Lie symmetries and particular solutions of Seiberg Witten equations in R 3

INSTITUTE OF PHYSICSPUBLISHING JOURNAL OFPHYSICSA: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 36 003 51 60 PII: S0305-44700354555-3 Lie symmetries and particular solutions of Seiberg Witten equations in R 3 DVAleynikovandEATolkachev B I Stepanov Institute of Physics, National Academy of Sciences of Belarus, 68 F Skaryna Avenue, 007 Minsk, Belarus E-mail: aleynik@dragon.bas-net.by and tea@dragon.bas-net.by Received 8 October 00, in final form 10 December 00 Published 19 February 003 Online at stacks.iop.org/jphysa/36/51 Abstract It is shown that the conformal group of three-dimensional Euclidean space SO4, 1isamaximal group of Lie symmetries of the Seiberg Witten and Freund s equations in R 3. Particular explicit solutions which are invariant under some subgroups of SO4, 1 are constructed. PACS numbers: 11.5.Hf, 11.30.Pb, 0.0.Rt 1. Introduction The role of Seiberg Witten equations [1] in bothhigh-energy physics and topology is common knowledge. On one hand, the Seiberg Witten equations describe the dynamics of twisted N supersymmetric Abelian gauge fields coupled to massless monopoles. The weak coupling limit of this theory is equivalent to the low-energy N super Yang Mills theory in the strongly coupled region of field space []. On the other hand, a twisted version of supersymmetric theories is applicable to a classification of four-dimensional manifolds [3, 4], as well as three-dimensional ones [5, 6]. New topological invariants defined by moduli space of Seiberg Witten equations were found [1, 4]. The particular solutions of Seiberg Witten equations reduced to three-dimensional flat space R 3 were constructed in [7 9]. Furthermore, the Freund equations were proposed, which differ from Seiberg Witten ones in the sign of the quadratic term [9]. This paper is devoted to a construction of group-invariant particular solutions of the Seiberg Witten equations and those of Freund. We also include into consideration zero modes of the three-dimensional massless Dirac operator without magnetic fields see, for instance, [10]. 0305-4470/03/0951+10$30.00 003 IOP Publishing Ltd Printed in the UK 51

5 D V Aleynikov and E A Tolkachev. Preliminaries Consider the equations which include a two-component spinor field ψ and a vector potential A in R 3 : ɛψ σ l ψ H l σ l l ia l ψ 0 ε lkm k A m H l 1 where ɛ ±1, 0; l,k,m 1,, 3; σ l are the Pauli matrices. For ɛ 1, these equations will be the Seiberg Witten ones. In the case ɛ 1, we will think of the equations 1 asthe Freund equations. For ɛ 0, we have the particular case of the zero-mode problem mentioned above. The solutions of the second equation 1 arethezero modes of the three-dimensional massless Dirac operator. The general ansatz of these which depends upon a three-dimensional vector function was constructed in [11]: ψ 1 f + fn3 A [ f] f + f fn 3 f f + fn3 fn 1 +ifn arctan fn fn 1 0 f 3 where f f ; n 1, n and n 3 are arbitrary orthogonal unit vectors in R 3.Strictly speaking, we ought to define the functions ψ and A on two overlapping regions of space. However, for our purpose, it is quite enough to deal with solutions and3 locally. If we substitute the solution intothe first equation of 1 andthesolution 3 into the third equation of 1, we obtain the second-order nonlinear differential equations in three independent and three dependent variables: f + Ɣf +ɛf f 0 f 0 4 f where Ɣf [ f [ f]] f 3 [ f 1 f ] f [ f 1 f 3 ]+f 1 [ f f 3 ] in short, we denote f 1 fn 1, f fn and f 3 fn 3 ; r is the independent vector and f fr is the dependent one. 3. Lie symmetries One ofthe most useful methods for determining particular explicit solutions to partial differential equations 4istoreducethemtoordinarydifferential equations which are invariant under some subgroups of the maximal Lie symmetry group of equations 4. For this purpose we need to construct the infinitesimal Lie algebra generators. Using the methods described in [1] gives see appendix A L [r r ] [f f ] 5 P r 6 K rr r r r 4rf f +[r[f f ]] 7 Here D r r f f. 8 r n 1 x + n y + n 3 f n 1 + n + n 3. z f 1 f f 3

