2.1 Double ramification cycles

For a pair of nonnegative integers $(g,n)$ in the stable range (i.e. satisfying $2g+2-n>0$ ), let ${\overline {\mathcal {M}}}_{g,n}$ be the moduli space of stable algebraic curves of genus g with n marked points labeled by the set $[n]:=\{1,\ldots ,n\}$ . Denote by $\psi _i\in H^2({\overline {\mathcal {M}}}_{g,n})$ the first Chern class of the line bundle $\mathcal {L}_i$ over ${\overline {\mathcal {M}}}_{g,n}$ formed by the cotangent lines at the i-th marked point on stable curves. The classes $\psi _i$ are called the psi classes. Denote by $\mathbb E$ the rank g Hodge vector bundle over ${\overline {\mathcal {M}}}_{g,n}$ whose fibers are the spaces of holomorphic one-forms on stable curves. Let $\lambda _j:=c_j(\mathbb E)\in H^{2j}({\overline {\mathcal {M}}}_{g,n})$ . Let ${\mathcal {M}}_{g,n}\subset {\overline {\mathcal {M}}}_{g,n}$ be the moduli space of smooth pointed curves and denote by ${\mathcal {M}}_{g,n}^{\mathrm {ct}}\subset {\overline {\mathcal {M}}}_{g,n}$ the locus of stable curves with no nonseparating nodes.

Consider an n-tuple of integers $A=(a_1,\ldots ,a_n)$ , such that $\sum a_i=0$ , it will be called a vector of double ramification data. Suppose first that not all the numbers $a_i$ are equal to zero. Let:

$$ \begin{align*}\mathcal{Z}_g(A)\subset{\mathcal{M}}_{g,n} \end{align*} $$

be the locus parameterising the isomorphism classes of pointed smooth curves $(C;p_1,\ldots ,p_n)$ satisfying the condition $\mathcal {O}_C(\sum _{i=1}^n a_ip_i)\cong \mathcal {O}_C$ , which is algebraic and defines $\mathcal {Z}_g(A)$ canonically as a substack of ${\mathcal {M}}_{g,n}$ of dimension $2g-3+n$ . Naively, the double ramification cycle $\mathrm {DR}_g(A)$ is defined as the cohomology class on ${\overline {\mathcal {M}}}_{g,n}$ that is Poincaré dual to a compactification of $\mathcal {Z}(A)$ in ${\overline {\mathcal {M}}}_{g,n}$ . A rigorous definition is the following (see, e.g. [Reference Janda, Pandharipande, Pixton and Zvonkine16]).

The positive parts of A define a partition $\mu =(\mu _1,\ldots ,\mu _{l(\mu )})$ . The negative parts of A define a second partition $\nu =(\nu _1,\ldots ,\nu _{l(\nu )})$ . Since the parts of A sum to $0$ , the partitions $\mu $ and $\nu $ must be of the same size. We now allow the case $|\mu |=|\nu |=0$ . Let $n_0:=n-l(\mu )-l(\nu )$ . The moduli space:

$$ \begin{align*}{\overline{\mathcal{M}}}_{g,n_0}(\mathbb P^1,\mu,\nu)^{\sim} \end{align*} $$

parameterises stable relative maps of connected algebraic curves of genus g to rubber $\mathbb P^1$ with ramification profiles $\mu ,\nu $ over the points $0,\infty \in \mathbb P^1$ , respectively. There is a natural map:

$$ \begin{align*}\mathrm{st}\colon {\overline{\mathcal{M}}}_{g,n_0}(\mathbb P^1,\mu,\nu)^{\sim}\to{\overline{\mathcal{M}}}_{g,n} \end{align*} $$

forgetting everything except the marked domain curve. The moduli space ${\overline {\mathcal {M}}}_{g,n_0}(\mathbb P^1,\mu ,\nu )^{\sim }$ possesses a virtual fundamental class $\left [{\overline {\mathcal {M}}}_{g,n_0}(\mathbb P^1,\mu ,\nu )^{\sim }\right ]^{\mathrm {vir}}$ , which is a homology class of degree $2(2g-3+n)$ . The double ramification cycle:

$$ \begin{align*}\mathrm{DR}_g(A)\in H^{2g}({\overline{\mathcal{M}}}_{g,n}) \end{align*} $$

is defined as the Poincaré dual to the push-forward $\mathrm {st}_*\left [{\overline {\mathcal {M}}}_{g,n_0}(\mathbb P^1,\mu ,\nu )^{\sim }\right ]^{\mathrm {vir}}\in H_{2(2g-3+n)}({\overline {\mathcal {M}}}_{g,n})$ .

Let us list some properties of the double ramification cycles (see, e.g. [Reference Janda, Pandharipande, Pixton and Zvonkine16]). In genus $0$ , we have:

$$ \begin{align*}\mathrm{DR}_0(A)=1\in H^0({\overline{\mathcal{M}}}_{g,n}). \end{align*} $$

If all the numbers $a_i$ are equal to zero, then we have:

$$ \begin{align*}\mathrm{DR}_g(0,\ldots,0)=(-1)^g\lambda_g\in H^{2g}({\overline{\mathcal{M}}}_{g,n}). \end{align*} $$

There is a very simple explicit formula for the restriction of the double ramification cycle to the moduli space ${\mathcal {M}}_{g,n}^{\mathrm {ct}}$ . For $J\subset [n]$ and $0\leq h\leq g$ in the stable range $2h-1+|J|>0$ and $2(g-h)-1+(n-|J|)>0$ , denote by $\delta ^J_h \in H^2({\overline {\mathcal {M}}}_{g,n})$ the Poincaré dual to the substack of ${\overline {\mathcal {M}}}_{g,n}$ formed by stable curves with a separating node at which two stable components meet, one of genus h and with marked points labeled by $|J|$ , and the other of genus $g-h$ and with marked points labeled by the complement $[n]\backslash J$ . We adopt the convention $\delta ^J_h:=0$ if at least one of the stability conditions $2h-1+|J|>0$ and $2(g-h)-1+(n-|J|)>0$ is not satisfied. Let $a_J:=\sum _{j\in J}a_j$ . Introduce a degree $2$ cohomology class $\theta _g(A)$ on ${\overline {\mathcal {M}}}_{g,n}$ by:

$$ \begin{align*}\theta_g(A):=\sum_{j=1}^n\frac{a_j^2\psi_j}{2}-\frac{1}{4}\sum_{h=0}^g\ \, \sum_{J\subset [n]}a_J^2\delta_h^J\in H^2({\overline{\mathcal{M}}}_{g,n}). \end{align*} $$

Then we have the formula:

(2.1) $$ \begin{align} \left.\mathrm{DR}_g(A)\right|{}_{{\mathcal{M}}_{g,n}^{\mathrm{ct}}}=\left.\frac{1}{g!}\theta_g(A)^g\right|{}_{{\mathcal{M}}_{g,n}^{\mathrm{ct}}}, \end{align} $$

which is called Hain’s formula. More properties of the double ramification cycles will be presented in Section 3.

