International Conference on

Secondary calculus and cohomological Physics, Moscow, August 1997

WDVV EQUATIONS AND THEIR SOLUTIONS FOR A (3, 2) AND (3, 4) SUPERMANIFOLD; THE ISOTROPIC AND NONISOTROPIC CASE

R. Martini and P.H.M. Kersten

University of Twente, Faculty of Applied Mathematics, P.O. Box 217, 7500 AE Enschede, The Netherlands E-mail: p.h.m.kersten@math.utwente.nl

Abstract. The WDVV equations for the isotropic and nonisotropic case on a supermanifold of dimension (3, 2) and (3, 4) will be derived. By the method of characteristics, the general solution is constructed after transforming the system of WDVV equations to appropriate coordinates.

Introduction

Recently a remarkable system of differential equations appeared in the physics of twodimensional topological field theory. Elaborating axioms describing a topological field theory led in a natural way to these very interesting but complicated and overdetermined system of equations [5].

Mathematically the system describes the conditions for a function Φ = Φ(x) of the variable x = (x^{1}, …, x^{n}) such that the third order derivatives

A_{abc}(x) = \frac{∂^{3}Φ(x)}{∂x^{a}∂x^{b}∂x^{c}}

for any x are the structure constants of an associative algebra. Appropriately this system is known in two-dimensional topological field theory as the equations of associativity or the Witten-Dijkgraaf-E.Verlinde-H.Verlinde (WDVV) system since it is first found in the papers [1,4].

In the physical setting solutions of the WDVV system describe moduli spaces of topological conformal field theories, whereas in the mathematical context the system applies to enumerative problems in algebraic geometry. For instance solutions of the system may be interpreted as generating functions of combinatorial identities counting numerical characteristics of spaces of algebraic curves in some complex manifold or algebraic variety.

In these applications solutions of the WDVV system determine an associative algebra structure on some cohomology space. Generally (and also for convenience) one is dealing only with even-dimensional cohomology spaces. So the resulting algebra structure turns out to be commutative.

Key words and phrases: Topological Field Theory, Equations of associativity, WDVV-euqations

However following Kontsevich-Manin [3] one may consider also odd-dimensional cohomology spaces. This automatically leads to ℤ^{2}-graded (super) spaces, equations and geometry.

For a comprehensive introduction to the WDVV-system we refer the very well-written survey of Dubrovin [2], where all necessary physical motivations and mathematical details can be found. In this paper [2] only the commutative non-graded case is considered and the WDVV-system is classified according to two essential different types. The first and second type correspond to the case where the identity of the algebra is respectively isotropic, or nonisotrpic.

In this paper we will study and solve the WDVV-system in three even and two odd variables (3, 2) and three even and four odd variables (3, 4). Both the isotropic and nonisotropic case are investigated.

1 General

In this section we shall recapitulate a number of notions which lead in the classical case [2] to the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations and for the super or graded situation to the super WDVV-equations [3].

Starting point in the considerations is a supermanifold M which is endowed with a tensor g of rank 2 and a tensor A of rank 3. Local coordinates are given by {x^{a}}, the basis of vector fields by ∂_{a} = ∂_{x^{a}}, and the basis of 1-forms by dx^{a}. Tensor components are g_{ab} and A_{ab}^{c} respectively. The degree of x^{a} i.e., |x^{a}| which will only appear in expressions of type p.e. (−1)^{|x^{a}| |x^{b}|} will be written as a i.e. (−1)^{|x^{a}||x^{b}|} = (−1)^{ab}. The tensor g, A are taken to be even i.e.

|g_{ab}| = (−1)^{a+b} |A_{ab}^{c}| = (−1)^{a+b+c}.  (1)

We further impose a number of requirements.

1. The existence of a \underline{pairing}

< ∂_{a}, ∂_{b} >= g_{ab}

which is symmetric, i.e.,

g_{ab} = (−1)^{ab}g_{ba}.  (2)

Furthermore g_{ab} is \underline{constant}, and \underline{invertible} (g^{ab}) = (g_{ab})^{−1}.

