Shifted quantum toroidal algebra of type $\mathfrak{gl}_{1|1}$ and the Pieri rule of the super Macdonald polynomials

Abstract: The super Macdonald polynomials indexed by the super partitions form a basis of the level zero super Fock module (combinatorial representation) of the quantum toroidal algebra $$\mathcal{U}<em>{q,t}(\widehat{\widehat{\mathfrak{gl}}}</em>{1|1})$$. The action of the super charges of $$\mathcal{U}<em>{q,t}(\widehat{\widehat{\mathfrak{gl}}}</em>{1|1})$$ implies the Pieri rule of the super Macdonald polynomials. We can express the Pieri rule in terms of differential operators in the power sums $p_k$ and the fermionic power sums $π_k$, which leads to the operators on the Fock space of a free boson and a free fermion. From the Pieri rule we compute the supersymmetric Hamiltonians given by the anti-commutator of the super charges and recover the results previously obtained in the literature. It is remarkable that we have to deal with a shifted quantum toroidal algebra.

• The paper introduces a shifted quantum toroidal algebra of gl(1|1) that underpins a combinatorial realization of super Macdonald polynomials.
• It derives explicit Pieri rules and differential operator formulas linking supercharges with the structure of a level zero super Fock space.
• The work provides new operator representations that enhance computational tools in supersymmetric integrable systems and gauge theories.

Shifted Quantum Toroidal Algebra of Type $\mathfrak{gl}_{1|1}$ and the Pieri Rule for Super Macdonald Polynomials

Introduction and Overview

This paper establishes a foundational connection between the shifted quantum toroidal algebra $\mathcal{U}_{q,t}(\widehat{\widehat{\mathfrak{gl}_{1|1}})$ and the structure of super Macdonald polynomials. The main achievements are threefold: (1) a precise combinatorial realization of the level zero super Fock representation via super partitions, (2) formulation and verification of Pieri rules for super Macdonald polynomials through explicit algebraic operators, and (3) derivation of supersymmetric Hamiltonians as anti-commutators of supercharges, fully consistent with algebraic and geometric expectations.

The results provide a rigorous framework relating quantum algebra representations and supersymmetric integrable systems. This has implications for the study of moduli spaces, instanton counting, BPS state enumeration, and the equivariant $K$-theory of various geometric objects associated with supersymmetric gauge theories. The necessity for a shifted quantum toroidal algebra to accommodate the super setting introduces nontrivial technical features absent in the purely bosonic case.

Super Macdonald Polynomials and Level Zero Fock Spaces

Super Macdonald polynomials $\mathcal{M}_{\Lambda}(x, \theta; q, t)$ generalize classical Macdonald polynomials by indexing with super partitions $\Lambda$ (elements of $\mathbb{Z}/2$ with prescribed $\mathbb{Z}_2$ grading and strictness for odd parts). They are defined as simultaneous eigenfunctions of commuting Hamiltonians in the level zero super Fock representation of $\mathcal{U}_{q,t}(\widehat{\widehat{\mathfrak{gl}_{1|1}})$. The basis elements are constructed from power sum generators $p_k$ (bosonic) and $\pi_k$ (fermionic), with a canonical isomorphism to the tensor product $\mathcal{U}_{q,t}(\widehat{\widehat{\mathfrak{gl}_{1|1}})$0 of free boson and fermion Fock spaces.

The super Macdonald polynomials exhibit invariance under simultaneous permutation of bosonic and fermionic coordinates, and are not invariant under the parameter involution $\mathcal{U}_{q,t}(\widehat{\widehat{\mathfrak{gl}_{1|1}})$1, in contrast to bosonic Macdonald polynomials. This leads to asymmetric eigenvalue spectra for associated Hamiltonians and underpins the technical necessity for a shifted toroidal algebra.

Shifted Quantum Toroidal Algebra Structure

The quantum toroidal algebra $\mathcal{U}_{q,t}(\widehat{\widehat{\mathfrak{gl}_{1|1}})$2 admits a $\mathcal{U}_{q,t}(\widehat{\widehat{\mathfrak{gl}_{1|1}})$3 grading, with odd generators $\mathcal{U}_{q,t}(\widehat{\widehat{\mathfrak{gl}_{1|1}})$4, and even generators $\mathcal{U}_{q,t}(\widehat{\widehat{\mathfrak{gl}_{1|1}})$5. Crucially, the algebra involves shift parameters $\mathcal{U}_{q,t}(\widehat{\widehat{\mathfrak{gl}_{1|1}})$6 in its commutation relations, leading to nontrivial mode expansions for the Cartan currents. The defining relations are formulated using generating currents and a structure function $\mathcal{U}_{q,t}(\widehat{\widehat{\mathfrak{gl}_{1|1}})$7 depending on specific parameterizations ($\mathcal{U}_{q,t}(\widehat{\widehat{\mathfrak{gl}_{1|1}})$8).

