Integrability, Quantization and Moduli Spaces of Curves

Integrability, Quantization
and Moduli Spaces of Curves††This paper is a contribution to the Special Issue
on Recent Advances in Quantum Integrable Systems. The full collection is
available at http://www.emis.de/journals/SIGMA/RAQIS2016.html

Paolo ROSSI

P. Rossi

IMB, UMR5584 CNRS, Université de Bourgogne Franche-Comté, F-21000 Dijon,
France<EMAIL_ADDRESS>http://rossi.perso.math.cnrs.fr/

Received February 28, 2017, in final form July 25, 2017; Published online July
29, 2017

This paper has the purpose of presenting in an organic way a new approach to
integrable $(1+1)$-dimensional field systems and their systematic quantization
emerging from intersection theory of the moduli space of stable algebraic
curves and, in particular, cohomological field theories, Hodge classes and
double ramification cycles. This methods are alternative to the traditional
Witten–Kontsevich framework and its generalizations by Dubrovin and Zhang and,
among other advantages, have the merit of encompassing quantum integrable
systems. Most of this material originates from an ongoing collaboration with
A. Buryak, B. Dubrovin and J. Guéré.

moduli space of stable curves; integrable systems; cohomological field
theories; double ramification cycle; double ramification hierarchy

14H10; 14H70; 37K10

##  1 Introduction

This paper deals with a novel construction that associates an integrable, tau-
symmetric hierarchy and its quantization to a cohomological field theory on
the moduli space of stable curves $\overline{\mathcal{M}}_{g,n}$, without the
semisemplicity assumption which is needed for the Dubrovin–Zhang hierarchy. It
is inspired by Eliashberg, Givental and Hofer’s symplectic field theory [17]
and is the fruit of a joint project of the author with A. Buryak and, more
recently, with J. Guéré and B. Dubrovin.

Since the construction makes explicit use of the intersection theory of the
double ramification cycle, we call this hierarchy the double ramification (DR)
hierarchy. It was in fact A. Buryak who introduced its classical version in
[1], where he also explicitly computed the first two examples (the classical
DR hierarchies of the trivial and Hodge CohFTs, corresponding to the KdV and
intermediate long wave hierarchies), thereby showing the interest and power of
this technique.

Its properties, quantization and relation with the DZ hierarchy were studied
and clarified in the series of joint papers [2, 3, 5, 8, 9], partly guided by
our previous investigations of the classical and quantum integrable systems
arising in SFT [19, 29, 30, 31, 32, 33].

The DR hierarchy has many interesting properties and even advantages over the
more classical Dubrovin–Zhang hierarchy, including a much more direct access
to the explicit form of the Hamiltonians and Poisson structure, a natural and
completely general technique to quantize the integrable systems thus produced,
recursion relations for the Hamiltonians that are reminiscent of genus $0$
TRRs in Gromov–Witten theory but work at all genera. When Dubrovin proposed to
me to work on a thesis on integrable systems arising in SFT, back in 2004, he
said he believed that was the actual correct approach to integrable
hierarchies from moduli spaces of curves. I believe that prediction has found
complete confirmation in the power of the DR hierarchy project.

Finally, one of the main parts of this project is the proof of the conjecture
(originally proposed in a weaker form by A. Buryak) that the DZ and DR
hierarchies for a semisimple CohFT are in fact equivalent under a change of
coordinates that preserves their tau-symmetry property (a normal Miura
transformation), and which we completely identified in [2]. While the general
proof of such conjecture is the object of an ongoing work, we managed to show
its validity in a number of examples and classes of interesting special cases.
Our present approach to the general statement reduces it to proving a finite
number of relations in the tautological ring of each
$\overline{\mathcal{M}}_{g,n}$ with $n\leq 2g$ [5].

After a self contained introduction to the language of integrable systems in
the formal loop space and the needed notions from the geometry of the moduli
space of stable curves we will explain the double ramification hierarchy
construction and present its main features, with an accent on the quantization
procedure, concluding with a list of examples worked out in detail. This paper
does not contain new results with respect to the series of papers [2, 3, 5, 8,
9]. It is however a complete reorganization and, in part, a rephrasing of
those results with the aim of showcasing the power of our methods and making
them more accessible to the mathematical physics community.

##  2 Integrable systems

In this section I will try to give, in a few pages, a precise idea of what an
integrable system is, in the context of evolutionary Hamiltonian PDEs. We will
introduce the minimal notions that will be used in what follows and assume a
certain familiarity with the finite-dimensional theory of Poisson manifolds,
to guide the reader in extending such notions to an infinite-dimensional
context.

###  2.1 Formal loop space

An evolutionary PDE is a system of differential equations of the form

$\displaystyle{\partial}_{t}u^{\alpha}=F^{\alpha}\big{(}u^{*},u^{*}_{1},u^{*}_{2},\dots\big{)},\qquad\alpha=1,\dots,N,$

where $u^{\alpha}_{k}={\partial}_{x}^{k}u^{\alpha}$ and, here and in what
follows, we use the symbol $*$ to indicate any value for the corresponding sub
or superscripts.

Such a system can be heuristically interpreted as a vector field on the
infinite-dimensional space of all loops $u\colon S^{1}\to V$, where $V$ is a
$N$-dimensional vector space with a basis $e_{1},\dots,e_{N}$ and $x$ is the
coordinate on $S^{1}$, so that $u^{\alpha}=u^{\alpha}(x)$ is the component
along $e_{\alpha}$ of such loop. This is just a heuristic interpretation as we
choose to work in a more formal algebraic setting by describing an appropriate
ring of functions for the loop space of $V$ as follows.

Consider the ring of differential polynomials
$\widehat{\mathcal{A}}=\mathbb{C}[[u^{*}]][u^{*}_{>0}][[\varepsilon]]$ and
endow it with the grading $\deg(u^{\alpha}_{k})=k$, $\deg(\varepsilon)=-1$. We
denote by $\widehat{\mathcal{A}}^{[d]}$ the degree $d$ part of
$\widehat{\mathcal{A}}$. The role of the parameter $\varepsilon$ and grading
will become clear shortly. The operator ${\partial}_{x}$ acts on
$\widehat{\mathcal{A}}$ in the obvious way, i.e.,
${\partial}_{x}=\sum\limits_{k\geq
0}u^{\alpha}_{k+1}\frac{{\partial}}{{\partial}u^{\alpha}_{k}}$ (we use the
convention of sum over repeated Greek indices, but not roman indices).

We define the space of local functionals as the quotient
$\widehat{\Lambda}=\widehat{\mathcal{A}}/(\operatorname{Im}{\partial}_{x}\oplus\mathbb{C}[[\varepsilon]])$
and denote by $\widehat{\Lambda}^{[d]}$ its degree $d$ part. The equivalence
class of $f(u^{*}_{*};\varepsilon)\in\widehat{\mathcal{A}}$ in this quotient
will be denoted suggestively as $\overline{f}=\int f(u^{*}_{*};\varepsilon)dx$
(hinting at the quotient with respect to $\operatorname{Im}{\partial}_{x}$ as
the possibility of integrating by parts on the circle $S^{1}$).

Local functionals in $\widehat{\Lambda}$ can hence be interpreted as functions
on our formal loop space of $V$ whose value on a given loop $u\colon S^{1}\to
V$ is the integral over $S^{1}$ of some differential polynomial in its
components $u^{\alpha}(x)$.

Changes of coordinates on the formal loop space will be described accordingly
as

$\displaystyle\widetilde{u}^{\alpha}=\widetilde{u}^{\alpha}(u^{*}_{*},\varepsilon)\in\widehat{\mathcal{A}}^{[0]},\qquad\det\left(\frac{{\partial}\widetilde{u}^{*}|_{\varepsilon=0}}{{\partial}u^{*}}\right)\neq
0.$

Notice here the importance of the parameter $\varepsilon$, whose exponent
counts the number of $x$-derivatives appearing in $\widetilde{u}^{\alpha}$.
Its importance lies in the fact that we can use the parameter $\varepsilon$ to
invert such change of coordinates: for fixed $\widetilde{u}^{\alpha}(x)$, we
just need to solve the ODE
$\widetilde{u}^{\alpha}=\widetilde{u}^{\alpha}(u^{*}_{*},\varepsilon)$ for the
functions $u^{\alpha}(x)$ order by order in $\varepsilon$ and we will obtain a
differential polynomial
$u^{\alpha}=u^{\alpha}(\widetilde{u}^{*}_{*};\varepsilon)$. The resulting
group is called the Miura group.

Differential polynomials and local functionals can also be described using
another set of formal variables, corresponding heuristically to the Fourier
components $p^{\alpha}_{k}$, $k\in\mathbb{Z}$, of the functions
$u^{\alpha}=u^{\alpha}(x)$. Let us, hence, define a change of variables

$\displaystyle
u^{\alpha}_{j}=\sum_{k\in\mathbb{Z}}(ik)^{j}p^{\alpha}_{k}e^{ikx},$

which is nothing but the $j$-th derivative of
$u^{\alpha}=\sum\limits_{k\in\mathbb{Z}}p^{\alpha}_{k}e^{ikx}$.

This allows us to express a differential polynomial
$f(u;u_{x},u_{xx},\dots;\varepsilon)\in\widehat{\mathcal{A}}^{[d]}$ as a
formal Fourier series $f=\sum
f_{\alpha_{1},\dots,\alpha_{n};s}^{k_{1},\dots,k_{n}}\varepsilon^{s}p^{\alpha_{1}}_{k_{1}}\dots
p^{\alpha_{n}}_{k_{n}}e^{i\big{(}\sum\limits_{j=1}^{n}k_{j}\big{)}x}$ where
the coefficient $f_{\alpha_{1},\dots,\alpha_{n};s}^{k_{1},\dots,k_{n}}$ is a
polynomial in the indices $k_{1},\dots,k_{n}$ of degree $s+d$. Moreover, the
local functional $\overline{f}$ corresponds to the constant term of the
Fourier series of $f$.

###  2.2 Poisson structures

In what follows we will be interested in Hamiltonian systems of evolutionary
PDEs. To this end we endow the space of local functionals with a Poisson
structure of the form

$\displaystyle\big{\\{}\overline{f},\overline{g}\big{\\}}_{K}:=\int\frac{\delta\overline{f}}{\delta
u^{\mu}}K^{\mu\nu}\frac{\delta\overline{g}}{\delta u^{\nu}}dx,\qquad
K^{\mu\nu}=\sum\limits_{j\geq 0}K^{\mu\nu}_{j}{\partial}_{x}^{j},\qquad
K^{\mu\nu}_{j}\in\widehat{\mathcal{A}}^{[-j+1]}.$

Given that the variational derivative $\frac{\delta}{\delta
u^{\alpha}}=\sum\limits_{k\geq
0}(-{\partial}_{x})^{k}\frac{{\partial}}{{\partial}u^{\alpha}_{k}}$ is the
natural extension to local functionals of the finite-dimensional notion of
partial derivative, the above formula seems quite natural. The differential
operator $K$ is called a Hamiltonian operator. Imposing antisymmetry and the
Jacobi identity for the Poisson brackets obviously imposes conditions on the
differential operator $K^{\mu\nu}$. For instance

$\displaystyle
K^{\alpha\beta}\big{|}_{\varepsilon=0}=g^{\alpha\beta}(u){\partial}_{x}+b^{\alpha\beta}_{\gamma}(u)u^{\gamma}_{x},$

and the matrix $(g^{\alpha\beta})$ is symmetric (and, for simplicity, we will
always assume it nondegenerate), the inverse matrix $(g_{\alpha\beta})$
defines a flat metric and the functions
$\Gamma_{\alpha\beta}^{\gamma}(u):=-g_{\alpha\mu}(u)b^{\mu\gamma}_{\beta}(u)$
are the coefficients of the Levi-Civita connection corresponding to this
metric (see [15]).

We also define the Poisson bracket between a differential polynomial
$f\in\widehat{\mathcal{A}}$ and a local functional
$\overline{g}\in\widehat{\Lambda}$ as follows

$\displaystyle\\{f,\overline{g}\\}_{K}=\sum_{s\geq
0}\frac{{\partial}f}{{\partial}u^{\mu}_{s}}{\partial}_{x}^{s}\left(K^{\mu\nu}\frac{\delta\overline{g}}{\delta
u^{\nu}}\right).$

Such formula, is compatible with the previous one in the sense that
$\int\\{f,\overline{g}\\}_{K}dx=\\{\overline{f},\overline{g}\\}_{K}$.

The action of a Miura transformation on the Poisson structure is given in
terms of Hamiltonian operators as follows

$\displaystyle K_{\widetilde{u}}^{\alpha\beta}=(L^{*})^{\alpha}_{\mu}\circ
K_{u}^{\mu\nu}\circ L^{\beta}_{\nu},$

where $(L^{*})^{\alpha}_{\mu}=\sum\limits_{s\geq
0}\frac{\partial\widetilde{u}^{\alpha}}{\partial
u^{\mu}_{s}}\partial_{x}^{s}$, $L^{\beta}_{\nu}=\sum\limits_{s\geq
0}(-\partial_{x})^{s}\circ\frac{\partial\widetilde{u}^{\beta}}{\partial
u^{\nu}_{s}}$.

The following Darboux-type theorem states that, up to change of coordinates,
there exists but one Poisson structure on the formal loop space.

######  Theorem 2.1 ([22]).

There exist a Miura transformation bringing any Poisson bracket to the
standard form

$\displaystyle K^{\mu\nu}=\eta^{\mu\nu}{\partial}_{x},\qquad\eta^{\mu\nu}\
\text{constant, symmetric and nondegenerate}.$

The standard Poisson bracket also has a nice expression in terms of the
variables $p^{\alpha}_{k}$:

$\displaystyle\big{\\{}p^{\alpha}_{k},p^{\beta}_{j}\big{\\}}_{\eta\partial_{x}}=ik\eta^{\alpha\beta}\delta_{k+j,0}.$

###  2.3 Integrable hierarchies

A Hamiltonian system is an evolutionary PDE of the form

$\displaystyle{\partial}_{t}u^{\alpha}=\big{\\{}u^{\alpha},\overline{h}\big{\\}}_{K}=K^{\alpha\nu}\frac{\delta\overline{h}}{\delta
u^{\nu}},\qquad\overline{h}\in\widehat{\Lambda}^{[0]},$

where $\overline{h}$ is called the Hamiltonian of the system.

An integrable system, or an integrable hierarchy, is an infinite system of
Hamiltonian evolutionary PDEs

$\displaystyle{\partial}_{t^{\beta}_{d}}u^{\alpha}=\big{\\{}u^{\alpha},\overline{h}_{\beta,d}\big{\\}}_{K}=K^{\alpha\mu}\frac{\delta\overline{h}_{\beta,d}}{\delta
u^{\mu}},\qquad\overline{h}_{\beta,d}\in\widehat{\Lambda}^{[0]},$ (2.1)

generated by Hamiltonians $\overline{h}_{\alpha,d}\in\widehat{\Lambda}^{[0]}$,
$\alpha=1,\dots,N$, $d\geq 0$ such that

$\displaystyle\big{\\{}\overline{h}_{\alpha,i},\overline{h}_{\beta,j}\big{\\}}_{K}=0.$

As in the finite-dimensional situation, the above Poisson-commutativity
condition for the Hamiltonians is equivalent to the compatibility of the
infinite system of PDEs they generate. A formal solution to the above
integrable hierarchy is given by a formal power series
$u^{\alpha}(x,t^{*}_{*};\varepsilon)\in\mathbb{C}[[x,t^{*}_{*},\varepsilon]]$
satisfying all the equations of the hierarchy simultaneously.

###  2.4 Tau-functions

Consider the Hamiltonian system (2.1). Let us assume that the Hamiltonian
$\overline{h}_{1,0}$ generates the spatial translations:

$\displaystyle{\partial}_{t^{1}_{0}}u^{\alpha}=K^{\alpha\mu}\frac{\delta\overline{h}_{1,0}}{\delta
u^{\mu}}=u^{\alpha}_{x}.$

A tau-structure for the hierarchy (2.1) is a collection of differential
polynomials $h_{\beta,q}\in\widehat{\mathcal{A}}^{[0]}_{N}$, $1\leq\beta\leq
N$, $q\geq-1$, such that the following conditions hold:

  1. 1)

$K^{\alpha\mu}\frac{\delta\overline{h}_{\beta,-1}}{\delta u^{\mu}}=0$,
$\beta=1,\dots,N$,

  2. 2)

the $N$ functionals $\overline{h}_{\beta,-1}$ are linearly independent,

  3. 3)

$\overline{h}_{\beta,q}=\int h_{\beta,q}dx$, $q\geq 0$,

  4. 4)

tau-symmetry:
$\frac{{\partial}h_{\alpha,p-1}}{{\partial}t^{\beta}_{q}}=\frac{{\partial}h_{\beta,q-1}}{{\partial}t^{\alpha}_{p}}$,
$1\leq\alpha,\beta\leq N$, $p,q\geq 0$.

