Abstract
We study the BPS spectrum of E-strings in a situation where the global E_8 symmetry is broken down to D_4 + D_4 by a certain twist. We find that the refined BPS index in this setup serves as a reduced BPS index of E-strings, which gives a novel trigonometric generalization of the Nekrasov partition function for four-dimensional N=2 supersymmetric SU(2) gauge theory with N_f=4 massless flavors. We determine the perturbative part of this index and also first few unrefined instanton corrections. By using these results together with the modular anomaly equation, the genus expansion of the free energy can be computed efficiently up to very high order. We also determine the unrefined elliptic genus of four E-strings under the twist.
Highlights
In this paper we focus on a particular twist of the E-string theory
We study the BPS spectrum of E-strings in a situation where the global E8 symmetry is broken down to D4 ⊕D4 by a certain twist
We find that the refined BPS index in this setup serves as a reduced BPS index of E-strings, which gives a novel trigonometric generalization of the Nekrasov partition function for four-dimensional N = 2 supersymmetric SU(2) gauge theory with Nf = 4 massless flavors
Summary
In this subsection we briefly review the refined BPS index of E-strings. Consider the E-string theory on R5 × S1. The BPS index Zgen can be viewed as the generating function for the sequence of elliptic genera of multiple E-strings. This setting of Wilson line parameters corresponds to breaking the global symmetry E8 down to D4 ⊕ D4 This is manifestly seen in the Seiberg-Witten description: the SeibergWitten curve for the E-string theory [18] with the above parameter values reduces to the following very simple form [19]. There are several known results about Zgen expressed in terms of these Jacobi forms [14, 16, 17] These results can be immediately translated for our reduced BPS index Z by substituting (2.11). This leads to substantial simplification of the refined BPS index
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