ as in Proposition 1.12 such that
$A_{t}=\Phi_{\gamma}(A)$.

######  Proposition 1.16.

Suppose that $(\mathcal{X},p)$ is a symplectic deformation of a
${\mathbb{Q}}\Gamma$-space $(X,\Sigma)$. Suppose that
$H_{1}(X_{t};{\mathbb{R}})=0$ for a general fiber $X_{t}$, $t\in
T\smallsetminus\\{0\\}$. Then $(\mathcal{X},p)$ is a
${\mathbb{Q}}\Gamma$-deformation if and only if the canonical class
$K_{X_{t}}\in H^{2}(X_{t};{\mathbb{Q}})$ vanishes on the kernel of the
inclusion homomorphism

(7) $H_{2}(X_{t};{\mathbb{R}})\to H_{2}(\mathcal{X};{\mathbb{R}})\approx
H_{2}(X;{\mathbb{R}}).$

###### Proof.

If the value of $K_{X_{t}}$ is not zero on an element of the kernel of (7)
then the value of $K_{\mathcal{X}^{\circ}}$ is also not zero on this element,
and thus $K_{\mathcal{X}^{\circ}}$ is not extendable to $\mathcal{X}$.

Consider the converse direction, i.e. suppose that $K_{X_{t}}$ vanishes on the
kernel of (7). Note that since $\mathcal{X}^{\circ}\supset X^{\circ}$ and
$(X,\Sigma)$ is a ${\mathbb{Q}}\Gamma$-space, the homomorphism

(8) $H_{2}(\mathcal{X}^{\circ};{\mathbb{R}})\to
H_{2}(\mathcal{X};{\mathbb{R}})\approx H_{2}(X;{\mathbb{R}})$

is surjective. The isomorphism here is provided by Proposition 1.11. Thus we
are left to prove that $K_{\mathcal{X}^{\circ}}$ vanishes on the kernel of
(8). Let $S\subset\mathcal{X}^{\circ}$ be a closed smooth surface representing
an element $\alpha$ of this kernel. We may assume that it is transversal to
$X^{\circ}\subset\mathcal{X}^{\circ}$. Since $p(S)\subset T$ is compact, the
intersection number of $X^{\circ}$ and $S$ in $\mathcal{X}^{\circ}$ vanishes,
and we may assume (after adding some handles to $S$) that $S\cap X=\emptyset$.

Consider a radius $R\subset T$, i.e. a half-open smooth path connecting $0$
and the boundary of $T$. We may assume that $S$ and $p^{-1}(R)$ are
transverse, and thus intersect along a smooth curve in $p^{-1}(R)\approx
X_{t}\times R$. Since $H_{1}(X_{t};{\mathbb{R}})=0$ we may find a
representative of $\alpha=[S]$ contained in $\mathcal{X}^{\circ}\smallsetminus
p^{-1}(R)\approx X_{t}\times(T\smallsetminus R)$ and thus in $X_{t}$. The
value of $K_{X_{t}}$ and thus of $K_{\mathcal{X}^{\circ}}$ on this
representative vanishes by assumption. ∎

######  Corollary 1.17.

Suppose that $(\mathcal{X},p)$ is a symplectic deformation of a
${\mathbb{Q}}\Gamma$-space $(X,\Sigma)$ with general fiber $X_{t}$. If
$H_{1}(X_{t};{\mathbb{R}})=0$ and $H_{2}(X_{t};{\mathbb{R}})\approx
H_{2}(X;{\mathbb{R}})$ then $(\mathcal{X},p)$ is a
${\mathbb{Q}}\Gamma$-deformation.

######  Proposition 1.18.

Suppose that $(\mathcal{X},\Sigma)$ is a ${\mathbb{Q}}\Gamma$-degeneration,
$L\subset X^{\circ}$ is a monotone Lagrangian submanifold of $X$, and
$\Gamma\subset T$ is a path as in Proposition 1.12. Then
$\Phi_{\gamma}(L)\subset X_{t}$ is a monotone Lagrangian submanifold.

###### Proof.

Given a membrane $B_{t}\subset X_{t}$, $\partial B_{t}\subset L_{t}$, we get a
new membrane $B\subset\mathcal{X}^{\circ}$, $\partial B\subset L$, by
attaching to $B_{t}$ a cylinder $\Gamma\times\partial B_{t}$ along the
symplectic connection defined by the path $\Gamma$. By construction we have
$\omega(B)=\omega(B_{t})$. The trivializations of the canonical classes of
$X^{\circ}$ and $X_{t}$ over (the orienting double cover of) $L$ and $L_{t}$
define the trivialization of the canonical class of $\mathcal{X}^{\circ}$ over
(the orienting double cover of) $L\cup L_{t}$. Thus we may speak of the values
of the relative canonical class and we get

(9) $\mu_{\mathcal{X}^{\circ}}(B_{t})=\mu_{\mathcal{X}^{\circ}}(B).$

But these values depend only on the classes of $B_{t}$ and $B$ in
$H_{2}(\mathcal{X},L_{t};{\mathbb{Q}})$ and
$H_{2}(\mathcal{X},L;{\mathbb{Q}})$ since $\mathcal{X}$ is a
${\mathbb{Q}}\Gamma$-space. Furthermore, since $B_{t}\subset X_{t}$ and
$X_{t}$ is a regular fiber of $p$, we have
$\mu_{\mathcal{X}^{\circ}}(B_{t})=\mu_{X_{t}}(B_{t})$.

To conclude the proof we note that the membrane $B$ must be homologous in
$H_{2}(\mathcal{X},L)$ to a membrane $B_{0}$ contained in $X$, since the
embedding $X\subset\mathcal{X}$ is a homotopy equivalence. We get
$\mu_{\mathcal{X}^{\circ}}(B_{0})=\mu_{X}(B_{0})$ since $X$ is a
${\mathbb{Q}}\Gamma$-space. Thus $\mu_{X}(B_{0})=\mu_{X_{t}}(B_{t})$. Since
$\omega(B_{0})=\omega(B)=\omega(B_{t})$ we conclude that $L_{t}\subset X_{t}$
is monotone with the same monotonicity coefficient as $L\subset X$. ∎

######  Remark 1.19.

For monotonicity of $L_{t}$ the hypothesis that $(\mathcal{X},\Sigma)$ is a
${\mathbb{Q}}\Gamma$-space is crucial. While $\Phi_{\gamma}(L)\subset X_{t}$
is Lagrangian for any symplectic smoothing $X_{t}$ of a symplectic singular
space $(X,\Sigma)$, it does not have to be monotone even if $(X,\Sigma)$ is a
${\mathbb{Q}}\Gamma$-space and $L\subset X^{\circ}$ is a monotone unless we
assume that $X_{t}$ is a ${\mathbb{Q}}\Gamma$-smoothing, i.e.
$(\mathcal{X},\Sigma)$ is also a ${\mathbb{Q}}\Gamma$-space.

