BCFG Drinfeld–Sokolov hierarchies and FJRW-theory

Appendix: Dubrovin–Zhang axiomatic theory of integrable hierarchies

As we explained in the introduction, Dubrovin–Zhang’s approach gives an alternative proof of the theorem of Fan–Jarvis–Ruan [15, 16] for the ADE integrable hierarchies conjecture. Their proof by-passed Givental’s higher genus theory and Kac-Wakimoto hierarchies. In a way, it is much more direct. Their proof has been written down explicitly in an extended version of [12] for the $A_n$ case which can be generalized directly to the other cases. For the reader’s convenience, we present their proof here.

For any semisimple Frobenius manifold, Dubrovin–Zhang developed an axiomatic theory of integrable hierarchies. Dubrovin–Zhang’s theory starts from a semisimple Frobenius manifold, to which they associate a bihamiltonian integrable hierarchy of hydrodynamic type, the so-called principal hierarchy. Then, they define the all genus integrable hierarchy as a topological deformation of the principal hierarchy. Under certain axioms, they are able to show that such a topological deformation is uniquely defined. The ADE Drinfeld–Sokolov hierarchies satisfy these axioms and hence coincide with Dubrovin–Zhang’s topological deformation of the principal hierarchies. The connection to FJRW-theory is through the Virasoro constraints, which are determined by the Frobenius manifold structure. More precisely, under the semisimple hypothesis, they show that the genus zero free energy $\mathcal {F}^0$ and the Virasoro constraints uniquely determine the higher genus free energies $\mathcal {F}^g$. Furthermore, $e^{\mathcal {F}}$ must be a tau function of their deformed hierarchy and hence a tau function of an ADE Drinfeld–Sokolov hierarchy. The only input data from the A-model are (1) the FJRW Frobenius manifold matches that of the mirror B-model; (2) FJRW-theory satisfies Virasoro constraints. Both are known for the examples of interest in this paper.

1.1 A.1: The principal hierarchy and the genus zero free energy

Let $\Lambda =\{\Lambda _{g,k}\}$ be a cohomological field theory. Its genus zero primary potential (or primary free energy)

$$\begin{aligned} F(v)=\sum _{k \ge 0}\frac{v^{\alpha _1} \dots v^{\alpha _k}}{k!}\int _{\overline{\mathcal {M}}_{0, k}}\Lambda _{0,k}(\phi _{\alpha _1},\dots ,\phi _{\alpha _k}) \end{aligned}$$

defines the potential of a Frobenius manifold. Namely,

• The matrix $\eta _{\alpha \beta }=\partial _1\partial _{\alpha }\partial _{\beta }F$ coincides with the pairing $\langle \phi _{\alpha }, \phi _{\beta }\rangle$. Here $\partial _{\alpha }=\frac{\partial }{\partial v^{\alpha }}$.

• The $(1,2)$ tensor $c^{\gamma }_{\alpha \beta }(v)=\eta ^{\gamma \gamma '}\partial _{\alpha }\partial _{\beta }\partial _{\gamma '}F$ with $(\eta ^{\alpha \beta })=(\eta _{\alpha \beta })^{-1}$ defines a family of commutative associative products in the following way:

  $$\begin{aligned} \phi _{\alpha }\circ _v \phi _{\beta }=c^{\gamma }_{\alpha \beta }(v)\phi _{\gamma }. \end{aligned}$$

  We also have

  $$\begin{aligned}&\phi _1\circ _v\phi _{\alpha }=\phi _{\alpha }\circ _v\phi _1=\phi _{\alpha },\\&\langle \phi _{\alpha }\circ _v\phi _{\beta }, \phi _{\gamma }\rangle =\langle \phi _{\alpha }, \phi _{\beta }\circ _v\phi _{\gamma }\rangle =\partial _{\alpha }\partial _{\beta }\partial _{\gamma }F. \end{aligned}$$

When $\{\Lambda _{g,k}\}$ comes from the FJRW theory of a quasi-homogeneous non-degenerate polynomial $W$, there exists a vector field

$$\begin{aligned} E=\sum _{\alpha }E^{\alpha }(v)\partial _{\alpha }=\sum _{\alpha } d_\alpha v^{\alpha }\partial _{\alpha } \end{aligned}$$

such that $d_1=1, d_{\alpha }=1-\deg (\phi _{\alpha })$ and $E(F)=(3-c_W)F$. The vector field $E$ is called the Euler vector field of $F$, and $c_W$ is called the charge of $F$. If $W$ is an ADE singularity, then $0<d_\alpha \le 1$ for all $1\le \alpha \le n$, and $0<c_W<1$. From now on, we assume that $F$ comes from the FJRW theory of an $ADE, D^T_n$ singularity. Then $F$ is always a polynomial, and it coincides with the Frobenius manifold potential obtained from the mirror Saito Frobenius manifold structure. The latter is known to be isomorphic to the Frobenius manifold of the corresponding Coxeter group [7, 38].

