# A Lagrangian Klein bottle you can’t squeeze

Jonny Evans

###### Abstract

Suppose you have a nonorientable Lagrangian surface $L$ in a symplectic
4-manifold. How far can you deform the symplectic form before the smooth
isotopy class of $L$ contains no Lagrangians? I solve this question for a
particular Lagrangian Klein bottle. I also discuss some related conjectures.

##  1 Introduction

Here are two overlapping questions:

######  Question 1.1 (Minimal nonorientable genus).

Given a symplectic 4-manifold $(X,\omega)$ and a $\mathbb{Z}/2$-homology class
$\beta\in H_{2}(X;\mathbb{Z}/2)$, what is the minimal nonorientable genus of a
nonorientable Lagrangian surface $L\subset X$ with $[L]=\beta$?

######  Question 1.2 (Nonsqueezing).

Given a symplectic 4-manifold $(X,\omega)$ and a nonorientable Lagrangian
surface $L\subset X$, how far can you deform $\omega$ in cohomology before
there is no Lagrangian smoothly isotopic to $L$?

If $L$ is orientable then these questions are less interesting: the genus is
determined by $[L]^{2}=-\chi(L)$ and, in Question 1.2, it is necessary to
deform $\omega$ subject to the cohomological condition $\int_{L}[\omega]=0$.
By contrast, if $L$ is nonorientable, we have $H^{2}(L;\mathbb{R})=0$, which
means that it is possible to deform $\omega$, keeping $L$ Lagrangian, in such
a way that $[\omega]$ ranges over an open set in $H^{2}(X;\mathbb{R})$.

I will give some general discussion of these questions in turn, then give a
concrete example of a Lagrangian Klein bottle for which Question 1.2 can be
answered completely (Theorem 3.1).

One running theme throughout the discussion is the use of visible and tropical
Lagrangians in almost toric 4-manifolds: these provide a rich source of
Lagrangian submanifolds coming respectively from straight lines and tropical
curves in integral affine surfaces. I have found them useful for thinking
about some of the phenomena under discussion, and for formulating conjectures.
Visible Lagrangians were introduced in Symington’s work [16]; tropical
Lagrangians were introduced independently by Mikhalkin [9] and Matessi [8].

###  1.1 Acknowledgements

I would like to thank Ivan Smith, Emily Maw, Georgios Dimitroglou Rizell, and
Daniel Cavey for helpful conversations around this topic. My research is
supported by EPSRC Grant EP/P02095X/2.

##  2 The minimal genus question

###  2.1 Review

######  Definition 2.1.

Define the nonorientable genus of the nonorientable surface
$\\#_{k}\mathbf{RP}^{2}$ to be $k$. Proposition 1.1 of [4] shows that any
$\mathbb{Z}/2$-homology class in a symplectic 4-manifold can be represented by
some embedded nonorientable Lagrangian, so Question 1.1 has a well-defined
answer, which I will denote111ŋ is the International Phonetic Alphabet symbol
for the “ng” sound. by $\textrm{{\tipaencoding\ng}}(X,\omega,\beta)$.

######  Remark 2.2.

Audin [1] showed that

$P_{2}(\beta)=\chi(L)=2-k\mod 4,$

where $P_{2}$ denotes the Pontryagin square operation and $\chi$ is the Euler
characteristic. If you find a Lagrangian with nonorientable genus $k$ then you
can perform a Hamiltonian finger move locally to introduce pairs of
intersections with index difference 1 and then perform Polterovich surgery
[13] on these self-intersections to get an embedded Lagrangian with
nonorientable genus $k+4$. This means that the set of genera which can be
realised is
$\\{\textrm{{\tipaencoding\ng}}(X,\omega,\beta),\textrm{{\tipaencoding\ng}}(X,\omega,\beta)+4,\ldots\\}$.

######  Remark 2.3.

The quantity $\textrm{{\tipaencoding\ng}}(X,\omega,\beta)$ is known in a small
range of cases, the lower bound being the principal difficulty.

  1. 1.

When $X$ satisfies $[\omega]\cdot c_{1}(X)>0$, we know that
$\textrm{{\tipaencoding\ng}}(X,\omega,0)=6$. This follows from Givental’s
construction [5] of a Lagrangian $\\#_{6}\mathbf{RP}^{2}$ in the 4-ball and
from the fact, proved by Shevchishin [14] that $X$ contains no nullhomologous
Lagrangian Klein bottles (see also the beautiful papers by Nemirovski [12,
11]).

  2. 2.

Let $X_{a,b,c}$ be the blow-up of the 4-ball in three subballs so that the
symplectic areas of the exceptional spheres $E_{1},E_{2},E_{3}$ are $a,b,c$.
Shevchishin and Smirnov [15] show that $E_{1}+E_{2}+E_{3}$ contains a
Lagrangian $\mathbf{RP}^{2}$ if and only if the following inequalities all
hold

$a<b+c,\quad b<c+a,\quad c<a+b.$

They call these the symplectic triangle inequalities. This gives the lower
bound $\textrm{{\tipaencoding\ng}}(X_{a,b,c},\omega,E_{1}+E_{2}+E_{3})\geq 5$
when $a,b,c$ violate the triangle inequalities.

######  Remark 2.4.

After the fact, we see that there is a tropical or almost toric motivation for
the Shevchishin-Smirnov triangle inequalities. The almost toric base diagram
in Figure 1 depicts the blow-up $X_{a,b,c}$; the affine lengths $a,b,c$
indicated correspond to the sizes of the exceptional spheres
$E_{1},E_{2},E_{3}$. In red you can see a tropical curve; using the ideas of
Mikhalkin [9] and Matessi [8], we can construct a Lagrangian submanifold $L$
living over a (small thickening of a) tropical curve. This tropical Lagrangian
is diffeomorphic to $\mathbf{RP}^{2}$ if and only if the inequalities all
hold: the preimage of the point marked with cross-hairs is a circle in $L$
whose neighbourhood is a Möbius strip.

$\times$$\times$$\oplus$$c$$c$$a$$b$$\times$$\times$$c$$c$$a$$b$ Figure 1:
Almost toric base diagrams for $X_{a,b,c}$ with a tropical curve in red. Left:
The symplectic triangle inequalities and the associated tropical Lagrangian is
diffeomorphic to $\mathbf{RP}^{2}$ (with the core circle of a cross-cap living
over the point marked by the cross-hair symbol). Right: The symplectic
triangle inequalities are violated and the associated tropical Lagrangian is
diffeomorphic to a disc.

###  2.2 $S^{2}\times S^{2}$

Let $X=S^{2}\times S^{2}$. Modulo an overall scale factor, any symplectic form
on $X$ is diffeomorphic to one from the family $\lambda
p_{1}^{*}\sigma+p_{2}^{*}\sigma$, where $p_{1},p_{2}\colon X\to S^{2}$ are the
two projections and $\sigma$ is an area form on $S^{2}$. We know that
$\textrm{{\tipaencoding\ng}}(X,\omega,0)=6$, which leaves two interesting
$\mathbb{Z}/2$-homology classes up to diffeomorphism: $\beta=[\star\times
S^{2}]$ and the class $\Delta$ of the diagonal. The Pontryagin squares are
$P_{2}(\beta)=0$ and $P_{2}(\Delta)=2$, so there is a chance to represent
$\beta$ by Lagrangian Klein bottles.