##  2\. Toric degenerations

In this section we apply the framework described in the previous section to
the case when the ambient space $Y={\mathbb{C}}{\mathbb{P}}^{N}$ is
projective, $X=X\subset Y$ is a normal toric variety whose embedding to $Y$ is
defined by a convex lattice polyhedron $\Delta\subset{\mathbb{R}}^{n}$ and
$\Sigma\subset X$ is the singular locus of $X$. In this case smoothings of $X$
can be produced in the framework of the patchworking construction introduced
by Viro [14] (see also its interpretation in [5] and [8]). We plan to give a
detailed description of applying patchworking in the context of symplectic
smoothings in our future paper [4].

###  2.1. Toric geometry set-up

Here we recall the basic language of toric geometry (cf. e.g. [1] and the
references therein). Let $M\approx{\mathbb{Z}}^{n}$ be an integer lattice of
rank $n$ and $N=M^{*}=\operatorname{Hom}(M,Z)$ its dual lattice. Denote by
$M_{{\mathbb{R}}}=M\otimes{\mathbb{R}}$ the corresponding real vector space
and by $N_{{\mathbb{R}}}=N\otimes{\mathbb{R}}=M_{{\mathbb{R}}}^{*}$ its dual
space. Let

$\Delta\subset N_{{\mathbb{R}}}$

be a convex polyhedron with integer vertices and non-empty interior. Denote by
$\mathcal{F}\subset M\smallsetminus\\{0\\}$ the set of the primitive outwards
supporting vectors for the facets of $\partial\Delta$.

The polyhedron $\Delta$ determines a toric Kähler manifold $X=X_{\Delta}$.
There are several constructions useful in different frameworks. As a
symplectic manifold, $X$ may be obtained from the quotient
$\Delta\times(M_{{\mathbb{R}}}/M)$ of the cotangent bundle
$T^{*}\Delta=\Delta\times M_{{\mathbb{R}}}$ by the lattice $M\subset
M_{{\mathbb{R}}}$. Namely, over a face $\Delta^{\prime}\subset\partial\Delta$
we take a quotient of the fiber torus $M_{{\mathbb{R}}}/M$ by the subtorus
annihilating the vector subspace of $N_{{\mathbb{R}}}$ parallel to
$\Delta^{\prime}$. The projection to $\Delta$ yields the moment map

$\mu_{\Delta}:X_{\Delta}\to\Delta.$

For $x\in\Delta$ the inverse image $\mu_{\Delta}^{-1}(x)$ is a real torus of
dimension equal to the dimension of the face of $\Delta$ containing $x$.

As a complex manifold $X$ can be obtained with the help of $\mathcal{F}$.
Namely, each face $\Delta^{\prime}\subset\partial\Delta$ gives us a subset
$\mathcal{F}_{\Delta^{\prime}}$ consisting of the supporting vectors to the
facets whose closure contains $\Delta^{\prime}$. The elements of
$\mathcal{F}_{\Delta^{\prime}}$ generate a cone in $M$, with the dual cone
$\sigma_{\Delta^{\prime}}\subset N$. The spectrum of the semigroup algebra
(over ${\mathbb{C}}$) of $\sigma_{\Delta^{\prime}}$ is a (possibly singular)
complex chart $U_{\Delta^{\prime}}$. The complex variety $X$ is obtained by
gluing these charts. All the charts contain the complex torus
$M\otimes{\mathbb{C}}^{\times}\approx({\mathbb{C}}^{\times})^{n}$. If
$\mathcal{F}_{\Delta^{\prime}}$ is an integer basis of the subspace
$M_{{\mathbb{R}}}^{\Delta^{\prime}}\subset M_{{\mathbb{R}}}$ it spans, then
the chart $U_{\Delta^{\prime}}$ is isomorphic to

$\tilde{U}_{\Delta^{\prime}}=({\mathbb{C}}^{\times})^{\dim\Delta^{\prime}}\times{\mathbb{C}}^{n-\dim\Delta^{\prime}}.$

If it is only a rational basis then the chart $U_{\Delta^{\prime}}$ is
isomorphic to the quotient of $\tilde{U}_{\Delta^{\prime}}$ by a finite group
(with the number of elements equal to the discriminant of
$\mathcal{F}_{\Delta^{\prime}}$). Considered together, symplectic and complex
constructions yield the following proposition.

######  Proposition 2.1.

The singular locus of $X_{\Delta}$ is given by

$\Sigma=\bigcup\mu_{\Delta}^{-1}(\Delta^{\prime}),$

where the union is taken over faces $\Delta^{\prime}\subset\partial\Delta$
such that $\mathcal{F}_{\Delta^{\prime}}\subset M$ is not an integer basis of
$M_{{\mathbb{R}}}^{\Delta^{\prime}}$.

If $\Delta$ is simple, i.e. each of its vertiecs is adjacent to $n$ facets,
then a neighborhood of each singular point of $\Sigma$ in $X$ can be presented
as the quotient singularity of ${\mathbb{C}}^{n}$ by a finite group.

In particular, $X_{\Delta}$ is smooth in complex codimension 2 and thus
$H_{2n}(X;{\mathbb{Q}})={\mathbb{Q}}$.

Yet another, the third toric variety construction exhibits $X_{\Delta}$ as a
subvariety of ${\mathbb{C}}{\mathbb{P}}^{m}$ with $m=\\#(\Delta\cap N)-1$,
thus giving its non-singular part a Kähler structure. For this purpose
consider a bijection between the $m+1$ homogeneous coordinates of
${\mathbb{C}}{\mathbb{P}}^{m}$ and the lattice points of $\Delta$. Each
lattice point of $\Delta$ is a multiplicatively linear functional on
$M_{{\mathbb{C}}}=M\otimes{\mathbb{C}}^{\times}\approx({\mathbb{C}}^{\times})^{n}$.
Taken together, they define a map to $({\mathbb{C}}^{\times})^{m+1}$ and,
after projectivization, to ${\mathbb{C}}{\mathbb{P}}^{m}$. The closure
$X_{\Delta}\subset{\mathbb{C}}{\mathbb{P}}^{m}$ of the image recovers at the
same time the symplectic toric variety of the first construction and the
complex toric variety from the second construction, see [5]. This implies the
following proposition.

######  Proposition 2.2.

For each lattice polytope $\Delta\subset N_{{\mathbb{R}}}$ the pair
$(X_{\Delta},\Sigma)$ is a singular symplectic space.

The smooth locus $X^{\circ}=X\smallsetminus\Sigma$ is a Kähler manifold. The
inverse image

$Z=\mu^{-1}_{\Delta}(\partial\Delta)$

is a complex (reducible) hypersurface. Smooth points of $Z$ are simple poles
of the meromorphic volume form

(10) $\Omega=\frac{1}{(2\pi
i)^{n}}\bigwedge\limits_{j=1}^{n}\frac{dz_{j}}{z_{j}}.$

Thus the homology class (with closed support) of $Z\cap X^{\circ}$ is dual to
minus the canonical class on $X^{\circ}$. Here $z_{j}$, $j=1,\dots,n$, are
basis coordinates in the complex torus
$U_{\Delta}=M\otimes{\mathbb{C}}^{\times}\approx({\mathbb{C}}^{\times})^{n}$.

