Conjugate Frobenius Manifold and Inversion Symmetry

Abstract

We give a conjugacy relation on certain type of Frobenius manifold structures using the theory of flat pencils of metrics. It leads to a geometric interpretation for the inversion symmetry of solutions to Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations.

1 Introduction

Boris Dubrovin introduced the notion of a Frobenius manifold as a geometric realization of a potential ${\mathbb {F}}$ which satisfies a system of partial differential equations known in topological field theory as Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations. More precisely, a Frobenius algebra is a commutative associative algebra with an identity e and a nondegenerate bilinear form $\Pi$ compatible with the product, i.e., $\Pi (a\circ b,c)=\Pi (a,b\circ c)$. A Frobenius manifold is a manifold with a smooth structure of a Frobenius algebra on the tangent space at any point with certain compatibility conditions. Globally, we require the metric $\Pi$ to be flat and the identity vector field e to be covariantly constant with respect to the corresponding Levi–Civita connection. Detailed information about Frobenius manifolds and related topics can be found in [7].

Let M be a Frobenius manifold. In flat coordinates $(t^1,\ldots ,t^r)$ for $\Pi$ where $e= \partial _{t^{r}}$ the compatibility conditions imply that there exists a function ${\mathbb {F}}(t^1,\ldots ,t^r)$ which encodes the Frobenius structure, i.e., the flat metric is given by

$$\begin{aligned} \Pi _{ij}(t)=\Pi (\partial _{t^i},\partial _{t^j})= \partial _{t^{r}} \partial _{t^i} \partial _{t^j} {\mathbb {F}}(t) \end{aligned}$$

(1.1)

and, setting $\Omega _1(t)$ to be the inverse of the matrix $\Pi (t)$, the structure constants of the Frobenius algebra are given by

$$\begin{aligned} C_{ij}^k(t)=\Omega _1^{kp}(t) \partial _{t^p}\partial _{t^i}\partial _{t^j} {\mathbb {F}}(t). \end{aligned}$$

Here, and in what follows, summation with respect to repeated upper and lower indices is assumed. The definition includes the existence of a vector field E of the form $E=(a_i^j t^i+b^j)\partial _{t^j}$ satisfying

$$\begin{aligned} E{\mathbb {F}}(t)= \left( 3-d \right) {\mathbb {F}}(t)+ \frac{1}{2}A_{ij} t^i t^j+B_i t^i+c \end{aligned}$$

(1.2)

where $a_i^j$, $b_j$, c, $A_{ij}$, $B_i$ and d are constants with $a_r^r=1$. The vector field E is called the Euler vector field and the number d is called the charge of the Frobenius manifold. The associativity of Frobenius algebra implies that the potential ${\mathbb {F}}(t)$ satisfies the WDVV equations

$$\begin{aligned} \partial _{t^i} \partial _{t^j} \partial _{t^k} {\mathbb {F}}(t)~ \Omega _1^{kp} ~\partial _{t^p} \partial _{t^q} \partial _{t^n} {\mathbb {F}}(t) = \partial _{t^n} \partial _{t^j} \partial _{t^k} {\mathbb {F}}(t) ~\Omega _1^{kp}~\partial _{t^p} \partial _{t^q} \partial _{t^i} {\mathbb {F}}(t),~~ \forall i,j,q,n. \end{aligned}$$

(1.3)

Conversely, an arbitrary potential ${\mathbb {F}}(t^1,\ldots , t^r)$ satisfying Eqs. (1.3) and (1.2) with (1.1) determines a Frobenius manifold structure on its domain [7]. Moreover, there exists a quasihomogenius flat pencil of metrics (QFPM) of degree d associated to the Frobenius structure on M which consists of the intersection form ${\Omega }_2$ and the flat metric $\Omega _1$ with the function $\tau =\Pi _{i1}t^i$ (see Definition 2.3 below). Here

$$\begin{aligned} \Omega _2^{ij}(t):=E(dt^i\circ dt^j) \end{aligned}$$

(1.4)

where the product $dt^i\circ dt^j$ is defined by lifting the product on TM to $T^*M$ using the flat metric $\Omega _1$. In this article we prove that, when $d\ne 1$, $e(\tau )=0$ and $E(\tau )=(1-d)\tau$, we can construct another QFPM of degree $2-d$ on M consisting of the intersection form $\Omega _2$ and a different flat metric ${\widetilde{\Omega }}_1$. We call it the conjugate QFPM. In particular, under a specific regularity condition, we get a conjugation between a certain type of Frobenius manifold structures on a given manifold. Precisely, we prove the following theorem.

Theorem 1.1

Let M be a Frobenius manifold with the Euler vector field E and the identity vector field e. Suppose the associated QFPM is regular of degree d with a function $\tau$. Assume that $e(\tau )=0$ and $E(\tau )=(1-d)\tau$. Then we can construct another Frobenius manifold structure on $M\backslash \{\tau =0\}$ of degree $2-d$. Moreover, we can apply the same method to the new Frobenius manifold structure and it leads to the original Frobenius manifold structure.

For a fixed Frobenius manifold the new structure that can be obtained using Theorem 1.1 will be called the conjugate Frobenius manifold structure.

Let us assume $\Pi _{i,j}=\delta _{i+j}^{r+1}$, i.e., the potential $\mathbb F$ has the standard form

$$\begin{aligned} \mathbb F(t) = \frac{1}{2} (t^r)^2 t^1 + \frac{1}{2} t^r \sum _{i=2}^{r-1} t^i t^{r-i+1} + G\left( t^1,\ldots ,t^{r-1}\right) \end{aligned}$$

(1.5)

and the quasihomogeneity condition (1.2) takes the form

$$\begin{aligned} E= d_i t^i\partial _{t^i},\quad E{\mathbb {F}}(t)= \left( 3-d \right) {\mathbb {F}}(t);\quad d_{r}=1. \end{aligned}$$

(1.6)

Here, the numbers $d_i$ are called the degrees of the Frobenius manifold. Recall that a symmetry of the WDVV equations is a transformation of the form

$$\begin{aligned} t^i\mapsto z^i, ~ {\Pi }\mapsto \widetilde{\Pi },~ {\mathbb {F}}\mapsto {\widetilde{\mathbb F}} \end{aligned}$$

such that ${\widetilde{\mathbb F}}$ satisfies the WDVV equations. The inversion symmetry ([7], Appendix B) is an involutive symmetry given by setting

$$\begin{aligned} z^1=-\frac{1}{t^1},\quad ~z^r=\Pi _{ij}(t)\frac{t^i t^j}{2 t^1},\quad ~ z^k=\frac{t^k}{t^1},\quad ~2\le k< r. \end{aligned}$$

(1.7)

Then

$$\begin{aligned} {\widetilde{\mathbb F}}(z) :=(t^1)^{-2}\left( \mathbb F(t)-\frac{1}{2}t^r \Pi _{ij}t^i t^j\right) \end{aligned}$$

(1.8)

is another solution to the WDVV equations with the flat metric ${\widetilde{\Pi }}_{ij}(z)=\delta _{i+j}^{r+1}$. The charge of the corresponding Frobenius manifold structure is $2-d$ and the degrees are

$$\begin{aligned} {\widetilde{d}}_1=-d_1,\quad \ \ {\widetilde{d}}_r=1,\quad \ \ {\widetilde{d}}_i = d_i-d_1 \ \ for \ \ 1< i < r. \end{aligned}$$

(1.9)

The inversion symmetry is obtained from a special Schlesinger transformation of the system of linear ODEs with rational coefficients associated to the WDVV equations. A geometric relation between Frobenius manifold structures correspond to $\mathbb F(t)$ and ${\widetilde{\mathbb F}} (z)$ was outlined through the sophisticated notion of Givental groups in [13]. In this article, we obtained a simple geometric interpertation and we report that ${\widetilde{\mathbb F}}(z)$ is the potential of the conjugate Frobenius manifold structure. In other words, we prove the following theorem.

