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39) Applying the point splitting regularization method, one therefore discovers that the gauge transformation has to be modified to remain a symmetry of the quantized theory. The new gauge transformation is δˆλ Aµ = ∂µ λ + iλ ∗ Aµ − iAµ ∗ λ . 36) with n > 1 are considered [158]. The consequences for the LEEA are now as follows. A background gauge field can be seen as a coherent state of open massless strings. The LEEA should therefore be invariant under the gauge transformations which transform the background fields.

11) refers to group and spacetime ‘indices’ which means more explicitly that one has to see its argument as a matrix carrying two independent pairs of ‘indices’ (a, b) and (x, y) δFa A[A0 (x), Λ(x)] . 10) and the obvious identity Aaµ [A0 , 0] = Aa0µ now allows one to write 1 = ∆[A0 ] 3 DΛ δ F A[A0 , Λ] , ∆[A0 ] = det Strictly speaking this is only true in a perturbative analysis. δF A[A0 , Λ] δΛ . 14) 42 Noncommutative Yang-Mills (NCYM) theories This is an obvious consequence of the functional form of the identity f (x)δ(x) = f (0)δ(x).

For the disk, three vertex operator positions have to be fixed. 26 Noncommutative geometry from string theory theory is formulated in terms of the open string metric GM N . Due to momentum conservation the prefactor is invariant under cyclic permutations of the pk . Performing the integrations of the above expression over (some of) the insertion points sk then produces contributions with different phase factors if the vertex operators exchange their positions in a non cyclic way. 22) as V1 (p1 , s1 ) · · · Vn (pn , sn ) G,θ = V1 (p1 , s1 ) ∗ · · · ∗ Vn (pn , sn ) G,θ=0 .