The Inversion Symmetry of the WDVV Equations and Tau Functions
Abstract
For two solutions of the WDVV equations that are related by the inversion symmetry, we show that the associated principal hierarchies of integrable systems are related by a reciprocal transformation, and the tau functions of the hierarchies are related by a Legendre type transformation. We also consider relationships between the Virasoro constraints and the topological deformations of the principal hierarchies.
1 Introduction
The WittenDijkgraafVerlindeVerlinde (WDVV) equations, which arise in the study of 2D topological field theory (TFT) in the beginning of 90’s of the last century, are given by the following system of PDEs for an analytic function :

The variable is specified so that
(1.1) 
The functions with
(1.2) yield the structure constants of an associative algebra for any fixed , i.e, they satisfy
(1.3) Here and in what follows summation with respect to repeated upper and lower indices is assumed.
In [6, 7] Dubrovin formulated the WDVV equations with an additional quasihomogeneity condition on , we will recall this condition in Sec. 4 and call a solution of (1.1)–(1.3) satisfying the quasihomogeneous condition a conformal solution of the WDVV equations. These equations together with the quasihomogeneity condition are satisfied by the primary free energy of the matter sector of a 2D TFT with primary fields as a function of the coupling constants [3, 4, 30]. In [6, 7] Dubrovin reformulated these equations in a coordinate free form by introducing the notion of Frobenius manifold structure on the space of the parameters and revealed significantly rich geometric structures of the WDVV equations, which have become important in the study of several different areas of mathematical research, including the theory of Gromov  Witten invariants, singularity theory and nonlinear integrable systems, see [7, 11, 12, 14] and references therein. In particular, such geometrical structures enable one to associate a solution of the WDVV equations with a hierarchy of bihamiltonian integrable systems of hydrodynamic type which is called the principal hierarchy in [14]. This hierarchy of integrable systems plays important role in the procedure of reconstructing a 2D TFT from its primary free energy as a solution of the WDVV equations. In this construction, the tau function that corresponds to a particular solution of the principal hierarchy serves as the genus zero partition function, and the full genera partition function of the 2D TFT is a particular tau function of an integrable hierarchy of evolutionary PDEs of KdV type which is certain deformation of the principal hierarchy, such a deformation of the principal hierarchy is call the topological deformation [14].
In this paper we are to interpret certain discrete symmetry of the WDVV equations in term of the associated principal hierarchy and its tau function. The discrete symmetry we consider here was given by Dubrovin in Appendix B of [7] and is called the inversion symmetry. This symmetry is obtained from a special Schlesinger transformation of the system of linear ODEs with rational coefficients associated to the deformed flat connection of the Frobenius manifolds (see Remark 4.2 of [8] for details). It turns out that in terms of the principal hierarchy and its tau function the inversion symmetry has a simple interpretation. On the principal hierarchy it acts as certain reciprocal transformation, and on the associated tau function it acts as a Legendre type transformation. In Appendix B of [7] there is given another class of discrete symmetries of the WDVV equations which are called the Legendretype transformations, the relation of such symmetries with the principal hierarchy and its tau function is given in [14]. Besides these discrete symmetries, the WDVV equations also possess continuous symmetries whose Lie algebra of infinitesimal generators (without the quasihomogeneity condition) was studied in [2, 21, 16, 19, 20].
Recall [7] that a symmetry of the WDVV equations is given by a transformation
(1.4) 
that preserves the WDVV equations. The inversion symmetry given in [7] has the following form:
(1.5) 
Here we assume that and the coordinates are normalized such that the constants take the values
(1.6) 
This can always be achieved by performing an invertible linear transformation
where . We call the solution of the WDVV equations (1.1)–(1.3) the inversion of the solution .
We arrange the content of the paper as follows. We first recall in Sec. 2 the definition of the principal hierarchy and its tau functions associated to a calibrated solution of the WDVV equations. We then show in Sec. 3 that the action of the inversion symmetry of the WDVV equations on principal hierarchies is given by certain reciprocal transformation, and we give the transformation rule of the associated tau functions, see Propositions 3.2 and 3.3. In Sec. 4 we impose the conformal condition on the solution of the WDVV equations and consider the transformation rule of the inversion symmetry on principal hierarchies and their bihamiltonian structures. These results will be used when we study the topological deformations of principal hierarchies. In Sec. 5 we consider transformation rule of the Virasoro constraints for tau functions of a principal hierarchy. In Sec. 6 we consider the action of the inversion symmetry on the topological deformations of the principal hierarchies and their tau functions.
