By D. E. Lerner, P. D. Sommers

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**Extra resources for Complex Manifold Techniques in Theoretical Physics (Research Notes in Mathematics) **

**Sample text**

Thus the reality of the expression (30) is quaranteed and qij contains k(k+l)/2 real parameters for a total of 25 2 3 k + 2 k--consistent with out previous result since the O(k) redundancy has not been removed. similar result can also be obtained for O(n), 4n~k, and SU(n), A 2n~k. For fixed n and general k it is not yet known how to write down the general solution. However, the 5k parameter solutions of 't Hooft can be easily obtained in this language. (the square root of the i th Choose the qli to be real and positive instanton scale) and bij to be diagonal (bii is the position of the i th instanton).

3 The reality of the kxk matrix (30) imposes lk(k-1) conditions leaving 4nk + ~(k+7) parameters. -G :-1) (n(2n+l) parameters). (31) Thus there remain 4(n+l)k-n(2n+l) meaningful parameters--the numbe= given previously [5] for general Sp(n) self-dual solutions when k>n. When k~n, the above counting must be modified since the nxn symplectic matrix G acts on only kxn independent columns of qij' Consequently q will be unaffected by a Sp(n-k) subgroup of Sp(n), yielding ij an additional (n-k)(2n-2k+l) parameters for a total of 2k 2 + 3k.

Holomorphic section i and let V(x) be the bundle of vertical vectors re- stricted to the submanifold A vector vET X a metric on T • X decompose T i. So V(x) is the normal bundle of in X corresponds to a section~ of V(i). i in X. We want to define To do so, it is sufficient (cf. 2) X and to define symplectic forms £ = V{x) 8 sphere i). Put W X Riemann Suppose that x E X corresponds to the X and £ 1 on S X X and S' respectively. X L(-1) (where L(-1) is-thought of as a bundle on the Thus W 8 L{l).