Seiberg-Witten Equations on R 8 Ayse Humeyra Bilge Department of Mathematics, Institute for Basic Sciences TUBITAK Marmara Research Center Gebze-Kocae

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1 Seiberg-Witten Equations on R 8 Ayse Humeyra Bilge Department of Mathematics Institute for Basic Sciences TUBITAK Marmara Research Center Gebze-Kocaeli Turkey E.mail : bilge@mam.gov.tr Tekin Dereli Department ofphysics Middle East Technical University Ankara Turkey E.mail : tekin@dereli.physics.metu.edu.tr Sahin Kocak Department of Mathematics Anadolu University Eskisehir Turkey E.mail : skocak@vm.baum.anadolu.edu.tr Abstract We show that there are no nontrivial solutions of the Seiberg- Witten equations on R 8 with constant standard spin c structure.

2 . Introduction The Seiberg-Witten equations are meaningful on any even-dimensional manifold. To state them let us recall the general set-up adopting the terminology of the forthcoming book by D.Salamon ([]). A spin c -structure on a 2n-dimensional real inner-product space V is a pair (W; ) where W is a 2 n -dimensional complex Hermitian space and :V!End(W) is a linear map satisfying (v) = (v); (v) 2 = kvk 2 for v 2 V.Globalizing this denes the notion of a spin c -structure : TX! End(W)ona2n-dimensional (oriented) manifold X W being a 2 n -dimensional complex Hermitian vector bundle on X. Such a structure exists i w 2 (X) has an integral lift. extends to an isomorphism between the complex Cliord algebra bundle C c (TX) and End(W)). There is a natural splitting W = W + W into the i n eigenspaces of (e 2n e 2n e ) where e ;e 2 ;;e 2n is any positively oriented local orthonormal frame of TX. The extension of to C 2 (X) gives via the identication of 2 (T X) with C 2 (X) a map : 2 (T X)!End(W) given by X X ( ij e i ^ e j)= ij (e i ) (e j ): i<j i<j The bundles W are invariant under () for 2 2 (T X). Denote () = ()j W. The map (and ) extends to : 2 (T X)C!End(W): (If 2 2 (T X) C is real-valued then () is skew-hermitian and if is imaginary-valued then () is Hermitian.) A Hermitian connection r on W is called a spin c connection (compatible with the Levi-Civita connection) if r v ( (w)) = (w)r v + (r v w) where is a spinor (section of W )v and w are vector elds on X and r v w is the Levi-Civita connection on X. r preserves the subbundles W.

3 There is a principal Spin c (2n) =fe i xj 2 R;x2Spin(2n)g C c (R 2n ) bundle P on X such that W and TX can be recovered as the associated bundles W = P Spin c (2n) C 2n ; TX = P Ad R 2n ; Ad being the adjoint action of Spin c (2n) on R 2n.We get then a complex line bundle L = P C using the map : Spin c (2n)! S given by (e i x)=e 2i. There is a one-to-one correspondence between spin c connections on W and spin c (2n) =Lie(Spin c (2n) =spin(2n)ir -valued connection--forms ^A 2 A(P ) (P; spin c (2n)) on P. Now consider the trace-part A of ^A: A = trace( ^A). 2 This is an imaginary valued -form A 2 (P; ir) which is equivariant and n satises A p (p ) = 2 ntrace() for v 2 T p P; g 2 Spin c (2n); 2 spin c (2n)(where p is the innitesimal action). Denote the set of imaginary valued -forms on P satisfying these two properties by A( ). There is a one-to-one correspondence between these -forms and spin c connections on W. Denote the connection corresponding to A by r A. A( ) is an ane space with parallel vector space (X; ir). For A 2 A( ) the -form 2A 2 (P; ir) represents a connection on the line bundle L. Because of this reason A is called a virtual connection on the virtual line bundle L =2. Let F A 2 2 (X; ir) denote the curvature of the -form A. Finally let D A denote the Dirac operator corresponding to A 2 A( ) C (X; W + )! C (X; W ) dened by D A () = 2nX i= (e i )r A;ei () where 2 C (X; W + ) and e ;e 2 ;;e 2n is any local orthonormal frame. The Seiberg-Witten equations can now be expressed as follows. Fix a spin c structure : TX! End(W) on X and consider the pairs (A; ) 2 A( ) C (X; W + ). The SW-equations read D A ()= ; + (F A ) = ( ) where ( ) 2 C (X; End(W + )) is dened by ( )() =<; > for 2 C (X; W + ) and ( ) is the traceless part of ( ). 2

