#
KP integrability of triple Hodge integrals. I.
From Givental group to hierarchy symmetries

Alexander Alexandrov
Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang
37673, Korea  E-mail: alexandrovsash at gmail.com

In this paper, we investigate a relation between the Givental group of rank
one and the Heisenberg-Virasoro symmetry group of the KP hierarchy. We prove,
that only a two-parameter family of the Givental operators can be identified
with elements of the Heisenberg-Virasoro symmetry group. This family describes
triple Hodge integrals satisfying the Calabi-Yau condition. Using the
identification of the elements of two groups we prove that the generating
function of triple Hodge integrals satisfying the Calabi-Yau condition and its
$\Theta$-version are tau-functions of the KP hierarchy. This generalizes the
result of Kazarian on KP integrability in the case of linear Hodge integrals.

MSC 2020 Primary: 37K10, 14N35, 81R10, 14N10; Secondary: 81T32.

###### Contents

  1. 1 Introduction
  2. 2 Two groups of symmetries
    1. 2.1 Givental group
    2. 2.2 Heisenberg-Virasoro subgroup of $GL(\infty)$
    3. 2.3 Quadratic part of the Virasoro group operators
    4. 2.4 Equivalence of Givental and Virasoro group elements
  3. 3 Hodge integrals
    1. 3.1 Triple Hodge integrals
    2. 3.2 Triple $\Theta$-Hodge integrals

##  1 Introduction

Since the works of ’t Hooft [35] and Brézin, Itzykson, Parisi, and Zuber [12],
it is known that the matrix models provide an efficient tool for the
investigation of enumerative geometry problems. Recent remarkable progress in
enumerative geometry/matrix model theory, in particular, is related to the
Chekhov-Eynard-Orantin topological recursion [15, 25] and Givental
decomposition [31, 32]. Both describe a non-trivial extension of the matrix
model theory, which ranges far beyond the scope of the classical matrix
integrals. Both topological recursion and Givental decomposition are naturally
described in terms of summation over graphs, thus formally can be represented
by infinite-dimensional Gaussian matrix integrals with integrands made of the
Kontsevich-Witten tau-function and its generalizations.

While a relationship between these two methods is known [21, 16], their
relation to other classical ingredients of the matrix model theory is not
always clear. Usually, matrix models satisfy two different families of
differential (or differential-difference) equations. One family, the linear
equations, is given by the Virasoro or, more generally, W-constraints. For
known examples, recursive solution of these equations naturally leads to
topological recursion and Givental-type decomposition, see, e.g., [1, 2, 3, 4,
5, 22]. Another family of equations is given by the bilinear Hirota equations
of the KP/Toda type integrable hierarchies. While the Virasoro constraints
constitute an inherent part of the Givental construction, the situation with
integrability is much less clear. Many examples of the enumerative geometry
generating functions, which can be described by topological recursion/Givental
decomposition (TR/GD), are certain tau-functions of integrable hierarchies,
however, a general relation between TR/GD and integrability is not known (see,
however, [30, 29, 28, 24]).

The main goal of this paper is to clarify the role of KP/Toda type
integrability in the general scheme of TR/GD. It is important, in particular,
because some elements of the integrable systems can be naturally identified
with the main components of the TR/GD. For example, the quantum spectral
curve, playing a central role in TR/GD construction, can be identified with a
specific Kac-Schwarz operator, while the wave function, annihilated by this
operator, is given by the Baker-Akhiezer function[6, 10, 7, 8, 9, 11].

In this paper, we address a question of the relation between the Givental
group and the symmetry groups of the integrable hierarchies. These are
different groups, acting on different spaces. However, sometimes it is
possible to identify the elements of two groups up to a linear change of
variables. An example of such identification was constructed in the work of
Liu and Wang [46]. They showed that the Givental operator associated with the
generating function of linear Hodge integrals is equal to an element of the
Heisenberg-Virasoro subgroup of $GL(\infty)$ up to a linear change of
variables.

Therefore, a question arises: are there any other elements of the Givental
group, which can be identified with the elements of the symmetry group of the
integrable hierarchy? In this paper, we give a complete answer to this
question for the rank one case of the Givental construction on one side and
the KP integrable hierarchy on another. By rank here we mean the rank of the
Frobenius manifold = the number of the KdV tau-functions in the Givental
formula. The first important result of this paper, Theorem 1, states that the
infinite-dimensional Givental group contains only a two-dimensional family of
such operators.

On the enumerative geometry side, this two-dimensional family is known to
describe a very interesting generating function, namely, a generating function
of triple Hodge integrals satisfying the Calabi-Yau condition. Hence, from the
identification of the Givental and Heisenberg-Virasoro group elements it
follows, that after a certain linear change of variables the generating
function of triple Hodge integrals satisfying the Calabi-Yau condition becomes
a tau-function of the KP hierarchy. Theorem 2, which describes this KP tau-
function, is the second important result of this paper. It generalizes the
result of Kazarian for linear Hodge integrals [36]. Theorem 3 extends this KP
integrability to the case with included Norbery’s $\Theta$-classes.

We restrict ourselfes to the simplest rank one case. However, we expect that
the results of this paper can be immediately generalized to the higher rank
Givental groups. This should lead to new interesting examples of integrable
generating functions of the cohomological field theory. In the companion paper
[11] we investigate a family of the tau-functions of the KP hierarchy, that
generalizes the two-parametric family constructed in this paper and
conjecturally describes interesting enumerative geometry invariants in the
$r$-spin case. This family can be described by a deformation of the
generalized Kontsevich model. The current project, in particular, was
motivated by [42], where a question about KP integrability of non-linear Hodge
integrals and their description in terms of generalized Kontsevich model was
raised.

It is known that enumerative geometry generating functions constitute only a
small class of tau-functions of integrable hierarchies. We hope that the
identification between the elements of the Givental group and symmetry groups
of integrable hierarchies will help us to classify all such tau-functions.

The present paper is organized as follows. In Section 2 we consider rank one
Givental operators, which after a linear change of variables can be identified
with the elements of the Heisenberg-Virasoro symmetry group of the KP
hierarchy. In Section 3 we identify this family with the generating functions
of triple Hodge integrals, which proves the KP integrability of the latter.

##  2 Two groups of symmetries

In this section we consider two different infinite-dimensional groups acting
on the bosonic Fock spaces. First of them, the Givental group of the quantized
symplectic transformations is a group of symmetries of semi-simple
cohomological field theory. Second is a Heisenberg-Virasoro subgroup of the
$GL(\infty)$ symmetry group of the KP hierarchy. We restrict the action of
both groups to the space of functions of odd times. We prove, that on this
space only a two-parametric family of the Givental operators can be
identified, up to a linear change of variables, with the elements of the
Heisenberg-Virasoro group.

###  2.1 Givental group

Let us consider the Givental quantization scheme for the rank one case. We
basically follow the presentation of [32, 40, 19]. However, we consider a
different normalization of the variables $T_{k}$, with an additional factor of
$\hbar$. This change of normalization corresponds to the transition from the
genus expansion of the generating function

$\begin{array}[]{c}\exp\left(\sum_{g=0}^{\infty}\sum_{n=1}^{\infty}\hbar^{2g-2}{\mathcal{F}}_{g,n}\right),\end{array}$
(2.1)

where $g$ is the genus, $n$ is the number of the marked points, to the
topological expansion

$\begin{array}[]{c}\exp\left(\sum_{g=0}^{\infty}\sum_{n=1}^{\infty}\hbar^{2g-2+n}{\mathcal{F}}_{g,n}\right),\end{array}$
(2.2)

because the latter is more convenient for the KP hierarchy description.

Let ${H}={\hbox{{\alef C}}}[\\![z,z^{-1}]\\!]$ be the space of formal Laurent
series in an indeterminate $z$. We introduce the symplectic form on this space

$\begin{array}[]{c}\Omega(f,g)=\frac{1}{2\pi i}\oint f(-z)g(z)dz.\end{array}$
(2.3)

For the natural decomposition

$\begin{array}[]{c}{H}={H}_{+}\oplus{H}_{-},\end{array}$ (2.4)

where ${H}_{+}={\rm span}\,_{\raise-1.0pt\hbox{$\mbox{\Bbbb
C}$}}\\{1,z,z^{2},\dots\\}$ and ${H}_{-}={\rm
span}\,_{\raise-1.0pt\hbox{$\mbox{\Bbbb
C}$}}\\{z^{-1},z^{-2},z^{-3},\dots\\}$, we introduce the Darboux coordinate
system $\\{p_{k},q_{m}\\}$ with

$\begin{array}[]{c}{\mathcal{J}}(z):=\sum_{k=0}^{\infty}\left(q_{k}z^{k}+(-z)^{-k-1}p_{k}\right).\end{array}$
(2.5)

After Givental, for any infinitesimal symplectic transformation $A$ on ${H}$
(that is, a transformation such that $\Omega(Af,g)+\Omega(f,Ag)=0$) we
consider

$\begin{array}[]{c}H_{A}:=\frac{1}{2}\Omega(A{\mathcal{J}},{\mathcal{J}}).\end{array}$
(2.6)

This defines a Lie algebra isomorphism:

$\begin{array}[]{c}H_{\left[A,B\right]}=\left\\{H_{A},H_{B}\right\\},\end{array}$
(2.7)

where the Poisson bracket is given by

$\begin{array}[]{c}\left\\{H_{A},H_{B}\right\\}:=\sum_{i=0}^{\infty}\left(\frac{\partial
H_{A}}{\partial p_{i}}\frac{\partial H_{B}}{\partial q_{i}}-\frac{\partial
H_{B}}{\partial p_{i}}\frac{\partial H_{A}}{\partial
q_{i}}\right).\end{array}$ (2.8)

Using the standard Weyl quantization we quantize these operators to order
$\leq 2$ linear differential operators

$\begin{array}[]{c}\widehat{A}=\widehat{H}_{A}:=\frac{1}{2}{\scriptstyle{{*}\atop{*}}}\Omega(A\widehat{\mathcal{J}},\widehat{\mathcal{J}}){\scriptstyle{{*}\atop{*}}},\end{array}$
(2.9)

where

$\begin{array}[]{c}\widehat{\mathcal{J}}(z)=\sum_{k=0}^{\infty}\left(\tilde{T}_{k}z^{k}+(-z)^{-k-1}\frac{\partial}{\partial
T_{k}}\right),\end{array}$ (2.10)

and ${\scriptstyle{{*}\atop{*}}}\dots\scriptstyle{{*}\atop{*}}$ denotes the
standard bosonic normal ordering, which puts all $\frac{\partial}{\partial
T_{k}}$ to the right of all $T_{k}$. The so-called dilaton shift is given by
$\tilde{T}_{k}=T_{k}-\hbar^{-1}\delta_{k,1}$. This gives us a central
extension of the original algebra, with the commutator

$\begin{array}[]{c}\left[\widehat{A},\widehat{B}\right]=\widehat{\left[A,B\right]}+\mathcal{C}(H_{A},H_{B}),\end{array}$
(2.11)

where the so-called 2-cocycle term satisfies

$\begin{array}[]{c}\mathcal{C}\left(p_{i}p_{j},q_{k}q_{m}\right)=-\mathcal{C}\left(q_{k}q_{m},p_{i}p_{j}\right)=\delta_{i,k}\delta_{j,m}+\delta_{i,m}\delta_{j,k}\end{array}$
(2.12)

and vanishes for all other elements $H_{A}$. For the finite symplectic
transformations we define

$\begin{array}[]{c}\widehat{e^{A}}:=e^{\widehat{A}}.\end{array}$ (2.13)

Below we consider only the symplectic transformations of the form
$R(z)=1+R_{1}z+R_{2}z^{2}+\dots\in 1+z{\hbox{{\alef C}}}[\\![z]\\!]$. They
satisfy the symplectic condition

$\begin{array}[]{c}R(z)R(-z)=1\end{array}$ (2.14)

if and only if $\log R(z)$ is odd function of $z$. Let us denote

$\begin{array}[]{c}\widehat{W}_{k}:=\widehat{z^{2k-1}},\,\,\,\,k\in{\hbox{{\alef
Z}}},\end{array}$ (2.15)

then from (2.9) we have [32, 40]

$\begin{array}[]{c}\widehat{W}_{k}=-\sum_{m}\tilde{T}_{m}\frac{\partial}{\partial
T_{m+2k-1}}+\frac{1}{2}\sum_{m=0}^{-2k}(-1)^{l+1}\tilde{T}_{m}\tilde{T}_{-2k-m}+\frac{1}{2}\sum_{m=0}^{2k-2}(-1)^{m}\frac{\partial^{2}}{\partial
T_{m}\partial T_{2k-m-2}}.\end{array}$ (2.16)

These operators satisfy the commutation relations

$\begin{array}[]{c}\left[\widehat{W}_{k},\widehat{W}_{m}\right]=-\frac{2k-1}{2}\delta_{k+m,1}.\end{array}$
(2.17)

After quantization, using (2.16) we obtain

$\begin{array}[]{c}\widehat{R}:=\exp\left(\widehat{\log
R(z)}\right).\end{array}$ (2.18)

Below we call the group of such operators the Givental group.

The Givental group acts of the space of cohomological field theories (CohFT)
with flat unit. If we relax the flat unit condition, we can include
translations of the times, see an example in Section 3.2. However,
translations are natural symmetries of the KP hierarchy, so this enrichment is
trivial from the point of view of the identification of operators of two
groups.

Operators $\widehat{R}$ can be factorized, namely, one can factor out the
part, corresponding to the linear change of variables and the translation of
variables. This factorization, except for simple extraction of the translation
operator, is given by Proposition 7.3 in [32]. Here we briefly remind the
reader this relation. Let us introduce a formal series in two variables

$\begin{array}[]{c}V^{R}(z,w):=\frac{1-R(-w)R(-z)}{w+z}.\end{array}$ (2.19)

The matrix of its coefficients

$\begin{array}[]{c}V^{R}(z,w)=\sum_{k,l=0}^{\infty}V^{R}_{kl}w^{k}z^{l}\end{array}$
(2.20)

is symmetric, $V_{kl}^{R}=V_{lk}^{R}$, and its entries are polynomials in
$R_{k}$. Let the linear change of the shifted variables from ${\bf T}$ to
${\bf T^{R}}$ be given by

$\begin{array}[]{c}\sum_{k=0}^{\infty}\tilde{T}^{R}_{k}z^{k}:=R(-z)\sum_{k=0}^{\infty}\tilde{T}_{k}z^{k}.\end{array}$
(2.21)

This transformation generates the change of the dilaton shift

$\begin{array}[]{c}\sum_{k=2}^{\infty}\delta_{k}z^{k}:=z(1-R(-z)).\end{array}$
(2.22)

Then, for any element of the Givental group (2.18) and any series $Z({\bf T})$
we have [32]

######  Lemma 2.1 (Givental).

$\begin{array}[]{c}\widehat{R}\cdot Z({\bf
T})=\,e^{\frac{1}{2}\sum_{i,j=0}^{\infty}{V_{ij}^{R}}\frac{\partial^{2}}{\partial
T_{i}\partial
T_{j}}}\,\left.e^{\hbar^{-1}\sum_{k=2}^{\infty}\delta_{k}\frac{\partial}{\partial
T_{k}}}\cdot Z({\bf T})\right|_{{T}_{k}\mapsto{T}^{R}_{k}}.\end{array}$ (2.23)

In this expression one acts by the operator on the function $Z({\bf T})$, and
then in the result substitutes ${T}_{k}$ with ${T}^{R}_{k}$.