Let us proceed to give the definition of the so-called principal hierarchy of the Frobenius manifold potential $F$ associated to a Coxeter group. For the definition of the principal hierarchy of a general Frobenius manifold, see [7, 12].

First we define $\mu _{\alpha }=1-\frac{1}{2}{c_{W}}-d_{\alpha }$, which are called the Hodge grading. Then we introduce a family of polynomials in $v^\alpha$, denoted by

$$\begin{aligned} \{\theta _{\alpha ,p}(v)\mid \alpha =1, \dots , n,\ p=0, 1, 2, \dots \}, \end{aligned}$$

via the following relations:

$$\begin{aligned} \theta _{\alpha ,p}(0)&=0,\quad \theta _{\alpha ,0}(v)=\eta _{\alpha \beta }v^{\beta },\quad p\ge 0,\\ \partial _{\alpha }\partial _{\beta }\theta _{\gamma ,p}(v)&=c^{\delta }_{\alpha \beta }(v)\partial _{\delta }\theta _{\gamma ,p-1}(v),\quad p\ge 1,\\ E(\partial _{\alpha }\theta _{\beta ,p})&=(\mu _{\alpha }+\mu _{\beta }+p)\partial _{\alpha }\theta _{\beta ,p},\quad p\ge 0. \end{aligned}$$

It is easy to see that the polynomials $\theta _{\alpha ,p}$ are uniquely determined by the above conditions, and they satisfy the normalization condition

$$\begin{aligned} \partial _\xi \theta _\alpha (z) \eta ^{\xi \zeta }\partial _\zeta \theta _\beta (-z)=\eta _{\alpha \beta }, \end{aligned}$$

(64)

where $\theta _\alpha (z)=\sum _{p\ge 0} \theta _{\alpha ,p} z^p$. The set $\{\theta _{\alpha ,p}\}$ is called the calibration of $F$.

Remark A.1

For the readers who are familiar with Givental’s notation, the matrix $S(z)$ appearing in Givental’s quantization formula is given by $S(z)=(S^\alpha _\beta (z))$, where

$$\begin{aligned} S^\alpha _\beta (z)=\eta ^{\alpha \gamma }\sum _{p\ge 0}\partial _{\gamma } \theta _{\beta ,p}z^{-p}. \end{aligned}$$

The principal hierarchy associated to $F$ is a hierarchy of partial differential equations of hydrodynamic type for $v^1,\dots , v^n$:

$$\begin{aligned} \frac{\partial v^{\beta }}{\partial t^{\alpha ,p}}=\eta ^{\beta \gamma }\partial _x\left( \frac{\partial \theta _{\alpha ,p+1}(v)}{\partial v^{\gamma }}\right) . \end{aligned}$$

(65)

Since the flow $\frac{\partial }{\partial t^{1,0}}$ equals $\frac{\partial }{\partial x}$, we will identify $t^{1,0}$ with $x$ in what follows. This hierarchy has a bihamiltonian structure

$$\begin{aligned} \frac{\partial v^{\beta }}{\partial t^{\alpha ,p}}=\{v^{\beta }, H_{\alpha ,p}\}_1=\frac{1}{p+\frac{1}{2}+\mu _{\alpha }}\{v^{\beta }, H_{\alpha ,p-1}\}_2, \end{aligned}$$

(66)

where the Hamiltonians $H_{\alpha ,p}=\int h_{\alpha ,p}dx$ are defined by the densities

$$\begin{aligned} h_{\alpha ,p}=\theta _{\alpha ,p+1}, \end{aligned}$$

(67)

and the pair of compatible Poisson brackets $\{\ ,\ \}_{1,2}$ are of hydrodynamic type and have the form of the leading terms of (21), i.e.

