Shifted Quantum Toroidal Algebra

• Shifted quantum toroidal algebra is a class of quantum algebras where integral shift data modify Cartan mode expansions and commutator supports while preserving key structures.
• In models of shifted quantum toroidal gl₁ and gl₂, modifications yield explicit finite-dimensional representations, such as pit modules with dimensions given by binomial coefficients.
• Advanced constructions leverage vertex operators, Hopf-type coproducts, and intertwiners, linking the theory to deformed W-algebras and supersymmetric gauge frameworks.

Shifted quantum toroidal algebra denotes a family of quantum toroidal-type algebras in which integral shift data deform the current formalism by modifying mode expansions, commutator support, or both. In recent work, this terminology encompasses shifted quantum toroidal $\mathfrak{gl}_1$, shifted quantum toroidal $\mathfrak{gl}_2$ obtained by gluing two commuting quantum toroidal $\mathfrak{gl}_1$ subalgebras, shifted quantum toroidal superalgebras such as $\mathfrak{gl}_{1|1}$, and shifted quiver quantum toroidal algebras acting on subcrystal representations (Bourgine, 2022, Feigin et al., 25 Dec 2025, Kanno et al., 16 May 2026, Noshita et al., 2021). The common theme is that the shift changes the toroidal current algebra at the Cartan–raising/lowering interface while preserving enough structure for Fock realizations, intertwiners, and coproduct-based constructions.

1. Algebraic meaning of the shift

The shift is implemented differently in different presentations, but in each case it enters in a controlled part of the current algebra rather than as a wholesale replacement of the defining relations. For shifted quantum toroidal $\mathfrak{gl}_1$, the algebra $$\ddot{U}_{q_1,q_2}^{\boldsymbol{\mu}}(\mathfrak{gl}(1))$$ depends on $\boldsymbol{\mu}=(\mu_+,\mu_-)\in\mathbb Z^2$, with currents

$$x^\pm(z)=\sum_{k\in\mathbb Z} z^{-k}x_k^\pm,\qquad \psi^\pm(z)=\sum_{\pm k\ge -\mu_\pm} z^{-k}\psi_k^\pm,$$

so the shift changes the domain of the Cartan mode expansions while leaving the current relations themselves unchanged (Bourgine, 2022).

In the family $\mathcal A_{M,N}$ introduced by Feigin, Jimbo, and Mukhin, the shift is encoded through a tuned pair of commuting quantum toroidal $\mathfrak{gl}_1$ algebras $\mathfrak{gl}_2$0 and $\mathfrak{gl}_2$1 with

$\mathfrak{gl}_2$2

and through the $\mathfrak{gl}_2$3 cross-commutator. The paper states explicitly that the shift parameter $\mathfrak{gl}_2$4 appears only in this commutator, and for $\mathfrak{gl}_2$5 one has

$\mathfrak{gl}_2$6

which is presented there as a hallmark of shifted quantum affine and toroidal algebras (Feigin et al., 25 Dec 2025).

In the shifted quantum toroidal superalgebra of type $\mathfrak{gl}_2$7, the shift appears directly in the super-commutator

$\mathfrak{gl}_2$8

and in the representation studied there the shift parameters are $\mathfrak{gl}_2$9 and $\mathfrak{gl}_1$0 (Kanno et al., 16 May 2026).

For shifted quiver quantum toroidal algebras, the shift vector $\mathfrak{gl}_1$1 is built into the positive Cartan current,

$\mathfrak{gl}_1$2

and the defining commutator becomes

$\mathfrak{gl}_1$3

The paper identifies $\mathfrak{gl}_1$4 with the difference between the numbers of zeros and poles of the vacuum charge function (Noshita et al., 2021).

Taken together, these constructions indicate that a shifted quantum toroidal algebra is not characterized by a single universal presentation. What is shared is the use of integral shift data to displace the current algebra in a way that changes representation-theoretic support conditions.

2. Shifted quantum toroidal $\mathfrak{gl}_1$5

The shifted quantum toroidal $\mathfrak{gl}_1$6 algebra provides the most explicit single-current model. Dominant shifts, defined there by $\mathfrak{gl}_1$7, are central for the existence of finite-dimensional highest $\mathfrak{gl}_1$8-weight representations. A principal family is the “pit” representation, whose basis is indexed by sub-Young diagrams of a fixed Young diagram $\mathfrak{gl}_1$9; the upward action vanishes when a shifting Laurent polynomial $\mathfrak{gl}_{1|1}$0 vanishes at the corresponding box content, thereby forbidding the addition of that box (Bourgine, 2022).

