Quarter BPS Dyons Research Articles

Refinement and modularity of immortal dyons

Extending recent results in mathcal{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} 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} = 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.

Lost chapters in CHL black holes: untwisted quarter-BPS dyons in the \u21242 model

Motivated by recent advances in Donaldson-Thomas theory, four-dimensional mathcal{N} = 4 string-string duality is examined in a reduced rank theory on a less studied BPS sector. In particular we identify candidate partition functions of “untwisted” quarter-BPS dyons in the heterotic ℤ2 CHL model by studying the associated chiral genus two partition function, based on the M-theory lift of string webs argument by Dabholkar and Gaiotto. This yields meromorphic Siegel modular forms for the Iwahori subgroup B(2) ⊂ Sp4(ℤ) which generate BPS indices for dyons with untwisted sector electric charge, in contrast to twisted sector dyons counted by a multiplicative lift of twisted-twining elliptic genera known from Mathieu moonshine. The new partition functions are shown to satisfy the expected constraints coming from wall-crossing and S-duality symmetry as well as the black hole entropy based on the Gauss-Bonnet term in the effective action. In these aspects our analysis confirms and extends work of Banerjee, Sen and Srivastava, which only addressed a subset of the untwisted sector dyons considered here. Our results are also compared with recently conjectured formulae of Bryan and Oberdieck for the partition functions of primitive DT invariants of the CHL orbifold X = (K3 × T2)/ℤ2, as suggested by string duality with type IIA theory on X.

Mock modularity from black hole scattering states

The exact degeneracies of quarter-BPS dyons in Type II string theory on K3 × T2 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 temperature-dependent 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.

BKM superalgebras from counting dyons in [formula omitted] supersymmetric type II compactifications

We study the degeneracy of quarter BPS dyons in N=4 type II compactifications of string theory. We find that the genus-two Siegel modular forms generating the degeneracies of the quarter BPS dyons in the type II theories can be expressed in terms of the genus-two Siegel modular forms generating the degeneracies of quarter BPS dyons in the CHL theories and the heterotic string. This helps us in understanding the algebra structure underlying the degeneracy of the quarter BPS states. The Conway group, Co0, plays a role similar to Mathieu group, M24, in the CHL models with eta quotients appearing in the place of eta products. We construct BKM Lie superalgebra structures for the ZN (for N=2,3,4) orbifolds of the type II string compactified on a six-torus.

Perturbative tests of non-perturbative counting

We observe that a class of quarter-BPS dyons in $ \mathcal{N} = 4 $ theories with charge vector (Q, P) and with nontrivial values of the arithmetic duality invariant I := gcd(Q∧P) are nonperturbative in one frame but perturbative in another frame. This observation suggests a test of the recently computed nonperturbative partition functions for dyons with nontrivial values of the arithmetic invariant. For all values of I, we show that the nonperturbative counting yields vanishing indexed degeneracy for this class of states everywhere in the moduli space in precise agreement with the perturbative result.

Arithmetic of quantum entropy function

Quantum entropy function is a proposal for computing the entropy associated with the horizon of a black hole in the extremal limit, and is related via AdS/CFT correspondence to the dimension of the Hilbert space in a dual quantum mechanics. We show that in N=4 supersymmetric string theories, quantum entropy function formalism naturally explains the origin of the subtle differences between the microscopic degeneracies of quarter BPS dyons carrying different torsion, i.e. different arithmetical properties. These arise from additional saddle points in the path integral -- whose existence depends on the arithmetical properties of the black hole charges -- constructed as freely acting orbifolds of the original AdS_2\times S^2 near horizon geometry. During this analysis we demonstrate that the quantum entropy function is insensitive to the details of the infrared cutoff used in the computation, and the details of the boundary terms added to the action. We also discuss the role of the asymptotic symmetries of AdS_2 in carrying out the path integral in the definition of quantum entropy function. Finally we show that even though quantum entropy function is expected to compute the absolute degeneracy in a given charge and angular momentum sector, it can also be used to compute the index. This can then be compared with the microscopic computation of the index.

