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Duality Transformations for Generalized WDVV equations in Seiberg-Witten theory

It is known that electric-magnetic duality transformations are symmetries of the generalized Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. In Seiberg-Witten theory the solutions to these equations come in certain sets according to the gauge group. We show that the duality transformations transform solutions within a set to another solution within the same set, by using a subset of the Picard-Fuchs equations on the Seiberg-Witten family of Riemann surfaces. The electric-magnetic duality transformations can be thought of as changes of a canonical homology basis on the surfaces which in our derivation is clearly responsible for the covariance of the generalized WDVV system.


Introduction
In 1994, Seiberg and Witten [1] solved the low energy behaviour of pure N=2 Super-Yang-Mills theory by giving the solution of the prepotential F . The essential ingredients in their construction are a family of Riemann surfaces Σ, a meromorphic differential λ SW on it and the definition of the prepotential in terms of period integrals of λ SW The cycles A I and B I belong to a subset of a canonical homology basis on the surface Σ and the a I are the moduli parameters of the family of surfaces. These formulae define the prepotential F (a 1 , ..., a r ) implicitly, where r denotes the rank of the gauge group under consideration. A link between the prepotential and the Witten-Dijkgraaf-Verlinde-Verlinde equations [2], [3] was proven to exist in [4] where it was found that the prepotential F (a 1 , ..., a r ) for pure N=2 SYM theory for classical Lie algebras (those of type A, B, C, D) satisfies the generalized WDVV equations where the F I are matrices given by (F I ) JK = ∂ 3 F ∂aI ∂aJ ∂aK and the γ M may depend on the a I . If F satisfies (1.2) for some set of γ M it will automatically satisfy them for any other set γ M as long as M γ M F M is invertible. These equations are indeed a generalization of the original WDVV equations, since we no longer demand the matrix M γ M F M to be flat and constant.
In an alternative approach to [4], Ito and Yang [5] give a proof which is valid 1 for Lie algebras of type A, D, E 6 and it was shown in [6] that this method can be adapted to give a proof also in the case of Lie algebras of type B, C. The approach used by Ito and Yang consists of two main ingredients: an associative algebra with structure constants C K IJ , together with a relation between the structure constants and the third order derivatives of the function F for some γ M . To derive this relation they used a subset of the Picard-Fuchs equations on the Riemann surfaces. If the matrix r M=1 γ M F M is invertible then we can substitute (1.3) into the WDVV equations (1.2) which then express associativity of the algebra through a relation on the structure constants: [C I , C J ] = 0.
In the physical context of Seiberg-Witten theory, electric-magnetic duality transformations are very important and therefore it is a natural question if these are symmetries of the generalized WDVV equations. The duality transformations form a subgroup of Sp(2r, Z) acting as linear transformations on the vector − → a , − → ∇ F . These transformations are indeed symmetries, as was shown in [7] for a pure S-duality transformation and recently in [8] for general symplectic transformations.
We will show in this Letter that the classes of solutions of the WDVV system coming from Seiberg-Witten theory are invariant under duality transformations, by giving the duality transformations their natural interpretation in the Seiberg-Witten context. The reason for the invariance is that the Picard-Fuchs equations leading to the relation (1.3) have solutions in terms of period integrals of λ SW . We require of our solutions (1.1) only that the cycles be canonical, so they are invariant under symplectic transformations of the cycles. These are precisely the duality transformations. In the derivation of (1.3) we will rely heavily on the connection between the Seiberg-Witten and Landau-Ginzburg theories which is quite straightforward in the simply laced (ADE) cases. For B and C Lie algebras this connection is less clear [6] but we will show that one can use the same method as in the other cases. For F 4 it is not rigorously proven that the prepotential satisfies the WDVV equations, so we cannot say anything definite about duality in this case 2 .

