Genus Two DAHA and Its Applications

• Genus two DAHA is a two-parameter algebraic structure defined on a closed genus-two surface using q-difference operators and trace multiplications, generalizing spherical A1-DAHA.
• It employs a rich operator model with Macdonald-type difference operators and a 15-generator presentation that establishes PBW and Gröbner basis properties.
• The framework connects algebra, cluster realizations, and skein theory, providing a quantization of the SL(2,C) character variety with explicit mapping class automorphisms.

Genus two DAHA denotes a family of DAHA-type structures attached to the closed genus-two surface $\Sigma_2$. In the literature, the term refers primarily to the Arthamonov–Shakirov algebra of genus-two knot operators, its generators-and-relations model $\mathcal A_{q,t}$, its commutative classical limit $\mathcal A_{q=1,t}$, and related cluster- and skein-theoretic realizations (Arthamonov et al., 2017, Arthamonov, 2023, Arthamonov et al., 2024, Hikami, 2019). The common theme is that genus two DAHA plays for $\Sigma_2$ a role analogous to that of spherical $A_1$-DAHA for the torus: it is generated by Macdonald-type $q$-difference operators and trace-like multiplication operators, carries mapping class group symmetry, specializes to the genus-two skein algebra at $q=t$, and has a classical limit identified with the $SL(2,\mathbb C)$-character variety of $\Sigma_2$ together with its Poisson deformation (Arthamonov et al., 2017, Arthamonov, 2023).

1. Foundational definitions and formulations

The foundational operator model is an algebra of $q$-difference operators in the three variables

$\mathcal A_{q,t}$0

with generic complex parameters $\mathcal A_{q,t}$1, acting on the $\mathcal A_{q,t}$2-invariant Laurent-polynomial space

$\mathcal A_{q,t}$3

It is generated by three multiplication operators $\mathcal A_{q,t}$4 and three commuting Macdonald-type $\mathcal A_{q,t}$5-difference operators $\mathcal A_{q,t}$6 (Arthamonov et al., 2017, Arthamonov, 2023). In the 2024 cluster paper this same object is denoted

$\mathcal A_{q,t}$7

the genus-2 spherical DAHA, and is defined as the subalgebra of $\mathcal A_{q,t}$8-difference operators on

$\mathcal A_{q,t}$9

generated by the operators $\mathcal A_{q=1,t}$0 and $\mathcal A_{q=1,t}$1 (Arthamonov et al., 2024).

A standard explicit description uses

$\mathcal A_{q=1,t}$2

while $\mathcal A_{q=1,t}$3 is written as a sum over $\mathcal A_{q=1,t}$4 of $\mathcal A_{q=1,t}$5-shifts $\mathcal A_{q=1,t}$6 with rational coefficients in the $\mathcal A_{q=1,t}$7; $\mathcal A_{q=1,t}$8 and $\mathcal A_{q=1,t}$9 are obtained by permuting indices (Arthamonov et al., 2024). The 2017 paper presents this as a genus-two analogue of spherical $\Sigma_2$0-DAHA because it contains genus-one rank-one slices, has commuting Macdonald-type Hamiltonians, and admits mapping class group automorphisms satisfying the defining relations of $\Sigma_2$1 (Arthamonov et al., 2017).

A distinct but related usage appears in Hikami’s skein-theoretic construction. There, “genus-two DAHA” refers to a representation of the genus-two skein algebra assembled from spherical $\Sigma_2$2-DAHA for $\Sigma_2$3 and spherical $\Sigma_2$4-DAHA for $\Sigma_2$5, glued by a quantum dilogarithmic factor (Hikami, 2019). This formulation is not an alternative abstract presentation of the Arthamonov–Shakirov algebra; rather, it is a concrete DAHA representation for $\Sigma_2$6.

