BPS Case Research Articles

A modified dyon solution in a non-Abelian gauge model

A modified SU(2) Georgi-Glashow model is considered here, and it is shown that besides the well-known Julia-Zee dyon solution, the model can also have a modified dyon solution. The properties of the modified dyon solution are studied using analytical and numerical methods. A comparative analysis of the modified dyon and the Julia-Zee dyon shows that their properties are significantly different. In particular, except for the BPS case, the energy and electric charge of the modified dyons exceed considerably those of the Julia-Zee dyons. At the same time, the energy and electric charge of the modified dyon are bounded for all admissible parameter values, whereas those of the Julia-Zee dyon can be arbitrarily large in the BPS case.

Integration theory for kinks and sphalerons in one dimension

The static kink, sphaleron and kink chain solutions for a single scalar field φ in one spatial dimension are reconsidered. By integration of the Euler–Lagrange equation, or through the Bogomolny argument, one finds that each of these solutions obeys a first-order field equation, an autonomous ODE that can always be formally integrated. We distinguish the BPS case, where the required integral is along a contour in the φ-plane, from the semi-BPS case, where the integral is along a contour in the Riemann surface double-covering the φ-plane, and is generally more complicated.

Quantum energies of BPS vortices in D=2+1 and D=3+1

We consider vortices in scalar electrodynamics and compute the leading quantum correction to their energies for the BPS case of identical classical masses of the Higgs and gauge fields. In particular, we focus on the winding number $n$ dependence of these corrections, from which we can extract the binding energies of configurations with larger $n$. For both dimensionalities, $D=2+1$ and $D=3+1$, we find that quantum corrections are negative and scale approximately linearly with $n$, so that combined vortices are favored over isolated ones.

Non-BPS black branes in M-theory over Calabi-Yau threefolds. (Non-)uniqueness and recombination of non-BPS black strings in single modulus CICY and THCY models

We study extremal solutions arising in M-theory compactifications on Calabi-Yau threefolds, focussing on non-BPS attractors for their importance in relation to the Weak Gravity Conjecture (WGC); M2 branes wrapped on two-cycles give rise to black holes, whereas M5 branes wrapped on four-cycles result in black strings. In the low-energy/field theory limit one obtains minimal N = 2, D = 5 supergravity coupled to Abelian vector multiplets. By making use of the effective black hole potential formalism with Lagrange multipliers and of the Attractor Mechanism, we obtain the explicit expressions of the attractor moduli for BPS and non-BPS solutions, and we compute the Bekenstein-Hawking black hole entropy and the black string tension. Furthermore, by focussing on one modulus complete intersection (CICY) or toric hypersurface (THCY) Calabi-Yau threefolds, we investigate the possible non-uniqueness of the attractor solutions, as well as the stability of non-BPS black holes and black strings (restricting to doubly-extremal solutions, for simplicity’s sake). In all models taken into consideration, we find that both BPS and non-BPS extremal black hole attractors are always unique for a given, supporting electric charge configuration; moreover, non-BPS black holes are always unstable, and thus they decay into constituent BPS/anti-BPS pairs: this confirms the WGC, for which macroscopic non-supersymmetric solutions are bound to decay. For what concerns extremal black strings, it is well known they are unique in the BPS case; we confirm uniqueness also for non-BPS strings in one-modulus CICY models. On the other hand, we discover multiple non-BPS extremal black string attractors (with different tensions) in most of the one-modulus THCY models, and we determine the corresponding magnetic configurations supporting them; this indicates the existence of volume-minimizing representatives in the same homology class having different values of their local minimal volume. Moreover, we find that non-BPS (doubly-) extremal black strings, both for single and multiple solutions, are kinematically stable against decay into their constituent BPS/anti-BPS pairs; in Calabi-Yau geometry, this means that the volume of the representative corresponding to the black string is less than the volume of the minimal piecewise-holomorphic representative, predicting recombination for those homology classes and thus leading to stable, non-BPS string solutions, which for the WGC are microscopic with small charges.

