On non-$L^2$ solutions to the Seiberg-Witten equations
We show that a previous paper of Freund describing a solution to the Seiberg-Witten equations has a sign error rendering it a solution to a related but different set of equations. The non-$L^2$ nature of Freund's solution is discussed and clarified and we
Onnon-L2solutionstotheSeiberg–Wittenequations
Adam,MuratoriandNash
whereΓiarethegammamatrices*satisfying{Γi,Γj}=2gijI,andDiandΓijaregivenby
1
Di= i+iAi,Γij=
DAA′MA
′
2
=0
B′+MB′M A′
MA′M
(2.3)
Wenowgiveashortsummaryoftherelevantpropertiesofthespinorformalismthatweneedhere.
WithaRiemannianmetricofsignature(+,+,+,+)the4componentsofa4-vectorva≡(v0,v1,v2,v3)arerepresentedbya2×2matrixwhichisdenotedbyvAA′andgivenby
1v0+iv3iv1+v2
vAA′=(2.4)
2iv1 v2v0 iv3ThisexpressionforvAA′canbewrittenasalinearcombinationofwhatareknownas
a
theInfeld–vanderWaerdenmatricesgAA′de nedby
ii110i0
,gAAσ,i=1,2,3(2.5)gAA′=′=
2012
whereσiaretheusualPaulimatricessothat 010 i
σ1=,σ2=,
10i0
σ3=
10
0 1
(2.6)
a
UsingthegAA′’sthelinearcombinationmentionedaboveisgivenby
a
vAA′=vagAA′
′′whereandmoregenerallyifwehaveatensorTa1a2...anitbecomesTA1A′
1A2A2...AnAn
(2.7)
a1a2an
′′=Taa...agTA1A′
12nA1A′gA2A′···gAnA′1A2A2...AnAnn
1
2
(2.8)
Inthisformalismspinorindicesareraisedandloweredwiththematrix ABde nedby 01
AB== AB(2.9)
10
