A new class of solutions to the WDVV equation

Physics Letters A 374 (2010) 504–506

Contents lists available at ScienceDirect

Physics Letters A www.elsevier.com/locate/pla

A new class of solutions to the WDVV equation Olaf Lechtenfeld a,∗ , Kirill Polovnikov b a b

Institut für Theoretische Physik, Leibniz Universität Hannover, 30167 Hannover, Germany Laboratory of Mathematical Physics, Tomsk Polytechnic University, 634050 Tomsk, Russia

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 20 July 2009 Accepted 13 November 2009 Available online 21 November 2009 Communicated by A.P. Fordy

The known prepotential solutions F to the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equation are = parametrized by a set {α } of covectors. This set may be taken to be indecomposable, since F {α }⊕{β} ˙ F {α } + F {β} . We couple mutually orthogonal covector sets by adding so-called radial terms to the standard form of F . The resulting reducible covector set yields a new type of irreducible solution to the WDVV equation. © 2009 Elsevier B.V. All rights reserved.

1. Introduction

this context, a slightly different but equivalent formulation of the WDVV equation appeared, namely

The Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equation

F i F k−1 F j = F j F k−1 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 (x1 , . . . , xn ) called the prepotential,

( F i ) pq = ∂i ∂ p ∂q F

(1.2)

(we abbreviate ∂i = ∂ i ). This equation was first introduced in ∂x the context of two-dimensional topological field theory [1,2] and N = 2 SUSY Yang–Mills theory [3] and was intensively studied during the following years. In 1999, it was shown [4] that one can construct solutions by taking the ansatz1

F =−

1 2 α

f α (α · x)2 ln |α · x|,

(1.3)

where {α } is the (positive) root system of some simple Lie algebra of rank n. This ansatz defines a constant metric given by the matrix

G = − xi F i =



f α α ⊗ α.

(1.4)

α

Shortly thereafter, it was proved [5] that certain deformations of such root systems still solve (1.1), and the 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 superconformal mechanics was discovered [7]. In

*

Corresponding author. E-mail address: [email protected] (O. Lechtenfeld). 1 The covector α evaluates on x ≡ (x1 , . . . , xn ) via α (x) = αi xi =: α · x. Positive coefficients f α may be absorbed in α . 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.11.052

[Fi, 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

− xi F i = 1

←→

−xi ∂i ∂ j ∂k F = δ jk .

(1.6)

The latter picks a Euclidean metric in R and determines a linear change of coordinates which relates the two formulations within the ansatz (1.3) by mapping G → 1 [8]. We also write x j = δ jk xk = x j . Rather recently, it was observed that the ansatz (1.3) can be successfully extended by adding a ‘radial term’ [9], n

F =−

1 2 α

f α (α · x)2 ln |α · x| −

1 2

f R R 2 ln R 2 ,

(1.7)

where {α } is some ∨-system and R denotes the radial coordinate in Rn ,

R 2 :=

n  i =1

G (x, x) =

x2i



so that f α (α ·x)2 = (1 − 2 f R ) R 2 .

(1.8)

α

The two types of term in (1.7) are extremal cases of a general ansatz employing arbitrary quadratic forms. For n > 2 only two choices for 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 → − f α . In this Letter we further generalize the ansatz (1.7) for the prepotential F and construct novel solutions to the WDVV equation.

O. Lechtenfeld, K. Polovnikov / Physics Letters A 374 (2010) 504–506



2. Relative radial terms Let us consider a reducible covector system L = L 1 ⊕ L 2 ⊕ · · · ⊕ L M , which is a direct sum of M irreducible covector collections with dimensions n1 , n2 , . . . , n M , respectively. Each subsystem L I ( I = 1, . . . , M ) is chosen to solve the WDVV equation via the ansatz (1.3) for α ∈ L I . Furthermore, we work in the ‘Euclidean co-

α∈L I

ordinates’ conforming to (1.6). The prototypical example is the root system of a non-simple but semi-simple Lie algebra. According to the covector set decomposition, we split the index set



∪ {n−n M +1, . . . , n}

X iI =

xi

if i ∈ S I ,

0

otherwise,

Xi =

xi

if i ∈ S I J , otherwise,

IJ δi j

(2.2) J δiIj +δi j ,

and likewise for = of relative radial coordinates

r 2I

=



x2i

=x

i

X iI ,

r 2I J

= r 2I

etc. An important concept is that

+ r 2J

=

i∈ S I 2

R =



+



x2i

=x

i

IJ Xi ,

Since all covector collections L I are ∨-systems, we must have2

f α αi α j =  I δiIj

..., (2.3)

2

α∈L

1 2

1 M

f α (α · x)2 ln |α · x| −

M

f r I J r 2I J ln r 2I J − · · · −

I< J

2 1 2



M 

frI J +

J ( = I )

frI J K + · · · + f R

J , K ( = I )

for  I = +1,

0

(2.8)

1 for  I = −1.