Existence of a tau-structure imposes non-trivial constraints on a Hamiltonian
hierarchy. A Hamiltonian hierarchy with a fixed tau-structure will be called
tau-symmetric.

The fact that $\\{\overline{h}_{\alpha,p-1},\overline{h}_{\beta,q}\\}=0$
implies $\int\frac{{\partial}h_{\alpha,p-1}}{{\partial}t^{\beta}_{q}}dx=0$.
Since
$\frac{{\partial}h_{\alpha,p-1}}{{\partial}t^{\beta}_{q}}\in\widehat{\mathcal{A}}^{[1]}$
has no constant term, there exists a unique differential polynomial
$\Omega_{\alpha,p;\beta,q}\in\widehat{\mathcal{A}}^{[0]}$ such that
${\partial}_{x}\Omega_{\alpha,p;\beta,q}=\frac{{\partial}h_{\alpha,p-1}}{{\partial}t^{\beta}_{q}}$
and $\left.\Omega_{\alpha,p;\beta,q}\right|_{u^{*}_{*}=0}=0$ (and hence, in
particular, $h_{\alpha,p-1}=\Omega_{\alpha,p;1,0}$).

Consider an arbitrary solution
$u^{\alpha}=u^{\alpha}(x,t^{*}_{*};\varepsilon)\in\mathbb{C}[[x,t^{*}_{*},\varepsilon]]$
of our hierarchy (2.1). Tau-symmetry guarantees that there exists a function
$F\in\mathbb{C}[[t^{*}_{*},\varepsilon]]$ such that

$\displaystyle\big{(}\Omega_{\alpha,p;\beta,q}(u(x,t;\varepsilon);u_{x}(x,t;\varepsilon),\dots)\big{)}\big{|}_{x=0}=\frac{{\partial}^{2}F}{{\partial}t^{\alpha}_{p}{\partial}t^{\beta}_{q}},$

for any $1\leq\alpha,\beta\leq N$ and $p,q\geq 0$. The function
$F(t^{*}_{*};\varepsilon)$ is called the tau-function of the given solution
(in fact, for historical reasons, the tau-function should correspond to the
exponential of $F$, but we will ignore this distinction here, calling $F$ tau-
function indistinctly). Tau-symmetric hierarchies hence have the property that
the evolution along a particular solution of any of their Hamiltonian
densities is subsumed under one single function $F(t^{*}_{*};\varepsilon)$.

Given a tau-structure, its system of normal coordinates is the system of
coordinates
$\widetilde{u}^{\alpha}=\eta^{\alpha\mu}h_{\mu,-1}(u^{*}_{*};\varepsilon)$.
The Hamiltonian operator takes the form
$K^{\alpha\beta}_{\widetilde{u}}=\eta^{\alpha\beta}{\partial}_{x}+O(\varepsilon)$,
$\eta$ being a constant symmetric nondegenerate matrix.

A class of Miura transformations preserving the tau structure is given by
_normal Miura transformations_. Let $u^{\alpha}$ already be normal coordinates
and $\mathcal{F}(u^{*}_{*};\varepsilon)\in\widehat{\mathcal{A}}^{[-2]}$. The
normal Miura transformation generated by $\mathcal{F}$ is given by

$\displaystyle\widetilde{u}^{\alpha}=u^{\alpha}+\eta^{\alpha\mu}{\partial}_{x}\big{\\{}\mathcal{F},\overline{h}_{\mu,0}\big{\\}}_{K}.$
(2.2)

Then the Hamiltonian densities
$\widetilde{h}_{\beta,q}=h_{\beta,q}+{\partial}_{x}\\{\mathcal{F},\overline{h}_{\beta,q+1}\\}_{K}$
form again a tau-structure and the coordinates $\widetilde{u}^{\alpha}$ are
normal for it. Moreover, for any solution of the system, its tau-function
changes in the following way under the normal Miura transformation:

$\displaystyle\widetilde{F}(t^{*}_{*};\varepsilon)=F(t^{*}_{*};\varepsilon)+\mathcal{F}(u^{*}_{*}(x,t^{*}_{*};\varepsilon);\varepsilon)\big{|}_{x=0}.$

###  2.5 Example: the KdV hierarchy

The Korteweg–de Vries equation is the most well known example of integrable
Hamiltonian PDEs. It is defined on the formal loop space of a one-dimensional
vector space $V=\mathbb{C}$, so we will suppress the Greek indices in all the
above notations. The metric on $V$ is simply $\eta=1$. The Poisson structure
is given by the Hamiltonian operator $K={\partial}_{x}$ (so it is in Getzler’s
standard form). The Hamiltonian is the following local functional in
$\widehat{\Lambda}^{[0]}$:

$\displaystyle\overline{h}_{\mathrm{KdV}}=\int\left(\frac{u^{3}}{6}+\frac{\varepsilon^{2}}{24}uu_{2}\right)dx.$

We can hence compute the Hamiltonian flow, i.e., the KdV equation

$\displaystyle u_{t}=uu_{1}+\frac{\varepsilon^{2}}{24}u_{3}.$

The KdV equation is one of the flows of an integrable hierarchy. There are
various ways to compute the other flows (or the other Hamiltonians) which
compose such hierarchy (see for instance [13]). Here I choose to construct
them by a recursive procedure that we discovered with A. Buryak in [9] and
which was not known before.

Let $g_{-1}=u\in\widehat{\mathcal{A}}^{[0]}$ and construct
$\overline{h}_{i}\in\widehat{\Lambda}^{[0]}$, $i\geq-1$ as
$\overline{h}_{i}=\int g_{i}dx$, where the differential polynomials
$g_{i}\in\widehat{\mathcal{A}}^{[0]}$ are produced by the recursive equation

$\displaystyle
g_{i+1}=(D-1)^{-1}{\partial}_{x}^{-1}\big{\\{}g_{i},\overline{h}_{\mathrm{KdV}}\big{\\}},\qquad
D:=\sum_{k\geq 0}(k+1)u_{k}\frac{{\partial}}{{\partial}u_{k}}.$

At each step, this equation produces a new Hamiltonian density whose Poisson
bracket with $\overline{h}_{\mathrm{KdV}}=\overline{h}_{1}$ is
${\partial}_{x}$-exact so that it makes sense to take the inverse
$x$-derivative. The operator $D-1$ is also easily inverted on each monomial of
the resulting differential polynomial ($D$ on $\widehat{\mathcal{A}}^{[0]}$
just counts the number of variables $u^{*}_{*}$ and $\varepsilon$). The reader
can promptly check that we obtain

$\displaystyle g_{-1}=u,$ $\displaystyle
g_{0}=\frac{u^{2}}{2}+\frac{\varepsilon^{2}}{24}u_{2},$ $\displaystyle
g_{1}=\frac{u^{3}}{6}+\frac{\varepsilon^{2}}{24}uu_{2}+\frac{\varepsilon^{4}}{1152}u_{4},$
$\displaystyle
g_{2}=\frac{u^{4}}{24}+\varepsilon^{2}\frac{u^{2}u_{2}}{48}+\varepsilon^{4}\left(\frac{7u_{2}^{2}}{5760}+\frac{uu_{4}}{1152}\right)+\varepsilon^{6}\frac{u_{6}}{82944}.$

The differential polynomials $g_{i}$ have the property that
$\frac{{\partial}g_{i}}{{\partial}u}=g_{i-1}$.

A tau structure is obtained simply by taking
$h_{i}=\frac{\delta\overline{h}_{i+1}}{\delta u}$. Indeed we have
$\overline{h}_{i}=\overline{g}_{i}$ and tau-symmetry holds. The coordinate $u$
is already a normal coordinate for this tau-structure.

###  2.6 Quantum Hamiltonian systems

We will need, first, to extend the space of differential polynomials to allow
for dependence on the quantization formal parameter $\hbar$.

The space of quantum differential polynomials is
$\widehat{\mathcal{A}}^{\hbar}:=\widehat{\mathcal{A}}[[\hbar]]$, where the new
formal variable $\hbar$ has degree $\deg(\hbar)=-2$.

The space of quantum local functionals is given, similarly to the classical
case, by
$\widehat{\Lambda}^{\hbar}:=\widehat{\mathcal{A}}^{\hbar}/(\operatorname{Im}{\partial}_{x}\oplus\mathbb{C}[[\varepsilon,\hbar]])$.

The change of variables

$\displaystyle
u^{\alpha}_{j}=\sum_{k\in\mathbb{Z}}(ik)^{j}p^{\alpha}_{k}e^{ikx},$

allows to express any quantum differential polynomial
$f=f(u^{*}_{*};\varepsilon,\hbar)\in\widehat{\mathcal{A}}^{\hbar}$ as a formal
Fourier series in $x$ with coefficients that are (power series in
$\varepsilon$ with coefficients) in $\mathbb{C}[p^{*}_{>0}][[p^{*}_{\leq
0}]][[\hbar]]$.

We can make $\mathbb{C}[p^{*}_{>0}][[p^{*}_{\leq 0}]][[\hbar]]$ a Weyl algebra
by endowing it with the “normal” $\star$-product

$\displaystyle f\star g=f\bigg{(}e^{\sum\limits_{k>0}i\hbar
k\eta^{\alpha\beta}\overleftarrow{\frac{\partial}{\partial
p^{\alpha}_{k}}}\overrightarrow{\frac{\partial}{\partial
p^{\beta}_{-k}}}}\bigg{)}g,$

and the commutator $[f,g]:=f\star g-g\star f$.

######  Remark 2.2.

We remark here that our notation differ from what might constitute the
standard in the (mathematical) physical literature: given two elements $f$ and
$g$ in the Weyl algebra $\mathbb{C}[p^{*}_{>0}][[p^{*}_{\leq 0}]][[\hbar]]$ we
have two different symbols for the commutative product $f\cdot g$ (or simply
$fg$) and the quantum non-commutative star-product $f\star g$, so we don’t
need any “normal ordering” symbol. Expressions of the normal ordered type
${:}fg{:}$ simply correspond to $fg$ in our notations, as customary in
deformation quantization.

These structures can then be translated to the language of differential
polynomials and local functionals. In [8] it was proved that, for any two
differential polynomials
$f(x)=f(u^{*},u^{*}_{x},u^{*}_{xx},\dots;\varepsilon,\hbar)$ and
$g(y)=g(u^{*},u^{*}_{y},u^{*}_{yy},\dots;\varepsilon,\hbar)$, we have

$\displaystyle f(x)\star g(y)=\sum_{\begin{subarray}{c}n\geq 0\\\
r_{1},\dots,r_{n}\geq 0\\\ s_{1},\dots,s_{n}\geq
0\end{subarray}}\frac{\hbar^{n}}{n!}\frac{\partial^{n}f}{\partial
u^{\alpha_{1}}_{s_{1}}\cdots\partial
u^{\alpha_{n}}_{s_{n}}}(x)\left(\prod_{k=1}^{n}(-1)^{r_{k}}\eta^{\alpha_{k}\beta_{k}}\delta_{+}^{(r_{k}+s_{k}+1)}(x-y)\right)$
$\displaystyle\hphantom{f(x)\star g(y)=}{}\times\frac{\partial^{n}g}{\partial
u^{\beta_{1}}_{r_{1}}\cdots\partial u^{\beta_{n}}_{r_{n}}}(y),$

where $\delta_{+}^{(s)}(x-y):=\sum\limits_{k\geq 0}(ik)^{s}e^{ik(x-y)}$,
$s\geq 0$, is the positive frequency part of the $s$-th derivative of the
Dirac delta distribution $\delta(x-y)=\sum\limits_{k\in\mathbb{Z}}e^{ik(x-y)}$
and

$\displaystyle[f(x),g(y)]=\sum_{\begin{subarray}{c}n\geq 1\\\
r_{1},\dots,r_{n}\geq 0\\\ s_{1},\dots,s_{n}\geq
0\end{subarray}}\frac{(-i)^{n-1}\hbar^{n}}{n!}\frac{\partial^{n}f}{\partial
u^{\alpha_{1}}_{s_{1}}\cdots\partial
u^{\alpha_{n}}_{s_{n}}}(x)(-1)^{\sum\limits_{k=1}^{n}r_{k}}\left(\prod_{k=1}^{n}\eta^{\alpha_{k}\beta_{k}}\right)$
$\displaystyle\hphantom{[f(x),g(y)]=}{}\times\sum_{j=1}^{2n-1+\sum\limits_{k=1}^{n}(s_{k}+r_{k})}C_{j}^{s_{1}+r_{1}+1,\dots,s_{n}+r_{n}+1}\delta^{(j)}(x-y)\frac{\partial^{n}g}{\partial
u^{\beta_{1}}_{r_{1}}\cdots\partial u^{\beta_{n}}_{r_{n}}}(y),$ (2.3)

where

$\displaystyle C_{j}^{a_{1},\dots,a_{n}}=\begin{cases}(-1)^{\frac{n-1+\sum
a_{i}-j}{2}}\widetilde{C}_{j}^{a_{1},\dots,a_{n}},&\text{if
$j=n-1+\sum\limits_{i=1}^{n}a_{i}\ (\mathrm{mod}\ 2)$},\\\
0,&\text{otherwise},\end{cases}$

and

$\displaystyle\prod_{i=1}^{k}\mathrm{Li}_{-d_{i}}(z)=\sum_{j=1}^{k-1+\sum
d_{i}}\widetilde{C}^{d_{1},\dots,d_{k}}_{j}\mathrm{Li}_{-j}(z),\qquad\mathrm{Li}_{-d}(z):=\sum_{k\geq
0}k^{d}z^{k}.$

In particular, for $f\in\widehat{\mathcal{A}}^{\hbar}$ and
$\overline{g}\in\widehat{\Lambda}^{\hbar}$, we get

$\displaystyle[f,\overline{g}]=\sum_{\begin{subarray}{c}n\geq 1\\\
r_{1},\dots,r_{n}\geq 0\\\ s_{1},\dots,s_{n}\geq
0\end{subarray}}\frac{(-i)^{n-1}\hbar^{n}}{n!}\frac{\partial^{n}f}{\partial
u^{\alpha_{1}}_{s_{1}}\cdots\partial
u^{\alpha_{n}}_{s_{n}}}(-1)^{\sum\limits_{k=1}^{n}r_{k}}\left(\prod_{k=1}^{n}\eta^{\alpha_{k}\beta_{k}}\right)$
$\displaystyle\hphantom{[f,\overline{g}]=}{}\times\sum_{j=1}^{2n-1+\sum\limits_{k=1}^{n}(s_{k}+r_{k})}C_{j}^{s_{1}+r_{1}+1,\dots,s_{n}+r_{n}+1}\partial_{x}^{j}\frac{\partial^{n}g}{\partial
u^{\beta_{1}}_{r_{1}}\cdots\partial u^{\beta_{n}}_{r_{n}}}.$

If $f$ and $\overline{g}$ are homogeneous, $[f,\overline{g}]$ is a non
homogeneous element of $\widehat{\mathcal{A}}^{\hbar}$ of top degree equal to
$\deg f+\deg\overline{g}-1$. Taking the classical limit of this expression one
obtains
$\big{(}\frac{1}{\hbar}[\overline{f},\overline{g}]\big{)}|_{\hbar=0}=\\{\overline{f}|_{\hbar=0},\overline{g}|_{\hbar=0}\\}$,
i.e., the standard hydrodynamic Poisson bracket on the classical limit of the
local functionals.

Notice that, given $\overline{g}\in\widehat{\Lambda}^{\hbar}$, the morphism
$[\cdot,\overline{g}]\colon\widehat{\mathcal{A}}^{\hbar}\to\widehat{\mathcal{A}}^{\hbar}$
is not a derivation of the commutative ring $\widehat{\mathcal{A}}^{\hbar}$
(while it is if we consider the non-commutative $\star$-product instead). This
means that, while it makes sense to describe the simultaneous evolution along
different time parameters $\mathrm{t}^{\alpha}_{i}$ (in the Heisenberg
picture, to use the physical language) of a quantum differential polynomial
$f\in\widehat{\mathcal{A}}^{\hbar}$ by a system of the form

$\displaystyle\frac{\partial f}{\partial
t^{\alpha}_{i}}=\frac{1}{\hbar}\big{[}f,\overline{h}_{\alpha,i}\big{]},\qquad\alpha=1,\dots,N,\qquad
i=0,1,2,\dots,$ (2.4)

where $\overline{h}_{\alpha,i}\in\widehat{\Lambda}^{\hbar}$ are quantum local
functionals with the compatibility condition
$[\overline{h}_{\alpha,i},\overline{h}_{\beta,j}]=0$, for
$\alpha,\beta=1,\dots,N$, $i,j\geq 0$, one should refrain from interpreting it
as the evolution induced by composition with
$\frac{{\partial}u^{\gamma}}{{\partial}t^{\alpha}_{i}}=\frac{1}{\hbar}[u^{\gamma},\overline{h}_{\alpha,i}]$,
as the corresponding chain rule does not hold: $\frac{{\partial}f}{\partial
t^{\alpha}_{i}}\neq\sum\limits_{k\geq
0}\frac{{\partial}f}{{\partial}u^{\gamma}_{k}}{\partial}_{x}^{k}\big{(}\frac{{\partial}u^{\gamma}}{{\partial}t^{\alpha}_{i}}\big{)}$.
This corresponds to the familiar concept that in quantum mechanics there are
no trajectories in the phase space along which observables evolve.

