Affine Laumon Partition Function

• Affine Laumon partition function is an equivariant generating function derived from moduli spaces of parabolic sheaves and quiver varieties with an affine type A structure.
• It employs localization on torus-fixed points labeled by partitions to compute instanton partition functions in 5d gauge theories and related geometric settings.
• It bridges representation theory, integrable systems, and conformal blocks by linking K-theoretic modules with quantum affine algebras and vertex operator algebras.

An affine Laumon partition function is an equivariant generating function attached to affine Laumon spaces, namely moduli spaces of parabolic sheaves on $\mathbb{P}^1\times\mathbb{P}^1$ or, equivalently in several formulations, quiver varieties with affine type $A$ structure. In the 5d gauge-theoretic setting it is the instanton partition function of a gauge theory with a surface defect, computed by localization on torus-fixed points labeled by partitions; in adjacent formulations it appears as a generating function of Euler characteristics, a generating function of Poincaré polynomials, or a character, trace, or matrix element in Verma-type modules of quantum affine algebras (Awata et al., 2022, Shiraishi, 2019, Creutzig et al., 2022, Shen, 2024).

1. Geometric origin

For fixed $n\ge 2$ and a degree vector $d=(d_1,\dots,d_n)\in \mathbb{Z}_{\ge 0}^n$, the affine Laumon space $\mathcal{M}_d$ is a moduli space of parabolic sheaves on $\mathbb{P}^1\times\mathbb{P}^1$, and equivalently a quiver variety defined by an ADHM-type moment-map equation and a stability condition (Shen, 2024). In another standard formulation, an affine Laumon space is the moduli space $\mathcal{M}_{\underline d}$ of rank $n$ degree $\underline d$ parabolic sheaves

$$\mathcal{F}_n(-D)\subset \mathcal{F}_1\subset \cdots \subset \mathcal{F}_{n-1}\subset \mathcal{F}_n$$

on $A$0, with framing along $A$1, and it is a smooth, quasi-projective variety of dimension $A$2 (Neguţ, 2011).

In gauge theory these spaces arise as moduli spaces of instantons in the presence of a surface operator. For $A$3, the starting physical theory in one formulation is a 5d $A$4 supersymmetric $A$5 gauge theory with four fundamental hypermultiplets and a surface defect; the affine Laumon space is the moduli space of parabolic sheaves on $A$6 with a parabolic structure along the divisor representing the defect (Awata et al., 2022). A related ADHM-orbifold description identifies affine Laumon spaces as $A$7-fixed components in instanton moduli spaces, equivalently as moduli spaces of framed parabolic torsion-free sheaves on $A$8 (Creutzig et al., 2022).

The localization formalism is controlled by a torus action. Fixed points are combinatorial. For type $A$9, the torus-fixed points are labeled by pairs of partitions $n\ge 2$0 (Awata et al., 2022). For general $n\ge 2$1, fixed points are parametrized by $n\ge 2$2-tuples of 2D partitions $n\ge 2$3, with colored boxes and degree vector determined by counting boxes of each color (Shen, 2024). This fixed-point structure is the source of the partition-function expansions.

2. Localization formulas and standard forms

The 5d affine Laumon instanton partition function for $n\ge 2$4 gauge theory with $n\ge 2$5 fundamentals and $n\ge 2$6 anti-fundamentals, in the presence of a surface defect, is the function