For a rectangular Young diagram $\mathfrak{gl}_{1|1}$1, the module dimension is stated to be $\mathfrak{gl}_{1|1}$2, and for the $\mathfrak{gl}_{1|1}$3 rectangle the construction yields a $\mathfrak{gl}_{1|1}$4-dimensional representation. These modules are described as highest $\mathfrak{gl}_{1|1}$5-weight modules, with diagonal Cartan action given by explicit rational functions determined by $\mathfrak{gl}_{1|1}$6 and $\mathfrak{gl}_{1|1}$7. The same paper presents shifted representation functors. For a Laurent polynomial

$\mathfrak{gl}_{1|1}$8

one defines

$\mathfrak{gl}_{1|1}$9

together with a right-shift analogue $\mathfrak{gl}_1$0 acting on $\mathfrak{gl}_1$1 and $\mathfrak{gl}_1$2 (Bourgine, 2022).

The same framework extends the Awata–Feigin–Shiraishi intertwiners to shifted representations. The vertical component in the shifted intertwiner acquires the factor

$\mathfrak{gl}_1$3

so the combinatorics of the allowed Young diagrams is absorbed directly into the intertwiner. A further structural feature is the limit $\mathfrak{gl}_1$4 with $\mathfrak{gl}_1$5 fixed, under which the shifted toroidal algebra degenerates to the shifted quantum affine algebra $\mathfrak{gl}_1$6, and finite-dimensional toroidal pit modules reduce to Kirillov–Reshetikhin modules in the affine setting (Bourgine, 2022).

The paper also introduces left-twisted and right-twisted Fock representations for infinitely shifted algebras. In particular, the left-twisted representation has $\mathfrak{gl}_1$7 and realizes only the negative Cartan and raising current. This suggests that the shifted $\mathfrak{gl}_1$8 theory already contains several of the characteristic phenomena later seen in higher-rank and super cases: truncated mode ranges, finite-dimensional highest $\mathfrak{gl}_1$9-weight modules, and nontrivial shifted intertwiners.

3. Shifted quantum toroidal $$\ddot{U}_{q_1,q_2}^{\boldsymbol{\mu}}(\mathfrak{gl}(1))$$0 and the family $$\ddot{U}_{q_1,q_2}^{\boldsymbol{\mu}}(\mathfrak{gl}(1))$$1

A major higher-rank development is the family $$\ddot{U}_{q_1,q_2}^{\boldsymbol{\mu}}(\mathfrak{gl}(1))$$2 of Feigin, Jimbo, and Mukhin. These algebras extend the quotient

$$\ddot{U}_{q_1,q_2}^{\boldsymbol{\mu}}(\mathfrak{gl}(1))$$3

by countably many gluing currents $$\ddot{U}_{q_1,q_2}^{\boldsymbol{\mu}}(\mathfrak{gl}(1))$$4. The index set is $$\ddot{U}_{q_1,q_2}^{\boldsymbol{\mu}}(\mathfrak{gl}(1))$$5 for even $$\ddot{U}_{q_1,q_2}^{\boldsymbol{\mu}}(\mathfrak{gl}(1))$$6 and $$\ddot{U}_{q_1,q_2}^{\boldsymbol{\mu}}(\mathfrak{gl}(1))$$7 for odd $$\ddot{U}_{q_1,q_2}^{\boldsymbol{\mu}}(\mathfrak{gl}(1))$$8. These currents satisfy adjoint covariance under both toroidal $$\ddot{U}_{q_1,q_2}^{\boldsymbol{\mu}}(\mathfrak{gl}(1))$$9 factors, quadratic exchange relations

$\boldsymbol{\mu}=(\mu_+,\mu_-)\in\mathbb Z^2$0

and a cross-commutator expressed through fused currents $\boldsymbol{\mu}=(\mu_+,\mu_-)\in\mathbb Z^2$1 and $\boldsymbol{\mu}=(\mu_+,\mu_-)\in\mathbb Z^2$2 (Feigin et al., 25 Dec 2025).