Theorem 1.2

Let M be a Frobenius manifold with charge $d\ne 1$. Suppose in the flat coordinates $(t^1,\ldots ,t^r)$, the potential ${\mathbb {F}}(t)$ has the standard form (1.5) and the quasihomogeneity condition takes the form (1.6) with $d_i\ne \dfrac{d_1}{2}$ for every i. Then we can construct the conjugate Frobenius manifold structure on $M\backslash \{t^1=0\}$. Moreover, flat coordinates for the conjugate Frobenius manifold are

$$\begin{aligned} s^1= -t^1 , \ \ s^i= t^i (t^1)^{\frac{d_1-2d_i}{d_1}}\ \ for \ \ 1<i< r, \ \ s^r= \frac{1}{2} \sum _{i=1}^{r} t^i t^{r-i+1} (t^1)^{\frac{-2}{d_1}-1}. \end{aligned}$$

(1.10)

In addition, the corresponding potential equals the potential obtained by applying the inversion symmetry to ${\mathbb {F}}(t)$ and it is given by

$$\begin{aligned} \widetilde{\mathbb F}(s) = (t^1)^{ \frac{-4}{d_1}} \left( \mathbb F(t^1,\ldots ,t^r) -\frac{1}{2} t^r \sum _{1}^{r} t^i t^{r-i+1}\right) . \end{aligned}$$

(1.11)

Examples of Frobenius manifolds satisfying the hypotheses of Theorem 1.2 include Frobenius manifold structures constructed on orbits spaces of standard reflection representations of irreducible Coxeter groups in [9, 22] and algebraic Frobenius manifolds constructed using classical W-algebras [5]. However, the result presented in this article is a consequence of the work [1, 6]. There, we investigated the existence of Frobenius manifold structures on orbits spaces of some non-reflection representations of finite groups and we noticed that certain structures appear in pairs. Analyzing such pairs led us to the notion of conjugate Frobenius manifold.

This article is organized as follows. In Sect. 2, we review the relation between Frobenius manifold, flat pencil of metrics and compatible Poisson brackets of hydrodynamic type. Then we introduce a conjugacy relation between certain class of quasihomogeneous flat pencils of metrics in Sect. 3. It can be interpreted as a conjugacy relation between certain class of compatible Poisson brackets of hydrodynamic type. We prove Theorem 1.1 in Sect. 3 and Theorem 1.2 in Sect. 4. In Sect. 5, we discuss the findings of this article on polynomial Frobenius manifolds. We end the article with some remarks.

2 Background

We review in this section the relation between flat pencil of metrics, compatible Poisson brackets of hydrodynamics type and Frobenius manifold. More details can be found in [8].

Let M be a smooth manifold of dimension r and fix local coordinates $(u^1, \ldots , u^r)$ on M.

Definition 2.1

A symmetric bilinear form (. , . ) on $T^*M$ is called a contravariant metric if it is invertible on an open dense subset $M_0 \subseteq M$. We define the contravariant Christoffel symbols $\Gamma ^{ij}_k$ for a contravariant metric (. , . ) by

$$\begin{aligned} \Gamma ^{ij}_k:=-\Omega ^{im} \Gamma _{mk}^j \end{aligned}$$

where $\Gamma _{mk}^j$ are the Christoffel symbols of the metric $<. ,.>$ defined on $TM_0$ by the inverse of the matrix $\Omega ^{ij}(u)=(du^i, du^j)$. We say the metric (., .) is flat if $<. ,.>$ is flat.

Let (. , . ) be a contraviariant metric on M and set $\Omega ^{ij}(u)=(du^i, du^j)$. Then we will use $\Omega$ to refer to the metric and $\Omega (u)$ to refer to its matrix in the coordinates. In particular, the Lie derivative of (. , . ) along a vector field X will be written $\mathrm {Lie}_X \Omega$ while $X\Omega ^{ij}$ means the vector field X acting on the entry $\Omega ^{ij}$. The Christoffel symbols given in Definition 2.1 determine for $\Omega$ the contravariant (resp. covariant) derivative $\nabla ^{i}$ (resp. $\nabla _{i}$) along the covector $du^i$ (resp. the vector field $\partial _{u^i}$). They are related by the identity $\nabla ^{i}=\Omega ^{ij}(u) \nabla _{j}$.

Definition 2.2

A flat pencil of metrics (FPM) on M is a pair $(\Omega _2,\Omega _1)$ of two flat contravariant metrics $\Omega _2$ and $\Omega _1$ on M satisfying

• 1. $\Omega _2+\lambda \Omega _1$ defines a flat metric on $T^*M$ for a generic constant $\lambda$,

• 2. the Christoffel symbols of $\Omega _2+\lambda \Omega _1$ are $\Gamma _{2k}^{ij}+\lambda \Gamma _{1k}^{ij}$, where $\Gamma _{2k}^{ij}$ and $\Gamma _{1k}^{ij}$ are the Christoffel symbols of $\Omega _2$ and $\Omega _1$, respectively.

Definition 2.3

A flat pencil of metrics $(\Omega _2,\Omega _1)$ on M is called quasihomogeneous flat pencil of metrics (QFPM) of degree d if there exists a function $\tau$ on M such that the vector fields E and e defined by

$$\begin{aligned} E= & {} \nabla _2 \tau ,\quad ~~E^i =\Omega _2^{ij}(u)\partial _{u^j}\tau \nonumber \\ e= & {} \nabla _1 \tau ,\quad ~~e^i = \Omega _1^{ij}(u)\partial _{u^j}\tau \end{aligned}$$

(2.1)

satisfy

$$\begin{aligned}{}[e,E]=e,\quad ~~ \mathrm {Lie}_E \Omega _2 =(d-1) \Omega _2,\quad ~~ \mathrm {Lie}_e \Omega _2 = \Omega _1~~\mathrm {and}~~ \mathrm {Lie}_e\Omega _1 =0. \end{aligned}$$

(2.2)

Such a QFPM is regular if the (1,1)-tensor

$$\begin{aligned} R_i^j = \frac{d-1}{2}\delta _i^j + {\nabla _1}_i E^j \end{aligned}$$

(2.3)

is nondegenerate on M.

Let $(\Omega _2,\Omega _1)$ be a QFPM of degree d. Then according to [8], we can fix flat coordinates $(t^1,t^2,\ldots ,t^r)$ for $\Omega _1$ such that

$$\begin{aligned}&\tau =t^1,\quad \ E^i= \Omega _2^{i1},\quad \ e^i= \Omega _1^{i1},\quad \Gamma _{1,k}^{ij}=0,\nonumber \\&\Gamma _{2,k}^{i1}= \frac{1-d}{2} \delta _k^i,\quad \Gamma _{2,k}^{1j}= \frac{d-1}{2}\delta _k^j+ \partial _{t^k} E^j,\quad \partial _{t^1} E^1=1-d. \end{aligned}$$

(2.4)

Moreover, if $(\Omega _2,\Omega _1)$ is regular then $d\ne 1$.

Consider the loop space $\mathfrak {L}(M)$ of M, i.e., the space of smooth maps from the circle $S^1$ to M. A local Poisson bracket on $\mathfrak {L}(M)$ is a Lie algebra structure on the space of local functionals on $\mathfrak {L}(M)$. Let $\{.,.\}$ be a local Poisson bracket of hydrodynamic type (PBHT), i.e., it has the following form in the local coordinates [8]

$$\begin{aligned} \{u^i(x),u^j(y)\}= \Omega ^{ij}(u(x))\delta ' (x - y) + \Gamma _k^{ij} (u(x)) u_x^k \delta (x-y),\quad i,j=1,\ldots ,r \end{aligned}$$

(2.5)

where $\delta (x-y)$ is the Dirac delta function defined by $\int _{S^1} f(y) \delta (x-y) dy=f(x)$. Then we say $\{.,.\}$ is nondegenerate if $\det \Omega ^{ij}\ne 0$ and the Lie derivative of $\{.,.\}$ along a vector field $X:=X^i\partial _{u^i}$ reads

$$\begin{aligned} \mathrm {Lie}_X\{.,.\}(u^i(x),u^j(y))&= (X^s\partial _{u^s}\Omega ^{ij}- \Omega ^{s j}\partial _{u^s}X^{i}-\Omega ^{is}\partial _{u^s} X^{j})\delta '(x-y)\\&\quad (X^s \partial _{u^s}\Gamma _k^{ij}-\Gamma _k^{sj} \partial _{u^s}X^i-\Gamma ^{i s}_k\partial _{u^s}X^j\\&\quad +\Gamma _s^{i j} \partial _{u^k}X^s-\Omega ^{i s}\partial _{u^s}\partial _{u^k} X^j)u_x^k\delta (x--y). \end{aligned}$$

We will use the following two theorems.

Theorem 2.4

[21] Let X be a vector field on M and $\{ . , .\}$ be a PBHT on $\mathfrak {L}(M)$. If $\mathrm {Lie}_X^2 \{.,.\}=0$, then $\mathrm {Lie}_X \{.,.\}$ is a PBHT and it is compatible with $\{.,.\}$, i.e., $\{.,.\} + \lambda \mathrm {Lie}_X \{. , .\}$ is a PBHT for every constant $\lambda$.

Theorem 2.5

[10] The form (2.5) defines a nondegenerate PBHT $\{ . , .\}$ if and only if the matrix $\Omega ^{i j}(u)$ defines a flat contravariant metric on M and $\Gamma _k^{i j}(u)$ are its Christoffel symbols.

