Cohomological Hall Induction

• Cohomological Hall induction is a framework that transfers cohomological data from simpler moduli spaces to larger ones using pull–push techniques along Hall correspondences.
• It employs Lagrangian correspondences and hyperbolic localization to construct associative multiplication structures, thereby unifying BPS state interactions and integrality theorems.
• The method extends to module constructions and functorial transfers across settings, linking quiver representations, DT invariants, and geometric Langlands analogies.

Cohomological Hall induction denotes a family of induction-type constructions in cohomological Hall algebra theory in which cohomology, mixed Hodge modules, or Donaldson–Thomas-theoretic objects are transported from smaller, graded, fixed-point, Levi, or torsion submoduli to larger moduli by pull–push along Hall correspondences. In its most elementary form it is the extension-theoretic convolution $m=p_*\circ q^*$; in shifted-symplectic and Donaldson–Thomas settings it is realized by Lagrangian correspondences and hyperbolic localization; and in recent work it organizes Hall multiplication, parabolic induction, module structures, integrality theorems, and comparisons between apparently different CoHA realizations (Davison, 2013, Kinjo et al., 2024, Hennecart et al., 25 Sep 2025).

1. Extension correspondences and the Hall-theoretic origin

The foundational mechanism is the extension correspondence. In the general sheaf-theoretic formulation, one starts from a space $X$ together with a correspondence

$$\mathrm{Ext}\xrightarrow{p}X,\qquad \mathrm{Ext}\xrightarrow{q}X\times X,$$

where $p$ is proper and quasismooth and $q$ yields pullback on cohomology. The cohomological Hall product is then

$m:=p_*\circ q^*:H(X\times X)\to H(X),$

and, via Künneth, an algebra structure $H(X)\otimes H(X)\to H(X)$. For quiver representations and for coherent sheaves on a smooth proper curve, the stack of short exact sequences supplies the required correspondence; in this sense the Hall product is already an induction from pairs of objects to their extensions (Latyntsev, 2021).

In the critical CoHA of a quiver with potential, the same pattern is upgraded to vanishing-cycle cohomology. For dimension vectors $\gamma_1,\gamma_2$, the product

$$m_{\gamma_1,\gamma_2}=\delta\circ \underline{\zeta}\circ \beta^{-1}\circ \alpha\circ \mathrm{TS}$$

is assembled from Thom–Sebastiani, pullback along an affine fibration, change of equivariance, pushforward along the inclusion of the extension locus, and finally pushforward from the parabolic quotient to the full moduli quotient. The 2013 construction also builds a localized coproduct and a $Q$-localized bialgebra structure, and proves a cohomological dimensional reduction theorem linking critical CoHAs to ordinary cohomology on zero loci; this already places induction and restriction in a single formalism (Davison, 2013).

Operator-theoretic versions make the same structure explicit. Quiver Schur algebras act on tensor powers of CoHA components by “merges” and “splits,” and are identified with algebras of multiplication and comultiplication operators on the CoHA. Their mixed analogues, attached to quivers with contravariant involution, act on cohomological Hall modules in the same way. In that realization, Hall induction is encoded by explicit Demazure-operator formulas and by convolution on Steinberg-type correspondences, so that induction and restriction become algebraic operators rather than only abstract correspondences (Przezdziecki, 2019).

2. Shifted-symplectic Hall induction for $X$0-Calabi–Yau categories

A major generalization replaces representation spaces by moduli of objects in a $X$1-Calabi–Yau dg-category. Let $X$2 be a stable dg-category of finite type with a $X$3-Calabi–Yau structure. Its derived moduli stack of objects $X$4 is a $X$5-Artin derived stack carrying a $X$6-shifted symplectic structure, and the construction is performed on an open substack $X$7 that is closed under direct sums and satisfies the usual finiteness and $X$8-reductivity hypotheses. The additional input is strong orientation data: an orientation on the $X$9-shifted symplectic stack $$\mathrm{Ext}\xrightarrow{p}X,\qquad \mathrm{Ext}\xrightarrow{q}X\times X,$$0 compatible with direct sums and satisfying a coherent associativity constraint along Hall-type correspondences. This compatibility is what makes induction and Hall multiplication associative at the Donaldson–Thomas level (Kinjo et al., 2024).