Each element $v\in M$ can be viewed by duality as a functional
$v:N_{{\mathbb{R}}}\to{\mathbb{R}}$. For a facet $\delta\subset\Delta$ denote
with $v_{\delta}\in\mathcal{F}\subset M$ its outward primitive integer
supporting vector. Then $v_{\delta}|_{\delta}$ is constant. Denote
$h_{\delta}=v_{\delta}(\delta)$. The condition that $0$ is contained in the
interior of $\Delta$ is equivalent to the condition that $h_{\delta}>0$ for
all facets $\delta\subset\Delta$. In this case the fiber

$L=\mu^{-1}_{\Delta}(0)\subset X$

is a Lagrangian torus. Furthermore, the restriction of $\Omega$ to $L$ is a
real volume form (of total volume 1). Thus the Maslov index $m(B)$ of a
membrane $B\subset X$, $\partial B\subset L$, coincides with twice the
intersection number $\\#(B\cap Z)$,

(11) $m(B)=2\\#(B\cap Z).$

######  Lemma 2.3.

The Lagrangian submanifold $L\subset X^{\circ}$ is monotone if and only if
$h=h_{\delta}$ does not depend on the choice of the facet
$\delta\subset\Delta$.

###### Proof.

Connect $0$ and a point inside $\delta$ with a simple path
$\gamma\subset\Delta$. Let $D_{\gamma}\subset X$ be the disk obtained from
$\gamma\times S\subset\Delta\times(M_{{\mathbb{R}}}/M)$ after the quotient.
Here $S\subset M_{{\mathbb{R}}}/M$ is a simple closed curve parallel to
$v_{\delta}$. By (11) we have $m(D_{\gamma})=2$. But
$\omega(D_{\gamma})=h_{\delta}$ by the construction of the symplectic form
$\omega$ on the cotangent space
$T^{*}\Delta=\Delta\times(M_{{\mathbb{R}}}/M)$. The lemma follows from the
observation that the disks $D_{\gamma}$ generate the group
$H_{2}(X^{\circ},L;{\mathbb{Q}})$. ∎

###  2.2. Toric ${\mathbb{Q}}\Gamma$-spaces and Maslov indices.

In general, $(X_{\Delta},\Sigma)$ does not have to be a
${\mathbb{Q}}\Gamma$-space.

######  Proposition 2.4.

A pair $(X_{\Delta},\Sigma)$ is a ${\mathbb{Q}}\Gamma$-space if and only if
for each vertex $\Delta^{\prime}\subset\Delta$ (a face of dimension 0) the set
$\mathcal{F}_{\Delta^{\prime}}\subset M$ is contained in a hyperplane.

###### Proof.

To compute the group $H_{2}(X_{\Delta};{\mathbb{R}})$ in terms of $\Delta$ we
consider the vector space $V_{\Delta}$ whose basis is in 1-1 correspondence
with the facets of $\Delta$ and the linear map $\lambda:V_{\Delta}\to
M_{{\mathbb{R}}}$ sending each facet to its outward normal vector. For the
dense torus $U_{\Delta}\approx({\mathbb{C}}^{\times})^{n}$ in $X_{\Delta}$ we
have

$H_{2}(X_{\Delta},U_{\Delta};{\mathbb{R}})=V_{\Delta}.$

The long exact sequence of $(X_{\Delta},U_{\Delta};{\mathbb{R}})$ implies that
$H_{2}(X_{\Delta};{\mathbb{R}})=\ker\lambda$. Let
$\operatorname{Sk}^{n-2}(\Delta)\supset X^{\circ}$ be the union of the faces
of $\Delta$ of dimension up to $n-2$. Since the inclusion homomorphism

$H_{2}(X\smallsetminus\mu_{\Delta}^{-1}(\operatorname{Sk}^{n-2}(\Delta));{\mathbb{R}})\to
H_{2}(X;{\mathbb{R}})$

is surjective, the inclusion homomorphism $H_{2}(X^{\circ};{\mathbb{R}})\to
H_{2}(X;{\mathbb{R}})$ is surjective as well.

Since $\Delta^{\prime}$ is a vertex, its inverse image
$\mu^{-1}(\Delta^{\prime})$ is a point. The chart $U_{\Delta^{\prime}}$ is
contractible and contains a fiber torus $L_{x}=\mu^{-1}(x)$,
$x\in\operatorname{Int}\Delta$. Connecting $x$ to a facet $\delta$ adjacent to
$\Delta^{\prime}$ with a simple path we get a disk $D_{\gamma}$ of Maslov
index 2 as in the proof of Lemma 2.3. The class

$[D_{\gamma}]\in
H_{2}(X_{\Delta},L_{x};{\mathbb{R}})=H_{2}(X_{\Delta},U_{\Delta};{\mathbb{R}})=V_{\Delta}$

is the basis vector corresponding to $\delta$. Thus $K_{X^{\circ}}$ has equal
values on all basis vectors in $V_{\Delta}$. But $D_{\gamma}$ is contained in
$U_{\Delta^{\prime}}$. Contractibility of $U_{\Delta^{\prime}}$ implies

$H_{2}(U_{\Delta^{\prime}},L_{x})=H_{1}(L_{x})=M\approx{\mathbb{Z}}^{n}.$

If $K_{X^{\circ}}$ is extendable to $X^{\circ}\cup U_{\Delta^{\prime}}$ then
any linear relation on the elements of $\mathcal{F}_{\Delta^{\prime}}$ has to
have zero sum of coefficients. Thus the elements of
$\mathcal{F}_{\Delta^{\prime}}$ must be contained in a hyperplane of
$M_{{\mathbb{R}}}$. Conversely, if they are contained in such a hyperplane, we
may find an extension of $K_{X^{\circ}}$ to $X^{\circ}\cup
U_{\Delta^{\prime}}$, and thus also to $X$. ∎

The hyperplane containement condition is vacuous in the case when each vertex
of $\Delta$ is adjacent to $n$ facets.

######  Corollary 2.5.

If $\Delta$ is simple then $(X_{\Delta},\Sigma)$ is a
${\mathbb{Q}}\Gamma$-space.

From now on we suppose that $0\in\Delta\subset N_{{\mathbb{R}}}$ is a simple
monotone convex lattice polyhedron, so that $L=L_{0}$ is a monotone Lagrangian
torus in the ${\mathbb{Q}}\Gamma$-space $(X_{\Delta},\Sigma)$.

Suppose that $B\subset X_{\Delta}$ is a membrane, i.e. the image of a smooth
surface map in $X$ such that $\partial B\subset L$. Suppose that $x\in B\cap
Z$ is isolated. By Proposition 2.1, a neighbourhood of $x$ is locally
isomorphic to the quotient of ${\mathbb{C}}^{n}$ by a group of order
$l\in{\mathbb{Z}}_{>0}$. We define the local intersection number
$\\#_{x}(B\cap Z)$ as $\frac{1}{l}$ times the local intersection number of the
inverse images of $B$ and $Z$ in ${\mathbb{C}}^{n}$. It is a rational number.

