On Symmetry Properties of Frobenius Manifolds and Related Lie-Algebraic Structures

Abstract

The aim of this paper is to develop an algebraically feasible approach to solutions of the oriented associativity equations. Our approach was based on a modification of the Adler–Kostant–Symes integrability scheme and applied to the co-adjoint orbits of the diffeomorphism loop group of the circle. A new two-parametric hierarchy of commuting to each other Monge type Hamiltonian vector fields is constructed. This hierarchy, jointly with a specially constructed reciprocal transformation, produces a Frobenius manifold potential function in terms of solutions of these Monge type Hamiltonian systems.

1. The Introductory Setting

Let us start with an interesting mathematical structure, suggested in [1,2,3,4,5], on the space of smooth functions: consider a real-valued $C^{\infty }$-smooth differentiable Frobenius manifold potential function $F\in C^{\infty }({\mathbb{R}}^n;\mathbb{R})$ and denote their partial derivatives as

$$F_{ij}(t):=\frac{{\partial }^{2}F(t)}{\partial t_i\partial t_j},F_{ijk}(t):=\frac{{\partial }^{3}F(t)}{\partial t_i\partial t_j\partial t_k}$$

(1)

for $i,j$, and $k=\overline{1,n},$ $n\in \mathbb{N}$. These partial derivatives are symmetrical, with respect to permutations of their indices. Let us assume additionally that the symmetric matrix $\eta :=\{{\eta }_{ij}(t):=F_{ij1}(t):i,j=\overline{1,n}\}$ is non-degenerate, and call it an induced metric on the ${\mathbb{R}}^n.$ In addition,

$$F_{ijk}(t)=\sum _{s\in \overline{1,n}}{\eta }_{is}(t)C_{ij}^s(t),$$

(2)

where, by definition,

$$C_{ij}^s(t):=\sum _{k\in \overline{1,n}}{F}_{ijk}(t){\eta }^{ks}(t),\sum _{k\in \overline{1,n}}{\eta }^{sk}(t){\eta }_{kj}(t)={\delta }_j^s$$

(3)

for all $i,j$, and $s\in \mathbb{N}.$ Assume now that the set ${\mathbb{R}}^n$ represents a local coordinate frame [6,7] of an a finite-dimensional manifold $M.$ Then its tangent space $T_t(M)$ at a point $t\in M$ is described by means of the local vector field system $\{\partial /\partial t_i\in T_t(M):i=\overline{1,n}\},$ which a priori commute to each other: $[\partial /\partial t_i,\partial /\partial t_j]=0$ for all $i,j=\overline{1,n}.$ Let us now assume that the manifold M is a Frobenius manifold [8,9,10], i.e., its tangent space $T_t(M)$ at any point $t\in M$ forms an associative Frobenius algebra $F_M$ with respect to some multiplication $“\circ ”$ on $F_M:$

$$\partial /\partial t_i\circ \partial /\partial t_j:={\sum }_{s\in \overline{1,n}}{C}_{ij}^s(t)\partial /\partial t_s,(\partial /\partial t_i\circ \partial /\partial t_j)\circ \partial /\partial t_s=\partial /\partial t_i\circ (\partial /\partial t_j\circ \partial /\partial t_s)$$

(4)

for any $i,j$ and $s=\overline{1,n}$ with the structure constants defined by the expression (3). Define now a set of matrices $C_i(t):=\{C_{ij}^k(t)=C_{ji}^k(t):j,k\in \overline{1,n}\},i=\overline{1,n}.$ Then, as it easily follows from (4), the structure constants (3) should satisfy the following additional constraints:

$$[C_i(t),C_j(t)]=0,\partial C_i(t)/\partial t_j=\partial C_j(t)/\partial t_i$$

(5)

for any $t\in M$ and all $i,j=\overline{1,n}.$ (5) are called the Witten–Dijkgraaf–Verlinde–Verlinde, or oriented associativity WDVV equations. These equations were first investigated in [11,12,13] for problems related with topological and string quantum field theory of elementary particles. A nice introduction into the topic can be found in B. Dubrovin Lecture Notes [2]. Lie-algebraic aspects of these equations and related integrability properties can be found in recent works [14,15].