Formulation	Basic data	Role
Operator algebra	$\Sigma_2$7	Foundational genus-two DAHA model
Abstract algebra $\Sigma_2$8	15 generators with normal-ordering, $\Sigma_2$9-, and Casimir relations	Flat two-parameter deformation and classical-limit framework
Cluster realization	$A_1$0	Faithful embedding into quantum cluster algebra
Skein representation	$A_1$1- and $A_1$2-DAHA glued on $A_1$3	DAHA realization of genus-two skein algebra

2. Spectral theory and genus-two Macdonald polynomials

A central structural feature is the existence of genus-two Macdonald polynomials, defined on admissible triples

$A_1$4

satisfying triangle inequalities and parity constraints (Arthamonov et al., 2017). In the 2017 formulation they are denoted $A_1$5, recursively characterized by genus-two Pieri rules involving multiplication by $A_1$6, $A_1$7, and $A_1$8. The principal eigenvalue statement is

$A_1$9

so the three $q$0-operators are simultaneously diagonalized by the genus-two Macdonald basis (Arthamonov et al., 2017).

The 2024 cluster paper reformulates the same spectral theory in the notation $q$1, where $q$2 is an admissible triple. These polynomials form a joint eigenbasis for the commuting operators $q$3: $q$4 That paper’s main spectral contribution is a nonrecursive coefficient formula: each monomial coefficient of the cluster-normalized genus-2 Macdonald polynomial is expressed as a weighted sum over lattice points in a convex polytope in $q$5 (Arthamonov et al., 2024).

The genus-two theory contains ordinary $q$6 Macdonald theory as a boundary case. The 2017 paper states

$q$7

with

$q$8

so ordinary rank-one Macdonald polynomials occur on the boundary faces of the admissible cone (Arthamonov et al., 2017). This is one of the mechanisms by which the genus-two construction generalizes the trigonometric $q$9 Ruijsenaars–Schneider model and $q=t$0 Macdonald polynomials.

3. The 15-generator algebra, flatness, and classical limit

The 2023 paper replaces the six-generator operator picture by a 15-generator algebra

$q=t$1

with generators

$q=t$2

defined over $q=t$3 (Arthamonov, 2023). The generators $q=t$4 and $q=t$5 are modeled on $q=t$6-commutators and iterated $q=t$7-commutators of the six basic generators. The defining system consists of normal-ordering relations, a family of 18 $q=t$8-relations obtained from triple-product ambiguities, and a single $q=t$9-Casimir relation. In this presentation, the genus two DAHA becomes an abstract associative algebra with a PBW/Gröbner-type structure rather than only a concrete operator algebra.

A principal theorem is that $SL(2,\mathbb C)$0 is flat and that the word problem is solved by reduction to a monomial basis. The paper constructs a set of $SL(2,\mathbb C)$1 noncommutative relations $SL(2,\mathbb C)$2, called a $SL(2,\mathbb C)$3-Gröbner basis, with unchanged leading monomials under specialization. The resulting irreducible monomials form a basis simultaneously for $SL(2,\mathbb C)$4, $SL(2,\mathbb C)$5, and $SL(2,\mathbb C)$6 (Arthamonov, 2023). This puts the earlier $SL(2,\mathbb C)$7-difference model on a firm algebraic footing and proves that the abstract algebra coincides with the Arthamonov–Shakirov algebra.

The specializations organize the geometric content: $SL(2,\mathbb C)$8 At $SL(2,\mathbb C)$9, one recovers the genus-two skein algebra. At $\Sigma_2$0, one obtains a commutative algebra $\Sigma_2$1 which is a one-parameter flat Poisson deformation of $\Sigma_2$2. At $\Sigma_2$3, one has the explicit isomorphism

$\Sigma_2$4

so the $\Sigma_2$5 fiber is the coordinate ring of the $\Sigma_2$6-character variety of the genus-two surface (Arthamonov, 2023).

The same paper proves that $\Sigma_2$7 is reduced, an integral domain, and of Krull dimension $\Sigma_2$8. It also shows that the Poisson bracket induced from the $\Sigma_2$9-deformation specializes at $q$0 to the Goldman Poisson bracket on the character variety (Arthamonov, 2023). This establishes the classical-limit meaning of genus two DAHA: it is not merely a higher-genus difference-operator algebra, but also a quantization and Poisson deformation of the genus-two character variety.

4. Mapping class symmetry, cluster realization, and skein-theoretic gluing

Mapping class group symmetry is one of the features that justify the DAHA terminology. The 2017 paper defines automorphisms $q$1 and $q$2, corresponding to Dehn twists along $q$3- and $q$4-cycles, and proves that they satisfy the Wajnryb relations for $q$5 (Arthamonov et al., 2017). A Fourier-like automorphism

$q$6

acts by the six-cycle

$q$7

providing the genus-two analogue of the DAHA Fourier transform (Arthamonov et al., 2017).

