We present a unified free field realization of the Ding-Iohara algebra at arbitrary levels, which satisfies a generalized form of the Serre relations. This realization, constructed using six free boson fields, arises from a specialized factorization of the structure function in the defining relations of the algebra. Based on this construction, we then develop intertwining operators for the Ding-Iohara algebra.

China

Zitao Chen

Xiang-Mao Ding

Free field realization of the Ding-Iohara algebra at general levels

The 2HDM+S is the singlet extension of the two-Higgs-doublet model (2HDM). The singlet field and its mixing with the 2HDM Higgs sector lead to new contributions to the electroweak precision observables, in particular, the oblique parameters. In this study, we performed a systematic investigation of the impacts of each mixing angle on the oblique parameters. We adopted the mixing angles and physical Higgs masses as our parameters, which allow a mapping when a specific symmetry structure of the Higgs potential and various theoretical considerations are taken into account. We identified five benchmark cases, where at most one mixing angle was nonzero, and analyzed the 95% C.L. allowed parameter space using the oblique parameters. In the alignment limit of the 2HDM, we find that, other than the usual mass relations of $m_H\sim m_{H^\pm}$ or $m_A\sim m_{H^\pm}$ , electroweak precision measurements also impose an upper limit on the neutral Higgs masses. In the cases with nonzero singlet mixing with the 2HDM Higgses $\textit{H}$ or $\textit{A}$, we find approximate mass relations of $c^2_{\alpha_{HS}} m_{H} + s^2_{\alpha_{HS}}m_{h_S} = m_{H^\pm}$ or $c^2_{\alpha_{AS}} m_{A} + s^2_{\alpha_{AS}}m_{A_S} = m_{H^\pm}$ . These relations are universal to the 2HDM+S models, with or without further symmetry assumption. We also studied the non-alignment limit of the 2HDM+S, which typically has tighter constraints on the masses and mixing angles. Finally, we examined the complementarity between the electroweak precision analyses and the Higgs coupling precision measurements.

Cheng Li

Juxiang Li

Shufang Su

Wei Su

Electroweak precision constraints of the 2HDM+S * Cheng Li, Juxiang Li, and Wei Su are supported by the Natural Science Foundation of China (NSFC) (12305115), Shenzhen Science and Technology Program (202206193000001, 20220816094256002), Guangdong Provincial Key Laboratory of Gamma-Gamma Collider and Its Comprehensive Applications (2024KSYS001), and Guangdong Provincial Key Laboratory of Advanced Particle Detection Technology (2024B1212010005). Juxiang Li is also supported by the Fundamental Research Funds for the Central Universities, and the Sun Yat-sen University Science Foundation. Shufang Su is supported by the Department of Energy (DEFG02-13ER41976/DE-SC0009913)

We investigated the impact of a constant external magnetic field on the dressed propagators of up-, down-, and strange quarks. In the weak field limit, we derive a general momentum-space representation for the propagator and numerically solve the corresponding gap equation. Our analysis reveals that the vector term of the propagator can be decomposed into components parallel and perpendicular to the magnetic field, resulting in anisotropic effective masses, with the transverse mass consistently exceeding the longitudinal mass. This mass disparity exhibits a power law dependence on the magnetic field strength and is less pronounced for the strange quark compared to up and down quarks. Additionally, the magnetic field induces axial-vector and tensor terms, highlighting the Zeeman effect, resulting from quark interactions with the magnetic field. These findings have important implications for (inverse) magnetic catalysis and phenomena such as vector meson and pion condensations.

Germany

Minghui Ding

Fei Gao

Sebastian M. Schmidt

Anisotropic quark propagation and Zeeman effect in an external magnetic field * Fei Gao is supported by the National Science Foundation of China (12305134). Minghui Ding appreciates the support received from the Helmholtz-Zentrum Dresden-Rossendorf High Potential Programme