Liesymmetries and particular solutions of Seiberg Witten equations in R 3 53 The commutation relations of generators 5 8are [L i,k j ] ε ijk K k [P i,k j ] δ ij D + ε ijk L k [K i,k j ] [L i,d] 0 [K i,d] K i [P i,d] P i. So, we have Lie algebra of the SO4, 1 group conformal group of three-dimensional Euclidean space. Generators L, P, K and D determine infinitesimal rotations, shifts, inversions and dilatations, respectively. Note in particular that 5 8 generate fibrepreserving transformations, meaning that the transformations in r do not depend on the coordinates f. The finite transformations of this group are rotations: r Or f Of 9 shifts: r r + b f f 10 dilatations: r expλr f exp λf 11 inversions: r f 1 rv + r v f + r r v 1 rv + r v 1 1+q [qf]+[q[qf]] q [vr] 1 vr where O is athree-dimensional orthogonal matrix; b and v are arbitrary vector parameters and λ is a scalar parameter. Suppose fr satisfies the equations 4, so does f Tgfr,whereTgis an arbitrary combination of the finite transformations 9 1 oftheso4, 1 group. For pure rotations, shifts, dilatations and inversions we obtain, respectively, f 1 TOfr OfO 1 r f TexpbPfr fr b f 3 TexpλDfr exp λfexp λr 1 1+rv + r v f 4 TexpvKfr fr + 1+q [q fr ]+[q [q fr ]] where r r + r v q [vr] 1+rv + r v 1+vr. Thus, a class of solutions can be associated with the initial particular solution. 4. Group-invariant solutions Consider, for instance, the solutions which are invariant under the SO3, SO, 1 and E subgroups of SO4, 1. These subgroups can be generated by the following combinations of vector fields 5 8: G 1 K 1 + ςp 1 / G K + ςp / G 3 L 3 13 where ς ±1, 0. In the cases ς 1andς 1, vector fields 13 generate the SO3 and SO, 1 groups, respectively. In the case ς 0, we have the E group. Usingthe methods [13], we find the group-invariant ansatz in the form see appendix B f ρ ς rn 3 3 rn 3r r + ςn 3. 14

54 D V Aleynikov and E A Tolkachev Here ρ ς and are group invariants ρ ς r + ς rn 3 rn 3 f fn 3 r rn 3. The ansatz 14 reduces the equations 4 totheordinary differential equations see appendix B for ρ ς which are compatible if ς ɛ. This implies that the Seiberg Witten equations admit a SO3-invariant solution, but do not admit a solution which is invariant under the SO, 1 or E group. A SO, 1-invariant solution satisfies the Freund equations only. For the massless Dirac operator without magnetic fields, we have a solution which is invariant under the E group. The reduced ordinary differential equations are easily solved and we get f ± rn 3r r + ɛn 3 4rn 3 3 ρɛ ɛ 3/. 15 For ɛ ς 0function15reproduces the special case of zero modes of the three-dimensional Dirac operator without magnetic field [10]. It should be noted that subgroups are determined up to inner automorphisms of the maximal symmetry group. So, a class of adjoint subgroups can be associated with a subgroup H: H ghg 1,whereg is an arbitrary element of the SO4, 1 group. Consider some adjoint subgroups to the SO3 group generated by vector fields 13with ς ɛ 1. Let the inner automorphism be generated by the element g exp π W 3 L 3, where W 3 K 3 + P 3 /. Then π exp W 3 L 3 G i exp π W 3 L 3 L i where i 1,, 3. We note that [W 3,L 3 ] 0. So, exp αw 3 L 3 expαw 3 exp αl 3 exp αl 3 expαw 3. The one-parameter finite transformations expαw 3 have the form r r 1 cos αrn 3 + r 1 sin αn 3 r +1 r 1 cos α rn 3 sin α f r +1 r 1 cos α rn 3 sin α 4 f + 1+q [qf]+[q[qf]] where 1 cos α[n 3 r] q 1 cos αn 3 r sin α. The solution which is invariant under the SO3 group generated by vector fields L can be obtained from 15: π f ±T exp W rn3 r r +1n 3 3 L 3 4rn 3 3 ρ1 1 r 3/ r. 17 3 So,wereproduce the well-known monopole-like solution [7, 8]. The next example is the SO3 group generated by vector fields exp vkl expvk L +[vk]. In this case the group-invariant solution has the form 16 r f TexpvK f TexpvK r r + vrr +[[vr]r] 3 r + vrr +[[vr]r]. 18 3

