Abstract
We study a family of 2d mathcal{N}=left(0, 4right) gauge theories which describes at low energy the dynamics of E-strings, the M2-branes suspended between a pair of M5 and M9 branes. The gauge theory is engineered using a duality with type IIA theory, leading to the D2-branes suspended between an NS5-brane and 8 D8-branes on an O8-plane. We compute the elliptic genus of this family of theories, and find agreement with the known results for single and two E-strings. The partition function can in principle be computed for arbitrary number of E-strings, and we compute them explicitly for low numbers. We test our predictions against the partially known results from topological strings, as well as from the instanton calculus of 5d Sp(1) gauge theory. Given the relation to topological strings, our computation provides the all genus partition function of the refined topological strings on the canonical bundle over frac{1}{2}K3 .
Highlights
We study a family of 2d N = (0, 4) gauge theories which describes at low energy the dynamics of E-strings, the M2-branes suspended between a pair of M5 and M9 branes
The gauge theory is engineered using a duality with type IIA theory, leading to the D2-branes suspended between an NS5-brane and 8 D8-branes on an O8-plane
A basic quantity one may wish to compute for a superconformal theory is its superconformal index, which involves the computation of its partition function on S1 × S5 with suitable fugacities turned on along S1
Summary
We consider the elliptic genus of the 2d (0, 4) O(n) gauge theory, constructed in the previous section. Multiplying all these factors, one has to integrate over the continuous parameters in u and sum over disconnected sectors of flat connections. Before explaining the Jeffrey-Kirwan residues (or JK-Res) of our integrand at u = u∗, let us first note that the results of [24] apply when the pole at u∗ is ‘projective.’ The pole is called projective when all the weight vectors ρi associated with the hyperplanes meeting at u = u∗ are contained in a half space. For n ≥ 8, we explain that there start to appear degenerate poles which are multiple poles Their residues are given by theta functions and their derivatives in the elliptic parameters
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