Quantum soliton scattering manifolds

We consider the quantum multisoliton scattering problem. For BPS theories one truncates the full field theory to the moduli space, a finite dimensional manifold of energy minimising field configurations, and studies the quantum mechanical problem on this. Non-BPS theories — the generic case — have no such obvious truncation. We define a quantum soliton scattering manifold as a configuration space which satisfies asymptotic completeness and respects the underlying classical dynamics of slow moving solitons. Having done this, we present a new method to construct such manifolds. In the BPS case the dimension of the n-soliton moduli space ℳn is n multiplied by the dimension of ℳ1. We show that this scaling is not necessarily valid for scattering manifolds in non-BPS theories, and argue that it is false for the Skyrme and baby-Skyrme models. In these models, we show that a relative phase difference can generate a relative size difference during a soliton collision. Asymptotically, these are zero and non-zero modes respectively and this new mechanism softens the dichotomy between such modes. Using this discovery, we then show that all previous truncations of the 2-Skyrmion configuration space are unsuitable for the quantum scattering problem as they have the wrong dimension. This gives credence to recent numerical work which suggests that the low-energy configuration space is 14- dimensional (rather than 12-dimensional, as previously thought). We suggest some ways to construct a suitable manifold for the 2-Skyrmion problem, and discuss applications of our new definition and construction for general soliton theories.

Circular Wilson loops in defect mathcal{N} = 4 SYM: phase transitions, double-scaling limits and OPE expansions

We consider circular Wilson loops in a defect version of mathcal{N} = 4 super-Yang- Mills theory which is dual to the D3-D5 brane system with k units of flux. When the loops are parallel to the defect, we can construct both BPS and non-BPS operators, depending on the orientation of the scalar couplings in the R-symmetry directions. At strong ’t Hooft coupling we observe, in the non supersymmetric case, a Gross-Ooguri-like phase transition in the dual gravitational theory: the familiar disk solution dominates, as expected, when the operator is far from the defect while a cylindrical string worldsheet, connecting the boundary loop with the probe D5-brane, is favourite below a certain distance (or equivalently for large radii of the circles). In the BPS case, instead, the cylindrical solution does not exist for any choice of the physical parameters, suggesting that the exchange of light supergravity modes always saturate the expectation value at strong coupling. We study the double-scaling limit for large k and large ’t Hooft coupling, finding full consistency in the non-BPS case between the string solution and the one-loop perturbative result. Finally we discuss, in the BPS case, the failure of the double-scaling limit and the OPE expansion of the Wilson loop, finding consistency with the known results for the one-point functions of scalar composite operators.

One-loop holography with strings in AdS4 \xd7 \u2102\u21193

We compute the one-loop effective action of string configurations embedded in AdS4 × ℂℙ3 which are dual to frac{1}{6} -BPS latitude Wilson Loops in the ABJM theory. To avoid ambiguities in the string path integral we subtract the frac{1}{2} -BPS case. The one-loop determinants are computed by Fourier-decomposing the two dimensional operators and then using the Gel’fand-Yaglom method. We comment on various aspects related to the regularization procedure, showing the cancellation of a hierarchy of divergences. After taking into account an IR anomaly from a change in topology, we find a precise agreement with the field theory result known from supersymmetric localization.

Supersymmetric near-horizon geometry and Einstein-Cartan-Weyl spaces

We show that the horizon geometry for supersymmetric black hole solutions of minimal five-dimensional gauged supergravity is that of a particular Einstein-Cartan-Weyl (ECW) structure in three dimensions, involving the trace and traceless part of both torsion and nonmetricity, and obeying some precise constraints. In the limit of zero cosmological constant, the set of nonlinear partial differential equations characterizing this ECW structure reduces correctly to that of a hyper-CR Einstein-Weyl structure in the Gauduchon gauge, which was shown by Dunajski, Gutowski and Sabra to be the horizon geometry in the ungauged BPS case.

