Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1161easy to see that a solution of them is given by [Sugino (2000)]Z mn = −i π 2 (1 − X) mn + iβ π 2 C mn, (m, n ≥ 0, except for m = n = 0),Z 00 = iβ π 2 ,if we pay attention to (6.265). Here β is an unknown real constant, which isnot fixed by the continuity conditions alone. This ambiguity of the solutioncomes from the property of the matrix X: XC = −CX. However it willbecome clear that the term containing the constant β does not contributeto the vertex | ˆV 4 〉 X .Therefore, we have the expression of the phase (6.261)θ ij (P i |Z| ¯P j ) = θ ij [i π 2 P 0,i ¯P 0,j + iβ π 2− i π 2∞∑m,n=0∞∑(−1) n P n,i ¯Pn,jn=0P m,i (1 − X) mn ¯Pn,j ].Then recalling (6.265) again, the last term in the r. h. s. can be discarded.Also we can rewrite the term containing β− θij4α ′ β(P i|C| ¯P j ) = + θij4α ′ β(P i|X T CX| ¯P j ) = + θij4α ′ β(P i|C| ¯P j ),on |V 4 〉 X . The above formula means that the term containing β can be setto zero on |V 4 〉 X . After all, the form of the 4-string vertex becomes[]| ˆV 4 〉 X 1234 = exp − θij4α ′ P ¯P 0,i 0,j |V 4 〉 X 1234.Note that the phase factor has the cyclic symmetric form− θij4α ′ P 0,i ¯P 0,j = i θij8α ′ (p(1) 0.i p(2) 0,j + p(2) 0.i p(3) 0,j + p(3) 0.i p(4) 0,j + p(4) 0.i p(1) 0,j ),which is a property the vertices should have 22 .22 Here the momentum p (r)0,i is given by p(r)0,i = √ 2α ′ p (r)i .