$n\ge 2$7

defined by the fixed-point sum

$n\ge 2$8

where $n\ge 2$9 and $d=(d_1,\dots,d_n)\in \mathbb{Z}_{\ge 0}^n$0, and $d=(d_1,\dots,d_n)\in \mathbb{Z}_{\ge 0}^n$1 is the $d=(d_1,\dots,d_n)\in \mathbb{Z}_{\ge 0}^n$2-orbifolded Nekrasov factor (Awata et al., 2022). In this notation, $d=(d_1,\dots,d_n)\in \mathbb{Z}_{\ge 0}^n$3 are masses of fundamental hypermultiplets, $d=(d_1,\dots,d_n)\in \mathbb{Z}_{\ge 0}^n$4 are masses of anti-fundamental hypermultiplets, $d=(d_1,\dots,d_n)\in \mathbb{Z}_{\ge 0}^n$5 are Coulomb parameters, and $d=(d_1,\dots,d_n)\in \mathbb{Z}_{\ge 0}^n$6 are expansion parameters with $d=(d_1,\dots,d_n)\in \mathbb{Z}_{\ge 0}^n$7 and $d=(d_1,\dots,d_n)\in \mathbb{Z}_{\ge 0}^n$8 (Awata et al., 2022).

A direct $d=(d_1,\dots,d_n)\in \mathbb{Z}_{\ge 0}^n$9 generalization is the 5d/K-theoretic partition function of type $\mathcal{M}_d$0,

$\mathcal{M}_d$1

defined as a sum over $\mathcal{M}_d$2-tuples of Young diagrams $\mathcal{M}_d$3, with weights built from $\mathcal{M}_d$4-orbifolded Nekrasov factors and color-counting exponents $\mathcal{M}_d$5 (Awata et al., 31 Oct 2025). In that formula the variables $\mathcal{M}_d$6 are graded by colored box numbers, while $\mathcal{M}_d$7 encode Coulomb moduli and mass parameters.

Other standard affine-Laumon generating functions differ by the cohomology theory or by the insertion:

Setting	Notation	Interpretation
5d defect instanton counting	$\mathcal{M}_d$8, $\mathcal{M}_d$9	K-theoretic Nekrasov partition function (Awata et al., 2022, Awata et al., 2023)
Euler characteristics	$\mathbb{P}^1\times\mathbb{P}^1$0	generating function for Euler characteristics (Shiraishi, 2019)
Poincaré polynomials	$\mathbb{P}^1\times\mathbb{P}^1$1	generating function of Poincaré polynomials (Creutzig et al., 2022)
Cohomological tangent-class integrals	$\mathbb{P}^1\times\mathbb{P}^1$2	generating function of equivariant integrals of $\mathbb{P}^1\times\mathbb{P}^1$3 (Neguţ, 2011)

This multiplicity of formulas is structural rather than accidental. The fixed-point sum over partitions is common, but the numerator, denominator, and grading depend on whether one is taking equivariant Euler characteristics, Poincaré polynomials, K-theoretic instanton weights, or cohomological characteristic-class integrals.

3. Representation-theoretic interpretation

A major development is the identification of equivariant K-theory of affine Laumon spaces with Verma-type modules of quantum affine algebras. The localized equivariant K-theory $\mathbb{P}^1\times\mathbb{P}^1$4 of affine Laumon spaces carries a geometric action of $\mathbb{P}^1\times\mathbb{P}^1$5, constructed by Hecke-type correspondences, and is isomorphic to the universal Verma module over the fraction field (Neguţ, 2018, Shen, 2024). A stronger statement holds integrally after specialization: away from the critical level, the direct sum

$\mathbb{P}^1\times\mathbb{P}^1$6

is identified, after a mild localization and specialization, with the contragredient dual Verma module of $\mathbb{P}^1\times\mathbb{P}^1$7 (Shen, 2024).

This identification is not merely formal. Fixed-point basis vectors are labeled by partitions, PBW bases are indexed by multipartitions, and the Shapovalov form is realized geometrically as an equivariant Euler characteristic pairing with a determinant twist (Shen, 2024). Stable envelopes in a variant of the Maulik–Okounkov formalism provide the triangular transition between geometric fixed-point bases and PBW dual bases, and prove that dual PBW vectors land in nonlocalized K-theory (Shen, 2024).

In this framework, the affine Laumon partition function becomes a character, trace, or matrix element. One description is a Nekrasov-type sum over fixed points with inverse Euler-class weights; another is a character of the module $\mathbb{P}^1\times\mathbb{P}^1$8, or, after specialization, of the contragredient dual Verma module $\mathbb{P}^1\times\mathbb{P}^1$9 (Shen, 2024). A closely related formulation identifies the non-stationary Ruijsenaars function $\mathcal{M}_{\underline d}$0 with the generating function for Euler characteristics of affine Laumon spaces, and its free-field construction uses affine screening operators and screened vertex operators (Shiraishi, 2019).

