arXiv subjects

Jean-Emile Bourgine

Publications and source records attributed to Jean-Emile Bourgine.

At least 19 recordsLinked to original sources

arXiv2026

Deformed W-algebras and chiralized cluster seeds: subregular W-algebras and Inverse Quantum Hamiltonian Reduction

The recently introduced formalism of chiral cluster seeds replaces quantum cluster variables with deformed vertex operators. In this framework, a decorated quiver associated with a seed encodes the operator product expansions of the corresponding vertex operators. This formalism is applied to several $(q,t)$-deformed W-algebras, including $\mathcal{W}_{\mathfrak{q},\mathfrak{t}}(\mathfrak{gl}(N|M))$, $U_q(\widehat{\mathfrak{sl}}_2)$, and the deformed Bershadsky--Polyakov algebra. In particular, it is shown that different free field realizations of the currents are related by mutations of the associated chiral cluster seed. The second part of the paper introduces a $(q,t)$-deformation of the subregular W-algebras, denoted by $\mathcal{W}_{\mathfrak{q},\mathfrak{t}}^{\text{sub}}(\mathfrak{sl}(N))$. All free field realizations obtainable through seed mutations are described. An embedding of $\mathcal{W}_{\mathfrak{q},\mathfrak{t}}^{\text{sub}}(\mathfrak{sl}(N))$ into the free field realization of $\mathcal{W}_{\mathfrak{q},\mathfrak{t}}(\mathfrak{sl}(N))$ tensored with a rank-two Heisenberg algebra is constructed. This embedding may be viewed as a deformed analogue of inverse quantum Hamiltonian reduction. The relation between the subregular algebras and $\mathcal{W}_{\mathfrak{q},\mathfrak{t}}(\mathfrak{gl}(1|N))$ is also discussed.

math.QA↗

arXiv2025

Generalized Macdonald functions and quantum toroidal gl(1) algebra

The Macdonald operator is known to coincide with a certain element of the quantum toroidal $\mathfrak{gl}(1)$ algebra in the Fock representation of levels $(1,0)$. A generalization of this operator to higher levels $(r,0)$ can be built using the coproduct structure, it is diagonalized by the generalized Macdonald symmetric functions, indexed by $r$-tuple partitions and depending on $r$ alphabets. In this paper, we extend to the generalized case some of the known formulas obeyed by ordinary Macdonald symmetric functions, such as the $e_1$-Pieri rule or the identity relating them to Whittaker vectors obtained by Garsia, Haiman, and Tesler. We also propose a generalization of the five-term relation, and the Fourier/Hopf pairing. In addition, we prove the factorized expression of the generalized Macdonald kernel conjectured previously by Zenkevich.

math-ph↗

arXiv2024

A $(q,t)$-deformation of the 2d Toda integrable hierarchy

A $(q,t)$-deformation of the 2d Toda integrable hierarchy is introduced by enhancing the underlying symmetry algebra $\mathfrak{gl}(\infty)\simeq \text{q-W}_{1+\infty}$ to the quantum toroidal $\mathfrak{gl}(1)$ algebra. The difference-differential equations of the hierarchy are obtained from the expansion of $(q,t)$-bilinear identities, and two equations refining the 2d Toda equation are found in this way. The derivation of the bilinear identities follows from the isomorphism between the Fock representation of level $(2,0)$ of the quantum toroidal $\mathfrak{gl}(1)$ algebra and the tensor product of the q-deformed Virasoro algebra with a $u(1)$ Heisenberg algebra. It leads to identify the $(q,t)$-deformed Casimir with the screening charges of the deformed Virasoro algebra. Due to the non-trivial coproduct, equations of the hierarchy no longer involve a single tau-function, but instead relate a set of different tau functions. We then define the universal refined tau function using the $L$-matrix of the quantum toroidal $\mathfrak{gl}(1)$ algebra and interpret it as the generating function of the deformed tau functions. The equations of the hierarchy, written in terms of the universal refined tau function, combine into two-term quadratic equations similar to the $RLL$ equations.

