Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1163which are rewritten as the relations between the modes(1 − Y )|Q i )| ˆV 3 〉 X = (1 − Y T )|Q i )| ˆV 3 〉 X = 0. (6.267)Recalling the conditions in the Neumann case(1 + Y )|P i )|V 3 〉 X = (1 + Y T )| ¯P i )|V 3 〉 X = 0, (6.268)(1 − Y )|Q i )|V 3 〉 X = (1 − Y T )| ¯Q i )|V 3 〉 X = 0,we end up with the following equations∑∞[(1 − Y ) m0 ( ¯Z 0n + π 2 ¯X∞∑ ∑∞0n )P n,j + (1 − Y ) mn¯Z nn ′P n ′ ,j]|V 3 〉 X = 0,n=0n=0n=1n ′ =0∑ ∞[(1 − Y T ) m0 (Z 0n − i π 2 X 0n) ¯P∞∑∑ ∞n,j + (1 − Y T ) mn Z nn ′¯Pn′ ,j]|V 3 〉 X = 0n=1n ′ =0for m ≥ 0. It can be easily found out that the expressionZ mn = −i π √3(1 + Y T ) mn (m, n ≥ 0, except for m = n = 0),Z 00 = 0,satisfies the above equations. It should be noted that in this case, becauseof CY C = Ȳ ≠ −Y , it does not contain any unknown constant differentlyfrom the 4–string case.Owing to the condition (6.268) we can write the phase factor only interms of the zero-modes. Finally we have [Sugino (2000)][]| ˆV 3 〉 X 123 = exp −θij4 √ 3α P ¯P ′ 0,i 0,j |V 3 〉 X 123[]= exp i θij12α ′ (p(1) 0,i p(2) 0,j + p(2) 0,i p(3) 0,j + p(3) 0,i p(1) 0,j ) |V 3 〉 X 123. (6.269)It is not clear whether the solutions we have obtained here are uniqueor not. However we can show that the phase factors are consistent with therelations between the overlaps which they should satisfy,3〈Î| ˆV 3 〉 123 = | ˆV 2 〉 12 , 4〈Î| ˆV 4 〉 1234 = | ˆV 3 〉 123 , 34〈 ˆV 2 || ˆV 3 〉 123 | ˆV 3 〉 456 = | ˆV 4 〉 1256 ,by using the momentum conservation on the vertices (p (1)i + · · · +p (N)i )| ˆV N 〉 X 1···N = 0. Furthermore we can see that the phase factors successfullyreproduce the Moyal product structures of the correlators among