Attractor flow trees, BPS indices and quivers

Alexandrov, Sergei; Pioline, Boris

doi:10.4310/atmp.2019.v23.n3.a2

Cited by 27 publications

(67 citation statements)

References 43 publications

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“…To this end, it suffices to consider all flow trees which start with the splitting γ → γ L + γ R at the root of the tree. It is also consistent with the general wall-crossing formula of [41], provided the tree index is computed for a small generic perturbation of the DSZ matrix γ ij [23]. Finally, it is useful to note that, assuming that all charges γ i are distinct, the sum over splittings and flow trees in (2.2) can be generated by iterating the quadratic equation [23] Ω…”

Section: )supporting

confidence: 68%

“…By definition, the attractor indices are of course moduli independent. The problem of expressing Ω(γ, z a ) in terms of attractor indices was addressed recently in [23], extending earlier work in [20,50]. Relying on the split attractor flow conjecture [22,6], it was argued that the rational BPS index…”

Section: Introductionmentioning

confidence: 93%

“…2. After expressing the DT invariants Ω(γ, z a ) through the moduli independent MSW invariants Ω MSW (γ) using the tree flow formula of [23], and expanding G in powers of Ω MSW (γ), each order in this expansion can be decomposed into a sum of products of certain indefinite theta series and of holomorphic generating functions h p i ,µ i of the invariants Ω MSW (γ) (see (4.2)), similarly to the usual decomposition of standard Jacobi forms. Thus, the modular properties of the indefinite theta series ϑ p,µ Φ tot n , n − 2 are tied with the modular properties of the generating functions h p i ,µ i , in order for G to be modular.…”

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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Section: )supporting

confidence: 68%

Section: Introductionmentioning

confidence: 93%

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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“…This problem is directly related to BPS counting in Calabi-Yau compactifications of M theory, and therefore is of relevance to questions in enumerative geometry [1][2][3][4][5]. Progress on Wall-Crossing over the past decade has yielded new insights into these questions, and led to developments in the study of BPS spectra both in the context of supersymmetric gauge theories and in string theory [6][7][8][9][10][11][12][13][14][15][16][17][18]. The present work focuses on the BPS spectra of five-dimensional gauge theories engineered by M theory on a toric Calabi-Yau threefold X.…”

Section: Contentsmentioning

confidence: 99%

“…so the two branches have different behavior here. 18 While λ − ∼ dx/x has a simple pole, λ + ∼ log x x dx has logarithmic branching and is therefore multi-valued around x = 0. This means that there is a logarithmic cut on sheet + of Σ starting above x = 0 and running to infinity.…”

Section: Covering Maps and Trivializationmentioning

confidence: 99%

Exploring 5d BPS Spectra with Exponential Networks

Banerjee

Longhi

Romo

2019

Ann. Henri Poincaré

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We develop geometric techniques for counting BPS states in five-dimensional gauge theories engineered by M theory on a toric Calabi-Yau threefold. The problem is approached by studying framed 3d-5d wall-crossing in presence of a single M5 brane wrapping a special Lagrangian submanifold L. The spectrum of 3d-5d BPS states is encoded by the geometry of the manifold of vacua of the 3d-5d system, which further coincides with the mirror curve describing moduli of the Lagrangian brane. The information about the BPS spectrum is extracted from the geometry of the mirror curve by construction of a nonabelianization map for the exponential networks. For the simplest Calabi-Yau, C 3 we reproduce the count of 5d BPS states encoded by the Mac Mahon function in the context of topological strings, and match predictions of 3d tt * geometry for the count of 3d-5d BPS states. We comment on applications of our construction to the study of enumerative invariants of toric Calabi-Yau threefolds.

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Refinement and modularity of immortal dyons

Alexandrov

Nampuri

2021

J. High Energ. Phys.

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Extending recent results in $$ \mathcal{N} $$ N = 2 string compactifications, we propose that the holomorphic anomaly equation satisfied by the modular completions of the generating functions of refined BPS indices has a universal structure independent of the number $$ \mathcal{N} $$ N of supersymmetries. We show that this equation allows to recover all known results about modularity (under SL(2, ℤ) duality group) of BPS states in $$ \mathcal{N} $$ N = 4 string theory. In particular, we reproduce the holomorphic anomaly characterizing the mock modular behavior of quarter-BPS dyons and generalize it to the case of non-trivial torsion invariant.

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Attractor flow trees, BPS indices and quivers

Cited by 27 publications

References 43 publications

Black Holes and Higher Depth Mock Modular Forms

Black Holes and Higher Depth Mock Modular Forms

Exploring 5d BPS Spectra with Exponential Networks

Refinement and modularity of immortal dyons

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