Black Holes and Higher Depth Mock Modular Forms

Abstract

By enforcing invariance under S-duality in type IIB string theory compactified on a Calabi-Yau threefold, we derive modular properties of the generating function of BPS degeneracies of D4-D2-D0 black holes in type IIA string theory compactified on the same space. Mathematically, these BPS degeneracies are the generalized Donaldson-Thomas invariants counting coherent sheaves with support on a divisor D, at the large volume attractor point. For D irreducible, this function is closely related to the elliptic genus of the superconformal field theory obtained by wrapping M5-brane on D and is therefore known to be modular. Instead, when D is the sum of n irreducible divisors D i , we show that the generating function acquires a modular anomaly. We characterize this anomaly for arbitrary n by providing an explicit expression for a non-holomorphic modular completion in terms of generalized error functions. As a result, the generating function turns out to be a (mixed) mock modular form of depth n − 1. BPS indices and wall-crossingIn this section, we review some aspects of BPS indices in theories with N = 2 supersymmetry, including the tree flow formula relating the moduli-dependent index Ω(γ, z a ) to the attractor index Ω ⋆ (γ). We then apply this formalism in the context of Calabi-Yau string vacua, and express the generalized DT invariants Ω(γ, z a ) in terms of their counterparts evaluated at the large volume attractor point (1.1), known as MSW invariants. Wall-crossing and attractor flowsThe BPS index Ω(γ, z a ) counts (with sign) micro-states of BPS black holes with total electromagnetic charge γ = (p Λ , q Λ ), for a given value z a of the moduli at spatial infinity. While Ω(γ, z a ) is a locally constant function over the moduli space, it can jump across real codimension one loci where certain bound states, represented by multi-centered black hole solutions of N = 2 supergravity, become unstable. The positions of these loci, known as walls of marginal stability, are determined by the central charge Z γ (z a ), a complex-valued linear function of γ whose modulus gives the mass of a BPS state of charge γ, while the phase determines the supersymmetry subalgebra preserved by the state. Since a bound state can only decay when its mass becomes equal to the sum of masses of its constituents, it is apparent that the walls correspond to hypersurfaces where the phases of two central charges, say Z γ L (z a ) and Z γ R (z a ), become aligned. The bound states which may become unstable are then those whose constituents have charges in the positive cone spanned by γ L and γ R . We shall assume that the charges γ L , γ R have non-zero Dirac-Schwinger-Zwanziger (DSZ) pairing γ L , γ R = 0, since otherwise marginal bound states may form, whose stability is hard to control.The general relation between the values of Ω(γ, z a ) on the two sides of a wall has been found in the mathematics literature by and Joyce-Song [42,43], and justified physically in a series of works [6,7,8,9]. However, in this work we require a somewhat differe...

Mock modularity from black hole scattering states

“…We note that a similar phenomenon arises in the context of N = 2 black holes [27,28], but in that context mock modular forms of higher depth are expected to arise due to the occurrence of BPS bound states involving an arbitrary number of constituents [29,30]. In this paper, inspired by earlier work [31,32] in the context of N = 2 black holes, we attempt to give a physical justification of this non-holomorphic correction from the macroscopic point of view in the N = 4 context, by computing the contribution of the continuum of scattering states in the quantum mechanics of two-centered BPS black holes.…”

“…In Section 4.5 we computed the index using localization, and found that the result (4.23) contains both a term proportional to the complementary error function, also present in [32], as well as a Gaussian term, which is in fact necessary for the modular covariance of the generating function of MSW invariants [27,28]. It would be interesting to apply similar localization techniques to the case of multi-centered black holes, where mock modular forms of higher depth are expected to occur [30]. Interestingly, such modular objects arise in the computation of elliptic genera of squashed toric manifolds [54], and presumably also in the context of higher rank monopole moduli spaces, which may provide a useful model for the dynamics of multi-centered black holes.…”

3d modularity

“…It would be interesting to find relations (dualities) to other physics problems where similar modular structures appeared, e.g. [88,[99][100][101][102].…”

Attractor flow trees, BPS indices and quivers

“…Since the latter are invariant under spectral flow, the resulting partition function can still be formally decomposed as a sum of indefinite theta series, with a kernel given by a sum over flow trees [36,17]. While the convergence and modular properties of this theta series are by now well understood when the reducible divisor D is the sum of two components [36,40], the representations of the tree index found in this paper will be instrumental in extending these results to a general reducible divisor [41].…”

“…The two representations may be useful for different purposes. For instance, the second representation is suitable for the proof of convergence of the BPS partition function, whereas the first representation is more convenient for analyzing its modular properties [41], this is why we give here both of them. Since their proofs are essentially identical, we only present the proof for the first representation.…”