From Theorems 2.5 and 2.4, we get the following corollary:

Corollary 2.6

Let $\{.,.\}_2$ and $\{.,.\}_1$ be two nondegenerate compatible PBHT on $\mathfrak {L}(M)$ having the form

$$\begin{aligned} \{u^i(x),u^j(y)\}_\alpha = \Omega _\alpha ^{ij} (u(x))\delta ' (x-y) + \Gamma _{\alpha ,k}^{ij} (u(x)) u_x^k \delta (x -- y),~ \alpha =1,2. \end{aligned}$$

Suppose $\{.,.\}_2+\lambda \{.,.\}_1$ is a nondegenerate PBHT for a generic constant $\lambda$. Then $(\Omega _2,\Omega _1)$ is a FPM on M. Conversely, a FPM on M determines nondegenerate compatible Poisson brackets of hydrodynamic type on $\mathfrak {L}(M)$.

As mentioned in the introduction, if M is a Frobenius manifold of charge d then there is an associated QFPM $(\Omega _2,\Omega _1)$ of degree d on M, where $\Omega _2$ is the intersection form and $\Omega _1$ is the flat metric. In the flat coordinates $(t^1,\ldots ,t^r)$ we have $\tau = \Pi _{i 1} t^i$. Then the Euler vector field E and the identity vector field e of the Frobenius manifold have the form (2.1) and satisfy equations (2.2). The following theorem give a converse statement.

Theorem 2.7

[8] Let M be a manifold carrying a regular QFPM $(\Omega _2,\Omega _1)$ of degree d. Then there exists a unique Frobenius manifold structure on M of charge d where $(\Omega _2,\Omega _1)$ is the associated QFPM.

3 Conjugate Frobenius Manifold

We fix a manifold M with a QFPM $T=(\Omega _2,\Omega _1)$ of degree $d\ne 1$. We fix a function $\tau$ for T which determines the vector fields E and e (see Definition 2.3). We suppose

$$\begin{aligned} e(\tau )=0 \ \ \text {and} \ \ E(\tau )= (1-d) \tau . \end{aligned}$$

(3.1)

We introduce the function $f(\tau ):=(\tau )^{\frac{2}{1-d}}$ and the vector field ${\widetilde{e}} := f(\tau ) e$. We define

$$\begin{aligned} \widetilde{\Omega }_1: = \mathrm {Lie}_{{\widetilde{e}}} \Omega _2 = f \Omega _1 -f'(E \otimes e + e \otimes E). \end{aligned}$$

(3.2)

Then

$$\begin{aligned} \mathrm {Lie}^2_{ {\widetilde{e}}} {\Omega }_2&= f^2 (\mathrm {Lie}_e^2 \Omega _2) + (2(f')^2 E(\tau ) -4f f')e\otimes e +f f'e(\tau ) \Omega _1\nonumber \\&\quad + ((f')^2- f f'') e(\tau )(E\otimes e+e\otimes E)=0 \end{aligned}$$

(3.3)

We fix flat coordinates $(t^1,\ldots ,t^r)$ leading to the identities (2.4). Considering the condition (3.1), we will further assume that $e=\partial _{t^r}$. Thus

$$\begin{aligned} \Omega _1^{i1}=\delta ^{i}_r, \ \ \partial _{t^r} \Omega _2^{i1}=\partial _{t^r}E^i=\delta ^{i}_r. \end{aligned}$$

(3.4)

Let $\{.,.\}$ denote the nondegenerate PBHT associated to $\Omega _2$. Then by Corollary 2.6, $\mathrm {Lie}_{e}\{.,.\}$ is the PBHT associated to $\Omega _1$ and $\mathrm {Lie}_{e}^2\{.,.\}=0$. We have a similar statement for ${{\widetilde{e}}}$.

Proposition 3.1

$\mathrm {Lie}_{{\widetilde{e}}}^2 \{.,.\}=0$. In particular, $\mathrm {Lie}_{{\widetilde{e}}} \{.,.\}$ is a PBHT compatible with $\{.,.\}$.

Proof

The PBHT associated to $\Omega _2$ has the form

$$\begin{aligned} \{t^\alpha (x),t^\beta (y)\} =\Omega _2^{\alpha \beta } \delta ' (x-y) + \Gamma _{2,\gamma }^{\alpha \beta } t^\gamma _x \delta (x-y). \end{aligned}$$

Here and in what follows, it is to be understood that all functions on the right hand side depend on t(x). Note that

$$\begin{aligned} \mathrm {Lie}_{{{\widetilde{e}}}}\{.,.\}(t^\alpha (x),t^\beta (y))={\widetilde{\Omega }}_1^{\alpha \beta }\delta '(x-y)+{\widetilde{\Gamma }}^{\alpha \beta }_{2,\gamma } t_x^\gamma \delta (x-y) \end{aligned}$$

where

$$\begin{aligned} {\widetilde{\Gamma }}_{2,\gamma }^{\alpha \beta }&= { {\widetilde{e}}}^\varepsilon \partial _\varepsilon \Gamma _{2,\gamma }^{\alpha \beta }- \Gamma _{2,\gamma }^{\varepsilon \beta } \partial _\varepsilon { {\widetilde{e}}}^\alpha -- \Gamma _{2,\gamma }^{\alpha \varepsilon } \partial _\varepsilon { {\widetilde{e}}}^\beta + \Gamma _{2,\varepsilon }^{\alpha \beta } \partial _\gamma { {\widetilde{e}}}^\varepsilon -- \Omega ^{\alpha \varepsilon }_2 \partial _{\varepsilon \gamma }^2 { {\widetilde{e}}}^\beta \\&= - \Gamma _{2,\gamma }^{\varepsilon \beta } \delta _r^\alpha \delta _\varepsilon ^1 f' -- \Gamma _{2,\gamma }^{\alpha \varepsilon } \delta _r^\beta \delta _\varepsilon ^1 f' + \Gamma _{2,\varepsilon }^{\alpha \beta } \delta _r^\varepsilon \delta _\gamma ^1 f' -- \Omega ^{\alpha \varepsilon }_2 \delta _r^\beta \delta _\gamma ^1 \delta _\varepsilon ^1 f''. \end{aligned}$$

From Eq. (3.3), the coefficients of $\delta '(x-y)$ of $\mathrm {Lie}_{ {\widetilde{e}}}^2 \{.,.\}$ vanish while the coefficients $\widetilde{\widetilde{\Gamma }}_{2,\gamma }^{\alpha \beta }$ of $\delta (x-y)$ have the form

$$\begin{aligned} \widetilde{\widetilde{\Gamma }}_{2,\gamma }^{\alpha \beta }=&-f f'' \partial _r \Omega _2^{\alpha \varepsilon } \delta _r^\beta \delta _\gamma ^1 \delta _\varepsilon ^1 + f'^2 \delta _r^\alpha \delta _r^\beta \delta _m^1 \delta _\varepsilon ^1 \Gamma _{2,\gamma }^{m \varepsilon } - f'^2 \delta _r^\beta \delta _r^m \delta _\gamma ^1 \delta _\varepsilon ^1 \Gamma _{2,m}^{\alpha \varepsilon } \\&+ f'^2 \delta _r^\beta \delta _r^\alpha \delta _\varepsilon ^1 \delta _m^1 \Gamma _{2,\gamma }^{\varepsilon m }- f'^2 \delta _r^\alpha \delta _r^m \delta _\gamma ^1 \delta _\varepsilon ^1 \Gamma _{2,m}^{\varepsilon \beta }+ f' f'' \Omega _2^{\varepsilon m}\delta _\varepsilon ^1 \delta _r^\alpha \delta _r^\beta \delta _\gamma ^1 \delta _m^1\\&- f'^2 \delta _\gamma ^1 \delta _r^\beta \delta _m^1 \delta ^\varepsilon _r \Gamma _{2,\varepsilon }^{\alpha m} - f'^2 \delta _\gamma ^1 \delta _r^\alpha \delta _m^1 \delta ^\varepsilon _r \Gamma _{2,\varepsilon }^{ m \beta } -\widetilde{\Omega }_2^{\alpha \varepsilon } \delta _r^\beta \delta _\gamma ^1 \delta _\varepsilon ^1 f''. \end{aligned}$$

Then from the identities (2.4) and the definition of $f(\tau )$, it follows that $\widetilde{\widetilde{\Gamma }}_{2,\gamma }^{\alpha \beta }=0$. For example,

$$\begin{aligned} \widetilde{\widetilde{\Gamma }}_{2,1}^{r r}&= - f \partial _r{\Omega _2^{r 1} f''} + f'^2 \Gamma _{2,1}^{1 1}- f'^2 \Gamma _{2,r}^{1 r}+ \Omega _2^{1 1} f'' f'\\&\quad + f'^2 \Gamma _{2,1}^{1 1}- f'^2 \Gamma _{2,r}^{r 1}- f'^2 \Gamma _{2,r}^{r 1} - f'^2 \Gamma _{2,r}^{r 1}- \widetilde{\Omega }_1^{r 1} f''\\&= -(d+1) f'^2 +(1-d) \tau f'f'' -f f'' -(-f) f'=0 \end{aligned}$$

and when $\gamma =1$, $\alpha =r$ and $\beta \ne r$

$$\begin{aligned} \widetilde{\widetilde{\Gamma }}_{2,1}^{r \beta }&= - 2 f'^2 \Gamma _{2,r}^{1 \beta }= - 2 f'^2 \left( \frac{d-1}{2}\delta ^\beta _r+ \partial _{t^r} E^\beta \right) =0. \end{aligned}$$

$\square$

Lemma 3.2

The pair ${{\widetilde{T}}}=(\Omega _2,\widetilde{\Omega }_1)$ form a QFPM of degree ${\widetilde{d}}=2-d$. Moreover, if T is regular then ${{\widetilde{T}}}$ is regular.

