Motivated by the recent interest in conformal and duality invariant nonlinear electrodynamics, we study the non-Abelian extension of ModMax electrodynamics. The theory is parameterized by a single dimensionless constant, and it is continuously connected to Yang-Mills theory in its vanishing limit. We show that the theory admits (anti-)self-dual instantons, despite the additional nonlinearities that characterize the non-Abelian ModMax theory. For SU(2), we construct the generalization of the BPST instanton and extend this solution to Euclidean de Sitter and anti-de Sitter backgrounds. In the latter case, the Chern-Pontryagin index depends on the instanton size since the configuration is not a pure gauge at infinity; a property already pointed out in Yang-Mills on negative-curvature backgrounds by Callan and Wilczek. We compute the contribution of the latter to the spectrum of the Dirac operator at the boundary, which is crucial for determining the non-local contributions to the Dirac index. Then, we show that the ansatz constructed with ’t Hooft symbols accommodates multi-instantons in the non-Abelian ModMax theory. The system of (anti-)self-dual equations reduces to a single nonlinear equation, which can be perturbatively solved order by order in the parameter that controls the nonlinearity. Following such a strategy, we provide a formal solution for the $\textit{N}$-instanton configuration to first order in the expansion. Then, we couple non-Abelian ModMax theory to gravity with a conformally coupled scalar field and construct new gravitating solutions that describe Euclidean wormholes and other smooth configurations with secondary hair.

Chile

Fabrizio Canfora

Cristóbal Corral

Borja Diez

Luis Guajardo

Julio Oliva

Self-dual instantons and gravitating dyons in non-Abelian ModMax theory

A novel Bogomol’nyi–Prasad–Sommerfield (BPS) bound for the Gross–Pitaevskii equations in two spatial dimensions is presented herein. The energy can be bounded from below in terms of the combination of two boundary terms, one related to the vorticity (but “dressed” by the condensate profile) and the second to the “skewness” of the configurations. The bound is saturated by configurations that satisfy a system of two first-order partial differential equations. When such a BPS system is satisfied, the Gross–Pitaevskii equations are also satisfied. The analytic solutions of this BPS system in the present manuscript represent configurations with fractional vorticity living in an annulus. Using these techniques, we present the first analytic examples of this kind. The hydrodynamic interpretation of the BPS system is discussed, and the implications of these results are outlined.

Czechia

Pablo Pais

A Bogomol’nyi–Prasad–Sommerfield bound with a first-order system in the 2$\textit{D}$ Gross–Pitaevskii equation

The nonlinear <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Σ</mi></math> model minimally coupled with Maxwell theory in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>3</mn><mo>+</mo><mn>1</mn></math> dimensions possesses a topologically nontrivial sector characterized by lasagnalike configurations. We demonstrate that, when a specific quantization condition is met, the associated second-order field equations admit a first-order Bogomol’nyi-Prasad-Sommerfield system. This discloses the existence of a (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math>)-dimensional supersymmetric extension with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>2</mn></math> supercharges. We examine its imprint on the time-dependent regime, with particular emphasis on the transition from integrability to chaos.

Argentina

Nicolás Grandi

Italy

Marcelo Oyarzo

Existence of a supersymmetric extension of the gauged nonlinear <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Σ</mi></math> model and the transition from integrability to chaos

The (generalized) Gross-Pitaevskii equation (GPE) for a complex scalar field in two spatial dimensions is analyzed. It is shown that there is an infinite family of self-interaction potentials which admit Bogomol'nyi-Prasad-Sommerfield (BPS) bounds together with the corresponding first-order BPS systems. For each member of this family, the solutions of the first-order BPS systems are automatically solutions of the corresponding second-order generalized GPE. The simplest topologically non-trivial solutions of these first-order BPS systems describe configurations with quantized fractional vorticity. The corresponding fraction is related to the degree of non-linearity. The case in which the self-interaction potential is of order six (namely <math altimg='si1.svg'><mo stretchy='false'>|</mo><mi mathvariant='normal'>&#936;</mi><msup><mrow><mo stretchy='false'>|</mo></mrow><mrow><mn>6</mn></mrow></msup></math>, which is a relevant theory both in relativistic quantum field theories in <math altimg='si2.svg'><mo stretchy='false'>(</mo><mn>2</mn><mo linebreak='badbreak' linebreakstyle='after'>+</mo><mn>1</mn><mo stretchy='false'>)</mo></math> dimensions in connection with the quantum Hall effect as well as in the theory of the supersolids) is analyzed in detail. Such formalism can also be extended to the case of quantum mixtures with multi-component GPEs. The relationship between these techniques and supersymmetry will be discussed. In particular, despite several common features, we will show that there are multi-component GPEs that are not supersymmetric (at least, not in the standard sense) and possess a BPS system of the above type.

Fractional vorticity, Bogomol'nyi-Prasad-Sommerfield systems and complex structures for the (generalized) spinor Gross-Pitaevskii equations

Euclidean anti–de Sitter (AdS) wormholes provide a natural setup for studying the anti–de Sitter/conformal field theory (AdS/CFT) correspondence with multiple boundaries. However, from a bottom-up perspective, they cannot be embedded in the four-dimensional Einstein-AdS-Maxwell theory if these boundaries have positive curvature. Nevertheless, Maldacena and Maoz showed that this obstruction could be circumvented by introducing merons in the four-dimensional Einstein-AdS-Yang-Mills theory. In this work, we show that Euclidean-AdS wormholes also exist in the four-dimensional Einstein-AdS-Skyrme theory, whose matter sector possesses a nontrivial baryonic charge. We compute its free energy and show that it does not depend on the integration constants whatsoever, resembling topological solitons. Additionally, we obtain its holographic stress tensor and show that it vanishes, allowing us to interpret this configuration as a holographic Bogomol’nyi-Prasad-Sommerfield (BPS) state. Other topologically nontrivial ground states in Einstein-Skyrme theory are found, such as gravitational instantons, representing the homotopically inequivalent vacua of the theory. We find that they develop Hawking-Page phase transitions above a critical temperature. Some of these solutions are periodic in Euclidean time, representing the gravitational analog of calorons in Yang-Mills theory.

