We study an inhomogeneous U(1) Chern-Simons Higgs model with a magnetic impurity in the BPS limit. The potential is sextic with both broken and unbroken phases, but its minimum varies spatially depending on the strength of the impurity. While the system lacks translation symmetry, it admits a supersymmetric extension. Depending on the sign of the impurity term, it has either a BPS sector or an anti-BPS sector (but not both), which satisfies the Bogomolny equations. The vacuum configuration of the broken phase is not simply determined by the minimum of the potential since it is no longer constant, but it becomes a nontrivial function satisfying the Bogomolny equations. Thus, the energy and angular momentum densities of the vacuum locally have nonzero distributions, although the total energy and angular momentum remain zero. As in the homogeneous case, the theory supports various BPS soliton solutions, including topological and nontopological vortices and Q-balls. The vorticities as well as the U(1) charges are exclusively positive or negative. For a Gaussian type impurity as a specific example, we obtain rotationally symmetric numerical solutions and analyze their detailed properties. We also discuss the case of a delta-function impurity as the infinitely thin limit of the Gaussian impurity which shows some nontrivial feature of BPS Chern-Simons Higgs theory.

South Korea

Yoonbai Kim

O-Kab Kwon

Hanwool Song

Chanju Kim

Inhomogeneous abelian Chern-Simons Higgs model with new inhomogeneous BPS vacuum and solitons

<p>The inhomogeneous Abelian Higgs model with a magnetic impurity in the Bogomolny-Prasad-Sommerfield (BPS) limit is studied for both relativistic and nonrelativistic regimes. Though the symmetry of spatial translation is broken by inhomogeneity, extension to an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>1</mn></math> supersymmetric theory is admitted. The quartic scalar potential has a minimum value depending on strength of the impurity but possesses a broken phase at a spatial asymptote. The vacuum configuration of a broken phase can be neither a constant nor the minimum of the scalar potential but is found as a nontrivial solution of the Bogomolny equations. While its energy density and magnetic field are given by the function of spatial coordinates, the energy and magnetic flux remain zero. The sign of the magnetic impurity term allows either a BPS sector or anti-BPS sector but not both. Thus, the obtained solution is identified as the new inhomogeneous broken vacuum of minimum zero energy. In the presence of rotationally symmetric Gaussian type inhomogeneity, topological vortex solutions are also obtained, and the effects of the impurity to the vortex are numerically analyzed.</p>

SeungJun Jeon

Vacuum and vortices in the inhomogeneous Abelian Higgs model

We study spontaneous supersymmetry breaking in inhomogeneous extensions of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 supersymmetric field theory models in 4-dimensions. The   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 Abelian Higgs model with the inhomogeneous mass parameter and the FI coefficient that are dependent on spatial coordinates, as well as the O’Raifeartaigh model with all its parameters being dependent on spatial coordinates, are studied in detail. In the presence of inhomogeneous parameters, half supersymmetry can be preserved by adding appropriate inhomogeneous deformations to the original Lagrangians. The inhomogeneous deformations often break the R-symmetry explicitly. In cases where the inhomogeneous deformations do not break the R-symmetry explicitly, we demonstrate that spontaneous breaking of the R-symmetry is infeasible. We argue that those models can not be spontaneous supersymmetry breaking models, according to the Nelson-Seiberg argument. We comment on this issue in the context of a generic   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 supersymmetric model as well.