A formal solution to the system (2.4) can be written in the form of an element
in $\widehat{\mathcal{A}}^{\hbar}[[t^{*}_{*}]]$:

$\displaystyle
f^{t^{*}_{*}}(u^{*}_{*};\varepsilon,\hbar):=\exp\left(\sum_{\begin{subarray}{c}1\leq\alpha\leq
N\\\ i\geq
1\end{subarray}}\frac{t^{\alpha}_{i}}{\hbar}\big{[}\cdot,\overline{h}_{\alpha,i}\big{]}\right)f(u^{*}_{*};\varepsilon,\hbar)$
$\displaystyle\hphantom{f^{t^{*}_{*}}(u^{*}_{*};\varepsilon,\hbar)}{}=\left(\prod_{\begin{subarray}{c}1\leq\alpha\leq
N\\\ i\geq
1\end{subarray}}\exp\left(\frac{t^{\alpha}_{i}}{\hbar}\big{[}\cdot,\overline{h}_{\alpha,i}\big{]}\right)\right)f(u^{*}_{*};\varepsilon,\hbar),$
(2.5)

where

$\displaystyle\exp\left(\frac{t^{\alpha}_{i}}{\hbar}\big{[}\cdot,\overline{h}_{\alpha,i}\big{]}\right):=\sum_{k\geq
0}\frac{(t^{\alpha}_{i})^{k}}{\hbar^{k}k!}\big{[}\big{[}\dots\big{[}\cdot,\overline{h}_{\alpha,i}\big{]},\dots,\overline{h}_{\alpha,i}\big{]},\overline{h}_{\alpha,i}\big{]},$

and $f\in\widehat{\mathcal{A}}^{\hbar}$ in the right-hand side of (2.5) is
interpreted as the initial datum. Lifting the quantum commutator
$[\cdot,\cdot]$ to $\widehat{\mathcal{A}}^{\hbar}[[t^{*}_{*}]]$, it is easy to
check that $f^{t^{*}_{*}}$ satisfies equation (2.4). We do insist that
$f^{t^{*}_{*}}(u^{*}_{*};\varepsilon,\hbar)\neq
f((u^{*}_{*})^{t^{*}_{*}},\varepsilon,\hbar)$.

###  2.7 Example: quantization of the KdV hierarchy

We present here a quantization of the KdV hierarchy described in Section 2.5.
The technique by which we will construct it is very general and works for
basically all integrable systems we know, see Section 5. We discovered it with
A. Buryak in [8].

First we consider the classical KdV Poisson bracket and we replace it with the
quantum commutator (2.3). Then we take the classical KdV Hamiltonian

$\displaystyle{\overline{H}}_{\mathrm{KdV}}:=\overline{h}_{\mathrm{KdV}}=\int\left(\frac{u^{3}}{6}+\frac{\varepsilon^{2}}{24}uu_{2}\right)dx$

and we consider it as an element of $\widehat{\Lambda}^{\hbar}$. In other
words the quantum local functional ${\overline{H}}_{\mathrm{KdV}}$ does not
explicitly depend on the parameter $\hbar$. However this is not true for the
other commuting quantum Hamiltonians. In order to write them all we use the
technique from [8].

Let $G_{-1}=u\in\widehat{\mathcal{A}}^{\hbar}$ and construct
${\overline{H}}_{i}\in\widehat{\Lambda}^{\hbar}$, $i\geq-1$ as
${\overline{H}}_{i}=\int G_{i}dx$, where the differential polynomials
$G_{i}\in\widehat{\mathcal{A}}^{\hbar}$ are produced by the recursive equation

$\displaystyle
G_{i+1}=(D-1)^{-1}{\partial}_{x}^{-1}\left(\frac{1}{\hbar}\big{[}G_{i},{\overline{H}}_{\mathrm{KdV}}\big{]}\right),\qquad
D:=\varepsilon\frac{{\partial}}{{\partial}\varepsilon}+2\hbar\frac{{\partial}}{{\partial}\hbar}+\sum_{k\geq
0}u_{k}\frac{{\partial}}{{\partial}u_{k}}.$

At each step, this equation produces a new Hamiltonian density whose Poisson
bracket with ${\overline{H}}_{\mathrm{KdV}}={\overline{H}}_{1}$ is
${\partial}_{x}$-exact so that it makes sense to take the inverse
$x$-derivative. The operator $D-1$ is also easily inverted on each monomial of
the resulting quantum differential polynomial ($D$ on
$\widehat{\mathcal{A}}^{\hbar}$ just counts the number of variables
$u^{*}_{*}$, $\varepsilon$ and, with weight $2$, $\hbar$). The reader can
promptly check that we obtain

$\displaystyle
G_{0}=\frac{u^{2}}{2}+\frac{\varepsilon^{2}}{24}u_{xx}-\frac{i\hbar}{24},$
$\displaystyle
G_{1}=\frac{u^{3}}{6}+\frac{\varepsilon^{2}}{24}uu_{xx}+\frac{\varepsilon^{4}}{1152}u_{xxxx}-i\hbar\frac{u+u_{xx}}{24}-\frac{i\hbar\varepsilon^{2}}{2880},$
$\displaystyle
G_{2}=\frac{u^{4}}{24}+\varepsilon^{2}\frac{u^{2}u_{2}}{48}+\varepsilon^{4}\left(\frac{7u_{2}^{2}}{5760}+\frac{uu_{4}}{1152}\right)+\varepsilon^{6}\frac{u_{6}}{82944}-i\hbar\frac{2uu_{2}+u^{2}}{48}-i\hbar\varepsilon^{2}\frac{u+5u_{2}+4u_{4}}{2880}$
$\displaystyle\hphantom{G_{2}=}{}-\frac{i\hbar\varepsilon^{4}}{120960}+(i\hbar)^{2}\frac{7}{5760}.$

The differential polynomials $G_{i}$ have the property that
$\frac{{\partial}G_{i}}{{\partial}u}=G_{i-1}$.

##  3 Cohomological field theories
and the double ramification cycle

In this section we introduce the notion of cohomological field theory, a
family of cohomology classes on the moduli spaces of stable curves which is
compatible with the natural maps and boundary structure [26], and the double
ramification cycle, another cohomology class representing (a compactification
of) the locus of curves whose marked points support a principal divisor. We
will assume a certain familiarity with the geometry of the moduli space
itself, referring to [37] for an excellent introductory exposition.

###  3.1 Moduli space of stable curves

Here we just recall the main definitions and fix the notations. In what
follows, by curve we mean a compact connected Riemann surface, smooth but for
a finite number of nodes with local model $xy=0$, with $n$ distinct marked
points labeled by $\\{1,\dots,n\\}$ and with genus $g$. A stable curve is a
curve for which the number of biholomorphic automorphisms, keeping the marked
points fixed and sending nodes to nodes, is finite.

This finiteness of the symmetry group can be translated into simple numerical
conditions: consider each irreducible component of the nodal curve as a marked
nodal curve itself. Suppose it carries $\nu$ of the original labeled markings
plus the $\mu$ nodes connecting it with the other irreducible components and
$\lambda$ further nodes that are double points. The numerical condition is
then $2\gamma-2+\nu+\mu+2\lambda>0$.

Given two integers $g,n\geq 0$ such that $2g-2+n>0$, the moduli space of
stable curves will be denoted by $\overline{\mathcal{M}}_{g,n}$. It is a
$(3g-3+n)$-dimensional compact complex orbifold (or smooth Deligne–Mumford
stack) parametrizing all possible stable curves with genus $g$ and $n$ marked
points. Each point in $\overline{\mathcal{M}}_{g,n}$ represents an equivalence
class of stable curves. Two stable curves with same $g$ and $n$ belong to the
same class if between them there exists a biholomorphisms sending nodes to
nodes and the $i$-th marked points to the $i$-th marked point, for all
$i=1,\dots,n$.

On $\overline{\mathcal{M}}_{g,n}$ there is a universal curve
$\overline{\mathcal{C}}_{g,n}\to\overline{\mathcal{M}}_{g,n}$, a morphism of
orbifolds whose fiber over a point $x\in\overline{\mathcal{M}}_{g,n}$ is
isomorphic to the curve $C_{x}$ represented by that point. Each fiber $C_{x}$
hence has $n$ marked numbered points which, varying
$x\in\overline{\mathcal{M}}_{g,n}$, form $n$ sections
$s_{i}\colon\overline{\mathcal{M}}_{g,n}\to\overline{\mathcal{C}}_{g,n}$,
$i=1,\dots,n$.

There are three natural morphisms among different moduli spaces. The forgetful
morphism
$\pi=\pi_{m}\colon\overline{\mathcal{M}}_{g,n+m}\to\overline{\mathcal{M}}_{g,n}$
forgets the last $m$ marked point on a curve (contracting all components of
the curve that might have thus become unstable). Notice that
$\pi\colon\overline{\mathcal{M}}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}$
coincides with the universal curve
$\overline{\mathcal{C}}_{g,n}\to\overline{\mathcal{M}}_{g,n}$.

The gluing morphism
$\sigma=\sigma_{(g_{1},n_{1};g_{2},n_{2})}\colon\overline{\mathcal{M}}_{g_{1},n_{1}+1}\times\overline{\mathcal{M}}_{g_{2},n_{2}+1}\to\overline{\mathcal{M}}_{g,n}$,
for $g_{1}+g_{2}=g$, $n_{1}+n_{2}=n$, glues two stable curves by identifying
the last marked point of the first one with the last marked point of the
second one, which become a node.

The loop morphism
$\tau=\tau_{g,n+2}\colon\overline{\mathcal{M}}_{g,n+2}\to\overline{\mathcal{M}}_{g+1,n}$
identifies the two last marked points on the same stable curve, hence forming
a non-separating node which increases the genus by $1$.

The union of the images of the maps $\sigma$ and $\tau$ (for all possible
stable choices of $(g_{1},n_{1};g_{2},n_{2})$ such that $g_{1}+g_{2}=g$,
$n_{1}+n_{2}=n$) is a divisor in $\overline{\mathcal{M}}_{g,n}$ with normal
crossings, called the boundary divisor. Each normal crossing of $k$ branches
of the boundary divisor is the moduli space of stable curves with at least $k$
distinct nodes and a given distribution of marked points among their
irreducible components.

On the total space of the universal curve there is a line bundle
$\omega_{g,n}\to\overline{\mathcal{C}}_{g,n}$. On the smooth points of the
fibers $C_{x}$ of $\overline{\mathcal{C}}_{g,n}$ it is defined as the relative
cotangent (canonical) bundle with respect to the projection
$\overline{\mathcal{C}}_{g,n}\to\overline{\mathcal{M}}_{g,n}$ and it extends
canonically to the singular points to give an actual line bundle on the full
$\overline{\mathcal{C}}_{g,n}$.

The tautological bundles $L_{i}\to\overline{\mathcal{M}}_{g,n}$, $i=1,\dots,n$
are defined as $L_{i}=s_{i}^{*}\omega_{g,n}$. The fiber of $L_{i}$ at the
point $x\in\overline{\mathcal{M}}_{g,n}$ is the cotangent line at the $i$-th
marked point of the curve $C_{x}$ represented by $x$. The first Chern class of
$L_{i}$ will be denoted by $\psi_{i}=c_{1}(L_{i})\in
H^{2}(\overline{\mathcal{M}}_{g,n},{\mathbb{Q}})$.

The Hodge bundle $\mathbb{H}\to\overline{\mathcal{M}}_{g,n}$ is the rank $g$
vector bundle over $\overline{\mathcal{M}}_{g,n}$ whose fiber over
$x\in\overline{\mathcal{M}}_{g,n}$ consists of the vector space of abelian
differentials on the curve $C_{x}$ represented by $x$. Its $g$ Chern classes
will be denoted by $\lambda_{i}=c_{i}(\mathbb{H})\in
H^{2i}(\overline{\mathcal{M}}_{g,n},{\mathbb{Q}})$, $i=1,\dots,g$, and
$\Lambda(s):=\sum\limits_{i=0}^{g}s^{i}\lambda_{i}$.

###  3.2 Cohomological field theories

Cohomological field theories (CohFTs) were introduced by Kontsevich and Manin
in [26] to axiomatize the properties of Gromov–Witten classes of a given
target variety. As it turns out this notion is actually more general, in the
sense that not all CohFTs come from Gromov–Witten theory. The main idea is to
define a family of cohomology classes on all moduli spaces
$\overline{\mathcal{M}}_{g,n}$, for all stable choices of $g$ and $n$,
parametrized by an $n$-fold tensor product of a vector space, in such a way
that they are compatible with the natural maps between moduli spaces we
considered above. Let us review their precise definition.

Let $g,n\geq 0$ such that $2g-2+n>0$. Let $V$ a $\mathbb{C}$-vector space with
basis $e_{1},\dots,e_{N}$ and endowed with a symmetric nondegenerate bilinear
form $\eta$. A cohomological field theory (CohFT) is a system of linear maps
$c_{g,n}\colon V^{\otimes n}\to
H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{C})$ such that

  * (i)

$c_{g,n}$ is $S_{n}$ equivariant (with respect to permutations of copies of
$V$ in $V^{\otimes n}$ and marked points on the curves),

  * (ii)

$c_{0,3}(e_{1}\otimes e_{\alpha}\otimes e_{\beta})=\eta_{\alpha\beta}$,

  * (iii)

$\pi^{*}c_{g,n}(e_{\alpha_{1}}\otimes\dots\otimes
e_{\alpha_{n}})=c_{g,n}(e_{\alpha_{1}}\otimes\dots\otimes
e_{\alpha_{n}}\otimes e_{1})$, where
$\pi\colon\overline{\mathcal{M}}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}$,

  * (iv)

$\sigma^{*}c_{g_{1}+g_{2},n_{1}+n_{2}}(e_{\alpha_{1}}\otimes\dots\otimes
e_{\alpha_{n_{1}+n_{2}}})=c_{g_{1},n_{1}+1}(e_{\alpha_{1}}\otimes\dots\otimes
e_{\alpha_{n_{1}}}\otimes
e_{\mu})\eta^{\mu\nu}c_{g_{2},n_{2}+1}(e_{\nu}\otimes
e_{\alpha_{n_{1}+1}}\otimes\dots\otimes e_{\alpha_{n_{1}+n_{2}}})$, where
$\sigma\colon\overline{\mathcal{M}}_{g_{1},n_{1}+1}\times\overline{\mathcal{M}}_{g_{2},n_{2}+1}\to\overline{\mathcal{M}}_{g_{1}+g_{2},n_{1}+n_{2}}$,

  * (v)

$\tau^{*}c_{g+1,n}(e_{\alpha_{1}}\otimes\dots\otimes
e_{\alpha_{n}})=c_{g,n+2}(e_{\alpha_{1}}\otimes\dots\otimes
e_{\alpha_{n}}\otimes e_{\mu}\otimes e_{\nu})\eta^{\mu\nu}$, where
$\tau\colon\overline{\mathcal{M}}_{g,n+2}\to\overline{\mathcal{M}}_{g+1,n}$.

In case the last axiom (the loop axiom) is not satisfied, we speak of
_partial_ CohFT instead.

The potential of the CohFT is defined as the generating function of the
intersection numbers of the CohFT with psi-classes, namely,

$\displaystyle F(t^{*}_{*};\varepsilon):=\sum_{g\geq
0}\varepsilon^{2g}F_{g}(t^{*}_{*}),$

where

$\displaystyle F_{g}(t^{*}_{*}):=\sum_{\begin{subarray}{c}n\geq 0\\\
2g-2+n>0\end{subarray}}\frac{1}{n!}\sum_{d_{1},\dots,d_{n}\geq
0}\left<\prod_{i=1}^{n}\tau_{d_{i}}(e_{\alpha_{i}})\right>_{g}\prod_{i=1}^{n}t^{\alpha_{i}}_{d_{i}},$
$\displaystyle\left<\tau_{d_{1}}(e_{\alpha_{1}})\cdots\tau_{d_{n}}(e_{\alpha_{n}})\right>_{g}:=\int_{\overline{\mathcal{M}}_{g,n}}c_{g,n}\big{(}{\otimes}_{i=1}^{n}e_{\alpha_{i}}\big{)}\prod_{i=1}^{n}\psi_{i}^{d_{i}}.$

Some examples of CohFTs are:

  * •

Trivial CohFT: $V=\mathbb{C}$, $\eta=1$, $c_{g,n}=1$.

  * •

Hodge CohFT: $V=\mathbb{C}$, $\eta=1$,
$c_{g,n}=\Lambda(s)=\sum\limits_{j=1}^{g}s^{j}\lambda_{j}$.

