We present a determination of the Cabibbo-Kobayashi-Maskawa matrix element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><msub><mi>V</mi><mrow><mi>c</mi><mi>b</mi></mrow></msub><mo stretchy="false">|</mo></math> from the decay <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><mi>D</mi><mo>ℓ</mo><msub><mi>ν</mi><mo>ℓ</mo></msub></math> using a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>365</mn><mtext> </mtext><mtext> </mtext><msup><mi>fb</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><mo stretchy="false">→</mo><mi mathvariant="normal">ϒ</mi><mo stretchy="false">(</mo><mn>4</mn><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mi>B</mi><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover></math> data sample recorded by the Belle II experiment at the SuperKEKB collider. The semileptonic decay of one <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi></math> meson is reconstructed in the modes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mn>0</mn></msup><mo stretchy="false">→</mo><msup><mi>D</mi><mo>−</mo></msup><mo stretchy="false">(</mo><mo stretchy="false">→</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><msup><mi>π</mi><mo>−</mo></msup><mo stretchy="false">)</mo><msup><mo>ℓ</mo><mo>+</mo></msup><msub><mi>ν</mi><mo>ℓ</mo></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mn>0</mn></msup><mo stretchy="false">(</mo><mo stretchy="false">→</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><mo stretchy="false">)</mo><msup><mo>ℓ</mo><mo>+</mo></msup><msub><mi>ν</mi><mo>ℓ</mo></msub></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>ℓ</mo></math> denotes either an electron or a muon. Charge conjugation is implied. The second <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi></math> meson in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">ϒ</mi><mo stretchy="false">(</mo><mn>4</mn><mi>S</mi><mo stretchy="false">)</mo></math> event is not reconstructed explicitly. Using an inclusive reconstruction of the unobserved neutrino momentum, we determine the recoil variable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>w</mi><mo>=</mo><msub><mi>v</mi><mi>B</mi></msub><mo>·</mo><msub><mi>v</mi><mi>D</mi></msub></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>v</mi><mi>B</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>v</mi><mi>D</mi></msub></math> are the four-velocities of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi></math> mesons. We measure the total decay branching fractions to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><msup><mrow><mi>B</mi></mrow><mrow><mn>0</mn></mrow></msup><mo stretchy="false">→</mo><msup><mrow><mi>D</mi></mrow><mrow><mo>−</mo></mrow></msup><msup><mrow><mo>ℓ</mo></mrow><mrow><mo>+</mo></mrow></msup><msub><mrow><mi>ν</mi></mrow><mrow><mo>ℓ</mo></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>2.06</mn><mo>±</mo><mn>0.05</mn><mo stretchy="false">(</mo><mrow><mi>stat</mi></mrow><mo stretchy="false">)</mo><mo>±</mo><mn>0.10</mn><mo stretchy="false">(</mo><mrow><mi>sys</mi></mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>%</mo></mrow></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mn>0</mn></msup><msup><mo>ℓ</mo><mo>+</mo></msup><msub><mi>ν</mi><mo>ℓ</mo></msub><mo stretchy="false">)</mo><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>2.31</mn><mo>±</mo><mn>0.04</mn><mo stretchy="false">(</mo><mrow><mi>stat</mi></mrow><mo stretchy="false">)</mo><mo>±</mo><mn>0.09</mn><mo stretchy="false">(</mo><mrow><mi>sys</mi></mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>%</mo></mrow></math>. We probe lepton flavor universality by measuring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">→</mo><mi>D</mi><mi>e</mi><msub><mi>ν</mi><mi>e</mi></msub><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">→</mo><mi>D</mi><mi>μ</mi><msub><mi>ν</mi><mi>μ</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mrow><mn>1.020</mn><mo>±</mo><mn>0.020</mn><mo stretchy="false">(</mo><mrow><mi>stat</mi></mrow><mo stretchy="false">)</mo><mo>±</mo><mn>0.022</mn><mo stretchy="false">(</mo><mrow><mi>sys</mi></mrow><mo stretchy="false">)</mo></mrow></math>. Fitting the partial decay branching fraction as a function of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>w</mi></math> and using the average of lattice QCD calculations of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><mi>D</mi></math> form factor, we obtain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><msub><mi>V</mi><mrow><mi>c</mi><mi>b</mi></mrow></msub><mo stretchy="false">|</mo><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>39.2</mn><mo>±</mo><mn>0.4</mn><mo stretchy="false">(</mo><mrow><mi>stat</mi></mrow><mo stretchy="false">)</mo><mo>±</mo><mn>0.6</mn><mo stretchy="false">(</mo><mrow><mi>sys</mi></mrow><mo stretchy="false">)</mo><mo>±</mo><mn>0.5</mn><mo stretchy="false">(</mo><mrow><mi>th</mi></mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow></math>.

