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

314 L.B. Drissi et al. / Nuclear Physics B 804 [PM] (2008) 307–341Fig. 4. A generic toric web-diagram of a 4-vertex. In toric geometry, this web-diagram corresponds to the real base of thelocal patch C 4 .The defining equation of the toric CY hypersurface H 3 is given by the field equations ofmotion of both the D- and the F -auxiliary fields,{ δLδDH 3 :a = 0, a = 1,...,r(3.5)δLδF α = 0, α= 1,...,m∣ ,with m = d − 3 and d 4. The first r equations, which are similar to Eq. (3.2), reduce thedimension down to (d −r). The second equations, which are gauge invariant constraint relations,⎧f ⎪⎨ α (z 1 ,...,z d ) = 0f α (λ 1 z 1 ,...,λ d z d ) = 0, α= 1,...,m,(3.6)⎪⎩λ j = e iqa j α a∣reduce the number of free field variables down to 3; say (w 1 ,w 2 ,w 3 ). Up to solving Eqs. (3.5),one can express all the z i field variables in terms of the w’s as shown belowz i = z i (w 1 ,w 2 ,w 3 ), i = 1,...,r + d.In the next subsection, we study in details the case d = 4.3.2. Results on non-planar vertex formalismThe results we will give below concern the following:(1) the toric realization of the local degenerate elliptic curve (1.2),(2) the set up of the non-planar 3-vertex formalism and(3) the computation of the partition function Z H3 .3.2.1. Local degenerate elliptic curveConsider the local Calabi–Yau threefold (1.2) and focus on the particular local degenerateelliptic curve,H 3 = O(+3) ⊕ O(−3) → E (t,∞) , m= 3.(3.8)The degenerate elliptic curve E (t,∞) is given by the toric boundary (divisor) of the complexprojective plane P 2 ,E (t,∞) = ∂ ( P 2) .This is just a compact divisor (hyperline) of P 2 . The toric web-diagram associated to (3.8) isgiven by Fig. 5.The non-compact toric 4-fold Y 4 of Eq. (3.4) is given byY 4 = O(−3) → WP 3 1,1,1,3 ,(3.7)(3.9)(3.10)