Bogomol'nyi-Prasad-Sommerfield Research Articles

Nuclear binding energies from a Bogomol'nyi-Prasad-Sommerfield Skyrme model

Recently, within the space of generalized Skyrme models, a submodel with a Bogomol'nyi-Prasad-Sommerfield (BPS) bound was identified that reproduces some bulk properties of nuclear matter already on a classical level and, as such, constitutes a promising field theory candidate for the detailed and reliable description of nuclei and hadrons. Here we extend and further develop these investigations by applying the model to the calculation of nuclear binding energies. Concretely, we calculate these binding energies by including the classical soliton energies, the excitation energies from the collective coordinate quantization of spin and isospin, the electrostatic Coulomb energies, and a small explicit isospin symmetry breaking, which accounts for the mass difference between proton and neutron. The integrability properties of the BPS Skyrme model allow, in fact, for an analytical calculation of all contributions, which may then be compared with the semi-empirical mass formula. We find that for heavier nuclei, where the model is expected to be more accurate on theoretical grounds, the resulting binding energies are already in excellent agreement with their physical values. This result provides further strong evidence for the viability of the BPS Skyrme model as a distinguished starting point and lowest-order approximation for the detailed quantitative investigation of nuclear and hadron physics.

BPS Invariants of Semi-Stable Sheaves on Rational Surfaces

Bogomolnyi-Prasad-Sommerfield (BPS) invariants are computed, capturing topo- logical invariants of moduli spaces of semi-stable sheaves on rational surfaces. For a suit- able stability condition, it is proposed that the generating function of BPS invariants of a Hirzebruch surface � � takes the form of a product formula. BPS invariants for other stability conditions and other rational surfaces are obtained using Harder-Narasimhan fil- trations and the blow-up formula. Explicit expressions are given for rank ≤3 sheaves on � � and the projective plane P 2 . The applied techniques can be applied iteratively to com- pute invariants for higher rank.

Skyrmions with vector mesons in the hidden local symmetry approach

The roles of light $\rho$ and $\omega$ vector mesons in the Skyrmion are investigated in a chiral Lagrangian derived from the hidden local symmetry (HLS) up to $O(p^4)$ including the homogeneous Wess-Zumino (hWZ) terms. We write a general "master formula" that allows us to determine the parameters of the HLS Lagrangian from a class of holographic QCD models valid at large $N_c$ and $\lambda$ ('t Hooft constant) limit by integrating out the infinite towers of vector and axial-vector mesons other than the lowest $\rho$ and $\omega$ mesons. Within this approach we find that the physical properties of the Skyrmion as the solitonic description of baryons are \textit{independent} of the HLS parameter $a$. Therefore the only parameters of the model are the pion decay constant and the vector meson mass. Once determined in the meson sector, we have a totally parameter-free theory that allows us to study unequivocally the role of light vector mesons in the Skyrmion structure. We find, as suggested by Sutcliffe, that inclusion of the $\rho$ meson reduces the soliton mass, which makes the Skyrmion come closer to the Bogomol'nyi-Prasad-Sommerfield (BPS) soliton, but the role of the $\omega$ meson is found to increase the soliton mass. In a stark contrast, the $\Delta$-$N$ mass difference, which is determined by the moment of inertia in adiabatic collective quantization of the Skyrmion, is increased by the $\rho$ vector meson, while it is reduced by the inclusion of the $\omega$ meson. All these observations show the importance of the $\omega$ meson in the properties of the nucleon and nuclear matter in the Skyrme model.