$$\begin{aligned}&\{v^\alpha (x), v^\beta (y)\}_1=\eta ^{\alpha \beta }\delta '(x-y), \end{aligned}$$

(68)

$$\begin{aligned}&\{v^\alpha (x), v^\beta (y)\}_2=g^{\alpha \beta }(v(x))\delta '(x-y)+\Gamma ^{\alpha \beta }_{\gamma }(v(x)) v^{\gamma }_x(x)\delta (x-y). \end{aligned}$$

(69)

Here $(g^{\alpha \beta })$ is the intersection form of the Frobenius manifold, and $\Gamma ^{\alpha \beta }_\gamma$ are the contravariant Christoffel coefficients of the associated metric [7].

The existence of a bihamiltonian structure implies the commutativity

$$\begin{aligned} \frac{\partial }{\partial t^{\alpha .p}}\left( \frac{\partial v^{\gamma }}{\partial t^{\beta .q}}\right) =\frac{\partial }{\partial t^{\beta ,q}}\left( \frac{\partial v^{\gamma }}{\partial t^{\alpha ,p}}\right) \end{aligned}$$

of the flows of the principal hierarchy. Hence, all these flows are integrable. To introduce the tau function of a solution of the principal hierarchy, we first define the functions $\Omega _{\alpha ,p;\beta ,q}(v)$ on the Frobenius manifold by the following generating function [7]:

$$\begin{aligned} \sum _{p, q\ge 0} \Omega _{\alpha ,p;\beta ,q}(v) z^p w^q=\frac{\partial _\xi \theta _\alpha (z) \eta ^{\xi \zeta }\partial _\zeta \theta _\beta (w)-\eta _{\alpha \beta }}{z+w}. \end{aligned}$$

(70)

We call these functions the two-point functions of $F$. They have the following symmetry property:

$$\begin{aligned} \Omega _{\alpha ,p;\beta ,q}=\Omega _{\beta ,q;\alpha ,p}. \end{aligned}$$

For any solution $v=v(t)$ of the principal hierarchy we have the identities

$$\begin{aligned} \frac{\partial \Omega _{\alpha ,p;\beta ,q}(v(t))}{\partial t^{\gamma ,r}}=\frac{\partial \Omega _{\gamma ,r;\beta ,q}(v(t))}{\partial t^{\alpha ,p}}. \end{aligned}$$

It follows that if $v(t)=\{v^{\alpha }(t)\}$ is a solution to the principal hierarchy (65), then there exists a function $\tau ^{0}(t)$ such that

$$\begin{aligned} \Omega _{\alpha ,p;\beta ,q}(v(t))=\frac{\partial ^2 \log \tau ^0(t)}{\partial t^{\alpha ,p}\partial t^{\beta ,q}}. \end{aligned}$$

(71)

The function $\tau ^0(t)$ is called the genus zero tau function corresponding to $v(t)$, and $\mathcal {F}^0(t)=\log \tau ^0(t)$ is called a genus zero free energy of $v(t)$.

Note that $\tau ^0(t)$ and $\mathcal {F}^0(t)$ are not uniquely defined. If $\mathcal {F}^0(t)$ is a free energy of $v(t)$, then $\tilde{\mathcal {F}}^0(t)=\mathcal {F}^0(t)+\sum _{\alpha ,p}c_{\alpha ,p}t^{\alpha ,p}+c$ is also a free energy of the same solution $v(t)$. The construction given in Section 6 of [7] (see also [12]) fixes this ambiguity and obtains the free energy $\mathcal {F}_{\Lambda }^0$ of the cohomological field theory from the topological solutions of the principal hierarchy.

Proposition A.2

([7])

• (a) Define recursively the following sequence in the space of formal power series in $\{t^{\alpha ,p}\}$:

  $$\begin{aligned}&v^\beta _{[0]}(t)=t^{\beta ,0},\\&v^\beta _{[k+1]}(t)=\eta ^{\beta \gamma }\sum _{\alpha ,p}t^{\alpha , p}\left. \frac{\partial \theta _{\alpha ,p}}{\partial v^{\gamma }}\right| _{v^\beta =v^\beta _{[k]}(t)},\quad k=1, 2, \dots . \end{aligned}$$

