Wall-crossing made smooth

Pioline, Boris

doi:10.1007/jhep04(2015)092

Cited by 16 publications

(25 citation statements)

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“…In the case with four supercharges, relevant for dyon dynamics in N = 2 gauge theories, the wave-function is a 4-component vector which decomposes under the rotation group SU (2) as 2 scalars and one doublet. This model is similar to the one studied in [32], in fact for λ = 0 it agrees with it upon rescaling the metric on R 4 by the harmonic function H. In addition to the bosonic part (3.3), the Hamiltonian also includes couplings between the spin and the magnetic field B = q r/r 3 sitting at the origin. After decomposing each mode into a radial and angular part using spin-weighted monopole harmonics and diagonalizing the resulting radial Hamiltonian, one finds that the energy levels and density of states in the continuum for a mode of helicity h ∈ {0, 0, ± 1 2 } are obtained from those of the bosonic model by replacing the relation j = |q| + , ν = j + 1 2 in (3.7), (3.11) by…”

Section: A Spectral Asymmetry and Helicity Partition Functionsupporting

confidence: 70%

“…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. The rest of the introduction contains a summary of the details of our problem and its proposed solution.…”

Section: Introductionmentioning

confidence: 52%

“…The quantum dynamics of the two-centered black hole bound state is not completely understood. In the context of black holes in N = 2 string vacua, it is well-described by the quiver quantum mechanics with 4 supercharges introduced in [46], or more simply by the supersymmetric quantum mechanics on R 3 which arises on its Coulomb branch [46], [47], [32]. In that case, 1 2 -BPS bound states arise from supersymmetric vacua in the quantum mechanics on R 3 describing the relative motion, while the 4 fermionic zero-modes come the center-of-motion degrees of freedom.…”

Section: Moduli Space Of Two-centered Black Holes and Continuum Contrmentioning

confidence: 99%

“…In the zero temperature limit β → ∞, I 2 (β) reduces to the helicity supertrace I 2 in (3.28), but it may also receive contributions from the continuum of scattering states due to a possible asymmetry between the bosonic and fermionic densities of states. These contributions are in fact necessary in order to ensure that I(β) is a smooth function of the parameter λ at finite β, as required by the Fredholm property of the Hamiltonian H. To compute the spectral asymmetry, we shall follow the same approach as in [32], with a shortcut to eschew a full analysis of the supersymmetric quantum mechanics. Unfortunately, applying the same shortcut to the computation of the refined indices introduced in (3.19), (3.20) does not seem to give a sensible result, so the results in this appendix should be viewed as heuristic.…”

Section: A Spectral Asymmetry and Helicity Partition Functionmentioning

confidence: 99%

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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: A Spectral Asymmetry and Helicity Partition Functionsupporting

confidence: 70%

Section: Introductionmentioning

confidence: 52%

Section: Moduli Space Of Two-centered Black Holes and Continuum Contrmentioning

confidence: 99%

Section: A Spectral Asymmetry and Helicity Partition Functionmentioning

confidence: 99%

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Mock modularity from black hole scattering states

Murthy

Pioline

2018

J. High Energ. Phys.

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“…It is a challenge to check these predictions from a direct computation of the difference of densities of bosonic and fermionic states of a system of n dyons. While the result near a wall of marginal stability can actually be deduced by analyzing the nonrelativistic electronmonopole system [43], the result (12) should hold throughout moduli space, where the constituents are relativistic.…”

