Abstract The open N = 2 string theory is defined on the 4D space-time with the split signature (+, +, −, −). The string field theory action of the open N = 2 string theory is described by the 4D Wess–Zumino–Witten (WZW$_{4}$) model. The equation of motion of the WZW$_{4}$ model is the Yang equation, which is equivalent to the anti-self-dual Yang–Mills equation. In this paper, we study soliton-type classical solutions of the WZW$_{4}$ model in the split signature by calculating the action density of the WZW$_{4}$ model. We find that the action density of the one-soliton solutions is localized on a 3D hyperplane. This shows that there would be codimension-one-solitonic objects, or equivalently some kind of three-branes in the open N = 2 string theory. We also prove that, in the asymptotic region of the space-time, the action density of the n-soliton solutions is a “non-linear superposition” of n one-solitons. This suggests the existence of n intersecting three-branes in the N = 2 strings. Finally we make a reduction to a (1 + 2)D real space-time to calculate the energy densities of the soliton solutions. We can successfully evaluate the energy distribution for the two-soliton solutions and find that there is no singularity in the interacting region. This implies the existence of smooth intersecting codimension-one branes in the whole region. Soliton solutions in the Euclidean signature are also discussed.

Japan

Masashi Hamanaka

Shan-Chi Huang

Hiroaki Kanno

Solitons in open = 2 String Theory Kahler-Chen-Simons theory A tour through N=2 strings On soliton solutions of the anti-self-dual Yang-Mills equations from the perspective of integrable systems Soliton solutions in an integrable chiral model in 2+1 dimensions On the space-time monopole equation A review on tachyon condensation in open string field theories Soliton equations as dynamical systems on infinite dimensional Grassmann manifold

We investigate the quasi-Hopf twist of the quantum toroidal algebra of   <math> <msub> <mi>gl</mi> <mn>1</mn> </msub> </math>  $$ {\mathfrak{gl}}_1 $$  as an elliptic deformation. Under the quasi-Hopf twist the underlying algebra remains the same, but the coproduct is deformed, where the twist parameter p is identified as the elliptic modulus. Computing the quasi-Hopf twist of the R matrix, we uncover the relation to the elliptic lift of the Nekrasov factor for instanton counting of the quiver gauge theories on ℝ4 × T 2. The same R matrix also appears in the commutation relation of the intertwiners, which implies an elliptic quantum KZ equation for the trace of intertwiners. We also show that it allows a solution which is factorized into the elliptic Nekrasov factors and the triple elliptic gamma function.

Panupong Cheewaphutthisakun

Quasi-Hopf twist and elliptic Nekrasov factor

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.

MacMahon KZ equation for Ding-Iohara-Miki algebra

We suggest a further generalization of the hypergeometric-like series due to M. Noumi and J. Shiraishi by substituting the Pochhammer symbol with a nearly arbitrary function. Moreover, this generalization is valid for the entire Shiraishi series, not only for its Noumi–Shiraishi part. The theta function needed in the recently suggested description of the double-elliptic systems [Awata et al. JHEP 2020:150, arXiv:2005.10563, (2020)], 6d N = 2* SYM instanton calculus and the doubly-compactified network models, is a very particular member of this huge family. The series depends on two kinds of variables,  $$\vec {x}$$  <math> <mover> <mi>x</mi> <mo>→</mo> </mover> </math>   and  $$\vec {y}$$  <math> <mover> <mi>y</mi> <mo>→</mo> </mover> </math>  , and on a set of parameters, which becomes infinitely large now. Still, one of the parameters, p is distinguished by its role in the series grading. When  $$\vec {y}$$  <math> <mover> <mi>y</mi> <mo>→</mo> </mover> </math>   are restricted to a discrete subset labeled by Young diagrams, the series multiplied by a monomial factor reduces to a polynomial at any given order in p. All this makes the map from functions to the hypergeometric-like series very promising, and we call it Shiraishi functor despite it remains to be seen, what are exactly the morphisms that it preserves. Generalized Noumi–Shiraishi (GNS) symmetric polynomials inspired by the Shiraishi functor in the leading order in p can be obtained by a triangular transform from the Schur polynomials and possess an interesting grading. They provide a family of deformations of Macdonald polynomials, as rich as the family of Kerov functions, still very different from them, and, in fact, much closer to the Macdonald polynomials. In particular, unlike the Kerov case, these polynomials do not depend on the ordering of Young diagrams in the triangular expansion.

Hidetoshi Awata

Russia

Andrei Mironov

Alexei Morozov

Shiraishi functor and non-Kerov deformation of Macdonald polynomials

As a development of [1], we note that the ordinary Shiraishi functions have an insufficient number of parameters to describe generic eigenfunctions of double elliptic system (Dell). The lacking parameter can be provided by substituting elliptic instead of the ordinary Gamma functions in the coefficients of the series. These new functions (ELS-functions) are conjectured to be functions governed by compactified DIM networks which can simultaneously play the three roles: solutions to non-stationary Dell equations, Dell conformal blocks with the degenerate field (surface operator) insertion, and the corresponding instanton sums in 6d SUSY gauge theories with adjoint matter. We describe the basics of the corresponding construction and make further conjectures about the various limits and dualities which need to be checked to make a precise statement about the Dell description by double-periodic network models with DIM symmetry. We also demonstrate that the ELS-functions provide symmetric polynomials, which are an elliptic generalization of Macdonald ones, and compute the generation function of the elliptic genera of the affine Laumon spaces. In the particular U(1) case, we find an explicit plethystic formula for the 6d partition function, which is a non-trivial elliptic generalization of the (q, t) Nekrasov-Okounkov formula from 5d.

