1 Introduction

Of the many ways to generalize the Korteweg–de Vries equation \(u_t=u_{xxx}+6 u u_x,\) the one that will be of most relevance to this paper is the matrix generalization (see, for example [3, 4])

$$\begin{aligned} {\mathcal {U}}_t = {\mathcal {U}}_{xxx} + 3 {\mathcal {U}} {\mathcal {U}}_x+ 3 {\mathcal {U}}_x {\mathcal {U}}, \end{aligned}$$
(1.1)

where the two first-derivative terms are required due to the non-commutativity of matrix multiplication. If one restricts such an equation to the space of commuting matrices, one arrives at the equation \({\mathcal {U}}_t = {\mathcal {U}}_{xxx} + 6 {\mathcal {U}} {\mathcal {U}}_x\) which is identical in form to the original KdV equation but with a matrix-valued, as opposed to a scalar-valued, field (see, for example [15, 23, 26]). The purpose of this paper is to construct \({\mathcal {A}}\)-valued, where \({\mathcal {A}}\) is a Frobenius algebra, generalizations of integrable systems, starting with those associated to an underlying Frobenius manifold and related dispersionless hierarchies, and extending the ideas to topological quantum field theories and dispersive hierarchies.

The structure of this paper may be summarized in the following diagram:

$$\begin{aligned} \begin{array}{ccc} \left\{ \begin{array}{c} {\mathcal {A}}-\mathrm{valued}\\ \mathrm{Frobenius~manifold~(\S 2)} \end{array} \right\} &{} \longrightarrow &{} \left\{ \begin{array}{c} {\mathcal {A}}-\mathrm{valued}\\ \mathrm{TQFT~(\S 3)} \end{array} \right\} \\ &{}&{}\\ \Big \downarrow &{}&{} \\ &{}&{}\\ \left\{ \begin{array}{c} {\mathcal {A}}-\mathrm{valued~bi-Hamiltonian}\\ \mathrm{dispersionless~systems~(\S 4)} \end{array} \right\} &{} \longrightarrow &{} \left\{ \begin{array}{c} {\mathcal {A}}-\mathrm{valued~bi-Hamiltonian}\\ \mathrm{dispersive~systems~(\S 5)} \end{array} \right\} \\ \end{array} \end{aligned}$$

The full reconstruction of a dispersive hierarchy (the missing vertical arrow in the above diagram) remains an open problem, even before one considers \({\mathcal {A}}\)-valued systems.

The starting point (Sect. 2) for the study of such \({\mathcal {A}}\)-valued hierarchies is the classical construction of Dubrovin [5] which associates to a Frobenius manifold a bi-Hamiltonian hierarchy of hydrodynamic type. By constructing the tensor product [13, 14] of such a manifold with a trivial Frobenius manifold (i.e., a fixed algebra), one automatically obtains a new Frobenius manifold and hence a bi-Hamiltonian hierarchy. The component fields of this new hierarchy can then be reassembled to form an \({\mathcal {A}}\)-valued hierarchy. The important feature of this construction is a simple, explicit, form of the new prepotential that defines the \({\mathcal {A}}\)-valued hierarchies.

More explicitly, given a Frobenius algebra \({\mathcal {A}}\) with basis \(e_i,\,i=1,\,\ldots ,\,n,\) one can replace the flat coordinates of a Frobenius manifold \({\mathcal {M}}\) with \({\mathcal {A}}\)-valued fields via the map

$$\begin{aligned} {\hat{}}\, : \, t^\alpha \mapsto \widehat{t^\alpha } = t^{(\alpha i)} e_i,\quad \alpha =1,\,\ldots ,\,m,\quad i=1,\,\ldots ,\,n \end{aligned}$$

and this action can be extended to functions, at least in the case of analytic Frobenius manifolds (and to wider classes of functions—see the ”Appendix”). Conversely, an \({\mathcal {A}}\)-valued field can be reduced to a scalar field via the Frobenius form (or trace form) \(\omega .\) This construction is described in Sect. 2. The main result is the following:

MainTheorem 1

(Theorem 2.9) Let F be the prepotential of a Frobenius manifold \({\mathcal {M}}\) and let \({\mathcal {A}}\) be a trivial Frobenius algebra with 1-form \(\omega .\) The function

$$\begin{aligned} {F}^{\mathcal {A}} = \omega \left( {\widehat{F}}\right) \end{aligned}$$

defines a Frobenius manifold, namely the manifold \({\mathcal {M}}\otimes {\mathcal {A}}.\)

Normally the prepotential of a tensor product of Frobenius manifolds bears little resemblance to the underlying prepotentials and in any case is only defined implicitly from the original prepotentials. However when one of the manifolds is trivial, the above closed form of the new prepotential exists and this enables the resulting hierarchies to be constructed explicitly.

In Sect. 3, we extend these ideas to a full topological quantum field theory on the big phase space \({\mathcal {M}}^\infty \), i.e., with gravitational descendent fields and with String and Dilaton vectors fields \({\mathcal {S}}\) and \({\mathcal {D}}\), respectively.

MainTheorem 2

(Theorem 3.2) Let \({\mathcal {F}}_{g\ge 0}\) be the prepotentials defining a TQFT, \({\mathcal {S}}\) and \({\mathcal {D}}\) the corresponding String and Dilaton vector fields and \({\mathcal {A}}\) be a trivial Frobenius algebra. Let f be an analytic function on \({\mathcal {M}}^\infty \) and define the \({\mathcal {A}}\)-valued function \({\hat{f}}\) to be:

$$\begin{aligned} {{\hat{f}}} = \left. f\right| _{t^\alpha _N \mapsto t^{(\alpha i)}_N e_i},\quad N\in {\mathbb {Z}}_{\ge 0},\quad \alpha =1,\,\ldots , \, m,\quad i=1,\,\ldots ,\,n. \end{aligned}$$

Then the functions

$$\begin{aligned} {\mathcal {F}}^{{\mathcal {A}}}_{g\ge 0} = \omega \left( {\widehat{ {\mathcal {F}}}_{g\ge 0} }\right) \end{aligned}$$

and vector fields

$$\begin{aligned} {\mathcal {S}}^{\mathcal {A}} = -\sum _{N,(\alpha i)} {\tilde{t}}^{(\alpha i)}_N \tau _{N-1,(\alpha i)},\quad {\mathcal {D}}^{\mathcal {A}} = -\sum _{N,(\alpha i)} {\tilde{t}}^{(\alpha i)}_N \tau _{N,(\alpha i)}, \end{aligned}$$

where \({\tilde{t}}^{(\alpha i)}_N = {t}^{(\alpha i)}_N - \delta _{N,1} \delta _{\alpha ,1} \delta _{i,1},\) satisfy the axioms of a topological quantum field theory.

In the remaining sections, a theory of \({\mathcal {A}}\)-valued integrable systems is developed, first for dispersionless systems and then for certain dispersive systems. More specifically, in Sect. 4 the construction of the \({\mathcal {A}}\)-valued dispersionless (or hydrodynamic) hierarchies is given. The deformed flat coordinates can be described very simply, and these form the Hamiltonian densities for the new evolution equations. By reassembling the fields, these equations can be written as \({\mathcal {A}}\)-valued evolution equations. To write these in Hamiltonian form requires the definition of a functional derivative with respect to an \({\mathcal {A}}\)-valued field, and such a derivative was defined in [19] and with this one can write the flow equations as \({\mathcal {A}}\)-valued bi-Hamiltonian evolution equations. These ideas are then extended to the dispersive case in Sect. 5.

2 Frobenius manifolds and their tensor products

2.1 Frobenius algebras and manifolds

We begin with the definition of a Frobenius algebra [5].

Definition 2.1

A Frobenius algebra \(\{ {\mathcal {A}},\circ ,e,\omega \}\) over \({\mathbb {R}}\) satisfies the following conditions:

(i) :

\(\circ : {\mathcal {A}} \times {\mathcal {A}} \rightarrow {\mathcal {A}}\) is a commutative, associative algebra with unity e;

(ii) :

\(\omega \in {\mathcal {A}}^\star \) defines a non-degenerate inner product \(\langle a,b \rangle = \omega (a \circ b).\)

Since \(\omega (a) = \langle e,a \rangle \) the inner product determines the form \(\omega \) and visa-versa. This linear form \(\omega \) is often called a trace form (or Frobenius form). One-dimensional Frobenius algebras are trivial: the requirement of an identity and the non-degeneracy of the inner product determines the algebra uniquely and the inner product up to a nonzero constant. Two-dimensional Frobenius algebra is easily classified: the requirement of an identity means there is only one non-trivial multiplication and the associativity condition is automatically satisfied in two-dimensions.

Example 2.2

Let \({\mathcal {A}}\) be a two-dimensional commutative and associative algebra with a basis \(e=e_1, e_2\) satisfying

$$\begin{aligned} e_1\circ e_1=e_1,\quad e_1\circ e_2=e_2, \quad e_2\circ e_2=\varepsilon e_1+\mu e_2,\quad \varepsilon ,\mu \in {\mathbb {R}}. \end{aligned}$$
(2.1)

Obviously, the algebra \({\mathcal {A}}\) has a matrix representation as follows

$$\begin{aligned} e_1 \mapsto \mathrm {I}_2=\left( \begin{array}{cc} 1&{}0\\ 0&{}1 \end{array}\right) ,\quad e_2 \mapsto \left( \begin{array}{cc} 0&{} {\varepsilon }\\ 1&{}\mu \end{array}\right) . \end{aligned}$$

It is easy to show that:

  1. (1)

    if \(\mu ^2=-4\varepsilon \), \({\mathcal {A}}\) is nonsemisimple, i.e., \(\exists \, {\widetilde{e}}=\mu e_1-2 e_2\) such that \({\widetilde{e}}\circ {\widetilde{e}}=0\);

  2. (2)

    if \(\mu ^2\ne -4\varepsilon \), then \({\mathcal {A}}\) is semisimple, i.e., for any nonzero element \({\widetilde{e}}=x e_1+y e_2\), \({\widetilde{e}}\circ {\widetilde{e}}\ne 0\).

Furthermore, we introduce two “basic” trace-type forms for \(a=a_1e_1+a_2e_2\in {\mathcal {A}}\) as follows

$$\begin{aligned} \omega _k(a)=a_k+a_2(1-\delta _{k,2})\delta _{\varepsilon ,0}, \quad k=1,\, 2, \end{aligned}$$
(2.2)

which induce two non-degenerate inner products on \({\mathcal {A}}\) given by

$$\begin{aligned} \langle a, b\rangle _k:=\omega _k(a\circ b),\quad a,\,b\in {\mathcal {A}},\quad k=1,2. \end{aligned}$$
(2.3)

The two Frobenius algebras \(\left\{ {\mathcal {A}},\circ , e, \omega _k\right\} \) will be denoted by \({\mathcal {Z}}_{2,k}^{\varepsilon ,\mu }\) for \(k=1,2\).

Example 2.3

Let \({\mathcal {A}}\) be an n-dimensional nonsemisimple commutative associative algebra \({\mathcal {Z}}_n\) over \({\mathbb {R}}\) with a unity e and a basis \(e_1=e, \ldots , e_n\) satisfying

$$\begin{aligned} e_i \circ e_j=\left\{ \begin{array}{ll} e_{i+j-1}, &{} i+j\le n+1,\\ 0,&{} i+j=n+2. \end{array}\right. \end{aligned}$$
(2.4)

Taking \(\Lambda =(\delta _{i,j+1})\in gl(m,{\mathbb {R}})\), one obtains a matrix representation of \({\mathcal {A}}\) as

$$\begin{aligned} e_j \mapsto \Lambda ^{j-1}, \quad j=1,\ldots , n. \end{aligned}$$

Similarly, for any \(a=\displaystyle \sum _{k=1}^{n} a_k e_k\in {\mathcal {A}}\), we introduce n trace-type forms, called “basic” trace-type forms, as follows

$$\begin{aligned} \omega _{k-1} (a)=a_k+a_{n}(1-\delta _{k,n}),\quad k=1,\ldots ,n. \end{aligned}$$
(2.5)

Every trace map \(\omega _k\) induces a non-degenerate symmetric bilinear form on \({\mathcal {A}}\) given by

$$\begin{aligned} \langle a, b\rangle _k:=\omega _k (a\circ b),\quad a,\,b\in {\mathcal {A}},\quad k=0,\ldots ,n-1. \end{aligned}$$
(2.6)

Thus, all of \(\left\{ {\mathcal {A}}, \circ , e, \omega _{k-1}\right\} \) are nonsemisimple Frobenius algebras, denoted by \({\mathcal {Z}}_{n,k-1}\) for \(k=1,\ldots ,n\). We remark that if we consider a linear combination of n “basic” trace-type forms as

$$\begin{aligned} \mathrm {tr}_n:=\displaystyle \sum _{s=0}^{n-1}\omega _{s}-(n-1)\,\omega _{n-1}, \end{aligned}$$

then \(\left\{ {\mathcal {A}}, \circ , e, \mathrm {tr}_n\right\} \) is also a Frobenius algebra which is exactly the algebra \(\left\{ {\mathcal {Z}}_n,\mathrm {tr}_n\right\} \) used in [26]Footnote 1.