$\ell$$\lambda$$\otimes$$\otimes$ Figure 2: A visible Lagrangian Klein bottle
in $(S^{2}\times S^{2},\omega_{\lambda})$ for $\lambda<2$. The cores of two
cross-caps are indicated with cross-hairs.

######  Lemma 2.5.

If $\lambda<2$ then $\beta$ is represented by a Lagrangian Klein bottle.

###### Proof.

The rectangle in Figure 2 is the toric moment polygon for the standard
Hamiltonian torus action on $S^{2}\times S^{2}$ with symplectic form
$\omega_{\lambda}$. There is a Lagrangian Klein bottle living over the line
$\ell$ (slope $1/2$) in the diagram. To see this, consider the two $S^{2}$
factors sitting inside $\mathbb{R}^{3}$ and let $(p_{j},\theta_{j})$ be
cylindrical coordinates on the $j$th factor ($j=1,2$). These are action-angle
coordinates, so $\omega=\sum dp_{j}\wedge d\theta_{j}$. The line $\ell$ is
given by $2p_{2}=p_{1}$ and the Lagrangian Klein bottle is cut out by this
equation together with $\theta_{2}=-2\theta_{1}$. This is certainly Lagrangian
for this symplectic form. To see that $L$ is a Klein bottle, notice that the
regular level sets of $p_{1}$ restricted to $L$ are circles
$\theta_{2}=-2\theta_{1}$ in the $(\theta_{1},\theta_{2})$-torus, which
collapse $2$-to-$1$ onto the circles of maxima and minima at
$p_{1}=\pm\lambda$ (as the torus collapses to the circle with coordinate
$\theta_{2}$). The projections of these circles are denoted with cross-hairs
in Figure 2. ∎

######  Remark 2.6.

This $L$ is a visible Lagrangian in the sense of Symington [16] as well as
being a tropical Lagrangian in the sense of Matessi [8] and Mikhalkin [9].
This Klein bottle is well-known: it appears in [3] as a Hamiltonian minimal
Lagrangian, in [6] as a Hamiltonian suspension, and in [4] as a fibre connect-
sum of $\mathbf{RP}^{2}$s. It has minimal Maslov number 1 and has a monotone
representative in its Lagrangian isotopy class if $\lambda=1$.

If $\lambda\geq 2$ then the line $\ell$ does not fit into the rectangle. The
following conjecture seems natural; while I cannot prove it, it inspired
Theorem 3.1 below.

######  Conjecture 2.7.

There is no Lagrangian Klein bottle in the class $\beta$ if $\lambda\geq 2$.

It is interesting to consider what happens for large $\lambda$. We have
essentially no tools to prove lower bounds when the Lagrangians are of high
genus and may be Floer-theoretically obstructed. The most pessimistic
conjecture is that Lagrangians with high genus become flexible enough that:

######  Conjecture 2.8.

$\lim_{\lambda\to\infty}\textrm{{\tipaencoding\ng}}(X,\omega_{\lambda},\beta)<\infty$.

The following lemma gives an upper bound on
$\textrm{{\tipaencoding\ng}}(X,\omega_{\lambda},\beta)$, but it goes to
infinity with $\lambda$.

######  Lemma 2.9.

We have $\textrm{{\tipaencoding\ng}}(X,\omega_{\lambda},\beta)\leq 20\ell+2$
when $\lambda<10\ell+2$.

###### Proof.

If $\lambda<10\ell+1$ then there is a tropical Lagrangian in the class $\beta$
with nonorientable genus $20\ell+2$. We show the tropical curve for $\ell=2$
in Figure 3 below; for general $\ell$ we simply repeat the pattern between the
vertical blue bars as often as required to get from the left-hand side to the
right-hand side of the rectangle.

$\otimes$$\otimes$$\otimes$$\otimes$$\otimes$pattern$\otimes$$\otimes$$\otimes$$\otimes$$\otimes$
Figure 3: A tropical curve giving a Lagrangian of genus $20\ell+2$ in the case
$\ell=2$.

The edges of this tropical curve are:

  * •

internal edges parallel to either $(3,1)$ or $(2,-1)$,

  * •

external edges parallel to $(2,-1)$ or $(1,2)$.

The corresponding tropical Lagrangian intersects the horizontal spheres with
even multiplicity and the vertical spheres with odd multiplicity, so it
inhabits the class $\beta$. The vertices of the tropical curve are not
smooth222At each vertex of a tropical curve, the outgoing edges
$v_{1},v_{2},v_{3}$ must sum to zero; if we write $m$ for the determinant
$|v_{1}\wedge v_{2}|=|v_{2}\wedge v_{3}|=|v_{3}\wedge v_{1}|$ then the self-
intersection of this vertex is defined to be $\frac{m-1}{2}$. Smoothness means
all vertices have self-intersection zero.: each has self-intersection equal to
2. By [9, Theorem 3.2], this tropical curve therefore yields an immersed
Lagrangian with $8\ell$ double points and $2+4\ell$ cross-caps where it hits
the boundary (marked with cross-hairs in Figure 3). When we perform
Polterovich surgery at the double points, we obtain a Lagrangian which is
topologically a surface of genus $8\ell$ with $4\ell+2$ cross-caps. This has
Euler characteristic $2-16\ell-4\ell-2=-20\ell$, so the nonorientable genus is
$2+20\ell$. ∎

######  Remark 2.10.

It seems harder to make the genus significantly smaller using tropical
Lagrangians, but there is no reason to believe that tropical Lagrangians
should give a sharp upper bound for ŋ.

##  3 Nonsqueezing

###  3.1 Statement

For each connected open interval $I\subset\mathbb{R}$ (length $|I|$), let
$C_{I}$ denote the cylinder $I\times(\mathbb{R}/2\pi\mathbb{Z})$ with
coordinates $(p,\theta)$, equipped with the symplectic form
$\frac{1}{2\pi}dp\wedge d\theta$; this has total area $|I|$. Let $S^{2}$
denote the 2-sphere equipped with its area form $\sigma$ satisfying
$\int_{S^{2}}\sigma=2$.

Let $U_{I}=S^{2}\times C_{I}$. Note that $U_{I}$ is obtained from
$(S^{2}\times S^{2},\omega_{|I|})$ by excising the spheres
$S^{2}\times\\{n,s\\}$, where $n,s$ denote the poles of the second factor.
Arguing as in Lemma 2.5, we see that if $|I|>1$, the only nontrivial class
$\beta\in H_{2}(U_{I};\mathbb{Z}/2)$ is represented by a Lagrangian Klein
bottle (see Figure 4).

$2$$|I|$$\otimes$$\otimes$ Figure 4: The visible Lagrangian Klein bottle in
$U_{I}$ when $|I|>1$.

######  Theorem 3.1.

Suppose that $|I|\leq 1$. If $\iota\colon K\to U_{I}$ is a Lagrangian
embedding of the Klein bottle in the class $\beta$ then
$\iota_{*}\colon\mathbb{Q}=H_{1}(K;\mathbb{Q})\to
H_{1}(U_{I};\mathbb{Q})=\mathbb{Q}$ is the zero map.

######  Remark 3.2.

The proof of Theorem 3.1 will occupy the rest of the paper. It uses SFT and
neck-stretching.

