Inverse algorithm and M2-brane theories

Abstract

Recent paper arXiv:1103.0553 studied the quiver gauge theories on coincident M 2 branes on a singular toric Calabi-Yau 4-folds which are complex cone over toric Fano 3-folds. There are 18 toric Fano manifolds but only 14 toric Fano were obtained from the forward algorithm. We attempt to systematize the inverse algorithm which helps in obtaining quiver gauge theories on M 2-branes from the toric data of the Calabi-Yau 4-folds. In particular, we obtain quiver gauge theories on coincident M 2-branes corresponding to the remaining 4 toric Fano 3-folds. We observe that these quiver gauge theories cannot be given a dimer tiling presentation.

Embedding and partial resolution of complex cones over Fano threefolds

“…We succeeded in finding the toric diagram of Fano P 3 embedded inside the toric diagrams of Fano B 2 and Fano B 3 . Moreover, performing partial resolution of the Fano B 2 and Fano B 3 , whose quiver Chern-Simons theories are known in [38], we also obtained the quiver Chern-Simons theory for Fano P 3 [42]. This quiver gauge theory for Fano P 3 matched with that obtained in [38], thus justifying the result of [38].…”

“…The quiver Chern-Simons theories corresponding to 14 of these smooth toric Fano threefolds were obtained in [37] using the forward algorithm approach. The quiver Chern-Simons theories for the remaining four toric Fanos, i.e., Fanos P 3 , B 1 , B 2 and B 3 was obtained in [38] by analyzing the patterns of Q F , Q D charge matrices and applying inverse algorithm. There is another quiver Chern-Simons theory known for Fano B 1 which was obtained in [39] and is different from the theory obtained in [38].…”

Is toric duality a Seiberg-like duality in (2 + 1)-d ?

“…In particular, we have focused on Fanos C 3 , C 5 and B 2 , which are the only Fanos in the literature, known to admit phases [18,41]. We try to find out whether these phases are also Seiberg-like duals.…”

“…The phases of C 5 have the same quiver diagram and superpotential as that of Fano C 3 but with different Chern-Simons levels [41]. Besides these two Fanos, we also have Fano B 2 whose phases [18] are shown in figure 5. In this paper, we will evaluate the indices for the phases of Fanos C 5 and B 2 to check for the Seiberg-like duality.…”

“…This theory was obtained in [18] and it does not admit a dimer tiling description. The theory has 2 toric phases whose quivers are shown in figure 5(a) and figure 5(b) respectively.…”