2. The existence of the structure of an \underline{C}_{M}-\underline{algebra}

∂_{a} ∘ ∂_{b} = ∑_{c} A_{ab}^{c}∂_{c}.  (3)

(Super)\underline{commutativity} results in

A_{ba}^{c} = (−1)^{ab}A_{ab}^{c}.  (4)

In order to have \underline{associativity} i.e.,

∂_{a} ∘ (∂_{b} ∘ ∂_{c}) = (∂_{a} ∘ ∂_{b}) ∘ ∂_{c}  (5)

we require

∑_{e} A_{ab}^{e}A_{ec}^{d} = (−1)^{a(b+c)} ∑_{e} A_{bc}^{e}A_{ea}^{d}.  (6)

Moreover we demand

∂_{d}A_{ab}^{c} = (−1)^{ad}∂_{a}A_{db}^{c}.  (7)

3. We now introduce

A_{abc} = ∑_{e} A_{ab}^{e}g_{ec}  (8)

and impose

A_{abc} = (−1)^{a(b+c)}A_{bca}.  (9)

Condition (9) reflects that

< ∂_{a} ∘ ∂_{b}, ∂_{c} >=< ∂_{a}, ∂_{b} ∘ ∂_{c} >  (10)

4. Existence of coordinate vector field ∂_{0}, is required, which is \underline{even} and

A_{ob}^{c} = δ_{bc}  (11)

or equivalently

A_{obc} = g_{bc}.  (12)

An \underline{even} function Φ defined on M is called a \underline{potential} for the (A, g)-structure if

A_{abc} = ∂_{a}∂_{b}∂_{c}Φ.  (13)

Conditions (2), (4), (7), (9) are immediate from (13). Crucial in this formulation is (6) which results in the remarkable WDVV equations

∑_{e,f} ∂_{a}∂_{b}∂_{e}Φg^{ef}∂_{f}∂_{c}∂_{d}Φ = (−1)^{a(b+c)} ∑_{e,f} ∂_{b}∂_{c}∂_{e}Φg^{ef}∂_{f}∂_{a}∂_{d}Φ.  (14)

It should be noted that (2), (3), (4), (6), (7), (9), (12), (14) hold for all appropriate values of indices a, . . ..

In section 2 and 3 we shall discuss the (A, g)-structures and the potential function Φ for the (3, 2)-case i.e. the super manifold M has local coordinates {x_{0}, x_{1}, x_{2}, ξ_{1}, ξ_{2}} where x_{0}, x_{1}, x_{2} are even and ξ_{1}, ξ_{2} odd, whereas in sections 4 and 5 we consider the (3,4) case.

We shall make particular choices for the (constant) g matrix, related to the isotropic and nonisotropic case.

In the classical case, no odd variables and dimension 3, the WDVV equations read

−f_{xyy}f_{xxx} + f_{xxy}^{2} − f_{yyy} = 0

for the \underline{isotropic} and

−f_{xyy}f_{xxy} + f_{xxx}f_{yyy} − 1 = 0

for the \underline{nonisotropic} case.

2 The isotropic case: n = 3, m = 2

In this section we derive the graded or super WDVV equations in the case of 3 even variables x_{0}, x_{1}, x_{2} and two odd variabales ξ_{1}, ξ_{2}; |x_{i}| = 0, |ξ_{i}| = 1, where |x| denotes the degree of x. We shall restrict here to the isotropic case. In the classical situation this results in the canonical form of the (3 × 3) matrix g

g = [ \begin{matrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} ]  (1)

In the graded case, due to the requirements (1), (2) that g_{ab} is constant and g being (graded) symmetric

g_{ab} = (−1)^{ab}g_{ba}

where a refers to the element x^{a}, ξ^{a}, we are led to a (5 × 5) matrix g

g = [ \begin{matrix}0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & α \\ 0 & 0 & 0 & −α & 0 \end{matrix} ] .  (2)

Starting from an even function Φ of variables x_{0}, x_{1}, x_{2}, ξ_{1}, ξ_{2} we have

Φ = F_{1}(x_{0}, x_{1}, x_{2}) + F_{2}(x_{0}, x_{1}, x_{2})ξ_{1}ξ_{2}.  (3)

From the definition of g by

g_{ab} = ∂_{0}∂_{a}∂_{b}Φ  (4)

we derive

Φ = G(x_{1}, x_{2}) + F (x_{1}, x_{2})ξ_{1}ξ_{2} + \frac{1}{2}x_{0}^{2}x_{2} + \frac{1}{2}x_{0}x_{1}^{2} − αx_{0}ξ_{1}ξ_{2}
+c_{1}x_{0}^{2} + c_{2}x_{0}x_{1} + c_{3}x_{0}x_{2} + c_{4}x_{0}.  (5)