First order flow equations for nonextremal black holes in AdS (super)gravity

We consider electrically charged static nonextremal black holes in d-dimensional Einstein-Maxwell-(A)dS gravity, whose horizon is a generic Einstein space in d − 2 dimensions. It is shown that for this system the Hamilton-Jacobi equation is exactly solvable and admits two branches of solutions. One of them exhibits a non-simply connected domain of integration constants and does not reduce to the well-known solution for the d = 4 BPS case. The principal functions generate two first order flows that are analytically different, but support the same general solution. One of the two sets of flow equations corresponds to those found by Lü, Pope and Vázquez-Poritz in hep-th/0307001 and (for d = 4 and Λ = 0) by Miller, Schalm and Weinberg in hep-th/0612308. This clarifies also the reason for the very existence of first order equations for nonextremal black holes, namely, they are just the expressions for the conjugate momenta in terms of derivatives of the principal function in a Hamilton-Jacobi formalism. In the last part of our paper we analyze how much of these integrability properties generalizes to matter-coupled N = 2, d = 4 gauged supergravity.

A construction of quarter BPS coherent states and Brauer algebras

BPS coherent states closely resemble semiclassical states and they have gravity dual descriptions in terms of semiclassical geometries. The half BPS coherent states have been well studied, however less is known about quarter BPS coherent states. Here we provide a construction of quarter BPS coherent states. They are coherent states built with two matrix fields, generalizing the half BPS case. These states are both the eigenstates of annihilation operators and in the kernel of dilatation operator. Another useful labeling of quarter BPS states is by representations of Brauer algebras and their projection onto a subalgebra $\mathbb{C}[S_n\times S_m]$. Here, the Schur-Weyl duality for the Walled Brauer algebra plays an important role in organizing the operators. One interesting subclass of these Brauer states are labeled by representations involving two Young tableaux. We obtain the overlap between quarter BPS Brauer states and quarter BPS coherent states, where the Schur polynomials are used. We also derive superposition formulas transforming a truncated version of quarter BPS coherent states to quarter BPS Brauer states. The entanglement entropy of Brauer states as well as the overlap between Brauer states and squeezed states are also computed.

Dualities near the horizon

Abstract In 4-dimensional supergravity theories, covariant under symplectic electricmagnetic duality rotations, a significant role is played by the symplectic matrix $ \mathcal{M} $ (φ), related to the coupling of scalars φ to vector field-strengths. In particular, this matrix enters the twisted self-duality condition for 2-form field strengths in the symplectic formulation of generalized Maxwell equations in the presence of scalar fields. In this investigation, we compute several properties of this matrix in relation to the attractor mechanism of extremal (asymptotically flat) black holes. At the attractor points with no flat directions (as in the $ \mathcal{N} $ = 2 BPS case), this matrix enjoys a universal form in terms of the dyonic charge vector $ \mathcal{Q} $ and the invariants of the corresponding symplectic representation $ {R_{\mathcal{Q}}} $ of the duality group G, whenever the scalar manifold is a symmetric space with G simple and non-degenerate of type E7. At attractors with flat directions, $ \mathcal{M} $ still depends on flat directions, but not $ \mathcal{M}\mathcal{Q} $ , defining the so-called Freudenthal dual of $ \mathcal{Q} $ itself. This allows for a universal expression of the symplectic vector field strengths in terms of $ \mathcal{Q} $ , in the near-horizon Bertotti-Robinson black hole geometry.

The 1/2 BPS Wilson loop in ABJ(M) at two loops: the details

We compute the expectation value of the 1/2 BPS circular Wilson loop operator in ABJ(M) theory at two loops in perturbation theory. Our result turns out to be in exact agreement with the weak coupling limit of the prediction coming from localization, including finite N contributions associated to non-planar diagrams. It also confirms the identification of the correct framing factor that connects framing-zero and framing one expressions, previously proposed. The evaluation of the 1/2 BPS operator is made technically difficult in comparison with other observables of ABJ(M) theory by the appearance of integrals involving the coupling between fermions and gauge fields, which are absent for instance in the 1/6 BPS case. We describe in detail how to analytically solve these integrals in dimensional regularization with dimensional reduction (DRED). By suitably performing the physical limit to three dimensions we clarify the role played by short distance divergences on the final result and the mechanism of their cancellation.