2.2 The noncommutative KdV hierarchy

The classical construction of the KdV hierarchy as the system of Lax equations (see, e.g. [Reference Dickey9]):

$$ \begin{align*}\frac{{\partial} L}{{\partial} t_n}=\frac{\varepsilon^{2n}}{(2n+1)!!}\left[\left(L^{n+1/2}\right)_+,L\right],\quad n\ge 1, \end{align*} $$

where $L:={\partial }_x^2+2\varepsilon ^{-2}u$ , u is a function of $x,t_1,t_2,\ldots, \varepsilon $ is a formal parameter and $(2n+1)!!:=(2n+1)\cdot (2n-1)\cdots 3\cdot 1$ admits generalisations, called noncommutative KdV hierarchies, where one doesn’t have the pairwise commutativity of the x-derivatives of the dependent variable u. In what follows, we will work with a specific example from the class of noncommutative KdV hierarchies.

Let $u_{k_1,k_2}$ , $k_1,k_2\in \mathbb Z_{\ge 0}$ , $\varepsilon $ and $\mu $ be formal variables and consider the space $\widehat {\mathcal A}:=\mathbb C[[u_{*,*},\varepsilon ,\mu ]]$ , whose elements will be called differential polynomials in two space variables. Consider a gradation on $\widehat {\mathcal A}$ given by:

$$ \begin{align*}\deg u_{k_1,k_2}:=(k_1,k_2),\qquad \deg\varepsilon:=(-1,0),\qquad \deg\mu:=(0,-1). \end{align*} $$

We will denote by $\widehat {\mathcal A}^{[(d_1,d_2)]}\subset \widehat {\mathcal A}$ the space of differential polynomials of degree $(d_1,d_2)$ . The space $\widehat {\mathcal A}$ is endowed with operators ${\partial }_x$ and ${\partial }_y$ of degrees $(1,0)$ and $(0,1)$ , respectively, defined by:

$$ \begin{align*} {\partial}_x:=\sum_{k_1,k_2\geq 0} u_{k_1+1,k_2} \frac{{\partial}}{{\partial} u_{k_1,k_2}},\qquad {\partial}_y:=\sum_{k_1,k_2\geq 0} u_{k_1,k_2+1} \frac{{\partial}}{{\partial} u_{k_1,k_2}}. \end{align*} $$

We see that $u_{k_1,k_2}={\partial }_x^{k_1}{\partial }_y^{k_2}u$ . We will denote $u_{0,0}$ simply by u.

The algebra $\widehat {\mathcal A}$ is also endowed with the Moyal star product defined by:

(2.2) $$ \begin{align} f*g:=f\ \exp\left(\frac{i \varepsilon\mu}{2}(\overleftarrow{{\partial}_x} \overrightarrow{{\partial}_y}-\overleftarrow{{\partial}_y} \overrightarrow{{\partial}_x})\right) g =\sum_{n\ge 0}\ \, \sum_{k_1+k_2=n}\frac{(-1)^{k_2}(i\varepsilon\mu)^n}{2^n k_1!k_2!}({\partial}_x^{k_1} {\partial}_y^{k_2} f) ({\partial}_x^{k_2}{\partial}_y^{k_1} g), \end{align} $$

where $f,g \in \mathbb C[[u_{*,*},\varepsilon ,\mu ]]$ . The Moyal star product is associative, and it is graded: if $\deg f = (i_1,i_2)$ and $\deg g = (j_1,j_2)$ , then $\deg {(f * g)} = (i_1+j_1,i_2+j_2)$ . Note also that when $\mu =0$ , the Moyal star product becomes the usual multiplication:

(2.3) $$ \begin{align} \left.(f*g)\right|{}_{\mu=0}=f|_{\mu=0}\cdot g|_{\mu=0}. \end{align} $$

Let us now consider the algebra of pseudo-differential operators of the form:

(2.4) $$ \begin{align} A=\sum_{i\le n}a_i*{\partial}_x^i,\quad n\in\mathbb Z,\quad a_i\in\mathbb C[[u_{*,*},\mu]][[\varepsilon,\varepsilon^{-1}], \end{align} $$

with the multiplication $\circ $ given by:

$$ \begin{align*}(a*{\partial}_x^i)\circ(b*{\partial}_x^j):=\sum_{k\ge 0}{i\choose k}(a*{\partial}_x^k b)*{\partial}_x^{i+j-k},\quad a,b\in \mathbb C[[u_{*,*},\mu]][[\varepsilon,\varepsilon^{-1}],\quad i,j\in\mathbb Z. \end{align*} $$

The positive part of a pseudo-differential operator (2.4) is defined by $A_+:=\sum _{0\le i\le n}a_i*{\partial }_x^i$ , and, as in the classical theory of pseudo-differential operators, a pseudo-differential operator A of the form ${\partial }_x^2+\sum _{i<2}a_i*{\partial }_x^i$ has a unique square root of the form ${\partial }_x+\sum _{i<1}b_i*{\partial }_x^i$ , which we denote by $A^{\frac {1}{2}}$ .

Consider the operator $L:={\partial }_x^2+2\varepsilon ^{-2}u$ . The noncommutative KdV hierarchy with respect to the Moyal star product (2.2) is defined by (see, e.g. [Reference Dimakis and Müller-Hoissen10, Reference Hamanaka15]):

(2.5) $$ \begin{align} \frac{{\partial} L}{{\partial} t_n}=\frac{\varepsilon^{2n}}{(2n+1)!!}\left[\left(L^{n+1/2}\right)_+,L\right],\quad n\ge 1. \end{align} $$

The ncKdV hierarchy is integrable in the sense that its flows pairwise commute. Explicitly, the first two equations of the hierarchy are:

$$ \begin{align*} \frac{{\partial} u}{{\partial} t_1}&={\partial}_x\left(\frac{u*u}{2}+\frac{\varepsilon^2}{12}u_{xx}\right),\\ \frac{{\partial} u}{{\partial} t_2}&={\partial}_x\left(\frac{u*u*u}{6}+\frac{\varepsilon^2}{24}(u*u_{xx}+u_x*u_x+u_{xx}*u)+\frac{\varepsilon^4}{240}u_{xxxx}\right). \end{align*} $$

For any $n\ge 1$ , the right-hand side of (2.5) has the form ${\partial }_x P_n$ , where $P_n\in \widehat {\mathcal A}^{[(0,0)]}$ . Moreover, $P_n=\sum _{g=0}^n P_{n,g}$ , where $P_{n,g}$ is a linear combination of the monomials $\varepsilon ^{2g}u_{d_1}*\cdots * u_{d_{n+1-g}}$ with $d_1+\cdots +d_{n+1-g}=2g$ . The leading term $P_{n,0}$ is equal to $\frac {u^{*(n+1)}}{(n+1)!}$ . The hierarchy:

(2.6) $$ \begin{align} \frac{{\partial} u}{{\partial} t_n}={\partial}_x\left(\frac{u^{*(n+1)}}{(n+1)!}\right),\quad n\ge 1, \end{align} $$

will be called the dispersionless noncommutative KdV (dncKdV) hierarchy.

