1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

116 M. HUTCHINGS AND C. H. TAUBES, SEIBERG-WITTEN EQUATIONS

and let h E SU(2) act on x E JR^3 by

h·x=hxh-^1.

Exercise 2.7. Check that this gives an isomorphism as in (2.4).

Now the map U(2) ---+ S0(3) sends [h, .-\] 1--+ ±h.

Finally, the fundamental representation of SO( 4) on JR^4 induces an orthogo-

nal representation of SO( 4) on A^2 1R^4 which preserves the decomposition A^2 JR^4 =

A:_JR^4 EB A~JR^4. The maps P-, P+ are given by the action of SO( 4) on A:_JR^4 , A~JR^4

respectively.

Exercise 2.8. Show that if appropriate identifications of A:_JR^4 and A~JR^4 with

JR^3 are chosen, then the diagram (2.1) commutes.

There is one more important map, not on the diagram:

(2.5)

Spinic(4) ------t U(l),

2.4. The spinor bundles

Let X be an oriented Riemannian 4-manifold with a Spine structure. Associated

to the representations S+ , s of SpinlC(4) are C^2 bundles S+, s. Sections of these

bundles are sometimes called spinors. Since the representations s+, s are unitary,
the spinor bundles S+ , S come with Hermitian metrics.
An important related bundle is the (Hermitian) line bundle L associated to the
representation (2.5).

Exercise 2.9. Show that A^2 S± = L. (This follows from a certain equality of

representations of Spin IC ( 4).)

2.5. Clifford multiplication

The spinor bundles are distinguished from arbitrary vector bundles by their relation
with the frame bundle. This relation manifests itself in the existence of Clifford
multiplication, a map

cl: T* X---+ End(S+ EB S_)

with the following properties:

• For v E T* X, cl( v) sends S+ to S_ and vice-versa.


• cl(v)^2 = -lvl^2 -

• If lvl = 1 then cl(v) is unitary.

We define cl : T X ® S+ ---+ S_ as follows. We begin by defining a map
JR^4 x C^2 ---+ C^2 : using matrix multiplication with the identification (2.3), we map
(x, 'lf;) 1--7 x'lj;. To show that this induces a map T X ® S+ ---+ S, by the def-
inition of associated vector bundle, we have to check that it commutes with the
representations of Spinic(4), i.e. that for [h, h+, ,] E Spinic(4),

(hxh+^1 )(h+,\'l/J) = h,\x'lj;.

This clearly holds. We define cl : T* X ® S_ ---+ S+ similarly: this time the map
JR^4 x C^2 ---+ C^2 sends (x, 'lj;) 1--7 -xt'lj;.

If a is a 1-form and 'lj; is a spinor then the notations cl(a)'lj;, cl( a® 'lj; ), a· 'lj; all

indicate Clifford multiplication by a on 'lj;.