A Frobenius manifold has such a structure on each tangent space.

Definition 2.4

[5] The set \(\{{\mathcal {M}},\circ ,e,\langle ~,~\rangle ,E \}\) is a Frobenius manifold if each tangent space \(T_t{\mathcal {M}}\) carries a smoothly varying Frobenius algebra with the properties:

(i):

\(\langle ~,~\rangle \) is a flat metric on \({\mathcal {M}}\);

(ii) :

\(\nabla e=0\), where \(\nabla \) is the Levi-Civita connection of \(\langle ,\rangle \);

(iii) :

the tensors \(c(u,v,w):=\langle u\circ v,w\rangle \) and \(\nabla _zc(u,v,w)\) are totally symmetric;

(iv) :

A vector field E exists, linear in the flat variables, such that the corresponding group of diffeomorphisms acts by conformal transformation on the metric and by rescalings on the algebra on \(T_t{\mathcal {M}}.\)

These axioms imply the existence of the prepotential F which satisfies the WDVV-equations of associativity in the flat coordinates of the metric (strictly speaking only a complex, non-degenerate bilinear form) on \({\mathcal {M}}.\) The multiplication is then defined by the third derivatives of the prepotential:

$$\begin{aligned} \frac{\partial ~}{\partial t^\alpha } \circ \frac{\partial ~}{\partial t^\beta } = c_{\alpha \beta }^{\phantom {\alpha \beta }\gamma }(\mathbf{t}) \frac{\partial ~}{\partial t^\gamma } \end{aligned}$$

where

$$\begin{aligned} c_{\alpha \beta \gamma } = \frac{ \partial ^3 F}{\partial t^\alpha \partial t^\beta \partial t^\gamma } \end{aligned}$$

and indices are raised and lowered using the metric \(\eta _{\alpha \beta } = \langle \dfrac{\partial ~}{\partial t^\alpha },\dfrac{\partial ~}{\partial t^\beta } \rangle \).

Example 2.5

Suppose \(c_{ij}^{~~k}\) are the structure constants for the Frobenius algebra \({\mathcal {A}}\), so \(e_i \circ e_j = c_{ij}^{~~k} e_k\) and \(\eta _{ij} = \langle e_i,e_j \rangle .\) For such an algebra, one obtains a cubic prepotential

$$\begin{aligned} F= & {} \frac{1}{6} c_{ijk} t^i t^j t^k,\\= & {} \frac{1}{6} \omega ( \mathbf{t}\circ \mathbf{t}\circ \mathbf{t}),\quad \mathbf{t} = t^i e_i. \end{aligned}$$

The Euler vector field takes the form \(E=\displaystyle \sum \nolimits _{i} t^i\frac{\partial }{\partial t^i}\) and \(E(F)=3F.\) The notation \({\mathcal {A}}\) will be used for both the algebra and the corresponding manifold.

Motivated by the classical Künneth formula in cohomology, Kaufmann, Kontsevich and Manin [13, 14] constructed the tensor product of two Frobenius manifolds \({\mathcal {M}}^\prime \) and \({\mathcal {M}}^{\prime \prime }\), denoted \({\mathcal {M}}^\prime \otimes {\mathcal {M}}^{\prime \prime }.\) The following formulation of this construction is taken from [6]. This formulation also gives criteria to check if a particular manifold is the tensor product of two more basic manifolds. For simplicity, we use the notation \(\partial _\alpha =\dfrac{\partial }{\partial t^\alpha }\) and \(\partial _{\alpha \beta }=\dfrac{\partial ~~}{\partial t^{(\alpha \beta )}}\).

Proposition 2.6

Let \({\mathcal {M}}^\prime \) and \({\mathcal {M}}^{\prime \prime }\) be two Frobenius manifolds of dimension \(n^\prime \) and \(n^{\prime \prime }.\) A Frobenius manifold \({\mathcal {M}}\) of dimension \(n^\prime n^{\prime \prime }\) is the tensor product \({\mathcal {M}} = {\mathcal {M}}^\prime \otimes {\mathcal {M}}^{\prime \prime }\) if the following conditions hold:

(i):

\(\{ T{\mathcal {M}}, \langle ,\rangle ,e\} = \{ T{\mathcal {M}}^\prime \otimes T{\mathcal {M}}^{\prime \prime }, \langle , \rangle ^\prime \otimes \langle ,\rangle ^{\prime \prime }, e^\prime \otimes e^{\prime \prime } \}.\) Flat coordinates are labeled by pairs \(t^{(\alpha ^\prime \alpha ^{\prime \prime })},\alpha ^\prime =1,\,\ldots ,\,n^\prime ,\,\alpha ^{\prime \prime } =1,\,\ldots ,\,n^{\prime \prime },\) and the unity vector field is

$$\begin{aligned} e=\dfrac{\partial ~}{\partial t^{(1 1)}} \end{aligned}$$

and the metric \(\langle ,\,\rangle \) has the form

$$\begin{aligned} \eta _{(\alpha ^\prime \alpha ^{\prime \prime })(\beta ^\prime \beta ^{\prime \prime })} = \eta _{\alpha ^\prime \beta ^\prime } \, \eta _{\alpha ^{\prime \prime } \beta ^{\prime \prime }}. \end{aligned}$$
(ii):

At a point \(t^{(\alpha ^\prime \alpha ^{\prime \prime })} =0,\alpha ^\prime>1,\alpha ^{\prime \prime }>1\) the algebra \(T_t{\mathcal {M}}\) is a tensor product

$$\begin{aligned} T_t{\mathcal {M}} = T_{t^\prime } {\mathcal {M}}^\prime \otimes T_{t^{\prime \prime }} {\mathcal {M}}^{\prime \prime }, \end{aligned}$$

that is:

$$\begin{aligned} c_{(\alpha ^\prime \alpha ^{\prime \prime })(\beta ^\prime \beta ^{\prime \prime })}^{\phantom {(\alpha ^\prime \alpha ^{\prime \prime })(\beta ^\prime \beta ^{\prime \prime })}(\gamma ^\prime \gamma ^{\prime \prime })}(t)= c_{\alpha ^\prime \beta ^\prime }^{\phantom {{\alpha ^\prime \beta ^\prime }}\gamma ^\prime }(t^\prime ) \, c_{\alpha ^{\prime \prime }\beta ^{\prime \prime }}^{\phantom {{\alpha ^{\prime \prime }\beta ^{\prime \prime }}} \gamma ^{\prime \prime }}(t^{\prime \prime }). \end{aligned}$$
(iii):

If the Euler vector fields of the two manifolds \({\mathcal {M}}\) and \({\mathcal {M}}^{\prime \prime }\) take the form

$$\begin{aligned} E^\prime= & {} \sum _{\alpha ^\prime } \left[ (1-q_{\alpha ^\prime }) t^{\alpha ^\prime } + r_{\alpha ^\prime }\right] \partial _{\alpha ^\prime },\\ E^{\prime \prime }= & {} \sum _{\alpha ^{\prime \prime }} \left[ (1-q_{\alpha ^{\prime \prime }}) t^{\alpha ^{\prime \prime }} + r_{\alpha ^{\prime \prime }}\right] \partial _{\alpha ^{\prime \prime }},\\ \end{aligned}$$

with scaling dimensions \(d^\prime \) and \(d^{\prime \prime }\), respectively, then the Euler vector field on \({\mathcal {M}}\) takes the form

$$\begin{aligned} E=\sum _{\alpha ^\prime ,\alpha ^{\prime \prime }} (1-q_{\alpha ^\prime }-q_{\alpha ^{\prime \prime }}) \partial _{(\alpha ^\prime \alpha ^{\prime \prime })} + \sum _{\alpha ^\prime } r_{\alpha ^\prime }\partial _{\alpha ^\prime 1^{\prime \prime }} + \sum _{\alpha ^{\prime \prime }} r_{\alpha ^{\prime \prime }} \partial _{{1^\prime \alpha ^{\prime \prime }}} \end{aligned}$$

and \(d=d^\prime +d^{\prime \prime }.\)

Such products describe the quantum cohomology of a product of varieties, and within singularity theory it appears when one takes the direct sum of singularities.

2.2 Tensor products with trivial algebras

We now take the tensor product of a Frobenius manifold \({\mathcal {M}}\) with a trivial manifold \({\mathcal {A}}\) defined by a Frobenius algebra (Example 2.5). To emphasize the different roles played by \({\mathcal {M}}\) and \({\mathcal {A}}\), we alter the general notation for tensor products as described above. The tensor product will be written as \({\mathcal {M}}_{{\mathcal {A}}},\) (so \({\mathcal {M}}_{{\mathcal {A}}} = {\mathcal {M}}\otimes {{\mathcal {A}}}\)). The basis \(e_i\) for \({\mathcal {A}}\) will be retained and the unity element denoted by \(e_1.\) Thus, notation such as \(e=\partial _{1}\) will not be used. Latin indices will be reserved for \({\mathcal {A}}\)-related objects, and Greek indices will be reserved for \({\mathcal {M}}\)-related objects. Thus, \(c_{\alpha \beta }^{~~\gamma }\) will denote the structure functions for the multiplication on \({\mathcal {M}}\) and \(c_{ij}^{~~k}\) will denote the structure constants for the multiplication on \({\mathcal {A}}.\) Coordinates on \({\mathcal {M}}_{{\mathcal {A}}}\) are denoted

$$\begin{aligned} \{ t^{(\alpha i)}, \alpha = 1,\,\ldots ,\, m=\mathrm{dim} {\mathcal {M}},\,\quad i=1,\,\ldots ,\, n=\mathrm{dim} {\mathcal {A}} \}. \end{aligned}$$

No confusion should arise with this notation.

We begin by constructing a lift of a scalar-valued function to an \({\mathcal {A}}\)-valued function and visa-versa.

Definition 2.7

Let f be an analytic function on \({\mathcal {M}}\) (that is, analytic in the flat coordinates for \({\mathcal {M}}\)). The \({\mathcal {A}}\)-valued function \({\hat{f}}\) is defined to be:

$$\begin{aligned} {{\hat{f}}} = \left. f\right| _{t^\alpha \mapsto t^{(\alpha i)} e_i} \end{aligned}$$

with \({\widehat{fg}}={\hat{f}}\circ {\hat{g}}\) and \({\hat{1}}=e_1.\) The evaluation \(f^{\mathcal {A}}\) of \({\hat{f}}\) is defined by

$$\begin{aligned} f^{\mathcal {A}}=\omega \left( {\hat{f}}\right) , \end{aligned}$$

where \(\omega \in {\mathcal {A}}^\star .\)

Since the function is analytic and the algebra \({\mathcal {A}}\) is commutative and associative, the above construction is well-defined.

Remark 2.8

This definition requires the existence of a distinguished coordinate system on \({\mathcal {M}}\) in which the function f is analytic. In the case of analytic Frobenius manifolds, one automatically has such a distinguished system of coordinates, namely the flat coordinates of the metric.

With these definitions one may construct a new prepotential from the original one.

