128 M. HUTCHINGS AND C .H. TAUBES, SEIBERG-WITTEN EQUATIONS4.2. The canonical Spine structure
An w-compatible almost complex structure is a map J : T X ___, T X such
that J^2 = -1 and such that
(4.1) g(v, w) = w(v, Jw)
is a Riemannian metric. Gromov [6] observed that as long as w is nondegenerate,
the space of such J 's is nonempty and contractible. (See also [12].)Exercise 4.2. Given a nondegenerate 2-form wand a Riemannian metric g , thereexists J satisfying ( 4.1) if and only if w is self-dual with respect to g and lw I = J2.
Choose an w-compatible J , and endow X with the Riemannian metric g given
by ( 4.1). The almost complex structure gives a decompositionA *T* x @ c = EB Tp,q'
p ,q
where Tp,q = f\PT^1 •^0 @ /\ qTO,l and T^1 •^0 (resp. T^0 •^1 ) is the holomorphic (resp.
antiholomorphic) part of T* X@ C.
Suppose a Spin IC structure is chosen. Clifford multiplication by w, cl+ ( w) :
S+ ___, S+ splits S+ into ±2i eigenspaces. (In the local model of the second lecture,w = dx^1 dx^2 +dx^3 dx^4 , so this claim follows from equation (2.10).) Let Ebe the -2i
eigenspace. This is a line bundle on X. We define the canonical Spine structure tobe the one for which Eis trivial. Thus the identification Sx ___, H^2 (X; Z) sends s
to c 1 (E).
Here are some key facts about the spin bundles and the Clifford action.
Lemma 4.3. There are natural identifications
( 4 .2)
(4.3)S+ = (To,o EB T^0 •^2 )@ E = E EB (K-^1 @ E),
s_ = r^0 •^1 @E
such that the formula for Clifford multiplication by v E T * X @ C acting on a ET^0 ·• @Eis
(4.4) cl(v) ·a= v'2(v^0 ·^1 /\a - i(v^1 ,0)a).
In particular, cl( w) equals -2i on E and + 2i on K-^1 E.
(Here v^0 •^1 is the T^0 •^1 component of v, and i denotes interior product.)Proof. Starting with a Spine structure as we defined it in the second lecture,
the isomorphisms ( 4.2), ( 4.3) are given by rescaled Clifford multiplication. More
precisely we extend Clifford multiplication to a complex linear map cl : A T X @
C __, End(S+ EB S) as in (2.9). Then there is an isomorphism from K-^1 @ E to
the +2i eigenspace of w on S+ given by ~cl. (We put in the factor of ~ to make
this an isometry.) We can see that this gives the desired isomorphism because in
the local model of the second lecture,
1 - ( 0 0)
2
cl(dzdw) = _ 2
0.
This explains (4.2). The isomorphism T^0 •^1 @E __, S in (4.3) is given similarly by
~cl. It is then straightforward to check (4.4). D