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By M. Aizenman (Chief Editor)

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21) Higher-Order Chern–Pontryagin Classes 53 which may be called the generalized Yang–Mills equations in 4m dimensions. When m = 1, it is the classical one, D(∗F ) = 0. , F (m) satisfies Eqs. 19). In the following, it will be convenient to work on the Euclidean space R4m instead of the sphere S 4m . Such a reduction is possible because, through a stereographic projection, R4m is conformal to a punctured sphere, say, S 4m \ {P }. 1), we know that the Hodge dual ∗F (m) is conformally invariant. Hence the Yang–Mills theory on R4m is identical to that on S 4m \ {P }.

Hence, it is easily seen that (69) is equivalent to  div(ϒ)n = 0    0  div(ϒ)δ = ϒ| = wgo |   h = w + ϒnn   η| = −ϒ(n, ·) (70) (71) and −τ = w 1 1 + S[∇ (ϒ(n, ·))] − div (ϒ(n, ·))go | − (∇n ϒ)| , 2 2 (72) where we replace ηn by w. ) So far our analysis has shown that ϕ − div(divϒ) = 0 ϒ =0 in M and     ∂ϕ    ∂n ϕ − ϒnn − div ϒ(n, ·) − (divϒ)n divϒ ϒ| (73) = 0 = 0 = 0 = wgo | (74) on , where w is a parameter function. Now it follows from (73) that thus integrating by parts and using the fact that (divϒ)i = O(r −δ−2 ), Dj (divϒi ) = O(r −δ−3 ) and δ > − (divϒ) = 0, 3 2 we see that divϒ ≡ 0 in M.

3) on the Poincar´e half-plane R2+ characterized by N zeros of the complex field φ. Higher-Order Chern–Pontryagin Classes 55 It is important to note that the curvature 2-form F is proportional to the interaction density terms fj k , Dj φ, and 1 − |φ|2 . In other words, F can be expressed in terms of a linear combination of the -matrices with fj k , Dj φ, and 1 − |φ|2 as coordinate-dependent coefficients [3]. 4. Asymptotic Estimates Let p1 , p2 , · · · , pN ∈ R2+ (with possible multiplicities) be the zeros of the complex field φ.

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