Contributions à l'Etude du Vertex Topologique en Théorie ... - Toubkal

330 L.B. Drissi et al. / Nuclear Physics B 801 [FS] (2008) 316–345or equivalently likeΩ − (1) =( −1 ∏t=−∞(Ψ− (1) ) q L 0( ∞)∏Ω + (1) = q L 0Ψ + (1) .t=0),(4.19)To check that these relations solve indeed the above constraint equations, let us give some explicitdetails.4d caseFirst consider the 4d-partition function Z 4d expressed in (4.15) which we rewrite by substituting(4.19) as follows,( −1)∏Z 4d =〈0|Ψ + (1)q L 0Ψ − (1)q L 0|0〉.(4.20)t=−∞By help of Eq. (4.14), it reads also like(∏ ∞(Z 4d =〈0|Ψ + (1) Ψ − ql )) |0〉.l=1Then commuting Ψ − (q l ) to the left by using the identity[ ∞()]∏ 1Ψ + (1)Ψ − (x) =(1 − xq k−1 ) k Ψ − (x)Ψ + (1), x < 1,k=1 l=1k=1see also Appendix A Eqs. (A.6) for general case, we get∞∏ ∞∏()1∞∏ s∏(Z 4d =(1 − q l+k−1 ) k =s=1 k=1)1(1 − q s ) kwhich, up on using ∑ sk=1 k = s(s+1)2, can be also put in the formZ 4d =∞∏(s=11(1 − q s ) s(s+1)2),(4.21)(4.22)(4.23)(4.24)that should be compared with Eq. (4.17). Notice that like for Z 3d , the 4d partition function canbe expressed in different, but equivalent, ways: We have the results⎧〈0|Ψ + (1)Ω − (q)|0〉,⎪⎨ 〈0|Ω + (q 1 Z 4d =)Ψ −(1)|0〉,(4.25)〈0|Γ + (1)Υ − (q)|0〉,⎪⎩〈0|Υ + (q 1 )Γ −(1)|0〉,where we have set