Topological duality between vortices and planar Skyrmions in BPS theories with area-preserving diffeomorphism symmetries

The Bogomol'nyi-Prasad-Sommerfield (BPS) baby Skyrme models are submodels of baby Skyrme models, where the nonlinear sigma model term is suppressed. They have Skyrmion solutions saturating a BPS bound, and the corresponding static energy functional is invariant under area-preserving diffeomorphisms (APDs). Here we show that the solitons in the BPS baby Skyrme model, which carry a nontrivial topological charge ${Q}_{b}\ensuremath{\in}{\ensuremath{\pi}}_{2}({S}^{2})$ (a winding number), are dual to vortices in a BPS vortex model with a topological charge ${Q}_{v}\ensuremath{\in}{\ensuremath{\pi}}_{1}({S}^{1})$ (a vortex number), in the sense that there is a map between the BPS solutions of the two models. The corresponding energy densities of the BPS solutions of the two models are identical. A further consequence of the duality is that the dual BPS vortex models inherit the BPS property and the infinitely many symmetries (APDs) of the BPS baby Skyrme models. Finally, we demonstrate that the same topological duality continues to hold for the $U(1)$ gauged versions of the models.

Asymptotic formulas for coefficients of inverse theta functions

We determine asymptotic formulas for the coefficients of a natural class of negative index and negative weight Jacobi forms. These coefficients can be viewed as a refinement of the numbers $p_k(n)$ of partitions of n into k colors. Part of the motivation for this work is that they are equal to the Betti numbers of the Hilbert scheme of points on an algebraic surface S and appear also as counts of Bogomolny-Prasad-Sommerfield (BPS) states in physics.

Integrality of relative BPS state counts of toric del Pezzo surfaces

Relative Bogomolny-Prasad-Sommerfield (BPS) state counts for log Calabi-Yau surface pairs were introduced by Gross-Pandharipande-Siebert in [4] and conjectured by the authors to be integers. For toric del Pezzo surfaces, we provide an arithmetic proof of this conjecture, by relating these invariants to the local BPS state counts of the surfaces. The latter were shown to be integers by Peng in [15]; and more generally for toric Calabi-Yau three-folds by Konishi in [8].

Semiclassical correlator for a1/4BPS Wilson loop and a chiral primary operator with a large R charge

We study a holographic description for correlation function of $1/4$ Bogomol'nyi-Prasad-Sommerfield (BPS) Wilson loop operator and $1/2$ BPS local operator carrying a large R charge of order $\sqrt{\ensuremath{\lambda}}$. We construct a rotating string solution that is extended in ${\mathrm{S}}^{5}$ as well as in ${\mathrm{AdS}}_{5}$. The string solution preserves the $1/8$ of the supersymmetry as expected from the gauge theory computation. By evaluating the string action including boundary terms, we show that the string solution reproduces correlation function in the large $J\ensuremath{\sim}\mathcal{O}(\sqrt{\ensuremath{\lambda}})$ limit. In addition, we found the second solution for which the ``size'' of the string becomes larger than the radius of ${\mathrm{S}}^{5}$. In the case $J=0$, this solution reduces to the previously known unstable string configuration. The gauge theory side also contains a saddle point that is not on the steepest descent path. We show that the saddle point value matches for this case as well.

Generalized BPS magnetic monopoles

We show the existence of Bogomol'nyi-Prasad-Sommerfield (BPS) magnetic monopoles in a generalized Yang-Mills-Higgs model which is controlled by two positive functions. This effective model, in principle, would describe the dynamics of the nonabelian fields in a chromoelectric media. We check the consistency of our generalized construction by analyzing an explicit case ruled by a real parameter. We also use the well-known spherically symmetric Ansatz to attain the corresponding self-dual equations describing the topological solutions. The overall conclusion is that the new solutions behave around the canonical one, with smaller or greater characteristic length.

