No more walls! A tale of modularity, symmetry, and wall crossing for 1/4 BPS dyons

Abstract

Abstract:We determine the generating functions of 1/4 BPS dyons in a class of 4d N = 4 string vacua arising as CHL orbifolds of K3×T 2 , a classification of which has been recently completed. We show that all such generating functions obey some simple physical consistency conditions that are very often sufficient to fix them uniquely. The main constraint we impose is the absence of unphysical walls of marginal stability: discontinuities of 1/4 BPS degeneracies can only occur when 1/4 BPS dyons decay into pairs of 1/2 BPS states. Formally, these generating functions in spacetime can be described as multiplicative lifts of certain supersymmetric indices (twining genera) on the worldsheet of the corresponding nonlinear sigma model on K3. As a consequence, our procedure also leads to an explicit derivation of almost all of these twining genera. The worldsheet indices singled out in this way match precisely a set of functions of interest in moonshine, as predicted by a recent conjecture.

The holographic landscape of symmetric product orbifolds

“…Further details of this example can be found in [41]. Generalizations of this constructions to CHL models are discussed in [43][44][45], which correspond to orbifolds acting only on K3.…”

Some comments on symmetric orbifolds of K3

“…Therefore, the twining genera of the symmetric orbifold Sym n (C) are completely determined in terms of the twining genera of the 'seed' K3 model C. The complete list of the possible twining genera for a K3 model C can be found in [11]. The twining genera φ g can be written as…”

“…Both Ψ g and Φ g can be defined explicitly in terms of infinite products, whose exponents depend on the Fourier coefficients of φ C K3 g ′r for all powers g ′r . The set of possible twining genera of NLSM on K3 are known [10,11] (although, for some symmetries g, there are more than one 'candidate' twining genus, and it is not known which one is actually realized in a given NLSM, see [11]), so the computation can be effectively performed. Standard arguments (reviewed in section 3.3) show that a twining genus φ K3 [n] g is invariant under deformations of the moduli that preserve the symmetry g, i.e.…”

“…In fact, this argument might be used to prove that the two meromorphic Siegel forms are the same up to a constant. Indeed, the location of the poles is strongly constrained by wall crossing [11], so it is not unlikely that the two forms have exactly the same poles. If this is the case, then the difference must be a holomorphic Siegel form of weight zero, which is necessarily a constant.…”

Calabi-Yau manifolds and sporadic groups

“…Since the elliptic genus only counts BPS states of the theory, one might have hoped that the BPS states have M 24 as symmetry group, but the full spectrum has a different symmetry group. By calculating explicit twined elliptic genera [18,19] one finds that there are more twined elliptic genera for K3 spaces than conjugacy classes of M 24 . The symmetry groups of the K3 manifold at different points in moduli space have found a beautiful string theory explanation in [20], where the authors study type IIA string theory on K3×T 3 .…”