BPS States 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.

Merging the weak gravity and distance conjectures using BPS extremal black holes

We analyze the charge-to-mass structure of BPS states in general infinite-distance limits of mathcal{N} = 2 compactifications of Type IIB string theory on Calabi-Yau three-folds, and use the results to sharpen the formulation of the Swampland Conjectures in the presence of multiple gauge and scalar fields. We show that the BPS bound coincides with the black hole extremality bound in these infinite distance limits, and that the charge-to-mass vectors of the BPS states lie on degenerate ellipsoids with only two non-degenerate directions, regardless of the number of moduli or gauge fields. We provide the numerical value of the principal radii of the ellipsoid in terms of the classification of the singularity that is being approached. We use these findings to inform the Swampland Distance Conjecture, which states that a tower of states becomes exponentially light along geodesic trajectories towards infinite field distance. We place general bounds on the mass decay rate λ of this tower in terms of the black hole extremality bound, which in our setup implies lambda ge 1/sqrt{6} . We expect this framework to persist beyond mathcal{N} = 2 as long as a gauge coupling becomes small in the infinite field distance limit.

Pair production and gravity as the weakest force

The Weak Gravity Conjecture (WGC) is usually formulated in terms of the stability of extremal black-holes or in terms of long distance Coulomb/Newton potentials. However one can think of other physical processes to compare the relative strength of gravity versus other forces. We argue for an alternative formulation in terms of particle pair production at threshold or, equivalently, pair annihilation at rest. Imposing that the production rate by any force mediator (photon or scalar) of pairs of charged particles be larger or equal to graviton production, we recover known conditions for the U(1) WGC and its extensions. Unlike other formulations though, threshold pair production is sensitive to short range couplings present in scalar interactions and gives rise to a Scalar WGC. Application to moduli scalars gives rise to specific conditions on the trilinear and quartic couplings which involve first and second derivatives of the WGC particle mass with respect to the moduli. Some solutions saturating equations correspond to massive states behaving like BPS, KK and winding states which feature duality invariance and are in agreement with the Swampland distance conjecture. Conditions for N = 2 BPS states saturate our bounds and we discuss specific examples of BPS states which become massless at large Kahler moduli in Type IIA N=2, D=4 CY and orbifold compactifications. We study possible implications for potentials depending on moduli only through WGC massive states. For some simple classes of potentials one recovers constraints somewhat similar but not equivalent to a Swampland dS conjecture.

Quiver Yangian from crystal melting

We find a new infinite class of infinite-dimensional algebras acting on BPS states for non-compact toric Calabi-Yau threefolds. In Type IIA superstring compactification on a toric Calabi-Yau threefold, the D-branes wrapping holomorphic cycles represent the BPS states, and the fixed points of the moduli spaces of BPS states are described by statistical configurations of crystal melting. Our algebras are “bootstrapped” from the molten crystal configurations, hence they act on the BPS states. We discuss the truncation of the algebra and its relation with D4-branes. We illustrate our results in many examples, with and without compact 4-cycles.

Duality and supersymmetry constraints on the weak gravity conjecture

Positivity bounds coming from consistency of UV scattering amplitudes are not always sufficient to prove the weak gravity conjecture for theories beyond Einstein-Maxwell. Additional ingredients about the UV may be necessary to exclude those regions of parameter space which are naïvely in conflict with the predictions of the weak gravity conjecture. In this paper we explore the consequences of imposing additional symmetries inherited from the UV theory on higher-derivative operators for Einstein-Maxwell-dilaton-axion theory. Using black hole thermodynamics, for a preserved SL(2, ℝ) symmetry we find that the weak gravity conjecture then does follow from positivity bounds. For a preserved O(d, d; ℝ) symmetry we find a simple condition on the two Wilson coefficients which ensures the positivity of corrections to the charge-to-mass ratio and that follows from the null energy condition alone. We find that imposing supersymmetry on top of either of these symmetries gives corrections which vanish identically, as expected for BPS states.

Lifting at higher levels in the D1D5 CFT

The D1D5P system has a large set of BPS states at its orbifold point. Perturbing away from this ‘free’ point leads to some states joining up into long supermultiplets and lifting, while other states remain BPS. We consider the simplest orbifold which exhibits this lift: that with N = 2 copies of the free c = 6 CFT. We write down the number of lifted and unlifted states implied by the index at all levels upto 6. We work to second order in the perturbation strength λ. For levels upto 4, we find the wavefunctions of the lifted states, their supermultiplet structure and the value of the lift. All states that are allowed to lift by the index are in fact lifted at order O(λ2). We observe that the unlifted states in the untwisted sector have an antisymmetry between the copies in the right moving Ramond ground state sector, and extend this observation to find classes of states for arbitrary N that will remain unlifted to O(λ2).

