By Jürgen Jost
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Additional resources for Harmonic Maps Between Surfaces (with a Special Chapter on Conformal Mappings)
We sum up our discussion with the basic definition. Definition 7. A noncommutative spin geometry is a real spectral triple (A, H, D; , J ) or (A, H, D; J ), according as its dimension is even or odd, that satisfies the seven axioms set out above. Riemannian spin manifolds provide the commutative examples. It is not hard to manufacture noncommutative examples with finite-dimensional matrix algebras , ; these are zero-dimensional geometries in the sense of Axiom 1. In the next chapter we study a more elaborate noncommutative example which, like the Riemann sphere, has dimension two.
Also, the coefficient of logarithmic divergence is / −2 ) σN (D = 2. N →∞ log N / −2 = lim −D As we shall see later on, this coefficient is 1/2π times the area for any 2-dimensional surface, so the area of the sphere is hereby computed to be 4π . 3 The first-order condition Axiom 2 (Order one). For all a, b ∈ A, the following commutation relation holds: [[D, a], J b∗ J † ] = 0. 4) This could be rewritten as [[D, a], b ] = 0 or as [[D, π(a)], π (b)] = 0. 2) and the Jacobi identity, we see that this condition is symmetric in the representations π and π , since [a, [D, b ]] = [[a, D], b ] + [D, [a, b ]] = −[[D, a], b ] = 0.
Now the Eisenstein series G2k (τ ) := m,n 1 , (m + nτ )2k with primed summation ranging over integer pairs (m, n) = (0, 0), converges absolutely for k > 1 and only conditionally for k = 1. We shall see below that −2 in fact diverges logarithmically, thereby establishing the two-dimenm,n |m+nτ | sionality of the geometry. The orientation cycle. In terms of the generators u = e2π iφ1 , v = e2π iφ2 of A0 , the volume form on the torus T2 is dφ1 ∧ dφ2 = (2π i)−2 u−1 v −1 du ∧ dv, with the corresponding Hochschild cycle: (2π i)−2 v −1 u−1 ⊗ u ⊗ v − u−1 v −1 ⊗ v ⊗ u .
Harmonic Maps Between Surfaces (with a Special Chapter on Conformal Mappings) by Jürgen Jost