Given an oriented $$\mathrm{Ext}\xrightarrow{p}X,\qquad \mathrm{Ext}\xrightarrow{q}X\times X,$$1-shifted symplectic stack $$\mathrm{Ext}\xrightarrow{p}X,\qquad \mathrm{Ext}\xrightarrow{q}X\times X,$$2, one has the Donaldson–Thomas perverse sheaf

$$\mathrm{Ext}\xrightarrow{p}X,\qquad \mathrm{Ext}\xrightarrow{q}X\times X,$$3

For $$\mathrm{Ext}\xrightarrow{p}X,\qquad \mathrm{Ext}\xrightarrow{q}X\times X,$$4, the cohomological DT invariant is

$$\mathrm{Ext}\xrightarrow{p}X,\qquad \mathrm{Ext}\xrightarrow{q}X\times X,$$5

with $$\mathrm{Ext}\xrightarrow{p}X,\qquad \mathrm{Ext}\xrightarrow{q}X\times X,$$6 and antisymmetric Euler pairing $$\mathrm{Ext}\xrightarrow{p}X,\qquad \mathrm{Ext}\xrightarrow{q}X\times X,$$7. Hall induction is encoded by the attractor correspondence

$$\mathrm{Ext}\xrightarrow{p}X,\qquad \mathrm{Ext}\xrightarrow{q}X\times X,$$8

where $$\mathrm{Ext}\xrightarrow{p}X,\qquad \mathrm{Ext}\xrightarrow{q}X\times X,$$9 is the stack of $p$0-graded objects and $p$1 is the stack of filtered objects. This correspondence is naturally $p$2-shifted Lagrangian.

The decisive statement is the hyperbolic-localization, or integral, isomorphism

$p$3

with $p$4 evaluation at the basepoint of $p$5 and $p$6 the index function. Restricting to the component of “sum of two objects” yields the Hall product

$p$7

Associativity is obtained from the $p$8-fold attractor correspondence and the compatibility of DT sheaves under iteration. In this framework, cohomological Hall induction includes what the paper calls internal induction: passage from graded pieces of an object to the object itself, implemented by $p$9, $q$0, and the vanishing-cycle formalism (Kinjo et al., 2024).

The same construction gives a precise realization of the “algebra of BPS states.” The BPS cohomology

$q$1

is the $q$2-graded piece of the CoHA, and the Hall product is the bound-state multiplication. A plausible implication is that Hall induction is best viewed not as an auxiliary operation but as the geometric mechanism by which BPS sectors interact.

3. Parabolic induction on character stacks and the $q$3-manifold Eisenstein analogue

For a connected reductive group $q$4 and a closed oriented $q$5-manifold $q$6, the derived character stack

$q$7

carries a $q$8-shifted symplectic structure when $q$9 is $m:=p_*\circ q^*:H(X\times X)\to H(X),$0-shifted symplectic. In the 2024 construction, a spin structure on $m:=p_*\circ q^*:H(X\times X)\to H(X),$1 is used to produce an orientation and hence a DT perverse sheaf on $m:=p_*\circ q^*:H(X\times X)\to H(X),$2; the 2025 integrality paper treats $m:=p_*\circ q^*:H(X\times X)\to H(X),$3 as a $m:=p_*\circ q^*:H(X\times X)\to H(X),$4-shifted symplectic stack with canonical orientation (Kinjo et al., 2024, Hennecart et al., 25 Sep 2025).

If $m:=p_*\circ q^*:H(X\times X)\to H(X),$5 is parabolic with Levi factor $m:=p_*\circ q^*:H(X\times X)\to H(X),$6, the attractor correspondence for $m:=p_*\circ q^*:H(X\times X)\to H(X),$7 decomposes into parabolic correspondences

$m:=p_*\circ q^*:H(X\times X)\to H(X),$8

which are Lagrangian in the shifted-symplectic sense. Hyperbolic localization then yields a map on DT cohomology

$m:=p_*\circ q^*:H(X\times X)\to H(X),$9

constructed as $H(X)\otimes H(X)\to H(X)$0 up to the index shifts and orientation twists appearing in the detailed construction. The same formalism produces a CoHA structure on

$H(X)\otimes H(X)\to H(X)$1

and CoHA-module structures on the analogous $H(X)\otimes H(X)\to H(X)$2- and $H(X)\otimes H(X)\to H(X)$3-theories. These are described as cohomological Hall induction modules (Kinjo et al., 2024).

The geometric Langlands analogy is explicit: the parabolic induction map is presented as a $H(X)\otimes H(X)\to H(X)$4-manifold analogue of the Eisenstein series functor. The 2025 integrality theorem sharpens this perspective by expressing critical cohomology of the character stack as a sum of Levi contributions: $H(X)\otimes H(X)\to H(X)$5 Thus the global DT theory of $H(X)\otimes H(X)\to H(X)$6-local systems is assembled from finite-dimensional BPS data attached to Levi subgroups and their centers (Hennecart et al., 25 Sep 2025).

