Symplectic duality for the constant term of the geometric Eisenstein series

Abstract: We study the cohomology of a quasimap space that categorifies the constant term of the geometric Eisenstein series for the mirabolic parabolic subgroup of $GL$ over the function field $\mathbb{F}<em>q(C)$ of a smooth projective curve $C$. This cohomology carries a natural action of an algebra of correspondences whose commutative subalgebra is the ring of regular functions on the Coulomb branch, which here is the $A</em>{n}$-surface singularity. A choice of rank-one local system on $C$ induces an action of the étale fundamental group on the Coulomb branch; the scheme-theoretic fixed locus carries a natural vector bundle. Our main result identifies the cohomology of the quasimap space with the local cohomology of this vector bundle, for a generic range of parameters.

• The paper establishes a symplectic duality framework linking the constant term of the geometric Eisenstein series to the Coulomb branch of a 3d N=4 supersymmetric gauge theory.
• The paper develops an algebra of correspondences with Clifford and Weyl-Clifford structures that act on the cohomology of quasimaps in moduli spaces.
• The paper provides explicit criteria for extension obstructions in local cohomology, paving the way for further categorification in automorphic forms and quantum geometry.

Symplectic Duality for the Constant Term of the Geometric Eisenstein Series

Eisenstein Series and Quasimap Cohomology: Geometric Framework

The paper investigates the categorification of the constant term in the geometric Eisenstein series for the mirabolic parabolic subgroup $P \subset GL_{n+1}$ over a smooth projective curve $C$ over $\mathbb{F}_q$. The key geometric object is the moduli stack $QM\vec{N}$ of quasimaps, defined as tuples $(\mathcal{V}^\bullet, \mathcal{L}, s)$ where $\mathcal{V}^\bullet$ is a flag of vector bundles built from successive extensions with prescribed graded pieces $(\mathcal{M}_1,\ldots,\mathcal{M}_{n+1})$, and $s$ is an injective sheaf map from a line bundle $\mathcal{L}$ into the total bundle. The cohomology $H_c^*(QM\vec N)$ is viewed as a categorified version of the constant term at the $C$0-bundle specified by the tuple.

Hecke-type modification correspondences act on the quasimaps via modifications of $C$1 at points of $C$2, yielding algebraic operators $C$3, $C$4 for cohomology classes $C$5. The resulting algebra $C$6 has a commutative degree-two subalgebra isomorphic to the coordinate ring of the $C$7-surface singularity: $C$8 Degree-one generators form a spin module, and degree-zero operators serve as first-order differential operators.

Symplectic Duality, Mirror Symmetry, and the Coulomb Branch

The algebraic structure of $C$9 is studied via symplectic duality: the commutative degree-two algebra of correspondences yields as its spectrum the Coulomb branch $\mathbb{F}_q$0 of a 3d $\mathbb{F}_q$1 SUSY gauge theory, with $\mathbb{F}_q$2-type singularity. Its minimal resolution $\mathbb{F}_q$3 is the symplectic dual of $\mathbb{F}_q$4, the Higgs branch. The natural torus action on $\mathbb{F}_q$5 induced by cohomological gradings preserves the symplectic form, and its attracting set $\mathbb{F}_q$6 is Lagrangian.

The main theorem asserts a torus-equivariant isomorphism between the shifted cohomology of quasimaps and the local cohomology of a specific sheaf supported on $\mathbb{F}_q$7: $\mathbb{F}_q$8 This result extends the $\mathbb{F}_q$9 case, and the proof utilizes explicit filtrations and purity properties: the Birula-type decomposition of $QM\vec{N}$0 and Lagrangian stratification of $QM\vec{N}$1 are related via smoothness arguments and the algebraic structure of the attractors.

Nontrivial Local Systems and Fundamental Group Actions

The theory is extended to quasimaps twisted by nontrivial rank-one local systems $QM\vec{N}$2 determined by characters of $QM\vec{N}$3 via geometric class field theory. In this case, $QM\vec{N}$4 (the geometric fundamental group) acts on the Coulomb branch algebra, and the scheme-theoretic fixed locus reduces to the fat point $QM\vec{N}$5 in the singular surface, while for its resolution, the fixed locus consists of $QM\vec{N}$6 reduced points.