Proof

The second term of the identity

$$\begin{aligned} \widetilde{\Omega }_1(t)= f \Omega _1 -f' E^i (\partial _{t^i} \otimes \partial _{t^r} + \partial _{t^r} \otimes \partial _{t^i}) \end{aligned}$$

contributes only to entries of the last row and last column of $\widetilde{\Omega }_1(t)$. From the normalization of $\Omega _1$, we get

$$\begin{aligned} \widetilde{ \Omega }_1^{i1}(t)=(f-f' E(\tau )) \delta ^i_r= (f- (1-d) \tau f') \delta ^i_r= (- f) \delta ^i_r. \end{aligned}$$

Therefore,

$$\begin{aligned} \det \widetilde{\Omega }_1(t)= f^r \det \Omega _1(t) \ne 0. \end{aligned}$$

Hence, using Proposition 3.1 and Corollary 2.6, ${{\widetilde{T}}}$ is a FPM. Let ${\widetilde{\nabla }}$ denote the contravariant (and also the covariant) derivative of ${\widetilde{\Omega }}_1$ and set $\widetilde{\tau }:=-\tau =-t^1$. Then the vector fields

$$\begin{aligned} {\widetilde{e}}:=\widetilde{\nabla }_1 \widetilde{\tau }, ~\text {and} \ \ {\widetilde{E}}:={\nabla }_2 \widetilde{\tau }=-E \end{aligned}$$

satisfy equations (2.2) and

$$\begin{aligned} \mathrm {Lie}_{{\widetilde{E}}} \Omega _2 = \mathrm {Lie}_{-E} \Omega _2= -(d-1)\Omega _2= ({\widetilde{d}}-1) \Omega _2. \end{aligned}$$

(3.5)

Hence, ${{\widetilde{T}}}$ is a QFPM of degree ${\widetilde{d}}=2-d$. For the regularity condition (2.3), we have

$$\begin{aligned} {{\widetilde{R}}}_i^j(t) = \frac{{\widetilde{d}}-1}{2}\delta _i^j + \widetilde{\nabla }_{1i} (-E^j)=\frac{1-d}{ 2} \delta _i^j-{\nabla }_{1 i} (E^j) =- R_i^j(t). \end{aligned}$$

(3.6)

Therefore, $\det ({{\widetilde{R}}}_i^j) \ne 0$ if and only if $\det (R_i^j) \ne 0$. $\square$

We keep the definitions ${\widetilde{\tau }}=-\tau$ and ${{\widetilde{E}}}=-E$ given in the proof of Lemma 3.2 and we call ${{\widetilde{T}}}=(\Omega _2,\widetilde{\Omega }_1)$ the conjugate QFPM of T. The name is motivated by the following corollary.

Corollary 3.3

${{\widetilde{T}}}$ has a conjugate and it equals T.

Proof

We observe that ${{\widetilde{d}}}=2-d\ne 1$ and the function $\widetilde{\tau }=-\tau$ satisfies the requirements (3.1) as

$$\begin{aligned} {\widetilde{e}} (\widetilde{\tau })=0 \ \ \text {and} \ \ {\widetilde{E}} ( \widetilde{\tau })= -E (-t^1)=(1-d) t^1= (1- {\widetilde{d}}) \widetilde{\tau }. \end{aligned}$$

(3.7)

However, applying Lemma 3.2 to ${{\widetilde{T}}}$, we get a QFPM $(\Omega _2,\mathrm {Lie}_{\widetilde{{\widetilde{e}}}}\Omega _2)$ where

$$\begin{aligned} \widetilde{ {\widetilde{e}}}=f( \widetilde{\tau }) {\widetilde{e}}= \widetilde{\tau }^{\frac{2}{1- {\widetilde{d}}}} \,{\widetilde{e}}= (t^1)^{\frac{2}{1- {\widetilde{d}}}}.(t^1)^{\frac{2}{1-{d}}} \partial _{t^r}=e. \end{aligned}$$

$\square$

Now we can prove Theorem 1.1.

Proof of Theorem 1.1

From the work in [8], regularity of the associated QFPM implies that the charge $d\ne 1$. Then the proof follows from applying Lemma 3.2, Corollary 3.3 and Theorem 2.7 to the associated regular QFPM. $\square$

For a fixed Frobenius manifold, the new Frobenius manifold structure constructed using Theorem 1.1 will be called the conjugate Frobenius manifold structure.

Example 3.4

We consider the Frobenius manifold structure of charge -1 defined by the following solution to the WDVV equations.

$$\begin{aligned} \mathbb F=\frac{1}{2} t_2^2 t_1 + t_1^2\log t_1 \end{aligned}$$

In the examples, we use subscript indices instead of superscript indices for convenience. Here, the identity vector field $e=\partial _{t_2}$ and the Euler vector field $E=2 t_1 \partial _{t_1}+t_2 \partial _{t_2}$. Note that $EF=(3-d) F + 2 t_1^2$. The corresponding regular QFPM consists of

$$\begin{aligned} \Omega _2(t) =\left( \begin{array}{cc} 2 t_1 &{}\quad t_2 \\ t_2 &{}\quad 4 \\ \end{array} \right) ,~\Omega _1(t) =\left( \begin{array}{cc} 0 &{}\quad 1 \\ 1 &{}\quad 0 \\ \end{array} \right) . \end{aligned}$$

(3.8)

The conjugate QFPM ${{\widetilde{T}}}=(\Omega _2,{\widetilde{\Omega }}_1)$ is of degree ${{\widetilde{d}}}=3$. In the coordinates

$$\begin{aligned} s_1=-t_1,\quad \ s_2=\frac{t_2}{t_1} \end{aligned}$$

we have

$$\begin{aligned}\Omega _2(s)=\left( \begin{array}{ccc} -2s_1 &{}\quad s_2 \\ s_2 &{}\quad \frac{4}{s_1^2}\\ \end{array} \right) , ~{\widetilde{\Omega }}_1(s)=\left( \begin{array}{ccc} 0 &{}\quad 1 \\ 1 &{}\quad 0\\ \end{array} \right) \end{aligned}$$

and the potential of the conjugate Frobenius manifold structure has the form

$$\begin{aligned} \widetilde{\mathbb F}=\frac{1}{2} s_1 s_2^2- \log s_1. \end{aligned}$$

Note that the Euler vector field ${\widetilde{E}}=-E(s)=-2s_1 \partial _{s_1}+s_2 \partial _{s_2}$ and ${{\widetilde{E}}} {\widetilde{\mathbb F}}=(3-{{\widetilde{d}}}){\widetilde{\mathbb F}}+2$. We observe that applying the inversion symmetry to the potential $\mathbb F(t)$, we get

$$\begin{aligned} {\widehat{\mathbb F}}(z)=\frac{1}{2} z_1 z_2^2- \log z_1+~ \text {constant} \end{aligned}$$

and ${\widehat{\mathbb F}}(z)$ defines the same conjugate Frobenius manifold structure. We prove this for certain type of Frobenius manifolds in next section.