The special case $\boldsymbol{\mu}=(\mu_+,\mu_-)\in\mathbb Z^2$3 is identified there with shifted quantum toroidal $\boldsymbol{\mu}=(\mu_+,\mu_-)\in\mathbb Z^2$4. For $\boldsymbol{\mu}=(\mu_+,\mu_-)\in\mathbb Z^2$5, the gluing currents correspond to the two non-abelian simple root currents of $\boldsymbol{\mu}=(\mu_+,\mu_-)\in\mathbb Z^2$6, while the subalgebra $\boldsymbol{\mu}=(\mu_+,\mu_-)\in\mathbb Z^2$7 plays the role of an extended Cartan. The case $\boldsymbol{\mu}=(\mu_+,\mu_-)\in\mathbb Z^2$8 is described as essentially isomorphic, on admissible modules, to a central extension $\boldsymbol{\mu}=(\mu_+,\mu_-)\in\mathbb Z^2$9 of the standard quantum toroidal $$x^\pm(z)=\sum_{k\in\mathbb Z} z^{-k}x_k^\pm,\qquad \psi^\pm(z)=\sum_{\pm k\ge -\mu_\pm} z^{-k}\psi_k^\pm,$$0 algebra, up to an extra invertible but non-central element $$x^\pm(z)=\sum_{k\in\mathbb Z} z^{-k}x_k^\pm,\qquad \psi^\pm(z)=\sum_{\pm k\ge -\mu_\pm} z^{-k}\psi_k^\pm,$$1 (Feigin et al., 25 Dec 2025).

A related but distinct realization appears in the affinization program for shifted quantum affine $$x^\pm(z)=\sum_{k\in\mathbb Z} z^{-k}x_k^\pm,\qquad \psi^\pm(z)=\sum_{\pm k\ge -\mu_\pm} z^{-k}\psi_k^\pm,$$2. There, $$x^\pm(z)=\sum_{k\in\mathbb Z} z^{-k}x_k^\pm,\qquad \psi^\pm(z)=\sum_{\pm k\ge -\mu_\pm} z^{-k}\psi_k^\pm,$$3 is a realization of the quantum toroidal algebra associated to $$x^\pm(z)=\sum_{k\in\mathbb Z} z^{-k}x_k^\pm,\qquad \psi^\pm(z)=\sum_{\pm k\ge -\mu_\pm} z^{-k}\psi_k^\pm,$$4 built from two commuting quantum toroidal $$x^\pm(z)=\sum_{k\in\mathbb Z} z^{-k}x_k^\pm,\qquad \psi^\pm(z)=\sum_{\pm k\ge -\mu_\pm} z^{-k}\psi_k^\pm,$$5 subalgebras and currents $$x^\pm(z)=\sum_{k\in\mathbb Z} z^{-k}x_k^\pm,\qquad \psi^\pm(z)=\sum_{\pm k\ge -\mu_\pm} z^{-k}\psi_k^\pm,$$6, $$x^\pm(z)=\sum_{k\in\mathbb Z} z^{-k}x_k^\pm,\qquad \psi^\pm(z)=\sum_{\pm k\ge -\mu_\pm} z^{-k}\psi_k^\pm,$$7, while $$x^\pm(z)=\sum_{k\in\mathbb Z} z^{-k}x_k^\pm,\qquad \psi^\pm(z)=\sum_{\pm k\ge -\mu_\pm} z^{-k}\psi_k^\pm,$$8, $$x^\pm(z)=\sum_{k\in\mathbb Z} z^{-k}x_k^\pm,\qquad \psi^\pm(z)=\sum_{\pm k\ge -\mu_\pm} z^{-k}\psi_k^\pm,$$9, is defined by shifting the central identification and modifying the $\mathcal A_{M,N}$0 relation. For odd $\mathcal A_{M,N}$1, the indices are relabeled so that half-integral and integral labels appear as needed. The paper emphasizes large families of representations for dominant shifts $\mathcal A_{M,N}$2, and notes that examples with even positive $\mathcal A_{M,N}$3 appear in extensions of deformed $\mathcal A_{M,N}$4-algebras of type $\mathcal A_{M,N}$5 (Feigin et al., 15 Nov 2025).

These two lines of work show that shifted quantum toroidal $\mathcal A_{M,N}$6 can be realized both as a gluing extension of commuting toroidal $\mathcal A_{M,N}$7 algebras and as an affinization of shifted quantum affine $\mathcal A_{M,N}$8. The common structural ingredient is an infinite family of additional currents whose cross-commutator carries the shift.

