Abstract
Motivated by recent advances in Donaldson-Thomas theory, four-dimensional mathcal{N} = 4 string-string duality is examined in a reduced rank theory on a less studied BPS sector. In particular we identify candidate partition functions of “untwisted” quarter-BPS dyons in the heterotic ℤ2 CHL model by studying the associated chiral genus two partition function, based on the M-theory lift of string webs argument by Dabholkar and Gaiotto. This yields meromorphic Siegel modular forms for the Iwahori subgroup B(2) ⊂ Sp4(ℤ) which generate BPS indices for dyons with untwisted sector electric charge, in contrast to twisted sector dyons counted by a multiplicative lift of twisted-twining elliptic genera known from Mathieu moonshine. The new partition functions are shown to satisfy the expected constraints coming from wall-crossing and S-duality symmetry as well as the black hole entropy based on the Gauss-Bonnet term in the effective action. In these aspects our analysis confirms and extends work of Banerjee, Sen and Srivastava, which only addressed a subset of the untwisted sector dyons considered here. Our results are also compared with recently conjectured formulae of Bryan and Oberdieck for the partition functions of primitive DT invariants of the CHL orbifold X = (K3 × T2)/ℤ2, as suggested by string duality with type IIA theory on X.
Highlights
Χ−101, extracting Fourier coefficients of this Sp4(Z)-Siegel modular form
The new partition functions are shown to satisfy the expected constraints coming from wall-crossing and S-duality symmetry as well as the black hole entropy based on the Gauss-Bonnet term in the effective action
The quarter-BPS index formula of [5] was generalized to dyons in N = 4 CHL orbifolds [34,35,36,37,38] in [39,40,41,42] with an appropriate Siegel modular form taking the role of χ10.3 These theories are obtained upon orbifolding heterotic strings on T 2 × T 4 by a 1/N shift along a circle in T 2 and a supersymmetry-preserving order N action on the internal CFT describing heterotic strings on T 4
Summary
For quarter-BPS dyons of unit-torsion and we expect that a finite number of discrete T-invariants provides a partition of the set (Q, P ) ∈ Λem gcd(Q ∧ P ) = 1. We remark that for any two of such disjoint charge sets Q, Q with quarter-BPS partition functions ZQ, ZQ , respectively, one can formally define the sum ZQ + ZQ. Extracting from ZQ + ZQ Fourier coefficients analogously to (2.32) in this case yields numbers for which the interpretation (2.29) does not hold, as there is no unique charge orbit (or orbit representative) given the quadratic invariants. Rather it is a sum of two BPS indices.
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