Example 3.5

We consider Frobenius manifold structures found recently in [3] on the orbits space of the reflection group of type $B_4$. It is provided to us by the anonymous reviewer of this article as an example of Frobenius manifold structure whose associated QFPM has a conjugate but it is not regular. The potential of this Frobenius manifold reads

$$\begin{aligned} \mathbb F=\frac{1}{2} t_4^2 t_1+t_2 t_3 t_4-\frac{1}{72}t_1^4+\frac{1}{2} t_3 t_1^2+\frac{1}{6} t_2^2 t_3 t_1-\frac{9 }{4}t_3^2+\frac{1}{108} t_2^4 t_3+\frac{3}{2} t_3^2 \log t_3. \end{aligned}$$

where the charge and degrees given by

$$\begin{aligned} d=\frac{1}{3}, \quad \ d_1=\frac{2}{3},\quad \ d_2=\frac{1}{3},\quad \ d_3=\frac{4}{3},\quad \ d_4=1. \end{aligned}$$

The action of the Euler vector field reads

$$\begin{aligned} E\ \mathbb F(t)= \left( 3-d \right) \mathbb F(t)+ \frac{1}{2}A_{ij} t^i t^j=\left( 3-d \right) \mathbb F(t)+2 t_3^2 \end{aligned}$$

(3.9)

and the intersection metric $\Omega _2$ will be

$$\begin{aligned} {\dot{\Omega }}_2^{i j}(t)= \Omega _2^{ij}(t)+ A^{i j},~~A^{i j}= {\Omega }_1^{i\alpha }(t) {\Omega }_1^{j \beta }(t) A_{\alpha \beta }. \end{aligned}$$

(3.10)

The associated QFPM $T=(\Omega _2,\Omega _1)$ is not regular. However, it has a conjugate QFPM ${{\widetilde{T}}}=(\Omega _2,{\widetilde{\Omega }}_1)$. Flat coordinates $(s_1,s_2,s_3,s_4)$ for ${\widetilde{\Omega }}_1$ are defined by

$$\begin{aligned} t_1= -s_1,\quad \ t_2= s_2,\quad \ t_3= -s_1^3 s_3,\quad \ t_4= -s_4 s_1^3-s_2 s_3 s_1^2 \end{aligned}$$

Note that one can still apply the inversion symmetry to the potential ${\mathbb {F}}$ to get a Frobenius manifold structure with a potential ${\widehat{\mathbb F}}(z)$ [7]. We checked that the QFPM obtained from ${\widehat{\mathbb F}}(z)$ agrees with ${{\widetilde{T}}}$. We do not consider this type of Frobenius manifolds in the next section as we will assume regularity condition (2.3) of the quasihomogeneous flat pencils of metrics.

Let us assume E has the form $E=d_i t^i \partial _{t^i}$. Then $d_1=1-d$ and we have the following standard results.

Corollary 3.6

T is regular QFPM if and only if $d_i \ne \frac{d_1}{2}$ for all i.

Proof

Applying the definition 2.3 to the matrix $R_i^j(t)=(\frac{d-1}{2}+d_i) \delta _i^j=(-\frac{d_1}{ 2}+d_i) \delta _i^j$. $\square$

Lemma 3.7

If $\Omega _1^{i j} \ne 0$, then $d_i+d_j =2-d$. Thus, if the numbers $d_i$ are all distinct then we can choose the coordinates $(t^1,\ldots ,t^r)$ such that $\Omega _1^{i j}=\delta ^{i+j}_{r+1}$.

Proof

Notice that using $[e,E]=e$, we get $\mathrm {Lie}_E \Omega _1 = (d-2) \Omega _1$. Then the statement follows from the equation

$$\begin{aligned} (2-d) \Omega _1^{ij}(t)=\mathrm {Lie}_E \Omega _1^{ij} (dt^i,dt^j) = -d_i \Omega _1 (dt^i,dt^j)-d_j \Omega _1 (dt^i,dt^j). \end{aligned}$$

$\square$

4 Relation with Inversion Symmetry

We continue using notations and assumptions given in the previous section, but we suppose that T is regular. Consider the Frobenius manifold structure defined on M by Theorem 2.7 and let $\mathbb F(t)$ be the corresponding potential. We assume $\Omega ^{ij}_1(t)=\delta ^{i+j}_{r+1}$ which is equivalent to requiring that $\mathbb F(t)$ has the standard form (1.5). We suppose further that the quasihomogeneity condition for $\mathbb F(t)$ takes the form (1.6). In this case the intersection form $\Omega _2$ satisfies [8]

$$\begin{aligned} {\Omega }_2^{ij}(t)=(d-1+d_i+d_j)\Omega ^{i\alpha }_1\Omega ^{j\beta }_1 \partial _{t^\alpha } \partial _{t^\beta } {\mathbb {F}}. \end{aligned}$$

(4.1)

Note that at this stage we are working under the hypothesis of Theorem 1.2.

Let us consider the coordinates (1.10) on $M\backslash \{t^1=0\}$. Then the nonzero entries of the Jacobian matrix are

$$\begin{aligned} \frac{\partial s^{i}}{\partial t^{1}}&=\frac{d_1-2d_i}{d_1} t^i (t^1)^{\frac{-2d_i}{d_1}},\ \ \frac{\partial s^{r}}{\partial t^{1}}= \left( \frac{-2 -d_1}{2 d_1}\right) \sum _2^{r-1} t^{i} t^{r-i+1} (t^1)^{\frac{-2 }{d_1}-2}\\&\quad -\frac{2}{d_1} t^r (t^1)^{\frac{-2}{d_1}-1}, \nonumber \\ \frac{\partial s^{i}}{\partial t^{i}}&= (t^1)^{\frac{d_1-2d_i}{d_1}}, \ \ \frac{\partial s^{r}}{\partial t^{i}}= t^{r-i+1} (t^1)^{\frac{-2 }{d_1}-1},\ \ \frac{\partial s^{r}}{\partial t^{r}}= (t^1)^{\frac{-2 }{d_1}}. \end{aligned}$$

Proposition 4.1

Consider the conjugate QFPM ${{\widetilde{T}}}=(\Omega _2,{\widetilde{\Omega }}_1)$. Then ${\widetilde{\tau }}=s^1$, ${\widetilde{\Omega }}_1^{ij}(s)=\delta ^{i+j}_{r+1}$, ${\widetilde{e}}=\partial _{s^r}$ and ${{\widetilde{E}}}={{\widetilde{d}}}_i s^i\partial _{s^i}$ where the numbers ${{\widetilde{d}}}_i$ are given in (1.9).

Proof

Using the duality between the degrees outlined in Lemma 3.7, we calculate the entries ${\widetilde{\Omega }}_1^{ij}(s)$ as follows.

• (I) For $i=1$

  $$\begin{aligned}\widetilde{\Omega }_1^{1j}(s)= - \frac{\partial s^{j}}{\partial t^{\alpha }} \widetilde{\Omega }_1^{1\alpha }= - \frac{\partial s^{r}}{\partial t^{r}}\widetilde{\Omega }_1^{1r} =- \frac{\partial s^{r}}{\partial t^{r}}(-(t^1)^{\frac{2}{d_1}}) \delta ^{1r}=\delta ^{1}_r.\end{aligned}$$

• (II) For $1< i < r$ and $1< j< r$

  $$\begin{aligned} \widetilde{\Omega }_1^{ij}(s)= & {} \frac{\partial s^{i}}{\partial t^{k}} \frac{\partial s^{j}}{\partial t^{k}} \widetilde{\Omega }_1^{k l}\\= & {} \frac{\partial s^{i}}{\partial t^{1}} \frac{\partial s^{j}}{\partial t^{1}} \widetilde{\Omega }_1^{11}+ \frac{\partial s^{i}}{\partial t^{i}} \frac{\partial s^{j}}{\partial t^{1}} \widetilde{\Omega }_1^{i1}+ \frac{\partial s^{i}}{\partial t^{1}} \frac{\partial s^{j}}{\partial t^{j}} \widetilde{\Omega }_1^{1j}+ \frac{\partial s^{i}}{\partial t^{i}} \frac{\partial s^{j}}{\partial t^{j}} \widetilde{\Omega }_1^{ij}\\= & {} \frac{\partial s^{i}}{\partial t^{i}} \frac{\partial s^{j}}{\partial t^{j}} \widetilde{\Omega }_1^{ij} \delta ^{i+j,r+1}\\= & {} (t^1)^{\frac{2d_1-2d_i-2d_{r-i+1}+2}{d_1}} \delta ^{i+j,r+1}\\= & {} \delta ^{i+j,r+1}. \end{aligned}$$

