K3 String Theory, Lattices and Moonshine

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

“…For the former two moonshines (of which Mathieu may be viewed as a special case of Umbral), twining genera associated to appropriate 4-plane preserving subgroups of Co 0 that are moreover subgroups of the Umbral groups have also been proposed in [24] (with twining genera in the M 24 case first computed in [37][38][39][40]). These Umbral genera coincide, in most cases, with those of [36] for compatible 4-plane preserving conjugacy classes; see [25] for details. Remarkably, our study singles out the proposed twining genera associated to the Mathieu, Umbral, and Conway moonshines from an infinite family of candidate Jacobi forms.…”

“…Returning to string theory on K3×T 2 , the twining genus is sufficient to determine the action of the corresponding symmetry on the sector of 1/4 BPS states. Finally, an explicit knowledge of all twining genera is expected to provide strong evidence for (or to disprove) the conjectural relations between K3 sigma models and Umbral moonshine [24,25]. Unfortunately, fundamental difficulties have, to date, prevented the determination of many twining genera, as we now describe.…”

“…An important step toward the completion of this program was made in [25], where it was proved that distinct twining genera -and, therefore, distinct CHL 1/4-BPS state counting functions -correspond to different conjugacy classes in the NLSM duality group.…”

“…Our strategy is to start from the results of [25] and, for each of the relevant classes of symmetries, to derive constraints on the twining genus both from the general properties of conformal field theory and from the full string theory on K3. General CFT arguments imply that the twining genus has suitable modular transformations (it is a weak Jacobi form of weight 0 and index 1) under certain subgroups of SL(2, Z), which were determined in [25]. The space of such Jacobi forms is always finite dimensional, so that the precise form of a twining genus can in principle be determined once a sufficient number of Fourier coefficients is known.…”

“…Thus, the first Fourier coefficient can always be computed, and this is sufficient to determine the twining genus in a few cases. This technique was already exploited in [25]. A straightforward generalization of this technique is the following.…”