We derive the explicit formula for fractional BPS lumps (or fractional instantons) in the $\textit{ℂP}$ nonlinear sigma model on a two-dimensional torus under various shift-clock twisted boundary conditions. After regularizing the $\textit{ℂP}$ model by an $\textit{N}$-component Abelian-Higgs model, those twisted boundary conditions introduce nontrivial ’t Hooft fluxes $\textit{p}$/$\textit{N}$ for the U(1) gauge field, and the topological charge becomes fractionalized as $\textit{k}$ + $\textit{p}$/$\textit{N}$ ∈ $\textit{ℤ}$ + $\textit{p}$/$\textit{N}$. The moduli space is globally determined as the $\textit{ℂP}$ -fiber bundle on a 2-torus, which is a Kähler manifold of complex dimension $\textit{Nk}$ + $\textit{p}$ as predicted by the index theorem. We present two different parametrizations of the moduli space: one of them immediately identifies the small-lump singularity appearing in the $\textit{ℂP}$ limit, while the other makes the modular invariance manifest.

Japan

Yui Hayashi

Tatsuhiro Misumi

Muneto Nitta

Keisuke Ohashi

Yuya Tanizaki

Fractional instantons in 2d $\textit{ℂP}$ $\textit{N}$ − 1 model and 4d Yang-Mills theory with ’t Hooft twists

<p>We numerically construct a stable three-string junction of a Y shape in a flag manifold nonlinear sigma model on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>/</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>.</p>

Yuki Amari

Toshiaki Fujimori

String junctions in flag manifold sigma models

In this paper, we discuss lumps (sigma model instantons) in flag manifold sigma models. In particular, we focus on the moduli space of BPS lumps in general Kähler flag manifold sigma models. Such a Kähler flag manifold, which takes the form   <math> <mfrac> <mrow> <mi>U</mi> <mfenced> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>n</mi> <mrow> <mi>L</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </mfenced> </mrow> <mrow> <mi>U</mi> <mfenced> <msub> <mi>n</mi> <mn>1</mn> </msub> </mfenced> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <mi>U</mi> <mfenced> <msub> <mi>n</mi> <mrow> <mi>L</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mfenced> </mrow> </mfrac> </math>  $$ \frac{\textrm{U}\left({n}_1+\cdots +{n}_{L+1}\right)}{\textrm{U}\left({n}_1\right)\times \cdots \times \textrm{U}\left({n}_{L+1}\right)} $$ , can be realized as a vacuum moduli space of a U(N 1) × ··· × U(N  L ) quiver gauged linear sigma model. When the gauge coupling constants are finite, the gauged linear sigma model admits BPS vortex configurations, which reduce to BPS lumps in the low energy effective sigma model in the large gauge coupling limit. We derive an ADHM-like quotient construction of the moduli space of BPS vortices and lumps by generalizing the quotient construction in U(N) gauge theories by Hanany and Tong. As an application, we check the dualities of the 2d models by computing the vortex partition functions using the quotient construction.

Moduli spaces of instantons in flag manifold sigma models. Vortices in quiver gauge theories

Abstract We derive an explicit form of the dilaton potential in improved holographic QCD (IHQCD) from the QCD lattice data of the chiral condensate as a function of the quark mass. This establishes a data-driven holographic modeling of QCD—machine-learning holographic QCD. The modeling consists of two steps for solving inverse problems. The first inverse problem is to find the emergent bulk geometry consistent with the lattice QCD simulation data at the boundary. We solve this problem with the refinement of the neural ordinary differential equation, a machine-learning technique. The second inverse problem is to derive a bulk gravity action with a dilaton potential such that its solution is the emergent bulk geometry. We solve this problem at non-zero temperature, and derive the explicit form of the dilaton potential. The dilaton potential determines the bulk action, the Einstein-dilaton system; thus we derive holographically the bulk system from the QCD chiral condensate data. The usefulness of the model is shown in the example of the prediction of the string breaking distance, whose value is found to be consistent with other lattice QCD data.

Koji Hashimoto

Takayuki Sumimoto

Deriving the dilaton potential in improved holographic QCD from the chiral condensate

We derive an explicit form of the dilaton potential in improved holographic QCD (IHQCD) from the experimental data of the <math><mi>ρ</mi></math> meson spectrum. For this purpose, we make use of the emergent bulk geometry obtained by deep learning from the hadronic data in the work by Akutagawa et al. [Phys. Rev. D 102, 026020 (2020)]. Requiring that the geometry is a solution of an IHQCD derives the corresponding dilaton potential backward. This determines the bulk action in a data-driven way, which enables us at the same time to ensure that the deep learning proposal is a consistent gravity. Furthermore, we find that the resulting potential satisfies the requirements normally imposed in IHQCD, and that the holographic Wilson loop for the derived model exhibits quark confinement.