Theorem 2.9

Let F be the prepotential of a Frobenius manifold \({\mathcal {M}}\) and let \({\mathcal {A}}\) be a Frobenius algebra with 1-form \(\omega .\) The function

$$\begin{aligned} F^{\mathcal {A}} = \omega \left( {\widehat{F}}\right) \end{aligned}$$

defines a Frobenius manifold, namely the manifold \({\mathcal {M}}_{\mathcal {A}}.\)

Note, one could ‘straighten out’  the coordinates \(t^{(\alpha i)}\) via the map

$$\begin{aligned} v^{i+(\alpha -1) n}=t^{(\alpha i)}, \quad 1\le i \le n,\quad 1\le \alpha \le m, \end{aligned}$$

and hence \(F^{\mathcal {A}}=F^{\mathcal {A}}(v^1,\ldots ,v^{mn})\). However, such a map is not unique and the tensor structure is lost.

Proof

The proof is in two parts: we first show that the prepotential \(F^{\mathcal {A}}\) defines a Frobenius manifold and then identify this with the tensor product \({\mathcal {M}} \otimes {\mathcal {A}}.\)

By construction, we have an nm-dimensional manifold with coordinates \(t^{(\alpha i)},\alpha =1,\,\ldots ,\,m=\mathrm{dim}{\mathcal {M}}, i=1,\,\ldots ,\,n=\mathrm{dim}{\mathcal {A}}.\) We begin with two simple results:

  • Because \(\eta _{ij} = \omega (e_i \circ e_j)\) it follows, since by definition, \((\eta ^{ij})=(\eta _{ij})^{-1},\) that

    $$\begin{aligned} \omega (e_i \circ e_r) \eta ^{rs} \omega (e_s\circ e_j) = \omega (e_i\circ e_j). \end{aligned}$$

    More generally, using the properties of the multiplication on \({\mathcal {A}},\)

    $$\begin{aligned} \omega (\ldots \circ e_i \circ e_r) \eta ^{rs} \omega (e_s\circ e_j\circ \ldots ) = \omega (\ldots \circ e_i\circ e_j \circ \ldots ). \end{aligned}$$
    (2.7)

  • The fundamental result that will be used extensively in the rest of the paper is the following:

    $$\begin{aligned} \frac{\partial {\hat{f}}}{\partial t^{(\alpha i)}} = \widehat{\frac{\partial f}{\partial t^\alpha }} \circ e_i. \end{aligned}$$
    (2.8)

    We introduce the notation \({\hat{f}} = [{\hat{f}}]^p e_p,\) so

    $$\begin{aligned} \frac{\partial {\hat{f}}}{\partial t^{(\alpha i)}} = \left[ \widehat{\frac{\partial f}{\partial t^\alpha }}\right] ^p e_p \circ e_i. \end{aligned}$$

    This will be used to separate out the \({\mathcal {A}}\)-valued part of various expressions.

With these,

$$\begin{aligned} \frac{\partial ^3 {\hat{F}}}{\partial t^{(\alpha i)} \partial t^{(\beta j)} \partial t^{(\gamma k)}} = \widehat{\left( \frac{\partial ^3 F}{\partial t^\alpha \partial t^\beta \partial t^\gamma }\right) } \circ e_i \circ e_j \circ e_k, \end{aligned}$$

so

$$\begin{aligned} \frac{\partial ^3 F^{\mathcal {A}}}{\partial t^{(\alpha i)} \partial t^{(\beta j)} \partial t^{(\gamma k)}}= & {} \omega \left( \widehat{c_{\alpha \beta \gamma }} \circ e_i \circ e_j \circ e_k\right) ,\\= & {} \left[ \widehat{c_{\alpha \beta \gamma }} \right] ^p \omega (e_p\circ e_i \circ e_j \circ e_k),\\= & {} c_{(\alpha i)(\beta j)(\gamma k)}. \end{aligned}$$

Normalization

We define \(\eta _{(\alpha i)(\beta j)}\) by

$$\begin{aligned} \eta _{(\alpha i)(\beta j)}= & {} c_{(11)(\alpha i)(\beta j)},\\= & {} \omega \left( \widehat{c_{1\alpha \beta }} \circ e_1 \circ e_i \circ e_j\right) ,\\= & {} \eta _{\alpha \beta }\, \eta _{ij} \end{aligned}$$

since \(\widehat{c_{1\alpha \beta }} = \widehat{\eta _{\alpha \beta }} = \eta _{\alpha \beta } e_1,\) and \(e_1\) is the unity for the multiplication on \({\mathcal {A}}.\)

This is non-degenerate (since by assumption \(\eta _{\alpha \beta }\) and \(\eta _{ij}\) are non-degenerate) and this will be taken to be the metric and used to raise and lower indices. In particular, \(\eta ^{(\alpha i)(\beta j)}=\eta ^{\alpha \beta }\, \eta ^{ij}.\)

Associativity

Using the metric to raise an index, one obtains

$$\begin{aligned} c_{(\alpha i)(\beta j)}^{\phantom {{(\alpha i)(\beta j)}}(\gamma k)} = \left[ \widehat{c_{\alpha \beta }^{~~\gamma }} \right] ^p \, c_{ij}^{~~q} c_{pq}^{~~k} \end{aligned}$$
(2.9)

and this defines a multiplication on \({\mathcal {M}}_{{\mathcal {A}}}.\) The structure of this multiplication may be made more transparent if one writes the basis for \(T{\mathcal {M}}_{{\mathcal {A}}}\) as a tensor product:

$$\begin{aligned} \frac{\partial ~}{\partial t^{(\alpha i)}} = \partial _\alpha \otimes e_i. \end{aligned}$$

With this, the multiplication may be written as:

$$\begin{aligned} \left( \partial _\alpha \otimes e_i\right) \circ \left( \partial _\beta \otimes e_j\right) = \left[ \widehat{ \partial _\alpha \circ \partial _\beta }\right] ^p \otimes e_p \circ e_i\circ e_j, \end{aligned}$$

where \( \widehat{f\partial _\alpha }=[{\hat{f}}]^p \partial _\alpha \otimes e_p,\) and hence \(\left[ \widehat{f\partial _\alpha }\right] ^p=[{\hat{f}}]^p \partial _\alpha .\) By construction, this multiplication defines a commutative multiplication with unity \(e=\dfrac{\partial ~}{\partial t^{(1 1)}}=\partial _1\otimes e_1.\)

To prove associativity, we first rewrite the equation that has to be satisfied by \(F^{\mathcal {A}}\), namely the WDVV equation:

$$\begin{aligned}&\dfrac{\partial ^3F^{\mathcal {A}}}{\partial t^{(\gamma k)}\,\partial t^{(\sigma s)}\, \partial t^{(\alpha i)}} {\eta }^{(\alpha i)(\beta j)} \dfrac{\partial ^3F^{\mathcal {A}}}{\partial t^{(\beta j)}\,\partial t^{(\delta p)}\, \partial t^{(\mu q)}}\\&\quad = \dfrac{\partial ^3F^{\mathcal {A}}}{\partial t^{(\mu q)}\,\partial t^{(\sigma s)}\, \partial t^{(\alpha i)}} {\eta }^{(\alpha i)(\beta j)} \dfrac{\partial ^3F^{\mathcal {A}}}{\partial t^{(\beta j)}\,\partial t^{(\delta p)}\, \partial t^{(\gamma k)}}. \end{aligned}$$

This is equivalent to

$$\begin{aligned}&\left[ \widehat{c_{\gamma \sigma \alpha }}\right] ^a \omega (e_a\circ e_k\circ e_s\circ e_i) \eta ^{\alpha \beta } \eta ^{ij} \omega (e_j\circ e_p\circ e_q\circ e_b) \left[ \widehat{c_{\beta \delta \mu }}\right] ^b\\&\quad = \left[ \widehat{c_{ \mu \sigma \alpha }}\right] ^a \omega (e_a\circ e_q\circ e_s\circ e_i) \eta ^{\alpha \beta } \eta ^{ij} \omega (e_j\circ e_p\circ e_k\circ e_b) \left[ \widehat{c_{\beta \delta \gamma }}\right] ^b, \end{aligned}$$

which becomes, on using Eq. (2.7),

$$\begin{aligned}&\left[ \widehat{c_{\gamma \sigma \alpha }}\right] ^a \eta ^{\alpha \beta } \omega (e_a\circ e_k\circ e_s\circ e_p\circ e_q\circ e_b) \left[ \widehat{c_{\beta \delta \mu }}\right] ^b\nonumber \\&\quad = \left[ \widehat{c_{ \mu \sigma \alpha }}\right] ^a \eta ^{\alpha \beta } \omega (e_a\circ e_q\circ e_s\circ e_p\circ e_k\circ e_b) \left[ \widehat{c_{\beta \delta \gamma }}\right] ^b. \end{aligned}$$
(2.10)

Since the prepotential F for the Frobenius manifold \({\mathcal {M}}\) satisfies the WDVV equation

$$\begin{aligned} \dfrac{\partial ^3 F}{\partial t^\gamma \,\partial t^\sigma \, \partial t^\alpha } {\eta }^{\alpha \beta } \dfrac{\partial ^3 {F}}{\partial t^\beta \,\partial t^\delta \, \partial t^\mu }= \dfrac{\partial ^3 {F}}{\partial t^\mu \,\partial t^\sigma \, \partial t^\alpha } {\eta }^{\alpha \beta } \dfrac{\partial ^3 {F}}{\partial t^\beta \,\partial t^\delta \, \partial t^\gamma }, \end{aligned}$$

it follows that

$$\begin{aligned} \widehat{\dfrac{\partial ^3 F}{\partial t^\gamma \,\partial t^\sigma \, \partial t^\alpha }} \circ \widehat{ {\eta }^{\alpha \beta }} \circ \widehat{ \dfrac{\partial ^3 {F}}{\partial t^\beta \,\partial t^\delta \, \partial t^\mu }}= \widehat{\dfrac{\partial ^3 {F}}{\partial t^\mu \,\partial t^\sigma \, \partial t^\alpha }} \circ \widehat{ {\eta }^{\alpha \beta }} \circ \widehat{\dfrac{\partial ^3 {F}}{\partial t^\beta \,\partial t^\delta \, \partial t^\gamma }}, \end{aligned}$$

where \(\widehat{ {\eta }^{\alpha \beta }}={\eta }^{\alpha \beta }\, e_1\). This reduces to

$$\begin{aligned} \left[ \widehat{c_{\gamma \sigma \alpha }}\right] ^a \eta ^{\alpha \beta } e_a\circ e_b \left[ \widehat{c_{\beta \delta \mu }}\right] ^b = \left[ \widehat{c_{ \mu \sigma \alpha }}\right] \eta ^{\alpha \beta } e_a \circ e_b \left[ \widehat{c_{\beta \delta \gamma }}\right] ^b. \end{aligned}$$
(2.11)

Thus we have, by multiplying by \(e_q\circ e_s\circ e_p\circ e_k\),

$$\begin{aligned}&\left[ \widehat{c_{\gamma \sigma \alpha }}\right] ^a \eta ^{\alpha \beta } e_a\circ e_k\circ e_s\circ e_p\circ e_q\circ e_b \left[ \widehat{c_{\beta \delta \mu }}\right] ^b\\&\quad = \left[ \widehat{c_{ \mu \sigma \alpha }}\right] ^a \eta ^{\alpha \beta } e_a\circ e_q\circ e_s\circ e_p\circ e_k\circ e_b \left[ \widehat{c_{\beta \delta \gamma }}\right] ^b, \end{aligned}$$

and evaluating the function with \(\omega ,\) gives the identity (2.10). Hence \(F^{\mathcal {A}}\) satisfies the WDVV equation in the flat coordinates of the metric \(\eta _{(\alpha i)(\beta j)}.\)