The notion of a Frobenius manifold was first axiomatized and thoroughly studied by B. Dubrovin [2,3,4,5] in the early nineties, and plays a central role in mirror field theory symmetry [16,17,18], theory of unfolding spaces of singularities [19], quantization theory [20,21], quantum cohomology [8], and integrability theory [1,19,22,23,24,25,26,27,28,29,30,31] of dispersion-less many-dimensional systems.

A full Frobenius structure on M consists of the data $(\circ ,e,\eta ,E).$ Here $\circ :T(M){\otimes }_{S}T(M)\to T(M)$ is an associative and commutative multiplication on the tangent sheaf, so that $T(M)$ becomes a sheaf of commutative algebras over the ring $\mathbb{R}\{t\}$ of convergent series with identity $e\in T(M),\eta$ is a metric on M (non-degenerate quadratic form $T(M){\otimes }_{S}T(M)$), and E is a so called Euler vector field. These structures are connected by various constraints and compatibility conditions, and are presented in [2,3] and [32,33]. For example, the metric $\eta$ must be flat and $“\circ ”$–invariant, i.e., ${⟨a|b\circ c⟩}_{\eta }={⟨a\circ b|c⟩}_{\eta }$ for the metric ${⟨·|·⟩}_{\eta }$ on M and any $a,b$, and $c\in T(M).$ Various weaker versions of the Frobenius structure are interesting in themselves and also appear in [19,20,21] in different contexts.

Let us also mention an additional notion of a unital Frobenius manifold $F_M,$ introduced in [10] and further studied in [9]. This structure consists of an associative and commutative multiplication $“\circ ”$ on the tangent sheaf as above, satisfying the following properties: $1^0$) a flat structure $T(M)$ on M subject to a flat connection $d_{\omega }:\mathsf{\Gamma }(\mathsf{\Lambda }(M)\otimes T(M))\to \mathsf{\Gamma }(\mathsf{\Lambda }(M)\otimes T(M)),d_{\omega }{d}_{\omega }=0,$ is compatible with a multiplication $“\circ ”,$ if in a neighborhood of any point there exists a vector field $C\in \mathsf{\Gamma }(T(M)),$ such that for arbitrary local flat vector fields $X,Y\in \mathsf{\Gamma }(T(M))$ one has

$$X\circ Y=[X,[Y,C]],$$

(6)

where $C\in \mathsf{\Gamma }(T(M))$ is called a local vector potential for ∘; $2^0)$ $T(M)$ is called compatible with $(\circ ,e),$ $e\in \mathsf{\Gamma }(T(M))$ is an identity element, if $1^0)$ holds and moreover, the identity element $e:=\partial /\partial t_1$ is flat, that is the corresponding covariant derivative ${\nabla }_X^{\omega }e=0$ for any $X\in \mathsf{\Gamma }(T(M)).$ From (6) one easily ensues the relationships (5), where

$$C_{ij}^k(t)=\partial /\partial t_i\partial /\partial t_j{C}^k(t),\partial /\partial t_1\circ \partial /\partial t_i=\partial /\partial t_i,$$

(7)

for any $i,j$, and $k=\overline{1,n}$ and $t\in M.$

As a very interesting example of the above construction can be obtained for the special case $n=3.$ We can take into account a reduction of the commuting matrices $C_j\in \mathrm{End}$ ${\mathbb{E}}^3,j=\overline{1,3},$ presented in [1,2,3]. Namely, assume that a smooth Frobenius manifold potential function $F\in C^{\infty }({\mathbb{R}}^n;\mathbb{R})$ is representable as

$$F(t)=\frac{1}{2}{t}_1^2{t}_3+\frac{1}{2}{t}_1{t}_2^2+f(t_1,t_2,t_3),$$

(8)

where a smooth mapping $f:{\mathbb{R}}^3\to \mathbb{R}$ satisfies, following from (4) in the form $(\partial /\partial t_2\circ \partial /\partial t_2)\circ \partial /\partial t_3=\partial /\partial t_2\circ (\partial /\partial t_2\circ \partial /\partial t_3),$ $\partial /\partial t_1\circ \partial /\partial t_j=\partial /\partial t_j,j=\overline{1,3},$ such a partial differential equation:

$$f_{t_2{t}_2{t}_3}^2-f_{t_3{t}_3{t}_3}-f_{t_2{t}_2{t}_2}{f}_{t_2{t}_3{t}_3}=0$$

(9)

for any $(t_1,t_2,t_3)\in {\mathbb{R}}^3$. In particular, as it was shown by B. Dubrovin and Y. Manin [2,3,32,33], the Equation (9) allows the following system of compatible (for any parameter $p\in \mathbb{C}\setminus \{0\}$) linear differential equations:

$$\frac{\partial x}{\partial t_1}=\frac{1}{p}{C}_{1}x,\frac{\partial x}{\partial t_2}=\frac{1}{p}{C}_{2}x,\frac{\partial x}{\partial t_3}=\frac{1}{p}{C}_{3}x$$