###  2.2 Heisenberg-Virasoro subgroup of $GL(\infty)$

In this section we briefly describe the Heisenberg-Virasoro subgroup of
$\operatorname{GL}(\infty)$ symmetry group of KP hierarchy, for more details
see the companion paper [11] and references therein. Let

$\begin{array}[]{c}\widehat{J}(z):=\sum_{k\in\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}}\frac{\widehat{J}_{k}}{z^{k+1}},\end{array}$ (2.24)

where

$\begin{array}[]{c}\widehat{J}_{k}=\begin{cases}\displaystyle{\frac{\partial}{\partial
t_{k}}\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\quad k>0},\\\\[2.0pt]
\displaystyle{0}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\quad
k=0,\\\\[2.0pt] \displaystyle{-kt_{-k}\,\,\,\,\,\mathrm{for}\quad
k<0.}\end{cases}\end{array}$ (2.25)

We consider the generating function of the Virasoro operators

$\begin{array}[]{c}{\scriptstyle{{*}\atop{*}}}\widehat{J}(z)^{2}{\scriptstyle{{*}\atop{*}}}=2\sum_{k\in\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}}\frac{\widehat{L}_{k}}{z^{k+2}}\end{array}$ (2.26)

or

$\begin{array}[]{c}\widehat{L}_{m}=\frac{1}{2}\sum_{a+b=-m}abt_{a}t_{b}+\sum_{k=1}^{\infty}kt_{k}\frac{\partial}{\partial
t_{k+m}}+\frac{1}{2}\sum_{a+b=m}\frac{\partial^{2}}{\partial t_{a}\partial
t_{b}},\end{array}$ (2.27)

where, depending on the sign of $m$, only the first or the last summation
appears. These formulas are the KP analogs of (2.10) and (2.9). The
Heisenberg-Virasoro group ${\mathcal{V}}$ is generated by the operators
$\widehat{J}_{k}$, $\widehat{L}_{k}$ and a unit. Let us consider a Virasoro
subgroup of ${\mathcal{V}}$ given by the operators of the form

$\begin{array}[]{c}\widehat{V}=\exp\left({\sum_{\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>0}}a_{k}\widehat{L}_{k}}\right)\in{\mathcal{V}}.\end{array}$ (2.28)

With any such operator one can associate the operator
$e^{{\sum_{\raise-1.0pt\hbox{$\mbox{\Bbbb Z}$}_{>0}}a_{k}{\mathtt{l}}_{k}}}$,
where

$\begin{array}[]{c}{\mathtt{l}}_{m}=-z^{m}\left(z\frac{\partial}{\partial
z}+\frac{m+1}{2}\right)\end{array}$ (2.29)

are the generators of the Witt subalgebra of the algebra of of diffeomorphisms
on the circle.

For any set $a_{k}$, $k\in{\hbox{{\alef Z}}}_{>0}$, consider the series

$\begin{array}[]{c}f(z):=e^{{\sum_{\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>0}}a_{k}{\mathtt{l}}_{k}}}\,z\,e^{-{\sum_{\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>0}}a_{k}{\mathtt{l}}_{k}}}\in z+z{\hbox{{\alef
C}}}[\\![z]\\!].\end{array}$ (2.30)

Then

$\begin{array}[]{c}h(z):=e^{-{\sum_{\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>0}}a_{k}{\mathtt{l}}_{k}}}\,z\,e^{{\sum_{\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>0}}a_{k}{\mathtt{l}}_{k}}}\in z+z{\hbox{{\alef
C}}}[\\![z]\\!]\end{array}$ (2.31)

is an inverse series, $f(h(z))=h(f(z))=z$. There is a one-to-one
correspondence between the space of series (2.30) and the subgroup (2.28) of
the Virasoro group. From the commutation relations of the Heisenberg-Virasoro
algebra,

$\displaystyle\left[\widehat{J}_{k},\widehat{J}_{m}\right]$
$\displaystyle=k\delta_{k,-m},$
$\displaystyle\left[\widehat{L}_{k},\widehat{J}_{m}\right]$
$\displaystyle=-m\widehat{J}_{k+m},$ (2.32)
$\displaystyle\left[\widehat{L}_{k},\widehat{L}_{m}\right]$
$\displaystyle=(k-m)\widehat{L}_{k+m}+\frac{1}{12}\delta_{k,-m}(k^{3}-k)$

it follows that

$\begin{array}[]{c}\widehat{V}\,\widehat{{J}}(z)\,\widehat{V}^{-1}=h^{\prime}(z)\widehat{{J}}(h(z)).\end{array}$
(2.33)

Let $\widehat{V}_{0}$ be a part of the element of the upper-triangular
subgroup (2.28) that describes the linear change of variables,

$\begin{array}[]{c}\widehat{V}_{0}:=\exp\left(\sum_{k=1}^{\infty}a_{k}\sum_{m=1}^{\infty}mt_{m}\frac{\partial}{\partial
t_{k+m}}\right).\end{array}$ (2.34)

With the help of the Campbell-Baker-Hausdorff formula operator $\widehat{V}$
can be factorized

$\begin{array}[]{c}\widehat{V}=\widehat{V}_{0}\,\exp\left(\frac{1}{2}\sum_{k,m=1}^{\infty}v_{km}\frac{\partial^{2}}{\partial
t_{k}\partial t_{m}}\right)\end{array}$ (2.35)

for some $v_{ij}\in{\hbox{{\alef C}}}[\\![a_{1},a_{2},a_{3},\dots]\\!]$. Below
we will describe the space of all $v_{ij}$ which can be obtained in this way.

Using the commutation relations of the Virasoro algebra we can always factor
out a term $\exp(a_{1}\widehat{L}_{1})$,

$\begin{array}[]{c}\exp\left({\sum_{\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>0}}a_{k}\widehat{L}_{k}}\right)=\exp(a_{1}\widehat{L}_{1})\,\exp\left({\sum_{\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>1}}\tilde{a}_{k}\widehat{L}_{k}}\right)\end{array}$ (2.36)

for some new $\tilde{a}_{k}$. Operator
${\widehat{L}}_{1}=\sum_{k=1}^{\infty}kt_{k}\frac{\partial}{\partial t_{k+1}}$
is a first order differential operator, so $\exp(a_{1}\widehat{L}_{1})$ gives
only a linear change of variables. As we are going to identify the Givental
operators with the elements of the Heisenberg-Virasoro group up to the linear
change of variables, there is a certain arbitrariness in the choice of
$a_{1}$. We call the transformation, given by $\exp(a_{1}\widehat{L}_{1})$,
the gauge transformation, and the choice of $a_{1}$ will be called the choice
of gauge. For the series (2.30) the gauge transformation is given by

$\begin{array}[]{c}f(z)\mapsto f(z/(1+a_{1}z)).\end{array}$ (2.37)

###  2.3 Quadratic part of the Virasoro group operators

To find the matrix $v_{km}$ in (2.35) it is enough to act by both sides of
(2.35) on the function

$\begin{array}[]{c}E:=\exp\left(\sum_{k=1}^{\infty}kt_{k}q_{k}\right),\end{array}$
(2.38)

where ${\bf q}$ are auxiliary variables, independent on ${\bf t}$. Let us show
that the coefficients $v_{nm}$ in (2.35) are the so-called Grunsky
coefficients of function $h(z)$. Consider

$\begin{array}[]{c}v(\eta_{1},\eta_{2}):=\sum_{k,m=1}^{\infty}v_{km}\eta_{1}^{k}\eta_{2}^{m}.\end{array}$
(2.39)

######  Lemma 2.2.

$\begin{array}[]{c}v(\eta_{1},\eta_{2})=\log\left(\frac{h(\eta_{1})-h(\eta_{2})}{\eta_{1}-\eta_{2}}\right).\end{array}$
(2.40)

######  Remark 2.1.

This formula was known to the experts a long time ago, and can be proven via
the Campbell-Baker-Hausdorff formula. Here we provide its proof for
completeness.

###### Proof.

On one hand, the action of the right hand side of (2.35) on $E$ yields

$\begin{array}[]{c}\widehat{V}\cdot
E=\widehat{V}_{0}\cdot\exp\left(\sum_{k=1}^{\infty}kt_{k}q_{k}+\frac{1}{2}\sum_{k,m=1}^{\infty}km\,v_{km}q_{k}q_{m}\right).\end{array}$
(2.41)

On the other hand

$\begin{split}\widehat{V}\cdot
E&=\exp\left(\sum_{k=1}^{\infty}k\left(\widehat{V}t_{k}\widehat{V}^{-1}\right)q_{k}\right)\cdot
1\\\
&=\exp\left(\sum_{k=1}^{\infty}\left(kt_{k}\tilde{q}_{k}+\tilde{q}_{-k}\frac{\partial}{\partial
t_{k}}\right)\right)\cdot 1,\end{split}$ (2.42)

where from (2.33) we have

$\begin{array}[]{c}\tilde{q}_{k}:=\frac{1}{2\pi
i}\oint_{\infty}\left(\sum_{m=1}^{\infty}\frac{q_{m}}{f(z)^{m}}\right)\frac{dz}{z^{k+1}},\,\,\,\,\,\,\,\,k\in{\hbox{{\alef
Z}}}.\end{array}$ (2.43)

Hence

$\begin{split}\widehat{V}\cdot
E&=\exp\left(\sum_{k=1}^{\infty}\left(kt_{k}\tilde{q}_{k}+\frac{1}{2}k\tilde{q}_{k}\tilde{q}_{-k}\right)\right)\\\
&=\widehat{V}_{0}\cdot\exp\left(\sum_{k=1}^{\infty}\left(kt_{k}q_{k}+\frac{1}{2}k\tilde{q}_{k}\tilde{q}_{-k}\right)\right).\end{split}$
(2.44)

Comparing this expression with (2.41) we conclude

$\begin{array}[]{c}\sum_{k,m=1}^{\infty}km\,\,v_{km}q_{k}q_{m}=\sum_{k=1}^{\infty}k\tilde{q}_{k}\tilde{q}_{-k}.\end{array}$
(2.45)

Therefore,

$\begin{split}v_{km}&=\frac{1}{km}\sum_{j=1}^{\infty}\frac{j}{(2\pi
i)^{2}}\oint_{\infty}\frac{dz}{z^{j+1}f(z)^{k}}\oint_{\infty}\frac{w^{j-1}dw}{f(w)^{m}}\\\
&=-\frac{1}{km}\sum_{j=1}^{\infty}\frac{1}{j}\frac{1}{(2\pi
i)^{2}}\oint_{\infty}\frac{dz^{-j}}{f(z)^{k}}\oint_{\infty}\frac{dw^{j}}{f(w)^{m}}\\\
&=\frac{1}{(2\pi
i)^{2}}\oint_{\infty}\frac{dz}{z^{k+1}}\oint_{\infty}\frac{dw}{w^{m+1}}\log\left(1-\frac{h(w)}{h(z)}\right)\\\
&=\frac{1}{(2\pi
i)^{2}}\oint_{\infty}\frac{dz}{z^{k+1}}\oint_{\infty}\frac{dw}{w^{m+1}}\log\left(\frac{h(z)-h(w)}{z-w}\right).\end{split}$
(2.46)

Here in the transition from the second to the third line we use the change of
variables from $z$ and $w$ to $f(z)$ and $f(w)$. To derive the last line we
use the positivity of $k$ and $m$, and the function there is chosen in such
way that it possess formal Taylor series expansion

$\begin{array}[]{c}\log\left(\frac{h(z)-h(w)}{z-w}\right)\in{\hbox{{\alef
C}}}[\\![z,w]\\!].\end{array}$ (2.47)

This completes the proof. ∎

###  2.4 Equivalence of Givental and Virasoro group elements

Let us compare the subgroup (2.28) of the Virasoro group and Givental group,
given by Lemma 2.1. First, we identify

$\begin{array}[]{c}T_{k}\equiv(2k+1)!!t_{2k+1}.\end{array}$ (2.48)

Second, we consider the action of two groups on the Fock space of functions of
odd variables ${\bf t}^{o}=\\{t_{1},t_{3},t_{5},\dots\\}$. On this space for
the element of the Virasoro group $\widehat{V}$ one can neglect all terms with
even $k$ or $m$ in (2.35) and all terms with even values of $k+m$ in (2.34).
Therefore, on this space the action of two operators will coincide, up to
linear change and translation of variables, if and only if

$\begin{array}[]{c}V_{km}^{R}=(2k+1)!!(2m+1)!!v_{{2k+1}{2m+1}}\end{array}$
(2.49)

for all $k,m\in{\hbox{{\alef Z}}}_{\geq 0}$.

######  Remark 2.2.

There is no restriction for the elements $v_{km}$ when $k$ or $m$ is even.
However, we will see that the constraint (2.49) is already very restrictive.

######  Remark 2.3.

Arguments of this section are closely related to the standard manipulations in
the context of TR/GD, often referred to as the Laplace transform, see, e.g.,
[26, 23, 21].

Let

$\begin{array}[]{c}x(z):=\frac{f(z)^{2}}{2}=\frac{z^{2}}{2}+z^{3}{\hbox{{\alef
C}}}[\\![z]\\!],\end{array}$ (2.50)

where $f(z)$ describes the Virasoro group element by (2.30) and

$\begin{array}[]{c}N(z):=\frac{z}{x^{\prime}(z)}=1+\sum_{k\in\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>1}}n_{k}z^{k}.\end{array}$ (2.51)

We fix the gauge by $a_{1}=0$, which immediately leads to $n_{1}=0$. Then we
have

######  Lemma 2.3.

From (2.49) it follows that $n_{k}=0$ for $k>3$.