######  Remark 3.3.

Note that if $|I|>1$ then $H_{1}(L;\mathbb{Q})\to H_{1}(U_{I};\mathbb{Q})$ is
an isomorphism for the Lagrangian Klein bottle $L$ coming from Lemma 2.5. To
see this, take either one of the circles living over the points marked with
cross-hairs in Figure 4; this is a generator for both $H_{1}(L;\mathbb{Q})$
and $H_{1}(U_{I};\mathbb{Q})$. We deduce:

######  Corollary 3.4.

The Lagrangian Klein bottle in $U_{\left(0,1+\epsilon\right)}$ from Lemma 2.5
cannot be squeezed into $U_{(0,1)}$.

######  Remark 3.5.

To reduce Conjecture 2.7 to this result, you would need to produce a pair of
symplectic spheres in the class $[S^{2}\times\star]$ which “link” your
Lagrangian Klein bottle in an appropriate way. Since this class has non-
minimal symplectic area, it is difficult to control the SFT limit of such
spheres.

We now proceed to the proof of Theorem 3.1.

###  3.2 Mohnke’s almost complex structure

Pick a flat metric $g$ on the Klein bottle. There is a contact form (the
canonical 1-form) on the unit cotangent bundle $M\subset T^{*}K$ whose closed
Reeb orbits correspond to closed geodesics on $K$. We will not distinguish
notationally between geodesics and the corresponding Reeb orbits and we will
write $-\gamma$ for the geodesic obtained by reversing $\gamma$. There are two
isolated simple geodesics $\gamma_{0},\gamma_{1}$ which are the core circles
for two disjoint embedded Möbius strips in $K$. Any isolated geodesic is a
multiple cover of one of these and all other geodesics occur in one-parameter
families. We call the isolated geodesics odd and the other geodesics even.

######  Theorem 3.6 (Mohnke [10, Section 2.1]).

There exists an almost complex structure $J^{-}$ on the cotangent bundle
$T^{*}K$ with the following properties:

  1. 1.

$J^{-}$ is cylindrical at infinity and suitable for neck-stretching.

  2. 2.

For any geodesic $\gamma$ there is a finite-energy $J^{-}$-holomorphic
cylinder $f_{\gamma}$ in $T^{*}K$ asymptotic to $\gamma$ and $-\gamma$.

  3. 3.

[10, Lemma 7(2)] Any $J^{-}$-holomorphic cylinder in $T^{*}K$ which intersects
the zero-section is one of these $f_{\gamma}$ for some closed geodesic
$\gamma$.

######  Remark 3.7.

If we let $W:=\overline{T^{*}K}$ denote the compactification of the cotangent
bundle obtained by gluing on its ideal contact boundary $M$ then there is a
well-defined intersection pairing $H_{2}(W,M;\mathbb{Z}/2)\otimes
H_{2}(W;\mathbb{Z}/2)\to\mathbb{Z}/2$. The cylinders $f_{\gamma}$ define
elements of $H_{2}(W,M;\mathbb{Z}/2)$ and we have [10, Lemma 7(3)]

$f_{\gamma}\cdot K=\begin{cases}1\mbox{ if }\gamma\mbox{ is odd}\\\ 0\mbox{ if
}\gamma\mbox{ is even.}\end{cases}$

######  Remark 3.8 ([10, Lemma 7(1)]).

Note that there are also no finite energy planes in $T^{*}K$, nor in the
symplectisation $\mathbb{R}\times M$, for any cylindrical almost complex
structure adapted to our chosen contact form. This is because there are no
contractible Reeb orbits, and a finite energy plane would provide a
nullhomotopy of its asymptote.

###  3.3 Neck-stretching

Let $I=(0,1)$ and $\bar{I}=[0,1]$. Suppose there is a Lagrangian Klein bottle
$K\subset U_{I}$ such that $\mathbb{Q}=H_{1}(K;\mathbb{Q})\to
H_{1}(U_{I};\mathbb{Q})=\mathbb{Q}$ is nonzero (in particular, it is
injective). Think of $K$ sitting inside $U_{\bar{I}}$ and make symplectic cuts
to $U_{\bar{I}}$ at $p=0,1$ to obtain a Lagrangian Klein bottle $K$ living in
the manifold $X=S^{2}\times S^{2}$ equipped with the product symplectic form
giving the factors areas $2$ and $1$ respectively. Crucially, the symplectic
cut introduces symplectic spheres $S_{0}$ and $S_{1}$ (at the $p=0,1$ cuts
respectively) which are disjoint from $K$.

Pick a sequence of almost complex structures $J_{t}$, $t\in\mathbb{R}$, on $X$
with the following properties:

  * •

on a Weinstein neighbourhood of $K$, $J_{t}$ coincides with Mohnke’s almost
complex structure $J^{-}$;

  * •

on a neck-stretching region $(a_{t},b_{t})\times M$ around $K$, $J_{t}$ is a
neck-stretching sequence;

  * •

the spheres $S_{0},S_{1}$ are $J_{t}$-holomorphic for all $t\in\mathbb{R}$.

Pick a point $k$ on $K$ which does not lie on any of the cylinders
$f_{\gamma}$ for an odd geodesic $\gamma$. Let $u_{t}\colon S^{2}\to X$ be a
$J_{t}$-holomorphic curve representing the class $\alpha=[\star\times S^{2}]$
and such that $u_{t}(0)=k$; there is a unique such $u_{t}$ up to
reparametrisation by a theorem of Gromov [7, 2.4.C], since $\alpha$ is a
minimal area sphere class in $X$.

By the SFT compactness theorem [2], there is a sequence $t_{i}$ such that
$u_{t_{i}}$ converges (after reparametrisations) to a holomorphic building
with components in $T^{*}K$ (the completion of the Weinstein neighbourhood of
$K$), components in $\mathbb{R}\times M$ (the completion of the neck) and
components in $X\setminus K$ (the completion of the complement of the
Weinstein neighbourhood).

###  3.4 SFT limit analysis

The components $v_{1},\ldots,v_{n}$ of the SFT limit building living in
$X\setminus K$ can be compactified, yielding topological surfaces in $X$ with
boundary on $K$; we will still denote these by $v_{1},\ldots,v_{n}$. The sum
of the $\omega$-areas of the $v_{i}$ (weighted by multiplicities if the SFT
limit involves a branched cover) equals the $\omega$-area of $\alpha$, which
is $1$.

######  Lemma 3.9.

There must be at least two planar components amongst the $v_{i}$, possibly
geometrically indistinct (i.e. having the same image).

###### Proof.

First note that the limit building intersects $K$ because $u_{t}(0)=k\in K$
for all $t$. It also necessarily has at least one component in $X\setminus K$
because $T^{*}K$ is exact and so contains no closed holomorphic curves. A
genus zero holomorphic building with at least two levels must have two planar
components (just for topological reasons) though these could be geometrically
indistinct. Any planar components live in $X\setminus K$. ∎

######  Lemma 3.10.

There are two components $v_{0},v_{1}$ of the limit building such that
$v_{i}\cdot S_{j}=\delta_{ij}$. These components are planar and there are no
further components of the limit building in $X\setminus L$.

###### Proof.

Since $\alpha$ intersects $S_{0}$ and $S_{1}$ there must be components of the
limit building which intersect $S_{0}$ and $S_{1}$. By positivity of
intersections, either:

  1. (A)

there is one component $v_{1}$ which hits both $S_{0}$ and $S_{1}$ once
transversely and all other components are disjoint from $S_{0},S_{1}$.