• (III) For $1< i < r$

  $$\begin{aligned} \widetilde{\Omega }_1^{i r}(s)= & {} (t^1)^{\frac{2}{d_1}} \frac{\partial s^{i}}{\partial t^{i}}\frac{\partial s^{r}}{\partial t^{r-i+1}}+ \left( -(t^1)^{\frac{2}{d_1}} \frac{\partial s^{i}}{\partial t^{1}}+\frac{-2 d_i}{d_1} t^i (t^1)^{\frac{2}{d_1}-1} \frac{\partial s^{i}}{\partial t^{i}}\right) .\frac{\partial s^{r}}{\partial t^{r}}\\= & {} (t^1)^{\frac{2}{d_1}} (t^1)^{\frac{d_1-2d_i}{d_1}}.t^{i} (t^1)^{\frac{-2 }{d_1}-1}+ \left( -\frac{d_1-2d_i}{d_1} (t^1)^{\frac{2}{d_1}} t^i (t^1)^{\frac{-2d_i}{d_1}}\right. \\&\left. +\frac{-2 d_i}{d_1} t^i (t^1)^{\frac{2}{d_1}-1} (t^1)^{\frac{d_1-2d_i}{d_1}}\right) (t^1)^{\frac{-2 }{d_1}}\\= & {} (t^1)^{\frac{-2d_i }{d_1}}t^{i} + \left( -\frac{d_1-2d_i}{d_1} (t^1)^{\frac{-2d_i}{d_1}} t^i +\frac{-2 d_i}{d_1} (t^1)^{\frac{-2d_i}{d_1}}t^i \right) \\= & {} (t^1)^{\frac{-2d_i }{d_1}}t^{i} - (t^1)^{\frac{-2d_i}{d_1}}t^{i} \\= & {} 0. \end{aligned}$$

• (IV) Finally,

  $$\begin{aligned}&\widetilde{\Omega }_1^{rr}(s)\\&= -(t^1)^{\frac{2}{d_1}} \frac{\partial s^{r}}{\partial t^{r}}.\frac{\partial s^{r}}{\partial t^{1}}+\sum _{i=2}^{r-1} \left( (t^1)^{\frac{2}{d_1}} \frac{\partial s^{r}}{\partial t^{r-i+1}} -\frac{2 d_i}{d_1} t^i (t^1)^{\frac{2}{d_1}-1} \frac{\partial s^{r}}{\partial t^{r}}\right) .\frac{\partial s^{r}}{\partial t^{i}}\\&+\left( -(t^1)^{\frac{2}{d_1}} \frac{\partial s^{r}}{\partial t^{1}}+\sum _{i=2}^{r-1}-\frac{2 d_i}{d_1} t^i (t^1)^{\frac{2}{d_1}-1} \frac{\partial s^{r}}{\partial t^{i}} + \frac{-4}{d_1} t^r (t^1)^{\frac{2}{d_1}-1} \frac{\partial s^{r}}{\partial t^{r}}\right) .\frac{\partial s^{r}}{\partial t^{r}}\\&=(\frac{2}{d_1}+1) \sum _2^{r-1} t^i t^{r-i+1} (t^1)^{\frac{-2}{d_1}-2} +\frac{4}{d_1} t^r (t^1)^{\frac{-2}{d_1}-1}+\sum _2^{r-1} t^i t^{r-i+1} (t^1)^{\frac{-2}{d_1}-2}\\&-\sum _2^{r-1} \frac{2d_i}{d_1}t^i t^{r-i+1} (t^1)^{\frac{-2}{d_1}-2}-\sum _2^{r-1}\frac{2d_{r-i+1}}{d_1} t^i t^{r-i+1} (t^1)^{\frac{-2}{d_1}-2}-\frac{4}{d_1} t^r (t^1)^{\frac{-2}{d_1}-1}\\&=\sum _2^{r-1} \left( \frac{2}{d_1}+2 -\frac{2d_i}{d_1}-\frac{2d_{r-i+1}}{d_1} \right) t^i t^{r-i+1} (t^1)^{\frac{-2}{d_1}-2}\\&=0. \end{aligned}$$

It is straightforward to show that ${\widetilde{e}}=\partial _{s^r}$. The vector field ${{\widetilde{E}}}=\Omega _2^{1j}(s)\partial _{s^j}$ while

$$\begin{aligned} {\Omega }_2^{1j}(s)&=\begin{pmatrix} d_1 t^1&-d_1 t^1 \frac{\partial s^2}{\partial t^1} -d_2 t^2 \frac{\partial s^2}{\partial t^2}&\,&-d_1 t^1 \frac{\partial s^3}{\partial t^1} -d_3 t^3 \frac{\partial s^3}{\partial t^3}&\,&\cdots&-d_1 t^1\frac{\partial s^r}{\partial t^1} -d_2 t^2\frac{\partial s^r}{\partial t^2}+\cdots -t^r \frac{\partial s^r}{\partial t^r} \end{pmatrix} \nonumber \\&= \begin{pmatrix} d_1 t^1&(d_2-d_1) t^2 (t^1)^{\frac{d_1-2d_2}{d_1}}&(d_3-d_1) t^3 (t^1)^{\frac{d_1-2d_3}{d_1}}&\cdots&\sum _{i=1}^{r} (-d_1 (\frac{-2-d_1}{2 d_1}) -d_i )t^i t^{r-i+1} (t^1)^{\frac{-2}{d_1}-1} \end{pmatrix} \nonumber \\&= \begin{pmatrix} d_1 t^1&(d_2-d_1) t^2 (t^1)^{\frac{d_1-2d_2}{d_1}}&(d_3-d_1) t^3 (t^1)^{\frac{d_1-2d_3}{d_1}}&\cdots&\frac{1}{2}\sum _{i=1}^{r} t^i t^{r-i+1} (t^1)^{\frac{-2}{d_1}-1}\end{pmatrix}\nonumber \\&= \begin{pmatrix} - d_1 s^1&\ \ \ (d_2-d_1) s^2&\ \ \ \ \ \ \ \ \ \ \ (d_3-d_1) s^3&\,&\ \ \ \ \ \ \cdots&\,&\ \ s^r \end{pmatrix}. \end{aligned}$$

(4.2)

$\square$

We observe that the inverse transformation of the inversion symmetry (1.7) is given by

$$\begin{aligned} t^1=\frac{-1}{z^1},\quad \ t^r=z^r + \frac{1}{2} \sum _2^{r-1} \frac{ z^i z^{r-i+1}}{z^1},\quad \ t^k= \frac{-z^k}{z^1}, ~2\le k\le r. \end{aligned}$$

Thus, the potential (1.8) obtained from applying the inversion symmetry to $\mathbb F(t)$ has the form

$$\begin{aligned} \widetilde{\mathbb F}(z) = (z^1)^{2} \mathbb F\left( \frac{-1}{z^1},\frac{-z^2}{z^1},\ldots ,\frac{-z^{r-1}}{z^1},\frac{1}{2} \sum _1^r \frac{z^i z^{r-i+1}}{z^1}\right) +\frac{1}{2} z^r \sum _{1}^{r} z^i z^{r-i+1}. \end{aligned}$$

Lemma 4.2

The potential ${\widetilde{\mathbb F}}(z)$ has the form

$$\begin{aligned} \widetilde{\mathbb F}(s) = (t^1)^{ \frac{-4}{d_1}} \left( \mathbb F(t^1,\ldots ,t^r) -\frac{1}{2} t^r \sum _{1}^{r} t^i t^{r-i+1}\right) , z^i\leftrightarrow s^i. \end{aligned}$$

(4.3)

Proof

We use the identities

$$\begin{aligned}&t^1=- s^1= (s^1)^2 (\frac{-1}{s^1}),~ t^r= (s^1)^{\frac{2}{d_1}} \left( \frac{1}{2} \sum _1^r \frac{s^i s^{r-i+1}}{s^1} \right) ,\\&t^i=(s^1)^{\frac{2 d_i}{d_1}} (\frac{-s^i}{s^1}), 1<i<r, \end{aligned}$$

and the quasihomogeneity of the potential $\mathbb F(t)$, i.e.,

$$\begin{aligned} \left( \frac{2}{d_1} E\right) \mathbb F(t)= \frac{2(3-d)}{d_1} \mathbb F(t) =\left( \frac{4}{d_1}+2\right) \mathbb F(t). \end{aligned}$$

(4.4)