Quasi-homogeneity

This follows immediately from the definition of \(F^{\mathcal {A}}\), but one can also derive the result by direct computation. The quasi-homogeneity of F is expressed by the equation

$$\begin{aligned} \sum _\alpha \left[ (1-q_\alpha ) t^\alpha + r_\alpha \right] \frac{\partial F}{\partial t^\alpha } = (3-d) F \end{aligned}$$

where quadratic terms will be ignored. On lifting this and using the evaluation map defined by \(\omega \), one obtains

$$\begin{aligned} \sum _{(\alpha i)} (1-q_\alpha ) t^{(\alpha i)} \omega \left( \widehat{\left( \frac{\partial F}{\partial t^\alpha }\right) } \circ e_i\right) + \sum _\alpha r_\alpha \omega \left( \widehat{\frac{\partial F}{\partial t^\alpha }} \right) = (3-d) F^{\mathcal {A}}. \end{aligned}$$

Using (2.8) yields the result \({E}^{\mathcal {A}}\left( F^{\mathcal {A}}\right) = (3-d) F^{\mathcal {A}}\) (again, up to quadratic terms) where

$$\begin{aligned} E^{\mathcal {A}} = \sum _{(\alpha i)} (1-q_\alpha ) t^{(\alpha i)} \frac{\partial ~}{\partial t^{(\alpha i)}} + \sum _\alpha r_\alpha \frac{\partial ~}{\partial t^{(\alpha 1)}}. \end{aligned}$$

These show that \(F^{\mathcal {A}}\) defines a Frobenius manifold. It remains to show that this is the tensor product \({\mathcal {M}} \otimes {\mathcal {A}}.\) In fact this is straightforward. Parts (i) and (iii) of Proposition 2.6 are immediate from above (since for the trivial Frobenius manifold \({\mathcal {A}}\), \(q_i=r_i=d=0\)), so it just remains to verify condition (ii). Since \(c_{\alpha \beta }^{\phantom {\alpha \beta }\gamma }\) is independent of \(t^1\) it follows that at points \(t^{(\alpha i)}=0,\alpha>1,i>1\) that \(\widehat{c_{\alpha \beta }^{\phantom {\alpha \beta }\gamma }}=c_{\alpha \beta }^{\phantom {\alpha \beta }\gamma } \left( t^{(\sigma 1)}\right) e_1\) and the result follows from Eq. (2.9).

Hence, the prepotential \(F^{\mathcal {A}}=\omega ({\widehat{F}})\) defines the Frobenius manifold structure on the tensor product \({\mathcal {M}}_{\mathcal {A}} = {\mathcal {M}}\otimes {\mathcal {A}}.\) If the multiplications on \({\mathcal {M}}\) and \({\mathcal {A}}\) are semisimple, then the multiplication on \({\mathcal {M}}_{\mathcal {A}}\) is also semisimple [13, 14]. \(\square \)

Remark 2.10

Note the existence of such a prepotential \(F^{\mathcal {A}}\) for such a tensor product follows from the original work of Kaufmann, Kontsevich and Manin. However, the explicit form for such an \(F^{\mathcal {A}}\) is not immediate from their construction. The above result gives an explicit and easily computable prepotential in the case when one of the manifolds is trivial.

Example 2.11

Let \({\mathcal {M}}\) be a one-dimensional Frobenius manifold

$$\begin{aligned}F(t^1)=\dfrac{1}{6}(t^1)^3,\, e=\partial _1, \, E=t^1\partial _1,\end{aligned}$$

so \({\mathcal {M}}_{{\mathcal {A}}}={\mathcal {A}}\) given in Example 2.5.

Example 2.12

Suppose \({\mathcal {A}}\) is a Frobenius algebra \({\mathcal {Z}}_{2,2}^{\varepsilon , 0}\) defined in Example 2.2. When \(\varepsilon \ne 0\), \({\mathcal {A}}\) is semisimple. When \(\varepsilon =0\), \({\mathcal {A}}\) is nonsemisimple and exactly the algebra \({\mathcal {Z}}_{2,2}\) given in Example 2.3. Let \({\mathcal {M}}\) be a two-dimensional Frobenius manifold with the flat coordinate \((t^1,t^2)\). We denote

$$\begin{aligned} \widehat{t^1}=v^1e_1+v^2 e_2,\quad \widehat{t^2}= v^3e_1+v^4 e_2. \end{aligned}$$

Case 1. \({\mathcal {M}}={\mathbb {C}}^2/W(A_2)\), i.e.,

$$\begin{aligned} F(t)=\frac{1}{2}(t^1)^2t^2-\frac{1}{72}(t^2)^4, \quad e=\frac{\partial }{\partial t^1},\quad E=t^1 \frac{\partial }{\partial t^1}+ \frac{2}{3}t^2 \frac{\partial }{\partial t^2}. \end{aligned}$$

The unity vector field and the Euler vector field of \({\mathcal {M}}_{\mathcal {A}}\) are given by, respectively,

$$\begin{aligned} e=\frac{\partial }{\partial v^1}, \quad E^{\mathcal {A}}=v^1\frac{\partial }{\partial v^1}+v^2\frac{\partial }{\partial v^2} +\frac{2}{3}v^3\frac{\partial }{\partial v^3}+\frac{2}{3}v^4\frac{\partial }{\partial v^4} \end{aligned}$$

and the potential of \({\mathcal {M}}_{\mathcal {A}}\) is given by

$$\begin{aligned} F^{\mathcal {A}}(v)=\frac{1}{2}(v^1)^2v^4+v^1v^2v^3-\frac{1}{18}(v^3)^3v^4 +\varepsilon \left( \frac{1}{2}(v^2)^2v^4-\frac{1}{18}v^3(v^4)^3\right) . \end{aligned}$$

We remark that when \(\varepsilon \ne 0\), \({\mathcal {M}}_{\mathcal {A}}\) is a polynomial semisimple Frobenius manifold. By a result of Hertling [11], the manifold \({\mathcal {M}}_{\mathcal {A}}\) decomposes into a direct product \({\mathcal {M}}_{A_2} \times {\mathcal {M}}_{A_2}\) of two \(A_2\)-Frobenius manifolds. The algebra \({\mathcal {A}}\) can be seen as controlling this decomposition.

Case 2. \({\mathcal {M}}=\mathrm {QH^*(\mathrm {CP}^1)}\), i.e.,

$$\begin{aligned} F(t)=\frac{1}{2}(t^1)^2t^2+e^{t^2}, \quad e=\dfrac{\partial }{\partial t^1}, \quad E=t^1\dfrac{\partial }{\partial t^1}+2\dfrac{\partial }{\partial t^2}. \end{aligned}$$

The unity vector field and the Euler vector field of \({\mathcal {M}}_{\mathcal {A}}\) are given by, respectively,

$$\begin{aligned} e=\dfrac{\partial }{\partial v^1},\quad E^{\mathcal {A}}=v^1\dfrac{\partial }{\partial v^1}+v^2\dfrac{\partial }{\partial v^2} +2\dfrac{\partial }{\partial v^3}+2\dfrac{\partial }{\partial v^4} \end{aligned}$$

and the potential of \({\mathcal {M}}_{\mathcal {A}}\) is given by

$$\begin{aligned} F^{\mathcal {A}}(v)=\left\{ \begin{array}{ll} \dfrac{1}{2}(v^1)^2v^4+v^1v^2v^3+\varepsilon (v^2)^2v^4+ \dfrac{\sinh (\sqrt{\varepsilon } v^4)}{\sqrt{\varepsilon }}\,e^{v^3},&{}\varepsilon \ne 0,\\ \dfrac{1}{2}(v^1)^2v^4+v^1v^2v^3+v^4\,e^{v^3}, &{}\varepsilon =0.\end{array}\right. \end{aligned}$$

3 \({\mathcal {A}}\)-valued topological quantum field theories

The ideas developed in the last section may be applied to the construction of \({\mathcal {A}}\)-valued topological quantum fields Theories on a suitably defined big phase space (i.e., with gravitational descendent fields). In fact, one could have started with this larger construction and obtained the results of the last section by restriction to the small phase space. Conversely, the reconstruction theorems which give big phase space structures from Frobenius manifold structures could be used to construct these \({\mathcal {A}}\)-valued TQFTs from the Frobenius manifold \({\mathcal {M}}_{\mathcal {A}}.\)

3.1 Background

A topological quantum field theory (or TQFT) is defined in terms of properties of certain correlators which are themselves defined in terms of prepotential \({\mathcal {F}}_{g\ge 0}\). For example, consider a smooth projective variety V with \(H^\mathrm{odd}(V;{\mathbb {C}})=0\), \(\{\gamma _1,\ldots ,\gamma _N\}\) a basis for the cohomology ring \(M:=H^{*}(V;{\mathbb {C}})\) and let

$$\begin{aligned} \eta _{\alpha \beta } = \eta (\gamma _\alpha ,\gamma _\beta ) = \int _V \gamma _\alpha \cup \gamma _\beta \end{aligned}$$

be the Poincaré pairing which defines a non-degenerate metric which may be used to raise and lower indices. Following the conventions of Liu and Tian [16, 17], a flat coordinate system \(\{t^\alpha _0, \alpha =1,\ldots ,N\}\) may be found on M so \(\gamma _\alpha =\frac{\partial ~}{\partial t^\alpha _0}\), and in which the components of \(\eta \) are constant.

The big phase space consists of an infinite number of copies of the M,  the small phase space, so

$$\begin{aligned} M^\infty = \prod _{n\ge 0} H^{*}(V;{\mathbb {C}}). \end{aligned}$$

The coordinate system \(\{ t_{0}^{\alpha }\}\) induces, in a canonical way, a coordinate system \(\{t^\alpha _n, n \in {\mathbb {Z}}_{\ge 0},\alpha =1,\ldots ,N\}\) on \(M^{\infty }.\) We denote by \(\tau _n(\gamma _\alpha ) = \frac{\partial ~}{\partial t^\alpha _n}\) (also abbreviated to \(\tau _{n,\alpha }\,)\) the associated fundamental vector fields, which represent various tautological line bundles over the moduli space of curves [16, 17].

The descendant Gromov–Witten invariants

$$\begin{aligned} \langle \tau _{n_1}(\gamma _{a_1}) \ldots \tau _{n_k}(\gamma _{a_k})\rangle _g \end{aligned}$$

may be combined into generating functions, called prepotentials, labeled by the genus g, 

$$\begin{aligned} {{\mathcal {F}}}_g=\sum _{k\ge 0} \frac{1}{k!} \sum _{n_1 ,\alpha _1\ldots n_k,\alpha _k} t^{\alpha _1}_{n_1} \ldots t^{\alpha _k}_{n_k} \langle \tau _{n_1}(\gamma _{\alpha _1}) \ldots \tau _{n_k}(\gamma _{\alpha _k})\rangle _g, \end{aligned}$$

and these in turn may be used to define k-tensor fields on the big phase space, via the formula

$$\begin{aligned} \langle \langle {{\mathcal {W}}}_{1}\ldots {\mathcal W}_{k}\rangle \rangle _{g} = \sum _{m_{1},\alpha _{1},\ldots , m_{k},\alpha _{k}} f^{1}_{m_{1},\alpha _{1}} \cdots f^{k}_{m_{k},\alpha _{k}} \frac{\partial ^{k} {{\mathcal {F}}_{g}}}{\partial t^{\alpha _{1}}_{m_{1}}\ldots \partial t^{\alpha _{k}}_{m_{k}} }, \end{aligned}$$
(3.1)

for any vector fields \({{\mathcal {W}}}_{i} = \sum _{m,\alpha } f^{i}_{m,\alpha } \frac{\partial }{\partial t^{\alpha }_{m}}\). The tensor field (3.1) has a physical interpretation as the k-point correlation function of the TQFT.

