Refined topological vertex with ON-planes

We propose refined topological vertex formalism for 5-brane systems with ON-planes by introducing a new vertex associated with reflection over an ON-plane, which gives rise to new vertex and edge factors. We test our proposal against various 5d N = 1 gauge theories which can be realized as 5-brane webs with ON-planes, which include D-type quiver theories. In particular, we compute the refined partition functions for 6d E-string theory on a circle as well as 5d SU(3) theory at the Chern-Simons level 9, which can be realized as 5-brane webs with two ON-planes. Our results completely agree with the known results.

Automated summary

This paper proposes a refined topological vertex formalism for 5-brane systems with ON-planes by introducing a new vertex associated with reflection over an ON-plane, resulting in new vertex and edge factors. The proposal is tested against various 5d N = 1 gauge theories, including D-type quiver theories, which can be realized as 5-brane webs with ON-planes. The refined partition functions for 6d E-string theory on a circle and 5d SU(3) theory at the Chern-Simons level 9, which can be realized as 5-brane webs with two ON-planes, are computed and found to be in complete agreement with known results.