The 2024 paper recasts this symmetry in cluster-algebraic terms. Its main theorem states that there exists a $q$8-equivariant injective algebra homomorphism

$q$9

where $\mathcal A_{q,t}$00 is the universally Laurent algebra of the exceptional finite mutation type $\mathcal A_{q,t}$01 (Arthamonov et al., 2024). The $\mathcal A_{q,t}$02-cycle twists are realized as cluster transformations such as

$\mathcal A_{q,t}$03

and a cluster modular involution

$\mathcal A_{q,t}$04

exchanges $\mathcal A_{q,t}$05- and $\mathcal A_{q,t}$06-cycle trace functions. This identifies the genus-two mapping class group action with explicit mutations and permutations in the $\mathcal A_{q,t}$07 cluster geometry (Arthamonov et al., 2024).

Hikami’s 2019 construction approaches the same topology from a different direction. It embeds $\mathcal A_{q,t}$08 into spherical $\mathcal A_{q,t}$09-DAHA and $\mathcal A_{q,t}$10 into spherical $\mathcal A_{q,t}$11-DAHA, identifies the DAHA $\mathcal A_{q,t}$12-actions with Dehn twists, and then glues the low-complexity pieces to obtain a representation of $\mathcal A_{q,t}$13 (Hikami, 2019). The gluing is controlled by the quantum-dilogarithmic factor

$\mathcal A_{q,t}$14

and the resulting genus-two operators represent curves $\mathcal A_{q,t}$15, together with the additional curve $\mathcal A_{q,t}$16, in a way compatible with the genus-two skein relations (Hikami, 2019). A plausible implication is that genus-two DAHA has two complementary realizations in current literature: as an intrinsic genus-two analogue of spherical $\mathcal A_{q,t}$17-DAHA, and as a gluing of lower-complexity DAHA blocks adapted to the topology of $\mathcal A_{q,t}$18.

5. Fixed loci, finite subgroup actions, and SCFT applications

The 2026 paper uses genus two DAHA as the main computational framework for finite-group actions on the genus-two $\mathcal A_{q,t}$19-character variety (Arthamonov et al., 8 Mar 2026). Its preferred coordinates are the $\mathcal A_{q,t}$20-generators

$\mathcal A_{q,t}$21

organized into orbits under the order-six symmetry $\mathcal A_{q,t}$22. In that paper, genus two DAHA plays three simultaneous roles: it is the noncommutative quantization $\mathcal A_{q,t}$23, it supplies explicit generators adapted to the surface topology, and it carries a mapping class group action by algebra automorphisms which descends in the classical limit to the natural action on the character variety (Arthamonov et al., 8 Mar 2026).

The classical comparison algebra is

$\mathcal A_{q,t}$24

described there as a one-parameter flat Poisson deformation of

$\mathcal A_{q,t}$25

The mapping class action is generated by a Dehn twist $\mathcal A_{q,t}$26 and the order-six automorphism $\mathcal A_{q,t}$27, with $\mathcal A_{q,t}$28, satisfying braid and commutation relations, while the hyperelliptic involution is trivial: $\mathcal A_{q,t}$29 This triviality is decisive for the fixed-point analysis, because subgroup actions differing by the hyperelliptic involution $\mathcal A_{q,t}$30 produce identical fixed loci (Arthamonov et al., 8 Mar 2026).

