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H. Awata

Publications and source records attributed to H. Awata.

At least 19 recordsLinked to original sources
arXiv2024

A quantum deformation of the ${\mathcal N}=2$ superconformal algebra

We introduce a unital associative algebra ${\mathcal{SV}ir\!}_{q,k}$, having $q$ and $k$ as complex parameters, generated by the elements $K^\pm_m$ ($\pm m\geq 0$), $T_m$ ($m\in \mathbb{Z}$), and $G^\pm_m$ ($m\in \mathbb{Z}+{1\over 2}$ in the Neveu-Schwarz sector, $m\in \mathbb{Z}$ in the Ramond sector), satisfying relations which are at most quartic. Calculations of some low-lying Kac determinants are made, providing us with a conjecture for the factorization property of the Kac determinants. The analysis of the screening operators gives a supporting evidence for our conjecture. It is shown that by taking the limit $q\rightarrow 1$ of ${\mathcal{SV}ir\!}_{q,k}$ we recover the ordinary ${\mathcal N}=2$ superconformal algebra. We also give a nontrivial Heisenberg representation of the algebra ${\mathcal{SV}ir\!}_{q,k}$, making a twist of the $U(1)$ boson in the Wakimoto representation of the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_2)$, which naturally follows from the construction of ${\mathcal{SV}ir\!}_{q,k}$ by gluing the deformed $Y$-algebras of Gaiotto and Rap$\check{\mathrm{c}}$\'{a}k.

math.QA↗
arXiv2020

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

As a development of arXiv:1912.12897, 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 $5d$.

hep-th↗
arXiv2020

Shiraishi functor and non-Kerov deformation of Macdonald polynomials

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, 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, $\vec x$ and $\vec y$, 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 $\vec y$ 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.

hep-th↗
arXiv2019

On complete solution of the quantum Dell system

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 $PQ$-duality fully explicit at the elliptic level.

hep-th↗
arXiv2019

Can tangle calculus be applicable to hyperpolynomials?

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 $L_{8n8}$, 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 $A^{\pm 1}\longrightarrow 0$. 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.

hep-th↗
arXiv2018

The MacMahon R-matrix

We introduce an $R$-matrix acting on the tensor product of MacMahon representations of Ding-Iohara-Miki (DIM) algebra $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$. 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 $\frac{t}{q}$. 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.

hep-th↗
arXiv2018

A non-torus link from topological vertex

The recently suggested tangle calculus for knot polynomials is intimately related to topological string considerations and can help to build the HOMFLY-PT invariants from the topological vertices. We discuss this interplay in the simplest example of the Hopf link and link $L_{8n8}$. It turns out that the resolved conifold with four different representations on the four external legs, on the topological string side, is described by a special projection of the four-component link $L_{8n8}$, which reduces to the Hopf link colored with two composite representations. Thus, this provides the first explicit example of non-torus link description through the topological vertex. It is not a real breakthrough, because $L_{8n8}$ is just a cable of the Hopf link, still, it can help to intensify the development of the formalism towards more interesting examples.

hep-th↗
arXiv2017

$(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces

We describe the general strategy for lifting the Wess-Zumino-Witten model from the level of one-loop Kac-Moody $U_q(\widehat{\mathfrak{g}})_k$ to generic quantum toroidal algebras. A nearly exhaustive presentation is given for the two series $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$ and $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_n)$, when screenings do not exist and thus all the correlators are purely algebraic, i.e. do not include additional hypergeometric type integrations/summations. Generalizing the construction of the intertwiner (refined topological vertex) of the Ding-Iohara-Miki (DIM) algebra, we obtain the intertwining operators of the Fock representations of the quantum toroidal algebra of type $A_n$. The correlation functions of these operators satisfy the $(q,t)$-Knizhnik-Zamolodchikov (KZ) equation, which features the ${\cal R}$-matrix. Matching with the Nekrasov function for the instanton counting on the ALE space is worked out explicitly. We also present an important application of the DIM formalism to the study of $6d$ gauge theories described by the double elliptic integrable systems. We show that the modular and periodicity properties of the gauge theories are neatly explained by the network matrix models providing solutions to the elliptic $(q,t)$-KZ equations.

hep-th↗
arXiv2016

Anomaly in RTT relation for DIM algebra and network matrix models

We discuss the recent proposal of arXiv:1608.05351 about generalization of the RTT relation to network matrix models. We show that the RTT relation in these models is modified by a nontrivial, but essentially abelian anomaly cocycle, which we explicitly evaluate for the free field representations of the quantum toroidal algebra. This cocycle is responsible for the braiding, which permutes the external legs in the q-deformed conformal block and its 5d/6d gauge theory counterpart, i.e. the non-perturbative Nekrasov functions. Thus, it defines their modular properties and symmetry. We show how to cancel the anomaly using a construction somewhat similar to the anomaly matching condition in gauge theory. We also describe the singular limit to the affine Yangian (4d Nekrasov functions), which breaks the spectral duality.

