Towards M2-brane theories for generic toric singularities

Abstract

We construct several examples of (2 + 1) dimensional N = 2 supersymmetric Chern-Simons theories, whose moduli space is given by non-compact toric Calabi-Yau four-folds, which are not derivable from any (3+1) dimensional CFT. One such example is the gauge theory associated with the cone over Q 111 . For several examples, we explicitly confirm the matter content, superpotential interactions and RG flows suggested by crystal models. Our results provide additional support to the idea that crystal models are relevant for describing the structure of these CFTs.

Integrability on the master space

“…These theories are CS gauge theories and they are toric if the four dimensional CY has a U (1) 4 isometry. In [27][28][29][30] many quiver gauge theories have been conjectured to describe this class of singularity. Extending the results of [6] in three dimensions have two non trivial problems.…”

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

“…There exists an algorithm for finding the toric data of Calabi-Yau threefold from the N = 1 quiver gauge theory, which is known in the literature as forward algorithm [2]. This algorithm can be extended in the context of M 2-branes to obtain the Calabi-Yau fourfold toric data from the (2 + 1) dimensional quiver supersymmetric Chern-Simons theories with Chern-Simons levels [3][4][5][6][7]. Conversely, we could obtain the matter content and superpotential of the quiver theories from the toric data using inverse algorithm [2].…”

“…For (2+1)-d quiver theories which have (3+1)-d parents [5], the Seiberg-like dual quiver diagram will be same as in (3+1)-d. Chern-Simons levels can be then appropriately assigned to each node consistent with the rules (1.1) so that the two Seiberg-like dual theories also share the same toric data [29]. Note that for non-chiral or vector-like theories in which number of incoming and number of outgoing arrows between any two pair of nodes is same, Seiberg-like duality can be seen from the brane picture [29].…”

“…For (2+1)-d chiral quivers which do not have (3+1)-d parent [5], the situation is more subtle. For example, let us consider the phases of Q 1,1,1 theory [5] as shown in figure 3.…”

“…For example, let us consider the phases of Q 1,1,1 theory [5] as shown in figure 3. The quivers 1, 2 and 3 shown in figure 3 are all toric duals or phases of Q 1,1,1 theory.…”