(10)

on vectors $x:=(x_1,x_2,x_3)\in {\mathbb{E}}^3,$ determined by matrices

$$C_1=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),C_2=\left(\begin{array}{ccc}0& b& c\\ 1& a& b\\ 0& 1& 0\end{array}\right),C_3=\left(\begin{array}{ccc}0& c& b^2-ac\\ 0& b& c\\ 1& 0& 0\end{array}\right),$$

(11)

where $a:=f_{t_2{t}_2{t}_2},b:=f_{t_2{t}_2{t}_3},c:=f_{t_2{t}_3{t}_3}$ and generating the corresponding loop $\widetilde{Diff}({\mathbb{R}}^3)$-group diffeomorphisms. It is easy also to check that matrices (11) satisfy the matrix Equation (5), that is

$$\begin{array}{ccc}\hfill [C_2,C_3]& =& 0=[C_1,C_j],\hfill \\ \hfill \partial C_3/\partial t_2& =& \partial C_2/\partial t_3,[C_2,C_3]=0=\partial C_j/\partial t_1,\hfill \end{array}$$

(12)

for $t\in M,j=\overline{1,3}.$ An effective Lie-algebraic analysis of the Dubrovin–Manin linear system (10) was recently presented in [14,15].

In the present work, based on a modification of the Adler–Kostant–Symes integrability scheme, applied to the co-adjoint orbits of the loop diffeomorphism group of circle, a new two-parametric hierarchy of commuting to each other Monge type Hamiltonian vector fields

$$u_{t_1}=u_x,v_{t_1}=v_x,u_{t_2}=-{(u^2+2v)}_x,v_{t_2}={(v^2-2uv)}_x,$$

(13)

and

$$u_{t_3}={(\frac{3}{2}{v}^2-6uv-u^3)}_x,v_{t_3}={(-v^3-3u^{2}v+3u{v}^2-3v^2)}_x,\dots ,$$

(14)

on a pair of smooth functions $(u,v)\in C^{\infty }(M;{\mathbb{R}}^2)$ is constructed. Making use of a suitably constructed reciprocal transformation, applied to this hierarchy, one gives rise to constructing a Frobenius manifold potential function in terns of solutions to these Hamiltonian systems. In particular, we succeeded in describing a class of Frobenius manifold structures, generated by the non-linear Monge type evolution systems (13) and (14).

Proposition 1.

Let a function $F:M\to \mathbb{R}$ be defined by the following differential relationships

$$\begin{array}{ccc}\hfill \frac{{\partial }^{2}F(t_1,t_2,t_3)}{\partial t_1\partial t_2}& =& v,\frac{{\partial }^2\mathbb{F}(t_1,t_2,t_3)}{\partial t_1\partial t_3}=v(2u-v),\hfill \\ \hfill \frac{{\partial }^{2}F(t_1,t_2,t_3)}{\partial t_1\partial t_4}& =& 2v[v^2+3v-3u(u-v)],\hfill \end{array}$$

(15)

where the pair of functions $(u,v)\in C^{\infty }(M;{\mathbb{R}}^2)$ satisfies the evolution flows (13) and (14). Then this function $F:M\to \mathbb{R}$ is a potential function of the Frobenius manifold $M,$ describing the related Frobenius manifold algebraic structures.