math-ph↗

arXiv2024

A Calogero model for the non-Abelian quantum Hall effect

A model of the non-Abelian fractional quantum Hall effect is obtained from the diagonalization of the matrix model proposed by Dorey, Tong, and Turner (DTT). The Hamiltonian is reminiscent of a spin Calogero-Moser model but involves higher-order symmetric representations of the non-Abelian symmetry. We derive the energy spectrum and show that the Hamiltonian has a triangular action on a certain class of wave functions with a free fermion expression. We deduce the expression of the ground states eigenfunctions and show that they solve a Knizhnik-Zamolodchikov equation. Finally, we discuss the emergence of Kac-Moody symmetries in the large $N$ limit using the level-rank duality and confirm the results obtained previously by DTT.

hep-th↗

arXiv2023

Shifted quantum groups and matter multiplets in supersymmetric gauge theories

The notion of shifted quantum groups has recently played an important role in algebraic geometry. This subtle modification of the original definition brings more flexibility in the representation theory of quantum groups. The first part of this paper presents new mathematical results for the shifted quantum toroidal $\mathfrak{gl}(1)$ and quantum affine $\mathfrak{sl}(2)$ algebras (resp. denoted $\ddot{U}_{q_1,q_2}^\boldsymbolμ(\mathfrak{gl}(1))$ and $\dot{U}_q^\boldsymbolμ(\mathfrak{sl}(2))$). It defines several new representations, including finite dimensional highest $\ell$-weight representations for the toroidal algebra, and a vertex representation of $\dot{U}_q^\boldsymbolμ(\mathfrak{sl}(2))$ acting on Hall-Littlewood polynomials. It also explores the relations between representations of $\dot{U}_q^\boldsymbolμ(\mathfrak{sl}(2))$ and $\ddot{U}_{q_1,q_2}^\boldsymbolμ(\mathfrak{gl}(1))$ in the limit $q_1\to\infty$ ($q_2$ fixed), and present the construction of several new intertwiners. These results are used in the second part to construct BPS observables for 5d $\mathcal{N}=1$ and 3d $\mathcal{N}=2$ gauge theories. In particular, it is shown that 5d hypermultiplets and 3d chiral multiplets can be introduced in the algebraic engineering framework using shifted representations, and the Higgsing procedure is revisited from this perspective.

hep-th↗

arXiv2022

Engineering 3D $\mathcal{N}=2$ theories using the quantum affine $\mathfrak{sl}(2)$ algebra

The algebraic engineering technique is applied to a class of 3D $\mathcal{N}=2$ gauge theories on the omega-deformed background $\mathbb{R}_ε^2\times S^1$. The vortex partition function and the fundamental qq-character are obtained from a network of intertwiners between representations of the shifted (or asymptotic) quantum affine $\mathfrak{sl}(2)$ algebra. This network involves two types of representations, the prefundamental representation of Hernandez-Jimbo, and a new vertex representation acting on a bosonic Fock space. The brane system associated to this network is identified: D3 branes carry the prefundamental module while NS5-branes (+D5) support the Fock module. In the process, we highlight the role of shifted quantum algebras in implementing the Higgsing procedure.

hep-th↗

arXiv2021

Intertwining operator and integrable hierarchies from topological strings

In [1], Nakatsu and Takasaki have shown that the melting crystal model behind the topological strings vertex provides a tau-function of the KP hierarchy after an appropriate time deformation. We revisit their derivation with a focus on the underlying quantum $W_{1+\infty}$ symmetry. Specifically, we point out the role played by automorphisms and the connection with the intertwiner - or vertex operator - of the algebra. This algebraic perspective allows us to extend part of their derivation to the refined melting crystal model, lifting the algebra to the quantum toroidal algebra of $\mathfrak{gl}(1)$ (also called Ding-Iohara-Miki algebra). In this way, we take a first step toward the definition of deformed hierarchies associated to A-model refined topological strings.