We now impose the associativity conditions (4) on (5) and we arrive at a system of partial differential equations for F, G i.e. (lower index i referring to differentiation with respect to x^{i}(i = 1, 2))

− G_{122}G_{111} + G_{112}^{2} − G_{222} = 0
−F_{1}^{2} + 2αF_{2} = 0
−αF_{1}G_{122} + α^{2}G_{222} − F_{2}^{2} = 0  (6)
−αF_{1}G_{112} + α^{2}G_{122} − F_{1}F_{2} = 0
−αF_{1}G_{111} + α^{2}G_{112} − F_{1}^{2} − αF_{2} = 0

Remark: The first equation (6) is a consequence of the other four equations.

We first solve the second equation in (6) by the method of characteristics and then substitute the general solution into the other ones to obtain G after transformation of the system to apropriate coordinates. In terms of x = x_{1}, y = x_{2}, u = F the equation reads

u_{y} = \frac{u_{x}^{2}}{2α}.  (7)

The contact ideal I associated to (7) is generated by the contact form ω and its exterior derivative i.e.

ω = du − u_{x}dx − \frac{u_{x}^{2}}{2α}dy

(8)

dω = −du_{x}dx − \frac{u_{x}}{α} du_{x}dy.

Now, a vector field X

X = X^{x}∂_{x} + X^{y}∂_{y} + X^{u}∂_{u} + X^{u_{x}}∂_{u_{x}}  (9)

is a Cauchy characteristics of I, generated by (8), in case

X⌟I ⊂ I  (10)

where ⌟ denotes contraction, from which we derive

X^{u} − u_{x}X^{x} − \frac{u_{x}^{2}}{2α}X^{y} = 0

and

−X^{u_{x}}(dx + \frac{u_{x}}{α} dy) + du_{x}(X^{x} + \frac{u_{x}}{α} X^{y}) = 0.

(10) leads to the Cauchy characteristic

X = −\frac{u_{x}}{α} ∂_{x} + ∂_{y} − \frac{u_{x}^{2}}{2α}∂_{u} + 0∂_{u_{x}}.  (11)

Starting from the initial value u(x, o) = f(x) where f is an arbitrary function of x we obtain the parametric solution

x_{1} = x = −\frac{f′(s)}{α}t + s

x_{2} = y = t  (12)

F = u = −\frac{f′(s)^{2}}{2α} t + f(s)

F_{1} = u_{x} = f′(s)

Equations (12) define a local 1 − 1 transformation (x_{1}, x_{2}) → (t, s) We now substitute (12) into (6) and obtain the following remarkable result i.e.,

G(x_{1}, x_{2}) ⇒ G(t, s)

(14)

G_{ttt} = \frac{3}{4} \frac{f′(s)^{4}}{α^{4}}

So G(t, s) is a polynomial of degree 3 in t. Finally, after a short computer calculation we have the final result

G = \frac{1}{8} \frac{f′(s)^{4}}{α^{4}} t^{3} + g_{3}(s)t^{2} + g_{2}(s)t + g_{1}(s)  (15)

where g_{1}(s) is an arbitrary function i.e., the initial condition of G at x_{2} = 0, g_{1}(s) = G(x_{1}, 0). The functions g_{2}(s), g_{3}(s) satisfy the following equations

g′′_{2}(s) = −2g′′_{1} · \frac{f′′}{α} − g′_{1} \frac{f′′′}{α} + \frac{3}{2} \frac{(f′)^{2}}{α^{2}}
g′_{3}(s) = −g′_{2} \frac{f′′}{α} − g′_{1} \frac{(f′′)^{2}}{α^{2}} − \frac{(f′)^{3}}{2α^{3}}  (16)

Equations (16) can be integrated straightforwardly, leading to expressions for g_{3}(s) and g_{2}(s).

The even function Φ (3) can be obtained from (5) (12) and (15), (16). The general solution can be written as an expression depending on the intitial data of F, G at (x_{1}, 0).

Example 2.1 Take f(s) = \frac{α}{2}s^{2} in (12).