Pure-Higgs states from the Lefschetz-Sommese theorem

We consider a special class of N=4 quiver quantum mechanics relevant in the description of BPS states of D4D0 branes in type II Calabi-Yau compactifications and the corresponding 4-dimensional black holes. These quivers have two abelian nodes in addition to an arbitrary number of non-abelian nodes and satisfy some simple but stringent conditions on the set of arrows, in particular closed oriented loops are always present. The Higgs branch can be described as the vanishing locus of a section of a vector bundle over a product of a projective space with a number of Grassmannians. The Lefschetz-Sommese theorem then allows to separate induced from intrinsic cohomology which leads to the notion of pure-Higgs states. We compute explicit formulae for an index counting these pure-Higgs states and prove — for this special class of quivers — some previously stated conjectures about them.

Dyonic black hole degeneracies in mathcal{N} = 4 string theory from Dabholkar-Harvey degeneracies

The degeneracies of single-centered dyonic frac{1}{4} -BPS black holes (BH) in Type II string theory on K3×T2 are known to be coefficients of certain mock Jacobi forms arising from the Igusa cusp form Φ10. In this paper we present an exact analytic formula for these BH degeneracies purely in terms of the degeneracies of the perturbative frac{1}{2} -BPS states of the theory. We use the fact that the degeneracies are completely controlled by the polar coefficients of the mock Jacobi forms, using the Hardy-Ramanujan-Rademacher circle method. Here we present a simple formula for these polar coefficients as a quadratic function of the frac{1}{2} -BPS degeneracies. We arrive at the formula by using the physical interpretation of polar coefficients as negative discriminant states, and then making use of previous results in the literature to track the decay of such states into pairs of frac{1}{2} -BPS states in the moduli space. Although there are an infinite number of such decays, we show that only a finite number of them contribute to the formula. The phenomenon of BH bound state metamorphosis (BSM) plays a crucial role in our analysis. We show that the dyonic BSM orbits with U-duality invariant ∆ < 0 are in exact correspondence with the solution sets of the Brahmagupta-Pell equation, which implies that they are isomorphic to the group of units in the order ℤ[ sqrt{left|Delta right|} ] in the real quadratic field ℚ( sqrt{left|Delta right|} ). We check our formula against the known numerical data arising from the Igusa cusp form, for the first 1650 polar coefficients, and find perfect agreement.

Representations of Cohomological Hall Algebras and Donaldson–Thomas Theory with Classical Structure Groups

We introduce a new class of representations of the cohomological Hall algebras of Kontsevich and Soibelman, which we call cohomological Hall modules, or CoHM for short. These representations are constructed from self-dual representations of a quiver with contravariant involution $$\sigma $$ and provide a mathematical model for the space of BPS states in orientifold string theory. We use the CoHM to define a generalization of the cohomological Donaldson–Thomas theory of quivers in which the quiver representations have orthogonal or symplectic structure groups. The associated invariants are called orientifold Donaldson–Thomas invariants. We prove the orientifold analogue of the integrality conjecture for $$\sigma $$ -symmetric quivers. We also formulate precise conjectures regarding the geometric meaning of orientifold Donaldson–Thomas invariants and the freeness of the CoHM of a $$\sigma $$ -symmetric quiver. We prove the freeness conjecture for disjoint union quivers, loop quivers and the affine Dynkin quiver of type $${\widetilde{A}}_1$$ . We also verify the geometric conjecture in a number of examples. Finally, we construct explicit Poincare–Birkhoff–Witt-type bases of the CoHM of finite type quivers.