###### Proof.

From (2.19) and Lemma 2.2 we see that the relation (2.49) is equivalent to

$\begin{array}[]{c}\frac{1-R(-w)R(-z)}{w+z}=\frac{1}{2\pi(zw)^{3/2}}\int_{\gamma^{2}}d\eta_{1}d\eta_{2}\,\eta_{1}\eta_{2}e^{-\frac{\eta_{1}^{2}}{2z}-\frac{\eta_{2}^{2}}{2w}}\log\left(\frac{h(\eta_{1})-h(\eta_{2})}{\eta_{1}-\eta_{2}}\right).\end{array}$
(2.52)

Here $\gamma$ is a local steepest descent contour in the vicinity of
$\eta_{1}=\eta_{2}=0$, and we consider the asymptotic expansion of the
integral at small values of $|z|$ and $|w|$. Let $x_{k}=x(\eta_{k})$. The
right hand side of (2.52) after the change of variables of integration
$\eta_{k}\mapsto f(\eta_{k})$ is equal to

$\begin{split}&\frac{1}{2\pi(zw)^{3/2}}\int_{\gamma^{2}}dx_{1}dx_{1}\log\left(\frac{\eta_{1}-\eta_{2}}{f(\eta_{1})-f(\eta_{2})}\right)e^{-\frac{x_{1}}{z}-\frac{x_{2}}{w}}\\\
&=-\frac{1}{2\pi(z+w)\sqrt{zw}}\int_{\gamma^{2}}dx_{1}dx_{2}\log\left(\frac{\eta_{1}-\eta_{2}}{f(\eta_{1})-f(\eta_{2})}\right)\left(\frac{\partial}{\partial
x_{1}}+\frac{\partial}{\partial
x_{2}}\right)e^{-\frac{x_{1}}{z}-\frac{x_{2}}{w}}\\\
&=\frac{1}{2\pi(z+w)\sqrt{zw}}\int_{\gamma^{2}}dx_{1}dx_{2}\,e^{-\frac{x_{1}}{z}-\frac{x_{2}}{w}}\left(\frac{\partial}{\partial
x_{1}}+\frac{\partial}{\partial
x_{2}}\right)\log\left(\frac{\eta_{1}-\eta_{2}}{f(\eta_{1})-f(\eta_{2})}\right),\end{split}$
(2.53)

where the last equality follows form the integration by parts. From the
identity

$\begin{array}[]{c}\frac{\partial x_{1}}{\partial\eta_{1}}\frac{\partial
x_{2}}{\partial\eta_{2}}\,\left(\frac{\partial}{\partial
x_{1}}+\frac{\partial}{\partial
x_{2}}\right)\log\left(\frac{\eta_{1}-\eta_{2}}{f(\eta_{1})-f(\eta_{2})}\right)=f^{\prime}(\eta_{1})f^{\prime}(\eta_{2})-\frac{\frac{\partial
x_{1}}{\partial\eta_{1}}-\frac{\partial
x_{2}}{\partial\eta_{2}}}{\eta_{1}-\eta_{2}}\end{array}$ (2.54)

it follows that the last line of (LABEL:newrr) is equal to

$\begin{array}[]{c}\frac{1}{2\pi(z+w)\sqrt{zw}}\int_{\gamma^{2}}\left(d\sqrt{2x_{1}}d\sqrt{2x_{2}}-\frac{dx_{1}d\eta_{2}-d\eta_{1}dx_{2}}{\eta_{1}-\eta_{2}}\right)\,e^{-\frac{x_{1}}{z}-\frac{x_{2}}{w}}.\end{array}$
(2.55)

Comparing it with (2.52), we see that the latter is equivalent to

$\begin{array}[]{c}R(-w)R(-z)=\frac{1}{2\pi\sqrt{zw}}\int_{\gamma^{2}}\frac{dx_{1}d\eta_{2}-d\eta_{1}dx_{1}}{\eta_{1}-\eta_{2}}e^{-\frac{x_{1}}{z}-\frac{x_{2}}{w}}.\end{array}$
(2.56)

The left hand side factorizes, so should the right hand side. The factors can
be easily found if one puts $w=0$,

$\begin{array}[]{c}R(-z)=\frac{1}{\sqrt{2\pi
z}}\int_{\gamma}\frac{dx(\eta)}{\eta}e^{-\frac{x(\eta)}{z}}.\end{array}$
(2.57)

Hence $x(\eta)$ should satisfy the equation

$\begin{array}[]{c}\frac{1}{2\pi\sqrt{zw}}\int_{\gamma^{2}}\left(\frac{dx_{1}d\eta_{2}-d\eta_{1}dx_{2}}{\eta_{1}-\eta_{2}}-\frac{dx_{1}dx_{2}}{\eta_{1}\eta_{2}}\right)e^{-\frac{x_{1}}{z}-\frac{x_{2}}{w}}=0.\end{array}$
(2.58)

Let us solve this equation for $x(\eta)$. First, we rewrite the left hand side
as

$\begin{split}&\frac{1}{2\pi\sqrt{zw}}\int_{\gamma^{2}}\frac{d\eta_{1}d\eta_{2}}{N(\eta_{1})N(\eta_{2})}\left(\frac{\eta_{1}N(\eta_{2})-\eta_{2}N(\eta_{1})}{\eta_{1}-\eta_{2}}-1\right)e^{-\frac{x_{1}}{z}-\frac{x_{2}}{w}}\\\
&=-\frac{1}{2\pi\sqrt{zw}}\int_{\gamma^{2}}d\eta_{1}d\eta_{2}\,\eta_{1}\eta_{2}\sum_{k=2}^{\infty}n_{k}\sum_{j+m=k-2}h(\eta_{1})^{j}h(\eta_{2})^{m}e^{-\frac{\eta_{1}^{2}}{2z}-\frac{\eta_{2}^{2}}{2w}},\end{split}$
(2.59)

where we apply the change of integration variables $\eta_{k}\mapsto
h(\eta_{k})$ and integrate by parts.

For any $g(\eta)\in\hbox{{\alef C}}[\\![\eta]\\!]$ let

$\begin{array}[]{c}\left[g(\eta)\right]_{\eta}:=\frac{1}{2}\left(g(\eta)-g(-\eta)\right)\end{array}$
(2.60)

be its odd part, and for a series of two variables
$g(\eta_{1},\eta_{2})\in{\hbox{{\alef C}}}[\\![\eta_{1},\eta_{2}]\\!]$

$\begin{array}[]{c}\left[g(\eta_{1},\eta_{2})\right]_{\eta_{1},\eta_{2}}:=\frac{1}{4}\left(g(\eta_{1},\eta_{2})-g(-\eta_{1},\eta_{2})-g(\eta_{1},-\eta_{2})+g(-\eta_{1},-\eta_{2})\right).\end{array}$
(2.61)

Then (2.58) is equivalent to

$\begin{array}[]{c}\left[\sum_{k=2}^{\infty}n_{k}\sum_{j+m=k-2}h(\eta_{1})^{j}h(\eta_{2})^{m}\right]_{\eta_{1},\eta_{2}}=0,\end{array}$
(2.62)

so we need to prove that it implies $n_{k}=0$ for $k>3$. It is obvious that it
implies $n_{4}=0$. Let

$\begin{array}[]{c}M_{j}:=\left[\sum_{k=5}^{\infty}n_{k}h(\eta_{1})^{k-j-2}\right]_{\eta_{1}}.\end{array}$
(2.63)

It is easy to find the first terms of the series $h(x)$:

$\begin{array}[]{c}h(\eta)=\eta+\frac{n_{2}}{4}\eta^{3}+\frac{n_{3}}{5}\eta^{4}+\frac{5n_{2}^{2}}{96}\eta^{5}+O(\eta^{6}).\end{array}$
(2.64)

Then

$\begin{array}[]{c}\left[\sum_{k=5}^{\infty}n_{k}\sum_{j+m=k-2}h(\eta_{1})^{j}h(\eta_{2})^{m}\right]_{\eta_{1},\eta_{2}}=M_{1}\eta_{2}^{1}+(M_{3}+\frac{n_{2}}{4}M_{1})\eta_{2}^{3}+O(\eta_{2}^{5}).\end{array}$
(2.65)

Hence, from (2.62) it follows that

$\begin{array}[]{c}M_{1}=0,\,\,\,\,\,M_{3}=0.\end{array}$ (2.66)

Let us prove by induction that these two equations have the only solution
$n_{k}=0$ for $k>4$. Assume that $n_{k}=0$ for all ${k=4,5,\dots,m}$. If $m$
is odd, then from the equation $M_{1}=0$ it follows that $n_{m+1}=0$, so we
can assume that $m$ is even, $m=2k_{0}$. Assume that $n_{2k_{0}+1}\neq 0$ for
some $k_{0}\geq 2$. There are two possibilities:

$\bullet$ Let $n_{3}=0$. Then

$\begin{array}[]{c}h(\eta)+h(-\eta)=2\frac{n_{2k_{0}+1}}{2k_{0}+3}\eta^{2k_{0}+2}+O(\eta^{2k_{0}+2})\end{array}$
(2.67)

and from the equation $M_{1}=0$ it follows that $n_{2k}=0$ for all
$k\in\\{3,\dots,2k_{0}\\}$. Then

$\begin{split}M_{1}&=\left(\frac{2k_{0}-2}{2k_{0}+3}n_{2k_{0}+1}^{2}+n_{4k_{0}+2}\right)\eta_{1}^{4k_{0}-1}+O(\eta_{1}^{4k_{0}+1}),\\\
M_{3}&=\left(\frac{2k_{0}-4}{2k_{0}+3}n_{2k_{0}+1}^{2}+n_{4k_{0}+2}\right)\eta_{1}^{4k_{0}-3}+O(\eta_{1}^{4k_{0}-1}).\end{split}$
(2.68)

Hence from (2.66) we have $n_{2k_{0}+1}=0$, which contradicts the assumption.

$\bullet$ Let $n_{3}\neq 0$. Then from the first equation in (2.66) we
immediately see that $n_{2k_{0}+2}=0$, and

$\begin{split}M_{1}&=\left(\frac{2k_{0}-2}{5}n_{3}n_{2k_{0}+1}+n_{2k_{0}+4}\right)\eta_{1}^{2k_{0}+1}+O(\eta_{1}^{2k_{0}+3}),\\\
M_{3}&=\left(\frac{2k_{0}-4}{5}n_{3}n_{2k_{0}+1}+n_{2k_{0}+4}\right)\eta_{1}^{2k_{0}-1}+O(\eta_{1}^{2k_{0}+2}).\end{split}$
(2.69)

Hence from (2.66) we have $n_{2k_{0}+1}=0$, which contradicts the assumption.
This completes the proof. ∎

Therefore, to identify the rank one Givental operators with the elements of
the Heisenberg-Virasoro group we can restrict ourself to a two-dimensional
family of the Virasoro operators (2.28), identification of the translation
part will be considered later. Let us choose a more convenient parametrization
of this family. For this purpose we use the gauge transformation (2.37). Let
us denote by $\widehat{V}_{n_{1},n_{2},n_{3}}$ the element of the Virasoro
subgroup (2.28) with function $f(z)$ given by

$\begin{array}[]{c}\frac{f(z)^{2}}{2}=\int_{0}^{z}\frac{\eta
d\eta}{1+n_{1}\eta+n_{2}\eta^{2}+n_{3}\eta^{3}}.\end{array}$ (2.70)

######  Lemma 2.4.

$\begin{array}[]{c}\widehat{V}_{n_{1},n_{2},0}=e^{\frac{n_{1}}{3}\widehat{L}_{1}}\widehat{V}_{0,n_{2}-n_{1}^{2}/3,-n_{1}n_{2}/3+2n_{1}^{3}/27}.\end{array}$
(2.71)

###### Proof.

Because of the one-to-one correspondence between the group (2.28) and the
space of formal series $z+z^{2}{\hbox{{\alef C}}}[\\![z]\\!]$, the statement
of the lemma follows from the identity

$\begin{array}[]{c}\int_{0}^{\frac{z}{1+n_{1}z/3}}\frac{\eta
d\eta}{1+(n_{2}-n_{1}^{2}/3)\eta^{2}+(2n_{1}^{3}/27-n_{1}n_{2}/3)\eta^{3}}=\int_{0}^{z}\frac{\eta
d\eta}{1+n_{1}\eta+n_{2}\eta^{2}}\end{array}$ (2.72)

and the transformation rule (2.37). ∎

Its easy to see, that for any $\tilde{n}_{2}$ and $\tilde{n}_{3}$ the
equations

$\begin{array}[]{c}\tilde{n}_{2}=n_{2}-n_{1}^{2}/3,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\tilde{n}_{3}=-n_{1}n_{2}/3+2n_{1}^{3}/27\end{array}$
(2.73)

have solutions. Hence, we can work with the 2-parameter subspace of the
Virasoro group, described by

$\begin{array}[]{c}\frac{f(z)^{2}}{2}=\int_{0}^{z}\frac{\eta
d\eta}{1+n_{1}\eta+n_{2}\eta^{2}}.\end{array}$ (2.74)

Let us parametrize it by $p$ and $q$

$\begin{array}[]{c}1+n_{1}z+n_{2}z^{2}=(1+\sqrt{p+q}z)(1+qz/\sqrt{p+q}),\end{array}$
(2.75)

or, equivalently,

$\begin{array}[]{c}q=n_{2},\,\,\,\,\,\
p=\frac{n_{1}^{2}-2n_{2}\pm\sqrt{n_{1}^{4}-4n_{1}^{2}n_{2}}}{2}.\end{array}$
(2.76)

We assume that $p+q\neq 0$, then $x$ is defined by

$\begin{array}[]{c}dx=f(z)df(z)=\frac{zdz}{(1+\sqrt{p+q}z)(1+qz/\sqrt{p+q})}.\end{array}$
(2.77)

In this parametrization, if $p\neq 0$ and $q\neq 0$ we have

$\begin{array}[]{c}x(z)=\frac{p+q}{pq}\log\left(1+\frac{qz}{\sqrt{p+q}}\right)-\frac{1}{p}\log(1+\sqrt{p+q}z).\end{array}$
(2.78)

For $q=0$ it degenerates to

$\begin{array}[]{c}x(z)=\frac{z}{\sqrt{p}}-\frac{1}{p}\log(1+\sqrt{p}z),\end{array}$
(2.79)

and for $p=0$ it degenerates to

$\begin{array}[]{c}x(z)=\frac{1}{q}\log(1+\sqrt{q}z)-\frac{1}{\sqrt{q}}\frac{z}{1+\sqrt{q}z}.\end{array}$
(2.80)

Let us denote

$\begin{array}[]{c}I(z):=\frac{1}{\sqrt{2\pi
z}}\int_{\gamma}\frac{dx(\eta)}{\eta}e^{-\frac{x(\eta)}{z}}.\end{array}$
(2.81)

It appears that if the factorization constraint (2.56) is satisfied, then
corresponding $R(z)$ satisfies the symplectic condition. Namely, let us
consider a two-parametric family of functions satisfying this condition,

$\begin{array}[]{c}R_{q,p}(z)=\exp\left({-\sum_{k=1}^{\infty}\frac{B_{2k}}{2k(2k-1)}\left(p^{2k-1}+q^{2k-1}-\left(\frac{pq}{p+q}\right)^{2k-1}\right)z^{2k-1}}\right).\end{array}$
(2.82)

Here $B_{2k}$ are the Bernoulli numbers

$\begin{array}[]{c}\frac{xe^{x}}{e^{x}-1}=1+\frac{x}{2}+\sum_{k=1}^{\infty}\frac{B_{2k}x^{2k}}{(2k)!}.\end{array}$
(2.83)

######  Lemma 2.5.

$\begin{array}[]{c}I(z)=R_{q,p}(-z).\end{array}$ (2.84)

###### Proof.

We need to consider three different cases.

