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}=left(0, 4right) 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.
Highlights
The diagonal subgroup SU(2)v of SU(2)R × SU(2)I, to which the chemical potential v couples, does not act chirally on the full spectrum of the CFT; it can be thought of as a chiral symmetry once we restrict to the chiral algebra underlying the elliptic genus; we find that shifting its fugacity v → q1/2/v implements a spectral flow which leads to a relation between the Ramond-Ramond elliptic genus EGn and the Neveu-Schwarz-Ramond elliptic genus EnG: EnG(mG, mF
The currents of the flavor symmetry of the chiral algebra are chiral operators of conformal weight 1, which we expect to see in the elliptic genera of the hGn CFTs; expanding the NS-R elliptic genera and extracting the states that contribute at energy (1, 0) above the NS-R vacuum,30 we find that q−
In this paper we have studied the properties of a BPS string of an arbitrary six-dimensional SCFT on the tensor branch from the point of view of the worldsheet N = (0, 4) CFT that lives on it
Summary
In this paper we take a different route and seek to reformulate the various known features of the BPS strings directly in the language of 2d conformal field theory The payoff of this approach is that we will find a very natural and uniform picture for how the global symmetries of the string are realized at the level of the CFT, which turns out to be a quite powerful asset in computing the strings’ elliptic genera. This is true first and foremost at the level of the Nekrasov partition function on the tensorial Coulomb branch, which, specializing to theories with one tensor multiplet, is given by ZNekrasov = Zpert · Zinst,.
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