This suggests a precise structural interpretation: in character-stack settings, cohomological Hall induction is both an algebraic operation and a Levi-to-$H(X)\otimes H(X)\to H(X)$7 assembly principle for BPS cohomology.

4. Symmetric stacks, graded points, and strong cohomological integrality

A second major development treats cohomological Hall induction as the engine of cohomological integrality for smooth symmetric stacks. Let $H(X)\otimes H(X)\to H(X)$8 be a smooth Artin stack with affine diagonal and good moduli space $H(X)\otimes H(X)\to H(X)$9, quasi-compact connected components and graded points, almost symmetric tangent representations at closed points, and a global equivariant parameter for each non-degenerate face of its component lattice. For a non-degenerate face $\gamma_1,\gamma_2$0 and chamber $\gamma_1,\gamma_2$1, one has a correspondence

$\gamma_1,\gamma_2$2

with $\gamma_1,\gamma_2$3 smooth, $\gamma_1,\gamma_2$4 proper, and $\gamma_1,\gamma_2$5 finite. This is the intrinsic Hall correspondence for stacks of graded points (Hennecart et al., 25 Sep 2025).

The relative cohomological Hall induction map is

$\gamma_1,\gamma_2$6

and after global sections one gets the absolute map

$\gamma_1,\gamma_2$7

Associativity takes the form

$\gamma_1,\gamma_2$8

so two-step induction equals induction along the induced chamber (Hennecart et al., 25 Sep 2025).

The BPS sheaf attached to a non-degenerate face is

$\gamma_1,\gamma_2$9

and the cohomological integrality map is

$$m_{\gamma_1,\gamma_2}=\delta\circ \underline{\zeta}\circ \beta^{-1}\circ \alpha\circ \mathrm{TS}$$0

The main theorem states that $$m_{\gamma_1,\gamma_2}=\delta\circ \underline{\zeta}\circ \beta^{-1}\circ \alpha\circ \mathrm{TS}$$1 is an isomorphism. In the strong form for symmetric stacks, the BPS sheaves are intersection complexes of good moduli spaces of stacks of graded points whenever they are nonzero, and the cohomology of the whole stack is recovered as cohomological Hall induction of those intersection complexes. The same formalism extends to $$m_{\gamma_1,\gamma_2}=\delta\circ \underline{\zeta}\circ \beta^{-1}\circ \alpha\circ \mathrm{TS}$$2-shifted symplectic stacks with DT mixed Hodge modules, to $$m_{\gamma_1,\gamma_2}=\delta\circ \underline{\zeta}\circ \beta^{-1}\circ \alpha\circ \mathrm{TS}$$3-shifted symplectic stacks via dimensional reduction, and to character stacks of compact oriented $$m_{\gamma_1,\gamma_2}=\delta\circ \underline{\zeta}\circ \beta^{-1}\circ \alpha\circ \mathrm{TS}$$4-manifolds; one consequence is a proof of Halpern–Leistner’s purity conjecture for $$m_{\gamma_1,\gamma_2}=\delta\circ \underline{\zeta}\circ \beta^{-1}\circ \alpha\circ \mathrm{TS}$$5-shifted symplectic stacks admitting proper good moduli spaces (Hennecart et al., 25 Sep 2025).

A plausible implication is that cohomological Hall induction supplies an integrality decomposition principle parallel to the role of primitive BPS generators in symmetric quiver CoHAs.

5. Modules, Hall representations, and induced operator algebras

Cohomological Hall induction is not confined to algebra structures; it also produces module and representation theories. For quivers with contravariant involution and duality structure, the cohomological Hall module

$$m_{\gamma_1,\gamma_2}=\delta\circ \underline{\zeta}\circ \beta^{-1}\circ \alpha\circ \mathrm{TS}$$6

is built from self-dual representations, and the CoHA acts by a Hall correspondence on isotropic subrepresentations: $$m_{\gamma_1,\gamma_2}=\delta\circ \underline{\zeta}\circ \beta^{-1}\circ \alpha\circ \mathrm{TS}$$7 The induced primitive quotient

$$m_{\gamma_1,\gamma_2}=\delta\circ \underline{\zeta}\circ \beta^{-1}\circ \alpha\circ \mathrm{TS}$$8

defines orientifold cohomological DT invariants. For $$m_{\gamma_1,\gamma_2}=\delta\circ \underline{\zeta}\circ \beta^{-1}\circ \alpha\circ \mathrm{TS}$$9-symmetric quivers these primitive pieces are finite-dimensional, giving the orientifold integrality theorem, and in several families—including disjoint union quivers, loop quivers, $Q$0, and finite type $Q$1—the paper proves PBW-type freeness statements for $Q$2 as a module over suitable CoHA subalgebras (Young, 2016).