D. Tolla

Spontaneous supersymmetry breaking in inhomogeneous supersymmetric field theories and BPS vacua

It has been known for some time that the cosmological bound on the invisible axion scale can be avoided by an early phase of strong QCD. While most approaches rely on theories where the strong coupling constant is determined through the expectation value of some scalar field, we show that this mechanism can also be implemented into the benchmark Kim-Shifman-Vainshtein-Zakharov and Dine-Fischler-Srednicki-Zhitnitsky models when the early phase of strong QCD emerges by the modification of the running coupling during inflation. For both models the physics that are responsible for making QCD strong do not displace the axions minimum by too much, so that the efficiency of the relaxation is controlled by parameters of the theory and the number of inflationary <math><mi>e</mi></math>-folds. In particular, we consider the case of very efficient relaxation where the axion abundance is dominated by inflationary quantum and postinflationary thermal fluctuations. Within this situation we identify the parameter space compatible with all cosmological constrains and derive conditions on the reheating temperature and the QCD scale during inflation that result in the axion making up all the dark matter. Because of duality below the Peccei-Quinn scale and a minor influence of the Kim-Shifman-Vainshtein-Zakharov and Dine-Fischler-Srednicki-Zhitnitsky fields on the running, our findings also apply to a minimal two-form implementation of the axion.

Germany

Emmanouil Koutsangelas

Removing the cosmological bound on the axion scale in the Kim-Shifman-Vainshtein-Zakharov and Dine-Fischler-Srednicki-Zhitnitsky models

We study supersymmetric inhomogeneous field theories in 1+1 dimensions which have explicit coordinate dependence. Although translation symmetry is broken, part of supersymmetries can be maintained. In this paper, we consider the simplest inhomogeneous theories with one real scalar field, which possess an unbroken supersymmetry. The energy is bounded from below by the topological charge which is not necessarily nonnegative definite. The bound is saturated if the first-order Bogomolny equation is satisfied. Non-constant static supersymmetric solutions above the vacuum involve in general a zero mode although the system lacks translation invariance. We consider two inhomogeneous theories obtained by deforming supersymmetric sine-Gordon theory and ϕ 6 theory. They are deformed either by overall inhomogeneous rescaling of the superpotential or by inhomogeneous deformation of the vacuum expectation value. We construct explicitly the most general supersymmetric solutions and obtain the BPS energy spectrum for arbitrary position-dependent deformations. Nature of the solutions and their energies depend only on the boundary values of the inhomogeneous functions. The vacuum of minimum energy is not necessarily a constant configuration. In some cases, we find a one-parameter family of degenerate solutions which include a non-vacuum constant solution as a special case.

Supersymmetric inhomogeneous field theories in 1+1 dimensions

We construct the 4-dimensional   <math> <mi>N</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math>  $$ \mathcal{N}=\frac{1}{2} $$  and   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 inhomogeneously mass-deformed super Yang-Mills theories from the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1* and   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2* theories, respectively, and analyse their supersymmetric vacua. The inhomogeneity is attributed to the dependence of background fluxes in the type IIB supergravity on a single spatial coordinate. This gives rise to inhomogeneous mass functions in the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super Yang-Mills theory which describes the dynamics of D3-branes. The Killing spinor equations for those inhomogeneous theories lead to the supersymmetric vacuum equation and a boundary condition. We investigate two types of solutions in the   <math> <mi>N</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math>  $$ \mathcal{N}=\frac{1}{2} $$  theory, corresponding to the cases of asymptotically constant mass functions and periodic mass functions. For the former case, the boundary condition gives a relation between the parameters of two possibly distinct vacua at the asymptotic boundaries. Brane interpretations for corresponding vacuum solutions in type IIB supergravity are also discussed. For the latter case, we obtain explicit forms of the periodic vacuum solutions.