2. Frobenius Manifolds, the Related Compatible Co-Adjoint Loop Lie Algebra and Integrability

Consider now the functional Lie algebra $\mathcal{G}\simeq (C^{\infty }(T^{*}({\mathbb{S}}^1);\mathbb{R});\{·,·\}),$ generated by special Hamiltonian vector fields on the cotangent space $T^{*}({\mathbb{S}}^1)$ to the circle ${\mathbb{S}}^1$ and endowed with the canonical Lie commutator

$$\{a,b\}(x;p):=\frac{\partial }{\partial p}a(x;p)\frac{\partial }{\partial x}b(x;p)-\frac{\partial }{\partial p}b(x;p)\frac{\partial }{\partial x}a(x;\lambda )$$

(16)

for any $a,b\in \mathcal{G}$ at point $(x,p)\in T^{*}({\mathbb{S}}^1).$ This algebra possesses the following symmetric and non-degenerate bi-linear form:

$$(a|b):={\int }_{\mathbb{R}}dp{\int }_{{\mathbb{S}}^1}dxa(x;p)b(x;p)dx,$$

(17)

with respect to which ${\mathcal{G}}^{*}\simeq$ $\mathcal{G}.$ Moreover, the Lie algebra is metrized with respect to the bilinear form (17) as it is $ad$-invariant: $(a|[b,c])=([a,b]|c)$ for any $a,b$, and $c\in \mathcal{G}.$

Below, we will consider the case when the Lie algebra $\mathcal{G}$ allows splitting into the direct sum of two sub-algebras: $\mathcal{G}$ $={\mathcal{G}}_{+}\oplus {\mathcal{G}}_{-},$ where

$${\mathcal{G}}_{+}:=\{a(x;p)=\sum _{j\in \mathbb{N}}{a}_j(x)p^j\in \mathcal{G}\}$$

(18)

and

$${\mathcal{G}}_{-}:=\{b(x;p)=\sum _{0\le j\ll \infty }{b}_j(x)p^{-j}\in \mathcal{G}\},$$

(19)

as $p\to \infty ,$ for which the following dual isomorphisms ${\mathcal{G}}_{+}^{*}\simeq {\mathcal{G}}_{-},$ ${\mathcal{G}}_{-}^{*}\simeq {\mathcal{G}}_{+}$ hold.

Proceed now to describing via the classical Adler–Kostant–Symes scheme [34,35,36,37,38,39] commuting co-adjoint orbits of the Lie algebra $\mathcal{G}$ on the adjoint space ${\mathcal{G}}^{*}\simeq \mathcal{G},$ generated by smooth Casimir functionals $h\in I({\mathcal{G}}^{*})$ with respect to the classical Lie-Poisson bracket on ${\mathcal{G}}^{*}\simeq \mathcal{G}:$

$$\{h(l),(l|a)\}:=(l|[\nabla h(l),a])=0$$

(20)

for $l\in {\mathcal{G}}^{*}$ and arbitrary $a\in \mathcal{G},$ where, by definition, $\frac{d}{d\epsilon }{h(l+\epsilon b)|}_{\epsilon =0}:=(\nabla h(l)|b)$ for any $b\in \mathcal{G}.$ Namely, the following Hamiltonian flows on ${\mathcal{G}}^{*}$

$$\partial l/\partial t_k=-a{d}_{\nabla h_{+}^{(k)}(l)}^{*}l=[\nabla h_{+}^{(k)}(l),l]=[l,\nabla h_{-}^{(k)}(l)],$$

(21)

where, by definition, $\nabla h_{\pm }^{(k)}(l):=\nabla h^{(k)}{(l)|}_{{\mathcal{G}}_{\pm }},$ are commuting to each other subject to the corresponding evolution parameters $t_k\in \mathbb{R},k\in {\mathbb{Z}}_{+},$ for arbitrary infinite hierarchy of smooth functionally independent Casimir functionals $h^{(k)}\in I({\mathcal{G}}^{*}),k\in {\mathbb{Z}}_{+}.$ The latter is, evidently, equivalent to the following Lax-Sato type vector field representations:

$$[\partial /\partial t_k+\widetilde{\nabla h_{+}^{(k)}}(l),\partial /\partial t_m+\widetilde{\nabla h_{+}^{(m)}}(l)]=0$$

(22)

for all $k,m\in {\mathbb{Z}}_{+},$ where, by definition, any element $a\in \mathcal{G}$ via the expression $\tilde{a}(x;p):=\frac{\partial a}{\partial p}\frac{\partial }{\partial x}-\frac{\partial a}{\partial x}\frac{\partial }{\partial p}$ $\in \mathsf{\Gamma }(T_{(x,p)}(T^{*}({\mathbb{S}}^1)))$ generates a canonical Hamiltonian vector field on $T^{*}({\mathbb{S}}^1)$ at point $(x;p)\in T^{*}({\mathbb{S}}^1).$