I. Adachi

K. Adamczyk

L. Aggarwal

H. Ahmed

Y. Ahn

H. Aihara

N. Akopov

S. Alghamdi

M. Alhakami

A. Aloisio

K. Amos

M. Angelsmark

N. Anh Ky

C. Antonioli

D.M. Asner

H. Atmacan

T. Aushev

V. Aushev

M. Aversano

R. Ayad

V. Babu

H. Bae

N.K. Baghel

S. Bahinipati

P. Bambade

Sw. Banerjee

M. Barrett

M. Bartl

J. Baudot

A. Baur

A. Beaubien

F. Becherer

J. Becker

J.V. Bennett

F.U. Bernlochner

V. Bertacchi

M. Bertemes

E. Bertholet

M. Bessner

S. Bettarini

B. Bhuyan

F. Bianchi

T. Bilka

D. Biswas

A. Bobrov

D. Bodrov

A. Bondar

G. Bonvicini

J. Borah

A. Boschetti

A. Bozek

M. Bračko

P. Branchini

R.A. Briere

T.E. Browder

A. Budano

S. Bussino

Q. Campagna

M. Campajola

L. Cao

G. Casarosa

C. Cecchi

M.C. Chang

P. Cheema

L. Chen

B.G. Cheon

K. Chilikin

J. Chin

K. Chirapatpimol

H.E. Cho

K. Cho

S.J. Cho

S.K. Choi

S. Choudhury

I. Consigny

L. Corona

J.X. Cui

E. De La Cruz-Burelo

S.A. De La Motte

G. De Pietro

R. de Sangro

M. Destefanis

A. Di Canto

J. Dingfelder

Z. Doležal

I. Domínguez Jiménez

T.V. Dong

X. Dong

K. Dugic

G. Dujany

P. Ecker

J. Eppelt

R. Farkas

P. Feichtinger

T. Ferber

T. Fillinger

C. Finck

G. Finocchiaro

F. Forti

A. Frey

Determination of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><msub><mi>V</mi><mrow><mi>c</mi><mi>b</mi></mrow></msub><mo stretchy="false">|</mo></math> using <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><mi>D</mi><mo>ℓ</mo><msub><mi>ν</mi><mi>l</mi></msub></math> decays at Belle II

We present a fully backreacted Einstein-Maxwell-Dilaton–flavor model with dynamical light and strange sectors, calibrated to lattice QCD using a machine-learning–assisted spectral method. The model reproduces the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>-flavor equation of state and chiral dynamics with quantitative accuracy, and maps the Columbia plot with a tricritical point at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>tri</mi></mrow></msubsup><mo>≃</mo><mn>21</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math> and a critical mass <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>c</mi></msub><mo>≃</mo><mn>0.785</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, consistent with lattice results. At finite density, it yields a crossover-to-first-order transition and predicts a critical endpoint at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mi>C</mi></msub><mo>=</mo><mn>75.4</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>μ</mi><mi>C</mi></msub><mo>=</mo><mn>768</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, within the reach of heavy-ion experiments. These findings establish a unified holographic framework for the QCD phase structure across quark masses and baryon density, providing the first consistent and quantitative description of both deconfinement and chiral transitions within a single holographic model.

China

Jin-Yang Shen

Xin-Yi Liu

Jin-Rui Wu

Yue-Liang Wu

Zhen Fang

Toward a holographic realization of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>-flavor QCD phase structure

We propose a new inverse seesaw model based on hidden local <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> symmetry framework where the inverse seesaw mechanism is induced at one loop level. A Majorana mass term of singlet fermion is forbidden by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> symmetry and it is generated at one-loop level by introducing relevant particle contents to get loop diagram, inducing inverse seesaw mechanism. The same particle contents also contribute to lepton magnetic(electric) dipole moment and lepton flavor violating decays without chiral suppression. We can then obtain sizable muon anomalous magnetic dipole moment that accommodate with deviation from the standard model prediction. The constraints from lepton flavor violating decays and electron magnetic(electric) dipole moment are also discussed to explore testability of the model.