Recall that $U_{\Delta}=\mu^{-1}_{\Delta}(\operatorname{Int}\Delta)$ has the
canonical complex structure of
$M\otimes{\mathbb{C}}^{\times}\approx({\mathbb{C}}^{\times})^{n}$. Thus it
comes with the natural map

$\operatorname{Log}:({\mathbb{C}}^{\times})^{n}\approx U_{\Delta}\to
M_{{\mathbb{R}}}\approx{\mathbb{R}}^{n},$

given in coordinates by
$\operatorname{Log}(z_{1},\dots,z_{n})=(\log|z_{1}|,\dots,\log|z_{n}|)$.

A non-zero element $v\in M$ generates a subgroup in
$M\otimes{\mathbb{C}}^{\times}$, and thus also a holomorphic cylinder
$C_{v}\subset({\mathbb{C}}^{\times})^{n}$ such that
$\operatorname{Log}(C_{v})\subset M_{{\mathbb{R}}}$ is the line passing
through the origin and parallel to $v$. We refer to any multiplicative
translate $aC_{v}$, $a\in M\otimes{\mathbb{C}}^{\times}$ as a holomorphic
cylinder parallel to $v$.

By a proper holomorphic curve in $X$ with boundary in $L$ we mean the image
$B=f(S)\subset X$ of a non-constant continuous map $f:S\to X$ from a compact
Riemann surface $S$ with boundary $\partial S$ such that
$f|_{S\smallsetminus\partial S}$ is holomorphic and $f(\partial S)\subset L$.
We say that $B$ is irreducible if $S$ is connected. We refer to an irreducible
proper holomorphic curve with boundary in $L$ (as well as to its image in $X$)
as a holomorphic membrane for $L$ if its boundary $\partial B=f(\partial S)$
is non-empty.

######  Proposition 2.6.

Suppose that $B\subset X_{\Delta}$ is a holomorphic membrane for $L_{x}$,
$x\in\operatorname{Int}\Delta$. Then $B\cap Z$ is a finite set and the Maslov
index of $B$ can be computed as

(12) $m(B)=2\sum\limits_{z\in B\cap Z}\\#_{z}(B\cap Z),$

and $\\#_{z}(B\cap Z)>0$ for any $z\in B\cap Z$.

###### Proof.

Since $Z$ is disjoint from a neighborhood of $\partial B\subset L$, the set of
isolated points of $Z\cap B$ must be finite, $l(\\#_{z}(B\cap Z))$ is the
local intersection number of a curve and a hypersurface in ${\mathbb{C}}^{n}$
and thus is a positive integer number. The formula (12) is a corollary of (11)
and the surjectivity of the homomorphism $H_{2}(X^{\circ},L)\to H_{2}(X,L)$. ∎

######  Lemma 2.7.

Suppose that $B=f(S)\subset X_{\Delta}$ is a holomorphic membrane for
$L_{x}=\mu_{\Delta}^{-1}(x)$, $x\in\operatorname{Int}\Delta$, such that
$f^{-1}(Z)\subset S$ consists of a single point.

Then $\nu_{{\mathbb{C}}}|_{B}$ is constant for any multiplicatively linear map

$\nu_{{\mathbb{C}}}:M\otimes{\mathbb{C}}^{\times}\to{\mathbb{C}}^{\times}$

defined as the pairing with an element $\nu\in N$ such that $\nu([\partial
B])=0$.

Here we view $[\partial B]$ as the element of
$H_{1}(L_{x};{\mathbb{R}})=M_{{\mathbb{R}}}$ (sitting on the lattice $M\subset
M_{{\mathbb{R}}}$).

###### Proof.

Note that
$\nu_{{\mathbb{C}}}:M\otimes{\mathbb{C}}^{\times}\to{\mathbb{C}}^{\times}$ is
a product of integer powers of the holomorphic coordinates in the dense torus
$U_{\Delta}\subset X_{\Delta}$. Let $w\in S\smallsetminus\partial S$ be the
(only) point of $f^{-1}(Z)$. Its image $f(w)\in Z$ is contained in the toric
chart $U_{\Delta^{\prime}}$ for a face $\Delta^{\prime}\subset\Delta$, and
thus the meromorphic function $\nu_{{\mathbb{C}}}\circ f$ near $w$ can be
expressed in a local coordinate $z$ near $w$ as

$c(z-w)^{k}+o((z-w)^{k}),$

$c\in{\mathbb{C}}^{\times}$, $k\in{\mathbb{Z}}$. If $k>0$ then a small
punctured disk centred at $w$ covers a small disk in ${\mathbb{C}}$ centred at
0. If $k<0$, we get such covering for $1/(\nu_{{\mathbb{C}}}\circ f)$. In both
cases we conclude that the linking number of $\nu_{\mathbb{C}}\circ f(\partial
S)$ and $0$ in ${\mathbb{C}}$ is $k\neq 0$. We get a contradiction since this
number is given by $\nu[\partial B]=0$. Thus $k=0$ and the function
$\nu_{{\mathbb{C}}}\circ f$ can be extended to a continuous function on $S$
holomorphic on $S\smallsetminus\partial S$ and constant on $\partial S$ (since
$f(\partial S)\subset L_{x}$). By the maximum principle,
$\nu_{{\mathbb{C}}}\circ f$ is constant. ∎

######  Corollary 2.8.

Suppose that $B=f(S)\subset X_{\Delta}$ is a holomorphic membrane for
$L_{x}=\mu_{\Delta}^{-1}(x)$, $x\in\operatorname{Int}\Delta$, such that
$f^{-1}(Z)\subset S$ consists of a single point.

Then $[\partial B]\neq 0\in H_{1}(L)=M$, and
$B\cap(M\otimes{\mathbb{C}}^{\times})$ is contained in a holomorphic cylinder
$a\lambda C_{[\partial B]}$, $a\in({\mathbb{C}}^{\times})^{n}$, parallel to
$[\partial B]\in M$.

###### Proof.

By Lemma 2.7, $\nu_{{\mathbb{C}}}$ is constant on the set
$B\cap(M\otimes{\mathbb{C}}^{\times})$ for any $\nu\in N$ with $\nu[\partial
B]=0$. Thus if $[\partial B]=0$ then $B$ must consist of a single point which
leads us to a contradiction. If $[\partial B]\neq 0$ then the multiplicative
projection of $B\cap(M\otimes{\mathbb{C}}^{\times})$ along $[\partial B]$
consists of a single point in $({\mathbb{C}}^{\times})^{n-1}$, and therefore,
$B\cap(M\otimes{\mathbb{C}}^{\times})$ is contained in $aC_{[\partial B]}$ for
some $a\in({\mathbb{C}}^{\times})^{n}$. ∎

##  3\. Holomorphic membranes in toric ${\mathbb{Q}}\Gamma$-degenerations

As it was suggested in [3], monotone Lagrarangian submanifolds may be
symplectically distinguished by the enumeration of holomorphic disks of the
smallest possible Maslov index. The disks are holomorphic with respect to a
pseudoholomorphic structure compatible with the symplectic structure. In the
case of orientable monotone Lagrangian with the positive proportionality
coefficient between the Maslov index and the area, any holomorphic membrane
has the Maslov index of at least 2, so no bubbling of disks is possible. By
the Riemann-Roch formula, the expected dimension of the space of Maslov index
2 disks is $n-1$. Thus by the Gromov compactness [6] we have finitely many of
such disks passing through a fixed point of the Lagrangian in the case if
these disks are regular.