  * •

GW theory of a smooth projective variety $X$: $V=H^{*}(X,\mathbb{C})$,
$\eta=\text{Poincar\'{e} pairing}$,
$c_{g,n}(\otimes_{i=1}^{n}e_{\alpha_{i}})=p_{*}{\rm
ev}^{*}(\otimes_{i=1}^{n}e_{\alpha_{i}})q^{\beta}$, where
$p\colon\overline{\mathcal{M}}_{g,n}(X,\beta)\to\overline{\mathcal{M}}_{g,n}$,
${\rm ev}\colon\overline{\mathcal{M}}_{g,n}(X,\beta)\to X^{n}$, where
$\overline{\mathcal{M}}_{g,n}(X,\beta)$ is the moduli space of stable maps $u$
from curves $C$ of genus $g$ with $n$ marked points to $X$ of degree
$u_{*}[C]=\beta\in H^{2}(X,\mathbb{Z})$. The projection $p$ forgets the map
$u$ and the evaluation map $ev$ evaluates the map $u$ on the $n$ marked
points.

Notice that, in order to perform the pushforward along $p$, a notion of
Poincaré duality must be used, which involves the virtual fundamental class of
$\overline{\mathcal{M}}_{g,n}(X,\beta)$.

  * •

Witten’s $r$-spin classes:

$\displaystyle V=\mathbb{C}^{r-1},\qquad r\geq
2,\qquad\eta_{\alpha\beta}=\delta_{\alpha+\beta,r},\qquad
c_{g,n}(e_{a_{1}+1},\dots,e_{a_{n}+1})\in
H^{*}(\overline{\mathcal{M}}_{g,n};{\mathbb{Q}})$

is a class of degree $\frac{(r-2)(g-1)+\sum\limits_{i=1}^{n}a_{i}}{r}$ if
$a_{i}\in\\{0,\dots,r-2\\}$ are such that this degree is a non-negative
integer, and vanishes otherwise. The class is constructed in [27] (see also
[11]) by pushing forward to $\overline{\mathcal{M}}_{g,n}$ Witten’s virtual
class on the moduli space of curves with $r$-spin structures. An $r$-spin
structure on a smooth curve $(C,x_{1},\dots,x_{n})$ is an $r$-th root $L$ of
the (twisted) canonical bundle $K(\sum a_{i}x_{i})$ of the curve, where
$a_{i}\in\\{0,\dots,r-1\\}$. Witten’s class is the virtual class of $r$-spin
structures with a holomorphic section (and vanishes when one of the $a_{i}$’s
equals $r-1$), but we will not go into the details of the construction here.
This is an example of CohFT that is not a Gromov–Witten theory.

  * •

Fan–Jarvis–Ruan–Witten (FJRW) theory: consider the data of $(W,G)$ where

    * –

$W=W(y_{1},\dots,y_{m})$ is a quasi-homogeneous polynomial with weights
$w_{1},\dotsc,w_{m}$ and degree $d$, which has an isolated singularity at the
origin,

    * –

$G$ is a group of diagonal matrices $\gamma=(\gamma_{1},\dotsc,\gamma_{m})$
leaving the polynomial $W$ invariant and containing the diagonal matrix
$\mathfrak{j}:=\big{(}e^{\frac{2\mathrm{i}\pi
w_{1}}{d}},\dotsc,e^{\frac{2\mathrm{i}\pi w_{m}}{d}}\big{)}$.

The vector space $V$ is given by

$\displaystyle V=\bigoplus_{\gamma\in G}\big{(}\mathcal{Q}_{W_{\gamma}}\otimes
d\underline{y}_{\gamma}\big{)}^{G},$

where $W_{\gamma}$ is the $\gamma$-invariant part of the polynomial $W$,
$\mathcal{Q}_{W_{\gamma}}$ is its Jacobian ring, the differential form
$d\underline{y}_{\gamma}$ is
$\bigwedge_{y_{j}\in(\mathbb{C}^{m})^{\gamma}}dy_{j}$, and the upper-script
$G$ stands for the invariant part under the group $G$. It comes equipped with
a bidegree and a pairing, see [12, equation (4)] or [28, equation (5.12)].

Roughly, the cohomological field theory [20, 21] is constructed using virtual
fundamental cycles of certain moduli spaces of stable orbicurves with one
orbifold line bundle $L_{i}$ for each variable $y_{i}$, $i=1,\dots,m$, such
that for each monomial $W_{j}$ in $W$,
$W_{j}(L_{1},\dots,L_{k})=K\big{(}\sum\limits_{i=1}^{n}x_{i}\big{)}$, where
$K$ is the canonical bundle of the curve and $x_{1},\dots,x_{n}$ are its
marked points.

###  3.3 Double ramification cycle

The double ramification cycle (or DR cycle)
$\mathrm{DR}_{g}(a_{1},\dots,a_{n})$ is defined as the Poincaré dual of the
push-forward to the moduli space of stable curves
$\overline{\mathcal{M}}_{g,n}$ of the virtual fundamental class of the moduli
space of rubber stable maps to $\mathbb{P}^{1}$ relative to $0$ and $\infty$,
with ramification profile (orders of poles and zeros) given by
$(a_{1},\dots,a_{n})\in\mathbb{Z}^{n}$, where $\sum\limits_{i=1}^{n}a_{i}=0$.
Here “rubber” means that we consider maps up to the $\mathbb{C}^{*}$-action in
the target $\mathbb{P}^{1}$ and a positive/negative coefficient $a_{i}$
indicates a pole/zero at the $i$-th marked point, while $a_{i}=0$ just
indicates an internal marked point (that is not a zero or pole).

We view the DR cycle as a cohomology class in
$H^{2g}(\overline{\mathcal{M}}_{g,n},{\mathbb{Q}})$ dual to the homology class
represented by some natural compactification of the locus, inside
$\mathcal{M}_{g,n}$, formed by complex curves with marked points
$x_{1},\dots,x_{n}$ such that $\sum\limits_{i=1}^{n}a_{i}x_{i}$ is the divisor
of the zeros and poles of a meromorphic function. Sometimes, however, we will
denote with the same symbol the Poincaré dual homology cycle instead. For
instance in what follows we often say, and write in formulae, that we
integrate over $\mathrm{DR}_{g}(a_{1},\dots,a_{n})$.

Recently Pixton conjectured an explicit formula for the DR cycle in terms of
$\psi$-classes and boundary strata of $\overline{\mathcal{M}}_{g,n}$, which
was then proven in [24]. The problem of expressing the DR cycle in terms of
other tautological classes has been known since around 2000 as Eliashberg’s
problem, as Yakov Eliashberg posed it as a central question in symplectic
field theory, and Pixton’s formula provides a surprisingly explicit answer. We
will not recall the full formula here, limiting ourselves to recalling instead
that the class $\mathrm{DR}_{g}(a_{1},\dots,a_{n})$ belongs to
$H^{2g}(\overline{\mathcal{M}}_{g,n},{\mathbb{Q}})$, is tautological, and is a
(non-homogeneous) polynomial class in the $a_{i}$’s formed by monomials of
even degree and top degree equal to $2g$.

In fact, the restriction of the DR cycle to the moduli space
$\mathcal{M}_{g,n}^{\text{ct}}\subset\overline{\mathcal{M}}_{g,n}$ of curves
of compact type (i.e., those stable curves having only separating nodes, where
a node is separating if removing it breaks the curve into two disjoint
components) is described by the simpler Hain’s formula [23]

$\displaystyle
H^{2g}\big{(}\mathcal{M}_{g,n}^{\text{ct}}\big{)}\ni\mathrm{DR}_{g}(a_{1},\dots,a_{n})\big{|}_{\mathcal{M}_{g,n}^{\text{ct}}}=\frac{1}{g!}\left(-\frac{1}{4}\sum_{J\subset\\{1,\dots,n\\}}\sum_{h=0}^{g}a_{J}^{2}\delta^{J}_{h}\right)^{g},$

where

$\displaystyle a_{J}:=\sum_{j\in
J}a_{j},\qquad\delta^{J}_{h}=\left\\{\raisebox{-0.42pt}{\includegraphics{Rossi-
Fig1}}\right\\},\qquad\delta^{\\{i\\}}_{0}=-\psi_{i}.$

From this formula it is apparent that
$\left.\mathrm{DR}_{g}(a_{1},\dots,a_{n})\right|_{\mathcal{M}_{g,n}^{\text{ct}}}$
is a polynomial class in the $a_{i}$’s homogeneous of degree $2g$. This
formula is useful for instance when computing the intersection in
$\overline{\mathcal{M}}_{g,n}$ of $\mathrm{DR}_{g}(a_{1},\dots,a_{n})$ with
the class $\lambda_{g}$, since the latter vanishes outside
$\mathcal{M}_{g,n}^{\text{ct}}$ anyway.

We close this section remarking that the $DR$ cycle can, in fact, be seen as a
partial CohFT with respect to the infinite-dimensional $\mathbb{C}$-vector
space $V$ generated by $\\{e_{i}\\}_{i\in\mathbb{Z}}$ with metric given by
$\eta(e_{i},e_{j})=\delta_{i+j,0}$ and unit $e_{0}$, via the identification
$\mathrm{DR}_{g}(a_{1},\dots,a_{n})=c_{g,n}(e_{a_{1}}\otimes\dots\otimes
e_{a_{n}})$.

##  4 The Dubrovin–Zhang hierarchy
of a cohomological field theory

Dubrovin and Zhang [16] give a construction of an integrable hierarchy
starting from a semisimple cohomological field theory. A CohFT is said to be
semisimple when the associative algebra with structure constants
$\eta^{\alpha\mu}\frac{{\partial}F_{0}}{{\partial}t^{\mu}_{0}{\partial}t^{\beta}_{0}{\partial}t^{\gamma}_{0}}\big{|}_{t^{*}_{>0}=0}$
is semisimple generically with respect to the variables $t^{*}_{0}$.

Dubrovin and Zhang’s framework gives, among other things, the language for
stating far reaching generalizations of Witten’s conjecture [36]. In this
section we briefly present their construction (following the clear exposition
of [6, 7]) and explain its relation to Witten-type conjectures.

###  4.1 DZ hierarchy

Consider the potential $F(t^{*}_{*};\varepsilon)=\sum\limits_{g\geq
0}\varepsilon^{2g}F_{g}(t^{*}_{*})$ of a semisimple CohFT. Denote
$\Omega_{\alpha,p;\beta,q}(t^{*}_{*};\varepsilon)=\frac{{\partial}^{2}F(t^{*}_{*};\varepsilon)}{{\partial}t^{\alpha}_{p}{\partial}t^{\beta}_{q}}=\sum\limits_{g\geq
0}\Omega^{[2g]}_{\alpha,p;\beta,q}(t^{*}_{*})\varepsilon^{2g}$.

The construction starts in genus $0$ and we use variables $v^{*}_{*}$ for the
fomal loops space. Here the hierarchy is given by the following Hamiltonian
densities and Poisson structure:

$\displaystyle
h_{\alpha,p}(v^{*})=\Omega^{[0]}_{\alpha,p+1;1,0}(t^{*}_{0}=v^{*},0,0,\dots),$
$\displaystyle\big{(}K^{\mathrm{DZ}}_{v}\big{)}^{\alpha\beta}=\eta^{\alpha\beta}{\partial}_{x}.$

Commutativity of these Hamiltonians is a simple consequence of the fact that
the nodal divisors $D_{(12|34)}$ and $D_{(13|24)}$ are equivalent in
$H^{*}(\overline{\mathcal{M}}_{0,4},{\mathbb{Q}})$,

Also, these Hamiltonian densities form a tau-structure by definition.

Let then $v^{\alpha}(x,t^{*}_{*})$, $\alpha,1,\dots,N$, be the solution to the
above integrable hierarchy with initial datum
$v^{\alpha}(x,t^{*}_{*}=0)=\delta^{\alpha}_{1}x$. We have, see, e.g., [6],

$\displaystyle
F_{g}(t^{*}_{0},t^{*}_{1},\dots)=F_{g}\big{(}P^{*}_{0}(v^{*}_{0},\dots,v^{*}_{3g-2}),\dots,P^{*}_{3g-2}(v^{*}_{0},\dots,v^{*}_{3g-2}),0,\dots\big{)}\big{|}_{x=0},$

where $P^{*}_{*}$ are in general rational functions, not differential
polynomials.

Consider the change of coordinates

$\displaystyle w^{\alpha}(v^{*}_{*};\varepsilon)=v^{\alpha}+\sum_{g\geq
1}\varepsilon^{2g}\frac{{\partial}^{2}F_{g}(v^{*}_{0},\dots,v^{*}_{3g-2})}{{\partial}t^{\alpha}_{0}{\partial}x}.$

It is a normal, but non-Miura, transformation, because the $P^{*}_{*}$ are not
differential polynomials.

The full Dubrovin–Zhang (DZ) hierarchy is just the transformation of the above
genus $0$ hierarchy with respect to the above non-Miura change of coordinates
$w^{\alpha}=w^{\alpha}(v^{*}_{*};\varepsilon)$, whose inverse we denote by
$v^{\alpha}=v^{\alpha}(w^{*}_{*};\varepsilon)$. In fact, in order to obtain a
tau-structure, we want to add a ${\partial}_{x}$-exact term to the transformed
Hamiltonians, as prescribed for a normal (albeit non-Miura) transformation, as
explained in Section 2.4:

$\displaystyle
h^{\mathrm{DZ}}_{\alpha,p}(w^{*}_{*};\varepsilon):=h_{\alpha,p}(v^{*}(w^{*}_{*};\varepsilon))+\sum_{g\geq
1}\varepsilon^{2g}\frac{{\partial}^{2}F_{g}(v^{*}_{*}(w^{*}_{*};\varepsilon))}{{\partial}t^{\alpha}_{p+1}{\partial}x},$
$\displaystyle\big{(}K^{\mathrm{DZ}}_{w}\big{)}^{\alpha\beta}=(L^{*})^{\alpha}_{\mu}\circ\big{(}K_{v}^{\mathrm{DZ}}\big{)}^{\mu\nu}\circ
L^{\beta}_{\nu},$

where $(L^{*})^{\alpha}_{\mu}=\sum\limits_{s\geq 0}\frac{\partial
w^{\alpha}}{\partial v^{\mu}_{s}}\partial_{x}^{s}$,
$L^{\beta}_{\nu}=\sum\limits_{s\geq 0}(-\partial_{x})^{s}\circ\frac{\partial
w^{\beta}}{\partial v^{\nu}_{s}}$.

The DZ hierarchy is an integrable tau-symmetric hierarchy whose tau-function
for the solution with initial datum
$w^{\alpha}(x,t^{*}_{*}=0;\varepsilon)=\delta^{\alpha}_{1}x$ (called the
topological solution) is, _by construction_ , the partition function of the
CohFT.

The technical hypothesis of semisimplicity of the CohFT is used in the proof
that, in spite of the fact that the transformation $v^{*}\mapsto w^{*}$ is not
Miura, the Hamiltonian densities
$h^{\mathrm{DZ}}_{\alpha,p}(w^{*}_{*};\varepsilon)$ and Poisson structure
$(K^{\mathrm{DZ}}_{w})^{\alpha\beta}$ are still of the correct differential
polynomial class.

###  4.2 Witten’s conjecture and its generalizations

In [36], Witten conjectured that the partition function of the trivial CohFT
is the tau-function of the topological solution to the KdV hierarchy.

Another way to state this, in light of the last section, is that the DZ
hierarchy of the trivial CohFT is the KdV hierarchy.

This conjecture was proved by Kontsevich in [25] and, after that, many similar
conjectures and results appeared in the literature, consisting in identifying
and controlling the DZ hierarchy of a given CohFT. For instance in [18],
Faber–Shadrin–Zvonkine proved that the DZ hierarchy of Witten’s $r$-spin class
(for $r\geq 2$ a CohFT that was defined in [27]) coincides with the $r$-KdV
Gelfand–Dickey hierarchy, another well known tau symmetric integrable system.

##  5 Double ramification hierarchies

In this section we introduce the main subject of this paper, the double
ramification hierarchy construction. We will give a self-contained exposition
of our main results, including an account of the progress made in proving the
conjecture that the DR and DZ hierarchy are actually equivalent up to a normal
Miura transformation, and our quantization technique for $(1+1)$-dimensional
integrable field theories.

###  5.1 The main idea

Symplectic field theory [17] is a large project attempting to provide a
unified view on established pseudoholomorphic curve theories in symplectic
topology like symplectic Floer homology, contact homology and Gromov–Witten
theory, leading to numerous new applications, including a construction of
quantum integrable systems from the geometry of the moduli spaces of
pseudoholomorphic curves in symplectic cobordisms between contact manifolds.

In a sense, the double ramification hierarchy arises from completely analogous
constructions in the complex algebraic setting and with the axiomatized
language of cohomological field theories replacing curve counting in target
varieties. In this sense the double ramification hierarchy is a quantum
integrable system, even if A. Buryak introduced first its classical version in
[1].