Asymptotic expansion of the 𝒩 = 4 dyon degeneracy

We study various aspects of power suppressed as well as exponentially suppressed corrections in the asymptotic expansion of the degeneracy of quarter BPS dyons in N=4 supersymmetric string theories. In particular we explicitly calculate the power suppressed corrections up to second order and the first exponentially suppressed corrections. We also propose a macroscopic origin of the exponentially suppressed corrections using the quantum entropy function formalism. This suggests a universal pattern of exponentially suppressed corrections to all extremal black hole entropies in string theory.

Genus two surface and quarter BPS dyons: the contour prescription

Following the suggestion of hep-th/0506249 and hep-th/0612011, we represent quarter BPS dyons in N=4 supersymmetric string theories as string network configuration and explore the role of genus two surfaces in determining the spectrum of such dyons. Our analysis leads to the correct contour prescription for integrating the partition function to determine the spectrum in different domains of the moduli space separated by the walls of marginal stability.

Wall crossing formula for 𝒩 = 4 dyons: a macroscopic derivation

We derive the wall crossing formula for the decay of a quarter BPS dyon into a pair of half-BPS dyons by analyzing the quantum dynamics of multi-centered black holes in = 4 supersymmetric string theories. Our analysis encompasses the cases where the final decay products are non-primitive dyons. The results are in agreement with the microscopic formula for the dyon spectrum in the special case of heterotic string theory on T6.

Generalities of quarter BPS dyon partition function and dyons of torsion two

We propose a general set of constraints on the partition function of quarter BPS dyons in any N=4 supersymmetric string theory by drawing insight from known examples, and study the consequences of this proposal. The main ingredients of our analysis are duality symmetries, wall crossing formula and black hole entropy. We use our analysis to constrain the dyon partition function for two hitherto unknown cases -- the partition function of dyons of torsion two (i.e. gcd(Q\wedge P)=2) in heterotic string theory on T^6 and the partition function of dyons carrying untwisted sector electric charge in Z_2 CHL model. With the help of these constraints we propose a candidate for the partition function of dyons of torsion two in heterotic string theory on T^6. This leads to a novel wall crossing formula for decay of quarter BPS dyons into half BPS dyons with non-primitive charge vectors. In an appropriate limit the proposed formula reproduces the known result for the spectrum of torsion two dyons in gauge theory.

Partition functions of torsion >1 dyons in heterotic string theory onT6

The original proposal of Dijkgraaf, Verlinde and Verlinde for the quarter BPS dyon partition function in heterotic string theory on T^6 is known to correctly produce the degeneracy of dyons of torsion 1, i.e. dyons for which gcd(Q\wedge P)=1. We propose a generalization of this formula for dyons of arbitrary torsion. Our proposal satisfies the constraints coming from S-duality invariance, wall crossing formula, black hole entropy and the gauge theory limit. Furthermore using our proposal we derive a general wall crossing formula that is valid even when both the decay products are non-primitive half-BPS dyons.

Black hole entropy function, attractors and precision counting of microstates

In these lecture notes we describe recent progress in our understanding of attractor mechanism and entropy of extremal black holes based on the entropy function formalism. We also describe precise computation of the microscopic degeneracy of a class of quarter BPS dyons in $${\mathcal{N}=4}$$ supersymmetric string theories, and compare the statistical entropy of these dyons, expanded in inverse powers of electric and magnetic charges, with a similar expansion of the corresponding black hole entropy. This comparison is extended to include the contribution to the entropy from multi-centred black holes as well.

Degeneracy of decadent dyons

A quarter-BPS dyon in $\mathcal{N}=4$ super Yang-Mills theory is generically `decadent' in that it is stable only in some regions of the moduli space and decays on submanifolds in the moduli space. Using this fact, and from the degeneracy of the system close to the decay, a new derivation for the degeneracy of such dyons is given. The degeneracy obtained from these very simple physical considerations is in precise agreement with the results obtained from index computations in all known cases. Similar considerations apply to dyons in $\mathcal{N}=2$ gauge theories. The relation between the $\mathcal{N} =4 $ field theory dyons and those counted by the Igusa cusp form in toroidally compactified heterotic string is elucidated.