Note that because of (2.3), the noncommutative KdV hierarchy becomes the classical KdV hierarchy when $\mu =0$ .

We are now ready to present our first conjecture. Let us introduce formal variables $t^a_d$ , $a\in \mathbb Z$ , $d\ge 0$ and consider the generating function:

$$ \begin{align*} \mathcal{F}^{\mathrm{DR}}(t^*_*,\varepsilon)&:=\sum_{\substack{g,n\ge 0\\2g-2+n>0}}\frac{\varepsilon^{2g}}{n!}\sum_{\substack{A=(a_1,\ldots,a_n)\in\mathbb Z^n\\\sum a_i=0}} \\[6pt] &\quad \sum_{d_1,\ldots,d_n\ge 0}\left(\int_{{\overline{\mathcal{M}}}_{g,n}}\mathrm{DR}_g(A)\prod_{i=1}^n\psi_i^{d_i}\right)\prod_{i=1}^n t^{a_i}_{d_i}\in\mathbb C[[t^*_*,\varepsilon]]. \end{align*} $$

Introduce a formal power series:

$$ \begin{align*}S(z):=\frac{e^{z/2}-e^{-z/2}}{z}=1+\frac{z^2}{24}+\frac{z^4}{1920}+O(z^6), \end{align*} $$

and let:

(2.7) $$ \begin{align} &(w^{\mathrm{DR}})^a:=\frac{{\partial}^2\mathcal{F}^{\mathrm{DR}}}{{\partial} t^0_0{\partial} t^{-a}_0}\in\mathbb C[[t^*_*,\varepsilon]],\quad a\in\mathbb Z,\notag\\[6pt] &w^{\mathrm{DR}}:=\sum_{a\in\mathbb Z}(w^{\mathrm{DR}})^a e^{iay}\in\mathbb C[[t^*_*,\varepsilon]][[e^{iy},e^{-iy}]],\notag\\[6pt] &u^{\mathrm{DR}}:=\frac{S(\varepsilon{\partial}_x)}{S(i\varepsilon{\partial}_x{\partial}_y)}w^{\mathrm{DR}}\in\mathbb C[[t^*_*,\varepsilon]][[e^{iy},e^{-iy}]]. \end{align} $$

Let us analyse the system of equations that this conjecture gives for the generating series $\mathcal {F}^{\mathrm {DR}}$ in a bit more detail. For any $a\in \mathbb Z$ , introduce a formal power series:

(2.8) $$ \begin{align} T(a,z):=\frac{S(z)}{S(az)}=1+\frac{1-a^2}{24}z^2+\frac{3-10a^2+7a^4}{5760}z^4+O(z^6)=\sum_{g\ge 0}Q_g(a)z^{2g}. \end{align} $$

Here, $Q_g(a)$ are polynomials in a. If we decompose:

(2.9) $$ \begin{align} u^{\mathrm{DR}}=\sum_{a\in\mathbb Z}(u^{\mathrm{DR}})^a e^{iay}, \end{align} $$

then the transformation (2.7) simply means that:

$$ \begin{align*}(u^{\mathrm{DR}})^a=(w^{\mathrm{DR}})^a+\sum_{g\ge 1}\varepsilon^{2g}Q_g(a){\partial}_x^{2g}(w^{\mathrm{DR}})^a,\quad a\in\mathbb Z. \end{align*} $$

Also, using the decomposition (2.9), we can rewrite the equations of the dncKdV hierarchy as a system of evolutionary PDEs with one spatial variable x and infinitely many times $t_d$ , $d\ge 1$ , for the functions $(u^{\mathrm {DR}})^a$ , $a\in \mathbb Z$ . For example, the first equation of the dncKdV hierarchy, $\frac {{\partial } u}{{\partial } t_1}={\partial }_x\left (\frac {u*u}{2}\right )$ , via Conjecture 1, gives the following PDEs for the functions $(u^{\mathrm {DR}})^a$ :

$$ \begin{align*}\frac{{\partial} (u^{\mathrm{DR}})^a}{{\partial} t_1}=\sum_{g\ge 0}\frac{\varepsilon^{2g}}{2^{2g}}\sum_{\substack{a_1,a_2\in\mathbb Z\\a_1+a_2=a}}\ \, \sum_{\substack{k_1,k_2\ge 0\\k_1+k_2=2g}}\frac{(-1)^{k_1}}{k_1!k_2!}a_1^{k_2}a_2^{k_1}{\partial}_x\left({\partial}_x^{k_1}(u^{\mathrm{DR}})^{a_1}{\partial}_x^{k_2}(u^{\mathrm{DR}})^{a_2}\right),\quad a\in\mathbb Z. \end{align*} $$

4.1 Partial cohomological field theories

Recall the following definition, which is a generalisation first considered in [Reference Liu, Ruan and Zhang19] of the notion of a cohomological field theory from [Reference Kontsevich and Manin18].

Definition 4.1. A partial cohomological field theory (CohFT) is a system of linear maps:

$$ \begin{align*} c_{g,n}\colon V^{\otimes n} \to H^{\mathrm{even}}({\overline{\mathcal{M}}}_{g,n}), \end{align*} $$

for $(g,n)$ in the stable range, where V is an arbitrary finite dimensional $\mathbb C$ -vector space called the phase space, together with a special element $e\in V$ called the unit and a symmetric nondegenerate bilinear form $\eta \in (V^*)^{\otimes 2}$ called the metric, such that, fixing a basis $e_1,\ldots ,e_{\dim V}$ in V, the following axioms are satisfied:

1. (i) The maps $c_{g,n}$ are equivariant with respect to the $S_n$ -action permuting the n copies of V in $V^{\otimes n}$ and the n marked points in ${\overline {\mathcal {M}}}_{g,n}$ , respectively.