The level zero representation (with central element $\mathcal{U}_{q,t}(\widehat{\widehat{\mathfrak{gl}_{1|1}})$9) is constructed by infinite tensor products of vector representations. States in the representation are indexed by super partitions, leading to a combinatorial super Fock space consistent with the fixed points of moduli space localization.

Pieri Rule and Differential Operator Realization

The action of supercharges $K$0 yields explicit Pieri rules for super Macdonald polynomials, expressed as differential operators in power sums $K$1. A central result is the identification:

$K$2

with $K$3 generated via: $K$4

Similarly, supercharges $K$5 and $K$6 act as: $K$7 where

$K$8

These rules hold across lower levels and are checked explicitly using algebraic and combinatorial expansions.

Supersymmetric Hamiltonians via Anti-Commutators

Hamiltonians $K$9 for super Macdonald polynomials are constructed as anti-commutators of supercharges, respecting and exploiting the shifted algebraic relations. The negative mode Hamiltonians admit an explicit expansion:

$\mathcal{M}_{\Lambda}(x, \theta; q, t)$0

with bi-linear fermion coefficients:

$\mathcal{M}_{\Lambda}(x, \theta; q, t)$1

The bosonic part reduces to the $\mathcal{M}_{\Lambda}(x, \theta; q, t)$2-inverted Ruijsenaars-Macdonald operator, confirming consistency with integrable system expectations.

Positive mode Hamiltonians ($\mathcal{M}_{\Lambda}(x, \theta; q, t)$3, $\mathcal{M}_{\Lambda}(x, \theta; q, t)$4) are more intricate due to shift dependence and lack simple involution relations. The paper develops explicit operator and integral representations, showing that their fermionic parts are not merely bi-linear but involve all higher-order terms.

Commutativity and Consistency Checks

Commutativity of the Hamiltonians is rigorously established, both algebraically and via operator identities. Comparison with prior results on super Macdonald polynomials and Hamiltonians [Alarie-Vezina et al., Galakhov et al.] confirms the correctness and universality of the approach. In particular, explicit checks demonstrate numerical agreement in eigenvalues and operator actions on explicit polynomial bases up to high levels.

Consistent with the shifted algebraic relations, the Hamiltonians are not related to each other by a simple parameter involution, and the super Macdonald polynomials themselves are not invariant under $\mathcal{M}_{\Lambda}(x, \theta; q, t)$5. This asymmetry is directly traced to the shifted structure of the quantum toroidal algebra.

Integral Operator Techniques

Integral representations for supercharges (e.g., $\mathcal{M}_{\Lambda}(x, \theta; q, t)$6, $\mathcal{M}_{\Lambda}(x, \theta; q, t)$7) are provided using vertex operator constructions, not only yielding concise computational tools but also enabling the derivation of fermionic non-bilinear expansions. These techniques are closely aligned with free field realizations and provide efficient checks of operator relations and commutator structures.

Implications and Future Directions

The synthesis achieved in this work bridges combinatorial representation theory, quantum integrable algebras, and the geometry of moduli spaces. Practically, the explicit differential operator and integral formulae enhance computational tractability in the calculation of superpolynomial spectra, instanton partition functions, and BPS state enumeration in supersymmetric field theories.

Theoretically, the identification of the shifted quantum toroidal algebra as the natural symmetry underlying super Macdonald polynomials opens pathways for further exploration of supersymmetric quantum algebras, generalizations of the Macdonald-Ruijsenaars hierarchies, and connections to supersymmetric vertex operator algebras (VOAs). There is scope for developing a geometric realization of the involution operator $\mathcal{M}_{\Lambda}(x, \theta; q, t)$8 as an algebra automorphism, and for understanding its action in higher-genus, elliptic, and more general superalgebraic contexts. The interplay between algebraic shifts and physical wall-crossing phenomena also merits deeper investigation.

Conclusion

This paper rigorously establishes the algebraic and combinatorial underpinnings of super Macdonald polynomials, identifying the shifted quantum toroidal algebra of type $\mathcal{M}_{\Lambda}(x, \theta; q, t)$9 as the organizing structure. The explicit formulation of Pieri rules, supercharges, and Hamiltonians as differential and integral operators augments both theoretical insight and computational capabilities for supersymmetric algebraic combinatorics and gauge theory applications. The results build a robust foundation for further developments in quantum algebras, supersymmetric integrable systems, and moduli space geometry.