Universal Features of BPS Strings in Six-dimensional SCFTs

TL;DR

This work develops a uniform, CFT-based description of BPS strings on the tensor branch of 6d SCFTs, showing that rank-one strings flow in the IR to ${\rm N}=(0,4)$ CFTs built from a nonlinear sigma model on the one-instanton moduli space and a chiral left-moving bundle encoding 6d matter. The authors demonstrate that the 6d flavor symmetry $F$ acts as a current algebra on the string, with levels determined by anomaly inflow, while the 6d gauge symmetry $G$ is realized through affine Kac–Moody characters at level $-n$ in many cases; a spectral flow relates RR and NS–R elliptic genera, simplifying spectrum constraints. They construct a modular bootstrap framework to determine the elliptic genera for many string CFTs, derive 5d Nekrasov one-instanton data from these genera, and illuminate how geometry from F-theory and M-/IIB-brane pictures underpins these universal features. The results yield new insights into 6d and 5d instanton partition functions, shed light on flavor-gauge current algebras on strings, and point to a chiral-algebra perspective on the BPS string CFTs, with several exceptional cases highlighted for future study.

Abstract

In theories with extended supersymmetry the protected observables of UV superconformal fixed points are found in a number of contexts to be encoded in the BPS solitons along an IR Coulomb-like phase. For six-dimensional SCFTs such a role is played by the BPS strings on the tensorial Coulomb branch. In this paper we develop a uniform description of the worldsheet theories of a BPS string for rank-one 6d SCFTs. These strings are the basic constituents of the BPS string spectrum of arbitrary rank six-dimensional models, which they generate by forming bound states. Motivated by geometric engineering in F-theory, we describe the worldsheet theories of the BPS strings in terms of topologically twisted 4d $\mathcal{N}=2$ theories in the presence of $1/2$-BPS 2d $(0,4)$ defects. As the superconformal point of a 6d theory with gauge group $G$ is approached, the resulting worldsheet theory flows to an $\mathcal{N}=(0,4)$ NLSM with target the moduli space of one $G$ instanton, together with a nontrivial left moving bundle characterized by the matter content of the six-dimensional model. We compute the anomaly polynomial and central charges of the NLSM, and argue that the 6d flavor symmetry $F$ is realized as a current algebra on the string, whose level we compute. We find evidence that for generic theories the $G$ dependence is captured at the level of the elliptic genus by characters of an affine Kac-Moody algebra at negative level, which we interpret as a subsector of the chiral algebra of the BPS string worldsheet theory. We also find evidence for a spectral flow relating the R-R and NS-R elliptic genera. These properties of the string CFTs lead to constraints on their spectra, which in combination with modularity allow us to determine the elliptic genera of a vast number of string CFTs, leading also to novel results for 6d and 5d instanton partition functions.

Figures (19)

• Figure 1: Schematic description of the geometric engineering of the tensor branch of a 6d SCFT in F-theory. The rational curves $\Sigma_I$ are in one-to-one correspondence with tensor multiplets, whose scalars' vacuum expectation values coincide with $\text{vol}(\Sigma_I)$. In red we have depicted the discriminant locus, corresponding to the divisor $\Delta$ of the F-theory base. In this example, we have a flavor divisor intersecting the curve $\Sigma_1$, carrying flavor symmetry $\mathfrak{f}$, while the compact curves $\Sigma_3$ and $\Sigma_4$ both support non-abelian gauge groups.
• Figure 2: Schematic description of the gauging of a (generalized) flavor symmetry in 6d SCFTs. Two models ${\mathcal{T}}_A$ and ${\mathcal{T}}_B$ that share the same flavor divisor can give rise to a consistent theory ${\mathcal{T}}_C$ by making the corresponding noncompact flavor curve (dotted blue line in the figure) into a compact one.
• Figure 3: The Higgsing tree for ${\bf 4}_{SO(8)}^{}$.
• Figure 4: The M-string Higgsing tree.
• Figure 5: The E-string Higgsing tree.