Higgsing towards E-strings

Abstract

We explore 6d (1, 0) superconformal field theories with SU(3) and SU(2) gauge symmetries which cascade after Higgsing to the E-string theory on a single M5 near an E8 wall. Specifically, we study the 2d $$ \mathcal{N} $$ N = (0, 4) gauge theories which describe self-dual strings of these 6d theories. The self-dual strings can be also viewed as instanton string solitons of 6d Yang-Mills theories. We find the 2d anomaly-free gauge theories for self-dual strings, amending the naive ADHM gauge theories which are anomalous, and calculate their elliptic genera. While these 2d theories respect the flavor symmetry of each 6d SCFT only partially, their elliptic genera manifest the symmetry fully as these functions as BPS index are invariant in strongly coupled IR limit. Our consistent 2d (0, 4) gauge theories also provide new insights on the non-linear sigma models for the instanton strings, providing novel UV completions of the small instanton singularities. Finally, we construct new 2d quiver gauge theories for the self-dual strings in 6d E-string theory for multiple M5-branes probing the E8 wall, and find their fully refined elliptic genera.

Cardy limits of 6d superconformal theories

“…Here, Z n is the elliptic genus of the self-dual strings with charge n. The elliptic genus can be computed from the 2d (0,4) gauge theories with gauge group U(n [40,56,57]. The fully refined elliptic genus can be computed from the following contour integral,…”

Elliptic quantum curves of class $$ {\mathcal{S}}_k $$

“…The 16 supersymmetries can be decomposed in representations of (SU(2) l , SU(2) r , SU(2) I ) ±± , where the two "±" label the chirality of world-sheet x 0 , x 1 and space along x 6 . The supersymmetries preserved by the NS5-D6-D2 brane system transform as (1, 2, 2) −+ , such that the D2 world-volume theory is a 2d N = (0, 4) quiver theory, see for instance [42]. The brane configuration allows one to read off the field content and the charges of the supermultiplets with respect to SU(2) l × SU(2) r × SU(2) I , labelled as (α,α, A).…”

“…From the decomposition, one can read off the charges of these N = (0, 2) multiplets as summarised in table 2. This 2d quiver gauge theory is known from [42] and reduces to the N = (0, 4) gauge theory description for M-strings introduced in [17] for case k = 1. For completeness, the 2d quiver gauge theory with multiplets (2.8) can be written as Table 2.…”

“…As expected from the duality between the 5d SU(2)×SU(2) affine quiver gauge theory and the 6d SCFT for 2 M5-branes on A 1 singularity, the full partition function for the 5d SU(2)×SU(2) affine quiver theory coincides with the partition function of the 6d SCFT given in section 2.1. Namely, 42) with the identification of the 5d/6d fugacities as…”

Bootstrapping ADE M-strings

“…Also, if one works out the 2d quiver gauge theories living on the self-dual strings, one can apply localization technique through the Jeffrey-Kirwan residue prescription [22,23]. In [19,[24][25][26][27][28][29][30], this was successfully carried out to tackle E-strings, M-strings, some minimal SCFTs [25] and many more. From the point of view of topological strings, traditional methods such as the topological vertex or mirror symmetry are not that straightforward to apply.…”