1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 2. SPIN AND THE SEIBERG-WITTEN EQUATIONS 117

Remark 2.10. A Spine structure is actually equivalent to the bundle S = S_ ffiS+

together with the Clifford action. From this point of view it easy to describe the
H^2 (X; Z) action on Sx: an element a E H^2 (X; Z) sends S to S © E, where. E is
the line bundle with c 1 (E) =a.

Digression: Recall that if E is a line bundle on X, the first Chern class

c 1 (E) E H^2 (X;Z) is (at least if Xis a closed oriented manifold) the Poincare dual

of the zero set of a generic section of E. On any X , c 1 gives a bijection between
the set of isomorphism classes of line bundles on X and H^2 (X; Z).

Exercise 2.11. (Chern-Weil theory) Prove: If A is a connection on a line bundle

E , then dFA = 0 and the cohomology class [FA] equals -27ric 1 (E).

2.6. The spin connection

Let A be a connection on L compatible with the metric. This A induces a "Spine
connection" A on S+ (and S_ too).
To really explain this requires a bit more background than we have given so
far. But the idea, abstractly, is that a connection on a principal G bundle P is a
1-form with values in Lie(G), the Lie algebra of G , satisfying certain conditions,
and this induces a connection on every associated vector bundle. So to define a
connection on S+ it is enough to construct a connection on the principal Spine ( 4)

bundle F. Now the maps

S0(4) U(l)

defined previously induce an isomorphism of Lie algebras

(2.6) Lie(Spine (4)) '.::::'. Lie(S0(4)) EB Lie(U(l)).

If[, is the principal U(l) bundle corresponding to L, then we have maps

p

/~

Fr £.

The Levi-Civita connection gives a Lie(S0(4))-valued 1-form on Fr, and the con-

nection A gives a Lie(U(l))-valued 1-form on£. Now pull these back to F via the

above maps, add them, and apply the isomorphism (2.6) to get the required 1-form

on F.

Explicitly, if X is Euclidean space, with F the trivial Spine ( 4) bundle, s~

the trivial C^2 bundle, and L the trivial line bundle, then the connection A is
an imaginary-valued 1-form Ajdxj (imaginary-valued, not C-valued, because the
connection is compatible with the metric), and