Consider a closed symplectic degeneration $p:\mathcal{X}\to T$,
$X_{t}=p^{-1}(t)$, $t\in T\ni 0$. Let $L=L_{0}\subset X_{0}$ be an orientable
Lagrangian submanifold (disjoint from the singular locus $\Sigma\subset
X_{0}\subset\mathcal{X}$). The image $L_{t}=\Phi_{\gamma}(L_{0})$ of $L_{0}$
under the symplectomorphism $\Phi_{\gamma}:X^{\circ}\to\Sigma_{\gamma}$
provided by Proposition 1.12 is an orientable Lagrangian in the smooth
symplectic manifold $X_{t}$. If $\mathcal{X}$ is a
${\mathbb{Q}}\Gamma$-degeneration and $L_{0}\subset X_{0}$ is monotone then by
Proposition 1.18 the Lagrangian $L_{t}\subset X_{t}$ is also monotone. In
other words, ${\mathbb{Q}}\Gamma$-smoothings of $(X,L)$ where $L$ is a
monotone Lagrangian variety of $X$ produce monotone Lagrangian varieties
$L_{t}$ diffeomorphic to $L=L_{0}$ in the smooth symplectic manifold $X_{t}$,
$t\neq 0$.

For the rest of the paper we assume that $(X_{t},L_{t})$ is a
${\mathbb{Q}}\Gamma$-smoothing of $(X_{0},L_{0})$ where $L_{0}\subset X_{0}$
is a monotone orientable Lagrangian. Thus for any relative homology class
$\beta_{t}\in H_{2}(X_{t},L_{t})$ with $m(\beta_{t})=2$ the number of
holomorphic disks $\\#_{\operatorname{disks}}(\beta_{t})\in{\mathbb{Z}}_{2}$
passing through a point $a\in L_{t}$ for a generic compatible almost complex
structure is well-defined. It is invariant under symplectomorphisms, and does
not depend on the choice of a generic compatible almost complex structure.
Here we reduce the number of disks mod 2 to avoid a need to discuss
orientation issues.

Since $L\subset X^{\circ}$, we have a canonical identification
$H_{1}(L_{t})=H_{1}(L)$ for any $t\in T$ by Proposition 1.12. For a class
$\alpha\in H_{1}(L;{\mathbb{R}})$ we define

(13)
$\\#_{\operatorname{disks}}(\alpha)=\sum\limits_{\beta_{t}}\\#_{\operatorname{disks}}(\beta_{t})\in{\mathbb{Z}}_{2},$

where the sum is taken over all $\beta_{t}\in H_{2}(X,L_{t})$ of Maslov index
2 and such that $\partial\beta_{t}=\alpha\in H_{1}(L_{t};{\mathbb{R}})$. By
the Gromov compactness and monotonicity, the sum on the right-hand side is
finite since the area of all holomorphic disks in the sum is the same (and
thus bounded). By Corollary 1.13, the sum does not depend on the choice of
$t\neq 0$. We define

(14) $\Delta^{\operatorname{disks}}(L)=\operatorname{ConvexHull}\\{\alpha\in
H_{1}(L;{\mathbb{R}})\ |\ \\#_{\operatorname{disks}}(\alpha)\neq 0\\}.$

This is a convex lattice polygon in the vector space $H_{1}(L;{\mathbb{R}})$
with the lattice $H_{1}(L)/\operatorname{Tor}H_{1}(L)$. Any symplectomorphism
of $(X_{t},L_{t})$ induces a lattice-preserving isomorphism of
$H_{1}(L;{\mathbb{R}})$. As a lattice polygon,
$\Delta^{\operatorname{disks}}(L)$ is a symplectic invariant of
$(X_{t},L_{t})$.

Following [10], [11], we consider the case when $L=\mu_{\Delta}^{-1}(0)$ is a
monotone Lagrangian torus in the toric variety $X_{\Delta}$ which is a
symplectic ${\mathbb{Q}}\Gamma$-space. Note that by Corollary 2.5 the latter
holds for any simple lattice polygon $\Delta\subset{\mathbb{R}}^{n}$. More
generally, the ${\mathbb{Q}}\Gamma$-condition coincides with the
${\mathbb{Q}}$-Gorenstein condition on the normal toric variety $X_{\Delta}$.
Denote

$\Delta^{*}=\operatorname{ConvexHull}(\mathcal{F})\subset M_{{\mathbb{R}}},$

where $\mathcal{F}\in M_{{\mathbb{R}}}$ is the set of primitive outward
supporting vectors for the facets of $\Delta$.

######  Theorem 1.

For any ${\mathbb{Q}}\Gamma$-smoothing $(X_{t},L_{t})$ of $(X_{\Delta},L)$ we
have

$\Delta^{\operatorname{disks}}(L)=\Delta^{*}.$

###### Proof.

Suppose that $\alpha\in H_{1}(L;{\mathbb{R}})$ is a class such that
$\\#_{\operatorname{disks}}(\alpha)\neq 0$. Consider the smooth embedded path
$\Gamma\subset T$ such that $L_{t}=\Phi_{\gamma}(L)$. For each
$s\in\Gamma\smallsetminus\\{0\\}$ there exists a holomorphic disk membrane in
$(X_{s},L_{s})$ whose boundary represents $\alpha$. All these disks are
contained in the symplectic product $Y\times S^{2}$, $S^{2}\supset T$, and
have the same symplectic area. The boundary of the disks is contained in the
Lagrangian submanifold $\bigcup_{s\in\Gamma}L_{s}\subset Y\times T$. By the
Gromov compactness we may extract a subsequence of holomorphic disk membranes
converging to a possibly reduced (bubbled) curve $B\subset X$ of Maslov index
2 whose boundary represents $\alpha$.