Duality transformations
Duality transformations play an important role in N = 2 Super-Yang-Mills theory, where in the classical theory they exchange the Bianchi identity and the equations of motion of the Yang-Mills field strength. In the quantum field theory, these transformations were studied extensively in the context of supergravity (for a review, see [9]) and they turn out to form a subgroup of Sp(2r, Z) where r is the rank of the gauge group. The action on (a I , F I ) of (1.1) is given by There is some terminology used by physicists for some special transformations • An example of S-duality is a transformation for which and in general S-duality transformations exchange the strong and weak coupling regimes of the physical theory.
• An example of T -duality is a tranformation for which and in general T -duality transformations are perturbative, so they go from weak coupling to weak coupling.
Seiberg and Witten used the fact that duality transformations can be thought of as changes of a canonical basis of a family of auxiliary Riemann surfaces in the following way. Let a canonical basis {A I , B I } of homology on a Riemann surface be given. The transformations that take this canonical basis into another one are known to be (see e.g. [10]) symplectic transformations Sp(2g, Z) acting like For Lie algebra A r the genus of the curves taken in Seiberg-Witten theory equals the rank of the gauge group, so (2.2) generate transformations on a I , F I since for example In other words, electric-magnetic duality transformations in the Seiberg-Witten context can be thought of as changes of canonical bases of homology on families of Riemann surfaces.
For other gauge groups, the rank is always less than the genus of the curve and we have to take a subset of the homology basis in such a way that this subset has canonical intersection numbers If we call the linear subspace of H 1 (Σ, Z) spanned by these cycles X, then duality transformations are generated by changes of the canonical homology basis which leave X invariant.

Picard-Fuchs equations and duality
In our derivation of the generalized WDVV equations, we consider a system of differential equations for the periods Γ λ SW (Picard-Fuchs like equations) and then substitute a I = AI λ SW and F I = BI λ SW in them. This way, the derivation of the WDVV equations is manifestly invariant under changes of the canonical homology basis and therefore under electric-magnetic duality transformations.
The Riemann surfaces for the ADE cases read [11] z where W can be though of as the one-dimensional version of the Landau-Ginzburg superpotential. In deriving the Picard-Fuchs equations, use is made of the flat coordinates of the corresponding Landau-Ginzburg theories. For the B, C Lie algebras, the methods of [5] are no longer directly applicable: we have to 'twist' the affine Lie algebra to construct the surfaces 3 . This leads to surfaces where W need not have a direct relation to Landau-Ginzburg theory. For the B, C cases, where the relation with the corresponding superpotential is still quite straightforward, it is shown in [6] that the method of using Picard-Fuchs equations can be adapted in order to give the WDVV equations.
In the next sections we will review the derivation of the WDVV equations in order to see that duality transformations are symmetries of them.