S-Duality and Refined BPS Indices

Whenever available, refined BPS indices provide considerably more information on the spectrum of BPS states than their unrefined version. Extending earlier work on the modularity of generalized Donaldson-Thomas invariants counting D4-D2-D0 brane bound states in type IIA strings on a Calabi-Yau threefold $\mathfrak{Y}$, we construct the modular completion of generating functions of refined BPS indices supported on a divisor class. Although for compact $\mathfrak{Y}$ the refined indices are not protected, switching on the refinement considerably simplifies the construction of the modular completion. Furthermore, it leads to a non-commutative analogue of the TBA equations, which suggests a quantization of the moduli space consistent with S-duality. In contrast, for a local CY threefold given by the total space of the canonical bundle over a complex surface $S$, refined BPS indices are well-defined, and equal to Vafa-Witten invariants of $S$. Our construction provides a modular completion of the generating function of these refined invariants for arbitrary rank. In cases where all reducible components of the divisor class are collinear (which occurs e.g. when $b_2(\mathfrak{Y})=1$, or in the local case), we show that the holomorphic anomaly equation satisfied by the completed generating function truncates at quadratic order. In the local case, it agrees with an earlier proposal by Minahan et al for unrefined invariants, and extends it to the refined level using the afore-mentioned non-commutative structure. Finally, we show that these general predictions reproduce known results for $U(2)$ and $U(3)$ Vafa-Witten theory on $\mathbb{P}^2$, and make them explicit for $U(4)$.

Quivers for 3-manifolds: the correspondence, BPS states, and 3d mathcal{N} = 2 theories

We introduce and explore the relation between quivers and 3-manifolds with the topology of the knot complement. This idea can be viewed as an adaptation of the knots-quivers correspondence to Gukov-Manolescu invariants of knot complements (also known as FK or hat{Z} ). Apart from assigning quivers to complements of T(2,2p+1) torus knots, we study the physical interpretation in terms of the BPS spectrum and general structure of 3d mathcal{N} = 2 theories associated to both sides of the correspondence. We also make a step towards categorification by proposing a t-deformation of all objects mentioned above.

Q-nonabelianization for line defects

We consider the q-nonabelianization map, which maps links L in a 3-manifold M to combinations of links tilde{L} in a branched N -fold cover tilde{M} . In quantum field theory terms, q-nonabelianization is the UV-IR map relating two different sorts of defect: in the UV we have the six-dimensional (2, 0) superconformal field theory of type mathfrak{gl} (N ) on M × ℝ2,1, and we consider surface defects placed on L × {x4 = x5 = 0}; in the IR we have the (2, 0) theory of type gl (1) on tilde{M} × ℝ2,1, and put the defects on tilde{L} × {x4 = x5 = 0}. In the case M = ℝ3, q-nonabelianization computes the Jones polynomial of a link, or its analogue associated to the group U(N ). In the case M = C × ℝ, when the projection of L to C is a simple non-contractible loop, q-nonabelianization computes the protected spin character for framed BPS states in 4d mathcal{N} = 2 theories of class S. In the case N = 2 and M = C × ℝ, we give a concrete construction of the q-nonabelianization map. The construction uses the data of the WKB foliations associated to a holomorphic covering tilde{C}to C .

5d SCFTs from decoupling and gluing

We systematically analyse 5d superconformal field theories (SCFTs) obtained by dimensional reduction from 6d mathcal{N} = (1, 0) SCFTs. Such theories have a realization as M-theory on a singular Calabi-Yau threefold, from which we determine the so-called combined fiber diagrams (CFD) introduced in [1–3]. The CFDs are graphs that encode the superconformal flavor symmetry, BPS states, low energy descriptions, as well as descendants upon flavor matter decoupling. To obtain a 5d SCFT from 6d, there are two approaches: the first is to consider a circle-reduction combined with mass deformations. The second is to circle-reduce and decouple an entire gauge sector from the theory. The former is applicable e.g. for very Higgsable theories, whereas the latter is required to obtain a 5d SCFT from a non-very Higgsable 6d theory. In the M-theory realization the latter case corresponds to decompactification of a set of compact surfaces in the Calabi-Yau threefold. To exemplify this we consider the 5d SCFTs that descend from non-Higgsable clusters and non-minimal conformal matter theories. Finally, inspired by the quiver structure of 6d theories, we propose a gluing construction for 5d SCFTs from building blocks and their CFDs.

Wall-crossing structures and application to SU(3) Seiberg–Witten integrable systems

We apply the wall-crossing structure formalism of Kontsevich and Soibelman to Seiberg–Witten integrable systems associated to pure SU(3). This gives an algorithm for computing the Donaldson–Thomas invariants, which correspond to BPS degeneracy of the corresponding BPS states in physics. The main ingredients of this algorithm are the use of split attractor flows and Kontsevich–Soibelman wall-crossing formulas. Besides the known BPS spectrum in pure SU(3) case, we obtain new family of BPS states with BPS-invariants equal to 2.

