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

In this work, we use the Born-Oppenheimer approximation, where the potential between atoms can be approximated as a function of distance between the two nuclei, to study the four-quark bound states. By this approximation, Heitler and London calculated the spectrum of the hydrogen molecule, which includes two protons (heavy) and two electrons (light). Generally, the observed exotic mesons Zb(10610), Zb(10650), Zc(3900) and Zc(4020) (Zc(4025)) may be molecular states made of two physical mesons and/or diquark-anti-diquark structures. Analogous to the Heitler-London method for calculating the mass of the hydrogen molecule, we investigate whether there exist energy minima for these two structures. Contrary to the hydrogen molecule case where only the spin-triplet possesses an energy minimum, there exist minima for both of these states. This implies that both molecule and tetraquark states can be stable objects. Since they have the same quantum numbers, however, the two states may mix to result in the physical states. A consequence would be that partner exotic states co-existing with Zb (10610), Zb (10650), Zc (3900) and Zc (4020) (Zc (4025)) are predicted and should be experimentally observed.

China

Xue-Wen Liu

Hong-Wei Ke

Yi-Bing Ding

Xue-Qian Li

Study of the structures of four-quark states in terms of the Born-Oppenheimer approximationSupported by National Natural Science Foundation of China (11375128)