2. (ii) Let $\pi \colon {\overline {\mathcal {M}}}_{g,n+1}\to {\overline {\mathcal {M}}}_{g,n}$ be the map that forgets the last marked point. Then

   $$ \begin{align*}\pi^* c_{g,n}( \otimes_{i=1}^n e_{\alpha_i}) = c_{g,n+1}(\otimes_{i=1}^n e_{\alpha_i}\otimes e),\quad 1 \leq\alpha_1,\ldots,\alpha_n\leq \dim V. \end{align*} $$
   Moreover, $c_{0,3}(e_{\alpha }\otimes e_{\beta } \otimes e) =\eta (e_{\alpha }\otimes e_{\beta }) =:\eta _{\alpha \beta }$ for $1\leq \alpha ,\beta \leq \dim V$ .

3. (iii) For decompositions $I \sqcup J = [n]$ , $|I|=n_1$ , $|J|=n_2$ and $g_1+g_2=g$ with ${2g_1-1+n_1>0}$ , $2g_2-1+n_2>0$ , let:

   (4.1) $$ \begin{align} \mathrm{gl}\colon{\overline{\mathcal{M}}}_{g_1,n_1+1}\times{\overline{\mathcal{M}}}_{g_2,n_2+1}\to {\overline{\mathcal{M}}}_{g_1+g_2,n_1+n_2} \end{align} $$
   be the corresponding gluing map. Then:
   (4.2) $$ \begin{align} \mathrm{gl}^* c_{g,n}( \otimes_{i=1}^n e_{\alpha_i}) &= c_{g_1,n_1+1}(\otimes_{i\in I} e_{\alpha_i} \otimes e_{\mu})\eta^{\mu \nu} c_{g_2,n_2+1}( \otimes_{j\in J} e_{\alpha_j}\otimes e_{\nu}), \nonumber \\ 1 & \leq\alpha_1,\ldots,\alpha_n\leq \dim V, \end{align} $$
   where $\eta ^{\alpha \beta }$ are the entries of the matrix $(\eta _{\alpha \beta })^{-1}$ .

Note that a notion of infinite rank partial CohFT (i.e. a partial CohFT with an infinite dimensional phase space V) requires some care. One needs to clarify what is meant by the matrix $(\eta ^{\alpha \beta })$ and to make sense of the, a priori infinite, sum over $\mu $ and $\nu $ , both appearing in Axiom (iii). One possibility is demanding that the image of the linear map $V^{\otimes (n-1)}\to H^*({\overline {\mathcal {M}}}_{g,n}) \otimes V^*$ induced by $c_{g,n}\colon V^{\otimes n}\to H^*({\overline {\mathcal {M}}}_{g,n})$ is contained in $H^*({\overline {\mathcal {M}}}_{g,n}) \otimes \eta ^{\sharp }(V)$ , where $\eta ^{\sharp }\colon V\to V^*$ is the injective map induced by the bilinear form $\eta $ . Then in Axiom (iii), instead of using an undefined bilinear form $(\eta ^{\alpha \beta })$ on $V^*$ , one can use the bilinear form on $\eta ^{\sharp }(V)$ induced by $\eta $ . This solves the problem with convergence.

A useful special case is the following. Consider a vector space V with a countable basis $\{e_{\alpha }\}_{\alpha \in \mathbb Z}$ , and suppose that for any $(g,n)$ in the stable range and each $e_{\alpha _1},\ldots ,e_{\alpha _{n-1}} \in V$ , the set $\{\beta \in \mathbb Z\, |\, c_{g,n}(\otimes _{i=1}^{n-1} e_{\alpha _i}\otimes e_{\beta })\neq 0\}$ is finite. In particular, this implies that the matrix $\eta _{\alpha \beta }$ is row- and column-finite (each row and each column have a finite number of nonzero entries), which is equivalent to $\eta ^{\sharp }(V)\subseteq \mathrm {span}(\{e^{\alpha }\}_{\alpha \in \mathbb Z})$ , where $\{e^{\alpha }\}_{\alpha \in \mathbb Z}$ is the dual ‘basis’. Let us further demand that the injective map $\eta ^{\sharp }\colon V \to \mathrm {span}(\{e^{\alpha }\}_{\alpha \in \mathbb Z})$ is surjective too (i.e. that a unique two-sided row- and column-finite matrix $(\eta ^{\alpha \beta })$ , inverse to $(\eta _{\alpha \beta })$ , exists; it represents the inverse map $(\eta ^{\sharp })^{-1}\colon \mathrm {span}(\{e^{\alpha }\}_{\alpha \in \mathbb Z})\to V$ ). Then the equation appearing in Axiom (iii) is well defined with the double sum only having a finite number of nonzero terms. Such a partial CohFT will be called a tame partial CohFT of infinite rank.

4.2 The DR/DZ equivalence conjecture

Let us fix a positive integer N.

4.2.1 Differential polynomials

Let us introduce formal variables $u^{\alpha }_i$ , $\alpha =1,\ldots ,N$ , $i=0,1,\ldots $ . Following [Reference Dubrovin and Zhang11] (see also [Reference Rossi21]), we define the ring of differential polynomials $\mathcal A_N$ in the variables $u^1,\ldots ,u^N$ as the ring of polynomials $f(u^*,u^*_1,u^*_2,\ldots )$ in the variables $u^{\alpha }_i$ , $i>0$ , with coefficients in the ring of formal power series in the variables $u^{\alpha }=u^{\alpha }_0$ :

$$ \begin{align*}\mathcal A_N:=\mathbb C[[u^*]][u^*_{\ge 1}]. \end{align*} $$

Remark 4.3. This way, we define a model of the loop space of a vector space V of dimension N by describing its ring of functions. In particular, it is useful to think of the variables $u^{\alpha }:= u^{\alpha }_0$ as the components $u^{\alpha }(x)$ of a formal loop $u\colon S^1\to V$ in a fixed basis $e_1,\ldots ,e_N$ of V. Then the variables $u^{\alpha }_{1}:= u^{\alpha }_x, u^{\alpha }_{2}:= u^{\alpha }_{xx},\ldots $ are the components of the iterated x-derivatives of a formal loop.

A gradation on $\mathcal A_N$ , which we denote by $\deg $ , is introduced by $\deg u^{\alpha }_i:= i$ . The homogeneous component of $\mathcal A_N$ of degree d is denoted by $\mathcal A_N^{[d]}$ . The operator:

$$ \begin{align*}\partial_x := \sum_{i\geq 0} u^{\alpha}_{i+1}\frac{\partial}{\partial u^{\alpha}_i} \end{align*} $$

increases the degree by $1$ .

Differential polynomials 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 $u^{\alpha }=u^{\alpha }(x)$ . We define a change of variables:

(4.3) $$ \begin{align} u^{\alpha}_j = \sum_{k\in\mathbb Z} (i k)^j p^{\alpha}_k e^{i k x}, \end{align} $$

which allows us to express a differential polynomial $f(u,u_x,u_{xx},\ldots )\in \mathcal A_N$ as a formal Fourier series in x. In the latter expression, the coefficient of $e^{i k x}$ is a power series in the variables $p^{\alpha }_j$ with the sum of the subscripts in each monomial in $p^{\alpha }_j$ equal to k.