  Then the limit $v^{\beta }(t)=\lim _{k\rightarrow \infty }v^{\beta }_{[k]}(t)$ gives the solution to the Euler-Lagrange equation

  $$\begin{aligned} v^\beta (t)=\eta ^{\beta \gamma }\sum _{\alpha ,p}t^{\alpha , p}\frac{\partial \theta _{\alpha ,p}}{\partial v^{\gamma }}. \end{aligned}$$

  (72)

• (b) Let $v(t)$ be the solution to the above Euler-Laguage equation. Then it is the unique solution to the principal hierarchy (65) satisfying the initial condition

  $$\begin{aligned} v^{\alpha }(t)|_{t^{\alpha ,p}=0\ (p>0)}=t^{\alpha ,0}. \end{aligned}$$

  (73)

  This solution is called the topological solution of the principal hierarchy.

• (c) The function

  $$\begin{aligned} \mathcal {F}^0_{\Lambda }=\frac{1}{2}\sum _{\alpha ,p;\beta ,q}\Omega _{\alpha ,p;\beta ,q}(v(t))\tilde{t}^{\alpha ,p}\tilde{t}^{\beta ,q},\quad \tilde{t}^{\alpha ,p}=t^{\alpha ,p}-\delta ^\alpha _1\delta ^p_1 \end{aligned}$$

  (74)

is a free energy of the topological solution of the principal hierarchy. It coincides with the genus zero free energy of the Frobenius manifold induced from the cohomological field theory $\Lambda$.

It is shown in [12] that for any Frobenius manifold one can obtain a dense subset of analytic monotonic solutions of the principal hierarchy by solving the following system of equations

$$\begin{aligned} \sum _{p\ge 0} \tilde{t}^{\alpha ,p} \frac{\partial \theta _{\alpha ,p}}{\partial v^\gamma }=0,\quad \gamma =1,\dots ,n, \end{aligned}$$

(75)

where $\tilde{t}^{\alpha ,p}=t^{\alpha ,p}-c^{\alpha ,p}$ and $c^{\alpha ,p}$ are some constants which vanish except for a finitely many of them. These constants are also required to satisfy a certain genericity condition which is omitted here. Note that the above topological solution corresponds to the case when $c^{\alpha ,p}=\delta ^\alpha _1\delta ^p_1$. For such a solution of the principal hierarchy, we can fix a free energy by using the formula (74). In what follows we only need to consider such a class of solutions of the principal hierarchy, and we fix their free energies $\mathcal {F}^0$ by using the formula given in (74).

1.2 A.2: The topological deformation and the higher genus free energies

As we explained in the last subsection, for any Frobenius manifold one can associate a free energy $\mathcal {F}^0$ to each solution $v(t)=\{ v^\alpha (t)\}$, obtained by solving the system (75), of the principal hierarchy. Now assume that the Frobenius manifold is semisimple. Dubrovin–Zhang provided an algorithm in [12] to define a certain topological deformation of the principal hierarchy and the higher genus free energies $\mathcal {F}^g(t) (g\ge 1)$ for solutions of the deformed integrable hierarchy.

The main tool in the construction of Dubrovin–Zhang is the Virasoro symmetries of the principal hierarchy (65). The action of Virasoro symmetries on the tau functions of the principal hierarchy can be represented by

$$\begin{aligned} \frac{\partial \tau ^0}{\partial s_m}=a_m^{\alpha ,p;\beta ,q} \frac{1}{\tau ^0}\frac{\partial \tau ^0}{\partial t^{\alpha ,p}} \frac{\partial \tau ^0}{\partial t^{\beta ,q}}+b^{\beta ,q}_{m;\alpha ,p} t^{\alpha ,p}\frac{\partial \tau ^0}{\partial t^{\beta ,q}}+c_{m;\alpha ,p;\beta ,q} t^{\alpha ,p} t^{\beta ,q}\tau ^0 \end{aligned}$$

for $m\ge -1$. Here the coefficients $a, b, c$ define an infinite number of linear operators $L_m, m\ge -1$ which satisfy the Virasoro commutation relations

$$\begin{aligned}{}[L_i, L_j]=(i-j) L_{i+j},\quad i, j\ge -1. \end{aligned}$$

We treat the Virasoro operators as a structure associated to a Frobenius manifold [11]. For our particular class of Frobenius manifolds associated to Coxeter groups, the Virasoro operators have the following expressions:

$$\begin{aligned} L_{-1}&=\sum _{p\ge 1}t^{\alpha ,p}\frac{\partial }{\partial t^{\alpha ,p-1}}+\frac{1}{2\hbar }\eta _{\alpha \beta }t^{\alpha ,0}t^{\beta ,0},\end{aligned}$$

