Ivancevic_Applied-Diff-Geom

1240 Applied Differential Geometry: A Modern Introductionto the action (6.347). We assume for simplicity that there is only a rankone gauge field; the extension to higher rank is straightforward. Comparing(6.347) and (6.355), we see that a constant B−field can be replaced by thegauge field A i = − 1 2 B ijx j , whose field strength is F = B. When we areworking on R n , we are usually interested in situations where B and F areconstant at infinity, and we fix the ambiguity be requiring that F is zeroat infinity.Naively, (6.355) is invariant under ordinary gauge transformations asδA i = ∂ i λ, (6.356)because (6.355) transforms by a total derivative∫∫∫δ dτA i (x)∂ τ x i = dτ∂ i λ∂ τ x i =dτ∂ τ λ.However, because of the infinities in quantum field theory, the theory hasto be regularized and we need to be more careful. We will examine a pointsplitting regularization, where different operators are never at the samepoint.Then expanding the exponential of the action in powers of A and usingthe transformation law (6.356), we find that the functional integral transformsby∫−∫dτA i (x)∂ τ x i ·dτ ′ ∂ τ ′λ, (6.357)plus terms of higher order in A. The product of operators in (6.357) canbe regularized in a variety of ways. We will make a point-splitting regularizationin which we cut out the region |τ − τ ′ | < δ and take the limitδ → 0. Though the integrand is a total derivative, the τ ′ integral contributessurface terms at τ − τ ′ = ±δ. In the limit δ → 0, the surface termscontribute∫−∫= −dτA i (x(τ))∂ τ x i (τ) ( λ(x(τ − )) − λ(x(τ + )) )dτ (A i (x) ∗ λ − λ ∗ A i (x)) ∂ τ x i .Here we have used the relation of the operator product to the ∗ product,and the fact that with the limit boundary propagator [Seiberg and Witten(1999)]〈x i (τ)x j (0)〉 = i 2 θij ɛ(τ),