Euclidean AdS wormholes and gravitational instantons in the Einstein-Skyrme theory

Recently, dynamical edge modes (DEM) in Maxwell theory have been constructed using a specific local boundary condition on the horizon. We discuss how to enforce this boundary condition on an infinite parallel plate in the QED vacuum by introducing Lagrange multiplier fields into the action. We carefully introduce appropriate boundary ghosts to maintain BRST invariance. Explicit correspondence of this BRST extended theory with the original DEM formulation is discussed, both directly, and through the correspondence between edge modes and Wilson lines attached to the boundary surface. We then use functional methods to calculate the Casimir energy for the first time with DEM boundary conditions imposed on two infinite parallel plates, both in generalized Coulomb and linear covariant gauge. Depending on the gauge, different fields are contributing, but, after correctly implementing the BRST symmetry, we retrieve the exact same Casimir energy as for two perfectly conducting parallel plates.

Belgium

David Dudal

Thomas Oosthuyse

Luigi Rosa

Sebbe Stouten

Dynamical edge modes in Maxwell theory from a BRST perspective, with an application to the Casimir energy

We derive a novel BPS bound from chiral perturbation theory minimally coupled to electrodynamics at finite isospin chemical potential. At a critical value of the isospin chemical potential, a system of three first-order differential field equations (which implies the second-order field equations) for the gauge field and the hadronic profile can be derived from the requirement to saturate the bound. These BPS configurations represent magnetic multi-vortices with quantized flux supported by a superconducting current. The corresponding topological charge density is related to the magnetic flux density, but is screened by the hadronic profile. Such a screening effect allows to derive the maximal value of the magnetic field generated by these BPS magnetic vortices, being B max = 2, 04 × 1014 G. The solution for a single BPS vortex is discussed in detail, and some physical consequences, together with the comparison with the magnetic vortices in the Ginzburg-Landau theory at critical coupling, are described.

Marcela Lagos

Aldo Vera

Superconducting multi-vortices and a novel BPS bound in chiral perturbation theory

We construct new exact solutions of the Georgi-Glashow model in <math><mn>3</mn><mo>+</mo><mn>1</mn></math> dimensions. These configurations are periodic in time but lead to a stationary energy density and no energy flux. Nevertheless, they possess a characteristic frequency which manifests itself through non-trivial resonances on test fields. This allows us to interpret them as non-Abelian self sustained coils. We show that for larger energies a transition to chaotic behavior takes place, which we characterize by Poincaré sections, Fourier spectra and exponential growth of the geodesic deviation in an effective Jacobi metric, the latter triggered by parametric resonances.

Nicolas Grandi

Exact oscillations and chaos on a non-Abelian coil

The first analytic solutions representing baryonic layers living at finite baryon density within a constant magnetic field in the gauged Skyrme model are constructed. A remarkable feature of these configurations is that, if the Skyrme term is neglected, then these baryonic layers in the constant magnetic background cannot be found analytically and their energies grow very fast with the magnetic field. On the other hand, if the Skyrme term is taken into account, the field equations can be solved analytically and the corresponding solutions have a smooth limit for large magnetic fields. Thus, the Skyrme term discloses the universal character of these configurations living at finite Baryon density in a constant magnetic field. The classical gran-canonical partition function of these configurations can be expressed explicitly in terms of the Pearcey integral. This fact allows us to determine analytically the Stokes lines of the partition function and the corresponding dependence on the baryonic chemical potential as well as on the external magnetic field. In this way, we can determine various critical curves in the (<math><msub><mrow><mi>μ</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>−</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>e</mi><mi>x</mi><mi>t</mi></mrow></msub></math>) plane which separates different physical behaviors. These families of inhomogeneous baryonic condensates can be also dressed with chiral conformal excitations of the solutions representing modulations of the layers themselves. Some physical consequences are analyzed.

Sergio L. Cacciatori

Federica Muscolino

Pearcey integrals, Stokes lines and exact baryonic layers in the low energy limit of QCD

A remarkable exact mapping, valid for low-enough energy scales and close to a sharp boundary distribution of hadronic matter, from the (<math><mrow><mn>3</mn><mo>+</mo><mn>1</mn></mrow></math>)-dimensional Skyrme model to the sine-Gordon theory in (<math><mn>1</mn><mo>+</mo><mn>1</mn></math>) dimensions in the attractive regime is explicitly constructed. Besides the intrinsic theoretical interest to be able to describe the prototype of nonintegrable theories (namely, quantum chromodynamics in the infrared regime) in terms of the prototype of integrable relativistic field theories (namely, sine-Gordon theory in (<math><mn>1</mn><mo>+</mo><mn>1</mn></math>) dimensions), we will show that this mapping can be extremely useful to analyze both equilibrium and out-of-equilibrium features of baryonic distributions in a cavity.

Exact mapping from the <math><mrow><mo>(</mo><mrow><mn>3</mn><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></math>-dimensional Skyrme model to the <math><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></math>-dimensional sine-Gordon theory and some applications