1) Let $p\neq 0$ and $q\neq 0$. Then $x(z)$ is given by (2.78). From the
integration by parts we have

$\begin{array}[]{c}\int_{\gamma}dx(\eta)\,e^{-\frac{x(\eta)}{z}}=0,\end{array}$
(2.85)

hence

$\begin{split}I(z)&=\frac{1}{\sqrt{2\pi
z}}\int_{\gamma}dx(\eta)\left(\frac{1}{\eta}-q\sqrt{p+q}\right)e^{-\frac{x(\eta)}{z}}\\\
&=\frac{1}{\sqrt{2\pi
z}}\int_{\gamma}\frac{d\eta}{(1+\sqrt{p+q}\eta)^{1-\frac{1}{pz}}(1+q\eta/\sqrt{p+q})^{\frac{p+q}{pqz}}}.\end{split}$
(2.86)

Let us make a change of integration variable
$\eta\mapsto\frac{p}{q\sqrt{p+q}}\eta-\frac{1}{\sqrt{p+q}}$, then

$\begin{array}[]{c}I(z)=\frac{1}{\sqrt{2\pi
z(p+q)}}\frac{(p+q)^{\frac{1}{pz}+\frac{1}{qz}}}{q^{\frac{1}{pz}}p^{\frac{1}{qz}}}\int_{\tilde{\gamma}}\frac{\eta^{\frac{1}{pz}-1}d\eta}{(1+\eta)^{\frac{1}{pz}+\frac{1}{qz}}}.\end{array}$
(2.87)

Here $\tilde{\gamma}$ is the new contour, corresponding to the saddle point at
$\eta=q/p$. We can deform the integration contour in such a way that the
asymptotic expansion does not change, which allows us to identify it with the
asymptotic expansion of the beta-function

$\begin{array}[]{c}B(a,b)=\int_{0}^{\infty}\frac{\eta^{a-1}dx}{(1+\eta)^{a+b}}\end{array}$
(2.88)

for $a=1/pz$ and $b=1/qz$. From Stirling’s expansion

$\begin{array}[]{c}\Gamma(z^{-1})=\sqrt{2\pi}e^{-z}z^{z-1/2}e^{\sum_{k=1}^{\infty}\frac{B_{2k}}{2k(2k-1)}z^{2k-1}}\end{array}$
(2.89)

one has the asymptotic expansion of the beta-function

$\begin{array}[]{c}B({(qz)}^{-1},{(pz)}^{-1})=\frac{\sqrt{2\pi
z(p+q)}q^{\frac{1}{pz}}p^{\frac{1}{qz}}}{(p+q)^{\frac{1}{pz}+\frac{1}{qz}}}e^{\sum_{k=1}^{\infty}\frac{B_{2k}}{2k(2k-1)}\left((pz)^{2k-1}+(qz)^{2k-1}-(\frac{pq}{p+q}z)^{2k-1}\right)}.\end{array}$
(2.90)

The statement of the lemma follows from the comparison of this expansion with
(2.87).

2) Let $q=0$. Then $x(z)$ is given by (2.79), and

$\begin{array}[]{c}I(z)=\frac{1}{\sqrt{2\pi
z}}\int_{\gamma}\frac{d\eta}{(1+\sqrt{p}\eta)^{1-\frac{1}{pz}}}e^{-\frac{\eta}{z\sqrt{p}}}.\end{array}$
(2.91)

After a change of the integration variable, $\eta\mapsto
z\sqrt{p}\eta-\frac{1}{\sqrt{p}}$, we get

$\begin{array}[]{c}I(z)=\frac{1}{\sqrt{2\pi
zp}}e^{\frac{1}{zp}}(zp)^{\frac{1}{zp}}\int_{\tilde{\gamma}}d\eta\eta^{\frac{1}{pz}-1}e^{-\eta}.\end{array}$
(2.92)

The asymptotic expansion of the integral coincides with the asymptotic
expansion of the gamma-function

$\begin{array}[]{c}\Gamma(z^{-1})=\int_{0}^{\infty}d\eta\eta^{\frac{1}{z}-1}e^{-\eta},\end{array}$
(2.93)

and the statement of lemma follows from the Stirling expansion (2.89).

3) Let $p=0$. This case can be reduced to the previous one. Namely, for $p=0$
we have

$\begin{array}[]{c}I(z)=\frac{1}{\sqrt{2\pi
z}}\int_{\gamma}\frac{d\eta}{(1+\sqrt{q}\eta)^{2+\frac{1}{qz}}}e^{\frac{1}{\sqrt{q}z}\frac{\eta}{1+\sqrt{q}\eta}}\end{array}$
(2.94)

which reduces to (2.79) after the change of integration variable
$\eta\mapsto-\eta/(1+\sqrt{q}\eta)$ and identification of $q$ and $p$ if we
apply (2.85). This completes the proof. ∎

Below we assume that coefficients ${\bf a}$ are fixed by (2.30), where the
function $f(z)$ is given by (2.77). Let us consider the operator

$\begin{array}[]{c}D_{q,p}=-e^{{\sum_{k\in\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>0}}a_{k}{\mathtt{l}}_{k}}}\,\frac{1}{z}\frac{\partial}{\partial
z}\,e^{-{\sum_{k\in\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>0}}a_{k}{\mathtt{l}}_{k}}}=-\frac{\partial}{\partial x},\end{array}$
(2.95)

or, equivalently,

$\begin{array}[]{c}D_{q,p}=-\frac{(1+\sqrt{p+q}z)(1+qz/\sqrt{p+q})}{z}\frac{\partial}{\partial
z}.\end{array}$ (2.96)

Then the functions

$\begin{array}[]{c}\phi_{k}(z):=D_{q,p}^{k}\cdot\frac{1}{z}\end{array}$ (2.97)

define a change of variables if we associate $kt_{k}$ with $z^{-k}$ and
$T_{k}^{q,p}$ with $\phi_{k}(z)$. This change of variables can be described by
the recursion

$\begin{array}[]{c}T_{0}^{q,p}({\bf t})=t_{1},\\\ T_{k}^{q,p}({\bf
t})=\left(q\widehat{L}_{0}+\frac{2q+p}{\sqrt{p+q}}\widehat{L}_{-1}+\left(\widehat{L}_{-2}-\frac{t_{1}^{2}}{2}\right)\right)\cdot
T_{k-1}^{q,p}.\end{array}$ (2.98)

Let us prove that the linear change of variables (2.98) gives a
transformation, required for identification of the Givental and Heisenberg-
Virasoro operators.

######  Lemma 2.6.

$\begin{array}[]{c}T^{R}_{k}({\bf T}_{q,p}({\bf
t}))=\widehat{V}_{0}\cdot(2k+1)!!t_{2k+1},\,\,\,\,\,\,k\in{\hbox{{\alef
Z}}_{\geq 0}}.\end{array}$ (2.99)

###### Proof.

Let us prove an equivalent relation

$\begin{array}[]{c}\widehat{V}_{0}^{-1}\cdot\sum_{k=0}^{\infty}T^{R}_{k}({\bf
T}_{q,p}({\bf t}))w^{k}=\sum_{k=0}^{\infty}(2k+1)!!t_{2k+1}w^{k}.\end{array}$
(2.100)

Let us associate $kt_{k}$ with $z^{-k}$. Then
$\sum_{k=0}^{\infty}T^{R}_{k}w^{k}$ is associated with

$\begin{array}[]{c}R(-w)\sum_{k=0}^{\infty}w^{k}\left(-\frac{1}{z}\frac{\partial}{\partial
z}\right)^{k}\cdot\frac{1}{z},\end{array}$ (2.101)

and, by (2.95), the sum $\sum_{m=0}^{\infty}T^{R}_{m}({\bf T}_{q,p}({\bf
t}))w^{m}$ is associated with

$\begin{array}[]{c}R(-w)\sum_{m=0}^{\infty}w^{m}D_{q,p}^{m}\cdot\frac{1}{z}.\end{array}$
(2.102)

Let us omit the factor $R(-w)$ for a moment and act on (2.102) by
$e^{-{\sum_{\raise-1.0pt\hbox{$\mbox{\Bbbb Z}$}_{>0}}a_{k}{\mathtt{l}}_{k}}}$:

$\begin{split}\left(e^{-{\sum_{\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>0}}a_{k}{\mathtt{l}}_{k}}}\cdot\sum_{m=0}^{\infty}w^{m}D_{q,p}^{m}\cdot\frac{1}{z}\right)_{-}&=\left(\sum_{m=0}^{\infty}w^{m}\left(-\frac{1}{z}\frac{\partial}{\partial
z}\right)^{m}e^{-{\sum_{\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>0}}a_{k}{\mathtt{l}}_{k}}}\cdot\frac{1}{z}\right)_{-}\\\
&=\left(\sum_{m=0}^{\infty}w^{m}\left(-\frac{1}{z}\frac{\partial}{\partial
z}\right)^{m}\frac{1}{h(z)}\right)_{-}\\\ &=\sum_{k,m=0}^{\infty}\frac{1}{2\pi
i}\frac{w^{m}}{z^{2k+1}}\oint\frac{d\eta}{h(\eta)}\left(\frac{\partial}{\partial\eta}\frac{1}{\eta}\right)^{m}\eta^{2k}\\\
&=\sum_{k,m=0}^{\infty}\frac{(2k-1)!!}{2\pi
i}\frac{w^{k+m}}{z^{2k+1}}\oint\frac{d\eta}{h(\eta)}\frac{(-1)^{m}(2m-1)!!}{\eta^{2m}}\\\
&=\sum_{k=0}^{\infty}(2k-1)!!\frac{w^{k}}{z^{2k+1}}\frac{1}{\sqrt{-2\pi
z}}\int_{\gamma}dx\frac{x}{h(x)}e^{\frac{x^{2}}{2z}}\\\
&=R(w)\sum_{k=0}^{\infty}(2k-1)!!\frac{w^{k}}{z^{2k+1}},\end{split}$ (2.103)

where for any series $\left(\sum_{k\in\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}}b_{k}z^{k}\right)_{-}:=\sum_{k\in\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{<0}}b_{k}z^{k}$. Hence,

$\begin{array}[]{c}\left(e^{-{\sum_{\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>0}}a_{k}{\mathtt{l}}_{k}}}\cdot
R(-w)\sum_{m=0}^{\infty}w^{m}D_{q,p}^{m}\cdot\frac{1}{z}\right)_{-}=\sum_{k=0}^{\infty}(2k-1)!!\frac{w^{k}}{z^{2k+1}},\end{array}$
(2.104)

and two linear transformations coincide. ∎

Let us also introduce

$\begin{array}[]{c}v_{k}:=[z^{k}]\int_{0}^{z}(f(\eta)-y(\eta))dx(\eta),\end{array}$
(2.105)

where

$\begin{array}[]{c}y(z)=\int^{z}\frac{dx(\eta)}{\eta}.\end{array}$ (2.106)

For $p\neq 0$ and $q\neq 0$

$\begin{array}[]{c}y(z)=\frac{\sqrt{p+q}}{p}\left(\log\left(1+\sqrt{p+q}z\right)-\log\left(1+\frac{qz}{\sqrt{p+q}}\right)\right).\end{array}$
(2.107)

For $q=0$ it reduces to

$\begin{array}[]{c}y(z)=\frac{1}{\sqrt{p}}\log\left(1+\sqrt{p}z\right)\end{array}$
(2.108)

and for $p=0$ it degenerates to

$\begin{array}[]{c}y(z)=\frac{z}{1+\sqrt{q}z}.\end{array}$ (2.109)

It is easy to see that $v_{k}=0$ for $k<4$.

Having in mind that the coefficients $v_{k}$ are given by (2.105), let us
remind the reader the main ingredients of the following theorem, given by
(2.27), (2.98), (2.30), (2.77), and (2.82), namely the Virasoro operators

$\begin{array}[]{c}\widehat{L}_{m}=\sum_{k=1}^{\infty}kt_{k}\frac{\partial}{\partial
t_{k+m}}+\frac{1}{2}\sum_{a+b=m}\frac{\partial^{2}}{\partial t_{a}\partial
t_{b}},\,\,\,\,\,\,\,\,\,m\geq 0,\end{array}$ (2.110)

the linear change of variables

$\begin{array}[]{c}T_{0}^{q,p}({\bf t})=t_{1},\\\ T_{k}^{q,p}({\bf
t})=\left(q\widehat{L}_{0}+\frac{2q+p}{\sqrt{p+q}}\widehat{L}_{-1}+\left(\widehat{L}_{-2}-\frac{t_{1}^{2}}{2}\right)\right)\cdot
T_{k-1}^{q,p},\end{array}$ (2.111)

and the coefficients $a_{k}$ are given implicitly through the formal series

$\begin{array}[]{c}f(z):=e^{{\sum_{k\in\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>0}}a_{k}{\mathtt{l}}_{k}}}\,z\,e^{-{\sum_{k\in\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>0}}a_{k}{\mathtt{l}}_{k}}}\in z+z{\hbox{{\alef
C}}}[\\![z]\\!],\end{array}$ (2.112)

which is defined by

$\begin{array}[]{c}f(z)df(z)=\frac{zdz}{(1+\sqrt{p+q}z)(1+qz/\sqrt{p+q})},\end{array}$
(2.113)

and a formal series

$\begin{array}[]{c}R_{q,p}(z)=\exp\left({-\sum_{k=1}^{\infty}\frac{B_{2k}}{2k(2k-1)}\left(p^{2k-1}+q^{2k-1}-\left(\frac{pq}{p+q}\right)^{2k-1}\right)z^{2k-1}}\right),\end{array}$
(2.114)

which defines an element of the Givental group. Then

######  Theorem 1.

Rank one Givental operator coincides up to a linear change of variables with
the element of the Heisenberg-Virasoro group of symmetries of KP hierarchy if
and only if $R(z)$ belongs to a family (2.114). For this family

$\begin{array}[]{c}\left.\widehat{R}_{q,p}\cdot Z({\bf t}^{o})\right|_{{\bf
T}={\bf T}^{q,p}({\bf
t})}=e^{\hbar^{-1}\sum_{k=4}v_{k}\frac{\partial}{\partial
t_{k}}}e^{\sum_{k\in\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>0}}a_{k}\widehat{L}_{k}}\cdot Z({\bf t}^{o})\end{array}$ (2.115)

for any function $Z({\bf t}^{o})$.