Super Yang-Mills theories with inhomogeneous mass deformations

We construct renormalized holographic entanglement entropy (HEE) and subregion complexity (HSC) in the CV conjecture for asymptotically AdS4 and AdS5 geometries under relevant perturbations. Using the holographic renormalization method developed in the gauge/gravity duality, we obtain counter terms which are invariant under coordinate choices. We explicitly define different forms of renormalized HEE and HSC, according to conformal dimensions of relevant operators in the d = 3 and d = 4 dual field theories. We use a general embedding for arbitrary entangling subregions and showed that any choice of the coordinate system gives the same form of the counter terms, since they are written in terms of curvature invariants and scalar fields on the boundaries. We show an explicit example of our general procedure. Intriguingly, we find that a divergent term of the HSC in the asymptotically AdS5 geometry under relevant perturbations with operators of conformal dimensions in the range 0 &lt; ∆ &lt;   <math> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math>  $$ \frac{1}{2} $$  and   <math> <mfrac> <mn>7</mn> <mn>2</mn> </mfrac> </math>  $$ \frac{7}{2} $$  &lt; ∆ &lt; 4 cannot be cancelled out by adding any coordinate invariant counter term. This implies that the HSCs in these ranges of the conformal dimensions are not renormalizable covariantly. We also write the plot of the renormalization procedure in the case of asymptotically AdS d+1 geometries, with d &gt; 4.

Dongmin Jang

Renormalized holographic subregion complexity under relevant perturbations

We investigate the gauge/gravity duality between the $$\mathcal{N} = 6$$ <math><mrow><mi>N</mi><mo>=</mo><mn>6</mn></mrow></math>  mass-deformed ABJ theory with $$\hbox {U}_k(N+l)\times \hbox {U}_{-k}(N)$$ <math><mrow><msub><mtext>U</mtext><mi>k</mi></msub><mrow><mo>(</mo><mi>N</mi><mo>+</mo><mi>l</mi><mo>)</mo></mrow><mo>×</mo><msub><mtext>U</mtext><mrow><mo>-</mo><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow></math>  gauge symmetry and the 11-dimensional supergravity on LLM geometries with SO(2,1) $$\times $$ <math><mo>×</mo></math>  SO(4)/$${\mathbb Z}_k\times $$ <math><mrow><msub><mi>Z</mi><mi>k</mi></msub><mo>×</mo></mrow></math>  SO(4)/$${{\mathbb {Z}}}_k$$ <math><msub><mi>Z</mi><mi>k</mi></msub></math>  isometry and the discrete torsion l. For chiral primary operators with conformal dimensions $$\Delta =1,2$$ <math><mrow><mi>Δ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></math> , we obtain the exact vacuum expectation values using the holographic method in 11-dimensional supergravity and show that the results depend on the shapes of droplet pictures in LLM geometries. The $$\frac{l}{\sqrt{N}}$$ <math><mfrac><mi>l</mi><msqrt><mi>N</mi></msqrt></mfrac></math>  contributions from the discrete torsion l for several simple droplet pictures in the large N limit are determined in holographic vacuum expectation values. We also explore the effects of the orbifolding $${{\mathbb {Z}}}_k$$ <math><msub><mi>Z</mi><mi>k</mi></msub></math>  and the asymptotic discrete torsion l, on the gauge/gravity duality dictionary and on the nature of the asymptotic limits of the LLM geometries.

Holography of massive M2-brane theory with discrete torsion

Recently,   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 3 mass-deformed ABJM model with arbitrary mass-function depending on a spatial coordinate was constructed. In addition to the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 3 case, we construct lower supersymmetric   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 and   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 inhomogeneously mass-deformed ABJM (ImABJM) models, which require three and two arbitrary mass-functions, respectively. We also construct general vacuum solutions of the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 3 ImABJM model for any periodic mass-function. There are two classes of vacua, which are diagonal type and GRVV type according to reference value of mass-functions. We provide explicit examples of the vacuum solutions and discuss related operators.

Kyung Kim

Aspects of massive ABJM models with inhomogeneous mass parameters

We study scalar perturbations of the five-dimensional rotating black holes and find an exact solution giving exact description of the Hawking radiation. Mathematically, the full solution for this spin-zero field is expressed in terms of the prolate spheroidal wave function with complex parameters. By using the spheroidal joining factor, we write the corresponding boundary condition and greybody factors. We also check that the exact result reproduces the low frequency limit of the greybody factor and shows good agreement with the known numerical results.