The basic relationships between these correlators may then be encapsulated in the following:

Definition 3.1

Let \({\tilde{t}}^\alpha _n=t^\alpha _n - \delta _{n,1} \delta _{\alpha ,1}\) and let

$$\begin{aligned} {\mathcal {S}}= & {} -\sum _{n,\alpha } {\tilde{t}}^\alpha _n \tau _{n-1}(\gamma _{\alpha } ),\\ {\mathcal {D}}= & {} -\sum _{n,\alpha } {\tilde{t}}^\alpha _n \tau _{n}(\gamma _{\alpha }) \end{aligned}$$

be the string and dilaton vector fields, respectively. Then the prepotentials \({{\mathcal {F}}}_g\) satisfy the following relations:

String equation

$$\begin{aligned} \langle \langle {\mathcal {S}} \rangle \rangle _g = \frac{1}{2} \delta _{g,0} \sum _{\alpha ,\beta } \eta _{\alpha \beta } t^\alpha _0 t^\beta _0\,; \end{aligned}$$

Dilaton equation

$$\begin{aligned} \langle \langle {\mathcal {D}} \rangle \rangle _g = (2g-2) {{\mathcal {F}}}_g - \frac{1}{24} \chi (V) \delta _{g,1}\,; \end{aligned}$$

Genus-zero topological recursion relation

$$\begin{aligned} \langle \langle \tau _{m+1}(\gamma _\alpha ) \tau _n(\gamma _\beta ) \tau _k(\gamma _\sigma ) \rangle \rangle _{{}_0} = \langle \langle \tau _{m}(\gamma _\alpha )\gamma _\mu \rangle \rangle _{{}_0} \langle \langle \gamma ^\mu \tau _n(\gamma _\beta ) \tau _k(\gamma _\sigma ) \rangle \rangle _{{}_0}. \end{aligned}$$

By restricting such theories to primary vector fields with coefficients in the small phase space, one recovers a Frobenius manifold structure [5, 6] on the small phase space, with

$$\begin{aligned} F_0(t_0^1,\ldots ,t_0^N) = \left. {\mathcal {F}}_0( \mathbf{t}) \right| _{t^\alpha _n=0,\,n>0} \end{aligned}$$

becoming the prepotential for the Frobenius manifold and multiplication given by

$$\begin{aligned} \tau _{0,\alpha }\circ \tau _{0,\beta } = \langle \langle \tau _{0,\alpha }\tau _{0,\beta } \gamma ^{\sigma }\rangle \rangle _{{}_0}\vert _{M} \gamma _{\sigma }. \end{aligned}$$

3.2 \({\mathcal {A}}\)-TQFT

Given such a theory, one may extend the previous construction to obtain a new TQFT. Again, the existence of such a result follows from various reconstruction theorems, but explicit formulae may be obtained when one tensors by a constant Frobenius algebra.

Theorem 3.2

Let \({\mathcal {F}}_{g\ge 0}\) be the prepotentials defining a TQFT, \({\mathcal {S}}\) and \({\mathcal {D}}\) the corresponding String and Dilaton vector fields and \({\mathcal {A}}\) be a trivial Frobenius algebra. Let f be an analytic function on \({\mathcal {M}}^\infty \) (that is, analytic in the flat coordinates \(t_N^\alpha \) for \({\mathcal {M}}^\infty \)) and define the \({\mathcal {A}}\)-valued function \({\hat{f}}\) to be:

$$\begin{aligned} {{\hat{f}}} = \left. f\right| _{t^\alpha _N \mapsto t^{(\alpha i)}_N e_i},\qquad N\in {\mathbb {Z}}_{\ge 0},\quad \alpha =1,\ldots , m,\quad i=1,\ldots ,n. \end{aligned}$$
(3.2)

Then the functions

$$\begin{aligned} {\mathcal {F}}^{{\mathcal {A}}}_{g\ge 0} = \omega \left( {\widehat{ {\mathcal {F}}}_{g\ge 0} }\right) \end{aligned}$$

and vector fields

$$\begin{aligned} {\mathcal {S}}^{\mathcal {A}}= & {} -\sum _{N,(\alpha i)} {\tilde{t}}^{(\alpha ,i)}_N \tau _{N-1,(\alpha i)},\\ {\mathcal {D}}^{\mathcal {A}}= & {} -\sum _{N,(\alpha i)} {\tilde{t}}^{(\alpha ,i)}_N \tau _{N,(\alpha i)} \end{aligned}$$

satisfy the axioms of a topological quantum field theory.

Proof

Genus-zero topological recursion relation

By repeating the construction in Theorem 2.9 (essentially using (2.8)), one easily obtains the equation

$$\begin{aligned} \langle \langle \tau _{M+1,(\alpha i)} \tau _{N,(\beta j)} \tau _{K,(\sigma k)}\rangle \rangle _{{}_0} = \omega \left( \langle \langle \tau _{M+1,\alpha } \tau _{N,\beta } \tau _{K,\sigma }\rangle \rangle _{{}_0}^{\hat{}} \circ e_i \circ e_j \circ e_k \right) \end{aligned}$$

(where we displace the \(\hat{}\) symbol for notational convenience, so \(f\,{}^{\hat{}}={\hat{f}}\)). On using the topological recursion relation, this decomposes as

$$\begin{aligned}&\langle \langle \tau _{M+1,(\alpha i)} \tau _{N,(\beta j)} \tau _{K,(\sigma k)}\rangle \rangle _{{}_0} = \eta ^{\mu \nu } \omega \left( \langle \langle \tau _{M,\alpha } \gamma _\mu \rangle \rangle _{{}_0}^{\hat{}} \circ e_i \circ e_j \circ \langle \langle \gamma _\mu \tau _{N,\beta } \tau _{K,\sigma } \rangle \rangle _{{}_0}^{\hat{}}\circ e_k \right) \,\\&\quad = \eta ^{\mu \nu } \omega \left( \langle \langle \tau _{M,\alpha } \gamma _\mu \rangle \rangle _{{}_0}^{\hat{}} \circ e_i \circ e_r\right) \eta ^{rs} \omega \left( e_s \circ e_j \circ \langle \langle \gamma _\mu \tau _{N,\beta } \tau _{K,\sigma } \rangle \rangle _{{}_0}^{\hat{}}\circ e_k \right) \end{aligned}$$

on using (2.7). Since

$$\begin{aligned} \langle \langle \tau _{M,(\alpha i)} \gamma _{(\mu r)} \rangle \rangle _0= & {} \omega \left( \langle \langle \tau _{M,\alpha } \gamma _\mu \rangle \rangle _{{}_0}^{\hat{}} \circ e_i \circ e_r\right) ,\\ \langle \langle \gamma _{(\mu s)} \tau _{N,(\beta j)} \tau _{K,(\sigma k)} \rangle \rangle _{{}_0}= & {} \omega \left( e_s \circ \langle \langle \gamma _\mu \tau _{N,\beta } \tau _{K,\sigma } \rangle \rangle _{{}_0}^{\hat{}}\circ e_s \circ e_j \circ e_k \right) , \end{aligned}$$

the result follows.

String equation

Again, on using (2.8) it follows that

$$\begin{aligned} \langle \langle {\mathcal {S}}^{\mathcal {A}} \rangle \rangle _g= & {} - \sum _{M,(\alpha i)} {{\tilde{t}}}^{(\alpha ,i)}_M \omega \left[ \widehat{\frac{\partial {\mathcal {F}}_g}{\partial t^\alpha _{M-1}}} \circ e_i\right] ,\\= & {} \omega \left( \langle \langle {\mathcal {S}} \rangle \rangle ^{\hat{}}_g \right) . \end{aligned}$$

Since \({\mathcal {S}}\) satisfies the string equation,

$$\begin{aligned} \langle \langle {\mathcal {S}}^{\mathcal {A}} \rangle \rangle _g= & {} \frac{1}{2} \delta _{g,0} \omega \left[ \sum _{\alpha ,\beta } \hat{t^\alpha _0} \circ \hat{t^\beta _0} \right] ,\\= & {} \frac{1}{2} \delta _{g,0} \sum _{(\alpha ,i),(\beta ,j)} \eta _{(\alpha i)(\beta j)} t^{(\alpha i)}_0 t^{(\beta j)}_0, \end{aligned}$$

using the definition of the lifting map and the fundamental property \(\omega (e_i \circ e_j)=\eta _{ij}.\)

Dilaton equation

Similarly, since \({\mathcal {D}}\) satisfies the Dilaton equation,

$$\begin{aligned} \langle \langle {\mathcal {D}}^{\mathcal {A}} \rangle \rangle _g= & {} \omega \left( \langle \langle {\mathcal {D}} \rangle \rangle _g^{\hat{}} \right) ,\\= & {} (2g-2) \omega (\hat{{\mathcal {F}}}_g) - \frac{1}{24} \delta _{g,1} \chi (V) \omega (e_1),\\= & {} (2g-2) {\mathcal {F}}^{\mathcal {A}}_g - \frac{1}{24} \delta _{g,1} \chi ^{\mathcal {A}}(V), \end{aligned}$$

where \(\chi ^{\mathcal {A}}(V)=\chi (V) \omega (e_1).\) \(\square \)

Remark 3.3

The above axioms do not include the big phase space counterpart to the Euler vector field, but the same ideas may be applied if such a field exists on the original TQFT.

The individual prepotentials may be combined into a single \(\tau \)-function

$$\begin{aligned} \tau (t^\alpha _N) = e^{\sum \hbar ^{g-1} {\mathcal {F}}_g}. \end{aligned}$$

In the simplest case, when \(\mathrm{dim} {\mathcal {M}}=1\), this defines a specific solution of the KdV hierarchy. The full connection between such \(\tau \)-functions and corresponding integrable hierarchies remains an important open problem.

Since each prepotential \({\mathcal {F}}_g\) lifts to prepotentials \({\mathcal {F}}^{\mathcal {A}}_g\), one may define a corresponding \(\tau \)-function

$$\begin{aligned} \tau ^{\mathcal {A}}(t^\alpha _N) = e^{\sum \hbar ^{g-1} {\mathcal {F}}^{\mathcal {A}}_g} \end{aligned}$$

and it is clear that \(\tau ^{\mathcal {A}} = \omega \left[ {\hat{\tau }}\right] .\) It seems natural to conjecture that such a function should define a solution to a corresponding \({\mathcal {A}}\)-valued dispersive integrable hierarchy. However, this first requires the development of a theory of such \({\mathcal {A}}\)-valued hierarchies.

3.3 The role of the Frobenius form \(\omega \)

The Frobenius form \(\omega \) plays a vital role in the above constructions; without it one only has \({\mathcal {A}}\)-valued objects. However, one can dispense with it and deal directly with such \({\mathcal {A}}\)-valued objects and derive relations satisfied by them. For example, using the lifting map (3.2), one can define \({\mathcal {A}}\)-valued ‘correlators’:

$$\begin{aligned} \langle \langle \tau _{N,(\alpha i)} \ldots \tau _{M,(\beta j)} \rangle \rangle _g^{{\mathcal {A}}}= & {} \left[ \frac{\partial ~}{\partial t^\alpha _N} \ldots \frac{\partial ~}{\partial t^\beta _M} {\mathcal {F}}_g\right] ^{\hat{}} \circ e_i \circ \ldots \circ e_j,\\= & {} \langle \langle \tau _{N,\alpha } \ldots \tau _{M,\beta } \rangle \rangle _g^{\hat{}}\circ e_i \circ \ldots \circ e_j. \end{aligned}$$

It is straightforward to derive the following recursion relation:

$$\begin{aligned}&\Omega \circ \langle \langle \tau _{M+1, (\alpha i)} \tau _{N, (\beta j)} \tau _{K, (\sigma k)} \rangle \rangle ^{{\mathcal {A}}}_{{}_0} \\&\quad = \eta ^{(\mu r)(\gamma s)} \langle \langle \tau _{M,(\alpha i)} \tau _{0,(\mu r)} \rangle \rangle ^{\mathcal {A}}_{{}_0} \circ \langle \langle \tau _{0, (\gamma s)} \tau _{N, (\beta j)} \tau _{K, (\sigma k)} \rangle \rangle ^{{\mathcal {A}}}_{{}_0}, \end{aligned}$$

where \(\Omega = \eta ^{rs} e_r \circ e_s.\) If this element is invertible, then one can obtain a bona fide \({\mathcal {A}}\)-valued recursion relation. We will not further develop such a theory here.