  2. (B)

there are two components $v_{0},v_{1}$ such that $v_{0}$ intersects $S_{0}$
once transversely and is disjoint from $S_{1}$ and vice versa for $v_{1}$.

Moreover, each of these components occurs with multiplicity one in the SFT
limit in order to get the correct intersection numbers $\alpha\cdot
S_{0},\alpha\cdot S_{1}$.

If $v_{2}$ is a component which does not intersect $S_{0}$ or $S_{1}$ then it
defines a class in $H_{2}(U_{I},K;\mathbb{Z})$. By assumption, the kernel of
the map $\mathbb{Z}\oplus\mathbb{Z}/2=H_{1}(K;\mathbb{Z})\to
H_{1}(U_{I};\mathbb{Z})=\mathbb{Z}$ is precisely the torsion part. Therefore
the long exact sequence

$\cdots\to H_{2}(U_{I};\mathbb{Z})\to H_{2}(U_{I},K;\mathbb{Z})\to
H_{1}(K;\mathbb{Z})\to H_{1}(U_{I};\mathbb{Z})\to\cdots$

splits off a sequence

$\cdots\to\mathbb{Z}\to H_{2}(U_{I},K;\mathbb{Z})\to\mathbb{Z}/2\to 0.$

This implies that the areas of classes in $H_{2}(U_{I},K;\mathbb{Z})$ are
half-integer multiples of the area of the generator $\beta\in
H_{2}(U_{I};\mathbb{Z})$, which is $2$. Therefore $v_{2}$ has integer area.

The area of $\alpha$ is $1$, so the (weighted) sum of the $\omega$ areas of
the $v_{i}$ equals 1. Since $v_{1}$ has positive area, $v_{2}$ must have
positive area strictly less than $1$, but this is not possible if $v_{2}$ has
integer area. Therefore there cannot be any component $v_{2}$ disjoint from
$S_{0}$ and $S_{1}$.

By Lemma 3.9, there are at least two planar components (or one planar
component with multiplicity two) in the limit building. This is not compatible
with Case (A), so we must be in Case (B) and $v_{0},v_{1}$ must additionally
be planes. ∎

######  Lemma 3.11.

  1. 1.

All the remaining parts of the limit building are cylinders.

  2. 2.

At least one of these cylinders lives in $T^{*}K$ and has the form
$f_{\gamma}$ for an odd geodesic $\gamma$.

  3. 3.

There are no other cylindrical components of the SFT limit building in
$T^{*}K$.

###### Proof.

  1. 1.

If a component has three or more punctures then the limit building must
contain at least three planar components (counted with multiplicity) but we
have seen that all the planar components must live in $X\setminus K$ (Remark
3.8) and that there are precisely two such components (Lemma 3.10).

  2. 2.

Since $u_{t}(0)=k$ for all $t$, the limit building contains a component in
$T^{*}K$, which must be a cylinder of the form $f_{\gamma}$ by Theorem 3.6(3).
At least one of these cylindrical components must correspond to an odd
geodesic because $\alpha$ has odd intersection with $K$ in
$H_{2}(X;\mathbb{Z}/2)$ and the intersection number picks up contributions
from each component of the building inside $T^{*}K$, which are nontrivial if
and only if $\gamma$ is odd (Remark 3.7).

  3. 3.

If there are two or more cylindrical components in $T^{*}K$ then there must be
a further cylindrical component in $\mathbb{R}\times M$ which connects the
asymptotes of two of these cylinders. Since this cylinder has no positive
asymptote, this cannot exist by the maximum principle. ∎

###### Proof of Theorem 3.1.

We chose $k\in K$ not to lie on any of the cylinders $f_{\gamma}$ for $\gamma$
an odd geodesic, but we have showed that these are the only cylinders which
can arise as components of the SFT limit building. Since the SFT limit
building must pass through $k$, we get a contradiction. ∎

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# Dressing operators in equivariant Gromov-Witten theory of $\mathbb{CP}^{1}$

Kanehisa Takasaki
Department of Mathematics, Kindai University
3-4-1 Kowakae, Higashi-Osaka, Osaka 577-8502, Japan E-mail:
<EMAIL_ADDRESS>
###### Abstract

Okounkov and Pandharipande proved that the equivariant Toda hierarchy governs
the equivariant Gromov-Witten theory of $\mathbb{CP}^{1}$. A technical clue of
their method is a pair of dressing operators on the Fock space of 2D charged
free fermion fields. We reformulate these operators as difference operators in
the Lax formalism of the 2D Toda hierarchy. This leads to a new explanation to
the question of why the equivariant Toda hierarchy emerges in the equivariant
Gromov-Witten theory of $\mathbb{CP}^{1}$. Moreover, the non-equivariant limit
of these operators turns out to capture the integrable structure of the non-
equivariant Gromov-Witten theory correctly.

2010 Mathematics Subject Classification: 14N35, 37K10

Key words: Gromov-Witten theory, Riemann sphere, Equivariant Toda hierarchy,
Dressing operators, Lax formalism

##  1 Introduction

The Gromov-Witten theory of the Riemann sphere $\mathbb{C}\mathbb{P}^{1}$ is
known to be related to integrable hierarchies of the Toda type. Such a link
was first observed by physicists employing random matrix models [1, 2]. This
discovery was enhanced to a mathematical statement called the Toda conjecture
[3, 4, 5]. The Toda conjecture was proved by several methods [6, 7, 8, 9, 10]
and generalized to $\mathbb{C}\mathbb{P}^{1}$ with orbifold points [11, 12,
13, 14]. The relevant integrable hierarchies are the 1D Toda hierarchy, the
bigraded Toda hierarchy [15, 16] and a kind of the Kac-Wakimoto hierarchies
[17]. Another direction of generalization is the equivariant Gromov-Witten
theory of $\mathbb{C}\mathbb{P}^{1}$ [7, 18, 19, 20]. The integrable
hierarchies emerging therein are the equivariant Toda hierarchy [21] and its
bigraded version [19].

The most exotic among these integrable hierarchies will be the equivariant
Toda hierarchy. This integrable hierarchy, like the 1D Toda hierarchy, is a
reduction of the 2D Toda hierarchy. Okounkov and Pandharipande [7] encode the
Gromov-Witten invariants into the vacuum expectation value of an operator
product on the fermionic Fock space of 2D charged free fermion fields. This
fermionic expression can be converted into a tau function of the 2D Toda
hierarchy, and eventually turns out to be a tau function of the equivariant
Toda hierarchy. Milanov [18] starts from a bosonic expression of the Gromov-
Witten invariants in the Givental theory [22], and derives the Hirota bilinear
equations of the equivariant Toda hierarchy. These results are further
generalized to $\mathbb{C}\mathbb{P}^{1}$ with two orbifold points [19, 20].