Then

\begin{matrix}x = −st + s & or & t = y \\ y = t & & s = \frac{x}{1−y} \end{matrix}

from which we have F = \frac{α}{2} \frac{x^{2}}{1−y} and G(x, y) is given by (2.14), (2.15).

If we choose

g_{1}(s) = \frac{c}{8}s^{4}

where c is some constant, then we arrive after a short computation at

G(x, y) = \frac{1}{8} \frac{x^{4}(y + c)}{(1 − y)^{2}}.

In the last case Φ, F, G become graded with respect to the following

deg x_{0} = 1, deg x_{1} = \frac{1}{2}, deg x_{2} = 0, deg(ξ_{1}) + deg(ξ_{2}) = 1

and we see that deg Φ = deg G = 2, deg F = 1.

3 The nonisotropic case: n = 3, m = 2

We shall now discuss the WDVV-equations (2) and their solutions for the nonisotropic situation. Here the matrix g_{ab}(a, b = 0, …, 4)

g_{ab} = ∂_{0}∂_{a}∂_{b}Φ  (1)

can be chosen in the following form

g = [ \begin{matrix}c_{1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & c_{2} \\ 0 & 0 & 0 & −c_{2} & 0 \end{matrix} ]  (2)

From (1) we already have, starting from

Φ = ϕ_{1}(x_{0}, x_{1}x_{2}) + ϕ_{2}(x_{0}, x_{1}, x_{2})ξ_{1}ξ_{2}  (3)

Φ = \frac{c_{1}}{6} x_{0}^{3} + x_{0}x_{1}x_{2} − c_{2}x_{0}ξ_{1}ξ_{2} + k_{1}x_{0}x_{2} + k_{2}x_{0} + k_{3}x_{0}x_{1} + k_{4}x_{0}^{2}  (4)
+F (x_{1}, x_{2})ξ_{1}ξ_{2} + G(x_{1}, x_{2}).

Substitution of (4) into the associativity conditions (14) leads to the following system of nonlinear partial differential equations for F and G, i.e.

− c_{1}G_{122}G_{112} + c_{1}G_{111}G_{222} − 1 = 0

c_{2}^{2} + 2c_{1}F_{1}F_{2} = 0
c_{2}F_{2}G_{122} + c_{2}F_{1}G_{222} + (F_{2})^{2} = 0  (5)

c_{2}F_{2}G_{122} + c_{2}F_{1}G_{222} + F_{1}F_{2} − \frac{c_{2}^{2}}{c_{1}} = 0

c_{2}F_{2}G_{111} + c_{2}F_{1}G_{112} + F_{1}^{2} = 0

It should be noted that the classical WDVV equations in the nonisotropic case is a consequence of the other four partial differential equations in (5). In a way, similar as in the previous case we construct the general solution of (5) and determine Φ (3).

Start from the second equation of (5) (F = u) i.e.

u_{1}u_{2} = \frac{−c_{2}^{2}}{2c_{1}} = α.

Contact ideal I is generated by

ω = du − u_{1}dx_{1} − u_{2}dx_{2} = du − u_{1}dx_{1} − \frac{α}{u_{1}} dx_{2}
dω  = du_{1}dx_{1} + α(u_{1})^{−2}du_{1}dx_{2}.  (6)

A Cauchy characteristic X

X = X^{1} \frac{∂_{x}}{∂x_{1}} + X^{2} \frac{∂}{∂x_{2}} + X^{u} \frac{∂}{∂u} + X^{u_{1}} \frac{∂}{∂u_{1}}  (7)

has to satisfy

X_{⌟}ω ∈ I and X_{⌟}dω ∈ I

i.e.

X^{u} − X^{1}u_{1} − X^{2}αu_{1}^{−1} = 0

and

−X^{u_{1}}dx_{1} + du_{1}X^{1} + α(u_{1})^{−2}X^{u_{1}}dx_{2} − α(u_{1})^{−2}du_{1}X^{2} ∈ I

from which we derive

X^{u_{1}} = 0 , X^{1} = α(u_{1})^{−2}X^{2}, X^{u} = u_{1}X_{1} + αu_{1}^{−1}X^{2}.