Consider the extension $\widehat {\mathcal A}_N:=\mathcal A_N[[\varepsilon ]]$ of the space $\mathcal A_N$ with a new variable $\varepsilon $ of degree $\deg \varepsilon :=-1$ . Abusing the terminology, we still call its elements differential polynomials. Let $\widehat {\mathcal A}_N^{[k]}\subset \widehat {\mathcal A}_N$ denote the subspace of differential polynomials of degree k.

4.2.2 The DR hierarchy of a partial CohFT

Consider an arbitrary partial CohFT:

$$ \begin{align*}c_{g,n}\colon V^{\otimes n} \to H^{\mathrm{even}}({\overline{\mathcal{M}}}_{g,n}). \end{align*} $$

Following [Reference Buryak1, Reference Buryak, Dubrovin, Guéré and Rossi2], we will present the construction of the DR hierarchy and the DR/DZ equivalence conjecture. Formally, the results presented here were obtained in [Reference Buryak1, Reference Buryak, Dubrovin, Guéré and Rossi2] for a CohFT, but, as it was already remarked in [Reference Buryak, Dubrovin, Guéré and Rossi2, Section 9.1], the construction of the DR hierarchy works without any change for an arbitrary partial CohFT, and all the results that we discuss here are true for an arbitrary partial CohFT with the same proofs.

Let $N:=\dim V$ , and let us fix a basis $e_1,\ldots ,e_N\in V$ . Introduce the following generating series:

(4.4) $$ \begin{align} P^{\alpha}_{\beta,d}&:=\sum_{\substack{g\ge 0\\n\ge 1}}\frac{(-\varepsilon^2)^g}{n!}\sum_{a_1,\ldots,a_n\in\mathbb Z}\left(\int_{\mathrm{DR}_g(-\sum_{i=1}^n a_i,0,a_1,\ldots,a_n)}\hspace{-1.2cm}\lambda_g\psi_2^d\eta^{\alpha\gamma} c_{g,n+2}(e_{\gamma}\otimes e_{\beta}\otimes_{i=1}^n e_{\alpha_i})\right) \nonumber \\[6pt] &\quad \prod_{i=1}^n p^{\alpha_i}_{a_i}e^{i(\sum_{j=1}^n a_j)x}, \end{align} $$

for $\alpha ,\beta =1,\ldots ,N$ and $d=0,1,2,\ldots $ . The expression on the right-hand side of (4.4) can be uniquely written as a differential polynomial from $\widehat {\mathcal A}^{[0]}_N$ using the change of variables (4.3). Concretely, it can be done in the following way. From Hain’s formula (2.1), it follows that the restriction $\mathrm {DR}_g\left (-\sum _{i=1}^n a_i,a_1,\ldots ,a_n\right )\big |_{{\mathcal {M}}_{g,n+1}^{\mathrm {ct}}}$ is a homogeneous polynomial in $a_1,\ldots ,a_n$ of degree $2g$ with the coefficients in $H^{2g}({\mathcal {M}}_{g,n+1}^{\mathrm {ct}})$ . This property, together with the fact that $\lambda _g$ vanishes on ${\overline {\mathcal {M}}}_{g,n}\backslash {\mathcal {M}}_{g,n}^{\mathrm {ct}}$ (see, e.g. [Reference Faber and Pandharipande12, Section 0.4]), implies that the integral:

(4.5) $$ \begin{align} \int_{\mathrm{DR}_g(-\sum_{i=1}^n a_i,0,a_1,\ldots,a_n)}\lambda_g\psi_2^d\eta^{\alpha\gamma} c_{g,n+2}(e_{\gamma}\otimes e_{\beta}\otimes_{i=1}^n e_{\alpha_i}) \end{align} $$

is a homogeneous polynomial in $a_1,\ldots ,a_n$ of degree $2g$ , which we denote by:

$$ \begin{align*} Q^{\alpha}_{\beta,d,g;\alpha_1,\ldots,\alpha_n}(a_1,\ldots,a_n)=\sum_{\substack{b_1,\ldots,b_n\ge 0\\b_1+\ldots+b_n=2g}}Q_{\beta,d,g;\alpha_1,\ldots,\alpha_n}^{\alpha;b_1,\ldots,b_n}a_1^{b_1}\ldots a_n^{b_n}. \end{align*} $$

Then we have:

$$ \begin{align*}P^{\alpha}_{\beta,d}=\sum_{\substack{g\ge 0\\n\ge 1}}\frac{\varepsilon^{2g}}{n!}\sum_{\substack{b_1,\ldots,b_n\ge 0\\b_1+\ldots+b_n=2g}}Q_{\beta,d,g;\alpha_1,\ldots,\alpha_n}^{\alpha;b_1,\ldots,b_n} u^{\alpha_1}_{b_1}\ldots u^{\alpha_n}_{b_n}. \end{align*} $$

The system of PDEs:

(4.6) $$ \begin{align} \frac{{\partial} u^{\alpha}}{{\partial} t^{\beta}_d}={\partial}_x P^{\alpha}_{\beta,d},\quad 1\le\alpha,\beta\le N,\quad d\ge 0 \end{align} $$

is called the DR hierarchy. The flows of the hierarchy pairwise commute. Let ${A^{\alpha } e_{\alpha }:=e\in V}$ . The flow $\frac {{\partial }}{{\partial } t^{1\!\! 1}}:=A^{\alpha }\frac {{\partial }}{{\partial } t^{\alpha }_0}$ is given by:

(4.7) $$ \begin{align} \frac{{\partial} u^{\alpha}}{{\partial} t^{1\!\! 1}_0}=u^{\alpha}_x. \end{align} $$

Remark 4.4. The DR hierarchy is actually Hamiltonian, and in [Reference Buryak1, Reference Buryak, Dubrovin, Guéré and Rossi2], it is introduced via a sequence of local functionals. However, since we don’t need the Hamiltonian structure in this paper, we introduce directly the equations of the DR hierarchy.