This paper reconsiders Okounkov and Pandharipande’s method for the equivariant
Gromov-Witten theory from the perspective of the Lax formalism of the 2D Toda
hierarchy. A central role in their method is played by what they call dressing
operators. 111This terminology is somewhat confusing, because dressing
operators of a different kind are already used in the Lax formalism of the 2D
Toda hierarchy. These operators on the fermionic Fock space are used to
convert the fermionic expression of the Gromov-Witten invariants into a
standard expression of tau functions of the 2D Toda hierarchy. It is, however,
not very clear from this expression that the tau function is indeed a tau
function of the equivariant Toda hierarchy. We reformulate Okounkov and
Pandharipande’s dressing operators as difference operators like the Lax
operators of the 2D Toda hierarchy. This enables us to explain the relation to
the equivariant Toda hierarchy in a more direct manner. Throughout this paper,
the ordinary equivariant Toda hierarchy and its orbifold generalizations are
treated on an equal footing.

##  2 Equivariant Toda hierarchy as reduction of 2D Toda hierarchy

Let us recall the Lax formalism of the 2D Toda hierarchy (see the recent
review [23]). $\boldsymbol{t}=\\{t_{k}\\}_{k=1}^{\infty}$ and
$\bar{\boldsymbol{t}}=\\{\bar{t}_{k}\\}_{k=1}^{\infty}$ are the two sets of
time variables. $s$ is the spatial coordinate, which is understood to be a
continuous variable throughout this paper. $\Lambda$ denotes the shift
operator

$\Lambda=e^{\partial_{s}},\quad\partial_{s}=\partial/\partial s.$

The Lax operators $L,\bar{L}$ of the 2D Toda hierarchy are difference (or
pseudo-difference) operators of the form

$\begin{gathered}L=\Lambda+\sum_{n=1}^{\infty}u_{n}\Lambda^{1-n},\quad\bar{L}^{-1}=\sum_{n=0}^{\infty}\bar{u}_{n}\Lambda^{n-1},\\\
u_{n}=u_{n}(s,\boldsymbol{t},\bar{\boldsymbol{t}}),\quad\bar{u}_{n}=\bar{u}_{n}(s,\boldsymbol{t},\bar{\boldsymbol{t}}),\quad\bar{u}_{0}(s,\boldsymbol{t},\bar{\boldsymbol{t}})\not=0,\end{gathered}$

and satisfy the Lax equations

$\begin{gathered}\frac{\partial L}{\partial
t_{k}}=[B_{k},L],\quad\frac{\partial
L}{\partial\bar{t}_{k}}=[\bar{B}_{k},L],\\\ \frac{\partial\bar{L}}{\partial
t_{k}}=[B_{k},\bar{L}],\quad\frac{\partial\bar{L}}{\partial\bar{t}_{k}}=[\bar{B}_{k},\bar{L}],\end{gathered}$

where

$B_{k}=(L^{k})_{\geq 0},\quad\bar{B}_{k}=(\bar{L}^{-k})_{<0}.$

$(\quad)_{\geq 0}$ and $(\quad)_{<0}$ denote the projection onto the non-
negative and negative powers of $\Lambda$:

$\left(\sum_{n\in\mathbb{Z}}a_{n}\Lambda^{n}\right)_{\geq 0}=\sum_{n\geq
0}a_{n}\Lambda^{n},\quad\left(\sum_{n\in\mathbb{Z}}a_{n}\Lambda^{n}\right)_{<0}=\sum_{n<0}a_{n}\Lambda^{n}.$

Let us mention that these equations should be formulated in the
$\hbar$-dependent form [24] to accommodate the $\hbar$-expansion (i.e., genus
expansion) of the Gromov-Witten theory. To avoid notational complexity,
however, we dare not to consider the $\hbar$-dependent form. We can move to
the $\hbar$-dependent formulation, at least formally, by rescaling the
variables as $t_{k}\to t_{k}/\hbar$, $\bar{t}_{k}\to\bar{t}_{k}/\hbar$ and
$s\to s/\hbar$.

The Lax operators can be expressed in a dressed form as

$L=W\Lambda W^{-1},\quad\bar{L}=\bar{W}\Lambda\bar{W}^{-1}$

with the dressing operators

$\begin{gathered}W=1+\sum_{n=1}^{\infty}w_{n}\Lambda^{-n},\quad\bar{W}=\sum_{n=0}^{\infty}\bar{w}_{n}\Lambda^{n},\\\
w_{n}=w_{n}(s,\boldsymbol{t},\bar{\boldsymbol{t}}),\quad\bar{w}_{n}=\bar{w}_{n}(s,\boldsymbol{t},\bar{\boldsymbol{t}}),\quad\bar{w}_{0}(s,\boldsymbol{t},\bar{\boldsymbol{t}})\not=0.\end{gathered}$

The dressing operators $W,\bar{W}$ satisfy the Sato equations

$\begin{gathered}\frac{\partial W}{\partial
t_{k}}=B_{k}W-W\Lambda^{k},\quad\frac{\partial
W}{\partial\bar{t}_{k}}=\bar{B}_{k}W,\\\ \frac{\partial\bar{W}}{\partial
t_{k}}=B_{k}\bar{W},\quad\frac{\partial\bar{W}}{\partial\bar{t}_{k}}=\bar{B}_{k}\bar{W}-W\Lambda^{-k}.\end{gathered}$

The logarithm of $L,\bar{L}$ can be defined with the dressing operators as

$\begin{gathered}\log L=W\log\Lambda W^{-1}=\partial_{s}-\frac{\partial
W}{\partial s}W^{-1},\\\
\log\bar{L}=\bar{W}\log\Lambda\bar{W}^{-1}=\partial_{s}-\frac{\partial\bar{W}}{\partial
s}\bar{W}^{-1}.\end{gathered}$

Note that $\log\Lambda=\partial_{s}$.

Let $a,b$ be positive integers. They are related to the orders of two orbifold
points of $\mathbb{C}\mathbb{P}^{1}$. The case of $a=b=1$ amounts to the
ordinary $\mathbb{C}\mathbb{P}^{1}$. The equivariant Toda hierarchy of type
$(a,b)$ can be obtained by imposing the reduction condition [5, 19]

$L^{a}-\nu\log L=\bar{L}^{-b}-\nu\log\bar{L}-\nu\log Q,$ (1)

where $\nu$ and $Q$ are parameters of the reduction. $\nu$ is called the
equivariant parameter. In the non-equivariant limit as $\nu\to 0$, the reduced
system turns into the bigraded Toda hierarchy of type $(a,b)$ [16]. $Q$ is
related to the particular solution that we shall consider later on.

(1) implies that both sides become an operator of the form

$\mathfrak{L}=B_{a}+\bar{B}_{b}-\nu\log\Lambda.$ (2)

The Lax equations of the 2D Toda hierarchy turn into the Lax equations

$\frac{\partial\mathfrak{L}}{\partial
t_{k}}=[B_{k},\mathfrak{L}],\quad\frac{\partial\mathfrak{L}}{\partial\bar{t}_{k}}=[\bar{B}_{k},\mathfrak{L}]$

for this reduced Lax operator. In the language of tau functions, (1) amounts
to the condition that the tau function depends on $s$ in the particular form

$\tau=Q^{s^{2}/2}f(\\{t_{k}+\delta_{ka}s/\nu,\bar{t}_{k}+\delta_{kb}s/\nu\\}_{k=1}^{\infty}).$
(3)

##  3 Dressing operators $V,\bar{V}$

Okounkov and Pandharipande’s dressing operators $V,\bar{V}$ [7] 222These
operators are denoted by $W,W^{*}$ in Okounkov and Pandharipande’s notation.
We have changed the notation to avoid confusion with the aforementioned
dressing operators $W,\bar{W}$ for the Lax operators. are elements of the
$GL(\infty)$ group acting on the Fock space of 2D charged free fermion fields.
The tau function of the equivariant Gromov-Witten theory, generalized to
$\mathbb{C}\mathbb{P}^{1}$ with two orbifold points of orders $a,b$ [20], can
be thereby expressed as

$\tau=\langle
s|\exp\left(\sum_{k=1}^{\infty}t_{k}J_{k}\right)g\exp\left(-\sum_{k=1}^{\infty}\bar{t}_{k}J_{-k}\right)|s\rangle,$
(4)

where

$g=V^{-1}e^{J_{a}/a}Q^{L_{0}}e^{J_{-b}/b}\bar{V}^{-1}.$ (5)

$J_{k}$’s are the generators of the $U(1)$ current algebra of the free fermion
system, and $L_{0}$ is the zero mode of the Virasoro algebra therein.

