The degeneracies of single-centered dyonic $$ \frac{1}{4} $$ 1 4 -BPS black holes (BH) in Type II string theory on K3×T2 are known to be coefficients of certain mock Jacobi forms arising from the Igusa cusp form Φ10. In this paper we present an exact analytic formula for these BH degeneracies purely in terms of the degeneracies of the perturbative $$ \frac{1}{2} $$ 1 2 -BPS states of the theory. We use the fact that the degeneracies are completely controlled by the polar coefficients of the mock Jacobi forms, using the Hardy-Ramanujan-Rademacher circle method. Here we present a simple formula for these polar coefficients as a quadratic function of the $$ \frac{1}{2} $$ 1 2 -BPS degeneracies. We arrive at the formula by using the physical interpretation of polar coefficients as negative discriminant states, and then making use of previous results in the literature to track the decay of such states into pairs of $$ \frac{1}{2} $$ 1 2 -BPS states in the moduli space. Although there are an infinite number of such decays, we show that only a finite number of them contribute to the formula. The phenomenon of BH bound state metamorphosis (BSM) plays a crucial role in our analysis. We show that the dyonic BSM orbits with U-duality invariant ∆ < 0 are in exact correspondence with the solution sets of the Brahmagupta-Pell equation, which implies that they are isomorphic to the group of units in the order ℤ[$$ \sqrt{\left|\Delta \right|} $$ Δ ] in the real quadratic field ℚ($$ \sqrt{\left|\Delta \right|} $$ Δ ). We check our formula against the known numerical data arising from the Igusa cusp form, for the first 1650 polar coefficients, and find perfect agreement.
We provide strong evidence that a large number of CY 3 manifolds are involved in an intricate way in Mathieu moonshine viz. their Gromov-Witten invariants are related to the expansion coefficients of the twined/twisted-twined elliptic genera of K3. We use the conjectured string duality between CHL orbifolds of heterotic string theory on K3 × T 2 and type IIA string theory on CY 3 manifolds to explicitly show this connection. We then work out two concrete examples where we exactly match the expansion coefficients on both sides of the duality.
We discuss ensemble averages of two-dimensional conformal field theories associated with an arbitrary indefinite lattice with integral quadratic form Q. We provide evidence that the holographic dual after the ensemble average is the three-dimensional Abelian Chern-Simons theory with kinetic term determined by Q. The resulting partition function can be written as a modular form, expressed as a sum over the partition functions of Chern-Simons theories on lens spaces. For odd lattices, the dual bulk theory is a spin Chern-Simons theory, and we identify several novel phenomena in this case. We also discuss the holographic duality prior to averaging in terms of Maxwell-Chern-Simons theories.
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