Quiver matrix model of ADHM type and BPS state counting in diverse dimensions

Abstract We review the problem of Bogomol’nyi–Prasad–Sommerfield (BPS) state counting described by the generalized quiver matrix model of Atiyah–Drinfield–Hitchin–Manin type. In four dimensions the generating function of the counting gives the Nekrasov partition function, and we obtain a generalization in higher dimensions. By the localization theorem, the partition function is given by the sum of contributions from the fixed points of the torus action, which are labeled by partitions, plane partitions and solid partitions. The measure or the Boltzmann weight of the path integral can take the form of the plethystic exponential. Remarkably, after integration the partition function or the vacuum expectation value is again expressed in plethystic form. We regard it as a characteristic property of the BPS state counting problem, which is closely related to the integrability.

Refined large N duality for knots

We formulate large [Formula: see text] duality of [Formula: see text] refined Chern–Simons theory with a torus knot/link in [Formula: see text]. By studying refined BPS states in M-theory, we provide the explicit form of low-energy effective actions of Type IIA string theory with D4-branes on the [Formula: see text]-background. This form enables us to relate refined Chern–Simons invariants of a torus knot/link in [Formula: see text] to refined BPS invariants in the resolved conifold. Assuming that the extra [Formula: see text] global symmetry acts on BPS states trivially, the duality predicts graded dimensions of cohomology groups of moduli spaces of M2–M5 bound states associated to a torus knot/link in the resolved conifold. Thus, this formulation can be also interpreted as a positivity conjecture of refined Chern–Simons invariants of torus knots/links. We also discuss about an extension to non-torus knots.

Modular forms as classification invariants of 4D mathcal{N} = 2 Heterotic-IIA dual vacua

We focus on 4D mathcal{N} = 2 string vacua described both by perturbative Heterotic theory and by Type IIA theory; a Calabi-Yau three-fold XIIA in the Type IIA language is further assumed to have a regular K3-fibration. It is well-known that one can assign a modular form Φ to such a vacuum by counting perturbative BPS states in Heterotic theory or collecting Noether-Lefschetz numbers associated with the K3-fibration of XIIA. In this article, we expand the observations and ideas (using gauge threshold correction) in the literature and formulate a modular form Ψ with full generality for the class of vacua above, which can be used along with Φ for the purpose of classification of those vacua. Topological invariants of XIIA can be extracted from Φ and Ψ, and even a pair of diffeomorphic Calabi-Yau’s with different Kähler cones may be distinguished by introducing the notion of “the set of Ψ’s for Higgs cascades/for curve classes”. We illustrated these ideas by simple examples.

BPS states for scalar field theories based on mathfrak{g} 2 and mathfrak{su} (4) algebras

We discuss two models in 1 + 1 dimensional space-time for real scalar field multiplets on the root space of mathfrak{g} 2 and mathfrak{su} (4) Lie algebras. The construction of these models is presented and the corresponding BPS solutions are found.

SU(2) channels the cancellation of K3 BPS states

The conformal field theoretic elliptic genus, an invariant for N = (2, 2) superconformal field theories, counts the BPS states in any such theory with signs, according to their bosonic or fermionic nature. For K3 theories, this invariant is the source of the Mathieu Moonshine phenomenon. There, the net number of frac{1}{4} - BPS states is positive for any conformal dimension above the massless threshold, but it may arise after cancellation of the contributions of an equal number of bosonic and fermionic BPS states present in non-generic theories, as is the case for the class of ℤ2-orbifolds of toroidal SCFTs. Never-theless, the space hat{mathrm{mathscr{H}}} of all BPS states that are generic to such orbifold theories provides a convenient framework to construct a particular generic space of states of K3 theories. We find a natural action of the group SU(2) on a subspace of hat{mathrm{mathscr{H}}} which is compatible with the cancellations of contributions from the corresponding non-generic states. In fact, we propose that this action channels those cancellations. As a by-product, we find a new subspace of the generic space of states in hat{mathrm{mathscr{H}}} .

BPS states in the Minahan-Nemeschansky E7 theory

We use the method of spectral networks to calculate BPS degeneracies in the Minahan-Nemeschansky E7 theory, as representations of the E7 flavor symmetry. Our results provide another example of a pattern noticed earlier in the Minahan-Nemeschansky E6 theory: when the electromagnetic charge is n times a primitive charge, the BPS index is a positive integer multiple of (−1)n+1n. We also calculate BPS degeneracies in the Minahan-Nemeschansky E6 theory for larger charges than were previously computed.