Then

$$\begin{aligned}&(t^1)^{ \frac{-4}{d_1}} \left[ \mathbb F(t^1,\ldots ,t^r) -\frac{1}{2} t^r \sum _{1}^{r} t^i t^{r-i+1}\right] \\&=(t^1)^{ \frac{-4}{d_1}} \left[ \mathbb F(t^1,\ldots ,t^r)+\big ( -t^1 (t^r)^2\big )-\frac{1}{2} t^r \sum _{2}^{r-1} t^i t^{r-i+1}\right] \\&=(s^1)^{ \frac{-4}{d_1}} \left[ \mathbb F\left( (s^1)^2 \left( \frac{-1}{s^1}\right) ,(s^1)^{\frac{2 d_2}{d_1}} \left( \frac{-s^2}{s^1}\right) ,\ldots ,(s^1)^{\frac{2 d_{r-1}}{d_1}} \left( \frac{-s^{r-1}}{s^1}\right) ,\right. \right. \\&\quad \left. \left. (s^1)^{\frac{2}{d_1}} \left( \frac{1}{2} \sum _1^r \frac{s^i s^{r-i+1}}{s^1}\right) \right) \right. \\&\quad \left. +\big ((s^r)^2 (s^1)^{\frac{4}{d_1}+1}\right. \big . \left. \big . +s^r \sum _2^{r-1} s^i s^{r-i+1} (s^1)^{\frac{4}{d_1}} + s^1 \left( \frac{1}{2}\sum _{2}^{r-1} (s^1)^{\frac{2}{d_1}-1} s^i s^{r-i+1}\right) ^2\Bigg )\right. \\&\left. \quad - \frac{1}{2}s^r (s^1)^\frac{4}{d_1} \sum _{2}^{r-1} s^i s^{r-i+1} - s^1 \left( \frac{1}{2}\sum _{2}^{r-1} (s^1)^{\frac{2}{d_1}-1} s^i s^{r-i+1}\right) ^2\right] \\&=(s^1)^{\frac{-4}{d_1}} \left[ (s^1)^{\frac{4}{d_1}+2} \mathbb F\left( \frac{-1}{s^1},- \frac{s^2}{s^1} ,-\frac{s^3}{s^1},\ldots , \frac{1}{2} \sum _{i=1}^{n} \frac{-s^i s^{n-i+1}}{s^1} \right) +(s^r)^2 (s^1)^{\frac{4}{d_1}+1}\right. \\&\left. \quad + \frac{1}{2} s^r \sum _2^{r-1} s^i s^{r-i+1} (s^1)^{\frac{4}{d_1}} \right] \\&=(s^1)^{2} \mathbb F\left( \frac{-1}{s^1},\frac{-s^2}{s^1},\ldots ,\frac{-s^{r-1}}{s^1},\frac{1}{2} \sum _1^r \frac{s^i s^{r-i+1}}{s^1}\right) + \frac{1}{2} s^r \sum _1^{r} s^i s^{r-i+1} \end{aligned}$$

which is the potential of the inversion symmetry by setting $s^i=z^i$. $\square$

Now we prove Theorem 1.2 stated in the introduction.

Proof of Theorem 1.2

By Corollary 3.6 and Theorem 1.1, we use the above notations and assume $T=(\Omega _2,\Omega _1)$ is the associated QFPM. We need to show that the conjugate QFPM ${{\widetilde{T}}}=(\Omega _2,{\widetilde{\Omega }}_1)$ equals the QFPM associated to the potential ${\widetilde{\mathbb F}}(s)$ given in (4.3). This leads to verifying that $\Omega _2 (s)$ equals the intersection form $\widehat{\Omega }_2(s)$ defined by ${\widetilde{\mathbb F}}(s)$. It is straightforward to show that ${{\widetilde{F}}}(s)$ is a quasihomogenius function, i.e., ${{\widetilde{E}}} {{\widetilde{F}}}=(3-{{\widetilde{d}}}){{\widetilde{F}}}$. Hence

$$\begin{aligned} {{\widehat{\Omega }}}_2^{ij}(s):=({{\widetilde{d}}}-1+{{\widetilde{d}}}_i+{{\widetilde{d}}}_j)\Omega ^{i\alpha }_1\Omega ^{j\beta }_1 \partial _{s^\alpha } \partial _{s^\beta } \widetilde{\mathbb F}. \end{aligned}$$

After long calculations we find that $\widetilde{\Omega }_2^{ij}(s) =\widehat{\Omega }_2^{ij}(s)$. For examples, we obtained the first row of ${\Omega }_2^{ij}(s)$ in (4.2) and for even r and $1< i,j< r$, we get by denoting $\partial _{t^i}\partial _{t^j} G(t)$ as $G_{i,j}$

$$\begin{aligned} {\Omega }_2^{ij} (s)&= \frac{\partial s^i}{\partial t^1} \frac{\partial s^j}{\partial t^1} \Omega _2^{1,1} +\frac{\partial s^i}{\partial t^i} \frac{\partial s^j}{\partial t^1} \Omega _2^{i,1} + \frac{\partial s^i}{\partial t^1} \frac{\partial s^j}{\partial t^j} \Omega _2^{1,j} +\frac{\partial s^i}{\partial t^i} \frac{\partial s^j}{\partial t^j} \Omega _2^{i,j} \nonumber \\&=d_1\left( 1-\frac{2d_i}{d_1}\right) \left( 1-\frac{2d_j}{d_1}\right) t^i t^{j} (t^1)^{1-\frac{2d_i}{d_1}-\frac{2d_j}{d_1}}\nonumber \\&\quad + d_i \left( 1-\frac{2d_j}{d_1}\right) t^i t^j (t^1)^{1-\frac{2d_i}{d_1}-\frac{2d_j}{d_1}}\nonumber \\&\quad + d_j \left( 1-\frac{2d_i}{d_1}\right) t^i t^j (t^1)^{1-\frac{2d_i}{d_1}-\frac{2d_j}{d_1}}\nonumber \\&\quad + (d-1+d_i+d_j) (t^1)^{2-\frac{2d_i}{d_1}-\frac{2d_j}{d_1}} (G_{r-i+1,n-j+1} + t^r \delta ^{r,i+j})\nonumber \\&=(d_1-d_i-d_j) t^i t^j (t^1)^{1-\frac{2d_i}{d_1}-\frac{2d_j}{d_1}}\nonumber \\&\quad + (-d_1+d_i+d_j) (t^1)^{2-\frac{2d_i}{d_1}-\frac{2d_j}{d_1}} \left( G_{r-i+1,r-j+1} + t^r \delta ^{r,i+j}\right) \nonumber \\&= (d_1-d_i-d_j) (t^1)^{1-\frac{2 d_i}{d_1}-\frac{2 d_j}{d_1}} \left( t^i t^j- t^1 G_{r-i+1,r-j+1} -t^1 t^r \delta ^{r, i+j} \right) . \end{aligned}$$

(4.5)

On the other hand

$$\begin{aligned}&\frac{\partial ^2 \widetilde{\mathbb F}}{\partial {s^{r-i+1}} \partial {s^{r-j+1}}}\nonumber \\&= \left( t^r \delta _{r,i+j} (t^1)^{1-\frac{2}{d_1}-\frac{2d_{r-i+1}}{d_1}}+ G_{r-i+1,r-j+1} (t^1)^{-1-\frac{4}{d_1}+\frac{2d_{r-i+1}}{d_1}} \right) \nonumber \\&\quad \times \left( -(s^1)^{\frac{2d_{r-j+1}}{d_1}-1} \right) + \left( t^i (t^1)^{1-\frac{2}{d_1}-\frac{2d_i}{d_1}} \right) \left( s^i (s^1)^{\frac{2}{d_1}-1} \right) \nonumber \\&=\left( t^r \delta _{r,i+j} (t^1)^{2-\frac{2d_i}{d_1}-\frac{2d_j}{d_1}}+ G_{r-i+1,r-j+1} (t^1)^{-2-\frac{4}{d_1}+\frac{2d_{r-i+1}}{d_1}+\frac{2d_{r-j+1}}{d_1}} \right) \nonumber \\&\quad - \left( t^i t^j (t^1)^{2-\frac{2d_i}{d_1}-\frac{2d_j}{d_1}} \right) \nonumber \\&=(t^1)^{1-\frac{2d_i}{d_1}-\frac{2d_j}{d_1}} \left( t^r \delta ^{r,i+j} t^1 + G_{r-i+1,r-j+1} t^1- t^i t^j \right) . \end{aligned}$$

(4.6)

Therefore,

$$\begin{aligned} \widehat{\Omega }_2^{ij}(s)= & {} (d_i+d_j-d_1)(t^1)^{1-\frac{2d_i}{d_1}-\frac{2d_j}{d_1}} \left( t^r t^1 \delta ^{r,i+j} + G_{r-i+1,r-j+1} t^1- t^i t^j \right) \nonumber \\= & {} \Omega _2^{ij}(s). \end{aligned}$$

(4.7)

$\square$

Example 4.3

Consider the following solution to WDVV equations

$$\begin{aligned} \mathbb F=\frac{t_1^3}{6}-\frac{1}{2} t_2^2 t_1+\frac{1}{2} t_2^2 t_3+\frac{1}{2} t_1 t_3^2. \end{aligned}$$

(4.8)