Liesymmetries and particular solutions of Seiberg Witten equations in R 3 55 Thus, the Seiberg Wittenequationsadmita class of SO3-invariantsolutions. In similar fashion we can obtain a class of SO, 1-invariant solutions satisfying the Freund equations as well as a class of E-invariant solutions which satisfy the massless Dirac equation without magnetic fields. We note in conclusion that solutions which are invariant under a two-parameter subgroups could be constructed. The most interesting example of these which satisfies the Seiberg Witten equations was constructed in [14]: 1 f 1+r [n 3r]+n 3 3 rr + 1 r n 3. This solution is invariant under a two-parameterabelian torus group generated by vector fields W 3 and L 3.Thegroup invariants are r +1 ρ r rn 3 1 r rn 3 n 3 [rf] f r rn 3 3 r 1 rn 3 rf + r +1rn 3 fn 3. Substitution of f into 1, and 3 gives 1 3 ρ 3 3 ρ 3 0. However, we were not able to construct group-invariant solutions in general, because equations 4 reduce to the system of ordinary differential equations for the three unknown functions. This system requires separate analysis. 19 5. Conclusions The conformal group of three-dimensional Euclidean space SO4, 1 is a maximal group of Lie symmetries of the Seiberg Witten and Freund equations in R 3.However, the Seiberg Witten equations admit a SO3-invariant solution, while the SO, 1-invariant solutions satisfy the Freund equations only. For massless Dirac operator without magnetic fields, we have a solutionwhich is invariant under E group. Appendix A The infinitesimal Lie group transformations have the form [1] v ξ x + η y + ζ z + ϕ + ψ + χ f 1 f f 3 where ξ, η, ζ, ϕ, ψ and χ depend on r, f. The second prolongation pr v of vector field A.1is pr v v + ϕ x + ϕ y + ϕ y + ψ x + ψ y f 1x f 1y f 1z f x f y + ψ y + χ x + χ y + χ y + ϕ xx + ϕ xy f z f 3x f 3y f 3z f 1xx + ϕ xz + ϕ yy + ϕ yz + ϕ zz + ψ xx f 1xz f 1yy f 1yz f 1zz f xx f 1xy A.1