From the above computation we derive the Cauchy characteristic

X = α(u_{1})^{−2} \frac{∂}{∂x_{1}} + \frac{∂}{∂x_{2}} + 2α(u_{1})^{−1} \frac{∂}{∂u} + 0\frac{∂}{∂u_{1}} .  (8)

Integration of this vectorfield leads to

\overline{x}_{1} = α((u_{1})_{0})^{−2}t + (x_{1})_{0}
\overline{x}_{2} = t (9)
\overline{u} = 2α((u_{1})_{0})^{−1}t + u_{0}
\overline{u}_{1} = (u_{1})_{0} (10)

We now choose initial condition at y = 0 i.e.

u(s, 0) = f(s).  (11)

Combination of (9), (11) then gives the general local parametric solution

x_{1} = α(f′(s))^{−2}t + s
x_{2} = t (12)
u = 2α(f′(s))^{−1}t + f(s)
u_{1}  = f′(s).

Remarkably (12) induces a transformation (x_{1}, x_{2}) ↔ (t, s) which is of course locally one to one, but very interesting for solving the other equations of (5)

(x_{1})_{t} = α(f′)^{−2}, (x_{1})_{s} = 1 − 2αt(f′)^{−3}f′′, (x_{2})_{t} = 1, (x_{2})_{s} = 0
and  (13)
(t)_{x_{1}}  = 0, (t)_{x_{2}} = 1
(s)_{x_{1}} = [1 − 2αt(f′)^{−3}f′′], (s)_{x_{2}} = −α(f′)^{−2}[1 − 2αt(f′)^{−3}f′′]^{−1}

Substitution of (12), (13) into the other equations of (5) yields after some calculations G ≡ G(t, s)

G_{ttt} = \frac{−2c_{2}^{3}}{c_{1}^{2}} (f′)^{−3}

from which we have that G is a polynomial of degree 3 in t. From this we obtain the final result

G = \frac{−c_{2}^{3}}{3c_{1}^{2}} (f′)^{−3}t^{3} + g_{3}(s)t^{2} + g_{2}(s)t + g_{1}(s)  (14)

where g_{1}(s) is an arbitrary function of s and g_{2}, g_{3} are obtained from the following equations by quadrature

(g_{3})′  = \frac{−c_{2}^{2}}{c_{1}^{2}} g′_{1}(f′′)^{2}(f′)^{−6} + \frac{c_{2}^{2}}{c_{1}} g′_{2}f′′(f′)^{−3} + \frac{c_{2}}{c_{1}} (f′)^{−1}
(g_{2})′′ = 2\frac{c_{2}^{2}}{c_{1}} g′′_{1}f′′(f′)^{−3} + \frac{c_{2}^{2}}{c_{1}} g′_{1}f′′′(f′)^{−3} − 3\frac{c_{2}^{2}}{c_{1}} g′_{1}(f′′)^{2}(f′)^{−4} − \frac{1}{c_{2}} f′

Summarizing the computations so far, we obtain the general solutions of (5) and by this function F̃ by the transformation in (x_{1}, x_{2})

\begin{aligned}x_{1} & = & −\frac{c_{2}^{2}}{c_{1}} (f′)^{−2} · t + s \\ x_{2} & = & t^{2} \end{aligned}

where F ≡ F (t, s), G ≡ G(t, s) are given by (13), (15)

while f(s) is the initial condition f(s) = F (s, 0)|(x = s, y = 0) and g_{1}(s) is an arbitrary function of course g_{1}(s) = G(s, 0)|(x = s, y = 0).

Example 3.1 We choose

f(s) = c_{2}s^{1/2}

and

g_{1}(s) = as^{3/2}

a an arbitrary constant, then we obtain from (13)

\begin{aligned}t & = & x_{2} \\ s & = & \frac{c_{1}x_{1}}{c_{1}−2x_{2}} \end{aligned}  (15)

and

\begin{aligned}g_{2}(s) & = & −\frac{2}{3} \frac{(c_{1}+6a)}{c_{1}}s^{3/2} \\ g_{3} & = & \frac{4}{3} \frac{(2c_{1}+3a)}{c_{1}^{2}}s^{3/2}, \end{aligned}

while F = u, G are obtained from (13), (15).