Takaaki Nomura

Japan

Hiroshi Okada

Radiative inverse seesaw model with hidden <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> gauge symmetry enhancing lepton <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>g</mi><mo>−</mo><mn>2</mn></math>

Within the framework of the thermal soft-wall model of AdS/QCD, we investigate phenomenological properties of pions at finite temperature. This includes the electromagnetic form factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mi>π</mi></msub><mo stretchy="false">(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo></math>, the thermal mass <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>M</mi><mi>π</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math>, charge radius <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>r</mi><mi>π</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math>, the generalized parton distribution (GPD) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>H</mi><mi>π</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo></math>, the charge density <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>ρ</mi><mi>π</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo></math> of the pion, the pion-nucleon coupling constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>g</mi><mrow><mi>π</mi><mi>N</mi><mi>N</mi></mrow></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math>, and pion-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Δ</mi></math> baryon coupling constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>g</mi><mrow><mi>π</mi><mi mathvariant="normal">Δ</mi><mi mathvariant="normal">Δ</mi></mrow></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math> coupling constant at finite temperature. The thermal pion form factor is extrapolated from the zero-temperature case. Subsequently, the GPD is obtained at finite temperature from this form factor. The above-mentioned quantities were analyzed using a thermal dilaton field in the five-dimensional anti–de Sitter action. Moreover, we determine the theoretical expression for the temperature-dependent pion-nucleon coupling constant, and the pion-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Δ</mi></math> baryon coupling constant using thermal profile functions of the nucleon, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Δ</mi></math> baryon and the pion. Our results show that the values of these quantities decrease with increasing temperature and vanish near the critical temperature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mi>c</mi></msub></math>, except for the pion radius, which diverges at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mi>c</mi></msub></math>. Our results reproduce expected features of low-energy hadron dynamics, thus validating the phenomenological utility of the thermal soft-wall model.

Azerbaijan

Narmin Nasibova

United States

Xerxes D. Arsiwalla

Pion phenomenology from the thermal soft-wall model of holographic QCD

We generalize the analysis of the asymptotic higher spin symmetries developed in the first three parts of this series by considering the minimal coupling of Einstein Gravity and Yang-Mills theory. We show that there exist symmetry parameters that satisfy a collection of dual equations of motion, which allow the construction of an infinite collection of charges that are conserved in the absence of radiation. These Noether charges act on the Einstein Yang-Mills phase space canonically and non-linearly. Their action defines a symmetry algebroid which reduces to a symmetry algebra at non-radiative cuts and generalizes the celestial $\textit{sw}$ algebra. The corresponding symmetry bracket is shown to satisfy the Jacobi identity and an interesting cross-product structure, which is analyzed in details.

Canada

Nicolas Cresto

Laurent Freidel

Asymptotic higher spin symmetries. Part IV. Einstein-Yang-Mills theory

We investigate the reconstruction of standard and generalized Rastall gravity inflationary models, using the scalar spectral index and the Rastall parameter expressed as functions of the number of <math altimg='si3.svg'><mrow><mi>e</mi><mo linebreak='goodbreak'>&#8722;</mo></mrow></math>folds <italic>N</italic>. Within a general formalism, we derive the effective potential in terms of the relevant cosmological parameters and the Rastall parameter for these gravity frameworks. As a specific example, we analyze the attractor <math altimg='si4.svg'><mrow><msub><mi>n</mi><mi>s</mi></msub><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow><mo linebreak='goodbreak'>&#8722;</mo><mn>1</mn><mo>&#8733;</mo><msup><mi>N</mi><mrow><mo>&#8722;</mo><mn>1</mn></mrow></msup></mrow></math>, first by considering constant values of the Rastall parameter to reconstruct the inflationary stage in standard Rastall gravity, and then by assuming a linear dependence on the number of <math altimg='si3.svg'><mrow><mi>e</mi><mo linebreak='goodbreak'>&#8722;</mo></mrow></math>folds <italic>N</italic> to reconstruct the inflationary model in generalized Rastall gravity. Thus, the reconstruction of the potential <italic>V</italic>( <italic>&#981;</italic>) is obtained for both standard and generalized Rastall gravity inflationary models. In both frameworks, we constrain key parameters of the reconstructed models during inflation using the latest observational data from Planck.

