5d E n Seiberg-Witten curve via toric-like diagram

Abstract

We consider 5d Sp(1) gauge theory with E N f +1 global symmetries based on toric(-like) diagram constructed from (p, q)-web with 7-branes. We propose a systematic procedure to compute the Seiberg-Witten curve for generic toric-like diagram. For N f = 6, 7 flavors, we explicitly compute the Seiberg-Witten curves for 5d Sp(1) gauge theory, and show that these Seiberg-Witten curves agree with already known E 7,8 results. We also discuss a generalization of the Seiberg-Witten curve to rank-N cases.

Topological vertex for Higgsed 5d T N theories

“…The factorisation of the partition function of the Sp(N ) gauge theories with massless anti-symmetric hypermultiplet is consistent with the fact that the Seiberg-Witten curves of the theories also factorise [43,44]. Although the product structure of the Seiberg-Witten curves imply that the prepotential is written by the sum of N copies of Sp(1) prepotential, our analysis shows that the full Nekrasov partition function itself also factorises.…”

Discrete theta angle from an O5-plane

“…The corresponding SU(2) brane configurations show a clear difference as in figure 1. The Seiberg-Witten curves for the E 1 theory and the E 1 theory hence show clear differences as written in [20,21]. Expressed in a brane web diagram with an O5-plane, on the other hand, the E 1 and E 1 configurations seem indistinguishable.…”

“…For example, overall rescaling is used to fix the constant to be 1, and a coordinate rescaling of t is used to fix the coefficient of t 2 to be 1. Note that asymptotic behaviors at large w determine the coupling of the theory or equivalently the instanton factor q [21]. We extrapolate the configuration in the large t ∼ w 3 region, which is originally the (3, 1) 5-brane, and that in the large t −1 ∼ w 3 region, which is originally the (3, −1) 5-brane, to the w = 1 axis and identify the distance between them as q −2 so that q is the same instanton factor for the 5d SU(2) theory [24].…”

“…It is also instructive to write down the corresponding toric-like diagram (also known as dot diagram) [21,27] which becomes largest at some region in (x 4 , x 6 ) space while white dot corresponds to A (m) k,l which does not become largest in any region in (x 4 , x 6 ) space. Therefore, each face of the 5-brane web corresponds to a black dot.…”

Equivalence of several descriptions for 6d SCFT

“…For instance, a suitable movement of 7-branes in the (x 6 , x 5 )-plane or the (p, q)-plane [9, 10] converts a naive brane configuration for such 5d theories given in 9(a) into a Tao web diagrams shown in figure 9(b). When the web diagram is reinterpreted [26] as a "toric-like" diagram [7,27], the distances between the branes are converted into the Kähler parameters of the corresponding 2-cycles in the Calabi-Yau geometry. Especially, the period of such spiral rotation in the Tao web diagram is expressed in terms of the Kähler parameters which is precisely the instanton factor q (2.24) obtained from the brane configuration in the previous section…”