Take now an analytic at the momentum $p\in \mathbb{R}$ element $l\in {\mathcal{G}}^{*}\simeq \mathcal{G}$ in the following asymptoptic as $p\to \infty$ form:

$$l(x;p)=p+u(x)+\sum _{j\in \mathbb{N}}{l}_j(x)p^{-j},$$

(23)

where the element $p\in {\mathcal{G}}^{*}$ is considered here as an infinitesimal Lie algebra $\mathcal{G}$ character, satisfying the conditions $[{\mathcal{G}}_{\pm },p]\in {\mathcal{G}}_{\pm },$ that can be easily checked by direct computations. The flows (21) are equivalent to the following co-adjoint action

$$\partial l_{-}/\partial t_k=-a{d}_{\nabla h_{+}^{(k)}(l)}^{*}{l}_{-}={[\nabla h_{+}^{(k)}(l),l_{-}]}_{-}$$

(24)

on ${\mathcal{G}}^{*}\simeq \mathcal{G}$ with respect to the evolution parameters $t_k\in \mathbb{R}$ for all $k\in {\mathbb{Z}}_{+}.$

It is worthy to observe now that in the case of the Casimir functionals $h^{(k)}:=\frac{1}{k+1}(l^k|l),k\in {\mathbb{Z}}_{+},$ the flows (24) can be equivalently rewritten as the Hamiltonian systems

$$\partial \tilde{l}(x;p)/\partial t_k=[\widetilde{l_{+}^k}(x;p),\tilde{l}(x;p)]$$

(25)

on ${\mathcal{G}}^{*}$ for all $k\in {\mathbb{Z}}_{+},$ where, by definition, $\tilde{l}(x;p):=\frac{\partial l}{\partial p}\frac{\partial }{\partial x}-\frac{\partial l}{\partial x}\frac{\partial }{\partial p}\in \mathsf{\Gamma }(T_{(x,p)}(T^{*}({\mathbb{S}}^1)))$ at point $(x;p)\in T^{*}({\mathbb{S}}^1).$ Using the Lie bracket (16), Equation (25) can be rewritten as the Hamiltonian flows on the cotangent space $T^{*}({\mathbb{S}}^1)$

$$\partial l(x;p)/\partial t_k=\{H_k(x,p),l(x;p)\},$$

(26)

where, by definitions, $H_k(x;p)=l_{+}^k(x;p)$ for any $k\in \mathbb{N},(x;p)\in T^{*}({\mathbb{S}}^1).$

Remark 1.

It is worth also to remark here that we can pose the following vector field iso-spectral problem

$$\tilde{l}(x;p)\psi (x;p|z)=z\psi (x;p|z),$$

(27)

where $\psi (·;z)\in C^{\infty }(T^{*}({\mathbb{S}}^1);\mathbb{C})$ is the eigenfunction corresponding to an eigenvalue $z\in \mathbb{C},$ which is a priori invariant with respect to all vector fields (25). The latter naturally allows to apply to (27) the modified inverse scattering transform technique developed in [40] and describe many classes of symbols $l\in \mathcal{G},$ generating important dispersion-less heavenly type [41] dynamical systems, important for applications in modern mathematical physics.

As the point variables $(x;p)\in T^{*}({\mathbb{S}}^1)$ are constant parameters for the evolution flows (25) on analytic at $p=\infty$ element $l\in {\mathcal{G}}^{*},$ one can put, by definition, $l(x;p)=z\in \mathbb{C}$ and resolve the functional equation $l(x;p)=z$ with respect to the symbol parameter $p\in \mathbb{R},$ obtaining the following expression:

$$p:=\xi (x;z)=z-u-\sum _{j\in \mathbb{N}}{\xi }_j(x)z^{-j}$$

(28)

with coefficients ${\xi }_j\in C^{\infty }({\mathbb{S}}^1;\mathbb{R}), j\in \mathbb{N},$ characterized by the following lemma.

Lemma 1.