###### Proof.

We have already identified the quadratic parts of two operators and linear
changes of variables in Lemmas 2.5 and 2.6. It remains to identify the
translations of the variables. Let us show that for any $Z({\bf t}^{o})$

$\begin{array}[]{c}\widehat{V}_{0}^{-1}\left(\sum_{k=4}^{\infty}v_{k}\frac{\partial}{\partial
t_{k}}\right)\widehat{V}_{0}\cdot Z({\bf
t}^{o})=\sum_{k=2}^{\infty}\delta_{k}\frac{\partial}{\partial T_{k}}Z({\bf
t}^{o}),\end{array}$ (2.116)

where $\delta_{k}$’s are given by (2.22). By definition,

$\begin{split}e^{-{\sum_{\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>0}}a_{k}{\mathtt{l}}_{k}}}\,\sum_{k=4}^{\infty}v_{k}z^{k}\,e^{{\sum_{\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>0}}a_{k}{\mathtt{l}}_{k}}}&=\int_{0}^{h(z)}(f(\eta)-y(\eta))dx(\eta)\\\
&=\int_{0}^{z}(\eta-y(h(\eta)))\eta d\eta.\end{split}$ (2.117)

Using integration by parts we have

$\begin{split}\frac{1}{\sqrt{2\pi z}}\int_{\gamma}d\eta(\eta-y(h(\eta)))\eta
e^{-\frac{\eta^{2}}{2z}}&=z-\frac{z}{\sqrt{2\pi
z}}\int_{\gamma}dy(\eta)e^{-\frac{x(\eta)}{z}}\\\ &=z(1-R(-z)).\end{split}$
(2.118)

This completes the proof. ∎

Let us stress that this theorem essentially describes the equality of two
operators, therefore the formula is true for any function $Z({\bf t}^{o})$ for
which both sides make sense, not only for the solutions of the KdV hierarchy.

######  Remark 2.4.

The identification of the Givental and Heisenberg-Virasoro operators (2.115)
for $p=0$ was established in [46].

It is natural to compare the higher rank Givental group with the symmetry
group of the multicomponent KP hierarchy. However, for higher rank Givental
groups there are other promising possibilities, and one can try to identify
the Givental operators with the operators from the symmetry groups of other
solitonic integrable hierarchies. In particular, for rank two, one can
consider 2D Toda lattice and extended Toda lattice. It would be interesting to
find a relation between our analysis and the vertex operator analysis of the
bilinear Hirota equations considered in [29, 30] for the ADE type
singularities.

######  Remark 2.5.

For any semisimple cohomological field theory one can derive the Virasoro
constraints by conjugation of the constraints for the KdV tau-functions with
the Givental group element. However, for the cases, when after a linear change
of variables the generating function coincides with a tau-function of
integrable hierarchy, these constraints gain additional properties. Namely,
after a linear change of variables, they belong to the symmetry algebra of the
hierarchy. Virasoro type constrains and their conjugation by the Givental
operators can provide alternative way for analysis of the integrability and
other interesting properties.

##  3 Hodge integrals

###  3.1 Triple Hodge integrals

Denote by $\mathcal{M}_{g,n}$ the moduli space of all compact Riemann surfaces
of genus $g$ with $n$ distinct marked points. Deligne and Mumford [18] defined
a natural compactification
$\mathcal{M}_{g,n}\subset\overline{\mathcal{M}}_{g,n}$ via stable curves with
possible nodal singularities. The moduli space $\overline{\mathcal{M}}_{g,n}$
is a non-singular complex orbifold of dimension $3g-3+n$. It is empty unless
the stability condition

$\displaystyle 2g-2+n>0$ (3.1)

is satisfied. We refer the reader to [18, 34] for the basic theory.

In his seminal paper [49], Witten initiated new directions in the study of
$\overline{\mathcal{M}}_{g,n}$. For each marking index $i$ consider the
cotangent line bundle ${\hbox{{\alef
L}}}_{i}\rightarrow\overline{\mathcal{M}}_{g,n}$, whose fiber over a point
$[\Sigma,z_{1},\ldots,z_{n}]\in\overline{\mathcal{M}}_{g,n}$ is the complex
cotangent space $T_{z_{i}}^{*}\Sigma$ of $\Sigma$ at $z_{i}$. Let $\psi_{i}\in
H^{2}(\overline{\mathcal{M}}_{g,n},\hbox{{\alef Q}})$ denote the first Chern
class of ${\hbox{{\alef L}}}_{i}$. We consider the intersection numbers

$\displaystyle\left<\tau_{a_{1}}\tau_{a_{2}}\cdots\tau_{a_{n}}\right>_{g}:=\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{a_{1}}\psi_{2}^{a_{2}}\cdots\psi_{n}^{a_{n}}.$
(3.2)

The integral on the right-hand side of (3.2) vanishes unless the stability
condition (3.1) is satisfied, all $a_{i}$ are non-negative integers, and the
dimension constraint $3g-3+n=\sum a_{i}$ holds true. Let $T_{i}$, $i\geq 0$,
be formal variables and let

$\begin{array}[]{c}\tau_{KW}:=\exp\left(\sum_{g=0}^{\infty}\sum_{n=0}^{\infty}\hbar^{2g-2+n}F_{g,n}\right),\end{array}$
(3.3)

where

$\begin{array}[]{c}F_{g,n}:=\sum_{a_{1},\ldots,a_{n}\geq
0}\left<\tau_{a_{1}}\tau_{a_{2}}\cdots\tau_{a_{n}}\right>_{g}\frac{\prod
T_{a_{i}}}{n!}.\end{array}$ (3.4)

Witten’s conjecture [49], proved by Kontsevich [38], states that the partition
function $\tau_{KW}$ becomes a tau-function of the KdV hierarchy after the
change of variables $T_{n}=(2n+1)!!t_{2n+1}$. Integrability immediately
follows [37] from Kontsevich’s matrix integral representation, for more
details see the companion paper [11].

There are different interesting deformations of this tau-function. Let us
consider the Hodge bundle 𝔼, a rank $g$ vector bundle over
${\mathcal{M}}_{g,n}$. Let ${\rm ch}_{k}\in
H^{2k}(\overline{\mathcal{M}}_{g,n})$ be the components of the Chern character
of 𝔼. According to Mumford, the even components of the Chern character of 𝔼
identically vanish. Then we consider

$\begin{array}[]{c}\tilde{\mathcal{F}}_{g,n}=\sum_{a_{1},\ldots,a_{n}\geq
0}\frac{\prod
T_{a_{i}}}{n!}\int_{\overline{\mathcal{M}}_{g,n}}e^{\sum_{k=1}^{\infty}s_{k}{\rm
ch}_{2k-1}}\psi_{1}^{a_{1}}\psi_{2}^{a_{2}}\cdots\psi_{n}^{a_{n}}.\end{array}$
(3.5)

We introduce a generating function

$\begin{array}[]{c}Z({\bf T};{\bf
s})=\exp\left(\sum_{g=0}^{\infty}\sum_{n=0}^{\infty}\hbar^{2g-2+n}\tilde{\mathcal{F}}_{g,n}\right).\end{array}$
(3.6)

In general, this deformation does not preserve integrability, because it is
easy to check that $Z({\bf T};{\bf s})$ is not a tau-function of the KdV
hierarchy in the variables $t_{k}$ anymore. Investigation of the integrable
properties of $Z({\bf t};{\bf s})$ is one of the main goals of this paper.

From Mumford’s theorem [47] it follows that

$\begin{array}[]{c}Z({\bf T};{\bf s})=\widehat{R}({\bf s})\cdot\tau_{KW}({\bf
T}),\end{array}$ (3.7)

where

$\begin{array}[]{c}\widehat{R}({\bf
s}):=\exp\left({\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}s_{k}\widehat{W}_{k}}\right)\end{array}$
(3.8)

and operators $\widehat{W}_{k}$ are given by (2.16).

For the Hodge bundle over $\overline{\mathcal{M}}_{g,n}$ let
$\lambda_{i}=c_{i}({\hbox{{\alef E}}})$. Then

$\begin{array}[]{c}\Lambda_{g}(u)=\sum_{i=0}^{g}u^{i}\lambda_{i}=e^{\sum_{m=1}^{\infty}(2m-2)!{\rm
ch}_{2m-1}u^{2m-1}}\end{array}$ (3.9)

satisfies $\Lambda_{g}(u)\Lambda_{g}(-u)=1$. Particular specifications of the
parameters ${\bf s}$ correspond to various families of Hodge integrals. The
generating function of the $m$-linear Hodge integrals

$\begin{array}[]{c}\int_{\overline{\mathcal{M}}_{g,n}}\Lambda_{g}(u_{1})\Lambda_{g}(u_{2})\dots\Lambda_{g}(u_{m})\psi_{1}^{a_{1}}\psi_{2}^{a_{2}}\cdots\psi_{n}^{a_{n}}\end{array}$
(3.10)

corresponds to the Miwa parametrization of ${\bf s}$ variables

$\begin{array}[]{c}s_{k}=(2k-2)!\sum_{j=1}^{m}u_{j}^{2k-1}.\end{array}$ (3.11)

In particular, the generating function of linear Hodge integrals corresponds
to the case $s_{k}=(2k-2)!u^{2k-1}$. Form Givental’s theory it follows that a
product of the Hodge classes $\Lambda_{g}(u)$ defines a CohFT with flat unit.

######  Remark 3.1.

We see that it is natural to consider a Miwa-like parametrization of the ${\bf
s}$ variables. It might be an indication of the existence of the integrable
structure in these variables, at least for certain values of ${\bf T}$
variables.

Let us consider the case of cubic Hodge integrals with an additional Calabi-
Yau condition

$\begin{array}[]{c}\frac{1}{u_{1}}+\frac{1}{u_{2}}+\frac{1}{u_{3}}=0.\end{array}$
(3.12)

It is convenient to use the parametrization

$\begin{array}[]{c}u_{1}=-p,u_{2}=-q,u_{3}=\frac{pq}{p+q}.\end{array}$ (3.13)

######  Remark 3.2.

Cubic Hodge integrals satisfying the Calabi-Yau condition are related to the
topological vertex, which plays an important role in the topological string
models on 3-dimensional Calabi-Yau manifolds. In this context the parameters
$p$ and $q$ are related to the local framings, for more detail see, e.g., [26]
and references therein.

Consider the generating function

$\begin{array}[]{c}Z_{q,p}({\bf
T})=e^{\sum_{g=0}^{\infty}\sum_{n=0}^{\infty}\hbar^{2g-2+n}{\mathcal{F}}_{g,n}},\end{array}$
(3.14)

where

$\begin{array}[]{c}{\mathcal{F}}_{g,n}=\sum_{a_{1},\dots,a_{n}}\frac{\prod
T_{a_{i}}}{n!}\int_{\overline{\mathcal{M}}_{g,n}}\Lambda_{g}(-q)\Lambda_{g}(-p)\Lambda_{g}(\frac{pq}{p+q})\psi_{1}^{a_{1}}\psi_{2}^{a_{2}}\cdots\psi_{n}^{a_{n}}.\end{array}$
(3.15)

Linear Hodge integrals appear as a specification at $p=u^{2}$, $q=0$ (or
$q=u^{2}$, $p=0$), because $\Lambda_{g}(0)=1$. From Theorem 1 and equation
(3.7) it immediately follows that

######  Theorem 2.

Generating function of triple Hodge integrals, satisfying the Calabi-Yau
condition, in the variables (2.98) is a tau-function of the KP hierarchy,

$\begin{array}[]{c}\tau_{q,p}({\bf t})=Z_{q,p}({{\bf T}^{q,p}(\bf
t)}).\end{array}$ (3.16)

It is related to the Kontsevich-Witten tau-function by an element of the
Heisenberg-Virasoro group

$\begin{array}[]{c}\tau_{q,p}({\bf
t})=e^{\hbar^{-1}\sum_{k=4}v_{k}\frac{\partial}{\partial
t_{k}}}e^{\sum_{k\in\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>0}}a_{k}\widehat{L}_{k}}\cdot\tau_{KW}({\bf t}).\end{array}$ (3.17)

For $p=0$ the KP integrability was proved by Kazarian [36].

######  Remark 3.3.

After announcement of the main results of this project, Kramer [39] found an
independent proof of KP integrability of triple Hodge integrals, satisfying
the Calabi-Yau condition. This prove uses the Mariño-Vafa formula (see [50]
for more details).

One can reverse the logic and prove the Mariño-Vafa formula using our proof of
Theorem 2. Indeed, the Mariño-Vafa formula describes the coefficents of
expansion of two KP tau-functions, related by an element of the Heisenberg-
Virasoro subgroup of the KP hierarchy symmetry group. Therefore, to prove the
Mariño-Vafa formula it is enough to prove the relation between two tau-
functions of the KP hierarchy, which can be done using the relations between
corresponding points of the Sato Grassmannian and Kac-Schwarz algebras. The
details will be given elsewhere.

######  Conjecture 3.1.

The case of triple Hodge integrals, satisfying the Calabi-Yau condition, is
the most general case of generating functions of Hodge integrals (3.6),
satisfying the KP hierarchy in the variables ${\bf T}$ after a linear change
of variables. A direct analog of this statement is also true for the
$\Theta$-Hodge integrals, see next section.

######  Conjecture 3.2.

Theorem 2 has a direct generalization for the $r$-spin case.

In a companion paper [11] we develop the methods of the KP hierarchy, suitable
for the investigation of the family of the tau-functions, conjecturally
describing a generalization of $\tau_{q,p}$ for the $r$-spin case.

######  Remark 3.4.

It would also be interesting to investigate KP type integrability of Hodge
integrals in the context of Gromov-Witten theory [17, 27]. Particularly
attracting is the Gromov-Witten theory of ${\hbox{{\alef P}}}^{1}$, where the
generating function is given by the tau-function of the extended Toda
hierarchy.

Hodge integrals, in particular, the representatives of the Calabi-Yau family
for the specific values of the parameters $q$ and $p$, are related to several
other interesting integrable structures, including the Dubrovin-Zhang,
Volterra or Hodge integrable hierarchies [14, 48, 19, 20, 41, 50]. Connection
between them and the KP integrability, investigated in this paper, will be
discussed elsewhere.

It is also interesting to consider the specific values of the parameters $q$
and $p$, for which the tau-function $\tau_{q,p}$ describe $k$-reductions of
the KP hierarchy.

######  Corollary 3.1.

For $p=-2q$ the tau-function $\tau_{q,p}$ does not depend on even times,

$\begin{array}[]{c}\frac{\partial}{\partial t_{2k}}\tau_{q,-2q}({\bf
t})=0,\,\,\,\,\,\,\,k\in{\hbox{{\alef Z}}}_{>0},\end{array}$ (3.18)

that is $\tau_{q,-2q}({\bf t})$ is a tau-function of the KdV hierarchy.

###### Proof.

It follows from the change of variables (2.98). Indeed, since $2q+p=0$ the
operator, in (2.98) does not contain $\widehat{L}_{-1}$, and the operators
$\widehat{L}_{0}-\frac{t_{1}^{2}}{2}$ and $\widehat{L}_{-2}$ preserve the
parity of the indices of time variables. ∎

It is easy to see that it is impossible to get higher $k$-reductions of the KP
hierarchy (aka Gelfand–Dickey hierarchies) this way.