Hidden local symmetry and infinite tower of vector mesons for baryons

In an effort to access dense baryonic matter relevant for compact stars in a unified framework that handles both single baryon and multibaryon systems on the same footing, we first address a holographic dual action for a single baryon focusing on the role of the infinite tower of vector mesons deconstructed from five dimensions. To leading order in 't Hooft coupling $\lambda=N_c g_{\rm YM}^2$, one has the Bogomol'nyi-Prasad-Sommerfield (BPS) Skyrmion that results when the warping of the bulk background and the Chern-Simons term in the Sakai-Sugimoto D4/D8-${\bar{\rm D8}}$ model are ignored. The infinite tower was found by Sutcliffe to induce flow to a conformal theory, i.e., the BPS. We compare this structure to that of the SS model consisting of a 5D Yang-Mills action in warped space and the Chern-Simons term in which higher vector mesons are integrated out while preserving hidden local symmetry and valid to $O(\lambda^0)$ and $O(p^4)$ in the chiral counting. We point out the surprisingly important role of the $\omega$ meson that figures in the Chern-Simons term that encodes chiral anomaly in the baryon structure and that may be closely tied to short-range repulsion in nuclear interactions.

Vector BPS Skyrme model

We analyze the vector meson formulation of the Bogomol'nyi-Prasad-Sommerfield (BPS) Skyrme model in ($3+1$) dimensions, where the term of sixth power in first derivatives characteristic for the original, integrable BPS Skyrme model (the topological or baryon current squared) is replaced by a coupling between the vector meson ${\ensuremath{\omega}}_{\ensuremath{\mu}}$ and the baryon current. We find that the model remains integrable in the sense of generalized integrability and almost solvable (reducible to a set of two first-order ordinary differential equations) for any value of the baryon charge. Further, we analyze the appearance of topological solitons for two one-parameter families of one-vacuum potentials: the old Skyrme potentials and the so-called BPS potentials. Depending on the value of the parameters, we find several qualitatively different possibilities. In the massless case, we have a parameter region with no Skyrmions, a unique compact Skyrmion with a discontinuous first derivative at the boundary (equivalently, with a source term located at the boundary, which screens the topological charge), and Coulomb-like localized solitons. For the massive vector meson, besides the no-Skyrmion region and a unique $C$-compact soliton, we find exponentially as well as power-like localized Skyrmions. Further, we find (for a specific potential) BPS solutions, i.e., Skyrmions saturating a Bogomolny bound (both for the massless and massive vector mesons), which are unstable for higher values of the baryon charge. The properties of the model are finally compared with its baby version in ($2+1$) dimensions and with the original BPS Skyrme model, contributing to a better understanding of the latter.

Sigma model BPS lumps on a torus

We study doubly periodic Bogomol'nyi-Prasad-Sommerfield (BPS) lumps in supersymmetric CP^{N-1} non-linear sigma models on a torus T^2. Following the philosophy of the Harrington-Shepard construction of calorons in Yang-Mills theory, we obtain the n-lump solutions on compact spaces by suitably arranging the n-lumps on R^2 at equal intervals. We examine the modular invariance of the solutions and find that there are no modular invariant solutions for n=1,2 in this construction.

Geodesic motion in the space-time of cosmic strings interacting via magnetic fields

We study the geodesic motion of test particles in the space-time of two Abelian-Higgs strings interacting via their magnetic fields. These bound states of cosmic strings constitute a field theoretical realization of p-q-strings which are predicted by inflationary models rooted in String Theory, e.g. brane inflation. In contrast to previously studied models describing p-q-strings our model possesses a Bogomolnyi-Prasad-Sommerfield (BPS) limit. If cosmic strings exist it would be exciting to detect them by direct observation. We propose that this can be done by the observation of test particle motion in the space-time of these objects. In order to be able to make predictions we have to solve the field equations describing the configuration as well as the geodesic equation numerically. The geodesics can then be classified according to the test particle's energy, angular momentum and momentum along the string axis. We find that the interaction of two Abelian-Higgs strings can lead to the existence of bound orbits that would be absent without the interaction. We also discuss the minimal and maximal radius of orbits and comment on possible applications in the context of gravitational wave emission.