hep-th↗
arXiv2011

Quantum Algebraic Approach to Refined Topological Vertex

We establish the equivalence between the refined topological vertex of Iqbal-Kozcaz-Vafa and a certain representation theory of the quantum algebra of type W_{1+infty} introduced by Miki. Our construction involves trivalent intertwining operators Phi and Phi^* associated with triples of the bosonic Fock modules. Resembling the topological vertex, a triple of vectors in Z^2 is attached to each intertwining operator, which satisfy the Calabi-Yau and smoothness conditions. It is shown that certain matrix elements of Phi and Phi^* give the refined topological vertex C_{lambda mu nu}(t,q) of Iqbal-Kozcaz-Vafa. With another choice of basis, we recover the refined topological vertex C_{lambda mu}^nu(q,t) of Awata-Kanno. The gluing factors appears correctly when we consider any compositions of Phi and Phi^*. The spectral parameters attached to Fock spaces play the role of the K"ahler parameters.

hep-th↗
arXiv2011

Notes on Ding-Iohara algebra and AGT conjecture

We study the representation theory of the Ding-Iohara algebra $\calU$ to find $q$-analogues of the Alday-Gaiotto-Tachikawa (AGT) relations. We introduce the endomorphism $T(u,v)$ of the Ding-Iohara algebra, having two parameters $u$ and $v$. We define the vertex operator $\Phi(w)$ by specifying the permutation relations with the Ding-Iohara generators $x^\pm(z)$ and $\psi^\pm(z)$ in terms of $T(u,v)$. For the level one representation, all the matrix elements of the vertex operators with respect to the Macdonald polynomials are factorized and written in terms of the Nekrasov factors for the $K$-theoretic partition functions as in the AGT relations. For higher levels $m=2,3,...$, we present some conjectures, which imply the existence of the $q$-analogues of the AGT relations.

math-ph↗
arXiv1999

Tachyon Condensation and Graviton Production in Matrix Theory

We study a membrane -- anti-membrane system in Matrix theory. It in fact exhibits the tachyon instability. By suitably representing this configuration, we obtain a (2+1)-dimensional U(2) gauge theory with a 't Hooft's twisted boundary condition. We identify the tachyon field with a certain off-diagonal element of the gauge fields in this model. Taking into account the boundary conditions carefully, we can find vortex solutions which saturate the Bogomol'nyi-type bound and manifest the tachyon condensation. We show that they can be interpreted as gravitons in Matrix theory.

hep-th↗
arXiv1997

Comments on the Problem of a Covariant Formulation of Matrix Theory

A possible avenue towards the covariant formulation of the bosonic Matrix Theory is explored. The approach is guided by the known covariant description of the bosonic membrane. We point out various problems with this particular covariantization scheme, stemming from the central question of how to enlarge the original U(N) symmetry of Matrix Theory while preserving all of its essential features in the infinite momentum frame.

hep-th↗
arXiv1997

Many-body Dynamics of D0--Branes

We show that the growth of the size with the number of partons holds in a Thomas-Fermi analysis of the threshold bound state of D0--branes. Our results sharpen the evidence that for a fixed value of the eleven dimensional radius the partonic velocities can be made arbitrarily small as one approaches the large N limit.

hep-th↗
arXiv1996

Virasoro-type Symmetries in Solvable Models

Virasoro-type symmetries and their roles in solvable models are reviewed. These symmetries are described by the two-parameter Virasoro-type algebra $Vir_{p,q}$ by choosing the parameters p and q suitably.

hep-th↗
arXiv1996

Quantum Deformation of the W_N Algebra

We review the W_N algebra and its quantum deformation, based on free field realizations. The (quantum deformed) W_N algebra is defined through the (quantum deformed) Miura transformation, and its singular vectors realize the Jack (Macdonald) polynomials. (Talk at the Nankai-CRM joint meeting on ``Extended and Quantum Algebras and their Applications to Physics'', Tianjin, China, August 19-24, 1996.)

q-alg↗
arXiv1996

Vertex Operators of the $q$-Virasoro Algebra; Defining Relations, Adjoint Actions and Four Point Functions

Primary fields of the $q$-deformed Virasoro algebra are constructed. Commutation relations among the primary fields are studied. Adjoint actions of the deformed Virasoro current on the primary fields are represented by the shift operator $Θ_ξ f(x)=f(ξx)$. Four point functions of the primary fields enjoy the connection formula associated with the Boltzmann weights of the fusion Andrews-Baxter-Forrester model.

q-alg↗