A new class of solutions to the WDVV equation
aa r X i v : . [ h e p - t h ] J u l ITP–UH–11/09
A new class of solutions to the WDVV equation
Olaf Lechtenfeld a and Kirill Polovnikov b a Institut f¨ur Theoretische Physik, Leibniz Universit¨at Hannover, 30167 Hannover, Germany b Laboratory of Mathematical Physics, Tomsk Polytechnic University, 634050 Tomsk, Russia
Abstract
The known prepotential solutions F to the Witten-Dijkgraaf-Verlinde-Verlinde(WDVV) equation are parametrized by a set { α } of covectors. This set may betaken to be indecomposable, since F { α } ˙ ⊕{ β } = F { α } + F { β } . We couple mutuallyorthogonal covector sets by adding so-called radial terms to the standard formof F . The resulting reducible covector set yields a new type of irreduciblesolution to the WDVV equation. Introduction
The Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation F i F − k F j = F j F − k F i with i, j, k = 1 , . . . , n (1.1)is a nonlinear constraint on a set of n n × n matrices F i whose entries are the third derivatives of a function F ( x , . . . , x n ) called the prepotential, ( F i ) pq = ∂ i ∂ p ∂ q F (1.2)(we abbreviate ∂ i = ∂∂x i ). This equation was first introduced in the context of two dimensional topologicalfield theory [1, 2] and N =2 SUSY Yang-Mills theory [3] and was intensively studied during the followingyears. In 1999, it was shown [4] that one can construct solutions by taking the ansatz F = − X α f α ( α · x ) ln | α · x | , (1.3)where { α } is the (positive) root system of some simple Lie algebra of rank n . This ansatz defines aconstant metric given by the matrix G = − x i F i = X α f α α ⊗ α . (1.4)Shortly thereafter, it was proved [5] that certain deformations of such root systems still solve (1.1), andthe list of possible collections of covectors { α } was extended to the so-called ∨ -systems [6].In 2005, an interesting connection between the WDVV equation and N =4 superconfermal mechanicswas discovered [7]. In this context, a slightly different but equivalent formulation of the WDVV equationappeared, namely[ F i , F j ] = 0 ←→ ( ∂ i ∂ k ∂ p F )( ∂ j ∂ l ∂ p F ) = ( ∂ j ∂ k ∂ p F )( ∂ i ∂ l ∂ p F ) , (1.5)supplemented by the homogeneity condition − x i F i = ←→ − x i ∂ i ∂ j ∂ k F = δ jk . (1.6)The latter picks a Euclidean metric in R n and determines a linear change of coordinates which relatesthe two formulations within the ansatz (1.3) by mapping G [8]. We also write x j = δ jk x k = x j .Rather recently, it was observed that the ansatz (1.3) can be successfully extended by adding a ‘radialterm’ [9], F = − X α f α ( α · x ) ln | α · x | − f R R ln R , (1.7)where { α } is some ∨ -system and R denotes the radial coordinate in R n , R := n X i =1 x i so that G ( x, x ) = X α f α ( α · x ) = (1 − f R ) R . (1.8)The two types of term in (1.7) are extremal cases of a general ansatz employing arbitrary quadratic forms.For n> f R are compatible with the WDVV equation, namely f R = 0 and f R = 1,which are related by flipping the sign of the metric G via f α
7→ − f α .In this Letter we further generalize the ansatz (1.7) for the prepotential F and construct novel solutionsto the WDVV equation. Let us consider a reducible covector system L = L ⊕ L ⊕ . . . ⊕ L M , which is a direct sum of M irreduciblecovector collections with dimensions n , n , . . . , n M , respectively. Each subsystem L I ( I = 1 , . . . , M ) ischosen to solve the WDVV equation via the ansatz (1.3) for α ∈ L I . Furthermore, we work in the‘Euclidean coordinates’ conforming to (1.6). The prototypical example is the root system of a non-simplebut semi-simple Lie algebra. According to the covector set decomposition, we split the index set { , , . . . , n } = { , . . . , n } ∪ { n +1 , . . . , n + n } ∪ · · · ∪ { n − n M +1 , . . . , n } = S ∪ S ∪ · · · ∪ S M (2.1) The covector α evaluates on x ≡ ( x , . . . , x n ) via α ( x ) = α i x i =: α · x . Positive coefficients f