Generalized self-dual Maxwell-Chern-Simons-Higgs model

We present a consistent Bogomol'nyi-Prasad-Sommerfield (BPS) framework for a generalized Maxwell-Chern-Simons-Higgs model. The overall model, including its self-dual potential, depends on three different functions, $h(|\ensuremath{\phi}|,N)$, $w(|\ensuremath{\phi}|)$, and $G(|\ensuremath{\phi}|)$, which are functions of the scalar fields only. The BPS energy is proportional to the magnetic flux when $w(|\ensuremath{\phi}|)$ and $G(|\ensuremath{\phi}|)$ are related to each other by a differential constraint. We present an explicit nonstandard model and its topologically nontrivial static configurations, which are described by the usual radially symmetric profile. Finally, we note that the nonstandard results behave in a similar way as their standard counterparts, as expected, reinforcing the consistence of the overall construction.

Brane Effective Actions, Kappa-Symmetry and Applications.

This is a review on brane effective actions, their symmetries and some of their applications. Its first part covers the Green-Schwarz formulation of single M- and D-brane effective actions focusing on kinematical aspects: the identification of their degrees of freedom, the importance of world volume diffeomorphisms and kappa symmetry to achieve manifest spacetime covariance and supersymmetry, and the explicit construction of such actions in arbitrary on-shell supergravity backgrounds.Its second part deals with applications. First, the use of kappa symmetry to determine supersymmetric world volume solitons. This includes their explicit construction in flat and curved backgrounds, their interpretation as Bogomol’nyi-Prasad-Sommerfield (BPS) states carrying (topological) charges in the supersymmetry algebra and the connection between supersymmetry and Hamiltonian BPS bounds. When available, I emphasise the use of these solitons as constituents in microscopic models of black holes. Second, the use of probe approximations to infer about the non-trivial dynamics of strongly-coupled gauge theories using the anti de Sitter/conformal field theory (AdS/CFT) correspondence. This includes expectation values of Wilson loop operators, spectrum information and the general use of D-brane probes to approximate the dynamics of systems with small number of degrees of freedom interacting with larger systems allowing a dual gravitational description.Its final part briefly discusses effective actions for N D-branes and M2-branes. This includes both Super-Yang-Mills theories, their higher-order corrections and partial results in covariantising these couplings to curved backgrounds, and the more recent supersymmetric Chern-Simons matter theories describing M2-branes using field theory, brane constructions and 3-algebra considerations.

Vortices and monopoles in mass-deformed SO and USp gauge theories

Effects of mass deformations on 1/2 Bogomol'nyi-Prasad-Sommerfield (BPS) non-Abelian vortices are studied in 4d N=2 supersymmetric U(1) \times SO(2n) and U(1) \times USp(2n) gauge theories, with Nf=2n quark multiplets. The 2d N=(2,2) effective worldsheet sigma models on the Hermitian symmetric spaces SO(2n)/U(n) and USp(2n)/U(n) found recently which describe the low-energy excitations of the orientational moduli of the vortices, are generalized to the respective massive sigma models. The continuous vortex moduli spaces are replaced by a finite number (2^{n-1} or 2^{n}) of vortex solutions. The 1/2 BPS kinks connecting different vortex vacua are magnetic monopoles in the 4d theory, trapped inside the vortex core, with total configurations being 1/4 BPS composite states. These configurations are systematically studied within the semi-classical regime.

Gravitating semilocal strings

We discuss the properties of semilocal strings minimally coupled to gravity. Semilocal strings are solutions of the bosonic sector of the Standard Model in the limit sin2 θW = 1 (where θW is the Weinberg angle) and correspond to embedded Abelian-Higgs strings for a particular choice of the scalar doublet. We focus on the limit where the gauge boson mass is equal to the Higgs boson mass such that the solutions fulfil the Bogomolnyi-Prasad-Sommerfield (BPS) bound.