4. Integrable systems and non-stationary equations

Affine Laumon partition functions occur as special functions of integrable systems. In the cohomological setting, the generating function

$\mathcal{M}_{\underline d}$1

was shown to coincide, up to an explicit factor, with the distinguished eigenfunction of the affine trigonometric Calogero–Moser Hamiltonian (Neguţ, 2011). This proves Braverman’s conjecture in that setting and identifies the affine Laumon partition function with an affine Calogero–Moser eigenfunction.

A different line of development begins with Shakirov’s non-stationary difference equation. For $\mathcal{M}_{\underline d}$2, the equation is gauge-equivalent to the Schrödinger form of qq-Painlevé VI, and the Hamiltonian factorizes because the time-translation element in the extended affine Weyl group admits a square-root factorization (Awata et al., 2022). The resulting coupled first-order system is conjecturally solved by two Weyl-related specializations $\mathcal{M}_{\underline d}$3 of the same affine Laumon partition function $\mathcal{M}_{\underline d}$4 (Awata et al., 2022).

That conjecture is established in the truncated case by identifying the K-theoretic Nekrasov partition function from the affine Laumon space with the Jackson-integral solution of the $\mathcal{M}_{\underline d}$5-KZ equation for $\mathcal{M}_{\underline d}$6 with generic spins (Awata et al., 2023). In this form, the truncated non-stationary equation becomes a $\mathcal{M}_{\underline d}$7-KZ equation, the Hamiltonian matrix agrees with the $\mathcal{M}_{\underline d}$8-matrix, and the affine Laumon partition function gives a fundamental solution (Awata et al., 2023).

The same theme extends to higher rank. A $\mathcal{M}_{\underline d}$9 non-stationary difference equation has been introduced with a Hamiltonian written in terms of $n$0-commuting variables, simple-root and higher-root factorizations, and a normal-ordered form. The conjecture is that the affine Laumon partition function of type $n$1 solves this equation, and the four-dimensional limit reduces to the Fuji–Suzuki–Tsuda system (Awata et al., 31 Oct 2025).

Another integrable interpretation is elliptic. The formal power series $n$2, called the non-stationary Ruijsenaars function, is identified with the generating function for the Euler characteristics of affine Laumon spaces. The Main Conjecture states that one can normalize it so that the limit $n$3 exists, and that the stationary limit gives the eigenfunction of the elliptic Ruijsenaars operator (Shiraishi, 2019).

5. Conformal blocks, Macdonald theory, and vertex algebras

Affine Laumon partition functions also provide explicit realizations of conformal blocks. In the $n$4 case, an AGT-type correspondence relates the four-point affine $n$5 conformal block to the Nekrasov instanton partition function of 4d $n$6 $n$7 gauge theory with a surface operator, computed on the Laumon space (Belavin, 2012). For $n$8 chiral fields in the Kazama–Suzuki realization of the $n$9 superconformal algebra, the corresponding $\underline d$0 construction uses the ordinary highest-weight representation, and the chiral four-point block reduces to a single-diagram combinatorial expression because the parameter choice forces one of the two Young diagrams to be empty (Belavin, 2012).

In a VOA direction, the generating function $\underline d$1 of Poincaré polynomials of affine Laumon spaces of type $\underline d$2 is given by a closed infinite-product formula, and this function is identified with the refined character of a universal Verma module of a vertex algebra constructed via iterated quantum Hamiltonian reduction (Creutzig et al., 2022). The paper formulates the associated iterated $\underline d$3-algebra conjecture and proves equality of characters.