<p>We construct relativized mock meson states for heavy quarkonium, where the Dirac spinors import kinetic relativistic correction and the dynamic wave functions are the same as those solved from the Schrödinger equation. We find that the relativistic correction imported by Dirac spinors is crucial to study the two-photon transition form factors. Using relativized mock meson states, we give credible predictions for the two-photon transition form factors and decay widths of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mrow><mi>c</mi><mo>,</mo><mi>b</mi></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><msub><mrow><mmultiscripts><mrow><mi>S</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>1</mn></mrow></mmultiscripts></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mo>,</mo><mi>b</mi></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><msub><mrow><mmultiscripts><mrow><mi>P</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi><mo>=</mo><mn>1</mn></math>, 2, 3).</p>

Tian-Cheng Ding

Jian Huang

Muyang Chen

Two-photon transition form factor of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mrow><mi>c</mi><mo>,</mo><mi>b</mi></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mi>S</mi><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mn>0</mn><mo>,</mo><mi>b</mi><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mi>P</mi><mo stretchy="false">)</mo></math> via relativized mock meson states

We notice that the famous pentagon identity for quantum dilogarithm functions and the five-term relation for certain operators related to Macdonald polynomials discovered by Garsia and Mellit can both be understood as specific cases of a general “master pentagon identity” for group-like elements in the Ding-Iohara-Miki (or quantum toroidal, or elliptic Hall) algebra. We prove this curious identity and discuss its implications.

Italy

Russia

Yegor Zenkevich

On pentagon identity in Ding-Iohara-Miki algebra

We derive a generalized Knizhnik-Zamolodchikov equation for the correlation function of the intertwiners of the vector and the MacMahon representations of Ding-Iohara-Miki algebra. These intertwiners are cousins of the refined topological vertex which is regarded as the intertwining operator of the Fock representation. The shift of the spectral parameter of the intertwiners is generated by the operator which is constructed from the universal R matrix. The solutions to the generalized KZ equation are factorized into the ratio of two point functions which are identified with generalizations of the Nekrasov factor for supersymmetric quiver gauge theories.

Japan

Panupong Cheewaphutthisakun

Hiroaki Kanno

MacMahon KZ equation for Ding-Iohara-Miki algebra

Reflection states are introduced in the vertical and horizontal modules of the Ding-Iohara-Miki (DIM) algebra (quantum toroidal g l 1 $$ \mathfrak{g}{\mathfrak{l}}_1 $$ ). Webs of DIM representations are in correspondence with ( p , q )-web diagrams of type IIB string theory, under the identification of the algebraic intertwiner of Awata, Feigin and Shiraishi with the refined topological vertex. Extending the correspondence to the vertical reflection states, it is possible to engineer the N = 1 $$ \mathcal{N}=1 $$ quiver gauge theory of D-type (with unitary gauge groups). In this way, the Nekrasov instanton partition function is reproduced from the evaluation of expectation values of intertwiners. This computation leads to the identification of the vertical reflection state with the orientifold plane of string theory. We also provide a translation of this construction in the Iqbal-Kozcaz-Vafa refined topological vertex formalism.

South Korea

J.E. Bourgine

M. Fukuda

Y. Matsuo

R.D. Zhu

Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver

The integrability of 4d gauge theories has been explored in various contexts, e.g., the Seiberg–Witten curve and its quantization. Recently, Maulik and Okounkov proposed that an integrable lattice model is associated with the gauge theory, through an R-matrix, which we refer to as the MO R-matrix in this paper, constructed in the instanton moduli space. In this paper, we study the R-matrix using the Ding–Iohara–Miki (DIM) algebra. We provide a concrete boson realization of the universal R-matrix in DIM and show that the defining conditions for the MO R-matrix can be derived from this free boson oscillator expression. Several consistency checks for the oscillator expression are also performed.

K. Harada

The Maulik–Okounkov R-matrix from the Ding–Iohara–Miki algebra

Ward identities in the most general “network matrix model” from [1] can be described in terms of the Ding–Iohara–Miki algebras (DIM). This confirms an expectation that such algebras and their various limits/reductions are the relevant substitutes/deformations of the Virasoro/W-algebra for (q,t) and (q1,q2,q3) deformed network matrix models. Exhaustive for these purposes should be the Pagoda triple-affine elliptic DIM, which corresponds to networks associated with 6d gauge theories with adjoint matter (double elliptic systems). We provide some details on elliptic qq -characters.

A. Mironov

A. Morozov

Y. Zenkevich

Ding–Iohara–Miki symmetry of network matrix models