2 Calibrations, Principal Hierarchies, and
Tau Functions
The notion of calibrations of a solution of the WDVV equations (or a Frobenius manifold) corresponds to the choice of a system of deformed flat coordinates on a Frobenius manifold [7], it was first introduced in [18] and then modified in [2]. In what follows we use the modified one.
Definition 2.1
Let be a solution of the WDVV equations, a family of functions
is called a calibration of , if their generating functions
satisfy the equations
(2.1) 
and the normalization conditions
(2.2)  
(2.3)  
(2.4)  
(2.5) 
The solution together with a calibration is called a calibrated solution of the WDVV equations.
Let be a calibrated solution of the WDVV equations, we introduce a hierarchy of evolutionary PDEs of hydrodynamic type:
(2.6) 
It is easy to see that
so in what follows we identify with . By using the WDVV equations one can prove the following results:

Each flow possesses a Hamiltonian formulism with the Hamiltonian operator and Hamiltonian ;

, where is the Poisson bracket defined by
(2.7) 
Denote by , then
The second property ii) also implies
which are easy corollaries of the properties of Poisson brackets.
Definition 2.2
The hierarchy (2.6) of integrable evolutionary PDEs is called the principal hierarchy associated to the calibration .
Remark 2.3
Now we are to define tau functions of the principal hierarchy as it is done in [7, 14]. First we recall the definition [7] of the family of functions
by the following generating functions
(2.8) 
Then one can prove that
(2.9) 
which imply that if is a solution of the principal hierarchy associated to certain calibration, then there exists a function such that
In particular, we have .
Definition 2.4
Let be a calibrated solution of the WDVV equations, be the associated principal hierarchy. A function is called a tau function of the principal hierarchy if
(2.10) 
where .
Note that if is a tau function of the principal hierarchy, then is a solution of the principal hierarchy. Indeed, by using the property of the following property of the functions [7]
(2.11) 
we have
On the other hand, the above argument shows that a solution of the principal hierarchy also defines a tau function.
3 The Actions of the Inversion Symmetry
In this section, we study the actions of the inversion symmetry on calibrations, principal hierarchies, and tau functions of a solution of the WDVV equations. We fix a pair of solutions of the WDVV equations that are related by the transformation (1.5).
Proposition 3.1
Let be a calibration of , then the following functions
(3.1)  
give a calibration of .
Proof It was shown in [7] that
One can also prove that
By using these identities, the proposition is proved straightforwardly.
Proposition 3.2
Let be the principal hierarchy associated to a calibration of . Introduce the following reciprocal transformation
(3.2)  
(3.3) 
and denote , then we have
(3.4) 
i.e. is the principal hierarchy of with calibration .
Proof From the definition of the reciprocal transformation we have
(3.5) 
The proposition can be proved by direct calculation.
Proposition 3.3
Let be a tau function of the principal hierarchy associated to a calibration of . Define
(3.6) 
then is a tau function of the principal hierarchy associated to .
Proof By definition the functions are given by (2.8) in terms of the functions . From the relation (1.5), (3.1) it follows that
(3.7) 
where , and we assume that , when . Then one can verify, by using (1.5), (3.1) and (2.10), that
where . The proposition is proved.
Let be a tau function of a principal hierarchy, then the reciprocal transformation (3.2) can be written as
It follows that up to the addition of a constant we have
(3.8) 
The constant can be absorbed by a translation of in the definition of the reciprocal transformation, so we will assume from now on the validity of (3.8). Thus in terms of a given tau function, the reciprocal transformation (3.2), (3.3) can be represented by (3.3), (3.6) and (3.8).
4 Conformal Case
In this section we are to include the quasihomogeneity condition into the WDVV equations as it is formulated in [7].
Definition 4.1
A solution of the WDVV equations is called conformal if there exists a vector field , called the Euler vector field, of the form
and some constants such that
It is often assumed that the matrix is diagnolizable and . The coordinates are normalized so that
and if . In this paper, we assume that
This assumption ensures that the solution of the WDVV equations obtained from by the action of the inversion symmetry also has a diagnolizable Euler vector field, while this is not always true without the above assumption, see [7] and Lemma 4.2. Then the Euler vector field can be written in the following form:
where the constants and are called the charge and the spectum of respectively [7].
Note that the WDVV equations only involve the third order derivatives of , so we can add certain quadratic functions of to such that the constants satisfy the following normalizing conditions
Further more, our assumption on and implies that
The following lemma is proved in [7].
