BKM Lie superalgebras from counting twisted CHL dyons

TL;DR

This paper explores the connection between twisted BPS state counting in CHL models and the structure of BKM Lie superalgebras, revealing a classification of these algebras via Siegel modular forms linked to string theory phenomena.

Contribution

It establishes a novel link between twisted BPS state generating functions and dd-modular forms, showing they correspond to Weyl-Kac-Borcherds denominator formulas of BKM Lie superalgebras.

Findings

Constructed Siegel modular forms for twisted BPS states

Identified dd-modular forms as BKM Lie superalgebra denominators

Mapped walls of marginal stability to Weyl chambers

Abstract

Following Sen[arXiv:0911.1563], we study the counting of (`twisted') BPS states that contribute to twisted helicity trace indices in four-dimensional CHL models with N=4 supersymmetry. The generating functions of half-BPS states, twisted as well as untwisted, are given in terms of multiplicative eta products with the Mathieu group, M_{24}, playing an important role. These multiplicative eta products enable us to construct Siegel modular forms that count twisted quarter-BPS states. The square-roots of these Siegel modular forms turn out be precisely a special class of Siegel modular forms, the dd-modular forms, that have been classified by Clery and Gritsenko[arXiv:0812.3962]. We show that each one of these dd-modular forms arise as the Weyl-Kac-Borcherds denominator formula of a rank-three Borcherds-Kac-Moody Lie superalgebra. The walls of the Weyl chamber are in one-to-one correspondence with the walls of marginal stability in the corresponding CHL model for twisted dyons as well as untwisted ones. This leads to a periodic table of BKM Lie superalgebras with properties that are consistent with physical expectations.