(76)

$$\begin{aligned} L_0&=\sum _{p\ge 0}\left( p+\frac{1}{2}+\mu _{\alpha }\right) t^{\alpha ,p}\frac{\partial }{\partial t^{\alpha ,p}}+\frac{1}{4}\sum _{\alpha =1}^n\left( \frac{1}{4}-\mu _\alpha ^2\right) ,\end{aligned}$$

(77)

$$\begin{aligned} L_m&=\frac{\hbar }{2} \sum _{p+q=m-1} (-1)^{q+1} \prod _{j=0}^m \left( \mu _\alpha +j-q-\frac{1}{2}\right) \eta ^{\alpha \beta } \frac{\partial ^2}{\partial t^{\alpha ,p}\partial t^{\beta ,q}}\nonumber \\&\quad + \sum _{p\ge 0} \prod _{j=0}^m\left( \mu _\alpha +p+\frac{1}{2}+j\right) t^{\alpha ,p}\frac{\partial }{\partial t^{\alpha ,p+m}},\quad m\ge 1. \end{aligned}$$

(78)

In particular, they depend only on the pairing (the flat metric) and grading (the Euler vector field) of the Frobenius manifold.

Note that the actions of the Virasoro symmetries on $\tau ^0$ are nonlinear. The idea of [12] is to make a “change of coordinates”

$$\begin{aligned} \tau ^0(t)\mapsto \tau (t)=e^{\hbar ^{-1}\mathcal {F}^0(t)+\Delta F(v, v_x,\dots )|_{v^\alpha =v^\alpha (t)}} \end{aligned}$$

(79)

with

$$\begin{aligned} \Delta F(v,v_x,\dots )=\sum _{g\ge 1}\hbar ^{g-1} F^g(v, v_x,\dots ), \end{aligned}$$

so that in terms of $\tau (t)$ the actions of the Virasoro symmetries are represented as

$$\begin{aligned} \frac{\partial \tau }{\partial s_m}=L_m\tau ,\quad m\ge -1. \end{aligned}$$

(80)

This linearization condition yields a system of linear equations for the gradients of the functions $F^g$, which is called the loop equation in [12] (see Theorem 3.10.31 there). It determines the functions $F^g$ recursively and uniquely up to the addition of constants. Here we assume that $F^g$ depends only on finitely many jet variables, which is a weaker version of Givental’s tameness condition [21]. If the function $\mathcal {F}^0(t)$ is the free energy of a solution $v(t)$ given by (75), then the tau function $\tau$ given in (79) satisfies the Virasoro constraints

$$\begin{aligned} L_m|_{t^{\alpha ,p}\rightarrow {{t}}^{\alpha ,p}-c^{\alpha ,p}}\tau (t)=0,\quad m\ge -1, \end{aligned}$$

and the function

$$\begin{aligned} \mathcal {F}^g(t)=F^g(v,v_x,\dots )|_{v^\alpha \rightarrow v^\alpha (t)},\quad g\ge 1 \end{aligned}$$

is called the genus $g$ free energy associated to $v(t)$. In particular, if we start from the genus zero free energy $\mathcal {F}_\Lambda ^0$ of a cohomological field theory $\Lambda$, then Theorem 3.10.31 of [12] shows that the higher genus free energies $\mathcal {F}_{\Lambda }^g$ are determined uniquely by $\mathcal {F}_\Lambda ^0$ and the Virasoro constraints

$$\begin{aligned} L_m|_{t^{1,1}\rightarrow {{t}}^{1,1}-1}\tau (t)=0,\quad m\ge -1. \end{aligned}$$

The functions $F^g(v, v_x,\dots )$ for $g\ge 1$ enable us to construct the topological deformation of the principal hierarchy [12]. We introduce the new dependent variables

$$\begin{aligned} w^\alpha&=v^\alpha +\sum _{g\ge 1} \hbar ^g A^\alpha _g(v, v_x,\dots )\nonumber \\&=v^\alpha +\hbar \eta ^{\alpha \gamma }\frac{\partial ^2\Delta F(v, v_x,\dots )}{\partial t^{1,0}\partial t^{\gamma ,0}}, \quad \alpha =1,\dots , n. \end{aligned}$$