Lemma 4.2 ([7])
Let be a conformal solution of the WDVV equations with charge and spectrum , and be its inversion, then is also conformal, whose charge and spectrum read
(4.1) 
Definition 4.3
Let be a conformal solution of the WDVV equations with spectrum , a calibration is called conformal if there exist constant matrices such that
(4.2) 
and
(4.3)  
(4.4) 
The property (4.3) implies that there is only finitely many nonzero matrices . These matrices and the metric , the spectrum form a representative of the monodromy data of at , and
form a system of flat coordinates for the deformed flat connection of the Frobenius manifold associated to the conformal solution of the WDVV equations. Here and . See [7, 8] for details.
Proposition 4.4
Let be a calibrated conformal solution of the WDVV equations, then the calibration of is also conformal.
Proof We only need to compute , then the matrices for can be obtained
The proposition is proved.
The principal hierarchy associated to a conformal calibtation has a very important additional structure – the bihamiltonian structure. We already know from Sec. 2 that the principal hierarchy has one Hamiltonian structure . When the calibrated solution is conformal, we have the following results.
Lemma 4.5 ([7])
Define a matrix differential operator , where
(4.5)  
(4.6) 
then is a Hamiltonian operator which is compatible with . Furthermore, for any conformal calibration of , we have
(4.7) 
where is the Poisson bracket defined by , see (2.7).
It has been shown in Proposition 3.2 that the inversion symmetry preserves the first Hamiltonian structure , then it is natural to ask: does it also preserve the second Hamiltonian structure ?
Proposition 4.6
We note that the action of reciprocal transformations of the form (3.2) and (3.3) on evolutionary PDEs of hydrodynamic type and their Hamiltonian structures of the form (4.5) was first investigated by Ferapontov and Pavlov in [17]. After the action of a reciprocal transformation a Hamiltonian operator of the form (4.5) becomes nonlocal in general, the nonlocal Hamiltonian operator is given by a differential operator of the form (4.5) plus an integral operator, in this case the metric is no longer flat. In [1] Abenda considered the conditions under which such a reciprocal transformation preserves the locality of a Hamiltonian structures of hydrodynamic type. In [23] we studied a general class of nonlocal Hamiltonian structures in terms of infinite dimensional Jacobi structures and gave the transformation rule of such Hamiltonian structures under certain reciprocal transformations, a criterion on whether a reciprocal transformation preserves the locality of a Hamiltonian structure was also given in [23].
Proof of Proposition 4.6 According to the general results of [23] (c.f. [17]), are Jacobi structures of hydrodynamic type. To prove the proposition, one need to show that they are both local, and their metrics coincide with the ones of .
The locality problem is well studied in Sec. 3.2 of [23]. We now give the proof of the locality of by using the criterion given in [23]. The proof of locality for is easier and we omit it here.
Let us denote . We first need to show that
(4.8) 
which is equivalent to say that there exists a constant such that
where is the LeviCivita connection of the metric (see (4.6)). By a straightforward calculation one can obtain that , thus (4.8) holds true.
5 Virasoro Constraints of the Tau Functions
There is an important class of solutions of the principal hierarchy (2.6) which can be obtained by solving the following system of equations [7, 14]:
(5.1) 
where and are some constants which are assumed to be zero except for finitely many of them. These constants are required to satisfy the genericity conditions that there exist constants such that
and the matrix
is invertible. Here and . One can obtain in this way a dense subset of the set of analytic monotone solutions of the principal hierarchy (2.6), see Sec. 3.6.4 of [14] for details. The tau function for the solution satisfying (5.1) can be chosen to be
(5.2) 
The validity of the defining relation (2.10) follows from (5.1) and the identity (2.11) for the functions .
Proposition 5.1
Proof To prove the validity of (5.3) let us consider the case when , the proof for other cases is similar. By using (1.5), (3.1) and (3.3) we have
Here we used the relation (3.8) and the fact that
(5.5) 
which follows from (2.11), (5.1) and (5.2). The validity of the relation (3.6) follows from (3.3), (3.7), (5.2) and (5.5). The proposition is proved.
In the case when the solution of the WDVV equations is conformal, the tau function (5.2) satisfies the Virasoro constraints [13, 15, 24]
(5.6) 
where , and the coefficients that appear in the above expressions are some constants determined by the monodromy data of the Frobenius manifold of , they define a set of linear differential operators
(5.7) 
which give a representation of the half branch of the Virasoro algebra