Okounkov and Pandharipande’s construction of $V,\bar{V}$ relies on the
correspondence between fermionic operators and $\mathbb{Z}\times\mathbb{Z}$
matrices. Those matrices can be further represented by difference operators on
the discrete space $\mathbb{Z}$, e.g.,

$J_{k}\longleftrightarrow\Lambda^{k},\quad L_{0}\longleftrightarrow H=s-1/2.$
(6)

Actually, $\Lambda^{k}$ and $H$ are meaningful in the continuous space as
well. We shall redefine $V$ and $\bar{V}$ as well to be difference operators
on the continuous space.

In our reformulation, $V$ and $\bar{V}$ are difference operators of the form
333This implies that their fermionic counterparts leave invariant the ground
state of the charge-$s$ sector in the Fock space as $\langle s|V=\langle s|$
and $\bar{V}|s\rangle=|s\rangle$.

$\begin{gathered}V=1+\sum_{n=1}^{\infty}v_{n}\Lambda^{-n},\quad\bar{V}=1+\sum_{n=1}^{\infty}\bar{v}_{n}\Lambda^{n},\\\
v_{n}=v_{n}(s),\quad\bar{v}_{n}=\bar{v}_{n}(s),\end{gathered}$

and satisfy the following intertwining relations:

$\displaystyle(\Lambda^{a}+H-\nu\log\Lambda)V$
$\displaystyle=V(\Lambda^{a}-\nu\log\Lambda),$ (7)
$\displaystyle\bar{V}(\Lambda^{-b}+H-\nu\log\Lambda)$
$\displaystyle=(\Lambda^{-b}-\nu\log\Lambda)\bar{V}.$ (8)

These operators can be constructed by power series expansion with respect to
$\nu$, see Section 5. We shall show in the next section that (7) and (8) lead
correctly to a solution of the reduction condition (1).

As further evidence for the validity of our reformation, let us mention that
(7) and (8) in the case of $a=b=1$ are consistent with Okounkov and
Pandharipande’s intertwining relations between two operators
$\mathcal{A}(z,w)$ and $\tilde{\mathcal{A}}(z,w)$ on the fermionic Fock space.
The counterparts of these operators on the difference operator side of the
correspondence (6) are

$\displaystyle\mathcal{A}(z,w)$
$\displaystyle=\left(\frac{\zeta(w)}{w}\right)^{z}\sum_{k\in\mathbb{Z}}\frac{\zeta(w)^{k}}{(1+z)_{k}}\mathcal{E}_{k}(w),$
$\displaystyle\tilde{\mathcal{A}}(z,w)$
$\displaystyle=\sum_{k\in\mathbb{Z}}\frac{w^{k}}{(1+z)_{k}}\Lambda^{k},$

where

$\mathcal{E}_{k}(z)=e^{z(H+k/2)}\Lambda^{k},\quad\zeta(z)=e^{z/2}-e^{-z/2},\quad(1+z)_{k}=\frac{\Gamma(1+z+k)}{\Gamma(1+z)}.$

For matching with Okounkov and Pandharipande’s notation, let us introduce the
new parameter $u=1/\nu$ and rewrite (7), specialized to $a=b=1$, as

$(u(\Lambda+H)-\log\Lambda)V=V(u\Lambda-\log\Lambda).$ (9)

######  Proposition 1.

(9) implies the intertwining relations

$\mathcal{A}(m,mu)V=V\tilde{\mathcal{A}}(m,mu),\quad m=1,2,\ldots.$ (10)

###### Proof.

Exponentiating both sides of (9), we have the identity

$e^{m(u\Lambda+uH-\log\Lambda)}V=Ve^{m(u\Lambda-\log\Lambda)}.$

The exponential on the right side boils down to

$e^{m(u\Lambda-\log\Lambda)}=e^{mu\Lambda}\Lambda^{-m}.$

Since $[\log\Lambda,H]=1$, we can use the Baker-Campbell-Hausdorff formula to
compute the exponential on the left side as

$\displaystyle e^{m(u\Lambda+uH-\log\Lambda)}$
$\displaystyle=e^{\Lambda}e^{m(uH-\log\Lambda)}e^{-\Lambda}$
$\displaystyle=e^{\Lambda}e^{-mu^{2}/2}e^{muH}\Lambda^{-m}e^{-\Lambda}$
$\displaystyle=e^{\Lambda}\mathcal{E}_{-m}(mu)e^{-\Lambda}.$

Therefore

$e^{\Lambda}\mathcal{E}_{-m}(mu)e^{-\Lambda}V=Ve^{mu\Lambda}\Lambda^{-m}.$

Since

$\displaystyle\mathcal{A}(m,mu)$
$\displaystyle=\frac{m!}{(mu)^{m}}e^{\Lambda}\mathcal{E}_{-m}(mu)e^{-\Lambda},$
$\displaystyle\tilde{\mathcal{A}}(m,mu)$
$\displaystyle=\frac{m!}{(mu)^{m}}e^{mu\Lambda}\Lambda^{-m},$

we find that (10) holds. ∎

(10) takes exactly the same form as the one presented by Okounkov and
Pandharipande [7]. We can derive a similar intertwining relation for $\bar{V}$
from (8) in much the same way. Thus Okounkov and Pandharipande’s intertwining
relations can be, at least partially, recovered in our reformulation of the
dressing operators $V,\bar{V}$.

##  4 Algebraic relation of Lax operators $L,\bar{L}$

The dressing operators $W,\bar{W}$ of the tau function (4) can be captured by
the factorization problem (see the review [23])

$\exp\left(\sum_{k=1}^{\infty}t_{k}\Lambda^{k}\right)U\exp\left(-\sum_{k=1}^{\infty}\bar{t}_{k}\Lambda^{-k}\right)=W^{-1}\bar{W},$
(11)

where $U$ is the difference operator

$U=V^{-1}e^{\Lambda^{a}/a}Q^{H}e^{\Lambda^{-b}/b}\bar{V}^{-1}$ (12)

that corresponds to the operator (5) on the fermionic Fock space. This
operator satisfies the following intertwining relation, which implies the
reduction condition (1) to the equivariant Toda hierarchy.