It corresponds to a trivial Frobenius manifold structure, i.e., Frobenius algebra structure does not depend on the point. Here the charge $d=0$, the Euler vector field $E=\sum t_i \partial _{t_i}$ and identity vector field $e=\partial _{t_3}$. The intersection form is

$$\begin{aligned} \Omega _2(t) =\left( \begin{array}{ccc} t_1 &{}\quad t_2 &{}\quad t_3 \\ t_2 &{}\quad t_3-t_1 &{}\quad -t_2 \\ t_3 &{}\quad -t_2 &{}\quad t_1 \\ \end{array} \right) \end{aligned}$$

Setting

$$\begin{aligned} s_1=-t_1,\ \ s_2=\frac{t_2}{t_1},\ \ s_3=\frac{t_2^2}{2 t_1^3}+\frac{t_3}{t_1^2} \end{aligned}$$

the conjugate QFPM has ${\widetilde{\Omega }}_1^{ij}(s)=\delta ^{i+j}_3$ and

$$\begin{aligned}\Omega _2(s)=\left( \begin{array}{ccc} -s_1 &{} 0 &{} s_3 \\ 0 &{} s_3+\frac{3 s_2^2}{2 s_1}+\frac{1}{s_1} &{} -\frac{s_2^3}{s_1^2}-\frac{2 s_2}{s_1^2} \\ s_3 &{} -\frac{s_2^3}{s_1^2}-\frac{2 s_2}{s_1^2} &{} \frac{3 s_2^4}{4 s_1^3}+\frac{3 s_2^2}{s_1^3}-\frac{1}{s_1^3} \\ \end{array} \right) \end{aligned}$$

The potential of the conjugate Frobenius manifold structure reads

$$\begin{aligned} \widetilde{\mathbb F}(s)=\frac{-1}{6 s_1} +\frac{s_2^2}{2 s_1} +\frac{s_2^4}{8 s_1}+\frac{1}{2} s_2^2 s_3+\frac{1}{2} s_1 s_3^2. \end{aligned}$$

One can check that this is the same potential obtained by applying the inversion symmetry to $\mathbb F(t)$. Note that ${{\widetilde{E}}}=-s_1 \partial _{s_1}+s_3 \partial _{s_3}$ and ${\widetilde{E}} \widetilde{\mathbb F}= \widetilde{\mathbb F}$.

5 The Conjugate of a Polynomial Frobenius Manifold

In this section, we recall the construction of Frobenius manifolds on the space of orbits of Coxeter groups given in [9] and we apply the results of this article.

We fix an irreducible Coxeter group ${{\mathcal {W}}}$ of rank r. We consider the standard real reflection representation $\psi : {{\mathcal {W}}}\rightarrow GL(V)$, where V is a complex vector space of dimension r. Then the orbits space $M=V/{{\mathcal {W}}}$ is a variety whose coordinate ring is the ring of invariant polynomials $\mathbb C[V]^{{\mathcal {W}}}$. Using the Shephard-Todd-Chevalley theorem, the ring $\mathbb C[V]^{{\mathcal {W}}}$ is generated by r algebraically independent homogeneous polynomials. Moreover, the degrees of a complete set of generators are uniquely specified by the group [16].

We fix a complete set of homogeneous generators $u^1,u^2,\ldots ,u^r$ for $\mathbb C[V]^{{\mathcal {W}}}$. Let $\eta _i$ be the degree of $u^i$. Here, we have

$$\begin{aligned} 2=\eta _1<\eta _2\le \eta _3\le \cdots \le \eta _{r-1}< \eta _r. \end{aligned}$$

It is known that $\eta _i+\eta _{r-i+1}=\eta _r+\eta _1$. Consider the invariant bilinear form on V under the action of ${{\mathcal {W}}}$. Then it defines a contravariant flat metric $\Omega _2$ on M and we let $u^1$ equals its quadratic form. We fix the vector field $e:=\partial _{u^r}$. There is another flat contravariant metric $\Omega _1:=\mathrm {Lie}_{e}\Omega _2$ on M, which was initially studied by K. Saito [19, 20] and it is called the Saito flat metric. Then $T:=(\Omega _2,\Omega _1)$ is a FPM and Dubrovin proved the following theorem.

Theorem 5.1

[8] $T=(\Omega _2,\Omega _1)$ is a regular QFPM of charge $\frac{\eta _r-2}{\eta _r}$ and leads to a polynomial Frobenius manifold structure on M, i.e., the corresponding potential is a polynomial function in the flat coordinates.

We observe that the polynomial Frobenius structure defined by T has $\tau =\frac{1}{\eta _r} u^1$, the Euler vector field $E=\frac{1}{\eta _r}\sum _i\eta _i u^i \partial _{u^i}$, the identity vector field e and degrees $\frac{\eta _i}{\eta _r}$. Note that E is independent of the choice of generators but e is defined up to a constant factor. Thus, changing the set of generators will lead to an equivalent Frobenius manifold structure [9]. The following theorem was conjectured by Dubrovin and proved by C. Hertling.

Theorem 5.2

[15] Any semisimple polynomial Frobenius manifold with positive degrees is isomorphic to a polynomial Frobenius structure constructed on the orbits space of the standard real reflection representation of a finite irreducible Coxeter group.

Clearly, T satisfies the hypotheses of Theorem 1.1 and we have a conjugate regular QFPM ${{\widetilde{T}}}:=(\Omega _2, \mathrm {Lie}_{{{\widetilde{e}}}} \Omega _2)$, where ${\widetilde{e}}= (\tau )^{\eta _r} e$. Moreover, from the work of K. Saito and his collaborators (see also [9]), we can fix $u^1,\ldots , u^r$ to be flat with respect to $\Omega _1$ and the potential of the polynomial Frobenius manifold will have the standard form (1.5). In particular ${{\widetilde{T}}}$ is the regular QFPM of the Frobenius manifold structure obtained by applying inversion symmetry to the polynomial Frobenius manifold on M. Considering Theorem 5.2, we wonder what is the intrinsic description for the conjugate Frobenius manifold as this may help in the classification of Frobenius manifolds.

In [1], we give a similar discussion for the r Frobenius manifold structures constructed in [22] on the orbits space M when ${{\mathcal {W}}}$ is of type $B_r$ or $D_r$.

6 Remarks

It is important to mention that the inversion symmetry of the WDVV equation can be applied to a solution $\mathbb F(t)$ in the standard form (1.5) under more general quasihomogeneity condition than condition (1.6) and without the regularity condition (2.3) of the associated QFPM . In this case, if the conjugate Frobenius manifold structure exists, we believe that it will be equivalent to Frobenius manifold structure obtained by applying the inversion symmetry, we confirm this by Example 3.4 and Example 3.5.

Note that Frobenius manifold structures which are invariant under inversion symmetry were studied in [18]. We did not consider these cases as the charge will equal 1.

It will be interesting to study the consequences of Theorem 1.2 on the interpretation of the inversion symmetry in terms of the action of the Givental groups obtained in [13] and the relation found in [17] between the principle hierarchies and tau functions of the two solutions to the WDVV equations related by the inversion symmetry. We also believe that the findings in this article can be generalized to the theory of bi-flat F-manifolds [2].

It is known that the leading term of a certain class of compatible local Poisson structures leads to a regular QFPM and thus to a Frobenius structure [8, 11]. Polynomial Frobenius manifolds obtained in [4] are constructed by fixing the regular nilpotent orbit in a simple Lie algebra and uses compatible local Poisson brackets obtained by Drinfeld-Sokolov reduction. In these cases, the Poisson brackets form an exact Poisson pencil, and thus their central invariants are constants [14]. If the Lie algebra is simply-laced, then the central invariants are equal [12] which means the Poisson structures are consistent with the principle hierarchy associated with the Frobenius manifold [11]. Fix one of these polynomial Frobenius structures and denote the associated local Poisson brackets by ${\mathbb {B}}_{2}$ and ${\mathbb {B}}_{1}$ (here ${\mathbb {B}}_{2}$ is the classical W-algebra). In the flat coordinates, these local Poisson brackets form an exact Poisson pencil under the identity vector field e, i.e., $\mathrm {Lie}_e{\mathbb {B}}_{2}={\mathbb {B}}_{1}$ and $\mathrm {Lie}_e{\mathbb {B}}_{1}=0$. Let us denote the leading term of ${\mathbb {B}}_{2}$ by $B_2$ and ${{\widetilde{e}}}$ is the vector field associated with the conjugate Frobenius manifold structure. We proved in this article that $\mathrm {Lie}_{{{\widetilde{e}}}}^2 B_2=0$. Then it is natural to ask if ${{\widetilde{e}}}$ also leads to an exact Poisson pencil, i.e., $\mathrm {Lie}_{{{\widetilde{e}}}}^2{\mathbb {B}}_{2}=0$. Our calculations for the simple Lie algebra of type $A_3$, shows that this is not true.