1  2

fα fβ

α ,β∈ L

+

M 

+

M 

α·β (α ∧ β)⊗2 α · xβ · x 



4 frI 1 − frI − 2

I =1

M 

I =1

−2

α∈L I

M 

frI

(2.4)





4 frI J 1 − frI J − 2

I =1

−2

M 

frI J

2 X iI X Ij X kI

I< J

− ··· − 2 fR



r 2I J xi δ jk + x j δki + xk δi j R2

−

−

2

2xi x j xk R4

=0

TIJ r 2I J (2.9)

I I T iIjkl = δik δ jl − δilI δ Ijk − δikI Xˆ Ij Xˆ lI + δilI Xˆ Ij Xˆ kI − δ Ijl Xˆ iI Xˆ kI + δ Ijk Xˆ iI Xˆ lI ,

··· T i jkl = δik δ jl − δil δ jk − δik xˆ j xˆ l + δil xˆ j xˆ k − δ jl xˆ i xˆ k + δ jk xˆ i xˆ l , (2.10) where

Xˆ iI ≡

IJ Xi

 .

IJ Xj r 4I J

X iI rI

,

Xˆ i ≡ IJ

fα fβ





 frI J 1 − frI J − 2

··· (2.5)

Contracting with xi should reproduce (1.6). Taking into account (2.3), one obtains a system of equations,

rI J

,

...,

xˆ i ≡

xi R

(2.11)

.

α·β (α ∧ β)⊗2 = 0, α · xβ · x

frI 1 − frI − 2

IJ Xk

IJ

Xi

Projecting onto the different (independent) poles in (2.9), the WDVV equation requires that





r 4I

 X I J δI J + X I J δI J + X I J δI J i j ki jk k ij

frI J K + · · · + f R

2 (α ∧ β)⊗ = (αi β j − α j βi )(αk βl − αl βk ), i jkl

α ,β∈ L I

−

R2

r 2I



M  K ( = I , J )

T

TI

with



αi α j αk fα α·x r 2I

frI J + · · · + f R

IJ IJ IJ IJ IJ IJ − δ jl Xˆ i Xˆ k + δ jk Xˆ i Xˆ l ,

f R R 2 ln R 2 ,

 X I δI + X I δI + X I δI i jk j ki k ij



J ( = I )

+ · · · + 4 f R {1− f R }

I =1

∂i ∂ j ∂k F M  

(2.7)

IJ IJ IJ IJ IJ IJ IJ IJ IJ IJ IJ T i jkl = δik δ jl − δil δ jk − δik Xˆ j Xˆ l + δil Xˆ j Xˆ k

f r I r 2I ln r 2I

defining a hierarchy f r I ⊂ f r I J ⊂ f r I J K ⊂ · · · ⊂ f R of radial couplings. We point out that this ansatz is no longer decomposable. In order to verify (2.4), we have to compute the WDVV coefficients

=−

M 

I< J

for the subspaces L I , L I J = L I ⊕ L J , L I J K = 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 by adding to the ansatz (1.3) not only the overall radial term as in (1.7) but also all possible relative radial terms,

−

 I ∈ {+1, −1},

with

α∈L I

i∈ S I J

x2i

1

(2.6)

J , K ( = I )

i

F =−

= δiIj .

frI J K + · · · + f R

This takes care of the homogeneity condition (1.6). We come to the WDVV equation (1.5), which reads

etc.,

for S I J = S I ∪ S J , etc.,

δiIj ,

frI J



M 

= 0

M 

frI +

J ( = I )

(2.1)



IJ

f α αi α

frI +

= S1 ∪ S2 ∪ · · · ∪ S M 

 I j + 2δ i j

hence

{1, 2, . . . , n} = {1, . . . , n1 } ∪ {n1 +1, . . . , n1 +n2 } ∪ · · ·

and introduce the notation

505

f R { 1− f R } = 0.

M 

 frI J + · · · + f R

= 0,

J ( = I )



M 

 frI J K + · · · + f R

= 0,

K ( = I , J )

(2.12)

2 This is not true for n I  2, because then f R may take any value since the WDVV equation is empty.

506

O. Lechtenfeld, K. Polovnikov / Physics Letters A 374 (2010) 504–506

−

Staring for a while at (2.12) while taking into account (2.8), one realizes that the only admissible radial couplings are

f r I , f r I J , f r I J K , . . . , f R ∈ {0, +1, −1}

but f R = −1.