4. Representation theory and explicit models

Representation theory is a central organizing principle for shifted quantum toroidal algebras. In the $\mathcal A_{M,N}$9 family, modules are constructed on direct sums of tensor products of Fock modules of $\mathfrak{gl}_1$0. A basic example for $\mathfrak{gl}_1$1 is

$\mathfrak{gl}_1$2

with gluing currents realized by single vertex operators,

$\mathfrak{gl}_1$3

More elaborate modules arise from iterating the coproduct construction, in which the $\mathfrak{gl}_1$4 become sums of vertex operators or contour-integral expressions (Feigin et al., 25 Dec 2025).

In shifted quantum toroidal $\mathfrak{gl}_1$5, the finite-dimensional pit modules and the shifted AFS intertwiners provide a complementary model. The basis by sub-Young diagrams, the explicit dependence on a shifting polynomial $\mathfrak{gl}_1$6, and the compatibility of intertwiners with shifted actions make these representations unusually concrete. The same paper uses them to model matter hypermultiplets in $\mathfrak{gl}_1$7d $\mathfrak{gl}_1$8 gauge theories and chiral multiplets in $\mathfrak{gl}_1$9d $\mathfrak{gl}_2$00 gauge theories, and revisits Higgsing in this shifted framework (Bourgine, 2022).

The super case brings a different representation-theoretic feature: the level zero super Fock module of $\mathfrak{gl}_2$01 is realized as a semi-infinite tensor product of vector representations with basis indexed by superpartitions, and it is isomorphic to a tensor product of a bosonic Fock space and a fermionic Fock space. The super Macdonald polynomials form a basis diagonalizing the Cartan modes, while the supercurrents admit a differential-operator realization on the supersymmetric power sums

$\mathfrak{gl}_2$02

The Pieri rule is encoded by operators such as

$\mathfrak{gl}_2$03

together with higher operators involving the coefficients $\mathfrak{gl}_2$04 and $\mathfrak{gl}_2$05 (Kanno et al., 16 May 2026).

Taken together, these constructions show that shifted quantum toroidal representation theory is not merely formal. It is realized through vertex operators, differential operators, Fock spaces, and finite-dimensional highest $\mathfrak{gl}_2$06-weight modules, depending on the presentation under consideration.

5. Coproducts, tensor products, and Hopf-type structures

Tensor-product theory in the shifted setting is subtle because the current relations often involve completions or infinite sums. For $\mathfrak{gl}_2$07, Feigin, Jimbo, and Mukhin formulate a conjectural coproduct

$\mathfrak{gl}_2$08

required to be coassociative and to restrict on the toroidal subalgebras to the standard tensor coproduct. On gluing currents,

$\mathfrak{gl}_2$09

$\mathfrak{gl}_2$10

where $\mathfrak{gl}_2$11 is the product of the universal $\mathfrak{gl}_2$12-matrices of the two toroidal subalgebras. The paper further states that, on Fock modules, this agrees with the explicit formulas for induced modules, and that for $\mathfrak{gl}_2$13 the conjecture would make $\mathfrak{gl}_2$14 into a Hopf algebra (Feigin et al., 25 Dec 2025).

Shifted quiver quantum toroidal algebras admit a generalized coproduct that is already part of the formalism. It maps

$\mathfrak{gl}_2$15

with

$\mathfrak{gl}_2$16

$\mathfrak{gl}_2$17

$\mathfrak{gl}_2$18

This coproduct is used to build $\mathfrak{gl}_2$19d subcrystal representations from $\mathfrak{gl}_2$20d ones and ensures that shifts add correctly under tensor product (Noshita et al., 2021).

A broader structural background comes from unshifted quantum toroidal theory. A new topological coproduct extending the Drinfeld–Jimbo coproduct on the horizontal quantum affine subalgebra was constructed from the “top” and “bottom” shuffle halves of the quantum toroidal algebra (Neguţ, 2019), and the quantum toroidal algebra of type $\mathfrak{gl}_2$21 was equipped with a deformed Drinfeld coproduct yielding a topological Hopf algebra structure (Chen et al., 2020). This suggests that completed or topological coproducts are not an ad hoc feature of shifted theories but a natural continuation of toroidal coproduct technology.

