Abstract
We present an explicit expression for the grand potential of the U(N )3 superconformal Chern-Simons theory with the Chern-Simons levels being (k, 0, −k). From the viewpoint of the Newton polygon, it is expected that the grand potential is given by the free energy of the topological string theory on the local D5 del Pezzo geometry, though the explicit identification was a puzzle for years. We show how the expectation is realized explicitly. As a bonus, we can also study the {mathbb{Z}}_2 orbifold of this theory and find the grand potential is now given in terms of the local E7 del Pezzo geometry.
Highlights
|N2 − N1| fractional M2-branes on the target space geometry C4/Zk is described by the N = 6 superconformal Chern-Simons theory with the gauge group U(N1)k×U(N2)−k and two pairs of bifundamental matters where the subscripts (k, −k) denote the Chern-Simons levels
From the viewpoint of the Newton polygon, it is expected that the grand potential is given by the free energy of the topological string theory on the local D5 del Pezzo geometry, though the explicit identification was a puzzle for years
We found that the free energy of the topological string theory (1.1) unifies the moduli space of the rank deformations of these two dual models with the six Kahler parameters of the local D5 del Pezzo geometry
Summary
In [26] we observed that when setting all the cosine functions in the numerators of the worldsheet instantons of the (2, 1) model in (A.1) to be 1 (with the replacement of k by 2k) we correctly reproduce the worldsheet instantons of the (2, 2) model for 1 ≤ d ≤ 5. This relation is not valid any more for higher instantons, though the expressions look close. In relating this model to the (2, 2) model by changing the brane configuration, we found a non-trivial cancellation of odd instantons, which is very similar to the cancellation of the linear μeff term here
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