The element $\xi \in C^{\infty }({\mathbb{S}}^1\times \mathbb{R};\mathbb{C})$ satisfies the following hierarchy of compatible evolution equations

$$\frac{\partial }{\partial t_k}\xi (x;z)=\frac{\partial {\mathcal{H}}_k(x;z)}{\partial x},$$

(29)

where the elements ${\mathcal{H}}_k(x;z):=l_{+}^k(x;\xi (x;z)),$ $k\in \mathbb{N},$ are determined, using the following simple algebraic expressions:

$${\mathcal{H}}_k(x;z):=H_k(x;\xi (x;z)),$$

(30)

which hold jointly with compatibility relationships

$$\frac{\partial {\mathcal{H}}_s(x;z)}{\partial t_k}=\frac{\partial {\mathcal{H}}_k(x;z)}{\partial t_s}$$

(31)

for all $k,s\in \mathbb{N}.$

Proof. 

Making use of the Equation (25), one can easily calculate for any $k\in \mathbb{N}$ the evolution equations

$$\frac{\partial }{\partial t_k}\left(\frac{1}{\xi (x;z)-p}\right):=\left\{H_k(x;p),\left(\frac{1}{\xi (x;z)-p}\right)\right\},$$

giving rise to the following expressions

$$\begin{array}{ccc}\hfill \frac{\partial \xi (x;z)}{\partial t_k}& =& \frac{\partial H_k(x;p)}{\partial x}+{\left.\frac{\partial H_k(x;p)}{\partial p}\right|}_{p=\xi (x;z)}\frac{\partial \xi (x;z)}{\partial x}=\hfill \\ & =& d{H}_k(x;\xi (x;z)/dx:=\frac{\partial {\mathcal{H}}_k(x;z)}{\partial x},\hfill \end{array}$$

(32)

which hold for all $k\in \mathbb{N}$ and all $z\in \mathbb{R}.$ The compatibility relationships are obvious, following from the commuting to each other flows (29). □

Consider now the functional identity

$$\frac{1}{\xi (x;z)-p}=\sum _{k\in \mathbb{N}}\frac{z^{-k}}{k}\frac{\partial }{\partial p}{H}_k(x;p),$$

(33)

which is satisfied as $z\to \infty ,$ owing to the following residuum calculation:

$$\begin{array}{c}\frac{1}{2\pi i}\oint \frac{z^{k-1}dz}{\xi (x;z)-p}=\frac{1}{2\pi i}\oint \frac{z^{k-1}dz}{\left[z-l(x;p)\right]\frac{\left[\xi (x;z)-p\right]}{\left[z-l(x;p)\right]}}=\\ ={\left.\frac{l{(x;p)}^{k-1}}{\partial \xi (x;z)/\partial z}\right|}_{z\to \infty }={\left.l{(x;p)}^{k-1}\partial l(x;p)/\partial p\right|}_{z\to \infty }=\frac{1}{k}\frac{\partial }{\partial p}{H}_k{(x;p)|}_{+},\end{array}$$

(34)

which holds for any $k\in \mathbb{N}.$ Consider now Hamiltonian functions $H_k:T^{*}({\mathbb{S}}^1)\to \mathbb{R},k\in \mathbb{N},$ and consider the related canonical Hamiltonian vector fields on the cotangent space $T^{*}(\mathbb{R}):$

$$\frac{\partial x}{\partial t_k}=\frac{\partial H_k(x;p)}{\partial p},\frac{\partial p}{\partial t_k}=-\frac{\partial H_k(x;p)}{\partial x}$$

(35)

with respect to a point $(x,p)\in T^{*}({\mathbb{S}}^1)$ subject to the evolution parameter $t_k\in \mathbb{R},k\in \mathbb{N}.$ Taking into account the evolution flows (35) and the fact that $\partial /\partial t_1=\partial /\partial x,$ the identity (33) can be rewritten as

$$\frac{1}{\xi (x;z)-p}=\sum _{k\in \mathbb{N}}\frac{z^{-k}}{k}\frac{\partial x}{\partial t_k}=D(z)x(t),$$

from which and the relationships (31) one ensues the functional representation

$$\xi (x;z)=z-\frac{\partial \mathcal{F}(t)}{\partial x}-D(z)\frac{\partial \mathcal{F}(t)}{\partial x}$$

(36)

for some smooth function $\mathcal{F}:M\to \mathbb{R}.$ Based now on Lemma 1 and relationships (33), (34) one can state now the following proposition.

Proposition 2.

