118 M. HUTCHINGS AND C. H. TAUBES, SEIBERG-WITTEN EQUATIONS
acting on C^2 -valued functions. (The factor of ~ co mes from the squaring of A in
the map (2.5).) Note in particular that the spin connection is compatible with the
metric on S+·
Exercise 2.12. Show that \J A is a module derivation: if a is a 1-form and 'l/; is
a section of S+ EB S_ then
(2.7)
in 01 (X, S+ EB S_). Here r is the Levi-Civita connection.
2. 7. The Dirac operator
Definition 2.13. If A is a compatible connection on L, the Dirac operator DA
is the composition
In the local model described above,DA = ('Vi+ i'\7 2
'\73+2'\7 4
where \J j = Oj + ~ Aj. If we introduce complex coordinates z = x^1 +ix^2 , w = x^3 +ix^4
and let 'Vz = ~('\71 - i'\72), etc., then
(2.8)
2.8. The Seiberg-Witten equations
Let X be a smooth oriented Riemannian 4-manifold with a Spine structure. Fix a
self-dual 2-form μ. The Seiberg-Witten equations are
DA'l/;=0,
F,t=q('l/;)+iμwhere A is a compatible connection on L and 'l/; is a section of S+·
Here F,t is the self-dual component of FA, which is an imaginary-valued 2-form.
Also
q : S+ __, iA~T* Xis a quadratic form defined as follows. We extend Clifford multiplication to a map
A^2 ---+ End ( S +) by defining
1(2.9) cl(v /\ w) =
2
(cl(v) cl(w) - cl(w) cl(v)).We extend complex linearly to A^2 ® C and restrict to get a map
cl+ : A~® C __, End(S+)·We then define