The components of $B$ consist of holomorphic disk membranes
$B_{j}=f_{j}(S_{j})$ for $(X_{\Delta},L)$ as well as closed rational curves
$A_{k}\subset X_{\Delta}$. We define the Maslov numbers of the closed
components $A_{k}$ by $m(A_{k})=-2K.[A_{k}]$. The Maslov numbers of $A_{k}$
and $B_{j}$ are positive rational numbers adding to $m(B)=2$. Suppose that
$C=f_{C}(S_{C})$ is a component of $B$ with the property that $C\smallsetminus
Z\neq 0$ (recall that $Z$ is union of toric divisors given by (10)). Any point
$z\in f^{-1}_{C}(S_{C})$ is isolated. Furthermore, the sum of the Maslov
numbers of all components $A_{k}$ bubbling at $z$ and twice the local
intersection number of $f_{C}$ and $Z$ at $z$ is a positive even number.
Indeed the half of this number coincides with the obstruction to extending the
trivialization of $\Lambda^{n}(T^{*}(X_{\Delta}-\Sigma))$ given by (10) from a
small positive loop around $z$ to a disk around $z$. Thus $C\subset
X_{\Delta}$ must be a disk component with $[\partial C]=\alpha$ and
$f_{C}^{-1}(Z)$ consists of a single point $z$.

The image $\mu_{\Delta}(z)$ is contained in the relative interior of a face of
$\Delta$. If $\Delta^{\prime}$ is any vertex of this face then $C\subset
U_{\Delta^{\prime}}$. Since the vertex chart $U_{\Delta^{\prime}}$ is
contractible, we have

$H_{2}(U_{\Delta^{\prime}},L;{\mathbb{R}})=H_{1}(L;{\mathbb{R}})=M_{{\mathbb{R}}}.$

The Maslov index of membranes in $(U_{\Delta^{\prime}},L)$ yields a linear
function $M_{{\mathbb{R}}}\to{\mathbb{R}}$ taking value 2 at all elements of
$\mathcal{F}_{\Delta^{\prime}}$ (spanning a hyperplane in $M_{{\mathbb{R}}}$
by the ${\mathbb{Q}}\Gamma$-condition). Since $z\in U_{\Delta^{\prime}}$, the
vector $\alpha\in M_{{\mathbb{R}}}$ belongs to the cone spanned by
$\mathcal{F}_{\Delta^{\prime}}$. If $m(C)=2$ then $\alpha$ belongs to the
convex hull of the set $\mathcal{F}_{\Delta^{\prime}}$. If $0<m(C)<2$ then
$\alpha$ belongs to the convex hull of the union of
$\mathcal{F}_{\Delta^{\prime}}$ and the origin. This implies
$\Delta^{\operatorname{disks}}(L)\subset\Delta^{*}$.

For the opposite inclusion it suffices to prove that
$\\#_{\operatorname{disks}}(\alpha)\neq 0$ if $\alpha$ is a vertex of
$\Delta^{*}$. Consider the holomorphic cylinder $aC_{\alpha}\subset
M\otimes{\mathbb{C}}^{\times}$ passing through $a\in L$. The complement
$aC_{\alpha}\smallsetminus L$ consists of two components. Let $B$ be the
component whose boundary orientation at $L$ agrees with $\alpha$. The closure
$\bar{B}\supset B$ in $X_{\Delta}$ is a regular holomorphic disk intersecting
$Z$ transversally at a single point $w$ (in the divisor corresponding to
$\alpha$). We have $m(\bar{B})=2$ by (11). Furthermore, for any other
(bubbled) holomorphic limit $C\cup\bigcup\limits_{k}A_{k}$ of holomorphic
disks in $(X_{s},L_{s})$ its membrane part $C$ must intersect $Z$
transversally at the same point $w$ by Corollary 2.8. Thus $m(C)=2$ and
$C=\bar{B}$, so no bubbling can appear and
$\\#_{\operatorname{disks}}(\alpha)=1$. ∎

Let $\Delta,\Delta^{\prime}\subset N_{{\mathbb{R}}}\approx{\mathbb{R}}^{n}$ be
two monotone lattice polytopes and $\Delta^{*},(\Delta^{\prime})^{*}\subset
M_{{\mathbb{R}}}\approx{\mathbb{R}}^{n}$ be their dual polytopes. Let
$X_{\Delta}$, $X_{\Delta^{\prime}}$ be the corresponding toric varieties with
the singular loci $\Sigma$, $\Sigma^{\prime}$ and the monotone Lagrangian tori
$L=\mu_{\Delta}^{-1}(0)\subset X_{\Delta}$,
$L^{\prime}=\mu_{\Delta^{\prime}}^{-1}(0)\subset X_{\Delta^{\prime}}$. Suppose
that $(X_{\Delta},\Sigma)$ and $(X_{\Delta^{\prime}},\Sigma^{\prime})$ admit
${\mathbb{Q}}\Gamma$-smoothings $X_{t}$ and $X^{\prime}_{t}$. Let
$L_{t}\subset X_{t}$ and $L^{\prime}_{t}\subset X^{\prime}_{t}$ be the
monotone Lagrangian tori given by Proposition 1.18.

######  Corollary 2.

If the pairs $(X_{t},L_{t})$ and $(X^{\prime}_{t},L^{\prime}_{t})$ are
symplectomorphic then the polytopes $\Delta^{*},(\Delta^{\prime})^{*}\subset
M_{{\mathbb{R}}}$ coincide up to an automorphism of the lattice $M$.

###### Proof.

The automorphism of $M=H_{1}(L_{t})=H_{1}(L^{\prime}_{t})$ is induced by the
restriction to $L_{t}$ of the symplectomorphism between $X_{t}$ and
$X^{\prime}_{t}$. ∎

In other words, if $X_{\Delta}$ and $X_{\Delta^{\prime}}$ are different toric
degenerations of the same symplectic manifold then the tori $L_{t}$ and
$L^{\prime}_{t}$ are symplectically different.

######  Corollary 3.

The number of non-symplectomorphic Lagrangian tori in a complex projective
variety is bounded from below by the number of its ${\mathbb{Q}}$-Gorenstein
toric degenerations.

In particular, the infinite number of different toric degenerations of Del
Pezzo surfaces (described in [7] for $\dim H_{2}(X_{\Delta};{\mathbb{R}})=1$
and in [2] in other cases) produces an infinite number of different Lagrangian
tori in ${\mathbb{C}}{\mathbb{P}}^{2}$ and all other Del Pezzo surfaces thus
recovering the results from [13] and [12]).

Acknowledgement We thank Joé Brendel and Felix Schlenk for their helpful
feedback to the first draft of this paper.