In [19] the relation between the generating function $Z_{q,-2q}$ and the
discrete KdV was established. Our results show that there is a simpler
relation between $Z_{q,-2q}$ and the ordinary KdV hierarchy, given by a linear
change of variables. We expect that this relation can help us better
understand the relation between the discrete and ordinary KdV hierarchies.

###  3.2 Triple $\Theta$-Hodge integrals

From Theorem 1 it follows that the action of Givental operator
$\widehat{R}_{q,p}$ on any KdV tau-function leads to the solution of the KP
hierarchy after a linear change of variables. The most natural alternative to
the KW tau-function here is the Brézin-Gross-Witten (BGW) tau-function. This
KdV tau-function governs the intersection theory with the insertions of the
fascinating Norbury’s $\Theta$-classes. We refer the reader to [44, 16, 45]
for a detailed description. The role of the BGW tau-function as the universal
building block of TR/GD was first observed by the author, Mironov and Morozov
[3, 4, 5] in the context of matrix models.

Norbury’s $\Theta$-classes are the cohomology classes, $\Theta_{g,n}\in
H^{4g-4+2n}(\overline{\mathcal{M}}_{g,n})$, described in [44]. Consider the
generating function of the intersection numbers of $\Theta$-classes and
$\psi$-classes

$\begin{array}[]{c}F^{\Theta}=\sum_{a_{1},\ldots,a_{n}\geq 0}\frac{\prod
T_{a_{i}}}{n!}\int_{\overline{\mathcal{M}}_{g,n}}\Theta_{g,n}\psi_{1}^{a_{1}}\psi_{2}^{a_{2}}\cdots\psi_{n}^{a_{n}}\end{array}$
(3.19)

then we have a direct analog of the Kontsevich-Witten tau-function [44]:

######  Theorem (Norbury).

Generating function

$\begin{array}[]{c}\tau_{\Theta}=\exp\left(\sum_{g=0}^{\infty}\sum_{n=0}^{\infty}\hbar^{2g-2+n}F_{g,n}^{\Theta}\right)\end{array}$
(3.20)

becomes a tau-function of the KdV hierarchy after the change of variables
$T_{n}=(2n+1)!!t_{2n+1}$.

Norbury also proved, that $\tau_{\Theta}$ is nothing but a tau-function of the
BGW model, described by a unitary matrix integral [33, 13]

$\begin{array}[]{c}\tau_{\Theta}=\tau_{BGW}.\end{array}$ (3.21)

KdV integrability of the BGW model follows from the relation to the
generalized Kontsevich model and was established by Mironov, Morozov and
Semenoff [43].

Insertion of Norbury’s $\Theta$-classes can be naturally absorbed by the
Givental formalism [45]. This allows us to relate integrable properties of
generating functions of intersection numbers with and without
$\Theta$-classes. Consider the symplectic transformation $R(z)$, introduced in
Section 2.1. For it we introduce the change of the dilaton shift associated to
the BGW tau-function

$\begin{array}[]{c}\sum_{k=1}^{\infty}\delta_{k}^{0}z^{k}:=1-R(-z).\end{array}$
(3.22)

Note, that the right hand side of this equation is equal to the right hand
side of (2.22), divided by $z$. For any symplectic transformation $R(z)$
consider the corresponding action of the Givental operator with new
translation part

$\begin{array}[]{c}\widehat{R}^{0}\cdot Z({\bf
T})=\,e^{\frac{1}{2}\sum_{i,j=0}^{\infty}{V_{ij}^{R}}\frac{\partial^{2}}{\partial
T_{i}\partial
T_{j}}}\,\left.e^{\hbar^{-1}\sum_{k=1}^{\infty}\delta_{k}^{0}\frac{\partial}{\partial
T_{k}}}\,Z({\bf T})\right|_{{T}_{k}\mapsto{T}^{R}_{k}}.\end{array}$ (3.23)

The classes $\Theta_{g,n}$ define a degenerate CohFT. It was shown by Norbury
that the multiplication of the CohFT by these classes corresponds to a simple
modification of the Givental operators, given by (3.23). It is described by
Proposition 3.9 of [45], let us formulate a version of this proposition for
the rank 1 case. We stress that the original proposition of Norbury describes
a relation between CohFT’s, and we consider only its corollary, a relation
between the partition functions of these CohFT’s. For more details see [45].

The action of the Givental group on the trivial CohFT, associated with the
Kontsevich–Witten tau-function, produces certain classes of cohomology
$\Omega_{g,n}\in H^{*}(\overline{\mathcal{M}}_{g,n})$. Let us consider it on
the level of partition functions. For any element of the Givental group
$\widehat{R}$ one has

$\begin{array}[]{c}\widetilde{Z}^{\Omega}({\bf
T})=\widehat{R}\cdot\tau_{KW},\end{array}$ (3.24)

where

$\begin{array}[]{c}\widetilde{Z}^{\Omega}({\bf
T})=\exp\left(\sum_{g=0}^{\infty}\sum_{n=0}^{\infty}\hbar^{2g-2+n}{\mathcal{F}}_{g,n}^{\Omega}\right),\end{array}$
(3.25)

with

$\begin{array}[]{c}{\mathcal{F}}_{g,n}^{\Omega}=\sum_{a_{1},\dots,a_{n}}\frac{\prod
T_{a_{i}}}{n!}\int_{\overline{\mathcal{M}}_{g,n}}\Omega_{g,n}\psi_{1}^{a_{1}}\psi_{2}^{a_{2}}\cdots\psi_{l}^{a_{n}}.\end{array}$
(3.26)

For Givental’s theory it follows that the classes $\Omega_{g,n}$ exist and
satisfy certain peculiar properties. In particular, for the rank one case
which we consider in this paper, they always can be expressed in terms of the
components of the Chern character of 𝔼, see (3.5).

Then the insertion of the $\Theta$-classes is described by

######  Proposition 3.2 (Norbury).

$\begin{array}[]{c}\widetilde{Z}^{\Theta\Omega}({\bf
T})=\widehat{R}^{0}\cdot\tau_{\Theta},\end{array}$ (3.27)

where $\Theta\Omega$ denotes the classes $\Theta_{g,n}\Omega_{g,n}$.

Let us apply this proposition for triple Hodge integrals, that is,
$\Omega_{g,n}=\Lambda_{g}(-q)\Lambda_{g}(-p)\Lambda_{g}(\frac{pq}{p+q})$. Let

$\begin{array}[]{c}{Z}_{q,p}^{\Theta}({\bf
T})=\exp\left(\sum_{g=0}^{\infty}\sum_{n=0}^{\infty}\hbar^{2g-2+n}{\mathcal{F}}_{g,n}^{\Theta}\right),\end{array}$
(3.28)

where

$\begin{array}[]{c}{\mathcal{F}}_{g,n}^{\Theta}=\sum_{a_{1},\dots,a_{n}}\frac{\prod
T_{a_{i}}}{n!}\int_{\overline{\mathcal{M}}_{g,n}}\Theta_{g,n}\Lambda_{g}(-q)\Lambda_{g}(-p)\Lambda_{g}(\frac{pq}{p+q})\psi_{1}^{a_{1}}\psi_{2}^{a_{2}}\cdots\psi_{l}^{a_{n}}.\end{array}$
(3.29)

From Proposition 3.2 it immediately follows that

######  Lemma 3.3.

$\begin{array}[]{c}{Z}_{q,p}^{\Theta}({\bf
T})=\widehat{R}^{0}_{q,p}\cdot\tau_{\Theta}.\end{array}$ (3.30)

Let us introduce the coefficients

$\begin{array}[]{c}\tilde{v}^{0}_{k}=[z^{k}]\int_{0}^{z}\left(df(\eta)-dy(\eta)\right).\end{array}$
(3.31)

They are related to the translations (3.22),

$\begin{array}[]{c}\widehat{V}_{0}^{-1}\left(\sum_{k=2}^{\infty}\tilde{v}^{0}_{k}\frac{\partial}{\partial
t_{k}}\right)\widehat{V}_{0}\cdot Z({\bf
t}^{o})=\sum_{k=1}^{\infty}\delta_{k}^{0}\frac{\partial}{\partial T_{k}}\cdot
Z({\bf t}^{o}).\end{array}$ (3.32)

Indeed

$\begin{split}e^{-{\sum_{\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>0}}a_{k}{\mathtt{l}}_{k}}}\sum_{k=2}^{\infty}v_{k}^{0}z^{k}e^{{\sum_{\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>0}}a_{k}{\mathtt{l}}_{k}}}&=\int_{0}^{h(z)}\left(df(\eta)-dy(\eta)\right)\\\
&=\int_{0}^{z}(d\eta-dy(h(\eta))),\end{split}$ (3.33)

and from integration by parts

$\begin{split}\frac{1}{\sqrt{2\pi z}}\int_{\gamma}(d\eta-
dy(h(\eta)))e^{-\frac{\eta^{2}}{2z}}&=1-\frac{1}{\sqrt{2\pi
z}}\int_{\gamma}dy(\eta)e^{-\frac{x(\eta)}{z}}\\\ &=1-R(-z).\end{split}$
(3.34)

Then we have

######  Theorem 3.

Generating function of triple $\Theta$-Hodge integrals, satisfying the Calabi-
Yau condition, in the variables (2.98) is a tau-function of the KP hierarchy,

$\begin{array}[]{c}{\tau}^{\Theta}_{q,p}({\bf t})={Z}_{q,p}^{\Theta}({{\bf
T}^{q,p}(\bf t)}).\end{array}$ (3.35)

It is related to the Brézin-Gross-Witten tau-function by an element of the
Heisenberg-Virasoro group

$\begin{array}[]{c}{\tau}^{\Theta}_{q,p}({\bf
t})=e^{\hbar^{-1}\sum_{k>0}\tilde{v}_{k}^{0}\frac{\partial}{\partial
t_{k}}}e^{\sum_{k\in\raise-1.0pt\hbox{$\mbox{\Bbbb
Z}$}_{>0}}a_{k}\widehat{L}_{k}}\cdot\tau_{BGW}({\bf t}).\end{array}$ (3.36)

######  Conjecture 3.3.

Theorem 3 has a direct generalization for the $r$-spin case.

Similar to Corollary 3.1 one has

######  Corollary 3.4.

For $p=-2q$ the tau-function $\tau_{q,p}$ does not depend on even times,

$\begin{array}[]{c}\frac{\partial}{\partial t_{2k}}\tau^{\Theta}_{q,-2q}({\bf
t})=0,\,\,\,\,\,\,\,k\in{\hbox{{\alef Z}}}_{>0},\end{array}$ (3.37)

that is $\tau^{\Theta}_{q,-2q}({\bf t})$ is a tau-function of the KdV
hierarchy.

The proof is completely analogous to the proof Corollary 3.1.

Generating functions ${Z}_{q,p}$, ${Z}_{q,p}^{\Theta}$ satisfy the Virasoro
constraints, and tau-functions ${\tau}_{q,p}$, ${\tau}^{\Theta}_{q,p}$ satisfy
the Heisenberg-Virasoro constraints. All these constraints can be obtained
from the constraints for $\tau_{KW}$ and $\tau_{BGW}$ by conjugation with
Givental (for $Z$’s) and Heisenberg-Virasoro (for $\tau$’s) group elements.
Moreover, all these generating functions possess simple cut-and-join
description. These topics, as well as matrix integral description of
${\tau}^{\Theta}_{q,p}({\bf t})$ and its generalization for the $r$-spin case,
will be described elsewhere.

## Acknowledgments

The author is grateful to G. Carlet, R. Kramer, and S. Shadrin for useful
discussions and the anonymous referees for the suggested improvements. This
work was supported by IBS-R003-D1 and by RFBR grant 18-01-00926. The author
would also like to thank Vasily Pestun for his hospitality at IHES supported
by the European Research Council under the European Union’s Horizon 2020
research and innovation programme, QUASIFT grant agreement 677368.