Section: -3mentioning

confidence: 99%

R3Index for Four-DimensionalN=2Field Theories

et al. 2015

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In theories with N ¼ 2 supersymmetry on R 3;1 , supersymmetric bound states can decay across walls of marginal stability in the space of Coulomb branch parameters, leading to discontinuities in the BPS indices Ωðγ; uÞ. We consider a supersymmetric index I which receives contributions from 1=2-BPS states, generalizing the familiar Witten index Trð−1Þ F e −βH . We expect I to be smooth away from loci where massless particles appear, thanks to contributions from the continuum of multiparticle states. Taking inspiration from a similar phenomenon in the hypermultiplet moduli space of N ¼ 2 string vacua, we conjecture a formula expressing I in terms of the BPS indices Ωðγ; uÞ, which is continuous across the walls and exhibits the expected contributions from single particle states at large β. This gives a universal prediction for the contributions of multiparticle states to the index I. This index is naturally a function on the moduli space after reduction on a circle, closely related to the canonical hyperkähler metric and hyperholomorphic connection on this space. DOI: 10.1103/PhysRevLett.114.121601 PACS numbers: 11.25.Mj, 11.25.Uv, 11.30.Pb It has been clear since the work of Seiberg and Witten [1] that extended supersymmetry gives enough control over four-dimensional quantum field theories to produce exact results on the dynamics of the theories, even when these theories are strongly interacting. Remarkably, such results are deeply related to some of the most interesting questions in the mathematics of algebraic geometry and differential geometry. As a significant example, the moduli space of a four-dimensional theory with N ¼ 2 supersymmetry on a circle is a hyperkähler manifold (a special class of manifolds satisfying Einstein's equations), whose metric encodes both instanton corrections to gauge couplings and the spectrum of Bogomol'nyi-Prasad-Sommerfield (BPS) states in the four-dimensional theory [2]. In this Letter, we reinforce this connection, and construct a canonical function on the aforementioned moduli space, which, on the one hand, generates a solution to the self-dual Yang-Mills equations on this manifold, and, on the other hand, purportedly encodes interactions of BPS states in four dimensions.BPS indices and the Witten index.-In four-dimensional field theories on R 3;1 with N ¼ 2 supersymmetry, the spectrum of BPS states in general strongly depends on the value of the Coulomb branch parameters. Part of this dependence can be removed by considering the BPS indexwhere H 1 ðγ; uÞ is the Hilbert space of one-particle states with electromagnetic charge γ ∈ Γ in the Coulomb vacuum u, J 3 is a component of the rotation group along a fixed axis, and ð−1Þ 2J 3 is the fermionic parity by virtue of the spin statistics theorem. The BPS index Ωðγ; uÞ, being sensitive only to short multiplets saturating the BPS bound [3], is a locally constant, integer valued function of u, but it is discontinuous across certain walls in moduli space, where some of the BPS bound states with charge γ decay into multiparticle B...

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Squashed toric manifolds and higher depth mock modular forms

Gupta

Murthy

Nazaroglu

2019

J. High Energ. Phys.

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Squashed toric sigma models are a class of sigma models whose target space is a toric manifold in which the torus fibration is squashed away from the fixed points so as to produce a neck-like region. The elliptic genera of squashed toric-Calabi-Yau manifolds are known to obey the modular transformation property of holomorphic Jacobi forms, but have an explicit non-holomorphic dependence on the modular parameter. The elliptic genus of the simplest one-dimensional example is known to be a mixed mock Jacobi form, but the precise automorphic nature for the general case remained to be understood. We show that these elliptic genera fall precisely into a class of functions called higher-depth mock modular forms that have been formulated recently in terms of indefinite theta series. We also compute a generalization of the elliptic genera of these models corresponding to an additional set of charges corresponding to the toric symmetries. Finally we speculate on some relations of the elliptic genera of squashed toric models with the Vafa-Witten partition functions of N = 4 SYM theory on CP 2 . The paper [8] began a systematic study of similar ideas for a class of theories which generalized the above story. On the geometric side, one now considered a class of SCFTs corresponding to squashed toric-Calabi-Yau manifolds, a certain deformation of toric CY manifolds, introduced in [9], which we explain below. The resulting elliptic genera, which were computed in [8] as a torus functional integral using the gauged linear sigma model (GLSM) description of the theory, obey the modular transformation properties of a Jacobi form as expected. The interesting feature is that they explicitly depend on τ , the complex conjugate of the modular parameter τ of the torus. The dependence on τ is intuitively understood by the fact that these target spaces are necessarily non-compact and that the density of bosonic and fermionic states in the corresponding continuum are not necessarily equal.This behavior is characteristic of a class of automorphic functions called mock modular forms [10,11], which, along with the closely related mock Jacobi forms, have been discussed with great interest in recent years in diverse contexts [12][13][14]. However, the details of the modular transformation properties of the elliptic genera of squashed toric models do not quite match the basic definitions of mock modular and mock Jacobi forms and suggest some sort of a generalization. The precise nature of their modular transformation properties, beyond the general properties obeyed by the partition function of a SCFT, was left open in [8]. This is the question that we address and answer in the current paper. The answer is related to a class of functions, called higher-depth mock modular forms, that have been formulated recently in terms of indefinite theta functions [15,16], and further developed in [17][18][19][20]. We show here that the elliptic genera of squashed toric sigma models belong precisely to this class. In the rest of this introduction we explain some...

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Wall-crossing made smooth

Cited by 16 publications

References 57 publications

Mock modularity from black hole scattering states

Mock modularity from black hole scattering states

R3Index for Four-DimensionalN=2Field Theories

Squashed toric manifolds and higher depth mock modular forms

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