Given a cohomological field theory $c_{g,n}\colon V^{\otimes n}\to
H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{C})$, at the heart of the
construction for the classical hierarchy lie its intersection numbers with the
DR cycle, the powers of one psi-class and the top Hodge class $\lambda_{g}$:

$\displaystyle
P_{\alpha,d;\alpha_{1},\dots,\alpha_{n}}^{g;a_{1},\dots,a_{n}}=\int_{\mathrm{DR}_{g}\big{(}{-}\sum
a_{i},a_{1},\dots,a_{n}\big{)}}\lambda_{g}\psi_{1}^{d}c_{g,n+1}\big{(}e_{\alpha}\otimes\otimes_{i=1}^{n}e_{\alpha_{i}}\big{)}.$

This is all the geometric content used in the definition of the DR hierarchy.

These intersection numbers are collected into generating functions
$g_{\alpha,d}$ depending on the indices $\alpha=1,\dots,N$ and $d\geq 0$ which
have the form of differential polynomials (see next section). The differential
polynomials $g_{\alpha,d}$ directly play the role of Hamiltonian densities for
a classical integrable system. The Poisson structure, on the other hand, and
contrary to what happens for the DZ hierarchy, does not depend on the
cohomological field theory and is always in Getzler’s standard form.

Notice that, because of the presence of the class $\lambda_{g}$, Hain’s
formula is sufficient to compute the above intersection numbers. This
advantage if often exploited in explicit computations.

###  5.2 DR hierarchy Hamiltonians

Because of the polynomiality properties of the DR cycle,
$P_{\alpha,d;\alpha_{1},\dots,\alpha_{n}}^{g;a_{1},\dots,a_{n}}$ is a
homogeneous polynomial in $a_{1},\dots,a_{n}$ of degree $2g$. So, if we write
it as such,

$\displaystyle
P_{\alpha,d;\alpha_{1},\dots,\alpha_{n}}^{g;a_{1},\dots,a_{n}}=\sum_{\sum
b_{i}=2g}\widetilde{P}_{\alpha,d;\alpha_{1},\dots,\alpha_{n}}^{g;b_{1},\dots,b_{n}}a_{1}^{b_{1}}\cdots
a_{n}^{b_{n}},$

we can give the following definition

$\displaystyle g_{\alpha,d}:=\sum_{\begin{subarray}{c}g\geq 0,n\geq 0\\\
2g-1+n>0\end{subarray}}\frac{(-\varepsilon^{2})^{g}}{n!}\sum_{a_{1},\dots,a_{n}\in\mathbb{Z}}P_{\alpha,d;\alpha_{1},\dots,\alpha_{n}}^{g;a_{1},\dots,a_{n}}\
p^{\alpha_{1}}_{a_{1}}\cdots p^{\alpha_{n}}_{a_{n}}\ e^{ix\sum a_{i}}$
$\displaystyle\hphantom{g_{\alpha,d}}{}=\sum_{\begin{subarray}{c}g\geq 0,n\geq
0\\\ 2g-1+n>0\end{subarray}}\frac{(-\varepsilon^{2})^{g}}{n!}\sum_{\sum
b_{i}=2g}\widetilde{P}_{\alpha,d;\alpha_{1},\dots,\alpha_{n}}^{g;b_{1},\dots,b_{n}}\
u^{\alpha_{1}}_{b_{1}}\cdots u^{\alpha_{n}}_{b_{n}},$

and hence we have two expressions for the DR Hamiltonian densities, in
variables $p^{*}_{*}$ and $u^{*}_{*}$ respectively. The second line, in
particular, is clearly a differential polynomial in
$\widehat{\mathcal{A}}^{[0]}$.

######  Remark 5.1.

Let us see, as an example using the definition and the pullback property of
cohomological field theories with respect to the forgetful morphism
$\pi\colon\overline{\mathcal{M}}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}$, how
to actually compute $\overline{g}_{1,0}$ for any CohFT. The involved
intersection numbers are
$P_{1,0;\alpha_{1},\dots,\alpha_{n}}^{g,a_{1},\dots,a_{n}}$, for $n>1$ and
$\sum a_{i}=0$ (this last condition is the effect of integrating $g_{1,0}$ in
$dx$), and these are integrals over $\overline{\mathcal{M}}_{g,n+1}$ of a
class pulled back from $\overline{\mathcal{M}}_{g,n}$. Namely, remembering
that the DR cycle is a partial CohFT too, so it has the pullback property with
respect to the coefficient $a=0$, we have
$\lambda_{g}\mathrm{DR}_{g}(0,a_{1},\dots,a_{n})c_{g,n+1}(e_{1}\otimes\otimes_{i=1}^{n}e_{\alpha_{i}})=\pi^{*}(\lambda_{g}\mathrm{DR}_{g}(a_{1},\dots,a_{n})c_{g,n+1}(\otimes_{i=1}^{n}e_{\alpha_{i}}))$.
This means that $P_{1,0;\alpha_{1},\dots,\alpha_{n}}^{g,a_{1},\dots,a_{n}}$
vanishes whenever the map $\pi$ exists, i.e., unless $g=0,n=2$, for which we
have $P_{1,0;\mu,\nu}^{0,a,-a}=\eta_{\mu\nu}$. This gives

$\displaystyle g_{1,0}=\int\frac{1}{2}\eta_{\mu\nu}u^{\mu}u^{\nu}dx.$ (5.1)

Commutativity $\\{\overline{g}_{\alpha,p},\overline{g}_{\beta,q}\\}=0$ with
respect to the standard Hamiltonian operator $(K^{\mathrm{DR}})^{\mu\nu}$
$=\eta^{\mu\nu}{\partial}_{x}$ (we omit the subscript $K$ in
$\\{\cdot,\cdot\\}_{K}$ when $K$ is in Getzler’s standard form), was proved in
[1]. Let’s give an idea of the proof.

In genus $0$, where the DR cycle is equal to $1$, this equation is basically
equivalent to the equivalence of boundary divisors $D_{(12|34)}$ and
$D_{(13|24)}$ in $H^{*}(\overline{\mathcal{M}}_{0,4},{\mathbb{Q}})$. The genus
$0$ argument can be ported to higher genus by working with images of the
curves of the DR cycle with respect to the meromorphic function (or more
precisely rubber map to $\mathbb{P}^{1}$) that is defined on them. This is a
general fact: we often find that genus $0$ properties of the DZ hierarchy have
all genera analogues on the DR hierarchy side.

Making this argument precise, one gets to prove the following equation for
products of double ramification cycles. For a subset
$I=\\{i_{1},i_{2},\dots\\},i_{1}<i_{2}<\cdots$, of the set $\\{1,\dots,n\\}$
let $A_{I}:=(a_{i_{1}},a_{i_{2}},\dots)$. Suppose the set $\\{1,2,\dots,n\\}$
is divided into two disjoint subsets, $I\sqcup J=\\{1,2,\dots,n\\}$, in such a
way that $\sum\limits_{i\in I}a_{i}>0$. Let us denote by
$\mathrm{DR}_{g_{1}}(0_{x_{1}},A_{I},-k_{1},\dots,-k_{p})\boxtimes\mathrm{DR}_{g_{2}}(0_{x_{2}},A_{J},k_{1},\dots,k_{p})$
the cycle in $\overline{\mathcal{M}}_{g_{1}+g_{2}+p-1,n+2}$ obtained by gluing
the two double ramification cycles at the marked points labeled by the
positive integers $k_{1},\dots,k_{p}$. Here $0_{x}$ indicates a coefficient
$0$ at the marked point $x$. Then

$\displaystyle\sum\frac{\prod\limits_{i=1}^{p}k_{i}}{p!}\mathrm{DR}_{g_{1}}(0_{x_{1}},A_{I},-k_{1},\dots,-k_{p})\boxtimes\mathrm{DR}_{g_{2}}(0_{x_{2}},A_{J},k_{1},\dots,k_{p})$
$\displaystyle\qquad{}-\sum\frac{\prod\limits_{i=1}^{p}k_{i}}{p!}\mathrm{DR}_{g_{1}}(0_{x_{2}},A_{I},-k_{1},\dots,-k_{p})\boxtimes\mathrm{DR}_{g_{2}}(0_{x_{1}},A_{J},k_{1},\dots,k_{p})=0.$
(5.2)

The sums are over $I$, $J$, $p>0$ $k_{1}>0,\dots,k_{p}>0$, $g_{1}\geq 0$,
$g_{2}\geq 0$.

If we intersect this relation with the class $\lambda_{g}$ (which kills the
terms with $p>1$) and with the $\psi$-classes and CohFT, and form the
corresponding generating function, we obtain precisely

$\displaystyle\sum_{k>0}\left(k\eta^{\mu\nu}\frac{{\partial}\overline{g}_{\alpha,p}}{{\partial}p^{\mu}_{k}}\frac{{\partial}\overline{g}_{\beta,q}}{{\partial}p^{\nu}_{-k}}-k\eta^{\mu\nu}\frac{{\partial}\overline{g}_{\beta,q}}{{\partial}p^{\mu}_{k}}\frac{{\partial}\overline{g}_{\alpha,p}}{{\partial}p^{\nu}_{-k}}\right)=\big{\\{}\overline{g}_{\alpha,p},\overline{g}_{\beta,q}\big{\\}}=0.$

In [1] Buryak computed the first two examples of DR hierarchies. For the
trivial CohFT he found the KdV hierarchy, the same result as for the DZ
hierarchy. For the Hodge CohFT he found the Intermediate Long Wave hierarchy
(ILW). When comparing this second case with the DZ hierarchy he realized that,
once more, the integrable systems were the same, but this time he had to
perform a Miura transformation to match them. This motivated him to propose
the following conjecture.

######  Conjecture 5.2 (weak DR/DZ equivalence [1]).

Given a semisimple CohFT, the associated DZ and DR hierarchy coincide up to a
Miura transformation.

###  5.3 Recursion relations

In [9], using results about the intersection of a $\psi$-class with the DR
cycle from [10], by analogy with [19], we found the following recursion
equations among the DR Hamiltonian densities.

######  Theorem 5.3 ([8]).

For all $\alpha=1,\dots,N$ and $p=-1,0,1,\dots$, let
$g_{\alpha,-1}=\eta_{\alpha\mu}u^{\mu}$. We have

$\displaystyle\partial_{x}(D-1)g_{\alpha,p+1}=\big{\\{}g_{\alpha,p},\overline{g}_{1,1}\big{\\}},$
(5.3) $\displaystyle\partial_{x}\frac{\partial g_{\alpha,p+1}}{\partial
u^{\beta}}=\big{\\{}g_{\alpha,p},\overline{g}_{\beta,0}\big{\\}},$ (5.4)

where $D:=\varepsilon\frac{\partial}{\partial\varepsilon}+\sum\limits_{s\geq
0}u^{\alpha}_{s}\frac{\partial}{\partial u^{\alpha}_{s}}$.

Equation (5.3) is especially striking. First of all it provides and effective
procedure to reconstruct the full hierarchy starting from the knowledge of
$\overline{g}_{1,1}$ only. Secondly, from the point of view of integrable
systems, such recursion was not known. Even in the simplest examples it does
not coincide with any previously known reconstruction techniques for the
symmetries of an integrable hierarchy (it is in fact this recursion that we
presented in Section 2.5 for the KdV equation). At the same time, its
universal form (its form is rigid, independent of the CohFT or the integrable
hierarchy) suggests that it should be regarded as some sort of intrinsic
feature of at least a class of integrable systems (see Section 5.7).

As a simple consequence of (5.4) for $\beta=1$, together with (5.1) we
immediately get

######  Corollary 5.4.

The DR Hamiltonian densities satisfy the string equation
$\frac{{\partial}g_{\alpha,d}}{{\partial}u^{1}}=g_{\alpha,d-1}$.

###  5.4 Tau-structure and the strong DR/DZ equivalence

In [2] we provide the DR hierarchy with a tau-structure and study its
topological tau-function.

######  Theorem 5.5.

The DR hierarchy is tau-symmetric. A tau-structure is given by
$h^{\mathrm{DR}}_{\alpha,p}=\frac{\delta\overline{g}_{\alpha,p+1}}{\delta
u^{1}}$.

###### Proof.

This is a general feature of integrable hierarchies with the standard
Hamiltonian operator $K^{\mu\nu}=\eta^{\mu\nu}{\partial}_{x}$ and satisfying
the string equation
$\frac{{\partial}\overline{g}_{\alpha,d}}{{\partial}u^{1}}=\overline{g}_{\alpha,d-1}$.
Indeed it suffices to apply the variational derivative $\frac{\delta}{\delta
u^{1}}$ to the commutation equation
$\\{\overline{g}_{\alpha,p},\overline{g}_{\beta,q}\\}=0$ to obtain
$\big{\\{}h^{\mathrm{DR}}_{\alpha,p-1},\overline{h}^{\mathrm{DR}}_{\beta,q}\big{\\}}=\big{\\{}h^{\mathrm{DR}}_{\beta,q-1},\overline{h}^{\mathrm{DR}}_{\alpha,p}\big{\\}}$.
∎

Consider the normal coordinates
$\widetilde{u}^{\alpha}=\eta^{\mu\nu}h^{\mathrm{DR}}_{\mu,-1}$. Let us write
the tau-function associated to the topological solution (with initial datum
$\widetilde{u}^{\alpha}(x,0;\varepsilon)=x\delta^{\alpha}_{1}$) as

$\displaystyle F^{\mathrm{DR}}(t^{*}_{*};\varepsilon)=\sum_{g\geq
0}\varepsilon^{2g}F^{\mathrm{DR}}_{g}(t^{*}_{*}),$

where

$\displaystyle F_{g}^{\mathrm{DR}}(t^{*}_{*})=\sum_{\begin{subarray}{c}n\geq
0\\\ 2g-2+n>0\end{subarray}}\frac{1}{n!}\sum_{d_{1},\dots,d_{n}\geq
0}\left<\prod_{i=1}^{n}\tau_{d_{i}}(e_{\alpha_{i}})\right>_{g}^{\mathrm{DR}}\prod_{i=1}^{n}t^{\alpha_{i}}_{d_{i}}.$

Notice that this _DR partition function_ has only an indirect geometric
meaning. Contrary to the correlators of the topological tau-function of the DZ
hierarchy (which coincide with the correlators of the CohFT), the correlators
$\Big{\langle}\prod\limits_{i=1}^{n}\tau_{d_{i}}(e_{\alpha_{i}})\Big{\rangle}_{g}^{\mathrm{DR}}$
are not a priori defined as intersection numbers in
$H^{*}(\overline{\mathcal{M}}_{g,n},{\mathbb{Q}})$, but only as the
coefficients of the series $F^{\mathrm{DR}}$. We can a posteriori try to study
their geometric meaning, and, as a consequence of certain properties of the DR
cycle, we find the following surprising selection rule.

######  Proposition 5.6 ([2]).

$\left<\tau_{d_{1}}(e_{\alpha_{1}})\cdots\tau_{d_{m}}(e_{\alpha_{m}})\right>^{\mathrm{DR}}_{g}=0$
when $\sum\limits_{i=1}^{m}d_{i}>3g-3+m$ or $\sum\limits_{i=1}^{m}d_{i}\leq
2g-2$.

In light of the conjectured equivalence with the DZ hierarchy, the first
selection rule looks like the corresponding vanishing property
$\left<\tau_{d_{1}}(e_{\alpha_{1}})\cdots\tau_{d_{m}}(e_{\alpha_{m}})\right>_{g}=0$
when $\sum\limits_{i=1}^{m}d_{i}>3g-3+m$, which just means that we cannot
integrate too many $\psi$-classes without surpassing the dimension of the
moduli space (for short, we say that correlators cannot be “too big”). But the
second selection rule actually says that the DR correlators cannot be too
small either! This rule one has no analogue in the DZ case and, as it turns
out, provides the key to a much deeper understanding of the DR/DZ equivalence.

The situation is that we are trying to compare two integrable tau-symmetric
hierarchies by a Miura transformation that is supposed to modify the tau-
function by killing all “small correlators” (which are present on the DZ side
and absent in the DR side). A natural candidate would then be a normal Miura
transformation (since they preserve tau-symmetry) generated by a differential
polynomial
$\mathcal{F}(w^{*}_{*};\varepsilon)\in\widehat{\mathcal{A}}^{[-2]}$,

$\displaystyle\widetilde{u}^{\alpha}=w^{\alpha}+\eta^{\alpha\mu}{\partial}_{x}\big{\\{}\mathcal{F},\overline{h}^{\mathrm{DZ}}_{\mu,0}\big{\\}}_{\mathrm{DZ}},$

and we know that such transformations modify the tau-function by

$\displaystyle\widetilde{F}(t^{*}_{*};\varepsilon)=F(t^{*}_{*};\varepsilon)+\mathcal{F}(w^{*}_{*}(x,t^{*}_{*};\varepsilon);\varepsilon)\big{|}_{x=0}.$

Can we find $\mathcal{F}(w^{*}_{*};\varepsilon)$ so that
$\widetilde{F}(t^{*}_{*};\varepsilon)$ satisfies the selection rule (i.e., has
no small correlators)? As it turns out, yes, and this selects a unique normal
Miura transformation!