F = u = (−\frac{2c_{2}}{c_{1}} t + c_{2})s^{1/2}  (16)

G = [−\frac{8}{3c_{1}^{2}} t^{3} + \frac{4}{3} \frac{(2c_{1} + 3a)}{c_{1}^{2}}t^{2} − \frac{2}{3} \frac{(c_{1} + 6a)}{c_{1}}t + a] s^{3/2}  (17)

t, s being given by (15). □

4 The isotropic case: n = 3, m = 4

In sections 2 we discussed the WDVV equations as they arise in the case of two odd variables. Here we shall investigate the influence in coupling and intertwining in the situation for four odd variables. Local coordinates will be x_{0}, x_{1}, x_{2}, ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}. The matrix g_{ab}(2) is of the form

g_{ab} = [ \begin{matrix}& & 1 & 0 & 0 & 0 & 0 \\ & 1 & & 0 & 0 & 0 & 0 \\ 1 & & & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & & & & −c_{2} \\ 0 & 0 & 0 & & & −c_{3} \\ 0 & 0 & 0 & & +c_{3} \\ 0 & 0 & 0 & +c_{2} \end{matrix} ]  (1)

The potential Φ from which g_{ab} is determined by

g_{ab} = ∂_{0}∂_{a}∂_{b}Φ  (2)

has the following structure

\begin{aligned}Φ & = & \frac{1}{2}x_{0}^{2}x_{2} & + & \frac{1}{2}x_{0}x_{1}^{2} & + & c_{2}x_{0}ξ_{1}ξ_{4} & + & c_{3}x_{0}ξ_{2}ξ_{3} \\ & + & k_{12}ξ_{1}ξ_{2} & + & k_{13}ξ_{1}ξ_{3} & + & k_{24}ξ_{2}ξ_{4} & + & k_{34}ξ_{3}ξ_{4} \\ & + & k_{02}x_{0}x_{2} & + & k_{01}x_{0}x_{1} & + & k_{00}x_{0}^{2} \\ & + & F ξ_{1}ξ_{4} & + & \overline{F} ξ_{2}ξ_{3} & + & Hξ_{1}ξ_{2}ξ_{3}ξ_{4} & + & G \end{aligned}  (3)

where k_{ij} are constant and F, \overline{F} , H, G are functions of x_{1}, x_{2}. The associativity conditions lead to the following system of partial differential equations for F, \overline{F} , H, G; i.e.,

2c_{2}F_{2} − (F_{1})^{2} = 0  (4)
2c_{3}\overline{F}_{2} − (\overline{F}_{1})^{2} = 0  (5)
c_{3}F_{2} + c_{2}\overline{F}_{2} + F_{1}\overline{F}_{1} = 0  (6)
c_{2}^{2}G_{222} + c_{2}F_{1}G_{122} − F_{2}^{2} = 0  (7)
c_{2}^{2}G_{122} + c_{2}F_{1}G_{112} − F_{1}F_{2} = 0  (8)
c_{2}^{2}G_{112} + c_{2}F_{1}G_{111} + c_{2}F_{2} − F_{1}^{2} = 0  (9)
c_{3}c_{2}H_{2} + c_{3}H_{1}F_{1} + H^{2} = 0  (10)
c_{3}c_{2}^{2}\overline{F}_{112} + c_{3}c_{2}H_{1}G_{111} + c_{3}c_{2}H_{2} + c_{3}c_{2}F_{1}\overline{F}_{111} − 2c_{3}H_{1}F_{1} + 2c_{2}\overline{F}_{11}H = 0  (11)
c_{3}^{2}G_{222} + c_{3}\overline{F}_{1}G_{122} − \overline{F} _{2}^{2} = 0  (12)
c_{3}^{2}G_{122} + c_{3}\overline{F}_{1}G_{112} − \overline{F}_{1}\overline{F}_{2} = 0  (13)
c_{3}^{2}G_{112} + c_{3}\overline{F}_{1}G_{111} + c_{3}\overline{F}_{2} − \overline{F} _{1}^{2} = 0  (14)
c_{3}c_{2}H_{2} + c_{2}H_{1}\overline{F}_{1} + H^{2} = 0  (15)
c_{3}^{2}c_{2}F_{112} + c_{3}c_{2}H_{1}G_{111} + c_{3}c_{2}H_{2} + c_{3}c_{2}\overline{F}_{1}F_{111} − 2c_{2}H_{1}\overline{F}_{1} + 2c_{3}F_{11}H = 0.  (16)