4 In dimension 2n =4; + (F A )= + (F + A )=(F+ A ) (where F + is the selfdual part of F and the second equality understood in the obvious sense) and therefore self-duality comes intimately into play. The rst problem in dimensions 2n >4is that there is not a generally accepted notion of selfduality. Although there are some meaningful denitions ([2][3][4][5][6]) (Equivalence of self-duality notions in [2][3][5][6] has been shown in [7] making them more relevant as they separately are) they do not assign a well-dened self-dual part to a given 2-form. Even though + (F A ) is still meaningful it is apparently less important due to the lack of an intrinsic self-duality of 2-forms in higher dimensions. The other serious problem in dimensions 2n >4is that the SW-equations as they are given above are overdetermined. So it is improbable from the outset to hope for any solutions. We verify below for 2n = 8 that there aren't indeed any solutions. In dimension 2n = 4 it is well-known that there are no nite-energy solutions ([]) but otherwise whole classes of solutions are found which are related to vortex equations ([8]). In the physically interesting case 2n = 8 wewill suggest a modied set of equations which is related to generalized self-duality referred to above. These equations include the 4-dimensional Seiberg-Witten solutions as special cases. 2. Seiberg-Witten Equations on R 8 We x the constant spin c structure : R 8! C 66 given by (e i )= (e i ) (e i ) (e i ;i=;2; :::; 8 being the standard basis for R 8 )where (e )= (e 2 )= i i i i i i i i 3

5 (e 3 )= (e 4 )= (e 5 )= (e 6 )= i i i i i i i i i i i i i i i i 4

6 (e 7 )= (e 8 )= i i i i i i i i (We obtain this spin c structure from the well-known isomorphism of the complex Cliord algebra C c (R 2n ) with End( C n ).) In our case X = R 8 ;W =R 8 C 6 ;W =R 8 C 8 and L = L =2 = R 8 C. Consider the connection -form A = 8X i=. A i dx i 2 (R 8 ;ir) on the line bundle R 8 C. Its curvature is given by F A = X i<j F ij dx i ^ dx j 2 2 (R 8 ;ir) where F ij j. The spin c connection r = r A on W + is given by (i =; :::; 8) where : R 8! C 8. r i + A i is given by + : 2 (T X)C!End(W + ) 5

7 + (F A )= G G 2 G 3 G 4 G 5 G 6 G 7 G 2 G 22 G 23 G 24 G 25 G 26 G 7 G 3 G23 G 33 G 34 G 35 G 26 G 6 G 4 G24 G34 G 44 G 35 G 25 G 5 G 5 G25 G35 G 44 G 34 G 24 G 4 G 6 G26 G35 G34 G 33 G 23 G 3 G 7 G26 G25 G24 G23 G 22 G 2 ; where G7 G6 G5 G4 G3 G2 G G = if 2 + if 34 + if 56 + if 78 ; G 2 = F 3 + if 4 + if 23 F 24 ; G 3 = F 5 + if 6 + if 25 F 26 ; G 4 = F 7 + if 8 + if 27 F 28 ; G 5 = F 35 + if 36 + if 45 F 46 ; G 6 = F 37 + if 38 + if 47 F 48 ; G 7 = F 57 + if 58 + if 67 F 68 ; G 22 = if 2 if 34 + if 56 + if 78 ; G 23 = F 35 if 36 + if 45 F 46 ; G 24 = F 37 if 38 + if 47 F 48 ; G 25 = F 5 + if 6 if 25 + F 26 ; G 26 = F 7 + if 8 if 27 + F 28 ; G 33 = if 2 + if 34 if 56 + if 78 ; G 34 = F 57 if 58 + if 67 F 68 ; G 35 = F 3 if 4 + if 23 F 24 ; G 44 = if 2 + if 34 + if 56 if 78 : For=( ; 2 ; :::; 8 ) 2 C (X; W + )=C (R 8 ;R 8 C 8 ) ( ) = =8 P i i 2 : : : =8 P i i : : : 2 8 : : : : : : : : : : : : : : : : : : : : : 8 8 =8 P i i It was remarked by Salamon([]p.87) that + (F A ) = implies F A =. (i.e.reducible solutions of 8-dim. SW-equations are at.) It can be explicitly veried that all solutions are reducible and at: Proposition: There are no nontrivial solutions of the Seiberg-Witten equations on R 8 with constant standard spin c structurei.e. + (F A ) = ( ) (alone) implies F A =and=. Proof: Trivial but tedious manipulation with the linear system. : Acknowledgement The abovework is based on a talk given at the 5th Gokova Geometry- Topology Conference held at Akyaka-Mugla Turkey during May

8 References [] D.Salamon Spin Geometry and Seiberg-Witten Invariants (April 996 version)(to appear). [2] A.Trautman Int.J.Theo.Phys.6(977)56. [3] D.H.Tchrakian J.Math.Phys.2(98)66. [4] E.Corrigan C.Devchand D.B.Fairlie J.Nuyts Nucl.Phys.B 24(983) 452. [5] B.Grossman T.W.Kephart J.D.Stashe Commun.Math.Phys.96(984)43 Erratum:ibid(985)3. [6] A.H.Bilge T.Dereli S.Kocak Lett.Math.Phys.36(996)3. [7] A.H.Bilge Self-duality in dimensions 2n >4 dg-ga/9642. [8] C.Taubes SW! Gr J. of the A.M.S.93(996). 7