Because of (4.7), as a solution of the DR hierarchy, we can consider an N-tuple of formal power series $u^{\alpha }(t^*_*,\varepsilon )\in \mathbb C[[t^*_*,\varepsilon ]]$ , $1\le \alpha \le N$ , satisfying the system (4.6) after the identification of the flows ${\partial }_x$ and $\frac {{\partial }}{{\partial } t^{1\!\! 1}_0}$ . The string solution $(u^{\mathrm {str}})^{\alpha }(t^*_*,\varepsilon )\in \mathbb C[[t^*_*,\varepsilon ]]$ of the DR heirarchy is defined as the unique solution satisfying the initial condition:

$$ \begin{align*} \left.(u^{\mathrm{str}})^{\alpha}\right|_{t^{\gamma}_n=\delta_{n,0}A^{\gamma} x}=A^{\alpha} x. \end{align*} $$

The potential of our partial CohFT is defined by:

$$ \begin{align*}\mathcal F(t^*_*,\varepsilon):=\sum_{\substack{g,n\ge 0\\2g-2+n>0}}\frac{\varepsilon^{2g}}{n!}\sum_{\substack{1\le\alpha_1,\ldots,\alpha_n\le N\\d_1,\ldots,d_n\ge 0}}\left(\int_{{\overline{\mathcal{M}}}_{g,n}}c_{g,n}\left(\otimes_{i=1}^n e_{\alpha_i}\right)\prod_{i=1}^n\psi_i^{d_i}\right)\prod_{i=1}^n t^{a_i}_{d_i}\in\mathbb C[[t^*_*,\varepsilon]]. \end{align*} $$

The exponent $\exp \left (\varepsilon ^{-2}\mathcal F\right )$ is traditionally called the partition function.

Define:

$$ \begin{align*}(w^{\mathrm{top}})^{\alpha}_n:=\eta^{\alpha\mu}\frac{{\partial}^n}{({\partial} t^{1\!\! 1}_0)^n}\frac{{\partial}^2\mathcal F}{{\partial} t^{\mu}_0{\partial} t^{1\!\! 1}_0},\quad 1\le\alpha\le N. \end{align*} $$

In [Reference Buryak, Dubrovin, Guéré and Rossi2, Proposition 7.2], the authors proved that there exists a unique differential polynomial $\mathcal P\in \widehat {\mathcal A}^{[-2]}_N$ , such that the power series $\mathcal F^{\mathrm {red}}\in \mathbb C[[t^*_*,\varepsilon ]]$ defined by:

$$ \begin{align*} \mathcal F^{\mathrm{red}}:=\mathcal F+\left.\mathcal P\right|_{u^{\gamma}_n=(w^{\mathrm{top}})^{\gamma}_n}, \end{align*} $$

satisfies the following vanishing property:

(4.8) $$ \begin{align} \mathrm{Coef}_{\varepsilon^{2g}}\left.\frac{{\partial}^n\mathcal F^{\mathrm{red}}}{{\partial} t^{\alpha_1}_{d_1}\ldots{\partial} t^{\alpha_n}_{d_n}}\right|{}_{t^*_*=0}=0,\quad\text{if}\quad \sum_{i=1}^n d_i\le 2g-2. \end{align} $$

The power series $\mathcal F^{\mathrm {red}}$ is called the reduced potential of our partial CohFT.

The differential polynomials ${\widetilde u}^{\alpha }\in \widehat {\mathcal A}^{[0]}_N$ defined by:

$$ \begin{align*} {\widetilde u}_{\alpha}:=\eta_{{1\!\! 1}\mu}P^{\mu}_{\alpha,0} \end{align*} $$

are called the normal coordinates of the DR hierarchy. The differential polynomials ${\widetilde u}^{\alpha }:=\eta ^{\alpha \nu }{\widetilde u}_{\nu }$ are also called the normal coordinates.

The following conjecture was presented in [Reference Buryak, Dubrovin, Guéré and Rossi2, Conjecture 7.5].

Conjecture 3. We have:

(4.9) $$ \begin{align} \frac{{\partial}^2\mathcal F^{\mathrm{red}}}{{\partial} t^{1\!\! 1}_0{\partial} t^{\alpha}_0}=\left.{\widetilde u}_{\alpha}\right|{}_{u^{\gamma}_n=(u^{\mathrm{str}})^{\gamma}_n},\quad 1\le\alpha\le N. \end{align} $$

Remark 4.5. To be precise, Conjecture 7.5 from [Reference Buryak, Dubrovin, Guéré and Rossi2] claims that:

(4.10) $$ \begin{align} \mathcal F^{\mathrm{DRH}}=\mathcal F^{\mathrm{red}}, \end{align} $$

where $\mathcal F^{\mathrm {DRH}}$ is the potential of the DR hierarchy, see Section 4.2 in [Reference Buryak, Dubrovin, Guéré and Rossi2] for the construction. Let us explain why it is equivalent to Conjecture 3. In one direction, equation (4.9) immediately follows from (4.10) and the definition of $\mathcal F^{\mathrm {DRH}}$ . Conversely, equation (4.9) implies that $\frac {{\partial }^2\mathcal F^{\mathrm {red}}}{{\partial } (t^{1\!\! 1}_0)^2}=\frac {{\partial }^2\mathcal F^{\mathrm {DRH}}}{{\partial } (t^{1\!\! 1}_0)^2}$ , which, using the string equations for $\mathcal F^{\mathrm {red}}$ [Reference Buryak, Dubrovin, Guéré and Rossi2, Proposition 7.2] and $\mathcal F^{\mathrm {DRH}}$ [Reference Buryak, Dubrovin, Guéré and Rossi2, Proposition 6.3], gives that $\mathcal F^{\mathrm {red}}=\mathcal F^{\mathrm {DRH}}$ (see [Reference Getzler14, Lemma 3.1]).

4.3 The Pixton class as a partial cohomological field theory

Proposition 4.6. The classes:

(4.11) $$ \begin{align} c_{g,n}(\otimes_{i=1}^n e_{a_i}) := \sum_{d=0}^g 2^{-d}\mu^{2d}P^d_g(a_1,\ldots,a_n) \end{align} $$

form an infinite rank tame partial cohomological field theory with the phase space $V=\mathrm {span}(\{e_a\}_{a \in \mathbb Z})$ , the unit $e_0$ and the metric given in the basis $\{e_a\}_{a \in \mathbb Z}$ by $\eta _{a b}=\delta _{a+b,0}$ .

Proof. Since $c_{g,n}(\otimes _{i=1}^n e_{a_i})=0$ unless $\sum _{i=1}^na_i=0$ , the tameness property is clear.