In system of equations (4) we presented the equations in a symmetric way in F and \overline{F}. It is a straightforward computation that a solution of system (4) ⋯ (23) solves the remaining system (13) ⋯ (16). In order to solve the system of WDVV equations (4) we proceed in a way similar to that in section 2. First of all, from (4-18) we derive

\begin{aligned}F & = & c_{2}(\frac{1}{2}(h′)^{2}t + h) \\ \overline{F} & = & c_{3}(\frac{1}{2}(h′)^{2}t + h + const ) \end{aligned}  (17)

where s, t satisfy

\begin{aligned}x_{1} & = & h′(s)t + s \\ x_{2} & = & t \end{aligned}  (18)

and h is an arbitrary function of s, h′ = \frac{dh(s)}{ds}.

We transform the system according to (17), (18) and derive the following result

G_{tttt} = 0  (19)

i.e., G is a polynomial of degree 3 with respect to t. So

G = g_{6}t^{3} + g_{5}t^{2} + g_{4}t + g_{3}  (20)

where g_{3}, …, g_{6} are functions dependent on s. Substitution of (20) leads to the following conditions.

H = \frac{c_{3}c_{2}}{c_{3}c_{2}\overline{h} + t}  (21)

where \overline{h} is another arbitrary function of s. Finally we arrive at the following expressions for g_{3}, …, g_{6}

\begin{aligned}g_{6} & = & \frac{1}{8}(h′)^{4} \\ g′_{5} & = & g′′_{4}h′′ − g′_{3}(h′′)^{2} + \frac{1}{2}(h′)^{3} \\ g′′_{4} & = & 2g′′_{3}h′′ + g′_{3}h′′′ + \frac{3}{2}(h′)^{2} \end{aligned}  (22)

c_{3}^{2}c_{2}^{2}(h′′)^{2}\overline{h}^{2} + c_{3}c_{2}\overline{h}′g′′′_{3} − 3c_{3}c_{2}\overline{h}′h′ − 2c_{3}c_{2}h′′\overline{h} + 1 = 0.  (23)

In case

\overline{h}′ = 0

i.e., \overline{h} = constant; then (23) reduces to

h′′ = \frac{−1}{c_{2}c_{3}\overline{h}}

from which h is obtained. In this situation there are no restrictions on g_{3}.

Otherwise, \overline{h}′ ≠ 0, g_{3}, g_{4}, g_{5} are obtained from (22) , (23) by quadrature.

(17), (20), (21), (22), (23) together with (18) present the general solution of the WDVV equations (4), and so lead to the potential Φ (3)

5 The nonisotropic case: n = 3, m = 4

Here we discuss first the graded WDVV equations as they result from the algebraic structure defined in section 1. The number of even variables is 3, (x_{0}, x_{1}, x_{2}) and the number of odd variables is 4, (ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}). The matrix g_{ab} (1) (2) is choosen to be

g_{ab} = [ \begin{matrix}c_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & c_{2} \\ 0 & 0 & 0 & 0 & 0 & c_{3} & 0 \\ 0 & 0 & 0 & 0 & −c_{3} & 0 & 0 \\ 0 & 0 & 0 & −c_{2} & 0 & 0 & 0 \end{matrix} ]  (1)

Due to the definition of g_{ab} (12) we arrive at the even potential Φ of the form

\begin{aligned}Φ & = & \frac{1}{6}c_{1}x_{0}^{3} − c_{2}x_{0}ξ_{1}ξ_{2} − c_{3}x_{0}ξ_{2}ξ_{3} + x_{0}x_{1}x_{2} \\ & + & k_{1}x_{0} + k_{2}x_{0}^{2} + k_{3}x_{0}x_{1} + k_{4}x_{0}x_{2} \\ & + & f_{12}ξ_{1}ξ_{2} + f_{13}ξ_{1}ξ_{3} + f_{24}ξ_{2}ξ_{4} + f_{34}ξ_{3}ξ_{4} \\ & + & F ξ_{1}ξ_{4} + \overline{F} ξ_{2}ξ_{3} + Hξ_{1}ξ_{2}ξ_{3}ξ_{4} + G \end{aligned}  (2)

where k_{1}, …, k_{4} are constants, f_{12}, f_{13}, f_{24}, f_{34}, F, \overline{F} , H, G being functions dependent on x_{1}, x_{2}. Then the associativity relations (6), (13) first lead to the conditions that f_{12}, f_{13}, f_{24}, f_{34} are constant and a system of partial differential equations for F, \overline{F} , H, G

