LECTURE 2. SPIN AND THE SEIBERG-WITTEN EQUATIONS 119
Let's work out the second of the Seiberg-Witten equations in our local model.
If { e^1 , ... , e^4 } is the standard basis for ffi.^4 then
(2.10) cl+(el /\ e2 + e3 /\ e4) = (-2i 2i)'
cl+(e1 /\ e3 - e2 /\ e4) = (-2 2) '
cl+ ( e^1 /\ e^4 + e^2 /\ e^3 ) = ( _ 2i -2i).
If ?P = ( ~) then ?P ®?Pt = c~r I~~) and thus
q (~) =i(lal^2 - lbl^2 )(e^1 /\ e^2 + e^3 /\ e^4 ) + 2ilm(ab)(e^1 /\ e^3 - e^2 /\ e^4 )
+ 2iRe(ab)(e^1 /\ e^4 + e^2 /\ e^3 ).
Let Ffj denote the ei /\ eJ component of FA - iμ. Then the second Seiberg-Witten
equation becomes
F{ 2 + F~ 4 = 2i(lal^2 - lbi2),
F{ 3 - F~ 4 - i(F{ 4 + F~ 3 ) = 4ab.
(2.11)
Remark 2 .14. Each paper on Seiberg-Witten theory seems to have a slightly dif-
ferent version of these equations. This is due to different conventions for Clifford
multiplication and things of this nature. For example, some authors define q us-
ing the inverse of cl+ instead of the adjoint, which makes their q half as large as
ours. The conventions in different papers should be isomorphic under appropriate
rescaling.
Remark 2.15. The Seiberg-Witten equations were first written down by Witten
[20]. See [10, 13, 9] for more details.