To prove the axioms from Definition 4.1, the crucial observation is that formula (3.1) is very close to the formula for the action of a Givental R-matrix on a topological field theory (see, e.g. [Reference Pandharipande, Pixton and Zvonkine20, Section 2] for an introduction to these techniques). Let $V_r:=\mathrm {span}(\{e_0,\ldots ,e_{r-1}\})$ and fix a bilinear form $\eta _r(e_a,e_b):=\frac {1}{r} \delta _{a+b=0\ \mod r}$ on $V_r$ . Starting with the topological field theory $\omega _{g,n}\colon V_r^{\otimes n}\to H^0({\overline {\mathcal {M}}}_{g,n})$ , where:

$$ \begin{align*} \omega_{g,n}(e_{a_1}\otimes\ldots\otimes e_{a_n}):=r^{2g-1}\delta_{a_1+\ldots+a_n=0\ \mod r}, \end{align*} $$

and applying the Givental R-matrix:

$$ \begin{align*} R(z):=\exp\left(-\mathrm{diag}_{a=0}^{r-1}(a^2)2^{-1} \mu^2 z \right), \end{align*} $$

we obtain the CohFT:

$$ \begin{align*} \begin{aligned} \Omega^r_{g,n}(e_{a_1}\otimes\ldots\otimes e_{a_n}) &=\sum_{\Gamma\in G_{g,n}}\ \, \sum_{w\in W_{\Gamma,r}}\frac{r^{2g-1-h^1(\Gamma)}}{|\operatorname{\mathrm{Aut}}(\Gamma)|} \xi_{\Gamma*}\left[\prod_{i=1}^n \exp(a_i^2 2^{-1} \mu^2 \psi_{l_i}) \right. \\[6pt] &\quad \times\left.\prod_{e=\{h,h'\}\in E(\Gamma)} 2^{-1}\mu^2\frac{1-\exp\left(2^{-1}\mu^2\left(w(h)^2 \psi_h+w(h')^2 \psi_{h'}\right)\right)}{ 2^{-1}\mu^2(\psi_h+\psi_{h'})}\right], \end{aligned} \end{align*} $$

whose unit is $e_0$ and where $W_{\Gamma ,r}$ are the same weightings appearing in formula (3.1). In particular, the factor $r^{2g-1-h^1(\Gamma )}$ comes from the product of the factors $r^{2g(v)-1}$ appended to each vertex $v\in V(\Gamma )$ times the factors r appended to each edge (from the $\eta _r^{-1}$ in the edge contributions), since:

$$ \begin{align*}&\sum_{v\in V(\Gamma)}(2g(v)-1)+|E| = |E|-|V|+2 \\[6pt] & \sum_{v\in V(\Gamma)}g(v) = (h^1(\Gamma)-1)+2(g-h^1(\Gamma)) = 2g-1-h^1(\Gamma). \end{align*} $$

Dividing the classes $\Omega ^r_{g,n}(\otimes _{i=1}^n e_{a_i})$ by $r^{2g-1}$ preserves the property of being a partial CohFT. Therefore, the classes:

(4.12) $$ \begin{align} \begin{aligned} \widetilde\Omega^r_{g,n}(e_{a_1}\otimes\ldots\otimes e_{a_n})&:=\sum_{\Gamma\in G_{g,n}}\ \, \sum_{w\in W_{\Gamma,r}}\frac{1}{|\operatorname{\mathrm{Aut}}(\Gamma)|}\frac{1}{r^{h^1(\Gamma)}}\xi_{\Gamma*}\left[\prod_{i=1}^n \exp(a_i^2 2^{-1} \mu^2 \psi_{l_i}) \right. \\[6pt] &\quad \times\left.\prod_{e=\{h,h'\}\in E(\Gamma)} 2^{-1}\mu^2\frac{1-\exp\left(2^{-1}\mu^2\left(w(h)^2 \psi_h+w(h')^2 \psi_{h'}\right)\right)}{ 2^{-1}\mu^2(\psi_h+\psi_{h'})}\right] \end{aligned} \end{align} $$

form a partial CohFT with the same phase space $V_r$ , the metric $\widetilde {\eta }_r(e_a,e_b) = \delta _{a+b=0\ \mod r}$ and the unit $e_0$ . Note that in this formula we have:

$$ \begin{align*}w(h)^2=-w(h)w(h')+r w(h),\qquad w(h')^2=-w(h)w(h')+r w(h'). \end{align*} $$

Note also that the class $\widetilde \Omega ^r_{g,n}(e_{a_1}\otimes \ldots \otimes e_{a_n})$ is zero unless $a_1+\ldots +a_n=0\ \mod r$ .

For an integer a, let us denote by $\widetilde {a}\in \{0,\ldots ,r-1\}$ the unique number, such that $a=\widetilde {a} \mod r$ . If $r>|a|$ , then, clearly,

$$ \begin{align*}\widetilde{a}= \begin{cases} a,&\text{if } a\ge 0,\\ r+a,&\text{if } a<0. \end{cases} \end{align*} $$

Consider an n-tuple $A=(a_1,\ldots ,a_n)\in \mathbb Z^n$ satisfying $a_1+\ldots +a_n=0$ , and let $\widetilde {A}:=(\widetilde {a}_1,\ldots ,\widetilde {a}_n)$ . Comparing formulas (3.1) and (4.12), and using Proposition 3” in [Reference Janda, Pandharipande, Pixton and Zvonkine16], we conclude that both classes $\widetilde \Omega ^r_{g,n}(\otimes _{i=1}^n e_{\widetilde {a}_i})$ and $\sum _{d\ge 0}2^{-d}\mu ^{2d}P_g^{d,r}(A)$ are polynomials in r (for r sufficiently large) having the same constant term, which is equal to the class $c_{g,n}(\otimes _{i=1}^n e_{a_i})$ (one should notice that the factors $2^{-1}\mu ^2$ appended to each psi class and each edge of a stable graph in (4.12) globally produce a factor $2^{-d}\mu ^{2d}$ ). The proposition can now be easily derived from that.

To prove Axiom (iii) from Definition 4.1 for the classes $c_{g,n}(\otimes _{i=1}^n e_{a_i})$ , consider the gluing map (4.1) with respect to the bilinear form $\widetilde {\eta _r}$ . We have:

$$ \begin{align*}\mathrm{gl}^*\widetilde\Omega^r_{g,n}(\otimes_{i=1}^n e_{\widetilde{a}_i})=\widetilde\Omega^r_{g_1,n_1+1}(\otimes_{i\in I} e_{\widetilde{a}_i}\otimes e_{\widetilde{-a_I}})\widetilde\Omega^r_{g_2,n_2+1}(\otimes_{j\in J} e_{\widetilde{a}_j}\otimes e_{\widetilde{-a_J}}). \end{align*} $$

Considering both sides as polynomials in r (for r sufficiently large) and taking the constant terms, we obtain:

$$ \begin{align*}\mathrm{gl}^*c_{g,n}(\otimes_{i=1}^n e_{a_i})=c_{g_1,n_1+1}(\otimes_{i\in I} e_{a_i}\otimes e_{-a_I})c_{g_2,n_2+1}(\otimes_{j\in J} e_{a_j}\otimes e_{-a_J}), \end{align*} $$

as required. Proofs of Axioms (i) and (ii) are the same, and we omit them.