hep-th↗

arXiv2021

Quantum $W_{1+\infty}$ subalgebras of BCD type and symmetric polynomials

The infinite affine Lie algebras of type ABCD, also called $\widehat{\mathfrak{gl}}(\infty)$, $\widehat{\mathfrak{o}}(\infty)$, $\widehat{\mathfrak{sp}}(\infty)$, are equivalent to subalgebras of the quantum $W_{1+\infty}$ algebras. They have well-known representations on the Fock space of either a Dirac fermion ($\hat A_\infty$), a Majorana fermion ($\hat B_\infty$ and $\hat D_\infty$) or a symplectic boson ($\hat C_\infty$). Explicit formulas for the action of the quantum $W_{1+\infty}$ subalgebras on the Fock states are proposed for each representation. These formulas are the equivalent of the \textit{vertical presentation} of the quantum toroidal $\mathfrak{gl}(1)$ algebra Fock representation. They provide an alternative to the fermionic and bosonic expressions of the \textit{horizontal presentation}. Furthermore, these algebras are known to have a deep connection with symmetric polynomials. The action of the quantum $W_{1+\infty}$ generators leads to the derivation of Pieri-like rules and q-difference equations for these polynomials. In the specific case of $\hat B_\infty$, a q-difference equation is obtained for $Q$-Schur polynomials indexed by strict partitions.

hep-th↗

arXiv2020

New quantum toroidal algebras from 5D $\mathcal{N}=1$ instantons on orbifolds

Quantum toroidal algebras are obtained from quantum affine algebras by a further affinization, and, like the latter, can be used to construct integrable systems. These algebras also describe the symmetries of instanton partition functions for 5D $\mathcal{N}=1$ supersymmetric quiver gauge theories. We consider here the gauge theories defined on an orbifold $S^1\times\mathbb{C}^2/\mathbb{Z}_p$ where the action of $\mathbb{Z}_p$ is determined by two integer parameters $(ν_1,ν_2)$. The corresponding quantum toroidal algebra is introduced as a deformation of the quantum toroidal algebra of $\mathfrak{gl}(p)$. We show that it has the structure of a Hopf algebra, and present two representations, called vertical and horizontal, obtained by deforming respectively the Fock representation and Saito's vertex representations of the quantum toroidal algebra of $\mathfrak{gl}(p)$. We construct the vertex operator intertwining between these two types of representations. This object is identified with a $(ν_1,ν_2)$-deformation of the refined topological vertex, allowing us to reconstruct the Nekrasov partition function and the $qq$-characters of the quiver gauge theories.

hep-th↗

arXiv2019

Fiber-base duality from the algebraic perspective

Quiver 5D $\mathcal{N}=1$ gauge theories describe the low-energy dynamics on webs of $(p,q)$-branes in type IIB string theory. S-duality exchanges NS5 and D5 branes, mapping $(p,q)$-branes to branes of charge $(-q,p)$, and, in this way, induces several dualities between 5D gauge theories. On the other hand, these theories can also be obtained from the compactification of topological strings on a Calabi-Yau manifold, for which the S-duality is realized as a fiber-base duality. Recently, a third point of view has emerged in which 5D gauge theories are engineered using algebraic objects from the Ding-Iohara-Miki (DIM) algebra. Specifically, the instanton partition function is obtained as the vacuum expectation value of an operator $\mathcal{T}$ constructed by gluing the algebra's intertwiners (the equivalent of topological vertices) following the rules of the toric diagram/brane web. Intertwiners and $\mathcal{T}$-operators are deeply connected to the co-algebraic structure of the DIM algebra. We show here that S-duality can be realized as the twist of this structure by Miki's automorphism.

hep-th↗

arXiv2018

A note on the algebraic engineering of 4D $\mathcal{N}=2$ super Yang-Mills theories