Elliptic lift of the Shiraishi function as a non-stationary double-elliptic function

Abstract We review the problem of Bogomol’nyi–Prasad–Sommerfield (BPS) state counting described by the generalized quiver matrix model of Atiyah–Drinfield–Hitchin–Manin type. In four dimensions the generating function of the counting gives the Nekrasov partition function, and we obtain a generalization in higher dimensions. By the localization theorem, the partition function is given by the sum of contributions from the fixed points of the torus action, which are labeled by partitions, plane partitions and solid partitions. The measure or the Boltzmann weight of the path integral can take the form of the plethystic exponential. Remarkably, after integration the partition function or the vacuum expectation value is again expressed in plethystic form. We regard it as a characteristic property of the BPS state counting problem, which is closely related to the integrability.

Quiver matrix model of ADHM type and BPS state counting in diverse dimensions

The mother functions for the eigenfunctions of the Koroteev-Shakirov version of quantum double-elliptic (Dell) Hamiltonians can be presented as infinite series in Miwa variables, very similar to the recent conjecture due to J. Shiraishi. Further studies should clear numerous remaining obstacles and thus solve the long-standing problem of explicitly constructing a Dell system, the top member of the Calogero-Moser-Ruijsenaars system, with the -duality fully explicit at the elliptic level.

On a complete solution of the quantum Dell system

We make a new attempt at the recently suggested program to express knot polynomials through topological vertices, which can be considered as a possible approach to the tangle calculus: we discuss the Macdonald deformation of the relation between the convolution of two topological vertices and the HOMFLY-PT invariant of the 4-component link <math><msub><mrow><mi>L</mi></mrow><mrow><mn>8</mn><mi>n</mi><mn>8</mn></mrow></msub></math>, which both depend on four arbitrary representations. The key point is that both of these are related to the Hopf polynomials in composite representations, which are in turn expressed through composite Schur functions. The latter are further expressed through the skew Schur polynomials via the remarkable Koike formula. It is this decomposition that breaks under the Macdonald deformation and gets restored only in the (large N) limit of <math><msup><mrow><mi>A</mi></mrow><mrow><mo>±</mo><mn>1</mn></mrow></msup><mo>⟶</mo><mn>0</mn></math>. Another problem is that the Hopf polynomials in the composite representations in the refined case are “chiral bilinears” of Macdonald polynomials, while convolutions of topological vertices involve “non-chiral combinations” with one of the Macdonald polynomials entering with permuted t and q. There are also other mismatches between the Hopf polynomials in the composite representation and the topological 4-point function in the refined case, which we discuss.

Can tangle calculus be applicable to hyperpolynomials?

We compute the topological partition function (twisted index) of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 U(N) Chern-Simons theory with an adjoint chiral multiplet on Σ g  × S 1. The localization technique shows that the underlying Frobenius algebra is the equivariant Verlinde algebra which is obtained from the canonical quantization of the complex Chern-Simons theory regularized by U(1) equivariant parameter t. Our computation relies on a Bethe/Gauge correspondence which allows us to represent the equivariant Verlinde algebra in terms of the Hall-Littlewood polynomials P  λ (x  B , t) with a specialization by Bethe roots x  B  of the q-boson model. We confirm a proposed duality to the Coulomb branch limit of the lens space superconformal index of four dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 theories for SU(2) and SU(3) with lower levels. In SU(2) case we also present more direct computation based on Jeffrey-Kirwan residue operation.

Katsuyuki Sugiyama

Yutaka Yoshida

Equivariant U(N) Verlinde algebra from Bethe/gauge correspondence

We introduce an R-matrix acting on the tensor product of MacMahon representations of Ding-Iohara-Miki (DIM) algebra   <math> <msub> <mi>U</mi> <mrow> <mi>q</mi> <mo>,</mo> <mi>t</mi> </mrow> </msub> <mfenced> <msub> <mover> <mover> <mrow> <mi>g</mi> <mi>l</mi> </mrow> <mo>^</mo> </mover> <mo>^</mo> </mover> <mn>1</mn> </msub> </mfenced> </math>  $$ {U}_{q,t}\left({\widehat{\widehat{\mathfrak{gl}}}}_1\right) $$ . This R-matrix acts on pairs of 3d Young diagrams and retains the nice symmetry of the DIM algebra under the permutation of three deformation parameters q, t −1 and   <math> <mfrac> <mi>t</mi> <mi>q</mi> </mfrac> </math>  $$ \frac{t}{q} $$ . We construct the intertwining operator for a tensor product of the horizontal Fock representation and the vertical MacMahon representation and show that the intertwiners are permuted using the MacMahon R-matrix.