######  Theorem 5.7 ([2]).

$\exists!\ \mathcal{F}(w^{*}_{*};\varepsilon)\in\widehat{\mathcal{A}}^{[-2]}$
such that $F^{\mathrm{red}}:=F+\mathcal{F}(w^{*}_{*};\varepsilon)|_{x=0}$
satisfies the above selection rules.

This makes Buryak’s conjecture much more precise.

######  Conjecture 5.8 (strong DR/DZ equivalence, [2]).

For any semisimple CohFT, the DR and DZ hierarchies coincide up to the normal
Miura transformation generated by the unique
$\mathcal{F}(w^{*}_{*};\varepsilon)$ found in Theorem 5.7. Even in the non-
semisimple case, we can state this conjecture as
$F^{\mathrm{red}}=F^{\mathrm{DR}}$.

When proven true, the conjecture would clearly state that, although equivalent
as integrable systems to the DZ hierarchy, the DR hierarchy contains strictly
less information than the DZ hierarchy. Indeed, starting from the DZ hierarchy
it is possible to construct the normal Miura transformation mapping to the DR
hierarchy, while the DR hierarchy does not contain this extra information.
This is perhaps not surprising given at least the presence of the class
$\lambda_{g}$ in the DR hierarchy intersection numbers.

From the point of view of integrable systems however, this is of great
interest. The fact the DR hierarchy is some sort of standard form of the DZ
hierarchy allows to study these systems ignoring complications that might just
come from the system of coordinates in which they are described. The presence
of powerful recursion relations for the Hamiltonians, for instance, seems to
rely precisely on this special standard form.

Finally we remark that the extra information that is killed by the above
normal Miura transformation, might be (maybe in part) recovered once we
consider the quantum DR hierarchy (which replaces $\lambda_{g}$ in the
construction by the full Hodge class $\Lambda(s)$), see below.

###  5.5 The proof of the strong DR/DZ conjecture

In [2] we prove the strong DR/DZ equivalence conjecture for a number of
CohFTs.

######  Theorem 5.9 ([2, 5]).

The strong DR/DZ equivalence conjecture holds in the following cases:

  * •

the trivial CohFT,

  * •

the full Hodge class,

  * •

Witten’s $3$-, $4$\- and $5$-spin classes,

  * •

the GW theory of $\mathbb{P}^{1}$,

  * •

up to genus $5$ for any rank $1$ CohFT,

  * •

up to genus $2$ for any semisimple CohFT.

However, in all these cases, the proof is either by direct computation or by
some ad hoc technique. A large and quite technical part of our project deals
with proving Conjecture 5.8 on completely general grounds.

The strategy of the proof for the general case which we are pursuing, in [3]
and our next paper in progress, is to give explicit geometric formulas for the
correlators appearing in both $F^{\mathrm{DZ}}$ and $F^{\mathrm{red}}$ in
terms of sums over certain decorated trees corresponding to cycles in the
$\overline{\mathcal{M}}_{g,n}$. This reduces the strong DR/DZ conjecture to a
family of relations in the tautological ring of
$\overline{\mathcal{M}}_{g,n}$. In particular we managed to further reduce
this family to a finite number (equal to the number of partitions of $2g$) of
relations for each genus [5].

###  5.6 Quantization

As we already remarked, the idea for the DR hierarchy came from symplectic
field theory where quantum integrable systems arise naturally. Let us see how
this happens in the language we used in this document, of cohomological field
theories in the complex algebraic category. The intersection numbers to be
considered look perhaps more natural,

$\displaystyle
P_{\alpha,d;\alpha_{1},\dots,\alpha_{n}}^{g;a_{1},\dots,a_{n}}(s)=\int_{\mathrm{DR}_{g}\left(-\sum
a_{i},a_{1},\dots,a_{n}\right)}\Lambda\left(s\right)\psi_{1}^{d}c_{g,n+1}\big{(}e_{\alpha}\otimes\otimes_{i=1}^{n}e_{\alpha_{i}}\big{)}.$

Indeed the product
$\Lambda(s)c_{g,n+1}\left(e_{\alpha}\otimes\otimes_{i=1}^{n}e_{\alpha_{i}}\right)$
is itself a CohFT (and every CohFT can be written this way), so we are simply
intersecting a CohFT, the $\psi$-classes and the DR cycle.

$P_{\alpha,d;\alpha_{1},\dots,\alpha_{n}}^{g;a_{1},\dots,a_{n}}(s)$ is a non-
homogeneous polynomial in $a_{1},\dots,a_{n}$ of top degree $2g$, so

$\displaystyle
P_{\alpha,d;\alpha_{1},\dots,\alpha_{n}}^{g;a_{1},\dots,a_{n}}(s)=\sum_{\sum
b_{i}=2k\leq
2g}\widetilde{P}_{\alpha,d;\alpha_{1},\dots,\alpha_{n}}^{g;b_{1},\dots,b_{n}}(s)a_{1}^{b_{1}}\cdots
a_{n}^{b_{n}},$

and we define

$\displaystyle G_{\alpha,d}:=\sum_{\begin{subarray}{c}g\geq 0,n\geq 0\\\
2g-1+n>0\end{subarray}}\frac{(i\hbar)^{g}}{n!}\sum_{a_{1},\dots,a_{n}\in\mathbb{Z}}P_{\alpha,d;\alpha_{1},\dots,\alpha_{n}}^{g;a_{1},\dots,a_{n}}\left(\frac{-\varepsilon^{2}}{i\hbar}\right)p^{\alpha_{1}}_{a_{1}}\cdots
p^{\alpha_{n}}_{a_{n}}\ e^{ix\sum a_{i}}$
$\displaystyle\hphantom{G_{\alpha,d}}{}=\sum_{\begin{subarray}{c}g\geq 0,n\geq
0\\\ 2g-1+n>0\end{subarray}}\frac{(i\hbar)^{g}}{n!}\sum_{\sum b_{i}\leq
2g}\widetilde{P}_{\alpha,d;\alpha_{1},\dots,\alpha_{n}}^{g;b_{1},\dots,b_{n}}\left(\frac{-\varepsilon^{2}}{i\hbar}\right)u^{\alpha_{1}}_{b_{1}}\cdots
u^{\alpha_{n}}_{b_{n}}.$

Notice how $(i\hbar)$ has replaced $(-\varepsilon^{2})$ as the genus parameter
and, at the same time, we have given the Hodge class parameter $s$ the value
$\left(\frac{-\varepsilon^{2}}{i\hbar}\right)$, so that these two choices
compensate in the limit $\hbar=0$ to give back the classical Hamiltonian
densities $g_{\alpha,p}$.

What about commutativity of these new Hamiltonians? We can again use equation
(5.2), but, because the top Hodge class $\lambda_{g}$ has now been replaced by
the full Hodge class $\Lambda(s)$, all values of $p>0$ will contribute to the
sum. This translates into the following equation

$\displaystyle\big{[}{\overline{G}}_{\alpha,p},{\overline{G}}_{\beta,q}\big{]}=0,$

where $[f,g]:=f\star g-g\star f$ with $f\star
g=f\Big{(}e^{\sum\limits_{k>0}i\hbar
k\eta^{\alpha\beta}\overleftarrow{\frac{\partial}{\partial
p^{\alpha}_{k}}}\overrightarrow{\frac{\partial}{\partial
p^{\beta}_{-k}}}}\Big{)}g$. The exponential here comes precisely from the fact
that double ramification cycles are now glued along any number of marked
points, not just one, as it was the case for the classical DR hierarchy.

From a mathematical physics viewpoint this is an entirely new and surprisingly
universal quantization technique for integrable field theories. We have
completely explicit formulas for the quantum versions of KdV, Toda, ILW,
Gelfand–Dickey and other integrable hierarchies, that, to our knowledge, were
either unknown or known in a much more indirect way. The reader will find
these examples in Section 6.

This explicit description also rests on the analogue of Theorem 5.3 which,
again, allows to reconstruct the full quantum hierarchy from the Hamiltonian
${\overline{G}}_{1,1}$ alone.

######  Theorem 5.10 ([8]).

For all $\alpha=1,\dots,N$ and $p=-1,0,1,\dots$, let
$G_{\alpha,-1}=\eta_{\alpha\mu}u^{\mu}$. We have

$\displaystyle\partial_{x}(D-1)G_{\alpha,p+1}=\frac{1}{\hbar}\big{[}G_{\alpha,p},{\overline{G}}_{1,1}\big{]},$
(5.5) $\displaystyle\partial_{x}\frac{\partial G_{\alpha,p+1}}{\partial
u^{\beta}}=\frac{1}{\hbar}\big{[}G_{\alpha,p},{\overline{G}}_{\beta,0}\big{]},$

where
$D:=\varepsilon\frac{\partial}{\partial\varepsilon}+2\hbar\frac{\partial}{\partial\hbar}+\sum\limits_{s\geq
0}u^{\alpha}_{s}\frac{\partial}{\partial u^{\alpha}_{s}}$.

Finally, in [3], we define and study the quantum analogue of the notion of
tau-structure and tau-functions and prove that the quantum DR hierarchy
satisfies tau-symmetry. This allows to define a quantum deformation of the DR
potential that clearly contains more geometric information on the associated
CohFT and needs to be investigated further.

###  5.7 Integrable systems of DR type

The recursion equation (5.5) or its classical version (5.3) are really
surprising from the point of view of integrable systems. No expert we talked
to was able to recognize them as something previously known.

Moreover we realized that one could interpret such equation as constraints for
the generating Hamiltonian ${\overline{G}}_{1,1}$ itself, just by imposing
that, at each step of the recursion, we still obtain a commuting quantity.
This technique proved fruitful to reproduce, for instance, the full DR
hierarchy starting from genus $0$ in the case of polynomial Frobenius
manifolds (i.e., those genus $0$ CohFT associated with Coxeter groups as in
[14]). In doing these computational experiments we realized that the
recursions (5.5), (5.3) were of independent value in the theory of integrable
systems.

Let us first state our result in the classical situation.

######  Theorem 5.11 ([3]).

Assume that a local functional $\overline{h}\in\widehat{\Lambda}^{[0]}$ is
such that the recursion

$\displaystyle\partial_{x}(D-1)g_{\alpha,p+1}=\big{\\{}g_{\alpha,p},\overline{h}\big{\\}},\qquad
g_{\alpha,-1}=\eta_{\alpha\mu}u^{\mu},\qquad
D=\varepsilon\frac{{\partial}}{{\partial}\varepsilon}+\sum_{k\geq
0}u^{\alpha}_{k}\frac{{\partial}}{{\partial}u^{\alpha}_{k}}$

produces, at each step, Hamiltonians that still commute with
$\overline{g}_{1,1}$ $($so that the recursion can go on indefinitely$)$.
Assume moreover that $\frac{\delta\overline{g}_{1,1}}{\delta
u^{1}}=\frac{1}{2}\eta_{\mu\nu}u^{\mu}u^{\nu}+{\partial}_{x}^{2}r$, where
$r\in\widehat{\mathcal{A}}^{[-2]}$.

Then, up to a triangular transformation of the form

$\displaystyle g_{\alpha,d}\mapsto
g_{\alpha,d}+\sum_{i=1}^{d+1}a^{\mu}_{\alpha,i}g_{\mu,d-i}+b_{\alpha,d},\qquad
a^{\mu}_{\alpha,i},b_{\alpha,d}\in\mathbb{C},\qquad d\geq 0,\qquad
1\leq\alpha\leq N,$

we have

  * $(i)$

$\overline{g}_{1,1}=\overline{h}$,

  * $(ii)$

$g_{1,0}=\frac{1}{2}\eta_{\mu\nu}u^{\mu}u^{\nu}+{\partial}_{x}^{2}(D-1)^{-1}r$,

  * $(iii)$

$\\{\overline{g}_{\alpha,p},\overline{g}_{\beta,q}\\}=0$,
$\alpha,\beta=1,\dots,N$, $p,q\geq-1$,

  * $(iv)$

$\\{g_{\alpha,p},\overline{g}_{\beta,0}\\}={\partial}_{x}\frac{{\partial}g_{\alpha,p+1}}{{\partial}u^{\beta}}$,
$\beta=1,\dots,N$, $p\geq-1,$

  * $(v)$

$\frac{{\partial}g_{\alpha,p}}{{\partial}u^{1}}=g_{\alpha,p-1}$,
$\alpha=1,\dots,N$, $p\geq-1$,

hence in particular we get an integrable tau-symmetric hierarchy.

This suggests that it is interesting to consider integrable systems
originating from local functionals satisfying the hypothesis of the above
theorem. We call them _integrable systems of DR type_.

In the quantum case the theorem is weaker, but only slightly: indeed it is not
automatic that a Hamiltonian
${\overline{H}}\in(\widehat{\Lambda}^{\hbar})^{[\leq 0]}$, for which the
recursion goes on indefinitely, fits into the recursion itself as
${\overline{G}}_{1,1}$, so one needs to impose it by hand (but from the
practical viewpoint it is just one extra explicit constraint on
${\overline{H}}$).

######  Theorem 5.12 ([3]).

Assume that a local functional
${\overline{H}}\in(\widehat{\Lambda}^{\hbar})^{[\leq 0]}$ is such that the
recursion

$\displaystyle\partial_{x}(D-1)G_{\alpha,p+1}=\frac{1}{\hbar}\big{[}G_{\alpha,p},{\overline{H}}\big{]},\qquad
G_{\alpha,-1}=\eta_{\alpha\mu}u^{\mu},\qquad
D=\varepsilon\frac{{\partial}}{{\partial}\varepsilon}+2\hbar\frac{{\partial}}{{\partial}\hbar}+\sum_{k\geq
0}u^{\alpha}_{k}\frac{{\partial}}{{\partial}u^{\alpha}_{k}}$

produces, at each step, Hamiltonians that still commute with ${\overline{H}}$
(so that the recursion can go on indefinitely). Assume moreover that
${\overline{G}}_{1,1}={\overline{H}}$ and that
$\frac{\delta{\overline{H}}}{\delta
u^{1}}=\frac{1}{2}\eta_{\mu\nu}u^{\mu}u^{\nu}+{\partial}_{x}R+C(\varepsilon,\hbar)$,
where $R\in\big{(}\widehat{\mathcal{A}}^{\hbar}\big{)}^{[\leq-1]}$ and
$c(\varepsilon,\hbar)\in\mathbb{C}[[\varepsilon,\hbar]]$.

Then, up to a triangular transformation of the form

$\displaystyle G_{\alpha,d}\mapsto
G_{\alpha,d}+\sum_{i=1}^{d+1}A^{\mu}_{\alpha,i}G_{\mu,d-i}+B_{\alpha,d}(\varepsilon,\hbar),$
$\displaystyle A^{\mu}_{\alpha,i}\in\mathbb{C},\qquad
B_{\alpha,d}\in\mathbb{C}[[\varepsilon,\hbar]],\qquad d\geq 0,\quad
1\leq\alpha\leq N,$

we have

  * $(i)$

${\overline{G}}_{1,0}=\int\big{(}\frac{1}{2}\eta_{\mu\nu}u^{\mu}u^{\nu}\big{)}dx$,

  * $(ii)$

$[{\overline{G}}_{\alpha,p},{\overline{G}}_{\beta,q}]=0$,
$\alpha,\beta=1,\dots,N$, $p,q\geq-1$,

  * $(iii)$

$\frac{1}{\hbar}[G_{\alpha,p},{\overline{G}}_{\beta,0}]={\partial}_{x}\frac{{\partial}G_{\alpha,p+1}}{{\partial}u^{\beta}}$,
$\beta=1,\dots,N$, $p\geq-1$,

  * $(iv)$

$\frac{{\partial}G_{\alpha,p}}{{\partial}u^{1}}=G_{\alpha,p-1}$,
$\alpha=1,\dots,N$, $p\geq-1$.

Since the hypothesis of the theorems above can be easily checked order by
order in $\hbar$ and $\varepsilon$, we were able to give a low order
classification of rank $1$ integrable systems of DR type. Both at the
classical and and at the quantum level it turns out that they correspond
precisely to DR hierarchies associated to rank $1$ cohomological field
theories.