Some BPS quantities of $\mathcal{N}=1$ 5D quiver gauge theories, like instanton partition functions or qq-characters, can be constructed as algebraic objects of the Ding-Iohara-Miki (DIM) algebra. This construction is applied here to $\mathcal{N}=2$ super Yang-Mills theories in four dimensions using a degenerate version of the DIM algebra. We build up the equivalent of horizontal and vertical representations, the first one being defined using vertex operators acting on a free boson's Fock space, while the second one is essentially equivalent to the action of Vasserot-Shiffmann's Spherical Hecke central algebra. Using intertwiners, the algebraic equivalent of the topological vertex, we construct a set of $\mathcal{T}$-operators acting on the tensor product of horizontal modules, and the vacuum expectation values of which reproduce the instanton partition functions of linear quivers. Analysing the action of the degenerate DIM algebra on the $\mathcal{T}$-operator in the case of a pure $U(m)$ gauge theory, we further identify the degenerate version of Kimura-Pestun's quiver W-algebra as a certain limit of q-Virasoro algebra. Remarkably, as previously noticed by Lukyanov, this particular limit reproduces the Zamolodchikov-Faddeev algebra of the sine-Gordon model.

hep-th↗

arXiv2018

Seiberg-Witten period relations in Omega background

Omega-deformation of the Seiberg-Witten curve is known to be written in terms of the qq-character, namely the trace of a specific operator acting in a Hilbert space spanned by certain Young diagrams. We define a differential form acting on this space and establish two discretised versions of the Seiberg-Witten expressions for the periods and related relations for the prepotential.

hep-th↗

arXiv2018

Quantum integrability of $\mathcal{N}=2$ 4d gauge theories

We provide a description of the quantum integrable structure behind the Thermodynamic Bethe Ansatz (TBA)-like equation derived by Nekrasov and Shatashvili (NS) for $\mathcal{N}=2$ 4d Super Yang-Mills (SYM) theories. In this regime of the background, -- we shall show --, the instanton partition function is characterised by the solution of a TQ-equation. Exploiting a symmetry of the contour integrals expressing the partition function, we derive a 'dual' TQ-equation, sharing the same T-polynomial with the former. This fact allows us to evaluate to $1$ the quantum Wronskian of two dual solutions (for $Q$) and, then, to reproduce the NS TBA-like equation. The latter acquires interestingly the deep meaning of a known object in integrability theory, as its two second determinations give the usual non-linear integral equations (nlies) derived from the 'dual' Bethe Ansatz equations.

hep-th↗

arXiv2017

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

Reflection states are introduced in the vertical and horizontal modules of the Ding-Iohara-Miki (DIM) algebra (quantum toroidal $\mathfrak{gl}_1$). 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 $\mathcal{N}=1$ 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.

hep-th↗

arXiv2017

Logarithmic Minimal Models with Robin Boundary Conditions

We consider general logarithmic minimal models ${\cal LM}(p,p')$, with $p,p'$ coprime, on a strip of $N$ columns with the $(r,s)$ Robin boundary conditions introduced by Pearce, Rasmussen and Tipunin. The associated conformal boundary conditions are labelled by the Kac labels $r\in{\Bbb Z}$ and $s\in{\Bbb N}$. The Robin vacuum boundary condition, labelled by $(r,s\!-\!\frac{1}{2})=(0,\mbox{$\textstyle \frac{1}{2}$})$, is given as a linear combination of Neumann and Dirichlet boundary conditions. The general $(r,s)$ Robin boundary conditions are constructed, using fusion, by acting on the Robin vacuum boundary with an $(r,s)$-type seam consisting of an $r$-type seam of width $w$ columns and an $s$-type seam of width $d=s-1$ columns. The $r$-type seam admits an arbitrary boundary field which we fix to the special value $ξ=-\tfracλ{2}$ where $λ=\frac{(p'-p)π}{2p'}$ is the crossing parameter. The $s$-type boundary introduces $d$ defects into the bulk. We consider the associated quantum Hamiltonians and calculate analytically the boundary free energies of the $(r,s)$ Robin boundary conditions. Using finite-size corrections and sequence extrapolation out to system sizes $N+w+d\le 26$, the conformal spectrum of boundary operators is accessible by numerical diagonalization of the Hamiltonians. Fixing the parity of $N$ for $r\ne 0$ and restricting to the ground state sequences $w=\big\lfloor\frac{|r|p'}{p}\big\rfloor$, $r\in{\Bbb Z}$ with the inverse $r=(-1)^{N+w+d}\big\lceil \frac{p w}{p'}\big\rceil$, we find that the conformal weights take the values $Δ^{p,p'}_{r,s-\frac12}$ where $Δ^{p,p'}_{r,s}$ is given by the usual Kac formula. The $(r,s)$ Robin boundary conditions are thus conjugate to scaling operators with half-integer values for the Kac label $s-\mbox{$\textstyle \frac{1}{2}$}$.