c_{2}^{2} + 2c_{1}F_{1}F_{2} = 0  (3)
c_{3}^{2} + 2c_{1}\overline{F}_{1}\overline{F}_{2} = 0  (4)
c_{2}c_{3} + c_{1}\overline{F}_{1}F_{2} + c_{1}\overline{F}_{2}F_{1} = 0  (5)
F_{1}G_{112} + F_{2}G_{111} + \frac{1}{c_{2}} (F_{1})^{2} = 0  (6)
F_{1}G_{122} + F_{2}G_{112} − \frac{c_{2}}{c_{1}} + \frac{1}{c_{2}} F_{1}F_{2} = 0  (7)
F_{1}G_{222} + F_{2}G_{122} + \frac{1}{c_{2}} (F_{2})^{2} = 0  (8)
H_{1}G_{222} + H_{2}G_{122} − \overline{F}_{12}F_{22} − \overline{F}_{22}F_{12} + \frac{1}{c_{2}} (H_{2}F_{2} − HF_{22}) + \frac{1}{c_{3}} (H_{2}\overline{F}_{2} − H\overline{F}_{22}) = 0 (9)

and

G_{122}G_{112} − G_{111}G_{222} + \frac{1}{c_{1}} = 0  (10)

It should be noted that (10) as well as all other equations resulting from the associativityconditions are consequences of (3) - (9) , which is the system of WDVV-equations in the nonisotropic (3,4)-case. We now construct the general solution for system (3) - (16) in order to construct the potential Φ. First of all we note that from equations (3), (11), (12) we derive in a similar way as in section 3 and 4

\begin{aligned}F & = & −\frac{c_{2}^{2}}{c_{1}} (h′)^{−1}t + h \\ \overline{F} & = & −\frac{c_{2}c_{3}}{c_{1}} (h′)^{−1}t + \frac{c_{3}}{c_{2}} h + const. \end{aligned}  (11)

where t, s are defines by

\begin{aligned}x_{1} & = & −\frac{c_{2}^{2}}{2c_{1}} (h′)^{−2}t + s \\ x_{2} & = & t \end{aligned}  (12)

whereas h is an arbitrary function of s, h′ = \frac{dh(s)}{ds}. We now transform the system of equations (3) according to (12) and derive

G_{tttt} = 0  (13)

i.e. G is a polynomial of degree 3 in t,

G(s, t) = g_{5}t^{3} + g_{4}t^{2} + g_{3}t + g_{2}  (14)

where g_{2}, …, g_{5} are functions of s. Proceeding with the result (14) we arrive at

g_{5} = −\frac{c_{2}^{3}}{3c_{1}} (h′)^{−3}

\begin{aligned}g′_{4} & = & \frac{c2}{c_{1}^{2}}[−c_{2}^{3}g′_{2}(h′′)^{2} + c_{2}c_{1}g′_{3}h′′(h′)^{3} + c_{1}(h′)^{5}](h′)^{−6} \\ g′′_{3} & = & \frac{1}{c_{2}c_{1}} [2c_{2}^{3}g′′_{2}h′′h′ + c_{2}^{3}g′_{2}h′′′h′ − 3c_{2}^{3}g′_{2}(h′′)^{2} \\ & − & c_{1}(h′)^{5}](h′)^{−4} \\ g′′′_{2} & = & \frac{c_{3}}{c_{2}} [c_{3}c_{2}^{2}(h′′)^{2}\overline{h}^{2} − 2c_{1}h′′(h′)^{2}\overline{h} − c_{1}(h′)^{3}\overline{h}′]/ \\ & & [c_{3}c_{2}^{2}\overline{h}′ + c_{1}h′] \end{aligned}  (15)

H = −c_{3}h′/(c_{3}h′\overline{h} + \overline{t})  (16)

In (15, (16) \overline{h} is an arbitrary function of s. (11), …, (16) represent the general solution for the potential Φ, expressed in two arbi-

trary function of s; h and \overline{h}.

Conclusion:

We derived the WDVV equations for a (3.2) and (3.4) dimensional manifold and constructed the general solutions for the isotropic and nonisotropic cases.

References

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