α may be absorbed in α . X Ii = ( x i if i ∈ S I , X IJi = ( x i if i ∈ S IJ S IJ = S I ∪ S J etc. , (2.2)and likewise for δ Iij , δ IJij = δ Iij + δ Jij etc.. An important concept is that of relative radial coordinates r I = X i ∈ S I x i = x i X Ii , r IJ = r I + r J = X i ∈ S IJ x i = x i X IJi , . . . , R = X i x i (2.3)for the subspaces L I , L IJ = L I ⊕ L J , L IJK = L I ⊕ L J ⊕ L K etc., all the way up to L .The key idea is to couple the mutually orthogonal components of this reducible covector system byadding to the ansatz (1.3) not only the overall radial term as in (1.7) but also all possible relative radialterms, F = − X α ∈ L f α ( α · x ) ln | α · x | − M X I =1 f r I r I ln r I − M X I 2, because then f R may take any value since the WDVV equation is empty. α ∧ β ) ⊗ ijkl = ( α i β j − α j β i )( α k β l − α l β k ) ,T Iijkl = δ Iik δ Ijl − δ Iil δ Ijk − δ Iik ˆ X Ij ˆ X Il + δ Iil ˆ X Ij ˆ X Ik − δ Ijl ˆ X Ii ˆ X Ik + δ Ijk ˆ X Ii ˆ X Il ,T IJijkl = δ IJik δ IJjl − δ IJil δ IJjk − δ IJik ˆ X IJj ˆ X IJl + δ IJil ˆ X IJj ˆ X IJk − δ IJjl ˆ X IJi ˆ X IJk + δ IJjk ˆ X IJi ˆ X IJl , (2.10) · · · T ijkl = δ ik δ jl − δ il δ jk − δ ik ˆ x j ˆ x l + δ il ˆ x j ˆ x k − δ jl ˆ x i ˆ x k + δ jk ˆ x i ˆ x l , where ˆ X Ii ≡ X Ii r I , ˆ X IJi ≡ X IJi r IJ , . . . , ˆ x i ≡ x i R . (2.11)Projecting onto the different (independent) poles in (2.9), the WDVV equation requires that X α,β ∈ L I f α f β α · βα · x β · x ( α ∧ β ) ⊗ = 0 ,f r I (cid:26) − f r I − (cid:16) M X J ( = I ) f r IJ + . . . + f R (cid:17)(cid:27) = 0 , (2.12) f r IJ (cid:26) − f r IJ − (cid:16) M X K ( = I,J ) f r IJK + . . . + f R (cid:17)(cid:27) = 0 , · · · f R (cid:8) − f R (cid:9) = 0 . Staring for a while at (2.12) while taking into account (2.8), one realizes that the only admissibleradial couplings are f r I , f r IJ , f r IJK , . . . , f R ∈ { , +1 , − } but f R = − . (2.13)The solutions are best described by a sequential procedure, starting from the ‘radial-free’ configuration f r I = f r IJ = . . . = f R = 0 ∀ I, J, . . . , corresponding to ǫ I = +1 ∀ I . Now, let us turn on someradial couplings of the first hierarchy level, f r I = +1 for some values of I , which flips the signs of thecorresponding ǫ I . On the next level, we may now switch on further radial couplings f IJ , but only if theydo not overlap. For each nonzero f IJ we must flip the signs of the corresponding ǫ I and ǫ J as well asthose of f I and f J . Continuing this scheme, we eventually arrive at the highest level, where activating f R will flip all signs in the hierarchy. In this way, a multitude of possible new WDVV solutions is generated. To illustrate the new possibilities for WDVV solutions, let us consider the semi-simple Lie algebra A ⊕ A .Our generalized ansatz (2.4) for this case ( M =2) reads F = − f ( x + x + x ) ln | x + x + x | − f X i We thank S. Krivonos for collaboration at various stages of this project. K.P. is grateful to the Institutf¨ur Theoretische Physik at the Leibniz Universit¨at Hannover for hospitality. The research was supportedby RF Presidential grant NS-2553.2008.2, RFBR grant 09-02-00078 and the Dynasty Foundation. References [1] E. Witten, Nucl. Phys. B (1990) 281.[2] R. Dijkgraaf, H. Verlinde, E. Verlinde, Nucl. Phys. B (1991) 59.[3] A. Marshakov, A. Mironov, A. Morozov, Phys. Lett. B (1996) 43 [hep-th/9607109].[4] R. Martini, P.K.H. Gragert, J. Nonlin. Math. Phys. (1999) 1.[5] A.P. Veselov, Phys. Lett. A (1999) 297 [hep-th/9902142].[6] A.P. Veselov, in: Integrability: The Seiberg-Witten and Whitham Equations, eds. H.W. Braden, I.M. Krichever, Gordon and Breach, 2000, p. 125 [hep-th/0105020].[7] S. Bellucci, A. Galajinsky, E. Latini, Phys. Rev. D (2005) 044023 [hep-th/0411232].[8] O. Lechtenfeld, in: Problems of Modern Theoretical Physics, ed. V. Epp, Tomsk State Pedagogical University Press, 2008, p. 256 [arXiv:0805.3245].[9] A. Galajinsky, O. Lechtenfeld, K. Polovnikov, JHEP0903