## References

  * [1] Ana Cannas da Silva.  Lectures on symplectic geometry, volume 1764 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 2001.
  * [2] John Alexander Cruz Morales and Sergey Galkin.  Upper bounds for mutations of potentials.  SIGMA Symmetry Integrability Geom. Methods Appl., 9:Paper 005, 13 pages, 2013.
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  * [4] Sergey Galkin and Grigory Mikhalkin.  Patchworking by mutations.  In preparation.
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  * [6] M. Gromov.  Pseudo holomorphic curves in symplectic manifolds.  Invent. Math., 82(2):307–347, 1985.
  * [7] Paul Hacking and Yuri Prokhorov.  Smoothable del Pezzo surfaces with quotient singularities.  Compos. Math., 146(1):169–192, 2010.
  * [8] Ilia Itenberg, Grigory Mikhalkin, and Eugenii Shustin.  Tropical algebraic geometry, volume 35 of Oberwolfach Seminars.  Birkhäuser Verlag, Basel, 2007.
  * [9] J. Kollár and N. I. Shepherd-Barron.  Threefolds and deformations of surface singularities.  Invent. Math., 91(2):299–338, 1988.
  * [10] Takeo Nishinou, Yuichi Nohara, and Kazushi Ueda.  Toric degenerations of Gelfand-Cetlin systems and potential functions.  Adv. Math., 224(2):648–706, 2010.
  * [11] Takeo Nishinou, Yuichi Nohara, and Kazushi Ueda.  Potential functions via toric degenerations.  Proc. Japan Acad. Ser. A Math. Sci., 88(2):31–33, 2012.
  * [12] Renato Vianna.  Infinitely many monotone Lagrangian tori in del Pezzo surfaces.  Selecta Math. (N.S.), 23(3):1955–1996, 2017.
  * [13] Renato Ferreira de Velloso Vianna.  Infinitely many exotic monotone Lagrangian tori in $\mathbb{CP}^{2}$.  J. Topol., 9(2):535–551, 2016.
  * [14] O. Ya. Viro.  Real plane algebraic curves: constructions with controlled topology.  Algebra i Analiz, 1(5):1–73, 1989.


# Conway’s Nightmare: Brahmagupta and Butterflies

Richard Evan Schwartz Supported by N.S.F. Grant DMS-2102802

A cyclic quad is a convex quadrilateral whose vertices all lie on the same
circle. Equivalently, opposite interior angles sum to $\pi$. That is, if we
let $\alpha,\beta,\gamma,\delta$ be the successive interior angles, then
$\alpha+\gamma=\beta+\delta=\pi$.

Let $A$ be the area of such a quadrilateral and let $a,b,c,d$ be the side
lengths. Let

$B=\sqrt{(s-a)(s-b)(s-c)(s-d)},\hskip 30.0pts=\frac{a+b+c+d}{2}.$ (1)

Brahmagupta’s formula says that $A=B$. I prefer to write

$A^{2}=B^{2}=(s-a)(s-b)(s-c)(s-d).$ (2)

Brahmagupta’s formula goes back $1400$ years. The historical paper [4]
discusses how Brahmagupta himself may have proved it. John Conway long sought
a simple and beautiful geometric proof of Brahmagupta’s formula, like Sam
Vandervelde’s recent proof [7]. Is Brahmagupta’s formula really a geometric
result? I am not so sure. When generalized to polygons with more sides, as in
[5], the discussion turns decidedly and deeply algebraic. In any case, I am
less interested in a geometric proof than I am in a proof that is short,
conceptual, and without calculation.

My proof only depends on basic facts about polynomials and continuity, but I
got the idea thinking about things in modern mathematics like flat cone
surfaces, control theory, and ergodicity. (It also helps to have smart
friends; see the acknowledgements at the end.) I call this proof “Conway’s
nightmare” because Sam had been calling his proof “Conway’s dream” in an early
draft of his paper. My proof probably would not have satisfied Conway. I will
set the stage for the proof, give the proof, then discuss the mathematics that
inspired it.

Setting the Stage: Let me explain my favorite proof of the Pythagorean
Theorem. If we fix the angles of a right triangle $T_{c}$ with sides $a,b,c$
and hypotenuse $c$, then ${\rm area\/}(T_{c})$ is proportional to $c^{2}$
because $a$ and $b$ are both proportional to $c$. We might write this as ${\rm
area\/}(T_{c})\propto c^{2}$.

Put another way, ${\rm area\/}(T_{c})=\lambda c^{2}$, where $\lambda$ is a
constant that depends on the angles. We can divide $T_{c}$ into two smaller
triangles $T_{a}$ and $T_{b}$ by dropping the altitude perpendicular to $c$.
Each smaller triangle has the same angles as $T_{c}$, so the same $\lambda$
works for all $3$ triangles. Since ${\rm area\/}(T_{c})={\rm
area\/}(T_{a})+{\rm area\/}(T_{b})$ we have $\lambda c^{2}=\lambda
a^{2}+\lambda b^{2}$. Cancelling $\lambda$ gives the result.

In this proof, we considered what happens when we vary a right triangle in a
special way: changing its size/position without changing the angles. Let us
call this operation morphing. The name will be more apt when we apply it to
cyclic quads. Noting how the relevant quantities associated to a right
triangle change when we morph, we get the desired relation up to a constant
that cancels out. This proves the Pythagorean Theorem one “angle-type” (a.k.a.
similarity class) at a time.

My proof of Brahmagupta’s formula has the same flavor. There is one part
(second paragraph) that examines how the relevant quantities vary when we
morph. – i.e., vary without changing the angles. This analysis alone gives a
useful partial result. The other part of the proof (first paragraph) combines
the morphing result with an obvious result concerning another operation,
recutting, to close the deal.

Let me recall two notions that arise in my proof. The signed distance between
two points $p,q$ on the same line $\ell$ is the dot product $(p-q)\cdot u$,
where $u$ is one of the two unit vectors parallel to $\ell$. The signed area
of a polygon is the sum of the areas of all the regions it bounds, weighted by
the number of times the polygon winds around the points in each region. This
quantity also has an algebraic expression, in terms of determinants. Both
quantities require choices; this amounts to choosing signs for some example.

The Proof: Let $C=A^{2}/B^{2}$. Let $X$ be the space of cyclic quads. To morph
a quad is to replace it by one with the same angles. To recut a quad is to cut
along a diagonal and reverse one triangle. Recutting preserves $C$. We claim
morphing does too. From any point in $X$ we can reach all nearby points by
morphs and recuts. (Recut, morph, re-recut to perturb one pair of opposite
angles; repeat using the other diagonal; morph one final time.) Hence $C$ is
constant on $X$. Since $X$ has squares, $C=1$.

Proof of Claim: Let $L(\rho,\sigma)$ be the space of lines
$\ell_{a},\ell_{b},\ell_{c},\ell_{d}$ with slopes $\rho,\sigma,-\rho,-\sigma$
and $\ell_{a}\cap\ell_{c}=(0,0)$. Parametrize
$L\cong\mbox{\boldmath{$R$}}^{2}$ by $(x,y)=\ell_{b}\cap\ell_{d}$. Let
$a,b,c,d$ be the signed distances between vertices of the associated quads,
and let $A$ be the signed area. Choose the signs so that $a,b,c,d,A>0$ in a
convex case. $B^{2}(x,y)$ is a degree $4$ polynomial as $a(x,y),...,d(x,y)$
are linear. $A(x,y)$, a sum of determinants of linear functions, is a degree
$2$ polynomial. If $xy=0$ the quads are butterflies, so $A=0$; also $|a|=|c|$
and $|b|=|d|$ and $ab=-cd$, so $2$ factors of $B^{2}$ vanish. Given their
degrees, $A(x,y)\propto xy$ and $B^{2}(x,y)\propto(xy)^{2}$. Hence $C|_{L}$ is
constant. Our claim follows: A (generic) quad and its morphs are all isometric
to quads in the same $L$. $\spadesuit$

Discussion: The main idea is that the roots, counted with multiplicity,
determine a real polynomial up to constants provided that the number of roots
equals the degree of the polynomial. A single evaluation then determines the
constant. My proof can be summarized like this: The function $C=A^{2}/B^{2}$
is invariant under recutting. It is also invariant under morphing because,
when analytically continued, $A^{2}$ and $B^{2}$ vanish to the same order on
the set of butterflies and nowhere else. The recutting/morphing process
spreads the constancy of $C$ through $X$ like a virus.