Let $F:M\to \mathbb{R}$ be a potential function on the Frobenius manifold $M,$ defined by means of the set of asymptotic relationship

$$D(y)F(t)+D(y)D(z)F(t)=-ln(1-z/y)-\sum _{k\in \mathbb{N}}\frac{y^{-k}}{k}{\mathcal{H}}_k(x;z)$$

(37)

where, by definition, the operator $D(\alpha )={\sum }_{k\in \mathbb{N}}\frac{{\alpha }^{-k}}{k}\frac{\partial }{\partial t_k},\alpha \in \mathbb{R},$ is the well known vertex operator. Then the element (28) satisfies the asymptotic representation (36) for all $x\in {\mathbb{S}}^1$ as $z\to \infty .$

Proof. 

The functional identity (37) easily reduces to the set of asymptotic expressions

$${\mathcal{H}}_k(x;z)=z^k-\partial F/\partial t_k-D(z)\partial F/\partial t_k$$

(38)

for all $k\in \mathbb{N}$ as $z\to \infty .$ Simultaneously one can observe that the expression (29) and (30) reduce to the representation (36), proving the proposition. □

This proposition is useful for constructing Frobenius manifolds, naturally related with some generating function $\mathcal{F}:M\to \mathbb{R},$ satisfying the relationship (36). As an example, we suggest the following element

$$l(x;p)=p+u(x)+ln\left(1+\frac{v(x)}{p}\right)\in {\mathcal{G}}^{*},$$

(39)

where $u,v\in C^{\infty }({\mathbb{S}}^1;\mathbb{R})$ are some functional parameters. The corresponding Casimir functions $h^{(t_1)}:=(l|l)/2,h^{(t_2)}:=(l^2|l)/3$ and $h^{(t_3)}:=(l^3|l)/4,h^{(t_4)}:=(l^4|l)/5$, etc., generate the following Hamiltonian flows on ${\mathcal{G}}^{*}\simeq \mathcal{G}:$

$$\partial l/\partial x=[l_{+},l],\partial l/\partial y=[l_{+}^2,l],\partial l/\partial t=[l_{+}^3,l],\partial l/\partial s=[l_{+}^4,l]$$

(40)

with respect to the evolution parameters $x=t_1\in \mathbb{R},t_2,t_3\in \mathbb{R}$, etc., where, for instance,

$$\begin{array}{ccc}\hfill l_{+}^2& :& =H_2(x;p)=p^2+2pu\in {\mathcal{G}}_{+},\hfill \\ \hfill l_{+}^3& :& =H_3(x;p)=p^3+3p^{2}u+3p{u}^2+3pv\in {\mathcal{G}}_{+}\hfill \end{array}$$

(41)

and so on. The above commutator expressions with respect to the evolution parameters $t_1,t_2$ and $t_3\in \mathbb{R}$ reduce to the next commuting to each other non-linear Monge type evolution systems

$$u_{t_1}=u_x,v_{t_1}=v_x,u_{t_2}=-{(u^2+2v)}_x,v_{t_2}={(v^2-2uv)}_x,$$

(42)

and

$$u_{t_3}={(\frac{3}{2}{v}^2-6uv-u^3)}_x,v_{t_3}={(-v^3-3u^{2}v+3u{v}^2-3v^2)}_x,$$

(43)

being also compatible dispersion-less Hamiltonian flows on the corresponding functional phase. Moreover, the evolution systems (42) and (43) are equivalent to the Lax-Sato vector field commutator representation (22), where

$$\begin{array}{ccc}\hfill \nabla h_{+}^{(t_1)}(\tilde{l})& =& (p+u)\frac{\partial }{\partial x}-u_{x}p\frac{\partial }{\partial p},\hfill \\ \hfill \nabla h_{+}^{(t_1)}(\tilde{l})& =& (p^2+2up+2v+u^2)\frac{\partial }{\partial x}-(u_x{p}^2+v_{x}p+2u{u}_{x}p)\frac{\partial }{\partial p}.\hfill \end{array}$$

(44)