hep-th↗

arXiv2017

(p,q)-webs of DIM representations, 5d N=1 instanton partition functions and qq-characters

Instanton partition functions of $\mathcal{N}=1$ 5d Super Yang-Mills reduced on $S^1$ can be engineered in type IIB string theory from the $(p,q)$-branes web diagram. To this diagram is superimposed a web of representations of the Ding-Iohara-Miki (DIM) algebra that acts on the partition function. In this correspondence, each segment is associated to a representation, and the (topological string) vertex is identified with the intertwiner operator constructed by Awata, Feigin and Shiraishi. We define a new intertwiner acting on the representation spaces of levels $(1,n)\otimes(0,m)\to(1,n+m)$, thereby generalizing to higher rank $m$ the original construction. It allows us to use a folded version of the usual $(p,q)$-web diagram, bringing great simplifications to actual computations. As a result, the characterization of Gaiotto states and vertical intertwiners, previously obtained by some of the authors, is uplifted to operator relations acting in the Fock space of horizontal representations. We further develop a method to build qq-characters of linear quivers based on the horizontal action of DIM elements. While fundamental qq-characters can be built using the coproduct, higher ones require the introduction of a (quantum) Weyl reflection acting on tensor products of DIM generators.

hep-th↗

arXiv2016

Coherent states in quantum $\mathcal{W}_{1+\infty}$ algebra and qq-character for 5d Super Yang-Mills

The instanton partition functions of $\mathcal{N}=1$ 5d super Yang-Mills are built using elements of the representation theory of quantum $\mathcal{W}_{1+\infty}$ algebra: Gaiotto state, intertwiner, vertex operator. This algebra is also known under the names of Ding-Iohara-Miki and quantum toroidal $\widehat{\mathfrak{gl}}(1)$ algebra. Exploiting the explicit action of the algebra on the partition function, we prove the regularity of the 5d qq-characters. These characters provide a solution to the Schwinger-Dyson equations, and they can also be interpreted as a quantum version of the Seiberg-Witten curve.

hep-th↗

arXiv2016

Holomorphic field realization of SH$^c$ and quantum geometry of quiver gauge theories

In the context of 4D/2D dualities, SH$^c$ algebra, introduced by Schiffmann and Vasserot, provides a systematic method to analyse the instanton partition functions of $\mathcal{N}=2$ supersymmetric gauge theories. In this paper, we rewrite the SH$^c$ algebra in terms of three holomorphic fields $D_0(z)$, $D_{\pm1}(z)$ with which the algebra and its epresentations are simplified. The instanton partition functions for arbitrary $\mathcal{N}=2$ super Yang-Mills theories with $A_n$ and $A^{(1)}_n$ type quiver diagrams are compactly expressed as a product of four building blocks: Gaiotto state, dilatation, flavor vertex operator and intertwiner which are written in terms of SH$^c$ and the orthogonal basis introduced by Alba, Fateev, Litvinov and Tarnopolskiy. These building blocks are characterized by new conditions which generalize the known ones on the Gaiotto state and the Carlsson-Okounkov vertex. Consistency conditions of the inner product give algebraic relations for the chiral ring generating functions defined by Nekrasov, Pestun and Shatashvili. In particular we show the polynomiality of the qq-characters which have been introduced as a deformation of the Yangian characters. These relations define a second quantization of the Seiberg-Witten geometry, and, accordingly, reduce to a Baxter TQ-equation in the Nekrasov-Shatashvili limit of the Omega-background.

hep-th↗