Theorem~\ref{thm:main} provides an explicit criterion for the duality isomorphism between $QM\vec{N}$7 and the local cohomology of the sheaf on the fixed locus of $QM\vec{N}$8. The isomorphism holds generically, except when precise extension obstructions (expressed via the degrees of the line bundles and cohomological dimensions) are present. These obstructions are characterized explicitly by the existence of indices $QM\vec{N}$9 satisfying intertwining conditions involving vanishing or surjectivity of cohomology and homomorphism groups, tying the nonreduced structure of the fixed locus to the failure of filtration splitting.

Structure and Computations: Correspondence Algebras and Clifford Modules

The algebra of correspondences $(\mathcal{V}^\bullet, \mathcal{L}, s)$0 and its descendants are equipped with explicit commutator and Casimir relations, giving rise to a superalgebra structure. Degree-two subalgebra is identified with the coordinate ring of $(\mathcal{V}^\bullet, \mathcal{L}, s)$1-surface, while degree-one generators, as modules over this ring, create a Clifford algebra structure. The Clifford algebra over the Coulomb branch corresponds under the symplectic duality to the action on quasimaps, and an explicit isomorphism is constructed.

For the fixed loci of torus actions, the algebra of correspondences is shown to be a Weyl-Clifford superalgebra acting on cohomology, and degree-zero generators are interpreted as global vector fields (first-order differential operators) on the resolution $(\mathcal{V}^\bullet, \mathcal{L}, s)$2. The transition maps between affine charts of $(\mathcal{V}^\bullet, \mathcal{L}, s)$3 derive from transformation laws induced by clutching functions determined by the degrees of the line bundles.

Filtration, Extensions, and Local Cohomology

Filtration of $(\mathcal{V}^\bullet, \mathcal{L}, s)$4 by Birula stratification aligns with the filtration on local cohomology of the sheaf over $(\mathcal{V}^\bullet, \mathcal{L}, s)$5 supported in the attracting sets. The associated graded modules correspond via explicit isomorphisms of $(\mathcal{V}^\bullet, \mathcal{L}, s)$6-modules. In the nontrivial character case, splitting of the filtration is equivalent to vanishing of multiplication by $(\mathcal{V}^\bullet, \mathcal{L}, s)$7, and criteria for obstruction are given by matching cohomological weights and extension dimensions. These criteria are shown to be necessary and sufficient using duality arguments in Borel–Moore homology and equivariant localization techniques.

Implications and Future Directions

This work provides a robust algebraic and geometric description of the constant term in the geometric Eisenstein series, elucidating its symplectic duality and mirror symmetry to the Coulomb branch of SUSY gauge theory. The explicit construction of correspondence algebras, Clifford structures, and differential operators enhances the understanding of categorified automorphic phenomena and quantum geometry. The results establish deep connections between moduli of bundles, derived algebraic geometry, and representation theory structures, notably in the context of the geometric Langlands program.

Practically, the methods enable computation of cohomological invariants for equivariantly stratified moduli spaces, facilitate virtual localization arguments in derived settings, and support generalizations to more involved parabolic subgroups and nontrivial twists. Theoretically, the extension criteria for fixed loci singularities provide guidance for further study in nonreduced geometric structures and categorification in arithmetic contexts.

Anticipated future developments include application to enumerative geometry on quiver and Higgs moduli, explicit realization of symplectic duality beyond $(\mathcal{V}^\bullet, \mathcal{L}, s)$8, and further advances in derived stack cohomology and equivariant techniques in geometric representation theory.

Conclusion

The paper rigorously establishes the symplectic dual description of the constant term of geometric Eisenstein series for mirabolic parabolic $(\mathcal{V}^\bullet, \mathcal{L}, s)$9 via local cohomology and correspondence algebras. It characterizes the interplay between quasimaps, Coulomb branches, and Clifford algebra actions, carefully delineates the extension conditions, and sets forth a foundational framework for geometric, algebraic, and representation-theoretic duality in the study of automorphic forms and moduli spaces (2606.20277).