56 D V Aleynikov and E A Tolkachev where + ψ xy + ψ xz + ψ yy + ψ yz + ψ zz f xy f xz f yy f yz f zz + χ xx + χ xy + χ xz + χ yy + χ yz + χ zz f 3xx f 3xy f 3xz f 3yy f 3yz f 3zz ϕ x D x ϕ ξf 1x ηf 1y ζf 1z + ξf 1xx + ηf 1xy + ζf 1xz ϕ x + ϕ f1 ξ x f 1x + ϕ f f x + ϕ f3 f 3x ξ f1 f 1x ξ f f 1x f x ξ f3 f 1x f 3x η x f 1y η f1 f 1x f 1y η f f x f 1y η f3 f 3x f 1y ζ x f 1z ζ f1 f 1x f 1z ζ f f x f 1z ζ f3 f 3x f 1z ϕ y D y ϕ ξf 1x ηf 1y ζf 1z + ξf 1xy + ηf 1yy + ζf 1yz ϕ z D z ϕ ξf 1x ηf 1y ζf 1z + ξf 1xz + ηf 1yz + ζf 1zz ψ x D x ψ ξf x ηf y ζf z + ξf xx + ηf xy + ζf xz χ z D z χ ξf 3x ηf 3y ζf 3z + ξf 3xz + ηf 3yz + ζf 3zz ϕ xx D xx ϕ ξf 1x ηf 1y ζf 1z + ξf 1xxx + ηf 1xxy + ζf 1xxz ϕ xy D xy ϕ ξf 1x ηf 1y ζf 1z + ξf 1xxy + ηf 1xyy + ζf 1xyz ψ xy D xy ψ ξf x ηf y ζf z + ξf xxy + ηf xyy + ζf xyz χ zz D zz χ ξf 3x ηf 3y ζf 3z + ξf 3xzz + ηf 3yzz + ζf 3zzz. Here D is the total derivative operator: D x Pr, f P x + f 1 P + f P + f 3 P P x + f 1x P f1 + f x P f + f 3x P f3 x f 1 x f x f 3 for arbitraryfunction Pr, f. It is convenient to denote the system of equations 4 by Rf,f ix,f iy,f iz,f ixx,f ixy,...f iyz,f izz 0. This system is invariant under infinitesimal transformation A.1 if pr v[rf,f ix,f iy,f iz,f ixx,f ixy,...,f iyz,f izz ] 0 A. for all solution of 4 i 1,, 3 [1]. The infinitesimal condition A. leads to the differential constraints ξ fi η fi ζ fi ϕ fi f j ψ fi f j χ fi f j 0 ϕ fi xx ϕ fi xy ϕ fi zz 0 ψ fi xx ψ fi xy ψ fi zz 0 χ fi xx χ fi xy χ fi zz 0 ϕ f1 ψ f χ f3 ϕf 1 + ψf + χf 3 f1 + f + f 3 ξ x η y ζ z ϕ f ξ y ϕ f3 ξ z ψ f1 η x ψ f3 η z χ f1 ζ x χ f ζ y ξ y η x ξ y ζ x η z ζ y ϕ f1 x ξ xx ϕ f1 y η yy ϕ f1 z ζ zz ϕ f1 x ϕ f y ϕ f3 z ψ f y ψ f1 x ψ f3 z χ f3 z χ f1 x χ f y ψ f3 x ϕ f z χ f1 y 0 ϕ f1 ξ x.

Liesymmetries and particular solutions of Seiberg Witten equations in R 3 57 The functions which satisfy these constraints are ξ a 1 z + y x / a xy a 3 xz a 4 x + a 5 y + a 6 z + a 8 η a z + x y / a 1 xy a 3 yz a 5 x a 4 y + a 7 z + a 9 ζ a 3 x + y z / a 1 xz a yz a 6 x a 7 y a 4 z + a 10 A.3 ϕ a 1 xf 1 + yf + zf 3 + a yf 1 xf + a 3 zf 1 xf 3 +a 4 f 1 + a 5 f + a 6 f 3 ψ a xf 1 +yf + zf 3 + a 1 xf yf 1 + a 3 zf yf 3 +a 4 f a 5 f 1 + a 7 f 3 A.4 χ a 3 xf 1 + yf +zf 3 + a 1 xf 3 zf 1 + a yf 3 zf +a 4 f 3 a 6 f 1 a 7 f. Substituting A.3andA.4 intoa.1gives v a 1 /K 1 a /K a 3 /K 3 a 4 D + a 5 L 3 a 6 L + a 7 L 1 + a 8 P 1 + a 9 P + a 10 P 3. Appendix B The vector fields 13are G 1 x y z + ς x + xy y + xz z xf 1 + yf + zf 3 f 1 + yf 1 xf f + zf 1 xf 3 f 3 G xy x + y x z + ς y + yz z + xf yf 1 f 1 xf 1 +yf + zf 3 f + zf yf 3 f 3 G 3 y x x y + f f 1. f 1 f Consider the matrices x y z +ς xy xz M y xy x z +ς yz y x 0 B.1 B. B.3 and M x y z +ς xy xz xf 1 + yf + zf 3 yf 1 xf zf 1 xf 3 y xy x z +ς yz xf yf 1 xf 1 +yf + zf 3 zf yf 3. y x 0 f f 1 0 The solutions which are invariant under group generated by vector fields B.1 B.3 existif rank M rank M <3. Condition B.4isequvalent to xf yf 1 xzf 3 z x y ςf 1. B.4 B.5 B.6