(81)

Such a change of dependent variables is called a quasi-Miura transformation in [12]. It differs from a Miura-type transformation. In fact, it does not depend polynomially on the jet variables $v^\alpha _x,\ 1\le \alpha \le n$. Note that the transformation (81) is invertible, so we can also represent $v^\alpha$ in terms of $w^\beta$ and their jet variables. Now let us write down the principal hierarchy (65) in terms of the new variables $w^\alpha$. We have an integrable hierarchy of the form

$$\begin{aligned} \frac{\partial w^\alpha }{\partial t^{\beta ,q}}&=\eta ^{\beta \gamma }\partial _x\left( \frac{\partial \theta _{\alpha ,p+1}(w)}{\partial w^{\gamma }}\right) +\sum _{g\ge 1} \hbar ^g R^\alpha _{g;\beta ,q}(w,w_x,\dots ,w^{(m_g)}),\nonumber \\&\alpha ,\beta =1,\dots ,n, \ q\ge 0. \end{aligned}$$

(82)

The above hierarchy is just the topological deformation of the principal hierarchy constructed in [12]. Since (81) is a quasi-Miura transformation, it is rather nontrivial that (82) belongs to the class of integrable hierarchies of KdV-type, i.e. the functions $R^\alpha _{g;\beta ,q}$ depend polynomially on $w^\beta _x, w^\beta _{xx}, \dots ,\partial _x^{2g+1} w^\beta$ as the Drinfeld–Sokolov hierarchies do. A proof of the polynomial dependence of $R^\alpha _{g;\beta ,q}$ on $w^\beta _x, w^\beta _{xx}, \dots ,\partial _x^{2g+1} w^\beta$ is given by Buryak, Posthuma and Shadrin in [2, 3]. They also proved the polynomial property of the first Hamiltonian structure of the deformed principal hierarchy, which is obtained from the Poisson bracket (68) and the first equation of (66) by the change of variables (81). In the new variable $w^\alpha$ the Poisson brackets $\{\,,\,\}_a, a=1,2$ have the forms

$$\begin{aligned} \{w^\alpha (x),w^\beta (y)\}_1&=\eta ^{\alpha \beta }\delta '(x-y)\nonumber \\&\quad +\sum _{g\ge 1}\sum _{k=0}^{2g+1} \hbar ^g P^{\alpha \beta }_{1;g,k}(w,w_x,\dots )\delta ^{(2g+1-k)}(x\!-\!y),\quad \end{aligned}$$

(83)

$$\begin{aligned} \{w^\alpha (x),w^\beta (y)\}_2&=g^{\alpha \beta }(w(x))\delta '(x-y)+\Gamma ^{\alpha \beta }_{\gamma }(w(x)) w^{\gamma }_x(x)\delta (x\!-\!y)\nonumber \\&\quad +\sum _{g\ge 1}\sum _{k=0}^{m_g} \hbar ^g P^{\alpha \beta }_{2;g,k}(w,w_x,\dots )\delta ^{(m_g-k)}(x\!-\!y).\quad \end{aligned}$$

(84)

As proved in [2, 3], the functions $P_{1;g,k}^{\alpha \beta }$ are homogeneous polynomials of $w^\gamma _x, w^\gamma _{xx},\dots$ of degree $k$. The Hamiltonian formalism of the topological deformation of the principal hierarchy is given by

$$\begin{aligned} \frac{\partial w^\alpha }{\partial t^{\beta ,q}}=\{w^\alpha (x),H_{\beta ,q}\}_1,\quad \alpha , \beta =1,\dots , n,\ q\ge 0. \end{aligned}$$

