Refining the Shifted Topological Vertex

We study aspects of the refining and shifting properties of the 3d MacMahon function $\mathcal{C}_{3}(q)$ used in topological string theory and BKP hierarchy. We derive the explicit expressions of the shifted topological vertex $\mathcal{S}_{\lambda \mu \nu}(q)$ and its refined version $\mathcal{T}_{\lambda \mu \nu}(q,t)$. These vertices complete results in literature.Comment: Latex, 14 pages, 2 figures. To appear in Jour Math Phy

Topological String on Toric CY3s in Large Complex Structure Limit

We develop a non planar topological vertex formalism and we use it to study the A-model partition function $\mathcal{Z}_{top}$ of topological string on the class of toric Calabi-Yau threefolds (CY3) in large complex structure limit. To that purpose, we first consider the $T^{2}\times R$ special Lagrangian fibration of generic CY3-folds and we give the realization of the class of large $\mu$ toric CY3-folds in terms of supersymmetric gauged linear sigma model with \emph{non zero} gauge invariant superpotentials $)$. Then, we focus on a one complex parameter supersymmetric $U(1)$ gauged model involving six chiral superfields ${\Phi_{i}}$ with $\mathcal{W}=\mu (\prod\nolimits_{i=0}^{5}\Phi_{i})$ and we use it to compute the function $\mathcal{Z}_{top}$ for the case of the local elliptic curve in the limit $\mu \to \infty$.Comment: Latex, 38 pages, 12 figures. To appear in Nucl Phys

Generalized MacMahon G(q) as q-deformed CFT Correlation Function

Using $\Gamma_{\pm}(z)$ vertex operators of the $c=1$ two dimensional conformal field theory, we give a 2d-quantum field theoretical derivation of the conjectured d- dimensional MacMahon function G$_{d}(q)$. We interpret this function G$_{d}(q)$ as a $(d+1)$- point correlation function $\mathcal{G}_{d+1}(z_{0},...,z_{d})$ of some local vertex operators $\mathcal{O}$. We determine these operators and show that they are particular composites of q-deformed hierarchical vertex operators $_{\pm}^{(p)}$, with a positive integer p. In agreement with literature's results, we find that G$_{d}(q)$, $d\geq 4$, cannot be the generating functional of all \textit{d- dimensional} generalized Young diagrams .Comment: 35 pages, Appendix B shortened, references updated, To appear in NP