Abstract
We study 6d E-string theory with defects on a circle. Our basic strategy is to apply the geometric transition to the supersymmetric gauge theories. First, we calculate the partition functions of the 5d SU(3)0 gauge theory with 10 flavors, which is UV-dual to the 5d Sp(2) gauge theory with 10 flavors, based on two different 5-brane web diagrams, and check that two partition functions agree with each other. Then, by utilizing the geometric transition, we find the surface defect partition function for E-string on ℝ4 × T2. We also discuss that our result is consistent with the elliptic genus. Based on the result, we show how the global symmetry is broken by the defects, and discuss that the breaking pattern depends on where/how we insert the defects.
Highlights
Between toric Calabi-Yau manifold and 5-brane web [14], we can translate the geometry into brane set up and vice versa
The defect of the E-string that we considered naturally has an SO(16) which can be understood as SO(20) global symmetry of the unHiggsed theory is broken to SO(16) along with the defect Higgsing
We referred to the one with O5-planes as Sp(2) gauge theory with 10 flavors just to distinguish natural 5-brane web, without orientifolds, for SU(3) gauge theory with 10 flavors
Summary
A Tao web diagram for 5d N = 1 SU(3)0 gauge theory with 10 flavors is depicted in figure 1(a) Though this 5-brane configuration, in principle, has infinitely many Kähler parameters, they are not independent due to the periodic structure of the web diagram, and only 13 parameters corresponding to the two Coulomb moduli parameters, ten mass parameters, and one instanton factor are independent. With the fundamental configuration as a strip-like diagram given, one performs the topological vertex computation with the following additional procedures: (i) the framing factor associated with a 5-brane crossing an O5-plane needs to be shifted by 1. Based on the 5-brane configuration, the partition function for 5d Sp(2) gauge theory with 10 flavors is expressed in terms of a Young diagram sum over μ1, μ2, μ3, as. Where Fn({A}, {M }) are exactly the same as those obtained from Tao diagram (2.8a), (2.8b), and (2.8c) since they are the dual to each other, the partition functions ZSU(3)+10F and Z Sp(2)+10F are derived from completely different diagrams. Due to computational complication, we only presented terms of quadratic order in q, but can be checked the equivalence to higher orders
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