Perturbative evaluation of circular 1/2 BPS Wilson loops in $ \mathcal{N}=6 $ super Chern-Simons theories

We present a complete two-loop analysis of the quantum expectation value for circular BPS Wilson loops in ABJ(M) theories. We examine in details the 1/2 BPS case, that requires non-trivial fermionic couplings with the contour, finding perfect agreement with the exact matrix model answer at zero framing. The result is obtained through a careful application of DRED regularization scheme, combined with a judicious rearrangement of the relevant perturbative contributions that reduces the computation to simple integrals. We carefully analyze the contribution of fermions that is crucial for the consistency with the localization procedure and point out the arising of pivotal evanescent terms, discussing their meaning in relation to Ward identities.

Nonextremal black holes in gauged supergravity and the real formulation of special geometry II

In Klemm and Vaughan (2012 arXiv:1207.2679), a new prescription for finding nonextremal black hole solutions to , D = 4 Fayet–Iliopoulos gauged supergravity was presented, and explicit solutions of various models containing one vector multiplet were constructed. Here, we use the same method to find new nonextremal black holes to more complicated models. We also provide a general recipe to construct non-BPS extremal solutions for an arbitrary prepotential, as long as an axion-free condition holds. These follow from a set of first-order conditions, and are related to the corresponding supersymmetric black holes by a multiplication of the charge vector with a constant field rotation matrix S. The fake superpotential driving this first-order flow is nothing else than Hamilton’s characteristic function in a Hamilton–Jacobi formalism, and coincides in the supersymmetric case (when S is plus or minus the identity) with the superpotential proposed by Dall’Agata and Gnecchi (2011 J. High Energy Phys.JHEP03(2011)037). For the nonextremal black holes that asymptote to (magnetic) AdS, we compute both the mass coming from holographic renormalization and the one appearing in the superalgebra. The latter correctly vanishes in the BPS case, but also for certain values of the parameters that do not correspond to any known supersymmetric solution of gauged supergravity. We finally show that the product of all horizon areas depends only on the charges and the asymptotic value of the cosmological constant.

More on ’t Hooft loops in $ \mathcal{N}=4 $ SYM

We study supersymmetric 't Hooft loop operators in N=4 super Yang-Mills, generalizing the well-known circular 1/2 BPS case and investigating their S-duality properties. We derive the BPS condition for a generic line operator describing pointlike monopoles and discuss its solutions in some particular case. In particular, we present the explicit construction of the magnetic counterpart of Zarembo and DGRT Wilson loops and provide the general dyonic configurations for an abelian gauge group. The quantum definition of these supersymmetric 't Hooft loop operators is carefully discussed and we attempt some computations to next-to-leading order in perturbation theory.

First-order flows and stabilisation equations for non-BPS extremal black holes

We derive a generalised form of flow equations for extremal static and rotating non-BPS black holes in four-dimensional ungauged N = 2 supergravity coupled to vector multiplets. For particular charge vectors, we give stabilisation equations for the scalars, analogous to the BPS case, describing full known solutions. Based on this, we propose a generic ansatz for the stabilisation equations, which surprisingly includes ratios of harmonic functions.