######  Proposition 2.

$(\Lambda^{a}-\nu\log\Lambda)U=U(\Lambda^{-b}-\nu\log\Lambda-\nu\log Q).$ (13)

###### Proof.

We use (7) and the operator identities

$\displaystyle e^{\Lambda^{a}/a}He^{-\Lambda^{a}/a}$
$\displaystyle=\Lambda^{a}+H,$ $\displaystyle
e^{-\Lambda^{-b}/b}He^{\Lambda^{-b}/b}$ $\displaystyle=\Lambda^{b}+H,$
$\displaystyle\log\Lambda\,Q^{H}$ $\displaystyle=Q^{H}(\log\Lambda+\log Q)$

to derive (13) from (12) as

$\displaystyle(\Lambda^{a}-\nu\log\Lambda)U$
$\displaystyle=V^{-1}(\Lambda^{a}+H-\nu\log\Lambda)e^{\Lambda^{a}/a}Q^{H}e^{\Lambda^{-b}/b}\bar{V}^{-1}$
$\displaystyle=V^{-1}e^{\Lambda^{a}/a}(H-\nu\log\Lambda)Q^{H}e^{\Lambda^{-b}/b}\bar{V}^{-1}$
$\displaystyle=V^{-1}e^{\Lambda^{a}/a}Q^{H}(H-\nu\log\Lambda-\nu\log
Q)e^{\Lambda^{-b}/b}\bar{V}^{-1}$
$\displaystyle=V^{-1}e^{\Lambda^{a}/a}Q^{H}e^{\Lambda^{-b}/b}(\Lambda^{-b}+H-\nu\log\Lambda-\nu\log
Q)\bar{V}^{-1}$ $\displaystyle=U(\Lambda^{-b}-\nu\log\Lambda-\nu\log Q).$

∎

######  Proposition 3.

The Lax operators obtained from the solution of the factorization problem (11)
satisfy the reduction condition (1).

###### Proof.

Let us rewrite (11) as

$U=\exp\left(-\sum_{k=1}^{\infty}t_{k}\Lambda^{k}\right)W^{-1}\bar{W}\exp\left(\sum_{k=1}^{\infty}\bar{t}_{k}\Lambda^{-k}\right)$

and plug it into (13). After some algebra, we find that

$W(\Lambda^{a}-\nu\log\Lambda)W^{-1}=\bar{W}(\Lambda^{-b}-\nu\log\Lambda-\nu\log
Q)\bar{W}^{-1}.$

This is nothing but (1). ∎

##  5 Construction of $V,\bar{V}$

We construct the dressing operators $V,\bar{V}$ by the power series expansion

$V=\sum_{k=0}^{\infty}\nu^{k}V_{k},\quad\bar{V}=\sum_{k=0}^{\infty}\nu^{k}\bar{V}_{k}$
(14)

with respect to $\nu$. As it turns out below, $V_{k}$’s become difference
operators of the form

$V_{0}=1+\sum_{n=1}^{\infty}v_{0n}\Lambda^{-n},\quad
V_{k}=\sum_{n=ka}^{\infty}v_{kn}\Lambda^{-n},\quad k\geq 1.$ (15)

$\bar{V}_{k}$’s, too, take a similar form. Since $\bar{V}$ can be obtained
from the formal adjoint (or transpose) $V^{*}$ of $V$ as

$\bar{V}=V^{*}|_{\nu\to-\nu,\,a\to b},$ (16)

we present the construction of $V$ only.

(7) splits into the following set of equations for $V_{k}$’s:

$\displaystyle(\Lambda^{a}+H)V_{0}$ $\displaystyle=V_{0}\Lambda^{a},$ (17)
$\displaystyle(\Lambda^{a}+H)V_{k}-\log\Lambda V_{k-1}$
$\displaystyle=V_{k}\Lambda^{k}-V_{k-1}\log\Lambda,\quad k\geq 1.$ (18)

Since $V_{0}$ is assumed to be invertible, see (15), we can rewrite (17) and
(18) as

$\displaystyle[\Lambda^{a},V_{0}]+HV_{0}$ $\displaystyle=0,$ (19)
$\displaystyle[\Lambda^{a},V_{0}^{-1}V_{k}]$
$\displaystyle=V_{0}^{-1}[\log\Lambda,V_{k-1}].\quad k\geq 1.$ (20)

######  Proposition 4.

There are difference operators of the form (15) with polynomial coefficients
$v_{kn}=v_{kn}(s)$ that satisfy (19) and (20).

###### Proof.

We first solve (19). This equation can be translated to the difference
equations

$v_{0,n+a}(s+a)-v_{0,n+a}(s)=-Hv_{0n}(s)$ (21)

for the coefficients $v_{0n}$ of $V_{0}$. Starting from $v_{00}=1$ and
$v_{01}=\cdots=v_{0,a-1}=0$, we can find the $v_{0n}$’s recursively with the
aid of the difference identity

$\displaystyle(s+a)s(s-a)\cdots(s-(k-2)a)-s(s-a)\cdots(s-(k-1)a)$
$\displaystyle=kas(s-a)\cdots(s-(k-2)a)$ (22)

among the $a$-step factorial products as follows. Let us examine the
difference equation

$v_{0a}(s+a)-v_{0a}(s)=-Hv_{00}(s)=-s+1/2$

at the first stage of the recursion. The identity (22) for $k=2$ and $k=1$
gives

$(s+a)s-s(s-a)=2as,\quad(s+a)s-s=a.$

Hence a polynomial solution of the difference equation can be obtained in the
form

$v_{0a}(s)=-\frac{1}{2a}(s+a)s+\frac{1}{2a}s.$

Since $v_{01}=\cdots=v_{0,a-1}=0$, the subsequent $a$ terms
$v_{0,a+1},\cdots,v_{0,2a-1}$ can be chosen to be equal to $0$. The next non-
trivial stage is the difference equation

$v_{0,2a}(s+a)-v_{0,2a}(s)=-Hv_{0a}(s).$

Expanding $-Hv_{0a}(s)$ into a linear combination of $s(s-a)(s-2a)$, $s(s-a)$
and $s$, we can apply the identity (22) for $k=2,1,0$ to find a polynomial
solution of this equation. Repeating this procedure, we obtain a set of
polynomials $v_{0a}$ that satisfy (21). We now turn to (20) and solve these
equations step-by-step with respect to $k$. Suppose that $V_{k-1}$ has been
constructed to be a difference operator of the form (15) with polynomial
coefficients. (20) consists of the difference equations

$v^{\prime}_{k,n+a}(s+a)-v^{\prime}_{k,n+a}(s)=f_{kn}(s)$

for the coefficients of

$V_{0}^{-1}V_{k}=\sum_{n=ka}^{\infty}v^{\prime}_{kn}\Lambda^{-n},\quad
V_{0}^{-1}[\log\Lambda,V_{k-1}]=\sum_{n=(k-1)a}^{\infty}f_{kn}\Lambda^{-n}.$

Since $f_{kn}$ is a polynomial in $s$, we can find a polynomial
$v^{\prime}_{k,n+a}$ that satisfies this difference equation. Thus
$V_{0}^{-1}V_{k}$, hence $V_{k}$, becomes a difference operator with
polynomial coefficients. ∎

######  Remark 1.

The intertwining relations (7) and (8) do not determine $V$ and $\bar{V}$
uniquely, leaving the gauge freedom

$V\to
V\left(1+\sum_{n=1}^{\infty}c_{n}\Lambda^{-n}\right),\quad\bar{V}\to\left(1+\sum_{n=1}^{\infty}\bar{c}_{n}\Lambda^{n}\right)\bar{V}$

of multiplying difference operators with constant coefficients
$c_{n},\bar{c}_{n}$.

