Multiple D3-Instantons and Mock Modular Forms I

Alexandrov, Sergei; Banerjee, Sibasish; Manschot, Jan; Pioline, Boris

doi:10.1007/s00220-016-2799-0

Cited by 32 publications

(91 citation statements)

References 60 publications

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“…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.…”

Section: Introductionsupporting

confidence: 60%

“…precisely the same result (4.20) as in the model with 8 supercharges, up to an overall factor of 1/4. In particular, in contrast to the model studied in [32], the contribution from the continuum produces both a term proportional to the complementary error function, as well as a Gaussian term, which is in fact necessary for the modular invariance of the generating function of MSW invariants [27,28].…”

Section: Supersymmetric Quantum Mechanics With Four Superchargesmentioning

confidence: 94%

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Mock modularity from black hole scattering states

Murthy

Pioline

2018

J. High Energ. Phys.

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The exact degeneracies of quarter-BPS dyons in Type II string theory on K3 × T 2 are given by Fourier coefficients of the inverse of the Igusa cusp form. For a fixed magnetic charge invariant m, the generating function of these degeneracies naturally decomposes as a sum of two parts, which are supposed to account for single-centered black holes, and two-centered black hole bound states, respectively. The decomposition is such that each part is separately modular covariant but neither is holomorphic, calling for a physical interpretation of the non-holomorphy. We resolve this puzzle by computing the supersymmetric index of the quantum mechanics of two-centered half-BPS black-holes, which we model by geodesic motion on Taub-NUT space subject to a certain potential. We compute a suitable index using localization methods, and find that it includes both a temperature-independent contribution from BPS bound states, as well as a temperaturedependent contribution due to a spectral asymmetry in the continuum of scattering states. The continuum contribution agrees precisely with the non-holomorphic completion term required for the modularity of the generating function of two-centered black hole bound states.

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Section: Introductionsupporting

confidence: 60%

Section: Supersymmetric Quantum Mechanics With Four Superchargesmentioning

confidence: 94%

Mock modularity from black hole scattering states

Murthy

Pioline

2018

J. High Energ. Phys.

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“…The case where D is the sum of two irreducible divisors was first considered in [20,24,25], and studied more recently in [14,26]. In that case, the generating function of MSW invariants turns out to be a mock modular form, with a specific non-holomorphic completion obtained by smoothing out the sign functions entering in the bound state contributions, recovering the prescription of [27,20].…”

Section: Introductionmentioning

confidence: 99%

“…In particular, D4-D2-D0 black holes in type IIA string theory compactified on a generic compact Calabi-Yau (CY) threefold Y can be lifted to an M5-brane wrapped on a divisor D ⊂ Y [2]. If the divisor D labelled by the D4-brane charge p a is irreducible, the indices Ω(γ, z a ) are independent of the moduli of Y, at least in the limit where the volume of Y is scaled to infinity, and their generating function is known to be a holomorphic (vector valued) modular form of weight − 1 2 b 2 (Y) − 1 [3,4,6,14]. This includes the case of vertical, rank 1 D4-D2-D0 branes in K3-fibered Calabi-Yau threefolds [15,16,17].…”

Section: Introductionmentioning

confidence: 99%

“…In order to establish this result, we follow the same strategy as in our earlier works [14,26] and analyze D3-D1-D(-1) instanton corrections to the metric on the hypermultiplet moduli space M H in type IIB string theory compactified on Y, at arbitrary order in the instanton expansion. After reducing on a circle and T-dualizing, this moduli space is identical to the vector multiplet moduli space in type IIA string theory compactified on Y × S 1 , where it receives instanton corrections from D4-D2-D0 black holes winding around the circle.…”

Section: Introductionmentioning

confidence: 99%

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Black Holes and Higher Depth Mock Modular Forms

Alexandrov

Pioline

2019

Commun. Math. Phys.

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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...

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Rigid limit for hypermultiplets and five-dimensional gauge theories

Alexandrov

Banerjee

Longhi

2018

J. High Energ. Phys.

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We study the rigid limit of a class of hypermultiplet moduli spaces appearing in Calabi-Yau compactifications of type IIB string theory, which is induced by a local limit of the Calabi-Yau. We show that the resulting hyperkähler manifold is obtained by performing a hyperkähler quotient of the Swann bundle over the moduli space, along the isometries arising in the limit. Physically, this manifold appears as the target space of the non-linear sigma model obtained by compactification of a five-dimensional gauge theory on a torus. This allows to compute dyonic and stringy instantons of the gauge theory from the known results on D-instantons in string theory. Besides, we formulate a simple condition on the existence of a non-trivial local limit in terms of intersection numbers of the Calabi-Yau, and find an explicit form for the hypermultiplet metric including corrections from all mutually non-local D-instantons, which can be of independent interest.

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Multiple D3-Instantons and Mock Modular Forms I

Cited by 32 publications

References 60 publications

Mock modularity from black hole scattering states

Mock modularity from black hole scattering states

Black Holes and Higher Depth Mock Modular Forms

Rigid limit for hypermultiplets and five-dimensional gauge theories

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