Macdonald-theoretic realizations are equally explicit. For $\underline d$4, the local generating series

$\underline d$5

built from based Laumon spaces is an eigenfunction of a Macdonald difference operator, while the global limits

$\underline d$6

are proportional to Macdonald polynomials $\underline d$7 by explicit $\underline d$8-Pochhammer factors (Braverman et al., 2012). That paper does not introduce the term “affine Laumon partition function,” but it produces such objects in the form of equivariant graded Euler characteristics attached to Laumon spaces and the arc space of the base affine space (Braverman et al., 2012).

6. Variants, limits, and scope of the term

The phrase “affine Laumon partition function” is not uniform across the literature. In some papers it means the 5d K-theoretic instanton partition function with a surface defect, written as a sum over partition data with orbifolded Nekrasov factors (Awata et al., 2022, Awata et al., 2023, Awata et al., 31 Oct 2025). In others it means the generating function of Euler characteristics (Shiraishi, 2019), the generating function of Poincaré polynomials (Creutzig et al., 2022), or the generating function of equivariant integrals of the Chern polynomial of the tangent bundle (Neguţ, 2011). There are also papers that do not use the term explicitly but provide the same object under a representation-theoretic or Macdonald-theoretic description (Shen, 2024, Braverman et al., 2012).

The limiting regimes are correspondingly varied. Along the “axes” $\underline d$9 or $$\mathcal{F}_n(-D)\subset \mathcal{F}_1\subset \cdots \subset \mathcal{F}_{n-1}\subset \mathcal{F}_n$$0, the function $$\mathcal{F}_n(-D)\subset \mathcal{F}_1\subset \cdots \subset \mathcal{F}_{n-1}\subset \mathcal{F}_n$$1 in the $$\mathcal{F}_n(-D)\subset \mathcal{F}_1\subset \cdots \subset \mathcal{F}_{n-1}\subset \mathcal{F}_n$$2 theory factorizes into one-variable $$\mathcal{F}_n(-D)\subset \mathcal{F}_1\subset \cdots \subset \mathcal{F}_{n-1}\subset \mathcal{F}_n$$3-hypergeometric series, and the Heine limit $$\mathcal{F}_n(-D)\subset \mathcal{F}_1\subset \cdots \subset \mathcal{F}_{n-1}\subset \mathcal{F}_n$$4 reduces the partition function to a Heine $$\mathcal{F}_n(-D)\subset \mathcal{F}_1\subset \cdots \subset \mathcal{F}_{n-1}\subset \mathcal{F}_n$$5-hypergeometric function (Awata et al., 2022). Special mass choices lead to Macdonald limits, including asymptotically free Macdonald functions of type $$\mathcal{F}_n(-D)\subset \mathcal{F}_1\subset \cdots \subset \mathcal{F}_{n-1}\subset \mathcal{F}_n$$6 and $$\mathcal{F}_n(-D)\subset \mathcal{F}_1\subset \cdots \subset \mathcal{F}_{n-1}\subset \mathcal{F}_n$$7 (Awata et al., 2022). For the non-stationary Ruijsenaars function, the limit $$\mathcal{F}_n(-D)\subset \mathcal{F}_1\subset \cdots \subset \mathcal{F}_{n-1}\subset \mathcal{F}_n$$8 gives dominant integrable characters of $$\mathcal{F}_n(-D)\subset \mathcal{F}_1\subset \cdots \subset \mathcal{F}_{n-1}\subset \mathcal{F}_n$$9 multiplied by $A$00; the limit $A$01 yields affine $A$02-difference Toda structures; and the limit $A$03 yields elliptic Calogero–Sutherland equations (Shiraishi, 2019). In the defect-gauge-theory setting, four-dimensional limits recover the KZ equation, quantum Painlevé VI, or the Fuji–Suzuki–Tsuda system, depending on the rank and the normalization (Awata et al., 2023, Awata et al., 31 Oct 2025).

A common structural feature across these variants is a localization sum over partition-labeled fixed points, together with a representation-theoretic reinterpretation as a Verma-module character or matrix element. What changes from paper to paper is the cohomology theory, the insertion, the normalization, and the integrable system to which the resulting function is attached.