Ivancevic_Applied-Diff-Geom

1238 Applied Differential Geometry: A Modern Introductionproduct would reduce to a ∗ product; we would gete ip·x (τ)e iq·x (τ ′ ) ∼ e ip·x ∗ e iq·x (τ ′ ).This is no coincidence. If the dimensions of all operators were zero, theleading terms of operator products O(τ)O ′ (τ ′ ) would be independent ofτ − τ ′ for τ → τ ′ , and would give an ordinary associative product of multiplicationof operators. This would have to be the ∗ product, since thatproduct is determined by associativity, translation invariance, and (6.351)(in the form x i ∗ x j − x j ∗ x i = iθ ij ).Now, consider an operator on the boundary of the disc that is of thegeneral form P (∂x, ∂ 2 x, . . . )e ip·x , where P is a polynomial in derivativesof x, and x are coordinates along the Dp−brane (the transverse coordinatessatisfy Dirichlet boundary conditions). Since the second term in thepropagator (6.350) is proportional to ɛ(τ −τ ′ ), it does not contribute to contractionsof derivatives of x. Therefore, the expectation value of a productof k such operators, of momenta p 1 , . . . , p k , satisfies〈∏ k〉P n (∂x(τ n ), ∂ 2 x(τ n ), . . . )e ipn·x(τ n)= (6.352)e − i 2n=1〈 k〉∏Pn>m pn i θij p m j ɛ(τ n−τ m) P n (∂x(τ n ), ∂ 2 x(τ n ), . . . )e ipn·x(τ n)n=1G,θG,θ=0where 〈. . . 〉 G,θ is the expectation value with the propagator (6.350)parametrized by G and θ. We see that when the theory is described interms of the open string parameters G and θ, rather than in terms of gand B, the θ dependence of correlation functions is very simple. Note thatbecause of momentum conservation ( ∑ m pm = 0), the crucial factor()exp − i ∑p n i θ ij p m j ɛ(τ n − τ m )(6.353)2n>mdepends only on the cyclic ordering of the points τ 1 , . . . , τ k around thecircle [Witten (1986b); Seiberg and Witten (1999)].The string theory S−matrix can be obtained from the conformal fieldtheory correlators by putting external fields on shell and integrating overthe τ’s. Therefore, it has a structure inherited from (6.352). To be veryprecise, in a theory with N × N Chan–Paton factors, consider a k pointfunction of particles with Chan–Paton wave functions W i , i = 1, . . . , k, momentap i , and additional labels such as polarizations or spins that we will,