## References

  * [1] A. S. Alexandrov, A. Mironov and A. Morozov, “Unified description of correlators in non-Gaussian phases of Hermitean matrix model,” Int. J. Mod. Phys. A 21 (2006) 2481.
  * [2] A. S. Alexandrov, A. D. Mironov and A. Y. Morozov, “Partition functions of matrix models as the first special functions of string theory: Finite Hermitian one-matrix model,” Theor. Math. Phys. 142 (2005) 349.
  * [3] A. S. Alexandrov, A. Mironov and A. Morozov, “M-theory of matrix models,” Theor. Math. Phys. 150 (2007) 153.
  * [4] A. S. Alexandrov, A. Mironov and A. Morozov, “Instantons and merons in matrix models,” Physica D 235 (2007) 126.
  * [5] A. Alexandrov, A. Mironov and A. Morozov, “BGWM as Second Constituent of Complex Matrix Model,” JHEP 0912 (2009) 053.
  * [6] A. Alexandrov, “Enumerative Geometry, Tau-Functions and Heisenberg-Virasoro Algebra,” Commun. Math. Phys. 338 (2015) no.1, 195.
  * [7] A. Alexandrov, “Cut-and-join description of generalized Brezin-Gross-Witten model,” Adv. Theor. Math. Phys. 22 (2018), 1347-1399.
  * [8] A. Alexandrov, “Open intersection numbers, Kontsevich-Penner model and cut-and-join operators,” JHEP 1508 (2015) 028.
  * [9] A. Alexandrov, D. Lewanski and S. Shadrin, “Ramifications of Hurwitz theory, KP integrability and quantum curves,” JHEP 1605 (2016) 124.
  * [10] A. Alexandrov, G. Chapuy, B. Eynard and J. Harnad, “Fermionic approach to weighted Hurwitz numbers and topological recursion,” Commun. Math. Phys. 360, no.2, 777-826 (2018).
  * [11] A. Alexandrov, A. Alexandrov, “KP integrability of triple Hodge integrals. II. Generalized Kontsevich matrix model,” Anal. Math. Phys. 11 (2021) no.1, 24.
  * [12] E. Brezin, C. Itzykson, G. Parisi and J. B. Zuber, “Planar Diagrams,” Commun. Math. Phys. 59 (1978) 35.
  * [13] E. Brezin and D. J. Gross, “The External Field Problem in the Large N Limit of QCD,” Phys. Lett. B 97 (1980) 120.
  * [14] A. Buryak, “Dubrovin-Zhang hierarchy for the Hodge integrals,” Commun. Num. Theor. Phys. 9 (2015) no.2, 239.
  * [15] L. Chekhov and B. Eynard, “Hermitean matrix model free energy: Feynman graph technique for all genera,” JHEP 0603 (2006) 014.
  * [16] L. Chekhov and P. Norbury, “Topological recursion with hard edges,” Int. J. Math. 30 (2019) no.03, 1950014.
  * [17] A. Chiodo and D. Zvonkine, “Twisted r-spin potential and Giventals quantization,” Adv. Theor. Math. Phys. 13 (2009) no.5, 1335.
  * [18] P. Deligne, D. Mumford, “The irreducibility of the space of curves of given genus,” Publications mathématiques de l’I.H.É.S. 36 (1969), 75–109.
  * [19] B. Dubrovin, S. Q. Liu, D. Yang and Y. Zhang, “Hodge–GUE Correspondence and the Discrete KdV Equation,” Commun. Math. Phys. 379 (2020) no.2, 461-490, doi:10.1007/s00220-020-03846-6
  * [20] B. Dubrovin, S. Q. Liu, D. Yang and Y. Zhang, “Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs,” Advances in Mathematics 293 (2016): 382-435.
  * [21] P. Dunin-Barkowski, N. Orantin, S. Shadrin and L. Spitz, “Identification of the Givental formula with the spectral curve topological recursion procedure,” Commun. Math. Phys. 328 (2014) 669.
  * [22] B. Eynard, “Topological expansion for the 1-Hermitian matrix model correlation functions,” JHEP 11 (2004), 031
  * [23] B. Eynard, “Invariants of spectral curves and intersection theory of moduli spaces of complex curves,” Commun. Num. Theor. Phys. 08 (2014) 541.
  * [24] B. Eynard, “The Geometry of integrable systems. Tau functions and homology of Spectral curves. Perturbative definition,” [arXiv:1706.04938 [math-ph]].
  * [25] B. Eynard and N. Orantin, “Invariants of algebraic curves and topological expansion,” Commun. Num. Theor. Phys. 1 (2007) 347.
  * [26] B. Eynard and N. Orantin, “Computation of Open Gromov-Witten Invariants for Toric Calabi-Yau 3-Folds by Topological Recursion, a Proof of the BKMP Conjecture,” Commun. Math. Phys. 337 (2015) no.2, 483-567.
  * [27] C. Faber and R. Pandharipande, “Hodge integrals and Gromov-Witten theory,” Inventiones mathematicae 139 (2000) no.1, 173.
  * [28] E. Feigin, J. van de Leur and S. Shadrin, “Givental symmetries of Frobenius manifolds and multi-component KP tau-functions,” Adv. Math. 224 (2010) no.3, 1031.
  * [29] E. Frenkel, A. Givental and T. Milanov, “Soliton equations, vertex operators, and simple singularities,” Funct. Anal. Other Math. 3 (2010) no.1, 47.
  * [30] A. Givental, and T. Milanov, ”Simple singularities and integrable hierarchies.” The breadth of symplectic and Poisson geometry. Birkhuser Boston, 2005. 173-201.
  * [31] A. Givental, “Semisimple Frobenius structures at higher genus,” Int. Math. Res. Not. 2001 (2001) no.23, 1265.
  * [32] A. B. Givental, “Gromov - Witten invariants and quantization of quadratic hamiltonians,” Moscow Math. J. 1 (2001) no.4, 551.
  * [33] D. J. Gross and E. Witten, “Possible Third Order Phase Transition in the Large N Lattice Gauge Theory,” Phys. Rev. D 21 (1980) 446.
  * [34] J. Harris, I. Morrison. Moduli of curves. Graduate Texts in Mathematics, 187. Springer-Verlag, New York, 1998.
  * [35] G. ’t Hooft, “A Planar Diagram Theory for Strong Interactions,” Nucl. Phys. B 72 (1974) 461.
  * [36] M. Kazarian, “KP hierarchy for Hodge integrals,” Adv. Math. 221 (2009) no.1, 1.
  * [37] S. Kharchev, A. Marshakov, A. Mironov, A. Morozov and A. Zabrodin, “Towards unified theory of 2-d gravity,” Nucl. Phys. B 380 (1992) 181.
  * [38] M. Kontsevich, “Intersection theory on the moduli space of curves and the matrix Airy function,” Commun. Math. Phys. 147, 1-23 (1992).
  * [39] R. Kramer, “Cycles of curves, cover counts, and central invariants,” PhD thesis, 2019.
  * [40] Y.-P. Lee, “Invariance of tautological equations II: Gromov-Witten theory,” Journal of the American Mathematical Society 22 (2008) no.2, 331.
  * [41] S. Q. Liu, D. Yang, Y. Zhang and C. Zhou, “The Hodge-FVH Correspondence,” Journal für die reine und angewandte Mathematik, 2021, https://doi.org/10.1515/crelle-2020-0051.
  * [42] A. Mironov and A. Morozov, “Virasoro constraints for Kontsevich-Hurwitz partition function,” JHEP 02, 024 (2009).
  * [43] A. Mironov, A. Morozov and G. W. Semenoff, “Unitary matrix integrals in the framework of generalized Kontsevich model. 1. Brezin-Gross-Witten model,” Int. J. Mod. Phys. A 11 (1996) 5031.
  * [44] P. Norbury, “A new cohomology class on the moduli space of curves,” [arXiv:1712.03662].
  * [45] P. Norbury, “Gromov-Witten invariants of $\hbox{{\alef P}}^{1}$ coupled to a KdV tau function” [arXiv:1812.04221].
  * [46] X. Liu and G. Wang, “Connecting the Kontsevich-Witten and Hodge Tau-functions by the ${\widehat{GL(\infty)}}$ Operators,” Commun. Math. Phys. 346 (2016) no.1, 143.
  * [47] D. Mumford, ”Towards an enumerative geometry of the moduli space of curves.” Arithmetic and geometry. Birkhauser Boston, 1983. 271-328.
  * [48] K. Takasaki, “Cubic Hodge integrals and integrable hierarchies of Volterra type,” [arXiv:1909.13095 [math-ph]].
  * [49] E. Witten, Two-dimensional gravity and intersection theory on moduli space. Surveys in differential geometry (Cambridge, MA, 1990), 243–310, Lehigh Univ., Bethlehem, PA, 1991.
  * [50] J. Zhou, “Hodge Integrals and Integrable Hierarchies,” Lett. Math. Phys. 93 (2010) no.1, 55-71.


††thanks: these authors contributed equally to this work††thanks: these
authors contributed equally to this work

# Free-Running Long-Distance Reference-Frame-Independent Quantum Key
Distribution

Bang-Ying Tang  College of Computer Science and Technology, National
University of Defense Technology, Changsha 410073, China  Huan Chen  College
of Liberal Arts and Sciences, National University of Defense Technology,
Changsha 410073, China  Ji-Peng Wang  School of Science and State Key
Laboratory of Information Photonics and Optical Communications, Beijing
University of Posts and Telecommunications, Beijing 100876, China  Hui-Cun Yu
Information and Navigation College, Air Force Engineering University, Xi’an
710077, China  College of Advanced Interdisciplinary Studies, National
University of Defense Technology, Changsha 410073, China  Lei Shi  Information
and Navigation College, Air Force Engineering University, Xi’an 710077, China
Shi-Hai Sun  School of Physics and Astronomy, Sun Yat-sen University, Zhuhai
519082, China  Wei Peng  College of Computer Science and Technology, National
University of Defense Technology, Changsha 410073, China  Bo Liu
<EMAIL_ADDRESS>College of Advanced Interdisciplinary Studies, National
University of Defense Technology, Changsha 410073, China  Wan-Rong Yu
<EMAIL_ADDRESS>College of Computer Science and Technology, National
University of Defense Technology, Changsha 410073, China

###### Abstract

Rapidly and randomly drifted reference frames will shorten the link distance
and decrease the secure key rate of realistic quantum key distribution (QKD)
systems. However, an actively or inappropriately implemented calibration
scheme will increase complexity of the systems and may open security
loopholes. In this article, we present a free-running reference-frame-
independent (RFI) QKD scheme, where measurement events are classified into
multiple slices with the same misalignment variation of reference frames and
each slice performs the post-processing procedure individually. We perform the
free-running RFI QKD experiment with a fiber link of
$100\text{\,}\mathrm{k}\mathrm{m}$ and the misalignment of the reference frame
between Alice and Bob is varied more than 29 periods in a 50.7-hour experiment
test. The average secure key rate is about
$734\text{\,}\mathrm{b}\mathrm{p}\mathrm{s}$ with a total loss of
$31.5\text{\,}\mathrm{d}\mathrm{B}$, which achieves the state-of-art
performance of the long-distance RFI QKD implementations. Our free-running RFI
scheme can be efficiently adapted into the satellite-to-ground and drone based
mobile communication scenarios, as it can be performed with rapidly varying
reference frame and a loss more than $40\text{\,}\mathrm{d}\mathrm{B}$, where
no secure key can be obtained by the original RFI scheme.

Quantum key distribution (QKD) has the potential to generate the information-
theoretical-secure keys between distant users (Alice and Bob) [1, 2, 3]. Since
the first BB84 protocol proposed [4], QKD has stepped into realistic
applications from the laboratory [5, 6, 7, 8, 9, 10, 11]. The communication
distance of QKD over different communication scenarios is significantly
improved, such as $509\text{\,}\mathrm{k}\mathrm{m}$ with fiber link [12],
$4600\text{\,}\mathrm{k}\mathrm{m}$ space-to-ground communication network
[13], $1\text{\,}\mathrm{k}\mathrm{m}$ optical-relayed entanglement
distribution over drones [14].

In general, the realistic QKD systems have a common need of establishing a
shared reference frame between distant users by characterizing and calibrating
the polarization or phase of photons. Nevertheless, an actively or
inappropriately implemented calibration scheme may increase the complexity of
the systems and may open an unpredictable security loophole [15, 16].

Reference-frame-independent (RFI) QKD protocol, proposed in 2010, is robust
for the slowly varying reference frames and suitable to the satellite-to-
ground and drone-based systems [17]. The secure key rate of RFI QKD systems
varies a lot with given misalignment ($\theta_{0}$) of reference frames [18].
Furthermore, F. Wang et al. shows the valid condition of RFI-QKD is that the
misalignment variation $\Delta\theta\leq\pi$ [19]. With the assumption of
$\Delta\theta\approx 0$, the fiber link distance of RFI QKD reaches to
$160\text{\,}\mathrm{k}\mathrm{m}$ implemented with measurement-device-
independent scheme [11, 20] and the free space RFI QKD experiment was
demonstrated with total loss of $38.0\text{\,}\mathrm{d}\mathrm{B}$ [9, 21].
However, the misalignment of reference frames in realistic RFI QKD scenarios
varies randomly, and the communication distance will be reduced heavily. W.
Liang et al. performed a proof-of-principle experiment of RFI QKD with random
$\theta_{0}$ and slowly varying phase environment, nevertheless, the
communication distance was limited to $65\text{\,}\mathrm{k}\mathrm{m}$ when
considering the finite-length effects [22, 23].

In this article, we proposed a free-running RFI scheme that can be adapted to
the long-distance QKD systems with randomly and rapidly varying misalignment
of reference frames. We implemented a free-running RFI QKD experiment over
$100\text{\,}\mathrm{k}\mathrm{m}$ fiber link with a total loss of
$31.5\text{\,}\mathrm{d}\mathrm{B}$ and the system repetition rate of
$80\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$. In the experiment, time-bin
encoded photons in $Z$ basis are used for generating secure keys and the
phase-encoded photons in the $X$ ($Y$) basis are used for quantifying the
characteristics of the quantum link. The experiment ran for
$50.7\text{\,}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{s}$ and the
misalignment of reference frame between Alice and Bob was more than 29
periods. The secure key rate of our implementation reaches
$734\text{\,}\mathrm{b}\mathrm{p}\mathrm{s}$, which achieves the state-of-art
performance of the long-distance RFI QKD systems.

Figure 1:  Illustration of the $100\text{\,}\mathrm{k}\mathrm{m}$ free-running
RFI QKD experiment. Two NI 6556 boards control the lasers and modulators. The
computers are used for data acquisition, post-processing, and classical
communication. APD is used to measure the intensity of signal and decoy states
so that NI 6556 adjusts the control signal of IM1. IM: intensity modulator,
BS: beam splitter, PM: phase modulator, PC: polarization controller, AMZI:
asymmetric Mach-Zehnder interferometer, SNSPD: superconducting nanowire single
photon detector, TDC: Time-to-Digital converter, APD: avalanche photon diode,
NI 6556: National Instruments 6556.

The setup of our free-running RFI QKD experiment is shown in FIG. 1. At
Alice’s side, photons are generated by a homemade laser with a repetition rate
of $80\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ and a pulse width of
$1.0\text{\,}\mathrm{n}\mathrm{s}$. An intensity modulator (IM1) and an
attenuator (ATT) are used to modulate quantum (signal and decoy) states of the
corresponding intensity. An asymmetric Mach–Zehnder interferometer (AMZI1)
divides each pulse into two adjacent pulses ($s$ and $l$) with a separation
time of $4.6\text{\,}\mathrm{n}\mathrm{s}$. The intensity modulator IM2
prepares the bit $0$ or $1$ of $Z$ basis by suppressing $s$ or $l$ randomly.
When both $s$ and $l$ pulses pass through IM2, a phase of
$\\{0,\pi/2,\pi,3\pi/2\\}$ will be randomly added to $s$ pulse with two phase
modulators (PM1 and PM2), where $0$ and $\pi$ indicate bit $0$ and $1$ in $X$
basis, $\pi/2$ and $3\pi/2$ indicate bit $0$ and $1$ in $Y$ basis. Then,
pulses are sent to Bob through a $100\text{\,}\mathrm{k}\mathrm{m}$ optical
fiber with a loss of
$0.235\text{\,}\mathrm{d}\mathrm{B}\mathrm{/}\mathrm{k}\mathrm{m}$.

At Bob’s side, the detection basis ($X$ and $Y$) is randomly chosen by the
phase modulator (PM3), where a phase of $0$ ($\pi/2$) is added to the $s$
pulse. Then another asymmetric Mach–Zehnder interferometer (AMZI2) which has
the same path difference with the AMZI1 divides each pulse into two pulses,
resulting four pulses, {$s+s$, $s+l$, $l+s$, $l+l$}. $s+l$ and $l+s$ pulses
will merge into one pulse due to the same arriving time and interfere at
AMZI2. The interference determines the exiting port of the $s+l$ ($l+s$) pulse
at the AMZI2, indicating the bit $0$ or $1$ of $X$($Y$) basis, respectively.
$s+s$ and $l+l$ pulses represent bit $0$ and $1$ of $Z$ basis. Then, photons
are coupled to two channels of the superconducting nanowire single photon
detector (SNSPD), and the arriving time is recorded with a time-to-digital
converter (TDC). The optimal detection efficiency, about 80%, is ensured by
the polarization controllers (PC2 and PC3). The dark count rate of the SNSPD
is about $200$ per second. The programmable modules (National Instruments
6556) are used to control the laser and modulators. The total loss of Bob’s
receiving module is around $10\text{\,}\mathrm{d}\mathrm{B}$.

Furthermore, an active feedback mechanism is performed to stabilize the
intensity of the quantum state, where a beam splitter (BS) and an avalanche
photon detector (APD) are performed to monitor the intensity of quantum
states. The optimal detection parameters are analyzed from the recorded events
of Bob’s side and then feedback to the programmable polarization controller
(PC1) and PM3.