The ADE cases
The family of Riemann surfaces associated with Seiberg-Witten theory with Lie algebras of A, D, E type is given by [11] z Here W can be thought of as a one variable Landau-Ginzburg superpotential for the corresponding Lie algebra. For instance, W Ar (x, u i ) = x r+1 − r−1 i=1 u i x r−i is the superpotential for Lie algebra A r . However W E6 contains a square root of a polynomial, but the structure constants of the chiral ring are the same as in the three-variable situation [12]. From the Landau-Ginzburg theory it is known (see e.g. [13] and references therein) that we can pass from the moduli u i to flat coordinates t i in terms of which the Gauss-Manin connection is set to zero. The Landau-Ginzburg product structure reads which leads to the algebra and the flat coordinates 4 satisfy Now that the Riemann surfaces (3.3) are introduced, we need a meromorphic differential on them whose derivatives with respect to the moduli are It is stated often in the literature that the forms − ∂W ∂t i dx z − 1 z are holomorphic and linearly independent. For all classical Lie algebras, the Riemann surfaces are hyperelliptic and this statement can be easily checked. For all exceptional Lie algebras however, the Riemann surfaces are not hyperelliptic and it is still not proven that the forms are holomorphic.
As a final ingredient for the Seiberg-Witten theory, we introduce a third set of coordinates a I on the moduli space of the family of surfaces and objects F I where we take a subset of a canonical basis for the homology. The holomorphic parts of the differentials ∂λSW ∂aI are canonical with respect to the cycles and therefore ∂FI ∂aJ is a submatrix of the period matrix, which is symmetric. So F I is a gradient and there exists a function F (a I ) with derivatives F I and this socalled prepotential solves the low energy behaviour of N = 2 super-Yang-Mills theory.
The product structure (3.4) can also be expressed in terms of the a I as follows where the structure constants are related through C K IJ (a) = i,j,k ∂ti ∂aI ∂tj ∂aJ C k ij (t) ∂aK ∂t k . Here we assume that the transformation from t i to a I is invertible, which can be justified: in the case of type A Lie algebras, the number of moduli equals the genus and therefore the Jacobian has rank r, and we can always find a square submatrix of rank r by choosing the proper cycles. Therefore we can always choose cycles in such a way that the transformation from t i to a I is invertible. In Seiberg-Witten theory, there is a specific prescription [11] of what cycles one has to take and invertibility of the corresponding submatrix should be checked. Following [5] we write down a subset of the Picard-Fuchs 5 equations which hold for integrals along any closed cycle Γ.
We will now prove the following theorem where we made use of the fact that the a I themselves are solutions to (3.12). Now we can substitute Γ = B K and find which is the relation (1.3) between the third order derivatives of F (a I ) and the structure constants C L IJ (a).
To obtain the generalized WDVV equations for the function F (a I ) we need the matrix r M=1 ∂aM ∂tr F M to be invertible, which is generically true. Otherwise small perturbations can make it so. Since the derivation of the theorem does not depend on the specific choice of canonical homology basis we started with, and since duality transformations are changes of canonical homology, we have the following Corollary 2 Take a set of solutions of the WDVV system coming from Seiberg-Witten theory with gauge group of ADE type. Duality transformations leave the set of solutions invariant.
In other words, after a duality transformation there exists a function F ( a I ) which is in general different 6 from F (a I ) but still satisfies the generalized WDVV system. 5 If a basis of cohomology on a family of Riemann surfaces is given by {ω i } where i = 1, ..., 2g then the Picard-Fuchs equations express where Γ is some closed cycle and the t i are the moduli of the family. By a subset of Picard-Fuchs equations, we mean that we can express derivatives of some (not all) period integrals into a linear combination of some others. 6 The transformations are often too difficult to perform explicitly.

The B and C cases
Let us stresss that for B, C Lie algebras, we no longer have with W a one variable version of the Landau-Ginzburg superpotential. This will be reflected in the Picard-Fuchs equations. In fact, if we still work with the flat coordinates of the Landau-Ginzburg superpotential W BC one can derive the Picard-Fuchs equations [5] where the C k ij are structure constants of the W BC Landau-Ginzburg theory, the d n are so-called degrees of the Lie algebra (the exponents +1), D k ij depend on the t i and h ∨ is the dual Coxeter number. Making a change of coordinates to the a I just like we did for ADE algebras and using the fact that the a I satisfy (3.14), we get  These equations are not of such a form that we can derive a relation between F IJK and the structure constants C k ij in the same way as before. However, in [6] it was shown that some other constants C k ij form structure constants of an associative algebra and are related to C k ij through where the second line is in matrix form. Substituting this into (3.15) we get and in [6] further information about the D l ij was used to conclude From this point we can proceed with the same reasoning as in the ADE case and conclude that the following theorem holds

Conclusion and outlook
It was shown in [8] that elements of the continuous symplectic group Sp(2n, C) are symmetries of the generalized WDVV system. Because of the relation between Seiberg-Witten theory and Riemann surfaces, this leads naturally to the study of the discrete subgroup Sp(2n, Z) in the Seiberg-Witten context, and we have shown how the interpretation of this discrete subgroup as changes of a canonical homology basis leads automatically to the invariance of classes of solutions to the WDVV system. Other solutions of the generalized WDVV system found so far either come from the original WDVV equations or from the context of tau functions of conformal mappings [14]. Interpretations of the symplectic transformations can be looked for in both contexts. This might show whether the discrete subgroup is a natural object there as well.
Other interesting symmetries of the WDVV system have been found [15] [16], but no systematic investigation of symmetries has been undertaken. We believe such an investigation would be interesting.