######  Remark 2.

Since the $V_{k}$’s take the particular form as shown in (15), $V$ itself can
be expressed as

$V=1+\sum_{n=1}^{\infty}\sum_{k=0}^{\infty}\nu^{k}v_{kn}\Lambda^{-n}.$

The sum over $k$ is actually a finite sum because $v_{kn}=0$ for $k>n/a$.
Therefore the coefficients $v_{n}$ of $V$ are polynomials in both $s$ and
$\nu$. Thus there is no divergence problem for the expansion into powers of
$\nu$. This is also the case for $\bar{V}$.

##  6 Non-equivariant limit

The leading terms $V_{0},\bar{V}_{0}$ in the expansion (14) may be thought of
as the non-equivariant limit

$V_{0}=\lim_{\nu\to 0}V,\quad\bar{V}_{0}=\lim_{\nu\to 0}\bar{V}$

of $V,\bar{V}$. These operators will play a role in the non-equivariant
Gromov-Witten theory of $\mathbb{C}\mathbb{P}^{1}$. Similar ideas can be found
in the recent papers of Chen and Guo [25] and Alexandrov [26]. Let us consider
this issue briefly.

In the non-equivariant limit, the operator $U$ in the factorization problem
(11) is replaced by

$U_{0}=V_{0}^{-1}e^{\Lambda^{a}/a}Q^{H}e^{\Lambda^{-b}/b}\bar{V}_{0}^{-1}.$
(23)

$V_{0}$ and $\bar{V}_{0}$ satisfy the intertwining relations

$(\Lambda^{a}+H)V_{0}=V_{0}\Lambda^{a},\quad\bar{V}_{0}(\Lambda^{-b}+H)=\Lambda^{-b}\bar{V}_{0}.$
(24)

(13) turns into

$\Lambda^{a}U_{0}=U_{0}\Lambda^{-b},$ (25)

which implies that $L$ and $\bar{L}$ satisfy the reduction condition

$L^{a}=\bar{L}^{-b}$ (26)

to the bigraded Toda hierarchy of type $(a,b)$ [16].

Actually, we have the refinement

$\Lambda^{a}U_{0}=U_{0}\Lambda^{-b}=V_{0}^{-1}e^{\Lambda^{a}/a}HQ^{H}e^{\Lambda^{-b}/b}\bar{V}_{0}^{-1}$
(27)

of (25), which can be derived in the same way as in the derivation of (13).
Exponentiating (27) gives

$\displaystyle\exp\left(\sum_{k=1}^{\infty}T_{k}\Lambda^{ka}\right)U_{0}=U_{0}\exp\left(\sum_{k=1}^{\infty}T_{k}\Lambda^{-kb}\right)$
$\displaystyle=V_{0}^{-1}e^{\Lambda^{a}/a}\exp\left(\sum_{k=1}^{\infty}T_{k}H^{k}\right)Q^{H}e^{\Lambda^{-b}/b}\bar{V}_{0}^{-1}.$
(28)

The new variables $T_{k}$ can be identified with coupling constants of the
non-equivariant Gromov-Witten theory [6, 25, 26]. In the language of fermions,
(28) amount to relations of the form

$\displaystyle\langle
s|\exp\left(\sum_{k=1}^{\infty}T_{k}J_{ka}\right)g_{0}|s\rangle=\langle
s|g_{0}\exp\left(\sum_{k=1}^{\infty}T_{k}J_{-kb}\right)|s\rangle$
$\displaystyle=\langle
s|e^{J_{a}/a}\exp\left(\sum_{k=1}^{\infty}T_{k}P_{k}\right)Q^{H}e^{J_{-b}/b}|s\rangle,$
(29)

where

$g_{0}=V_{0}^{-1}e^{J_{a}/a}Q^{H}e^{J_{-b}/b}\bar{V}_{0}^{-1}$ (30)

and $P_{k}$’s are fermion operators that correspond to $H^{k}$ by the
correspondence (6). Note that $V_{0}^{-1}$ and $\bar{V}_{0}^{-1}$ disappear in
the last line of (29) because $\langle s|V_{0}^{-1}=\langle s|$ and
$\bar{V}_{0}^{-1}|s\rangle=|s\rangle$. Thus the coupling constants $T_{k}$ can
be identified with part of the time variables of the bigraded Toda hierarchy.

Let us mention that this construction can capture the extended (logarithmic)
flows of the 1D/bigraded Toda hierartchy [15, 16] as well. Let
$\tilde{\boldsymbol{T}}=\\{\tilde{T}_{k}\\}_{k=1}^{\infty}$ be time variables
of the extended flows and deform $U_{0}$ as

$U_{0}(\tilde{\boldsymbol{T}})=\exp\left(\sum_{k=1}^{\infty}\tilde{T}_{k}\Lambda^{ka}\log\Lambda\right)U_{0}\exp\left(-\sum_{k=1}^{\infty}\tilde{T}_{k}\Lambda^{-kb}\log\Lambda\right).$
(31)

By the intertwining relation (25) of $U_{0}$, the deformed operator
$U_{0}(\tilde{\boldsymbol{T}})$ satisfies the differential equations

$\frac{\partial
U_{0}(\tilde{\boldsymbol{T}})}{\partial\tilde{T}_{k}}=\Lambda^{ka}[\log\Lambda,U_{0}(\tilde{\boldsymbol{T}})]=\Lambda^{ka}\frac{\partial
U_{0}(\tilde{\boldsymbol{T}})}{\partial s}$ (32)

hence turns out to be a genuine difference operator (i.e., does not contain
$\log\Lambda$). Thus the factorization problem (11) persists to be meaningful.
The associated dressing operators $W,\bar{W}$ satisfy the Sato equations

$\frac{\partial
W}{\partial\tilde{T}_{k}}=C_{k}W-W\Lambda^{ka}\log\Lambda,\quad\frac{\partial\bar{W}}{\partial\tilde{T}_{k}}=C_{k}\bar{W}-\bar{W}\Lambda^{-kb}\log\Lambda$

of the extended flows [15, 16]. The $C_{k}$’s are defined as

$C_{k}=\mathcal{L}^{k}\log\Lambda-(W\Lambda^{ka}W^{-1}\frac{\partial
W}{\partial s}W^{-1})_{\geq
0}-(\bar{W}\Lambda^{-kb}\bar{W}^{-1}\frac{\partial\bar{W}}{\partial
s}\bar{W}^{-1})_{<0},$

where $\mathcal{L}$ denotes the difference operator of finite order defined by
both sides of the reduction condition (26).

##  7 Conclusion

We have reformulated Okounkov and Pandharipande’s dressing operators as
difference operators that satisfy the intertwining relations (7) and (8). This
formulation fits well into the Lax formalism of the 2D Toda hierarchy. These
dressing operators are building blocks of the factorization problem (11) that
captures Okounkov and Pandharipande’s tau function in the Lax formalism. It is
a rather immediate consequence of the factorization problem that the Lax
operators satisfy the reduction condition (1) to the equivariant Toda
hierarchy. We have thus found a new explanation to the question of why the
equivariant Toda hierarchy emerges in the Gromov-Witten theory of
$\mathbb{C}\mathbb{P}^{1}$.

### Acknowledgements

This work was partly supported by the JSPS Kakenhi Grant JP18K03350.

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