To make this idea work, we have to enlarge the space $X$ so that it includes
some nonconvex quads, especially butterflies. We’ll explain it from another
point of view here. First of all, let us modify $X$ so that we consider cyclic
quads modulo isometry. Call quads cousins if they are morphs of each other. We
think of $X$ as a fiber bundle, where the fibers are the cousin families. Each
fiber is a convex cone in $\mbox{\boldmath{$R$}}^{2}$. We create a new space
$X^{*}$ by replacing these cones by the copies of $\mbox{\boldmath{$R$}}^{2}$
which contain them. The space $X^{*}$ is a plane bundle with the same base.
The fibers are our $L$ spaces.

The fact that $A$ is a degree $2$ polynomial on the fibers is a key idea of
Bill Thurston’s paper Shapes of Polyhedra [6]. In Thurston’s work, he
introduces local complex linear coordinates on the space of flat cone spheres
with prescribed cone angles. Prescribing the cone angles is like restricting
to a fiber. Thurston’s coordinates are like my $(x,y)$ coordinates. He shows
the area of a flat cone surface with fixed cone angles is the diagonal part of
a Hermitian form in his coordinates. There is also a real valued version of
this theory which is even closer to my proof, exposited recently in the A.M.S.
Notices [2] by Danny Calegari. The same ideas also arise in translation
surfaces. The main point is that if you fix the slopes of the lines (or the
cone angles, in Thurston’s case), various algebraic functions are simplified
and become linear.

Now we discard $X^{*}$ and go back to $X$. Once we know that $C$ is fiberwise
constant on $X$ how do we get $C=1$? Let me mention two alternate approaches
first. Each fiber contains triangles (i.e. degenerate quads) and then we could
deduce $C=1$ by Heron’s formula, a degenerate version of Brahmagupta’s formula
which is somewhat easier to prove. (There are other reductions of Brahmagupta
to Heron, e.g. [1].) However, you would still need to prove Heron’s formula.
Better yet, Peter Doyle noticed that each fiber contains a (perhaps nonconvex)
quad whose diagonals are perpendicular, and that for such quads Brahmagupta’s
formula can be verified with some clever but ultimately easy algebra. I’ll
leave this as a challenge.

My inspiration for the morphing/recutting proof came from control-theory
flavored proofs of ergodicity. The prototypical example is E. Hopf’s proof [3]
that the geodesic flow on a hyperbolic surface is ergodic, meaning that any
invariant (measurable) function is (almost everywhere) constant. This may seem
far-fetched, but consider the picture.

The geodesic flow lives on a $3$-manifold, the unit tangent bundle of the
surface. This $3$ manifold has $2$ invariant codimension $1$ foliations, the
stable foliation and the unstable foliation. The first step in Hopf’s proof is
to use the expansion/contraction properties of the flow along the leaves of
the foliations to establish the (almost everywhere) constancy on each leaf of
the stable foliation and each leaf of the unstable foliation. The next step is
to walk around, going from a stable leaf to an unstable leaf to a stable leaf,
etc, to spread this constancy around over the whole $3$-manifold. Our space
$X$ has $2$ codimension $1$ foliations, each consisting of quads sharing a
pair of opposite interior angles. The method I gave, in the first paragraph of
the proof, for moving around in $X$ is the same kind of alternating walk
through the leaves of these foliations.

I can’t resist explaining $2$ other ways I might have done the control theory
part. The first approach minimizes “effort” and the second approach minimizes
exposition length, but both make extra demands on the reader.

  1. 1.

Using a finite sequence of morphs and recuts one can start with some point in
$X$ and reach an open subset of $X$. Hence $C$ is constant on an open subset
of $X$. Since $X$ is connected and $C$ is an analytic function, $C$ is
constant on $X$. Since $X$ has squares, $C=1$.

  2. 2.

One can join an arbitrary cyclic quad to a square by a finite sequence of
morphs and recuts. Hence $C=1$.

Here is an algorithm for the second approach. To make it as clean as possible
we note that both morphing and recutting extend to degenerate cyclic quads –
i.e. triangles with one marked point. Morph so as to maximize the intersection
angle between the diagonals, recut along the longest diagonal, repeat until
done. Let $\phi=(d_{1}^{2}+d_{2}^{2})/D^{2}$ where $d_{1},d_{2}$ are the
diagonal lengths and $D$ is the diameter of the circle containing the
vertices. The algorithm produces a finite number of marked triangles,
increasing $\phi$ by a factor of at least $3/2$ each step, until $\phi>1$.
Then we get one or two quads with perpendicular diagonals, the last being a
square.

Acknowledgements: Peter Doyle rekindled my interest in Brahmagupta’s formula
by showing me Sam Vandervelde’s proof. Then Peter went on to explain how one
can rotate a cyclic quad so that it is a quad of $L(\rho,\sigma)$. Something
about this rang a bell, and once I realized that this was just like Thurston’s
paper the rest fell into place. Peter’s insight about the connection to
$L(\rho,\sigma)$ was my main inspiration. After I explained my proof to Jeremy
Kahn, he suggested the idea of taking the quotient by translation and working
in $\mbox{\boldmath{$R$}}^{2}$ rather than $\mbox{\boldmath{$R$}}^{4}$. This
simplified the algebra. Danny Calegari, Dan Margalit, Javi Gomez-Serrano, and
Joe Silverman all had helpful expository suggestions.

## References

  * [1] A. Hess, A highway from Heron to Brahmagupta, Forum Geometricorum 12 (2012) pp 121–192.
  * [2] D. Calegari, Pentagonum Pentagonorum, A.M.S. Notices, Sept 2021
  * [3] E. Hopf, Ergodic theory and the geodesic flow on surfaces of constant negative curvature, B. A. M. S. 77(6), (1971) pp 863-877
  * [4] S. Kichenassamy, Brahmagupta’s derivation of the area of a cyclic quadrilateral, Historica Mathematica, Elsevier, 2010, 37(1), pp 28–61
  * [5] F. M. Maley, D. P. Robbins, J. Roskies, On the Areas of Cyclic and Semicyclic Polygons, arXiv: 0407300
  * [6] W. Thurston, Shapes of polyhedra and triangulations of the sphere, Geometry and Topology Monographs, Volume 1: The Epstein Birthday Schrift, pp 511-549
  * [7] S. Vandervelde, In Search of Conway’s Proof, preprint 2022