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

L.B. Drissi et al. / Nuclear Physics B 804 [PM] (2008) 307–341 313This local threefold is obtained by gluing three 3-vertices. For simplicity, we consider herealso the case where there is no boundaries. The corresponding partition function reads as follows:Z X3 (q, t) = ∑C ∅μν T (q)C λμ T ∅ (q)C λ T ∅ν (q)( −e −t) |ν|+|λ|+|ν| qκ(λ)+κ(μ)+κ(ν)(2.7)withλ,μ,νκ(λ) = 2 [( ‖λ‖ 2 −|λ| ) − (∥ ∥λ T ∥ ∥ 2 − ∣ ∣λ T ∣ ∣ )] ,being the Casimir of the 2d partition.(2.8)3. Beyond the planar vertex methodFirst, we describe briefly the field theory setting of the local elliptic curve geometry leavingtechnical details for next sections. Then we give our first results regarding the topological nonplanar3-vertex formalism and the explicit expression of the partition function associated withEq. (1.2).3.1. Field theory set upThe local CY3 examples we have described above have toric web-diagrams involving planar3-vertices; see Figs. 1–3. These toric threefolds X 3 have a very remarkable field theory setup; they describe supersymmetric vacua of 2D N = 2 linear sigma model with U r (1) gaugesymmetry and (r + 3) matter multiplets Φ i ,U r (1): Φ j ≡ e iqa j Φ j , a = 1,...,r.The defining equation of X 3 is given by the field equation of motion of the D a auxiliary fields,X 3 :r+3δLδD a = ∑qi a |z i| 2 = 0,i=1where the field coordinates z i are the leading components of the chiral superfields Φ i and wherer+3∑qi a = 0,i=1a = 1,...,r,stands for the Calabi–Yau condition.Toric Calabi–Yau threefolds can be also realized as hypersurfaces H 3 in higher d dimensioncomplex Kähler toric manifolds Y d ,H 3 ⊂ Y d , d 4.Locally, the Kähler toric d-fold Y d may be imagined as given by the toric fibration R d × T d or asR d × F d with fiber F d = R × T d−1 . The toric web-diagram of Y d involve d-dimensional verticeswhere shrink all the 1-cycles of the toroidal fibration (see Fig. 4).The toric CY3 hypersurfaces H 3 have also a supersymmetric field theory setting. It will bedeveloped in details in forthcoming sections, see also the analysis of [47]. TheH 3 ’s describe aswell supersymmetric vacua of 2D N = 2 linear sigma model with gauge {V a } and matter {Φ i }multiplets, i = 1,...,r + d.(3.1)(3.2)(3.3)(3.4)