6. Super, quiver, and geometric incarnations

The shifted quantum toroidal superalgebra of type $\mathfrak{gl}_2$22 shows that the shift can have direct combinatorial and spectral consequences. In the relevant level zero super Fock module, the natural annihilation operators are shifted modes such as $\mathfrak{gl}_2$23 rather than zero modes, and the Hamiltonians are constructed as anti-commutators of supercharges with mode choices dictated by the shift. The super Macdonald polynomials are simultaneous eigenfunctions of the commuting Hamiltonians, but the paper emphasizes that the fermionic part does not enjoy the same $\mathfrak{gl}_2$24 symmetry as in the bosonic case; this asymmetry is attributed there to the shift (Kanno et al., 16 May 2026).

Shifted quiver quantum toroidal algebras supply a different geometric incarnation. They act on $\mathfrak{gl}_2$25d and $\mathfrak{gl}_2$26d subcrystals obtained from $\mathfrak{gl}_2$27d BPS crystals by removing arrows from the quiver. Detailed constructions are given for $\mathfrak{gl}_2$28, $\mathfrak{gl}_2$29 with $\mathfrak{gl}_2$30, the conifold, the suspended pinch point, and $\mathfrak{gl}_2$31. The shift vector is determined by the altered vacuum charge function; explicitly,

$\mathfrak{gl}_2$32

The generalized coproduct then assembles higher-dimensional crystal modules from vector-type $\mathfrak{gl}_2$33d modules (Noshita et al., 2021).

Shifted toroidal representations also appear in supersymmetric gauge theory through algebraic engineering. Shifted $\mathfrak{gl}_2$34 intertwiners and representations are used to include $\mathfrak{gl}_2$35d hypermultiplets and $\mathfrak{gl}_2$36d chiral multiplets, while the shifted framework provides an algebraic interpretation of Higgsing and related brane transitions (Bourgine, 2022). A plausible implication is that shifted quantum toroidal structures are best viewed not only as algebraic deformations but also as devices for encoding boundaries, defects, and truncated growth conditions in geometric and physical models.

7. Distinctions, misconceptions, and current directions

A recurrent misconception is that every toroidal algebra with extra discrete data should be regarded as a shifted quantum toroidal algebra in the same sense. The orbifold algebras attached to $\mathfrak{gl}_2$37 and parameters $\mathfrak{gl}_2$38 provide a counterexample: they are described as deformations of quantum toroidal $\mathfrak{gl}_2$39 with a non-symmetrizable generalized Cartan matrix, and the paper states explicitly that for general $\mathfrak{gl}_2$40 they are not a simple shift of the kind appearing in shifted toroidal algebras (Bourgine et al., 2019).

A second misconception is that the shift is implemented uniformly across all constructions. The literature instead exhibits several distinct mechanisms: truncation of Cartan mode ranges in shifted $\mathfrak{gl}_2$41, support displacement in the $\mathfrak{gl}_2$42–$\mathfrak{gl}_2$43 commutator for $\mathfrak{gl}_2$44, explicit powers $\mathfrak{gl}_2$45 in the super and quiver Cartan currents, and half-integral relabeling of gluing-current indices for odd shifts in certain $\mathfrak{gl}_2$46 realizations (Bourgine, 2022, Feigin et al., 25 Dec 2025, Kanno et al., 16 May 2026, Feigin et al., 15 Nov 2025).

A third point concerns Hopf structures. Some settings admit full or formal Hopf data, such as shifted quiver quantum toroidal algebras and several unshifted toroidal models, whereas for $\mathfrak{gl}_2$47 the coproduct remains conjectural and the module theory is restricted to “admissible” representations because some commutator relations involve infinite sums through fusion (Noshita et al., 2021, Feigin et al., 25 Dec 2025). This is not a deficiency of a single presentation so much as a sign that shifted toroidal algebras sit at the boundary between algebraic and topological tensor structures.

Current directions are therefore shaped by structural questions rather than by the mere existence of examples. The $\mathfrak{gl}_2$48 program explicitly points toward PBW-type properties, graded characters, categorification, and geometric realizations via quivers or sheaf-theoretic constructions (Feigin et al., 25 Dec 2025). The shifted $\mathfrak{gl}_2$49 affinization program isolates dominant-shift representation families and links even positive shifts to extensions of deformed $\mathfrak{gl}_2$50-algebras of type $\mathfrak{gl}_2$51 (Feigin et al., 15 Nov 2025). The cumulative picture is that shifted quantum toroidal algebra is not a single object but a research area organized around a shared principle: toroidal current algebras remain tractable, and often highly explicit, after integral shift data are incorporated into their defining current geometry.