Duality orbits, dyon spectrum and gauge theory limit of heterotic string theory onT6

For heterotic string theory compactified on T6, we derive the complete set of T-duality invariants which characterize a pair of charge vectors (Q, P) labelling the electric and magnetic charges of the dyon. Using this we can identify the complete set of dyons to which the previously derived degeneracy formula can be extended. By going near special points in the moduli space of the theory we derive the spectrum of quarter BPS dyons in = 4 supersymmetric gauge theory with simply laced gauge groups. The results are in agreement with those derived from field theory analysis.

Spectrum of dyons and black holes in CHL orbifolds using Borcherds lift

The degeneracies of supersymmetric quarter BPS dyons in four dimensions and of spinning black holes in five dimensions in a CHL compactification are computed exactly using Borcherds lift. The Hodge anomaly in the construction has a physical interpretation as the contribution of a single M-theory Kaluza-Klein 6-brane in the 4d-5d lift. Using factorization, it is shown that the resulting formula has a natural interpretation as a two-loop partition function of left-moving heterotic string, consistent with the heuristic picture of dyons in the M-theory lift of string webs.

Rare decay modes of quarter BPS dyons

The degeneracy of quarter BPS dyons in = 4 supersymmetric string theories is known to jump across walls of marginal stability on which a quarter BPS dyon can decay into a pair of half BPS dyons. We show that as long as the electric and magnetic charges of the original dyon are primitive elements of the charge lattice, the subspaces of the moduli space on which a quarter BPS dyon becomes marginally unstable against decay into a pair of quarter BPS dyons or a half BPS dyon and a quarter BPS dyon are of codimension two or more. As a result any pair of generic points in the moduli space can be connected by a path avoiding these subspaces and there is no jump in the spectrum associated with these subspaces.

Walls of marginal stability and dyon spectrum in Script N = 4 supersymmetric string theories

The spectrum of quarter BPS dyons in N=4 supersymmetric string theories can change as the asymptotic moduli cross walls of marginal stability on which the dyon can break apart into a pair of half BPS states. In this paper we classify these marginal stability walls and examine this phenomenon in the context of exact dyon spectrum found in a class of N=4 supersymmetric string theories. We argue that the dyon partition functions in different domains separated by marginal stability walls are the same, but the choice of integration contour needed for extracting the degeneracies from the partition function differ in these different regions. We also find that in the limit of large charges the change in the degeneracy is exponentially suppressed compared to the leading contribution. This is consistent with the fact that in the computation of black hole entropy we do not encounter any change as the asymptotic moduli fields move across the walls of marginal stability. Finally we carry out some tests of S-duality invariance in the theory.

Dyon spectrum in generic Script N = 4 supersymmetric Bbb ZNorbifolds

We find the exact spectrum of a class of quarter BPS dyons in a generic N=4 supersymmetric Z_N orbifold of type IIA string theory on K3\times T^2 or T^6. We also find the asymptotic expansion of the statistical entropy to first non-leading order in inverse power of charges and show that it agrees with the entropy of a black hole carrying same set of charges after taking into account the effect of the four derivative Gauss-Bonnet term in the effective action of the theory.

Dyon spectrum in Script N = 4 supersymmetric type II string theories

We compute the spectrum of quarter BPS dyons in freely acting Z_2 and Z_3 orbifolds of type II string theory compactified on a six dimensional torus. For large charges the result for statistical entropy computed from the degeneracy formula agrees with the corresponding black hole entropy to first non-leading order after taking into account corrections due to the curvature squared terms in the effective action. The result is significant since in these theories the entropy of a small black hole, computed using the curvature squared corrections to the effective action, fails to reproduce the statistical entropy associated with elementary string states.

Dyon spectrum in CHL models

We propose a formula for the degeneracy of quarter BPS dyons in a class of CHL models. The formula uses a modular form of a subgroup of the genus two modular group Sp(2,Z). Our proposal is S-duality invariant and reproduces correctly the entropy of a dyonic black hole to first non-leading order for large values of the charges.