Walls in a supersymmetric massive nonlinear sigma model on a complex quadric surface

The Bogomol'nyi-Prasad-Sommerfield (BPS) multiwall solutions are constructed in a massive K\"ahler nonlinear sigma model on the complex quadric surface, ${Q}^{N}=\frac{SO(N+2)}{SO(N)\ifmmode\times\else\texttimes\fi{}SO(2)}$ in 3-dimensional space-time. The theory has a nontrivial scalar potential generated by the Scherk-Schwarz dimensional reduction from the massless nonlinear sigma model on ${Q}^{N}$ in 4-dimensional space-time and it gives rise to $2[N/2+1]$ discrete vacua. The BPS wall solutions connecting these vacua are obtained based on the moduli matrix approach. It is also shown that the moduli space of the BPS wall solutions is the complex quadric surface ${Q}^{N}$.

Magnetic bion condensation: A new mechanism of confinement and mass gap in four dimensions

In recent work, we derived the long-distance confining dynamics of certain QCD-like gauge theories formulated on small ${S}^{1}\ifmmode\times\else\texttimes\fi{}{\mathbb{R}}^{3}$ based on symmetries, an index theorem, and Abelian duality. Here, we give the microscopic derivation. The solution reveals a new mechanism of confinement in QCD(adj) in the regime where we have control over both perturbative and nonperturbative aspects. In particular, consider $SU(2)$ QCD(adj) theory with $1\ensuremath{\le}{n}_{f}\ensuremath{\le}4$ Majorana fermions, a theory which undergoes gauge symmetry breaking at small ${S}^{1}$. If the magnetic charge of the Bogomol'nyi-Prasad-Sommerfield (BPS) monopole is normalized to unity, we show that confinement occurs due to condensation of objects with magnetic charge 2, not 1. Because of index theorems, we know that such an object cannot be a two identical monopole configuration. Its net topological charge must vanish, and hence it must be topologically indistinguishable from the perturbative vacuum. We construct such non-self-dual topological excitations, the magnetically charged, topologically null molecules of a BPS monopole and $\overline{\mathrm{KK}}$ antimonopole, which we refer to as magnetic bions. An immediate puzzle with this proposal is the apparent Coulomb repulsion between the BPS-$\overline{\mathrm{KK}}$ pair. An attraction which overcomes the Coulomb repulsion between the two is induced by $2{n}_{f}$-fermion exchange. Bion condensation is also the mechanism of confinement in $\mathcal{N}=1$ SYM on the same four-manifold. The $SU(N)$ generalization hints a possible hidden integrability behind nonsupersymmetric QCD of affine Toda type, and allows us to analytically compute the mass gap in the gauge sector. We currently do not know the extension to ${\mathbb{R}}^{4}$.

Cosmic strings interacting with dark strings

Motivated by astrophysical observations of excess electronic production in the galaxy, new theoretical models of the matter sector have been proposed in which the Standard model couples to the matter sector through an attractive interaction. The coupling of the Standard model to dark strings, which are solutions of the low energy sector has been investigated recently. Here, we discuss the interaction between strings and standard cosmic strings and show that they can form bound states. In the presence of the interaction term, a Bogomolny-Prasad-Sommerfield (BPS) bound exists that depends on the interaction parameter and we observe that the attractive interaction between strings and cosmic strings is most efficient if the two strings are identical. Moreover, our model allows for string solution that can lower their energy by coupling to an electromagnetic field. We also investigate the gravitational properties of these solutions and show that they become supermassive with a singular space-time for values of the gravitational coupling larger as compared to the non-interacting case. Moreover, the deficit angle of the solutions decreases with increasing interaction.

Multiple layer structure of non-Abelian vortex

Bogomolny–Prasad–Sommerfield (BPS) vortices in U(N) gauge theories have two layers corresponding to non-Abelian and Abelian fluxes, whose widths depend nontrivially on the ratio of U(1) and SU(N) gauge couplings. We find numerically and analytically that the widths differ significantly from the Compton lengths of lightest massive particles with the appropriate quantum number.