4 \({\mathcal {A}}\)-valued dispersionless integrable systems

It was shown by Dubrovin that, given a Frobenius manifold \({\mathcal {M}}\), one can construct an associated bi-Hamiltonian hierarchy of hydrodynamic type, known as the principal hierarchy, with the geometry of the manifold encoding the various components required in its construction. This hierarchy may be written as

$$\begin{aligned} \begin{array}{lcr} \displaystyle {\frac{\partial t^\alpha }{\partial T^{(N,\sigma )}}} &{} = &{} \displaystyle {{\mathcal {P}}_1^{\alpha \beta } \, \frac{\partial h_{(N,\sigma )}}{\partial t^\beta }},\\ &{}&{}\\ &{} = &{}\displaystyle {{\mathcal {P}}_2^{\alpha \beta } \, \frac{\partial h_{(N-1,\sigma )}}{\partial t^\beta }} \end{array} \end{aligned}$$
(4.1)

with (compatible) Hamiltonian operators

$$\begin{aligned} {\mathcal {P}}^{\alpha \beta }_1=\eta ^{\alpha \beta } \frac{\mathrm{d}~}{\mathrm{d}X}, \qquad {\mathcal {P}}^{\alpha \beta }_2 = \, g^{\alpha \beta } \frac{\mathrm{d}~}{\mathrm{d}X} + \Gamma ^{\alpha \beta }_\gamma t^\gamma _X, \end{aligned}$$

where \(g^{\alpha \beta }=c^{\alpha \beta }_{\phantom {\alpha \beta }\gamma } E^\gamma \) is the intersection form on \({\mathcal {M}}\) (and \(\Gamma ^{\alpha \beta }_\gamma =-g^{\alpha \mu } \Gamma ^\beta _{\mu \gamma }\)). The Hamiltonian densities \(h_{(N,\sigma )}\) come from the coefficients in the expansion of the deformed flat coordinates for the Dubrovin connection,

$$\begin{aligned} t_\alpha (\lambda ) = \sum _{N=0}^\infty h_{(N,\alpha )} \lambda ^N,\quad h_{(0,\alpha )} = \eta _{\alpha \beta } t^\beta , \end{aligned}$$

and these satisfy the recursion relation

$$\begin{aligned} \frac{\partial ^2 h_{(N,\sigma )}}{\partial t^\alpha \partial t^\beta } = c_{\alpha \beta }^{\phantom {\alpha \beta }\mu }(\mathbf{t}) \frac{\partial h_{(N-1,\sigma )}}{\partial t^\mu } \end{aligned}$$
(4.2)

(together with certain normalization conditions).

The Frobenius manifold \({\mathcal {M}}_{\mathcal {A}}\) will automatically inherit such a hierarchy by the very nature of it being a Frobenius manifold. However, such a hierarchy is best written as an \({\mathcal {A}}\)-valued system, with m-\({\mathcal {A}}\)-valued dependent fields rather than mn-scalar-valued dependent fields.

We begin by showing how the deformed flat variables on \({\mathcal {M}}_{\mathcal {A}}\) may be constructed from those on \({\mathcal {M}}.\) This is achieved by lifting and evaluation the Hamiltonian densities for \({\mathcal {M}}.\)

Lemma 4.1

Let \(h_{N,\sigma }\) be the coefficients in the deformed flat connection on \({\mathcal {M}}.\) Then the functions

$$\begin{aligned} {\mathfrak {h}}_{(N,\sigma ,r)} = \omega \left( \widehat{h_{(N,\sigma )}} \circ e_r \right) \end{aligned}$$

satisfy the recursion relation

$$\begin{aligned} \frac{\partial ^2 {\mathfrak {h}}_{(N,\sigma ,r)}}{\partial t^{(\alpha i)} \partial t^{(\beta j)}} = c_{(\alpha i)(\beta j)}^{\phantom {{(\alpha i)(\beta j)}}(\gamma k)} \frac{\partial {\mathfrak {h}}_{(N-1,\sigma ,r)}}{\partial t^{(\gamma k)}}. \end{aligned}$$

and the initial conditions \({\mathfrak {h}}_{(0,\sigma ,r)} = \eta _{(\sigma r)(\mu s)} t^{(\mu s)}\) and hence define the deformed flat coordinates on \({\mathcal {M}}_{\mathcal {A}}.\)

Proof

This is a straightforward calculation (we drop the \(\sigma \)-label on the various h’s for clarity): We have

$$\begin{aligned} \frac{\partial \widehat{h_N}}{\partial t^{(\alpha i)}} = \widehat{\left( \frac{\partial h_N}{\partial t^\alpha }\right) } \circ e_i \end{aligned}$$

and hence

$$\begin{aligned} \frac{\partial ^2 \widehat{h_N}}{\partial t^{(\alpha i)} \partial t^{(\beta j)}}= & {} \widehat{\left( \frac{\partial ^2 h_N}{\partial t^\alpha \partial t^\beta }\right) } \circ e_i \circ e_j,\\= & {} \widehat{\left( c_{\alpha \beta }^{~~\gamma }\right) } \circ \widehat{ \frac{\partial h_{N-1}}{\partial t^\gamma } } \circ e_i \circ e_j. \end{aligned}$$

Thus, using \(\omega \) to evaluate this \({\mathcal {A}}\)-valued expression gives

$$\begin{aligned} \frac{\partial ^2 {\mathfrak {h}}_{(N,r)}}{\partial t^{(\alpha i)} \partial t^{(\beta j)}}= & {} \omega \left( \widehat{\left( \frac{\partial ^2 h_N}{\partial t^\alpha \partial t^\beta }\right) } \circ e_i \circ e_j \circ e_r\right) ,\\= & {} \left[ \widehat{ c_{\alpha \beta }^{~~\gamma } }\right] ^p c_{ij}^{~~q} \omega \left( \widehat{ \left( \frac{\partial h_{N-1}}{\partial t^\gamma }\right) } \circ e_p \circ e_q \circ e_r \right) ,\\= & {} \underbrace{\left[ \widehat{ c_{\alpha \beta }^{~~\gamma } }\right] ^p c_{ij}^{~~q} c_{pq}^{~~k} }_{c_{(\alpha i)(\beta j)}^{\phantom {(\alpha i)(\beta j)} (\gamma k)}} \omega \left( \frac{\partial \widehat{h_{N-1}}}{\partial t^{(\gamma k)}} \circ e_r \right) ,\\= & {} c_{(\alpha i)(\beta j)}^{\phantom {(\alpha i)(\beta j)} (\gamma k)} \frac{\partial {\mathfrak {h}}_{(N-1,r)}}{\partial t^{(\gamma k)}}. \end{aligned}$$

If \(N=0\), then, since \(\widehat{t^\mu }=t^{(\mu s)} e_s,\)

$$\begin{aligned} {\mathfrak {h}}_{(0,\sigma ,r)}= & {} \omega \left( \widehat{ h_{(0,\sigma )}} \circ e_r\right) ,\\= & {} \eta _{\sigma \mu } \eta _{rs} t^{(\mu s)} \omega \left( e_s \circ e_r\right) ,\\= & {} \eta _{(\sigma r)(\mu s)} t^{(\mu s)}, \end{aligned}$$

which is, as required, a Casimir function on \({\mathcal {M}}_{{\mathcal {A}}}.\) \(\square \)

In the obvious way, one can lift the operators \({\mathcal {P}}_1,{\mathcal {P}}_2\) to \({\mathcal {A}}\)-valued operators and obtain the following theorem:

Theorem 4.2

The principal hierarchy on \({\mathcal {M}}_{\mathcal {A}}\) may be written in terms of \({\mathcal {A}}\)-valued fields, densities and operators, as

$$\begin{aligned} \begin{array}{lcr} \displaystyle { \frac{\partial {{{\widehat{t}}}^\alpha }}{\partial T^{(N,\sigma , r)}}} &{} = &{} \displaystyle {\widehat{{\mathcal {P}}_1^{\alpha \beta }} \circ \, \frac{\partial \widehat{h_{(N,\sigma )}}}{\partial t^{(\beta r)}}},\\ &{}&{}\\ &{} = &{} \displaystyle {\widehat{{\mathcal {P}}_2^{\alpha \beta }} \circ \, \frac{\partial \widehat{h_{(N-1,\sigma )}}}{\partial t^{(\beta r)}}}. \end{array} \end{aligned}$$
(4.3)

Proof

First Hamiltonian structure

By definition, and on using previous results,

$$\begin{aligned} \frac{\partial t^{(\alpha i)}}{\partial T^{(N,\sigma ,r)}}= & {} \eta ^{(\alpha i)(\beta j)} \frac{\mathrm{d}~}{\mathrm{d}X} \frac{\partial {\mathfrak {h}}_{(N,\sigma ,r)}}{\partial t^{(\beta j)}},\\= & {} \eta ^{\alpha \beta } \eta ^{ij} \frac{\mathrm{d}~}{\mathrm{d}X} \left[ \widehat{ \frac{\partial h_{(N,\sigma )}}{\partial t^\beta }}\right] ^k \omega (e_k\circ e_j\circ e_r). \end{aligned}$$

Since \(\widehat{t^\alpha } = t^{(\alpha i)} e_i\) by definition, one obtains

$$\begin{aligned} \frac{\partial \widehat{t^\alpha } }{\partial T^{(N,\sigma ,r)}}= & {} \eta ^{\alpha \beta } \frac{\mathrm{d}~}{\mathrm{d}X} \left[ \widehat{ \frac{\partial h_{(N,\sigma )}}{\partial t^\beta }}\right] ^k \, \eta ^{ij} \omega (e_k\circ e_j\circ e_r) e_i,\\= & {} \eta ^{\alpha \beta } \frac{\mathrm{d}~}{\mathrm{d}X} \left\{ \widehat{ \frac{\partial h_{(N,\sigma )}}{\partial t^\beta }} \circ e_r \right\} ,\\= & {} \widehat{\eta ^{\alpha \beta }}\circ \frac{\mathrm{d}~}{\mathrm{d}X} \widehat{ \frac{\partial h_{(N,\sigma )}}{\partial t^{(\beta r)}}}, \end{aligned}$$

since as the components of \(\eta \) are constants, \(\widehat{\eta ^{\alpha \beta }} = \eta ^{\alpha \beta } e_1.\)

Second Hamiltonian structure

The second Hamiltonian operator \({\mathcal {P}}^{\alpha \beta }_2\) on \({\mathcal {M}}\) takes the formFootnote 2

$$\begin{aligned} {\mathcal {P}}^{\alpha \beta }_2 = g^{\alpha \beta } \frac{{rm d}~}{\mathrm{d}X} + \left( \frac{d+1}{2} - q_\beta \right) c^{\alpha \beta }_{\phantom {\alpha \beta }\gamma } t^\gamma _X \end{aligned}$$

and hence on \({\mathcal {M}}_{\mathcal {A}},\)

$$\begin{aligned} \frac{\partial t^{(\alpha i)}}{\partial T^{(N,\sigma ,r)}} = \left[ g^{(\alpha i)(\beta j)} \frac{\mathrm{d}~}{\mathrm{d}X} + \left( \frac{d+1}{2} - q_\beta \right) c^{(\alpha i)(\beta j)}_{\phantom {(\alpha i)(\beta j)}(\gamma k)} t^{(\gamma k)}_X \right] \frac{ \partial {\mathfrak {h}}_{(N-1,\sigma , r)}}{\partial t^{(\beta j)}}.\nonumber \\ \end{aligned}$$
(4.4)

Note, since the Euler vector field on \({\mathcal {A}}\) is trivial (\(q_i=r_i=d_{\mathcal {A}}=0\)), it follows that \(q_{(\beta j)} = q_\beta \) and d is the same on both \({\mathcal {M}}\) and \({\mathcal {M}}_{\mathcal {A}}.\) Also, by definition,

$$\begin{aligned} g^{(\alpha i)(\beta j)}= & {} c^{(\alpha i)(\beta j)}_{\phantom {(\alpha i)(\beta j)}(\gamma k)} E^{(\gamma k)},\\= & {} \eta ^{\beta \mu } \eta ^{js} \left[ \widehat{ c_{\mu \gamma }^{\phantom {\mu \gamma }\alpha } } \right] ^p c_{sk}^{\phantom {sk}q} c_{pq}^{\phantom {pq}i} (1-q_\gamma ) t^{(\gamma k)}. \end{aligned}$$