A different induction formalism starts from torsion pairs. Given a torsion pair $Q$3 in the heart of a stable $Q$4-category, the torsion part carries a categorified CoHA and the torsion-free part carries canonical left and right module structures. On Borel–Moore homology and $Q$5-theory this yields actions of the CoHA of $Q$6 on $Q$7, and hence canonical subalgebras of the endomorphism ring of the homology or $Q$8-theory of torsion-free moduli. The construction is applied to Nakajima quiver varieties, moduli of torsion-free sheaves on surfaces, Pandharipande–Thomas stable pairs, and relative Hilbert schemes; in this sense cohomological Hall induction extends the “positive part” of a CoHA into a larger operator algebra generated by left and right actions (Diaconescu et al., 2022).

Further operator-theoretic realizations arise from quiver Schur algebras and their mixed analogues. The total quiver Schur algebra embeds into $Q$9, sending elementary merges to CoHA multiplication operators and elementary splits to CoHA comultiplication operators. Mixed quiver Schur algebras similarly embed into $X$00, packaging CoHA multiplication, comultiplication, action, and coaction in a single algebraic object (Przezdziecki, 2019).

An additional variant concerns cotangent representations of reductive groups. For a cocharacter relation $X$01, the paper defines Hall induction maps

$X$02

via dimensional reduction from vanishing-cycle constructions. In this setting the equivariant Borel–Moore homology of $X$03 is torsion-free under the stated torus hypotheses, and the image of the restriction map in $X$04-theory satisfies generalized wheel conditions (Gubarevich, 31 Dec 2025).

6. Transport, comparison, and functorial transfer

One broad role of cohomological Hall induction is transport of Hall structures across moduli problems, families, and geometric operations. A particularly explicit instance is the framed CoHA of a quiver. Here the moduli stack of framed representations yields a framed CoHA, and the equivariant cohomology of the disjoint union of Nakajima varieties

$X$05

embeds as a subalgebra of the framed CoHA. Restricted to this subalgebra, the algebra multiplication is identified with the stable envelope map, and one obtains an inductive formula

$X$06

which computes stable envelopes inductively in tautological classes (Botta, 2022).

Transport across equivalences is equally prominent. For the Dolbeault, de Rham, Betti, and Hodge moduli stacks on a smooth projective curve, relative and absolute CoHA structures are constructed on Borel–Moore homology, and the nonabelian Hodge and Riemann–Hilbert correspondences identify the resulting CoHAs: $X$07 The same comparison carries the BPS algebra and BPS Lie algebra structures. In this setting, one may regard the relative Hodge–Deligne CoHA as inducing the Dolbeault and de Rham CoHAs, and derived Riemann–Hilbert as transporting the de Rham CoHA to the Betti side (Hennecart, 2023).

The derived McKay correspondence produces another transport theorem. For the minimal resolution $X$08 of a Kleinian singularity and the affine preprojective algebra $X$09 of the corresponding affine Dynkin quiver, the derived equivalence identifies semistable sheaf moduli on $X$10 with semistable $X$11-representation moduli of slope zero for a suitable choice of stability vector $X$12. This induces equivalences of categorified Hall algebras and, after decategorification, isomorphisms of K-theoretic Hall algebras and cohomological Hall algebras (Diaconescu et al., 2020).

Functorial transfer also appears under combinatorial operations on quivers with potential. Edge contraction along an arrow produces a contracted quiver with potential $X$13, and there is an induced algebra homomorphism

$X$14

that preserves the Hopf algebra structure, Drinfeld double, mutation, and dimensional reduction, and relates the associated scattering diagrams and DT series (2401.04839). For character varieties of surface groups, one can likewise compare the $X$15D CoHA multiplication coming from the standard presentation with that obtained from a brane-tiling presentation (Mistry, 2022).

Taken together, these constructions indicate that cohomological Hall induction has become a unifying principle for Hall multiplication, parabolic Eisenstein-type functors, BPS-integrality decompositions, module structures, stable envelopes, and the functorial transfer of CoHA structures across equivalences and geometric transformations.