######  Proposition 5.13.

Rescaling $\varepsilon^{2}\to\varepsilon^{2}\gamma$ and $\hbar\to\hbar\gamma$
to keep track of the genus, the most general rank $1$ hierarchy of DR type is
uniquely determined up to genus $4$ by the Hamiltonian

$\displaystyle{\overline{G}}_{1}=\int\Bigg{[}\frac{u^{3}}{6}+\left(\left(-\frac{\varepsilon^{2}}{24}-\frac{s_{1}}{2}i\hbar\right)u_{1}^{2}-\frac{i\hbar}{24}u\right)\gamma$
$\displaystyle\hphantom{{\overline{G}}_{1}=\int\Bigg{[}}{}+\left(\left(-\frac{s_{1}}{120}\varepsilon^{4}-\frac{s_{1}^{2}}{10}i\hbar\varepsilon^{2}-\frac{24s_{1}^{3}+5s_{2}}{60}(i\hbar)^{2}\right)u_{2}^{2}\right)\gamma^{2}$
$\displaystyle\hphantom{{\overline{G}}_{1}=\int\Bigg{[}}{}+\left(\left(-\frac{s_{1}^{3}}{360}\varepsilon^{6}-\frac{s_{2}}{1728}\varepsilon^{6}-\frac{24s_{1}^{4}+5s_{1}s_{2}}{720}i\hbar\varepsilon^{4}-\frac{4608s_{1}^{5}+2400s_{2}s_{1}^{2}+35s_{3}}{28800}(i\hbar)^{2}\varepsilon^{2}\\!\right.\right.$
$\displaystyle\left.\hphantom{{\overline{G}}_{1}=\int\Bigg{[}}{}-\frac{2304s_{1}^{6}+2400s_{2}s_{1}^{3}+105s_{3}s_{1}-500s_{2}^{2}}{7200}(i\hbar)^{3}\right)u_{2}^{3}$
$\displaystyle\hphantom{{\overline{G}}_{1}=\int\Bigg{[}}{}+\left(-\frac{s_{1}^{2}}{420}\varepsilon^{6}-\frac{96s_{1}^{3}+5s_{2}}{2520}i\hbar\varepsilon^{4}-\frac{24s_{1}^{4}+5s_{2}s_{1}}{105}(i\hbar)^{2}\varepsilon^{2}\right.$
$\displaystyle\left.\left.\hphantom{{\overline{G}}_{1}=\int\Bigg{[}}{}-\frac{4608s_{1}^{5}+2400s_{2}s_{1}^{2}+35s_{3}}{8400}(i\hbar^{3})\right)u_{3}^{2}\right)\gamma^{3}+O\big{(}\gamma^{4}\big{)}\Bigg{]}dx.$

It coincides with the DR hierarchy associated with the most general rank $1$
CohFT, the class $e^{-\sum\limits_{i\geq
1}\frac{(2i)!}{B_{2i}}s_{i}\mathrm{Ch}_{2i-1}(\mathbb{H})}\in
H^{*}(\overline{\mathcal{M}}_{g,n},{\mathbb{Q}})$, where by
$\mathrm{Ch}_{2i-1}(\mathbb{H})$ we denote the Chern characters of the Hodge
bundle on $\overline{\mathcal{M}}_{g,n}$.

Tests in rank $2$ show the emergence of classical integrable systems of more
general origin. However this was expected from geometry too. Indeed the
construction of the classical DR hierarchy also works for partial CohFTs,
i.e., CohFTs that do not satisfy the loop gluing axiom. It would appear from
computations that classical integrable systems of DR type are classified by
partial CohFTs but only those coming from actual CohFTs possess a DR type
quantization.

##  6 Examples

In this section we list and work out in detail some of the examples of quantum
integrable systems we were able to compute in [1, 2, 3, 4, 8, 9]. The explicit
formula for the Hamiltonians can either be computed from the intersection
numbers with the double ramification cycle, as the definition of the DR
hierarchy prescribes, or by imposing that ${\overline{G}}_{1,1}$ satisfies the
hypothesis of Theorem 5.12, together with homogeneity with respect to grading
of the variables, when applicable.

###  6.1 Korteweg–de Vries

The KdV hierarchy is the DR hierarchy of the trivial CohFT, with $V$ one-
dimensional and generated by $e_{1}$, so we can suppress the corresponding
index $\alpha=1$, $\eta_{11}=\eta=1$ and $c_{g,n}(\otimes_{i=1}^{n}e_{1})=1\in
H^{0}(\overline{\mathcal{M}}_{g,n},{\mathbb{Q}})$. It is uniquely determined
by the Hamiltonian [8]

$\displaystyle{\overline{G}}_{1}=\int\left(\frac{u^{3}}{6}+\frac{\varepsilon^{2}}{24}uu_{xx}-\frac{i\hbar}{24}u\right)dx.$

In [8] we found a closed form for the generating series
$G(z)=\sum\limits_{d\geq-1}z^{d}G_{d}$ for the densities of its symmetries
$G_{d}$ produced by the recursion (5.5) in the dispersionless limit
$\varepsilon=0$,

$\displaystyle G(z)|_{\varepsilon=0}=\frac{1}{z^{2}S(\sqrt{i}\lambda
z)}e^{zS\big{(}\frac{\lambda}{\sqrt{i}}z{\partial}_{x}\big{)}u}-z^{-2},$

where

$\displaystyle
S(y)=\frac{e^{\frac{y}{2}}-e^{-\frac{y}{2}}}{y},\qquad\lambda^{2}=\hbar.$

###  6.2 Intermediate long wave

The full Chern class of the Hodge bundle
$c_{g,n}(\otimes_{i=1}^{n}e_{1})=\Lambda(\mu)=\sum\limits_{j=1}^{g}\mu^{j}\lambda_{j}\in
H^{*}(\overline{\mathcal{M}}_{g,n},{\mathbb{Q}})$ is a mixed degree
deformation with parameter $\mu$ of the trivial CohFT, defined on the same $V$
with the same metric. The corresponding hierarchy is uniquely determined by
the Hamiltonian [8]

$\displaystyle{\overline{G}}_{1}=\int\left(\frac{u^{3}}{6}+\sum_{g\geq
1}\varepsilon^{2g}\mu^{g-1}\frac{|B_{2g}|}{2(2g)!}uu_{2g}-\frac{i\hbar}{24}u-i\hbar\sum_{g\geq
1}\varepsilon^{2g-2}\mu^{g}\frac{|B_{2g}|}{2(2g)!}uu_{2g}\right)dx,$

where $B_{2g}$ are Bernoulli numbers: $B_{0}=1$, $B_{2}=\frac{1}{6}$,
$B_{4}=-\frac{1}{30}$, $\dots$.

At the classical limit $\hbar=0$ we also have
$\overline{g}_{0}=\int\frac{u^{2}}{2}dx$, therefore $h_{-1}^{\mathrm{DR}}=u$,
so we see that the coordinate $u$ is normal. In [1] it is proved that the
Miura transformation

$\displaystyle u\mapsto w(u)=u+\sum_{g\geq
1}\frac{2^{2g-1}-1}{2^{2g-1}}\frac{|B_{2g}|}{(2g)!}\varepsilon^{2g}\mu^{g}u_{2g}$
(6.1)

maps the Hamiltonians and the Hamiltonian operator of this DR hierarchy to the
Hamiltonians and the Hamiltonian operator of the Dubrovin–Zhang hierarchy. It
is easy to see that the transformation (6.1) has the form (2.2) if we put

$\displaystyle\mathcal{F}=\sum_{g\geq
1}\varepsilon^{2g}\frac{2^{2g-1}-1}{2^{2g-1}}\frac{|B_{2g}|}{(2g)!}\mu^{g}u_{2g-2}.$

In particular, the standard Hamiltonian operator $\partial_{x}$ is transformed
to the Hamiltonian operator

$\displaystyle K=\partial_{x}+\sum_{g\geq
1}\varepsilon^{2g}\mu^{g}\frac{(2g-1)|B_{2g}|}{(2g)!}\partial_{x}^{2g+1}.$

In [1] it is also explained how this DR hierarchy is related to the hierarchy
of the conserved quantities of the intermediate long wave (ILW) equation (see,
e.g., [35]):

$\displaystyle w_{\tau}+2ww_{x}+T(w_{xx})=0,$ $\displaystyle
T(f):=\mathrm{p.v.}\int_{-\infty}^{+\infty}\frac{1}{2\delta}\left(\mathrm{sgn}(x-\xi)-\coth\frac{\pi(x-\xi)}{2\delta}\right)f(\xi)d\xi.$

The ILW equation can be transformed into the first equation of our DR
hierarchy by setting $w=\frac{\sqrt{\mu}}{\varepsilon}u$,
$\tau=-\frac{1}{2}\frac{\varepsilon}{\sqrt{\mu}}t_{1}$,
$\delta=\frac{\varepsilon\sqrt{\mu}}{2}$ (indeed $T(f)=\sum\limits_{n\geq
1}\delta^{2n-1}2^{2n}\frac{|B_{2n}|}{(2n)!}\partial_{x}^{2n-1}f$).

This means that our methods give a way to determine the symmetries of the ILW
equation (alternative to [35]) and its quantization.

###  6.3 Extended Toda

Consider the cohomological field theory associated to the Gromov–Witten theory
of ${{\mathbb{C}}{\mathbb{P}}}^{1}$. We have
$V=H^{*}({{\mathbb{C}}{\mathbb{P}}}^{1},\mathbb{C})=\left<1,\omega\right>$,
where $1$ and $\omega$ is the unit and the class dual to a point respectively.
The matrix of the metric in this basis is given by

$\displaystyle\eta_{11}=\eta_{\omega\omega}=0,\qquad\eta_{1\omega}=\eta_{\omega
1}=1.$

The DR hierarchy is uniquely determined by

$\displaystyle{\overline{G}}_{1,1}=\int\left(\frac{(u^{1})^{2}u^{\omega}}{2}+\sum_{g\geq
1}\varepsilon^{2g}\frac{B_{2g}}{(2g)!}u^{1}u^{1}_{2g}+q\left(\frac{e^{\frac{\varepsilon{\partial}_{x}}{2}}+e^{-\frac{\varepsilon{\partial}_{x}}{2}}}{2}u^{\omega}-2\right)e^{S(\varepsilon{\partial}_{x})u^{\omega}}+qu^{\omega}\right.$
$\displaystyle\left.\hphantom{{\overline{G}}_{1,1}=\int}{}-\frac{i\hbar}{12}u^{1}+i\hbar\sum_{g\geq
1}\varepsilon^{2g-2}\frac{B_{2g}}{(2g)!}u^{\omega}_{2g}u^{1}\right)dx,$

where $q$ can be considered as a parameter (it is in fact the formal variable
keeping track of the degree of the covers of $\mathbb{P}^{1}$ enumerated by
this Gromov–Witten theory).

At the classical level we have

$\displaystyle\overline{g}_{1,0}=\int
u^{1}u^{\omega}dx\qquad\text{and}\qquad\overline{g}_{\omega,0}=\int\left(\frac{(u^{1})^{2}}{2}+q\left(e^{S(\varepsilon{\partial}_{x})u^{\omega}}-u^{\omega}\right)\right)dx,$

where $S(z):=\frac{e^{\frac{z}{2}}-e^{-\frac{z}{2}}}{z}$. Therefore,
$h^{\mathrm{DR}}_{1,-1}=u^{\omega}$ and $h^{\mathrm{DR}}_{\omega,-1}=u^{1}$.
Thus, the coordinates $u^{\alpha}$ are normal,
${\widetilde{u}}^{\alpha}=u^{\alpha}$. In [9] we proved that the Miura
transformation

$\displaystyle u^{\alpha}\mapsto
w^{\alpha}(u)=\frac{\varepsilon{\partial}_{x}}{e^{\frac{\varepsilon{\partial}_{x}}{2}}-e^{-\frac{\varepsilon{\partial}_{x}}{2}}}u^{\alpha}=u^{\alpha}+\sum_{g\geq
1}\varepsilon^{2g}\frac{1-2^{2g-1}}{2^{2g-1}}\frac{B_{2g}}{(2g)!}u^{\alpha}_{2g}$
(6.2)

maps the Hamiltonians and the Hamiltonian operator of the double ramification
hierarchy to the Hamiltonians and the Hamiltonian operator of the
Dubrovin–Zhang hierarchy. It is easy to see that the transformation (6.2) has
the form (2.2) if we put

$\displaystyle\mathcal{F}=\sum_{g\geq
1}\varepsilon^{2g}\frac{1-2^{2g-1}}{2^{2g-1}}\frac{B_{2g}}{(2g)!}u^{\omega}_{2g-2}.$

The relation with the extended Toda hierarchy follows, at this point, from a
result of [16]. Indeed, consider formal loop space variables $v^{1}$, $v^{2}$
and the formal series

$\displaystyle
a=\sum_{k\in\mathbb{Z}}a_{k}(v^{*}_{*};\varepsilon;q)e^{k\varepsilon{\partial}_{x}},\qquad
a_{k}\in\widehat{\mathcal{A}}\otimes\mathbb{C}\big{[}q,q^{-1}\big{]},$

let $a_{+}:=\sum\limits_{k\geq 0}a_{k}e^{k\varepsilon{\partial}_{x}}$ and
$\operatorname{Res}(a):=a_{0}$. Consider the operator

$\displaystyle
L=e^{\varepsilon{\partial}_{x}}+v^{1}+qe^{v^{2}}e^{-\varepsilon{\partial}_{x}}.$

The equations of the extended Toda hierarchy look as follows

$\displaystyle\frac{{\partial}L}{{\partial}t^{1}_{p}}=\varepsilon^{-1}\frac{2}{p!}\big{[}\big{(}L^{p}(\log
L-H_{p})\big{)}_{+},L\big{]},$
$\displaystyle\frac{{\partial}L}{{\partial}t^{\omega}_{p}}=\varepsilon^{-1}\frac{1}{(p+1)!}\big{[}\big{(}L^{p+1}\big{)}_{+},L\big{]}.$

We refer the reader to [16] for the precise definition of the logarithm $\log
L$. The Hamiltonian structure of the extended Toda hierarchy is given by the
operator

$\displaystyle K^{\rm
Td}=\begin{pmatrix}0&\varepsilon^{-1}\big{(}e^{\varepsilon{\partial}_{x}}-1\big{)}\\\
\varepsilon^{-1}\big{(}1-e^{-\varepsilon{\partial}_{x}}\big{)}&0\end{pmatrix},$

and the Hamiltonians

$\displaystyle\overline{h}^{\rm
Td}_{1,p}[v]=\int\left(\frac{2}{(p+1)!}\operatorname{Res}\big{(}L^{p+1}(\log
L-H_{p+1})\big{)}\right)dx,$ $\displaystyle\overline{h}^{\rm
Td}_{\omega,p}[v]=\int\left(\frac{1}{(p+2)!}\operatorname{Res}\big{(}L^{p+2}\big{)}\right)dx.$

So the equations of the extended Toda hierarchy can be written as follows

$\displaystyle\frac{{\partial}v^{\alpha}}{{\partial}t^{\beta}_{p}}=\big{(}K^{\rm
Td}\big{)}^{\alpha\mu}\frac{\delta\overline{h}^{\rm Td}_{\beta,p}[v]}{\delta
v^{\mu}}.$

For $k\geq 1$, let

$\displaystyle(S_{2k-1})^{\alpha}_{\beta}=\begin{cases}\displaystyle\frac{1}{k!(k-1)!}q^{k},&\text{if
$\alpha=1$, $\beta=\omega$},\\\
\displaystyle-\frac{2H_{k-1}}{((k-1)!)^{2}}q^{k-1},&\text{if $\alpha=\omega$,
$\beta=1$},\\\ 0,&\text{otherwise},\end{cases}$
$\displaystyle(S_{2k})^{\alpha}_{\beta}=\begin{cases}\displaystyle\left(\frac{1}{(k!)^{2}}-\frac{2H_{k}}{k!(k-1)!}\right)q^{k},&\text{if
$\alpha=\beta=1$},\\\ \displaystyle\frac{1}{(k!)^{2}}q^{k},&\text{if
$\alpha=\beta=\omega$},\\\ 0,&\text{otherwise}.\end{cases}$

Here $H_{k}:=1+\frac{1}{2}+\dots+\frac{1}{k}$, if $k\geq 1$, and $H_{0}:=0$.
For convenience, let us also introduce local functionals $\overline{h}^{\rm
Td}_{\alpha,-1}[w]:=\int\eta_{\alpha\mu}w^{\mu}dx$. For the operator $S_{i}$,
denote by $S^{*}_{i}$ the adjoint operator with respect to the metric $\eta$.

######  Theorem 6.1 ([16]).

The change of coordinates

$\displaystyle
w^{1}(v)=\frac{\varepsilon{\partial}_{x}}{e^{\varepsilon{\partial}_{x}}-1}v^{1},\qquad
w^{\omega}(v)=\frac{\varepsilon^{2}{\partial}_{x}^{2}}{e^{\varepsilon{\partial}_{x}}+e^{-\varepsilon{\partial}_{x}}-2}v^{2}$

maps the Hamiltonian operator $K^{\rm Td}$ to $K^{\mathrm{DZ}}$, while for the
Hamiltonians we have

$\displaystyle\overline{h}^{\mathrm{DZ}}_{\alpha,p}[w]=\sum_{i=0}^{p+1}(-1)^{i}(S^{*}_{i})^{\mu}_{\alpha}\overline{h}^{\rm
Td}_{\mu,p-i}[v(w^{*}_{*};\varepsilon)].$

###  6.4 Gelfand–Dickey

Let $r\geq 3$ and consider the cohomological field theory formed by Witten’s
$r$-spin classes (see Section 3.2 or, e.g., [4]). In this case we have
$V=\left<e_{i}\right>_{i=1,\dots,r-1}$ and the metric is given by
$\eta_{\alpha\beta}=\delta_{\alpha+\beta,r}$. Moreover, from dimension
counting, we obtain that ${\overline{G}}^{r\text{-spin}}_{1,1}$ is a
homogeneous local functional of degree $2r+2$ with respect to the grading
$|u^{a+1}_{k}|=r-a$, $|\varepsilon|=1$, $|\hbar|=r+2$.