For simplicity, we will consider the first term in (4.4) only, and the corresponding proof of the second term follows practically verbatim the proof of the first. Thus,

$$\begin{aligned}&g^{(\alpha i)(\beta j)} \frac{\mathrm{d}~}{\mathrm{d}X} \frac{ \partial {\mathfrak {h}}_{(N-1,\sigma , r)}}{\partial t^{(\beta j)}} \\&\quad = \left[ c^{\alpha \beta }_{\phantom {\alpha \beta }\gamma }\right] ^p c_{pk}^{\phantom {pk}q} (1-q_\gamma ) t^{(\gamma k)} \frac{\mathrm{d}~}{\mathrm{d}X} \left[ \widehat{ \frac{\partial h_{(N-1,\sigma )}}{\partial t^\beta }} \right] ^d \omega (e_d \circ e_j \circ e_r),\\&\quad = \left[ \widehat{g^{\alpha \beta }} \right] ^q c_q^{\phantom {q}ij} \frac{\mathrm{d}~}{\mathrm{d}X} \left[ \widehat{ \frac{\partial h_{(N-1,\sigma )}}{\partial t^\beta }} \right] ^d c_{dr}^{\phantom {dr}s} \eta _{sj}, \end{aligned}$$

since \(\widehat{g^{\alpha \beta }} = \widehat{c^{\alpha \beta }_{\phantom {\alpha \beta }\gamma }} \circ (1-q_\gamma ) t^{(\gamma q)} e_q.\) On using the associative and commutative properties of the multiplication, and on contracting with \(e_i\), one obtains

$$\begin{aligned} g^{(\alpha i)(\beta j)} \frac{\mathrm{d}~}{\mathrm{d}X} \frac{ \partial {\mathfrak {h}}_{(N-1,\sigma , r)}}{\partial t^{(\beta j)}} e_i= & {} \left[ \widehat{g^{\alpha \beta }} \right] ^q c_{qs}^{\phantom {qs}i} \frac{\mathrm{d}~}{\mathrm{d}X} \left[ \widehat{ \frac{\partial h_{(N-1,\sigma )}}{\partial t^\beta }} \circ e_r \right] ^s e_i,\\= & {} \widehat{g^{\alpha \beta }} \circ \frac{\mathrm{d}~}{\mathrm{d}X} \left[ \widehat{ \frac{ \partial h_{(N-1,\sigma )}}{\partial t^\beta }} \circ e_r \right] ,\\= & {} \widehat{g^{\alpha \beta }} \circ \frac{\mathrm{d}~}{\mathrm{d}X} \widehat{ \frac{\partial h_{(N-1,\sigma )}}{\partial t^{(\beta r)}} }. \end{aligned}$$

Note that these flows on \({\mathcal {M}}_{\mathcal {A}}\) simplify if \(r=1.\) \(\square \)

Example 4.3

If \(\dim {\mathcal {M}}=1\) and \(r=1\), one obtains the bi-Hamiltonian structures from the \({\mathcal {A}}\)-valued Mongé equation

$$\begin{aligned} {\mathcal {U}}_T= {\mathcal {U}}\circ {\mathcal {U}}_X \end{aligned}$$

with conserved densities

$$\begin{aligned} {\mathfrak {h}}_N = \frac{1}{(N+1)!} \omega ( \underbrace{{\mathcal {U}} \circ \cdots \circ {\mathcal {U}}}_{N+1\mathrm {~terms}} ). \end{aligned}$$

The form of the flows in Theorem 4.2 is somewhat hybrid in nature, and to rewrite them as a genuine \({\mathcal {A}}\)-valued bi-Hamiltonian system one must introduce the variational derivative with respect to an \({\mathcal {A}}\)-valued field. Such a derivative was introduced in [19] and is defined by the equation

$$\begin{aligned} \langle \delta {\mathcal {H}}; v \rangle = \left. \frac{\mathrm{d}~}{\mathrm{d}\epsilon } {\mathcal {H}}\left[ \widehat{u^\alpha } + \epsilon \widehat{v^\alpha } \right] \right| _{\epsilon =0}, \end{aligned}$$
(4.5)

where

$$\begin{aligned} {\mathcal {H}}= \int \omega ({\widehat{h}}) \,\mathrm{d}X. \end{aligned}$$

With this the flows may be written as an \({\mathcal {A}}\)-valued bi-Hamiltonian system.

Corollary 4.4

The flows given in Theorem 4.2 may be written as

$$\begin{aligned} \begin{array}{lcr} \displaystyle { \frac{\partial {{{\widehat{t}}}^\alpha }}{\partial T^{(N,\sigma , r)}}} &{} = &{} \displaystyle {\widehat{{\mathcal {P}}_1^{\alpha \beta }} \circ \, \frac{\delta {\mathcal {H}}_{(N,\sigma ,r)}}{\delta \widehat{t^\beta }}},\\ &{}&{}\\ &{} = &{} \displaystyle {\widehat{{\mathcal {P}}_2^{\alpha \beta }} \circ \, \frac{\delta {\mathcal {H}}_{(N-1,\sigma ,r)}}{\delta \widehat{t^\beta }}}, \end{array} \end{aligned}$$
(4.6)

where

$$\begin{aligned} {\mathcal {H}}_{(N,\sigma ,r)} = \int \omega \left( \widehat{h_{(N,\sigma ,r)}}\right) \, \mathrm{d}X. \end{aligned}$$

Proof

From (4.5),

$$\begin{aligned} \langle \delta {\mathcal {H}}_{(N,\sigma ,r)} ; \widehat{v^\beta } \rangle= & {} \int \omega \left( \frac{ \partial \widehat{h_{(N,\sigma ,r)}}}{\partial t^{(\beta j)}} v^{(\beta j)} \circ e_r \right) \, \mathrm{d}X,\\= & {} \int \omega \left( \widehat{ \frac{\partial h_{(N,\sigma ,r)}}{\partial t^\beta } } \circ e_r \circ \underbrace{v^{(\beta j)} e_j}_{\widehat{v^\beta }} \right) \, \mathrm{d}X, \end{aligned}$$

and hence

$$\begin{aligned} \frac{\delta {\mathcal {H}}_{(N,\sigma ,r)}}{\delta \widehat{t^\beta }} = \frac{ \partial \widehat{h_{(N,\sigma )}}}{\partial t^{\beta }} \circ e_r . \end{aligned}$$

With this, the result follows immediately. \(\square \)

4.1 Polynomial (inverse)-metrics and bi-Hamiltonian structures

Since all one-dimensional metrics are flat, it follows immediately from the Dubrovin-Novikov [7] Theorem that the operator

$$\begin{aligned} {{\mathcal {P}}} = f(u) \frac{\mathrm{d}~}{\mathrm{d}X} + \frac{1}{2} f^\prime (u) \end{aligned}$$

is Hamiltonian. In this section, we study the case where f is a polynomial.

Example 4.5

Applying the lifting procedures to the operator \({\mathcal {P}}\) defined by the linear function \(f(u)=u+\lambda \) results in the linear operator

$$\begin{aligned} {{\mathcal {P}}^{ij}} = \left\{ c^{ij}_k u^k \frac{\mathrm{d}~}{\mathrm{d}X} + \frac{1}{2} c^{ij}_k u^k_X \right\} + \lambda \frac{\mathrm{d}~}{\mathrm{d}X} \end{aligned}$$
(4.7)

defined on the Frobenius algebra \({\mathcal {A}}.\)

This is the Hamiltonian operator first constructed by Balinski and Novikov [2]. Similarly, more complicated examples may be obtained by starting with more general polynomials and applying the same procedure.

These more general examples appear to be in contradiction to an alternative method of constructing Hamiltonian operators via bi-Hamiltonian recursion. The recursion operator constructed from the bi-Hamiltonian pencil (4.7) takes the form

$$\begin{aligned} {{\mathcal {R}}}^i_j = c^i_{jk} u^k + \frac{1}{2} c^i_{jk} u^k_X \left( \frac{\mathrm{d}~}{\mathrm{d}X} \right) ^{-1}. \end{aligned}$$

Suppose one has a (local) Hamiltonian operator

$$\begin{aligned} {\mathcal {P}}_n = g^{ij}_{(n)}(u) \frac{\mathrm{d}~}{\mathrm{d}X} + \Gamma ^{ij}_{(n)k}(u) u^k_X \end{aligned}$$

with \(g^{ij}_{(0)} = \eta ^{ij}, \Gamma ^{ij}_{(0)k}=0.\) Applying the operator \({\mathcal {R}}\) gives

$$\begin{aligned} \left( {\mathcal {R}}{\mathcal {P}}_{(n)}\right) ^{ij} = \left\{ g^{ij}_{(n+1)}(u) \frac{\mathrm{d}~}{\mathrm{d}X} + \Gamma ^{ij}_{(n+1)k}(u) u^k_X\right\} + {\mathrm{non}\text {-}\text {local~terms}} \end{aligned}$$

and we now define \({\mathcal {P}}_{(n+1)}\) to be the local-term in the above expression. This gives the recursion scheme:

$$\begin{aligned} g^{ij}_{(n+1)}= & {} 2 c^{ip}_r u^r \eta _{pq} g^{qj}_{(n)},\\ \Gamma ^{ij}_{(n+1)k}= & {} 2 c^{ip}_r u^r \eta _{pq} \Gamma ^{qj}_{(n)k} + c^{ip}_k \eta _{pq} g^{qj}_{(n)}. \end{aligned}$$

It is a tedious, through straightforward exercise to show that if the pair \(\{g_{(n)},\Gamma _{(n)}\}\) defines a flat metric, then so does \(\{g_{(n+1)},\Gamma _{(n+1)}\}\), and hence \({\mathcal {P}}_{(n)}\) is a local Hamiltonian operator for all n. The above lifting procedure circumvents such a direct computational approach. The fact that the local (if the metric defining the local part is flat) and non-local parts of the Hamiltonian operator define separate, compatible, Hamiltonian operator is of course, well known (see, for example, [10]).

5 \({\mathcal {A}}\)-valued dispersive integrable systems

In this section, the above ideas are extended to include dispersive, higher-order, dispersive systems.

5.1 \({\mathcal {A}}\)-valued dispersive integrable systems

The main result of this section is the following theorem:

Theorem 5.1

Let \(u=\{u^\alpha (x,t)|\alpha =1,\ldots ,n \}\). Let

$$\begin{aligned} u^\alpha _t=K^\alpha (u,u_x,\ldots ) \end{aligned}$$
(5.1)

be a Hamiltonian system with the Hamiltonian H[u], then the corresponding \({\mathcal {A}}\)-valued system

$$\begin{aligned} \widehat{u^\alpha _t}=\widehat{K^\alpha (u,u_x,\ldots )} \end{aligned}$$
(5.2)

is also Hamiltonian with the Hamiltonian \({\mathcal {H}}[{\widehat{u}}]=\omega \,\left( \widehat{H[u]}\right) \).