58 D V Aleynikov and E A Tolkachev Let x r cos θ,y r sin θ,z, f 1 f sin θ 1 cosθ + γ, f f sin θ 1 sinθ + γ,f 3 f cos θ 1.Then x sin θ r y cos θ r cosθ + γ f 1 f sinθ + γ θ +cosθ + sin θ r r θ +sinθ r cos θ sin θ 1 r γ γ f + cos θ 1 f θ 1 sin θ 1 f + cos θ 1 f cos θ 1 f 3 f sin θ 1. f θ 1 Rewrite B.1 B.3innewcoordinates, θ 1 sinθ + γ f sin θ 1 cosθ + γ + f sin θ 1 γ γ G 1 r + z ς sin θ r θ + r z + ς cos θ + r z cos θ r z r + z + ς sin θ + z cot θ 1 sinθ + γ r γ r f cos θ z cosθ + γ f θ 1 G r + z ς cos θ r θ + r z + ς sin θ + r z sin θ r z r z + ς cos θ + z cot θ 1 cosθ + γ r γ r f sin θ z sinθ + γ f θ 1 G 3 θ. B.7 B.8 B.9 Equations B.5, B.6 inthesecoordinates are The characteristic equations sin γ 0 γ 0 B.10 r z cos θ 1 z r ς sin θ 1. dz r z dr r z + ς give two group invariants r + z + ς z and dz z df f C 1 and fz C B.11

Liesymmetries and particular solutions of Seiberg Witten equations in R 3 59 respectively. Using B.10andB.11 gives xc f 1 f sin θ 1 cos θ z C 1 ς yc f f sin θ 1 sin θ z C 1 ς or where z r f 3 f cos θ 1 ς C z C 3 1 ς f 1 x z ρ C 1 f y f z 3 z x y ς z 3 C C 1 ς. We note that >0. Taking into account that f 1 x x z 3 ρ + f 1 z y f x xy z 3 ρ f y y z 3 ρ + f z z f 3 y yz ρ z 3 yz ρ z 3 ρ y f 3 z 3 z f 1 z ρ y f 3 z 3 x z ρ z 3 ρ xz ρ z 3 ρ x z 3 + 3ρ z z 3 xz ρ z 3 ρ x z 3 Equations 4reduce to the ordinary differential equations ρ ς +5ρ ρ ρ +3 ς ρ ς +ɛ ρ ς 0 B.1 ρ ς +3ρ 0. ρ B.13 Equation B.13admits the solution C B.14 ρ ς 3 where C is a positive real constant. Substituting B.14 intob.1 givesc 1andς ɛ. References [1] Witten E 1994 Math. Res. Lett. 1 769 96 [] Iga K 00 Int. J. Mod. Phys. APreprint hep-th/00771 submitted [3] Witten E 1988 Commun. Math. Phys. 117 353 86 [4] Donaldson S 1996 Bull. Am. Math. Soc. 36 45 70 [5] Taubes C 1990 J. Diff. Geom. 31 547 99 [6] Marino M and Moore G 1999 Nucl. Phys. 547 569 98 [7] Freund P G O 1995 J. Math. Phys. 36 673 4 [8] Naber G L 1999 Proc. Int. Conf. on Geometry, Integrability and Quantization Bulgaria pp 181 99 [9] Adam C, Muratori B and Nash C 000 Preprint hep-th 000315

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