Nearing extremal intersecting giants and new decoupled sectors in 𝒩 = 4 SYM

We study near-horizon limits of near-extremal charged black hole solutions to five-dimensional U(1)3 gauged supergravity carrying two charges, extending the recent work of Balasubramanian et al. [1]. We show that there are two near-horizon decoupling limits for the near-extremal black holes, one corresponding to the near-BPS case and the other for the far from BPS case. Both of these limits are only defined on the 10d IIB uplift of the 5d black holes, resulting in a decoupled geometry with a six-dimensional part (conformal to) a rotating BTZ × S3. We study various aspects of these decoupling limits both from the gravity side and the dual field theory side. For the latter we argue that there should be two different, but equivalent, dual gauge theory descriptions, one in terms of the 2d CFT's dual to the rotating BTZ and the other as certain large R-charge sectors of d = 4, = 4 U(N) SYM theory. We discuss new BMN-type sectors of the = 4 SYM in the N → ∞ limit in which the engineering dimensions scale as N3/2 (for the near-BPS case) and as N2 (for the far from BPS case).

A note on Chern–Simons solitons—A type III vortex from the wall vortex

We study some properties of topological Chern–Simons vortices in 2 + 1 dimensions. As has already been understood in the past, in the large magnetic flux limit, they are well described by a Chern–Simons domain wall, which has been compactified on a circle with the symmetric phase inside and the asymmetric phase on the outside. Our goal is two-fold. First we want to explore how the tension depends on the magnetic flux discretized by the integer n. The BPS case is already known, but not much has been explored about the non-BPS potentials. A generic renormalizable potential has two dimensionless parameters that can be varied. Variation of only one of them leads to a type I and type II vortex, very similar to the Abrikosov–Nielsen–Olesen (ANO) case. Variation of both the parameters leads to a much richer structure. In particular we have found a new type of vortex, which is type I-like for small flux and then turns type II-like for larger flux. We could tentatively denote it a type III vortex. This results in a stable vortex with number of fluxes which can be greater than one. Our second objective is to study the Maxwell–Chern–Simons theory and understand how the large n limit of the CS vortex is smoothly connected with the large n limit of the ANO vortex.

Bubbling AdS and droplet descriptions of BPS geometries in IIB supergravity

This paper focuses on supergravity duals of BPS states in = 4 super Yang-Mills. In order to describe these duals, we begin with a sequence of breathing mode reductions of IIB supergravity: first on S3, then S3 × S1, and finally on S3 × S1 × CP1. We then follow with a complete supersymmetry analysis, yielding 1/8, 1/4 and 1/2 BPS configurations, respectively (where in the last step we take the Hopf fibration of S3). The 1/8 BPS geometries, which have an S3 isometry and are time-fibered over a six-dimensional base, are determined by solving a non-linear equation for the Kahler metric on the base. Similarly, the 1/4 BPS configurations have an S3 × S1 isometry and a four-dimensional base, whose Kahler metric obeys another non-linear, Monge-Ampere type equation. Despite the non-linearity of the problem, we develop a universal bubbling AdS description of these geometries by focusing on the boundary conditions which ensure their regularity. In the 1/8 BPS case, we find that the S3 cycle shrinks to zero size on a five-dimensional locus inside the six-dimensional base. Enforcing regularity of the full solution requires that the interior of a smooth, generally disconnected five-dimensional surface be removed from the base. The AdS5 × S5 ground state corresponds to excising the interior of an S5, while the 1/8 BPS excitations correspond to deformations (including topology change) of the S5 and/or the excision of additional droplets from the base. In the case of 1/4 BPS configurations, by enforcing regularity conditions, we identify three-dimensional surfaces inside the four-dimensional base which separate the regions where the S3 shrinks to zero size from those where the S1 shrinks. We discuss a large class of examples to show the emergence of a universal bubbling AdS picture for all 1/2, 1/4 and 1/8 BPS geometries.

Script N=4 superconformal mechanics as a non linear realization

An action for a superconformal particle is constructed using the non linear realization method for the group PSU(1,1|2), without introducing superfields. The connection between PSU(1,1|2) and black hole physics is discussed. The lagrangian contains six arbitrary constants and describes a non-BPS superconformal particle. The BPS case is obtained if a precise relation between the constants in the lagrangian is verified, which implies that the action becomes kappa-symmetric.