Deriving the dilaton potential in improved holographic QCD from the meson spectrum

Abstract We discuss a supersymmetric extension of a non-relativistic Chern–Simons matter theory, known as the supersymmetric Jackiw–Pi model, in a harmonic trap. We show that the non-relativistic version of the superconformal symmetry, called the super-Schrödinger symmetry, is not spoiled by an external field including the harmonic potential. It survives as a modified symmetry whose generators have explicit time dependences determined by the strength of the trap, the rotation velocity of the system, and the fermion number chemical potential. We construct 1/3 Bogomol'nyi–Prasad–Sommerfield (BPS) states of trapped Jackiw–Pi vortices preserving part of the modified superconformal symmetry and discuss fluctuations around static BPS configurations. In addition to the bosonic massive Nambu–Goldstone modes, we find that there exist massive Nambu–Goldstone fermions associated with broken generators of the modified super-Schrödinger symmetry. Furthermore, we find that eigenmodes form supermultiplets of a modified supersymmetry preserved by the static BPS backgrounds. As a consequence of the modified supersymmetry, infinite towers of explicit spectra can be found for eigenmodes corresponding to bosonic and fermionic lowest Landau levels.

Massive Nambu–Goldstone fermions and bosons for non-relativistic superconformal symmetry: Jackiw–Pi vortices in a trap

In this paper we examine analytically the large-N gap equation and its solution for the 2D ℂℙ N −1 sigma model defined on a Euclidean spacetime torus of arbitrary shape and size (L, β), β being the inverse temperature. We find that the system has a unique homogeneous phase, with the ℂℙ N −1 fields n  i  acquiring a dynamically generated mass (λ) ≥ Λ2 (analogous to the mass gap of SU(N ) Yang-Mills theory in 4D), for any β and L. Several related topics in the recent literature are discussed. One concerns the possibility, which turns out to be excluded according to our analysis, of a “Higgs-like” — or deconfinement — phase at small L and at zero temperature. Another topics involves “soliton-like” (inhomogeneous) solutions of the generalized gap equation, which we do not find. A related question concerns a possible instability of the standard ℂℙ N −1 vacuum on R2, which is shown not to occur. In all cases, the difference in the conclusions can be traced to the existence of certain zeromodes and their proper treatment. The ℂℙ N −1 model with twisted boundary conditions is also analyzed. The θ dependence and different limits involving N , β and L are briefly discussed.

Italy

Stefano Bolognesi

Sven Gudnason

Kenichi Konishi

Large-N ℂℙ N −1 sigma model on a Euclidean torus: uniqueness and stability of the vacuum

This is the third of the series of articles on the large-N two-dimensional ℂℙ N − 1 sigma model, defined on a finite space interval L with Dirichlet boundary conditions. Here the cases of the general Dirichlet boundary conditions are studied, where the relative ℂℙ N − 1 orientations at the two boundaries are generic, and numerical solutions are presented. Distinctive features of the ℂℙ N − 1 sigma model, as compared e.g., to an O(N) model, which were not entirely evident in the basic properties studied in the first two articles in the large N limit, manifest themselves here. It is found that the total energy is minimized when the fields are aligned in the same direction at the two boundaries.

China

Large-N ℂℙ N − 1 sigma model on a finite interval: general Dirichlet boundary conditions

f (R) supergravity is known to contain a ghost mode associated with higher-derivative terms if it contains R  n  with n greater than two. We remove the ghost in f (R) supergravity by introducing auxiliary gauge field to absorb the ghost. We dub this method as the ghostbuster mechanism [1]. We show that the mechanism removes the ghost super-multiplet but also terms including R  n  with n ≥ 3, after integrating out auxiliary degrees of freedom. For pure supergravity case, there appears an instability in the resultant scalar potential. We then show that the instability of the scalar potential can be cured by introducing matter couplings in such a way that the system has a stable potential.

United States

Yusuke Yamada

Ghostbusters in f (R) supergravity

We continue the analysis started in a recent paper of the large-N two-dimensional  <math><mi>ℂ</mi><msup><mi>ℙ</mi><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msup></math> $$ \mathbb{C}{\mathrm{\mathbb{P}}}^{N-1} $$ sigma model, defined on a finite space interval L with Dirichlet (or Neumann) boundary conditions. Here we focus our attention on the problem of the renormalized energy density  <math><mi>ℰ</mi></math> $$ \mathrm{\mathcal{E}} $$ (x, Λ, L) which is found to be a sum of two terms, a constant term coming from the sum over modes, and a term proportional to the mass gap. The approach to  <math><mi>ℰ</mi><mfenced><mrow><mi>x</mi><mo>,</mo><mi>Λ</mi><mo>,</mo><mspace width="0.25em"></mspace><mi>L</mi></mrow></mfenced><mo>→</mo><mfrac><mi>N</mi><mrow><mn>4</mn><mi>π</mi></mrow></mfrac><msup><mi>Λ</mi><mn>2</mn></msup></math> $$ \mathrm{\mathcal{E}}\left(x,\varLambda,\ L\right)\to \frac{N}{4\pi }{\varLambda}^2 $$ at large LΛ is shown, both analytically and numerically, to be exponential: no power corrections are present and in particular no Lüscher term appears. This is consistent with the earlier result which states that the system has a unique massive phase, which interpolates smoothly between the classical weakly-coupled limit for LΛ → 0 and the “confined” phase of the standard  <math><mi>ℂ</mi><msup><mi>ℙ</mi><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msup></math> $$ \mathbb{C}{\mathrm{\mathbb{P}}}^{N-1} $$ model in two dimensions for LΛ → ∞.

Alessandro Betti

Large-N  <math><mi>ℂ</mi><msup><mi>ℙ</mi><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msup></math> $$ \mathbb{C}{\mathrm{\mathbb{P}}}^{\mathrm{N}-1} $$ sigma model on a finite interval and the renormalized string energy