Chile

Ramón Herrera

Carlos Ríos

Reconstructing inflation in a generalized Rastall theory of gravity

Motivated by integrating the dilaton field (as a UV correction) with dRGT-like massive gravity (as an IR correction) into Einstein gravity, we investigate the thermodynamic and optical properties of black holes within this gravitational framework. We begin by reviewing the black hole solutions in Maxwell–dilaton–dRGT-like massive gravity, followed by an analysis of how various parameters influence on the asymptotical behavior of the spacetime and the event horizon of these black holes. In the subsequent section, we examine the conserved and thermodynamic quantities associated with these black holes, paying particular attention to the effects of parameters like $$\beta ,$$ $$\alpha ,$$ and the massive parameters $$(\eta _{1}$$ and $$\eta _{2})$$ on their local stability by simultaneously evaluating the heat capacity and temperature. We also adopt an alternative method to study phase transitions using geometrothermodynamics. Furthermore, we explore how the parameters of Maxwell–dilaton–dRGT-like massive gravity impacts the optical characteristics and radiative behavior of black holes. In particular, we analyze the effects of the dilaton coupling constant $$(\alpha ),$$ charge ($\textit{q}$), the massive gravity parameter $$(\eta _1),$$ and the graviton mass $$(m_g)$$ on the radius of the photon sphere and the resulting black hole shadow. Moreover, the theoretical shadow radius is compared to the observational data from $$Sgr A^*.$$ Additionally, we investigate the energy emission rate of these black holes, revealing that these parameters substantially influence the emission peak.

Iran

B. Panah

N. Heidari

M. Soleimani

Some perspective of thermodynamical and optical properties of black holes in Maxwell–dilaton–dRGT-like massive gravity

Recent developments for the consistent embedding of general 4D static and spherically-symmetric spacetimes in arbitrary single-brane braneworld models in the form of the General Embedding Algorithm (GEA) [$\textit{Phys.Rev. D}$ (2024) L041501], initiated the program of studying the bulk structure of braneworld wormholes. In this article, adopting a completely generic approach, we derive the general conditions that the metric functions of any braneworld spacetime must satisfy to describe a wormhole structure in the bulk. Particular emphasis is placed on clarifying the proper uplift of 4D spacetimes, expressed in terms of arbitrary radial coordinates on the brane, and we demonstrate the important role of the circumferential radius metric function $\textit{r}$($\textit{u}$) for the embedding. To ensure applicability even when $\textit{r}$($\textit{u}$) is non-invertible, we develop an extended formulation of the GEA. Additionally, the flare-out conditions for braneworld wormholes are presented for the first time and are found to differ from the case of flat extra dimensions. To illustrate the method, we first perform the uplift into both thin (Randall-Sundrum II) and thick braneworld models for four well-known 4D wormhole spacetimes: the effective braneworld wormhole solutions of Casadio-Fabbri-Mazzacurati and Bronnikov-Kim, the Simpson-Visser spacetime, and the Ellis-Bronnikov or “anti-Fisher” solution. Subsequently, we study their bulk features by means of curvature invariants, flare-out conditions, energy conditions and embedding diagrams. Our analysis reveals that the assumption of a warped extra dimension has non-trivial implications for the structure of 5D wormholes.

Czechia

Thomas Pappas

South Korea

Theodoros Nakas

On the uplift of 4D wormholes in braneworld models and their 5D structure

Through the holographic correspondence, we compute the entanglement entropy of the gluonic field sourced by a quark-antiquark pair undergoing uniform back-to-back acceleration. Previous calculations had obtained this only for the case where the entanglement surface is located midway between the quark and antiquark. Here, we consider the more general case with a relative lateral displacement, and determine the entanglement entropy as a function of the distance between the quark and the entanglement surface. This setup is of interest because it departs from the usual simplifying conditions of staticity and thermality, and because it yields more information about the entanglement pattern in the gluonic field and about the possibility of eventually developing a purely worldsheet interpretation for said entanglement.

Mexico

Andrés Argandoña

Alberto Güijosa

Position dependence of the holographic entanglement entropy for an accelerating quark-antiquark pair

We consider generalisations of the elliptic Calogero-Moser systems associated to complex crystallographic groups in accordance to [1]. In our previous work [2], we proposed these systems as candidates for Seiberg-Witten integrable systems of certain SCFTs. Here we examine that proposal for complex crystallographic groups of rank one. Geometrically, this means considering elliptic curves $\textit{T}$ with $${\mathbb{Z}}_{m}$$ -symmetries, $\textit{m}$ = 2, 3, 4, 6, and Poisson deformations of the orbifolds ( $${T}^{2}\times {\mathbb{C}})/{\mathbb{Z}}_{m}$$ . The $\textit{m}$ = 2 case was studied in [2], while $\textit{m}$ = 3, 4, 6 correspond to Seiberg-Witten integrable systems for the rank 1 Minahan-Nemeschansky SCFTs of type $\textit{E}$ . This allows us to describe the corresponding elliptic fibrations and the Seiberg-Witten differential in a compact elegant form. This approach also produces quantum spectral curves for these SCFTs, which are given by Fuchsian ODEs with special properties.

Philip Argyres

United Kingdom

Oleg Chalykh

Yongchao Lü

Complex crystallographic reflection groups and Seiberg-Witten integrable systems: rank 1 case