The vector fields (44), being considered as elements of the Lie algebra $\tilde{\mathcal{G}}\simeq diff({\mathbb{S}}^1\times \mathbb{C})$ of holomorphic with respect to the variable $p\in \mathbb{C}$ vector fields on ${\mathbb{S}}^1\times \mathbb{C},$ naturally splits into the direct sum of two sub-algebras $\tilde{\mathcal{G}}={\tilde{\mathcal{G}}}_{+}\oplus$ ${\tilde{\mathcal{G}}}_{-},$ holomorphic in the parameter $p\in \mathbb{C}$ inside ${\mathbb{D}}_{+}^1(0)$ of the unit circle ${\mathbb{D}}_{+}^1(0)\subset \mathbb{C}$ and outside ${\mathbb{D}}_{-}^1(0)$ of this disk, respectively, appear to be generated by the corresponding Casimir functionals on the adjoint space ${\tilde{\mathcal{G}}}^{*}\simeq {\mathsf{\Omega }}^1({\mathbb{S}}^1\times \mathbb{C})$ at some root element $\tilde{l}\in$ ${\tilde{\mathcal{G}}}^{*}$ subject to the following canonical non-degenerate bi-linear form on ${\tilde{\mathcal{G}}}^{*}\times \tilde{\mathcal{G}}:$

$$(\tilde{l}|\tilde{a}):={\int }_0^{2\pi }re{s}_p⟨l|a⟩dx,$$

(45)

where we put, by definition, $\tilde{l}:=⟨l|d\mathrm{x}⟩,\tilde{a}:=⟨a|\partial /\partial \mathrm{x}⟩,$ $\mathrm{x}:=(p;x)\in \mathbb{C}\times {\mathbb{S}}^1.$ Based on the definition of Casimir functionals, one easily enough obtains that this root element equals

$$\begin{array}{ccc}\hfill \tilde{l}& =& (u_x{p}^2+{(v+u^2)}_{x}p)dx+(p^2+2up+v+u^2)dp=\hfill \\ & =& d\left(\frac{1}{3}{p}^3+u{p}^2+(v+u^2)p\right),\hfill \end{array}$$

(46)

being a complete derivative of the scalar element $\tilde{\eta }=\frac{1}{3}{p}^3+u{p}^2+(v+u^2)p\in {\mathsf{\Omega }}^0({\mathbb{S}}^1\times \mathbb{C}),\tilde{l}=d\tilde{\eta },$ for all $(p;x)\in \mathbb{C}\times {\mathbb{S}}^1.$ Moreover, the system of evolution equations (42) and (43) becomes equivalent to the following co-adjoint flows

$$\partial \tilde{l}/\partial y=-a{d}_{\nabla h_{+}^{(t_2)}(\tilde{l})}^{*}\tilde{l},\partial \tilde{l}/\partial t=-a{d}_{\nabla h_{+}^{(t_3)}(\tilde{l})}^{*}\tilde{l}$$

(47)

on the adjoint space ${\tilde{\mathcal{G}}}^{*},$ generated by the corresponding Casimir functionals $h^{(t_2)},h^{(t_3)}\in I({\tilde{\mathcal{G}}}^{*})$ and satisfying the determining relationships $a{d}_{\nabla h^{(t_2)}(\tilde{l})}^{*}\tilde{l}=0,a{d}_{\nabla h^{(t_3)}(\tilde{l})}^{*}\tilde{l}=0.$ As now the basic Lie algebra $\tilde{\mathcal{G}}\simeq diff({\mathbb{S}}^1\times \mathbb{C})$ of holomorphic vector fields on ${\mathbb{S}}^1\times \mathbb{C}$ is not, evidently, metrized, the flows (47) on ${\tilde{\mathcal{G}}}^{*}$ do not possess the standard Lax type commutator representation.

Taking into account the expressions (36) and (39), one can formulate the following proposition.

Proposition 3.

Let a function $F:M\to \mathbb{R}$ be defined by the following differential relationships

$$\begin{array}{ccc}\hfill \frac{{\partial }^{2}F(t_1,t_2,t_3)}{\partial t_1\partial t_2}& =& v,\frac{{\partial }^{2}F(t_1,t_2,t_3)}{\partial t_1\partial t_3}=v(2u-v),\hfill \\ \hfill \frac{{\partial }^{2}F(t_1,t_2,t_3)}{\partial t_1\partial t_3}& =& 2v[v^2+3v-3u(u-v)],\hfill \end{array}$$

(48)

where the pair of functions $(u,v)\in C^{\infty }(M;{\mathbb{R}}^2)$ satisfies the evolution flows (42) and (43). Then it is a potential function of the Frobenius manifold $M,$ describing the related Frobenius manifold algebraic structures.

This result makes it possible to describe a wide variety of Frobenius manifold potential functions in terns of solutions to these Monge type Hamiltonian systems (42) and (43).