Lectures on Seiberg-Witten Invariants by John D. Moore PDF

By John D. Moore

ISBN-10: 3540614559

ISBN-13: 9783540614555

ISBN-10: 3540685928

ISBN-13: 9783540685920

Within the fall of 1994, Edward Witten proposed a collection of equations which offer the most result of Donaldson idea in a much easier manner than have been idea attainable. the aim of those notes is to supply an common creation to the equations that Witten proposed. they're directed in the direction of graduate scholars who've already taken a easy path in differential geometry and topology.

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5. THE SPIN CONNECTION 51 However, we want a connection on W. A connection dA on W is called a Spin(4)-connection if it can be expressed in terms of each local trivialization as ( d ~ z ) ~ = da~ + r where Ca is a one-form with values in the Lie algebra of Spin(4) = SU+(2) • SU_(2). Now the Lie algebra of Spin(4) is A 2 T M which is generated by e~ 9 ej for i < j. Thus the condition that dA is a Spin(4)-connection is simply Ca : E dgcxijei . e j , i

To get an idea of how this action works, consider the element A= ei~ 0 ) 0 e-ie ESU(2). Then +iz+ x+ I)(eO f e-ie(t +iz) = \e-,O(x+iy) e,oO) ~) ' a rotation through the angle 0 in the same direction in the (t, z)- and (x, y)planes. On the other hand, one checks that p(I, A) rotates the (t, z)- and (x, y)-planes in opposite directions. 1. EUCLIDEAN GEOMETRY AND THE SPIN GROUPS 41 More generally, any element A E SU(2) is conjugate to an element of the form 0 ) (o 0 e-iO , and hence there is a positively oriented orthonormal basis (el, e2, ca, e4) of V such that p(A, I) rotates the planes el A e2 and e3 A e4 through an angle 0 in the same direction, while ,o(I, A) rotates el A e2 and e3 A e4 in opposite directions.

We claim t h a t F is an isomorphism. To see t h a t F is surjective, choose a u n i t a r y connection A0 on L to serve as base point. If F is any element of g, we can write F - FAo = d a , where a is a real-valued one-form on M . T h e n d A = dAo -- i a is a u n i t a r y connection on L which has curvature - i F . To see t h a t F is injective, note t h a t if dAo - i a and dAo -- i(t are two u n i t a r y connections on L which have the same curvature, then d a = d~t. Since H 1 (M; R) = 0 by hypothesis, a - ~ = dr for some r : M -+ R such t h a t r = 0.

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