Abstract
We consider the spectrum of BPS states of the heterotic sigma model with (0, 8) supersymmetry and T8 target, as well as its second-quantized counterpart. We show that the counting function for such states is intimately related to Borcherds’ automorphic form Φ12, a modular form which exhibits automorphy for O(2, 26; ℤ). We comment on possible implications for Umbral moonshine and theories of AdS3 gravity.
Highlights
JHEP10(2017)121 automorphic form for the modular group SP (2, Z)
We consider the spectrum of BPS states of the heterotic sigma model with (0, 8) supersymmetry and T 8 target, as well as its second-quantized counterpart
We show that the counting function for such states is intimately related to Borcherds’ automorphic form Φ12, a modular form which exhibits automorphy for O(2, 26; Z)
Summary
The hero of our story will be the Borcherds modular form Φ12 [10]. A nice description of the relevant aspects of this form can be found in the work of Gritsenko [11], from which we borrow heavily. This function is a weakly holomorphic Jacobi form of weight zero and index one for the lattice N. The Niemeier and Leech points define cusps in the domain of definition of Φ12 These formulae should be thought of as expansions of the modular form around the cusps. These Jacobi forms, for suitable choices of δ, are the BPS counting functions discussed in [9] That is, they control the coefficients in the expansion of a certain “F 4” term in the low-energy effective action of heterotic string compactification to three dimensions, when the moduli are deformed a slight distance away from a point with Niemeier symmetry (the enhanced symmetry point itself having singular couplings). For the group SO+(Lδ), where Lδ is the lattice of signature (2,3) and with quadratic form
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