For each orientation-preserving finite subgroup $\mathcal A_{q,t}$31, the paper imposes fixed-point equations in the $\mathcal A_{q,t}$32-presentation and computes the radical ideal of the fixed locus, usually together with a primary decomposition. The resulting fixed loci have dimensions $\mathcal A_{q,t}$33, $\mathcal A_{q,t}$34, $\mathcal A_{q,t}$35, and $\mathcal A_{q,t}$36. The trivial hyperelliptic case $\mathcal A_{q,t}$37 fixes the entire six-dimensional DAHA/character variety. For $\mathcal A_{q,t}$38 and $\mathcal A_{q,t}$39, the fixed locus is four-dimensional, and the radical ideals are equivalent because the actions differ by $\mathcal A_{q,t}$40. For $\mathcal A_{q,t}$41, the $\mathcal A_{q,t}$42 fixed set decomposes as $\mathcal A_{q,t}$43 with dimensions $\mathcal A_{q,t}$44, while in the $\mathcal A_{q,t}$45-deformed case it becomes $\mathcal A_{q,t}$46 with dimensions $\mathcal A_{q,t}$47. For $\mathcal A_{q,t}$48, the $\mathcal A_{q,t}$49 fixed locus has two two-dimensional components, and in the $\mathcal A_{q,t}$50-deformed case there are two two-dimensional components plus two isolated points (Arthamonov et al., 8 Mar 2026).

A headline observation is the existence of nontrivial coincidences between different subgroup actions: $\mathcal A_{q,t}$51 Some of these equivalences are explained by the trivial action of the hyperelliptic involution; others are described as more subtle and tied to “genus/irregularity transitions” (Arthamonov et al., 8 Mar 2026). The same paper proposes the resulting fixed subvarieties as symmetry-reduced moduli spaces relevant to $\mathcal A_{q,t}$52 $\mathcal A_{q,t}$53 SCFTs, especially of Argyres–Douglas type, via the nonabelian Hodge correspondence between the genus-two character variety and the Betti moduli space of the corresponding Hitchin system.

6. Interpretive status, scope, and limitations

The literature is explicit that genus two DAHA is not introduced as a standard Cherednik algebra attached to a root system in the usual sense. The 2017 paper argues from a topological perspective that the algebra is a genus-two generalization of $\mathcal A_{q,t}$54 spherical DAHA, but it does not construct a full non-spherical genus-two DAHA, does not provide a complete abstract presentation by generators and relations, and does not prove a PBW theorem (Arthamonov et al., 2017). Those missing algebraic foundations are supplied only later by the 2023 generators-and-relations treatment (Arthamonov, 2023).

Conversely, the 2023 paper is careful to present $\mathcal A_{q,t}$55 as a DAHA-type algebra by structural analogy: it is realized by Macdonald-type $\mathcal A_{q,t}$56-difference operators, carries mapping class group symmetry, specializes to the skein algebra at $\mathcal A_{q,t}$57, and has a classical limit equal to a Poisson deformation of the character variety (Arthamonov, 2023). The 2024 cluster paper strengthens this interpretation by producing a faithful $\mathcal A_{q,t}$58-cluster model, but it still does not reframe genus two DAHA as a traditional braid-group/Hecke quotient; rather, it shows that the Arthamonov–Shakirov genus-2 knot-operator algebra behaves like a spherical DAHA and admits a $\mathcal A_{q,t}$59-equivariant cluster realization (Arthamonov et al., 2024).

The subject therefore has several interacting layers. One layer is algebraic: commuting Hamiltonians, $\mathcal A_{q,t}$60-difference realizations, $\mathcal A_{q,t}$61-relations, Gröbner bases, and flat deformations. A second is geometric: skein algebras, cluster Poisson varieties of type $\mathcal A_{q,t}$62, and $\mathcal A_{q,t}$63-character varieties. A third is topological and representation-theoretic: explicit mapping class group actions by automorphisms. A fourth, developed most fully in 2026, is the use of genus two DAHA coordinates to compute symmetry-reduced fixed loci and to propose moduli spaces relevant to Hitchin systems and $\mathcal A_{q,t}$64 $\mathcal A_{q,t}$65 SCFTs (Arthamonov et al., 8 Mar 2026).

This suggests a stable current consensus. “Genus two DAHA” is best understood as the genus-two analogue of rank-one spherical DAHA in a structural sense: a two-parameter algebra tied to Macdonald-type operators on $\mathcal A_{q,t}$66, equipped with mapping class group symmetry, compatible with skein quantization, and admitting both cluster and character-variety limits (Arthamonov et al., 2017, Arthamonov, 2023, Arthamonov et al., 2024). At the same time, the terminology remains deliberately cautious. The papers do not claim a full higher-genus Cherednik theory; they establish a concrete and computable genus-two framework whose strength lies in explicit formulas, explicit automorphisms, and explicit degenerations.