4.4 The DR/DZ equivalence conjecture implies Conjectures 1 and 2

Consider the partial CohFT given by the Pixton class,

$$ \begin{align*}c_{g,n}(\otimes_{i=1}^n e_{a_i}) := \sum_{d=0}^g 2^{-d}\mu^{2d}P^d_g(a_1,\ldots,a_n) \end{align*} $$

and the corresponding DR hierarchy.

Proposition 4.8. The normal coordinates of the DR hierarchy are given by:

(4.13) $$ \begin{align} {\widetilde u}^{\alpha}=u^{\alpha}+\sum_{g\ge 1}\varepsilon^{2g}\frac{(\mu\alpha)^{2g}}{2^{2g}(2g+1)!}u^{\alpha}_{2g},\quad \alpha\in\mathbb Z. \end{align} $$

Proof. To compute the normal coordinates ${\widetilde u}^{\alpha }$ , one has to compute the integrals:

$$ \begin{align*} \int_{{\overline{\mathcal{M}}}_{g,n+2}}\mathrm{DR}_g\left(-\sum_{i=1}^n a_i,0,a_1,\ldots,a_n\right)\lambda_g 2^{-d}P^d_g(0,-\alpha,\alpha_1,\ldots,\alpha_n), \end{align*} $$

where $n\ge 1$ , $a_1,\ldots ,a_n,\alpha _1,\ldots ,\alpha _n\in \mathbb Z$ , $0\le d\le g$ , which by degree reasons can be nonzero only if $g-1+n=d$ . Therefore, only the integrals with $n=1$ and $d=g$ give a nontrivial contribution, that is, the integrals:

which gives formula (4.13).

Proposition 4.9. The reduced potential $\mathcal F^{\mathrm {P},\mathrm {red}}$ of our partial CohFT is equal to:

$$ \begin{align*}\mathcal F^{\mathrm{P},\mathrm{red}}=\mathcal F^{\mathrm{P}}+\sum_{g\ge 1}\frac{(\varepsilon\mu)^{2g}}{2^{2g}(2g+1)!}(w^{\mathrm{top}})^0_{2g-2}. \end{align*} $$

Proof. Equivalently, we have to check that:

$$ \begin{align*}\mathcal F^{\mathrm{P},\mathrm{red}}=S(\varepsilon\mu{\partial}_x)\mathcal F^{\mathrm{P}}, \end{align*} $$

where we identify $x=t^0_0$ .

For $d\in \mathbb Z$ , denote by $S_d\subset \mathbb C[[t^*_*,\varepsilon ]]$ the space of formal power series F satisfying the condition:

$$ \begin{align*} \mathrm{Coef}_{\varepsilon^{2g}}\left.\frac{{\partial}^n F}{{\partial} t^{\alpha_1}_{d_1}\ldots{\partial} t^{\alpha_n}_{d_n}}\right|_{t^*_*=0}=0,\quad\text{if}\quad\sum_{i=1}^n d_i\le 2g+d. \end{align*} $$

By degree reasons, we have:

$$ \begin{align*}\mathcal F^{\mathrm{P}}-\sum_{g\ge 1}(\varepsilon\mu)^{2g} \bigg((-1)^g\underbrace{\int_{{\overline{\mathcal{M}}}_{g,1}}\lambda_g\psi_1^{2g-2}}_{=:b_g}\bigg)t^0_{2g-2}\in S_{-2}. \end{align*} $$

Let $b_0:=1$ . Using the string equation:

$$ \begin{align*} \frac{{\partial}\mathcal F^{\mathrm{P}}}{{\partial} t^0_0}=\sum_{a\ge 0}t^{\alpha}_{a+1}\frac{{\partial}\mathcal F^{\mathrm{P}}}{{\partial} t^{\alpha}_a}+\frac{1}{2}\sum_{\alpha\in\mathbb Z}t^{\alpha}_0 t^{-\alpha}_0, \end{align*} $$

we obtain:

$$ \begin{align*} &{\partial}_x^{2h}\mathcal F^{\mathrm{P}}-\sum_{g\ge 0}(\varepsilon\mu)^{2g}(-1)^g b_g t^0_{2g+2h-2}\in S_{2h-2}\quad\text{for } h\ge 1 \\[6pt] & \Rightarrow \quad S(\varepsilon\mu{\partial}_x)\mathcal F^{\mathrm{P}}-\sum_{g\ge 1}(\varepsilon\mu)^{2g}t^0_{2g-2}\left(\sum_{g_1+g_2=g}\frac{(-1)^{g_1}b_{g_1}}{2^{2g_2}(2g_2+1)!}\right)\in S_{-2}. \end{align*} $$

By [Reference Faber and Pandharipande13, Theorem 2], the numbers $b_g$ are given by:

$$ \begin{align*}1+\sum_{g\ge 1}b_g z^{2g}=\frac{iz}{e^{iz/2}-e^{-iz/2}}=\frac{1}{S(iz)}, \end{align*} $$

which implies that $S(\varepsilon \mu {\partial }_x)\widehat {\mathcal {F}}^{\mathrm {DR}}\in S_{-2}$ , as required.

Proof. Conjecture 3, together with Proposition 4.8, implies that:

$$ \begin{align*}\frac{{\partial}^2\mathcal F^{\mathrm{P},\mathrm{red}}}{{\partial} t^0_0{\partial} t^{-\alpha}_0}=S(\mu\varepsilon\alpha{\partial}_x)(u^{\mathrm{str}})^{\alpha}. \end{align*} $$

On the other hand, from Proposition 4.9, it follows that:

$$ \begin{align*}\frac{{\partial}^2\mathcal F^{\mathrm{P},\mathrm{red}}}{{\partial} t^0_0{\partial} t^{-\alpha}_0}=S(\varepsilon\mu{\partial}_x)(w^{\mathrm{P}})^{\alpha}. \end{align*} $$

Therefore,

$$ \begin{align*}(u^{\mathrm{str}})^{\alpha}=\frac{S(\varepsilon\mu{\partial}_x)}{S(i\varepsilon\mu\alpha{\partial}_x)}(w^{\mathrm{P}})^{\alpha}\quad\Rightarrow\quad u^{\mathrm{P}}=\sum_{\alpha\in\mathbb Z}(u^{\mathrm{str}})^{\alpha} e^{i\alpha y}. \end{align*} $$

In [Reference Buryak and Rossi6, Theorem 4.1], the author proved that $\sum _{\alpha \in \mathbb Z}(u^{\mathrm {str}})^{\alpha } e^{i\alpha y}$ satisfies the noncommutative KdV hierarchy. This implies that $u^{\mathrm {P}}$ satisfies the noncommutative KdV hierarchy, as required.