Proof

The proof is very similar to those done in Sect. 4. Without loss of generality, we assume that the system (5.1) can be written as

$$\begin{aligned} u^\alpha _t=\{u^\alpha , H[u]\}={\mathcal {P}}^{\alpha \beta }\dfrac{\delta h}{\delta u^\beta }, \quad H[u]=\int h(u)\mathrm{d}x, \end{aligned}$$
(5.3)

where \({\mathcal {P}}^{\alpha \beta }\) is a Hamiltonian operator. So the system (5.2) reads

$$\begin{aligned} \widehat{u^\alpha _t}=\widehat{{\mathcal {P}}^{\alpha \beta }}\circ \widehat{\dfrac{\delta h}{\delta u^\beta }}. \end{aligned}$$
(5.4)

Let

$$\begin{aligned} {\mathcal {H}}[{\widehat{u}}]=\int {\mathfrak {h}}({\widehat{u}})\mathrm{d}x,\quad {\mathfrak {h}}({\widehat{u}})=\omega \,\left( \widehat{h(u)}\right) . \end{aligned}$$
(5.5)

With respect to an \({\mathcal {A}}\)-valued field, the variational derivative \(\dfrac{\delta {\mathfrak {h}}}{\delta \widehat{u^\beta }}\) is defined by the formula, essentially due to [19],

$$\begin{aligned} \omega \,\int \left( \dfrac{\delta {\mathfrak {h}}}{\delta \widehat{u^\beta }}\circ \widehat{\delta u^\beta } \right) \mathrm{d}x = \left. \frac{\mathrm{d}}{\mathrm{d}\epsilon }\right| _{\epsilon =0} {\mathcal {H}}\left[ \widehat{u^\beta } + \epsilon \widehat{\delta u^\beta } \right] . \end{aligned}$$
(5.6)

Observe that

$$\begin{aligned}&\left. \frac{\mathrm{d}}{\mathrm{d}\epsilon }{\mathcal {H}}\left[ \widehat{u^\beta } + \epsilon \widehat{\delta u^\beta } \right] \right| _{\epsilon =0}= \left. \frac{\mathrm{d}}{\mathrm{d}\epsilon }\right| _{\epsilon =0}\omega \,\left( \int h(\widehat{u^\beta }+ \epsilon \widehat{\delta u^\beta }) \mathrm{d}x\right) ,\nonumber \\&\quad =\omega \,\left( \widehat{\left. \frac{\mathrm{d}}{\mathrm{d}\epsilon }\right| _{\epsilon =0} H[{u^\beta }+ \epsilon {\delta u^\beta }]}\right) =\omega \,\left( \int \left( \widehat{\dfrac{\delta h}{\delta u^\beta }}\circ \widehat{\delta u^\beta }\right) \mathrm{d}x \right) \end{aligned}$$
(5.7)

from which follows

$$\begin{aligned} \dfrac{\delta {\mathfrak {h}}}{\delta \widehat{u^\beta }}=\widehat{\dfrac{\delta h}{\delta u^\beta }}. \end{aligned}$$
(5.8)

For two functionals

$$\begin{aligned} {\mathcal {F}}[{\widehat{u}}]=\int {\mathfrak {f}}({\widehat{u}})\mathrm{d}x,\quad {\mathcal {G}}[{\widehat{u}}]=\int {\mathfrak {g}}({\widehat{u}})\mathrm{d}x, \end{aligned}$$
(5.9)

with \({\mathfrak {f}}({\widehat{u}})=\omega \,\left( \widehat{f(u)}\right) \) and \({\mathfrak {g}}({\widehat{u}})=\omega \,\left( \widehat{g(u)}\right) \), we define a bilinear bracket as

$$\begin{aligned} \left\{ {\mathcal {F}}[{\widehat{u}}], {\mathcal {G}}[{\widehat{u}}]\right\} _{{\mathcal {A}}}=\omega \,\left( \int \dfrac{\delta {\mathfrak {g}}}{\delta \widehat{u^\alpha }}\circ \widehat{{\mathcal {P}}^{\alpha \beta }} \circ \dfrac{\delta {\mathfrak {g}}}{\delta \widehat{u^\beta }} \mathrm{d}x\right) . \end{aligned}$$
(5.10)

By using the definition of the hat map and (5.8), we rewrite the bracket (5.10) as

$$\begin{aligned} \left\{ {\mathcal {F}}[{\widehat{u}}], {\mathcal {G}}[{\widehat{u}}]\right\} _{{\mathcal {A}}}=\omega \,\widehat{\{F[u], G[u]\}}, \end{aligned}$$
(5.11)

where \(F[u]=\int f(u)\mathrm{d}x\) and \(G[u]=\int g(u)\mathrm{d}x\). Consequently, we conclude that the bracket \(\{~,~\}_{{\mathcal {A}}}\) is also a Poisson bracket. Furthermore using (5.8), the system (5.4) could be written as

$$\begin{aligned} u^{(\alpha ,i)}_t=\{u^{(\alpha ,i)}, {\mathcal {H}}[{\widehat{u}}]\}_{{\mathcal {A}}},\quad {\mathcal {H}}[{\widehat{u}}]=\int \omega \,\left( \widehat{h(u)}\right) \mathrm{d}x. \end{aligned}$$

We thus complete the proof of the theorem. \(\square \)

Corollary 5.2

The \({\mathcal {A}}\)-valued version of the Hamiltonian system \(u^\alpha _t=\{u^\alpha , H[u]\}\) is also Hamiltonian and given by

$$\begin{aligned} u^{(\alpha ,i)}_t=\{u^{(\alpha ,i)}, {\mathcal {H}}[{\widehat{u}}]\}_{{\mathcal {A}}},\quad {\mathcal {H}}[{\widehat{u}}]=\omega \,\left( \widehat{H[u]}\right) . \end{aligned}$$

These results extend naturally to the lifts of bi-Hamiltonian structures, yielding \({\mathcal {A}}\)-valued bi-Hamiltonian operators.

5.2 mKdV and (modified)-Camassa–Holm bi-Hamiltonian structures

The celebrated Miura transformation maps the second Hamiltonian operator of the KdV hierarchy to constant form. Explicitly, if

$$\begin{aligned} {\mathcal {H}}^\mathrm{KdV}_1 = -D, \qquad {\mathcal {H}}^\mathrm{KdV}_2 = -D^3 + 2 u D + u_X \end{aligned}$$

(in this section we write D in place of \(\frac{\mathrm{d}~}{\mathrm{d}X}\)). Then applying the Miura map \(u=-v_X+\frac{1}{2} v^2\) gives

$$\begin{aligned} {\mathcal {H}}^\mathrm{KdV}_2 = {\mathcal {H}}^{m\mathrm{KdV}}_1=D, \end{aligned}$$

and the second mKdV structure is then obtained by applying the same map to the third KdV Hamiltonian structure defined by bi-Hamiltonian recursion (\({\mathcal {H}}_3={\mathcal {H}}_2 {\mathcal {H}}^{-1} {\mathcal {H}}_2\)), yielding the non-local operator

$$\begin{aligned} {\mathcal {H}}_2^{m\mathrm{KdV}} = D^3 -D v D^{-1} v D. \end{aligned}$$

Just as the Balinski–Novikov structures on the Frobenius algebra \({\mathcal {A}}\) may be obtained by lifting, so \({\mathcal {A}}\)-valued non-local operators may be found by using the above results.

Proposition 5.3

The \({\mathcal {A}}\)-valued operators, defined by lifting \({\mathcal {H}}^{m\mathrm{KdV}}_1\) and \({\mathcal {H}}^{m\mathrm{KdV}}_2\) to the Frobenius algebra \({\mathcal {A}}\) are:

$$\begin{aligned} \left( {\mathcal {H}}^{m\mathrm{KdV}}_1\right) ^{ij}= & {} \eta ^{ij} D,\\ \left( {\mathcal {H}}^{m\mathrm{KdV}}_2\right) ^{ij}= & {} \eta ^{ij} D^3 - c^{ij}_p c^p_{mn} D v^m D^{-1} v^n D. \end{aligned}$$

These may also be obtained using the \({\mathcal {A}}\)-valued Miura map

$$\begin{aligned} u = -v_x + \frac{1}{2} v \circ v. \end{aligned}$$

Proof

These results follow directly by applying the results in Sect. 5.1. They may also be obtained by direct (but tedious) calculation. The form of the \({\mathcal {A}}\)-valued Miura map is obvious and again can be verified by direct calculations. While not developed here, one should be able to applying lifting results directly to scalar-Miura maps, with all the actions commuting. \(\square \)

\({\mathcal {A}}\)-valued KdV and mKdV equations can now easily be constructed, the KdV examples coinciding with the examples constructed in [21]. Here we construct \({\mathcal {A}}\)-valued modified Camassa–Holm equations.

Example 5.4

One may apply the standard tri-Hamiltonian ‘tricks’[9] to obtain the \({\mathcal {A}}\)-valued bi-Hamiltonian pair:

$$\begin{aligned} {\mathcal {C}}^{ij}_1= & {} \eta ^{ij} (D^3+D),\\ {\mathcal {C}}^{ij}_2= & {} c^{ij}_p c^p_{mn} D v^m D^{-1} v^n D. \end{aligned}$$

Starting with the lifted Casimir of the scalar operator \({\mathcal {C}}_1\), one obtains the multi-component modified Camassa–Holm equation

$$\begin{aligned} v_T + v_{XXT}= & {} \phantom {+} \frac{1}{2} v_{XXX} \circ v_X \circ v_X + v_{XX} \circ v_{XX} \circ v_x\\&+ \frac{1}{2} v_{XXX} \circ v \circ v + 2 v_{XX} \circ v_X \circ v + \frac{1}{2} v_X \circ v_X \circ v_X\\&+ \frac{3}{2} v_X \circ v \circ v. \end{aligned}$$

Note we use the adjective ‘modified’ in the original, strict, sense of equations obtained from an original, unmodified, equation via the action of a Miura map, rather than in the looser sense of just modifying ‘by-hand’ the terms that appear in the equation. Two-component examples may easily be found using one of the algebras constructed in Example 2.2.

6 Conclusions

Central to the results of this paper is the use of a distinguished coordinate system, namely the flat coordinates of the Frobenius manifold \({\mathcal {M}}.\) But the lifting procedure may be applied to any geometric structure which is analytic in some fixed coordinate system. However, such results loose some of their coordinate free character: one is using a specific coordinate system to define new objects then relying in their tensorial properties to define then properly in an arbitrary system of coordinates. As an example of this, one can apply the idea to F-manifolds defined by Hertling and Manin [12].

Proposition 6.1

Consider an F-manifold with structure functions \(c_{\alpha \beta }^{\phantom {\alpha \beta }\gamma }(t)\) analytic in the coordinates \(\{ t^\alpha \}.\) Let \({\mathcal {A}}\) be an arbitrary Frobenius algebra. Then the structure functions defined by the lifted multiplication (2.9)

$$\begin{aligned} c_{(\alpha i)(\beta j)}^{\phantom {{(\alpha i)(\beta j)}}(\gamma k)} = \left[ \widehat{c_{\alpha \beta }^{~~\gamma }} \right] ^p \, c_{ij}^{~~q} c_{pq}^{~~k} \end{aligned}$$

define an F-manifold.

The proof is straightforward and will be omitted. The link between F-manifolds and equations of hydrodynamic type has been explored by a number of authors [18, 22], so one should be able to apply the idea of this paper to construct their \({\mathcal {A}}\)-valued counterparts.

In quantum cohomology, the tensor product of Frobenius manifolds generalizes the classical Künneth product formula. In singularity theory it corresponds to the direct sum of singularities. If one of the manifolds is trivial, then this descriptions degenerates—there is no parameter space of versal deformations. However, one could try to construct an \({\mathcal {A}}\)-valued singularity theory. This is purely speculative, but Arnold has constructed a theory of versal deformations of matrices [1] but it remains to see if this is what would be required.

As remarked earlier, since \({\mathcal {M}}_{\mathcal {A}}\) is a Frobenius manifold in its own right, one can apply the deformation theory developed by Dubrovin and Zhang [8] directly to the hydrodynamic flows given in Theorem 4.2. But central to this approach is the existence of a single \(\tau \)-function. However, the deformations/dispersive systems constructed in Sects. 4 and 5 have \({\mathcal {A}}\)-valued \(\tau \)-functions. Thus, we have two distinct deformation procedures, unless they are connected by some set of transformations. It may be possible to construct a deformation theory along the lines of [8] but with an \({\mathcal {A}}\)-valued \(\tau \)-function.

This paper has concentrated on Frobenius algebra-valued integrable systems, via their Hamiltonian structure. Other approaches to integrability—the structure of \({\mathcal {A}}\)-valued Lax equations, for example, have not been considered here. Part of such a theory have been constructed by the authors in [23] where an \({\mathcal {A}}\)-valued KP hierarchy is constructed via such \({\mathcal {A}}\)-valued Lax equations and operators. In a different direction, there are many other algebra-valued generalizations of KdV equation, from Jordan algebra to Novikov algebra-valued systems [20, 21, 24, 25]. Whether such algebra-valued systems can be combined with the theory of Frobenius manifolds remains an open question. Developing a theory which encompasses the non-commutative/non-local hierarchies, such as the original matrix KdV equation (1.1), would be of considerable interest and would encompass the theory developed in this paper.