Here the densities of the Hamiltonians $H_{\beta , q}$ are taken as

$$\begin{aligned} \tilde{h}_{\beta ,q}&= h_{\beta ,q}(v)+\hbar \frac{\partial ^2\Delta F(v, v_x,\dots )}{\partial t^{1,0}\partial t^{\beta ,q}} \\&= h_{\beta ,q}(w)+\sum _{g\ge 1}\hbar ^g Q_{\beta , q; g}(w,w_x,\dots ), \end{aligned}$$

and $Q_{\beta , q; g}$ are homogeneous polynomials of $w^\gamma _x, w^\gamma _{xx},\dots$ of degree $2g$. Moreover, the functions

$$\begin{aligned} \tilde{\Omega }_{\alpha ,p;\beta ,q}=\hbar \frac{\partial ^2\log \tau }{\partial t^{\alpha ,p}\partial t^{\beta ,q}} =\Omega _{\alpha ,p;\beta ,q}(w)+\sum _{g\ge 1}\hbar ^g W_{\alpha ,p;\beta ,q; g}(w,w_x,\dots ) \end{aligned}$$

provide the tau-structure of the topological deformation of the principal hierarchy. Here $W_{\alpha ,p;\beta ,q; g}$ are polynomials of $w^\gamma _x, w^\gamma _{xx},\dots$ of degree $2g$.

It is conjectured in [12] that in the second Poisson bracket (84), the coefficients $P^{\alpha \beta }_{2;g,k}$ also have the polynomial property. If this conjecture is valid, then the topological deformation of the principal hierarchy has a bihamiltonian structure, a property that is possessed by the KdV hierarchy and, more generally, by the Drinfeld–Sokolov hierarchy associated to untwisted affine Lie algebras.

For the semisimple Frobenius manifolds associated to the Coxeter groups of ADE type or ADE singularities, we have a full picture of the topological deformations of the principal hierarchies, including their bihamiltonian structures. Namely, these integrable hierarchies coincide with the Drinfeld–Sokolov hierarchies associated to the untwisted affine Lie algebras of ADE type, which we described in Sect. 4. To prove this assertion, we need to use the following results:

• 1. The dispersionless ADE Drinfeld–Sokolov hierarchies (22) coincide with the principal hierarchies of the Frobenius manifolds associated to the Coxeter groups of ADE type or, equivalently, to the ADE singularities, as it is proved in [9].

• 2. A result of C.-Z. Wu [45] shows that for an ADE Drinfeld–Sokolov hierarchy, the actions of the Virasoro symmetries on the tau functions defined in (19) are linear. Furthermore, they are given by the Virasoro operators (76)–(78).

• 3. In an extended version of [12], Dubrovin–Zhang proved that for a semisimple Frobenius manifold, any deformation of the principal hierarchy that possesses a bihamiltonian structure and a tau structure is quasi-trivial. Moreover, the quasi-Miura transformation has the form (81).

As it is shown in Sect. 4, an ADE Drinfeld–Sokolov hierarchy has a bihamiltonian structure together with a tau structure. Hence, it is obtained from the principal hierarchy of the Frobenius manifold associated to the corresponding Coxeter group of ADE type by a quasi-Miura transformations of the form (81). Now the property of linear actions of the Virasoro symmetries on the tau functions requires that the function $\Delta F$ satisfies the loop equation of the corresponding Frobenius manifold. Then using the uniqueness of solution of the loop equation we obtain the following theorem:

Theorem A.3

For a semisimple Frobenius manifold associated to a Coxeter group of ADE type, or to an ADE singularity, the topological deformation (82) of the principal hierarchy (65) coincides with the ADE Drinfeld–Sokolov hierarchy described in Sect. 4.

The above theorem provides an alternative proof of the ADE Witten conjecture by directly connecting the FJRW invariants to the Drinfeld–Sokolov hierarchies.

Theorem A.4

(ADE Witten Conjecture)

• (A) The generating function of the FJRW invariants for an ADE, $D^T_n$ singularity with the maximal diagonal symmetry group is the logarithm of a tau function of the mirror Drinfeld–Sokolov hierarchy.

• (B) The generating function of the FJRW invariants for $(D_{2n}, \langle J\rangle )$ is the logarithm of a tau function of the $D_{2n}$-Drinfeld–Sokolov hierarchy.

In particular, this tau function is uniquely determined by the Drinfeld–Sokolov hierarchy and the string equation $L_{-1}\tau =0$.

Proof

According to [15–17], the generating function of the genus zero FJRW invariants of an ADE singularity coincides with the genus zero free energy of the semisimple Frobenius manifold of the mirror singularity. Furthermore, the all genus generating function satisfies the Virasoro constraints. By the result of [12] that the genus zero free energy and the Virasoro constraints uniquely determine the full genus free energy, the exponential of this function must be a tau function of the topological deformation of the principal hierarchy associated to the ADE singularity and satisfies the Virasoro constraints. The theorem then follows from Theorem A.3. $\square$