The following formula can be deduced from the recursion 5.3,

$\displaystyle{\widetilde{u}}^{\alpha}=h^{\mathrm{DR}}_{r-\alpha,-1}=\frac{\delta}{\delta
u^{1}}\frac{{\partial}}{{\partial}u^{r-\alpha}}(D-2)^{-1}\overline{g}_{1,1}=D^{-1}\frac{{\partial}}{{\partial}u^{r-\alpha}}\frac{\delta\overline{g}_{1,1}}{\delta
u^{1}},$ (6.3)

and will be useful in what follows.

For the $3$-spin theory we have (see [3])

$\displaystyle{\overline{G}}_{1,1}^{3\text{-spin}}=\int\Bigg{[}\left(\frac{1}{2}\big{(}u^{1}\big{)}^{2}u^{2}+\frac{\big{(}u^{2}\big{)}^{4}}{36}\right)+\left(-\frac{1}{12}\big{(}u_{1}^{1}\big{)}^{2}-\frac{1}{24}u^{2}\big{(}u_{1}^{2}\big{)}^{2}\right)\varepsilon^{2}$
$\displaystyle\hphantom{{\overline{G}}_{1,1}^{3\text{-spin}}=\int\Bigg{[}}{}+\frac{1}{432}\big{(}u_{2}^{2}\big{)}^{2}\varepsilon^{4}-\frac{i\hbar}{12}u^{1}\Bigg{]}dx.$

Therefore, for the classical limit,

$\displaystyle\frac{\delta\overline{g}^{3\text{-spin}}_{1,1}}{\delta
u^{1}}=u^{1}u^{2}+\frac{\varepsilon^{2}}{6}u^{1}_{xx}.$

Using (6.3), we can easily see that the coordinate $u^{\alpha}$ is normal,
${\widetilde{u}}^{\alpha}=u^{\alpha}$.

For the $4$-spin theory we have (see [3])

$\displaystyle{\overline{G}}_{1,1}^{4\text{-spin}}=\int\Bigg{[}\left(\frac{u^{1}\big{(}u^{2}\big{)}^{2}}{2}+\frac{\big{(}u^{1}\big{)}^{2}u^{3}}{2}+\frac{\big{(}u^{2}\big{)}^{2}\big{(}u^{3}\big{)}^{2}}{8}+\frac{\big{(}u^{3}\big{)}^{5}}{320}\right)$
$\displaystyle\hphantom{{\overline{G}}_{1,1}^{4\text{-spin}}=\int\Bigg{[}}{}+\left(-\frac{\big{(}u_{1}^{1}\big{)}^{2}}{8}-\frac{u^{3}\big{(}u_{1}^{2}\big{)}^{2}}{16}-\frac{u^{3}u_{1}^{1}u_{1}^{3}}{32}+\frac{3}{64}\big{(}u^{2}\big{)}^{2}u_{2}^{3}+\frac{1}{192}\big{(}u^{3}\big{)}^{3}u_{2}^{3}\right)\varepsilon^{2}$
$\displaystyle\hphantom{{\overline{G}}_{1,1}^{4\text{-spin}}=\int\Bigg{[}}{}+\left(\frac{1}{160}\big{(}u_{2}^{2}\big{)}^{2}+\frac{3}{640}u_{2}^{1}u_{2}^{3}+\frac{5\big{(}u^{3}\big{)}^{2}u_{4}^{3}}{4096}\right)\varepsilon^{4}-\frac{\big{(}u_{3}^{3}\big{)}{}^{2}\varepsilon^{6}}{8192}$
$\displaystyle\hphantom{{\overline{G}}_{1,1}^{4\text{-spin}}=\int\Bigg{[}}{}+\left(\frac{1}{96}\big{(}u^{3}_{1}\big{)}^{2}-\frac{1}{96}\big{(}u^{3}\big{)}^{2}-\frac{1}{8}u^{1}\right)i\hbar-\frac{1}{1280}u^{3}i\hbar\varepsilon^{2}\Bigg{]}dx.$

Therefore, for the classical limit,

$\displaystyle\frac{\delta\overline{g}^{4\text{-spin}}_{1,1}}{\delta
u^{1}}=u^{1}u^{3}+\frac{\big{(}u^{2}\big{)}^{2}}{2}+\varepsilon^{2}\left(\frac{1}{4}u^{1}_{xx}+\frac{1}{64}{\partial}_{x}^{2}\big{(}\big{(}u^{3}\big{)}^{2}\big{)}\right)+\varepsilon^{4}\frac{3}{640}u^{3}_{4},$

and the normal coordinates are given by

$\displaystyle{\widetilde{u}}^{1}=u^{1}+\frac{\varepsilon^{2}}{96}u^{3}_{xx},\qquad{\widetilde{u}}^{2}=u^{2},\qquad{\widetilde{u}}^{3}=u^{3}.$

For the $5$-spin theory we content ourselves to write the classical
Hamiltonian (see [4]),

$\displaystyle\overline{g}^{5\text{-spin}}_{1,1}=\int\Bigg{[}\frac{\big{(}u^{1}\big{)}^{2}u^{4}}{2}+u^{1}u^{2}u^{3}+\frac{\big{(}u^{2}\big{)}^{3}}{6}+\frac{\big{(}u^{3}\big{)}^{4}}{30}+\frac{u^{2}\big{(}u^{3}\big{)}^{2}u^{4}}{5}+\frac{\big{(}u^{2}\big{)}^{2}\big{(}u^{4}\big{)}^{2}}{10}$
$\displaystyle\hphantom{\overline{g}^{5\text{-spin}}_{1,1}=\int\Bigg{[}}{}+\frac{\big{(}u^{3}\big{)}^{2}\big{(}u^{4}\big{)}^{3}}{50}+\frac{\big{(}u^{4}\big{)}^{6}}{3750}+\varepsilon^{2}\left(\frac{1}{6}u^{1}u^{1}_{2}+\frac{3}{20}u^{2}u^{3}u^{3}_{2}+\frac{1}{10}u^{2}\big{(}u^{3}_{1}\big{)}^{2}+\frac{1}{20}u^{1}_{2}u^{3}u^{4}\right.$
$\displaystyle\hphantom{\overline{g}^{5\text{-spin}}_{1,1}=\int\Bigg{[}}{}+\frac{1}{10}u^{2}u^{2}_{2}u^{4}+\frac{1}{40}\big{(}u^{2}_{1}\big{)}^{2}u^{4}+\frac{1}{50}u^{2}u^{4}\big{(}u^{4}_{1}\big{)}^{2}+\frac{1}{75}u^{2}\big{(}u^{4}\big{)}^{2}u^{4}_{2}+\frac{1}{75}(u^{3})^{2}u^{4}u^{4}_{2}$
$\displaystyle\left.\hphantom{\overline{g}^{5\text{-spin}}_{1,1}=\int\Bigg{[}}{}+\frac{1}{50}u^{3}u^{3}_{2}\big{(}u^{4}\big{)}^{2}+\frac{1}{1200}\big{(}u^{4}\big{)}^{4}u^{4}_{2}\right)+\varepsilon^{4}\left(\frac{7}{600}u^{2}u^{2}_{4}+\frac{11}{900}u^{1}u^{3}_{4}+\frac{7}{1200}u^{2}u^{4}u^{4}_{4}\right.$
$\displaystyle\hphantom{\overline{g}^{5\text{-spin}}_{1,1}=\int\Bigg{[}}{}+\frac{17}{1200}u^{2}u^{4}_{1}u^{4}_{3}+\frac{71}{7200}u^{2}\big{(}u^{4}_{2}\big{)}^{2}+\frac{31}{3600}u^{3}u^{3}_{4}u^{4}+\frac{7}{450}u^{3}_{1}u^{3}_{3}u^{4}+\frac{91}{7200}\big{(}u^{3}_{2}\big{)}^{2}u^{4}$
$\displaystyle\left.\hphantom{\overline{g}^{5\text{-spin}}_{1,1}=\int\Bigg{[}}{}+\frac{13}{12000}\big{(}u^{4}_{2}\big{)}^{2}\big{(}u^{4}\big{)}^{2}+\frac{3}{4000}u^{4}_{2}\big{(}u^{4}_{1}\big{)}^{2}u^{4}\right)+\varepsilon^{6}\left(\frac{53}{108000}u^{3}u^{3}_{6}+\frac{11}{18000}u^{2}u^{4}_{6}\right.$
$\displaystyle\left.\hphantom{\overline{g}^{5\text{-spin}}_{1,1}=\int\Bigg{[}}{}+\frac{1397}{6480000}\big{(}u^{4}_{3}\big{)}^{2}u^{4}+\frac{617}{1620000}u^{4}_{4}u^{4}_{2}u^{4}\right)+\varepsilon^{8}\frac{107}{10800000}u^{4}u^{4}_{8}\Bigg{]}dx.$

Therefore, for the classical limit,

$\displaystyle\frac{\delta\overline{g}^{5\text{-spin}}_{1,1}}{\delta
u^{1}}=u^{1}u^{4}+u^{2}u^{3}+\varepsilon^{2}\left(\frac{1}{3}u^{1}_{xx}+\frac{1}{20}{\partial}_{x}^{2}\big{(}u^{3}u^{4}\big{)}\right)+\varepsilon^{4}\frac{11}{900}u^{3}_{4},$

and the normal coordinates are given by

$\displaystyle{\widetilde{u}}^{1}=u^{1}+\frac{\varepsilon^{2}}{60}u^{3}_{xx},\qquad{\widetilde{u}}^{2}=u^{2}+\frac{\varepsilon^{2}}{60}u^{4}_{xx},\qquad{\widetilde{u}}^{3}=u^{3},\qquad{\widetilde{u}}^{4}=u^{4}.$

In [4] it was proved that the Hamiltonians and the Hamiltonian operator of
these DR hierarchies in the coordinates ${\widetilde{u}}^{\alpha}$ coincide
with the Hamiltonians and the Hamiltonian operator of the corresponding
Dubrovin–Zhang hierarchies, without further normal Miura transformation, i.e.,
in our previous notations for the loop space variables of the DZ hierarchy, we
have $w^{\alpha}={\widetilde{u}}^{\alpha}$.

This is in agreement with Conjecture 5.8 as, by simple degree reasons, the
CohFT correlators
$\left<\tau_{d_{1}}(e_{\alpha_{1}})\cdots\tau_{d_{n}}(e_{\alpha_{n}})\right>_{g}$
already satisfy the selection rule of Proposition 5.6. This fact is true, more
in general, for any Fan–Jarvis–Ruan–Witten CohFT of $ADE$ type and gives a
powerful way of computing the DZ hierarchy for these CohFTs via the much more
explicit DR hierarchy construction.

The relation with the Gelfand–Dickey hierarchies is described as follows.
First let us recall the definition of the Gelfand–Dickey hierarchies. Consider
formal loop space variables $f_{0},\dots,f_{r-2}$ and let

$\displaystyle
L:=D_{x}^{r}+f_{r-2}D_{x}^{r-2}+\dots+f_{1}D_{x}+f_{0},\qquad\text{where}\quad
D_{x}:=\varepsilon{\partial}_{x}.$

The $r$-th Gelfand–Dickey hierarchy is the following system of partial
differential equations

$\displaystyle\varepsilon\frac{{\partial}L}{{\partial}T_{m}}=\big{[}\big{(}L^{m/r}\big{)}_{+},L\big{]},\qquad
m\geq 1.$ (6.4)

We immediately see that $\frac{{\partial}L}{{\partial}T_{rk}}=0$, so we can
omit the times $T_{rk}$. Since $(L^{1/r})_{+}=D_{x}$, we have
$\frac{{\partial}f_{i}}{{\partial}T_{1}}=(f_{i})_{x}$.

The Hamiltonian structure of the Gelfand–Dickey hierarchy is defined as
follows. Consider differential polynomials
$X_{0},X_{1},\dots,X_{r-2}\in\widehat{\mathcal{A}}$ in the formal loop
variables $f_{0},\dots,f_{r-2}$ and a pseudo-differential operator

$\displaystyle X:=D_{x}^{-(r-1)}\circ X_{r-2}+\dots+D_{x}^{-1}\circ X_{0}.$

It is easy to see that the positive part $[X,L]_{+}$ of the commutator has the
following form

$\displaystyle[X,L]_{+}=\sum_{0\leq\alpha,\beta\leq
r-2}\big{(}\big{(}K^{\mathrm{GD}}\big{)}^{\alpha\beta}X_{\beta}\big{)}D_{x}^{\alpha},$

where

$\displaystyle\big{(}K^{\mathrm{GD}}\big{)}^{\alpha\beta}=\sum_{i\geq
0}\big{(}K^{\mathrm{GD}}\big{)}^{\alpha\beta}_{i}{\partial}_{x}^{i},\qquad\big{(}K^{\mathrm{GD}}\big{)}^{\alpha\beta}_{i}\in\widehat{\mathcal{A}},$

are differential operators and the sum is finite. The operator
$K^{\mathrm{GD}}=\big{(}\big{(}K^{\mathrm{GD}}\big{)}^{\alpha\beta}\big{)}_{0\leq\alpha,\beta\leq
r-2}$ is Hamiltonian. Consider local functionals

$\displaystyle\overline{h}_{m}^{\mathrm{GD}}:=-\frac{r}{m+r}\int\mathop{\mathrm{res}}\nolimits
L^{(m+r)/r}dx,\qquad m\geq 1.$

We have

$\displaystyle\big{\\{}\overline{h}^{\mathrm{GD}}_{m},\overline{h}^{\mathrm{GD}}_{n}\big{\\}}_{K^{\mathrm{GD}}}=0.$

For a local functional $\overline{h}\in\widehat{\Lambda}$ define a pseudo-
differential operator $\frac{\delta\overline{h}}{\delta L}$ by

$\displaystyle\frac{\delta\overline{h}}{\delta
L}:=D_{x}^{-(r-1)}\circ\frac{\delta\overline{h}}{\delta
f_{r-2}}+\dots+D_{x}^{-1}\circ\frac{\delta\overline{h}}{\delta f_{0}}.$

Then the right-hand side of (6.4) can be written in the following way

$\displaystyle\big{[}\big{(}L^{m/r}\big{)}_{+},L\big{]}=\left[\frac{\delta\overline{h}^{\mathrm{GD}}_{m}}{\delta
L},L\right]_{+}=\sum_{0\leq\alpha,\beta\leq
r-2}\left(\big{(}K^{\mathrm{GD}}\big{)}^{\alpha\beta}\frac{\delta\overline{h}^{\mathrm{GD}}_{m}}{\delta
f_{\beta}}\right)D_{x}^{\alpha}.$

Therefore, the sequence of local functionals $\overline{h}^{\mathrm{GD}}_{m}$
together with the Hamiltonian operator $K^{\mathrm{GD}}$ define a Hamiltonian
structure of the Gelfand–Dickey hierarchy (6.4).

The DZ hierarchy is related to the Gelfand–Dickey hierarchy as follows.
Introduce the Miura transformation

$\displaystyle{\widetilde{u}}^{\alpha}=\frac{1}{(r-\alpha)(-r)^{\frac{r-\alpha-1}{2}}}\mathop{\mathrm{res}}\nolimits
L^{(r-\alpha)/r}.$

Then we have

$\displaystyle
K^{\mathrm{DZ}}:=(-r)^{\frac{r}{2}}K^{\mathrm{GD}}_{\widetilde{u}},$
$\displaystyle\overline{h}_{\alpha,d}^{\mathrm{DZ}}:=\frac{1}{(-r)^{\frac{r+k-1}{2}-d}k!_{r}}\overline{h}_{k}^{\mathrm{GD}}[v({\widetilde{u}}^{*}_{*};\varepsilon)],$

where $k:=\alpha+rd$ and $k!_{r}:=\prod\limits_{i=0}^{d}(\alpha+ri)$.

### Acknowledgements

This paper originates in part from my Habilitation mémoire [34] and in part
from an introductory talk I gave at the RAQIS’16 conference held at Geneva,
Switzerland, in August 2016. I would like to express my gratitude to its
organizers. Moreover I would like to thank my direct collaborators on the DR
hierarchy project, A. Buryak, B. Dubrovin and J. Guéré, and the people who
supported us with advice and insight, among the others Dimitri Zvonkine, Rahul
Pandharipande and Yakov Eliashberg. During this work I was partially supported
by a Chaire CNRS/Enseignement superieur 2012–2017 grant.

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