In the RFI QKD system, the reference frame of $X$ ($Y$) is defined as

$\displaystyle X_{B}$ $\displaystyle=X_{A}\cos\theta+Y_{A}\sin\theta,$ (1)
$\displaystyle Y_{B}$ $\displaystyle=Y_{A}\cos\theta-X_{A}\sin\theta,$

where $\theta$ is the misalignment angle of reference frames between Alice and
Bob and $\theta\in[0,2\pi)$. The quality of the quantum channel $C$ can be
estimated as [17]

$\displaystyle C$ $\displaystyle=\left\langle
X_{A}X_{B}\right\rangle^{2}+\left\langle
Y_{A}Y_{B}\right\rangle^{2}+\left\langle
X_{A}Y_{B}\right\rangle^{2}+\left\langle Y_{A}X_{B}\right\rangle^{2}.$ (2)

Given initial $\theta_{0}$, the RFI-QKD can be performed with the varying
$\Delta\theta$ which satisfies

$\frac{3\times 10^{4}}{\eta N_{0}}\omega\leq\Delta\theta\leq\pi,$ (3)

where $\eta$ is the total efficiency of system, $\omega$ denotes the intense
constant of noise and $N_{0}$ is the photon-generating rate [19].

Assume $E_{k}^{\alpha,\beta}$ is the QBER, where $k\in\\{\mu,\nu\\}$ is the
mean photon number of signal and decoy states, $\alpha,\beta\in\\{X,Y,Z\\}$
are the basis chosen by Alice and Bob, respectively. And the total pulses sent
by Alice is $N_{\textrm{t}}$.

In the free-running RFI scheme, we split the misalignment angle $\theta$ into
$m$ slices as

$\Theta_{i}=\left\\{\begin{matrix}\\{\theta|(2-\frac{1}{m})\pi\leq\theta<2\pi~{}or~{}0\leq\theta<\frac{\pi}{m}\\},&i=0\\\
\\{\theta|\frac{2i-1}{m}\pi\leq\theta<\frac{2i+1}{m}\pi\\},&i>0\end{matrix}\right.,$
(4)

where $i=0,1,\dots,m-1$.

According to Equ (3), the interval sampling time $T$ for quantifying the
quantum channel should be less than $\pi/\omega$.

Given $T$, $\theta$ of each interval can be calculated as

$\theta=\left\\{\begin{array}[]{cc}\arccos\left(1-2E_{\mu}^{XX}\right)&E_{\mu}^{XY}<0.5\\\
2\pi-\arccos\left(1-2E_{\mu}^{XX}\right)&E_{\mu}^{XY}\geq
0.5\end{array}\right..$ (5)

Thus, the measurement events can be split into $m$ slices according to the
measured QBER in XY-basis and then generate the secure keys for each slice
respectively.

Figure 2: The measured QBER in each bases, and the count rate in Z-basis. Each
point is measured within 5s. (a) [(b)] shows the QBER of signal [decoy] state
on $X$, $Y$ basis for the total
$50.7\text{\,}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{s}$, (c) shows
the QBER of signal state and decoy state on $Z$ basis: $E_{\mu}^{ZZ}$ and
$E_{\mu}^{ZZ}$, and (d) shows the count rate of signal and decoy state on $Z$
basis: $N_{\mu}^{ZZ}$ and $N_{\nu}^{ZZ}$. Figure 3: The QBER of the merged
sets with the experimental measured data. (a) shows the QBER of signal state
on $X$,$Y$ basis, (b) shows the QBER of decoy state on $X$,$Y$ basis, (c)
shows the QBER on $Z$ basis, and each point represents the merged result of
each set. Figure 4:  The $C$ value, secure key rate of merged sub-block and
the simulated result with $N_{\textrm{t}}=10^{12}$ and $\Delta\theta=\pi/8$.
The $C$ value and secure key rate varies with $\theta$ because the
misalignment of reference frame affects the leaked information [18].

In the experiment, the $Z$ basis is sent with the probability of 50%, $X$ and
$Y$ basis are sent with the probability of 25%. The signal, decoy, and vacuum
states are sent with the probability of 50%, 40%, and 10%, respectively. The
mean photon number of signal and decoy states are $0.722$ and $0.104$. The
experiment is free-running for about
$50.7\text{\,}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{s}$. The
sampling interval $T$ is set to 5 seconds. The slice number $m$ is set as $16$
and the error correction efficiency is $1.16$ [24].

Figure 5: The simulated secure key rate of our free-running RFI scheme with
$N_{\textrm{t}}=10^{12},10^{13}$. Our experimental result and the simulated
optimal secure key rate of the original RFI scheme are also shown here. In the
simulation, $\omega=6.9\times 10^{-3}$ rad/s and $m=16$.

FIG. 2 shows the QBER of signal and decoy states for each bases and the count
rate of $Z$ basis. The QBER of the X, Y bases varied periodically with the
time due to the environmental noise which affects the phase of the two
interferometers, while the QBER of the Z basis maintained stable with small
fluctuations. The average QBER of the signal state on the $Z$ basis is about
0.6%. The average count rate of the signal and decoy states of $Z$ basis is
about $2.6\text{\,}\mathrm{k}\mathrm{b}\mathrm{p}\mathrm{s}$ and
$0.4\text{\,}\mathrm{k}\mathrm{b}\mathrm{p}\mathrm{s}$. During the whole
experiment, the reference frame varied for more than 29 periods and the noise
$\omega$ reaches an upper bound of $6.9\times 10^{-3}$ rad/s.

Without loss of generality, we use $\\{0,2\pi/m,\cdots,2(m-1)\pi/m\\}$ to
represent the misalignment angle of the slices. FIG. 3 shows the average QBER
of each slice. The QBERs of $X$, $Y$ basis vary against the rotation $\theta$
while the QBERs of signal and decoy state on $Z$ basis are stabilized at
around 0.6% and 2.3%. The corresponding $C$ value and secure key rate of each
slice are shown in FIG. 4 with red circles. The blue dash lines in FIG. 4
illustrate the simulated results with $N_{\textrm{t}}=10^{13}$ [10, 5, 25,
18]. In principle, the $C$ value and the secure key rate reach the maximum
value with $\theta\in\\{0,\pi/2,\pi,3\pi/2\\}$ and drop to the minimum value
with $\theta\in\\{\pi/4,3\pi/4,5\pi/4,7\pi/4\\}$. The experimental $C$ value
and secure key rate are slightly less than the simulated results due to the
imperfect preparation and measurement of photons. The secure key rate is
varying between $603\text{\,}\mathrm{b}\mathrm{p}\mathrm{s}$ and
$943\text{\,}\mathrm{b}\mathrm{p}\mathrm{s}$, in slice $[15\pi/16,17\pi/16)$
and $[3\pi/16,5\pi/16)$ respectively. Finally, the measured average secure key
rate in our experiment is $734\text{\,}\mathrm{b}\mathrm{p}\mathrm{s}$.

Furthermore, we simulate the average secure key rate of the free-running RFI
QKD versus the total loss with $N_{\textrm{t}}=10^{12},10^{13}$ and the noise
$\omega=6.9\times 10^{-3}$ rad/s. The results are shown in FIG. 5. The total
loss tolerance of the original RFI QKD protocol is limited to
$32.6\text{\,}\mathrm{d}\mathrm{B}$, which can be increased to around
$43.3\text{\,}\mathrm{d}\mathrm{B}$ with our free-running RFI scheme.
Meanwhile, the free-running RFI scheme is conducted after the basis shifting
procedure, and the number $m$ of slices can be optimized to further improve
the performance. Especially, our free-running RFI scheme can be performed into
the satellite-to-ground and drone-based QKD scenarios, which avoids the
complicated alignment of unpredictable varying reference frames.

This work was supported by National Natural Science Foundation of China under
Grant No. 61972410 and No. 61971436, the Research Plan of National University
of Defense Technology under Grant No. ZK19-13 and No. 19-QNCXJ-107 and the
Postgraduate Scientific Research Innovation Project of Hunan Province under
Grant No.CX20200003.

## References

  * Lo  _et al._ [2014] H.-K. Lo, M. Curty, and K. Tamaki, Secure quantum key distribution, Nature Photonics 8, 595 (2014).
  * Shannon [1949] C. E. Shannon, Communication theory of secrecy systems, Bell system technical journal 28, 656 (1949).
  * Shannon [1948] C. E. Shannon, A mathematical theory of communication, Bell system technical journal 27, 379 (1948).
  * Bennett and Brassard [1984] C. H. Bennett and G. Brassard, Quantum cryptography: Public key distribution and coin tossing, Proceedings of IEEE International Conference on Computers, Systems and Signal Processing 1, 175 (1984).
  * Lo  _et al._ [2005] H.-K. Lo, X. Ma, and K. Chen, Decoy state quantum key distribution, Physical Review Letters 94, 230504 (2005).
  * Zhang  _et al._ [2017a] Z. Zhang, Q. Zhao, M. Razavi, and X. Ma, Improved key-rate bounds for practical decoy-state quantum-key-distribution systems, Physical Review A 95, 012333 (2017a).
  * Kupko  _et al._ [2020] T. Kupko, M. von Helversen, L. Rickert, J.-H. Schulze, A. Strittmatter, M. Gschrey, S. Rodt, S. Reitzenstein, and T. Heindel, Tools for the performance optimization of single-photon quantum key distribution, npj Quantum Information 6, 29 (2020).
  * Wei  _et al._ [2020] K. Wei, W. Li, H. Tan, Y. Li, H. Min, W.-J. Zhang, H. Li, L. You, Z. Wang, X. Jiang, T.-Y. Chen, S.-K. Liao, C.-Z. Peng, F. Xu, and J.-W. Pan, High-speed measurement-device-independent quantum key distribution with integrated silicon photonics, Physical Review X 10, 031030 (2020).
  * Chen  _et al._ [2020a] H. Chen, J. Wang, B. Tang, Z. Li, B. Liu, and S. Sun, Field demonstration of time-bin reference-frame-independent quantum key distribution via an intracity free-space link, Optics Letters 45, 3022 (2020a).
  * Wang  _et al._ [2019] J. Wang, H. Liu, H. Ma, and S. Sun, Experimental study of four-state reference-frame-independent quantum key distribution with source flaws, Physical Review A 99, 032309 (2019).
  * Liu  _et al._ [2018] H. Liu, J. Wang, H. Ma, and S. Sun, Polarization-multiplexing-based measurement-device-independent quantum key distribution without phase reference calibration, Optica 5, 902 (2018).
  * Chen  _et al._ [2020b] J.-P. Chen, C. Zhang, Y. Liu, C. Jiang, W. Zhang, X.-L. Hu, J.-Y. Guan, Z.-W. Yu, H. Xu, J. Lin, M.-J. Li, H. Chen, H. Li, L. You, Z. Wang, X.-B. Wang, Q. Zhang, and J.-W. Pan, Sending-or-not-sending with independent lasers: Secure twin-field quantum key distribution over 509 km, Physical Review Letters 124, 070501 (2020b).
  * Chen  _et al._ [2021] Y.-A. Chen, Q. Zhang, T.-Y. Chen, W.-Q. Cai, S.-K. Liao, J. Zhang, K. Chen, J. Yin, J.-G. Ren, Z. Chen, S.-L. Han, Q. Yu, K. Liang, F. Zhou, X. Yuan, M.-S. Zhao, T.-Y. Wang, X. Jiang, L. Zhang, W.-Y. Liu, Y. Li, Q. Shen, Y. Cao, C.-Y. Lu, R. Shu, J.-Y. Wang, L. Li, N.-L. Liu, F. Xu, X.-B. Wang, C.-Z. Peng, and J.-W. Pan, An integrated space-to-ground quantum communication network over 4,600 kilometres, Nature 10.1038/s41586-020-03093-8 (2021).
  * Liu  _et al._ [2021] H.-Y. Liu, X.-H. Tian, C. Gu, P. Fan, X. Ni, R. Yang, J.-N. Zhang, M. Hu, J. Guo, X. Cao, X. Hu, G. Zhao, Y.-Q. Lu, Y.-X. Gong, Z. Xie, and S.-N. Zhu, Optical-relayed entanglement distribution using drones as mobile nodes, Physical Review Letters 126, 020503 (2021).
  * Jain  _et al._ [2011] N. Jain, C. Wittmann, L. Lydersen, C. Wiechers, D. Elser, C. Marquardt, V. Makarov, and G. Leuchs, Device calibration impacts security of quantum key distribution, Physical Review Letters 107, 110501 (2011).
  * Dong  _et al._ [2014] Z.-Y. Dong, N.-N. Yu, Z.-J. Wei, J.-D. Wang, and Z.-M. Zhang, An attack aimed at active phase compensation in one-way phase-encoded qkd systems, The European Physical Journal D 68, 230 (2014).
  * Laing  _et al._ [2010] A. Laing, V. Scarani, J. G. Rarity, and J. L. O’Brien, Reference-frame-independent quantum key distribution, Physical Review A 82, 012304 (2010).
  * Zhang  _et al._ [2018] C.-M. Zhang, J.-R. Zhu, and Q. Wang, Practical reference-frame-independent quantum key distribution systems against the worst relative rotation of reference frames, Journal of Physics Communications 2, 055029 (2018).
  * Wang  _et al._ [2016] F. Wang, P. Zhang, X. Wang, and F. Li, Valid conditions of the reference-frame-independent quantum key distribution, Phys.rev.a 94, 062330 (2016).
  * Zhang  _et al._ [2017b] C.-M. Zhang, J.-R. Zhu, and Q. Wang, Practical decoy-state reference-frame-independent measurement-device-independent quantum key distribution, Physical Review A 95, 032309 (2017b).
  * Li  _et al._ [2019] Y.-P. Li, W. Chen, F.-X. Wang, Z.-Q. Yin, L. Zhang, H. Liu, S. Wang, D.-Y. He, Z. Zhou, G.-C. Guo, and Z.-F. Han, Experimental realization of a reference-frame-independent decoy bb84 quantum key distribution based on sagnac interferometer, Optics Letters 44, 4523 (2019).
  * Liang  _et al._ [2014] W.-Y. Liang, S. Wang, H.-W. Li, Z.-Q. Yin, W. Chen, Y. Yao, J.-Z. Huang, G.-C. Guo, and Z.-F. Han, Proof-of-principle experiment of reference-frame-independent quantum key distribution with phase coding, Scientific Reports 4, 3617 (2014).
  * Sheridan  _et al._ [2010] L. Sheridan, T. P. Le, and V. Scarani, Finite-key security against coherent attacks in quantum key distribution, New Journal of Physics 12, 123019 (2010).
  * Tang  _et al._ [2021] B.-Y. Tang, B. Liu, W.-R. Yu, and C.-Q. Wu, Shannon-limit approached information reconciliation for quantum key distribution, Quantum Information Processing 20, 113 (2021).
  * Lim  _et al._ [2014] C. C. W. Lim, M. Curty, N. Walenta, F. Xu, and H. Zbinden, Concise security bounds for practical decoy-state quantum key distribution, Physical Review A 89, 022307 (2014).