The solutions are best described by a sequential procedure, starting from the ‘radial-free’ configuration f r I = f r I J = · · · = f R = 0 ∀ I , J , . . . , corresponding to  I = +1 ∀ I . Now, let us turn on some radial couplings of the first hierarchy level, f r I = +1 for some values of I , which flips the signs of the corresponding  I . On the next level, we may now switch on further radial couplings f I J , but only if they do not overlap. For each nonzero f I J we must flip the signs of the corresponding  I and  J as well as those 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 1 ⊕ A 2 . Our generalized ansatz (2.4) for this case (M = 2) reads

1 2

−

f 0 (x1 + x2 + x3 )2 ln |x1 + x2 + x3 |

1 2

3  1 1 (xi − x j )2 ln |xi − x j | − f r r 2 ln r 2 − f R R 2 ln R 2 f

2

i< j

with r 2 =

2

3

2

x21 + x22 + x23 − x1 x2 − x1 x3 − x2 x3

R 2 = x21 + x22 + x23 .



and (3.1)

The last term in (3.1) couples the center of mass (the A 1 part) to the relative motion (the A 2 part). Since both subsystems are at most two-dimensional, the WDVV equation is empty, being a consequence of the homogeneity condition (1.6). Hence, we only have to fulfill (2.6), which yields

3 f 0 + 2 f R = 1 and



fR =

3 f + 3 fr + 2 f R = 1

0

⇒

f 0 = f + f r = + 13 ,

1

⇒

f 0 = f + f r = − 13 ,

−→

F =−

2





f1

(xi + x j )2 ln |xi + x j |

1i < j 3



+

 2

(xi − x j ) ln |xi − x j |

−

2





f2

2

(xi + x j ) ln |xi + x j |

4i < j 6



+



(xi − x j )2 ln |xi − x j |

4i < j 6

− +

1 2

2

2 2 ln r12 − f r12 r12 2 2 ln r23 − f r23 r23

where f I =

r12 =

3 

2

f r2 r22 ln r22 − 1 2 1 2

1

f r3 r32 ln r32

2

2 2 ln r13 f r13 r13

f R R 2 ln R 2 ,

(3.3)

1 4 I

 and

x2i ,

r22 =

i =1

6 

x2i ,

r32 =

i =4

r i2j = r i2 + r 2j ,

9 

x2i ,

i =7

R 2 = r12 + r22 + r33 =

9 

x2i .

(3.4)

i =1





f3

(xi + x j )2 ln |xi + x j |

7i < j 9





(xi − x j )2 ln |xi − x j |

7i < j 9

1

2

3

+1 −1 −1 −1 −1 +1 +1 −1 +1 +1 −1 +1 +1 +1 +1 −1 −1 +1 −1 −1

+1 +1 −1 −1 −1 −1 +1 −1 −1 +1 −1 −1 +1 +1 +1 +1 −1 +1 +1 −1

+1 +1 +1 −1 +1 +1 +1 −1 −1 −1 −1 −1 −1 +1 −1 −1 −1 +1 +1 +1

f r1

f r2

f r3

0

0 0 +1 +1 0 0 −1 0 0 −1 0 0 −1 −1 0 0 +1 0 0 +1

0 0 0 +1 0 0 0 +1 +1 +1 0 0 0 −1 0 0 0 −1 −1 −1

+1 +1 +1 0

−1 −1 0

−1 −1 0

−1 −1 −1 0

+1 +1 0

+1 +1

f r12 0 0 0 0 +1 +1 +1 +1 +1 +1 0 0 0 0 −1 −1 −1 −1 −1 −1

f r13

f r23

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

fR 0 0 0 0 0 0 0 0 0 0 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1

Acknowledgements We thank S. Krivonos for collaboration at various stages of this project. K.P. is grateful to the Institut für Theoretische Physik at the Leibniz Universität Hannover for hospitality. The research was supported by RF Presidential grant NS-2553.2008.2, RFBR grant 09-02-00078 and the Dynasty Foundation. References

1i < j 3

1

2 1

1

(3.2)

giving us a one-parameter family of solutions. To see the power of the WDVV equation, we present a more generic M = 3 example, based on the Lie algebra D 3 ⊕ D 3 ⊕ D 3 :

1

−

2 1

f r1 r12 ln r12 −

We list here (up to permutations) all possible radial coupling configurations:

3. Examples

F =−

−

(2.13)

1

[1] E. Witten, Nucl. Phys. B 340 (1990) 281. [2] R. Dijkgraaf, H. Verlinde, E. Verlinde, Nucl. Phys. B 352 (1991) 59. [3] A. Marshakov, A. Mironov, A. Morozov, Phys. Lett. B 389 (1996) 43, hep-th/ 9607109. [4] R. Martini, P.K.H. Gragert, J. Nonlinear Math. Phys. 6 (1999) 1. [5] A.P. Veselov, Phys. Lett. A 261 (1999) 297, hep-th/9902142. [6] A.P. Veselov, in: H.W. Braden, I.M. Krichever (Eds.), Integrability: The Seiberg– Witten and Whitham Equations, Gordon and Breach, 2000, p. 125, hep-th/ 0105020. [7] S. Bellucci, A. Galajinsky, E. Latini, Phys. Rev. D 71 (2005) 044023, hep-th/ 0411232. [8] O. Lechtenfeld, in: V. Epp (Ed.), Problems of Modern Theoretical Physics, Tomsk State Pedagogical University Press, 2008, p. 256, arXiv:0805.3245. [9] A. Galajinsky, O. Lechtenfeld, K. Polovnikov, JHEP 0903 (2009) 113, arXiv:0802. 4386.