On the Higgsing and UnHiggsing of Fano 3-folds

Abstract

We study the Higgsing and UnHiggsing of M2-brane theories that probe cones over smooth toric Fano 3-folds, via brane tilings. We find many new examples of M2-brane gauge theories not studied previously in the literature, including those that do not correspond to cones over Fano varieties. Our analysis also provides, upon unHiggsing the known Fano varieties, new examples of quiver gauge theories that describe the same toric variety, but with external point multiplicities. As a byproduct of our results, we discuss an example of a CY 4-fold that does not have a tiling description, and study the Higgsing of this theory. *

Embedding and partial resolution of complex cones over Fano threefolds

“…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].…”

“…It is interesting to mention that we tried removing various possible points in the toric diagrams of each of the 18 Fano threefolds and were able to detect the following embeddings (some of these embeddings are already known, in the references given): [42] , Fano B 2 [42] and Fano C 5 . [39] and Fano C 4 [39].…”

“…It is an interesting exercise to obtain the embeddings of toric diagram of the complex cones over smooth toric Fano threefolds, into a larger toric diagram of some other Calabi-Yau fourfold where the corresponding quiver gauge theory for the latter is known. Some of these embeddings were discussed in [39]. Using the methods of higgsing [40,41] and unhiggsing [34], it was shown in [39], that the toric diagram of the complex cones over Fano threefolds can be embedded into Calabi-Yau fourfolds which may or may not be complex cones over Fano threefolds.…”

“…Some of these embeddings were discussed in [39]. Using the methods of higgsing [40,41] and unhiggsing [34], it was shown in [39], that the toric diagram of the complex cones over Fano threefolds can be embedded into Calabi-Yau fourfolds which may or may not be complex cones over Fano threefolds. We also made a small progress in [42], where we studied the partial resolution of Fanos B 1 , B 2 , B 3 , B 4 .…”

“…In this work, we study the embedding of complex cones over the 18 smooth toric Fano threefolds, to find if they are embedded inside the complex cones of any other Fano threefold. Note that some of these embeddings such as embedding of Fano E 1 inside Fano F 2 ; Fano B 4 inside Fano D 2 and Fano C 4 ; Fano C 3 inside Fano E 3 have been already done in [39]. Moreover, once we have obtained a possible embedding, we apply partial resolution approach to obtain the Q F and Q D charge matrices.…”