Critical Cohomological Hall Algebras

• Critical Cohomological Hall algebras are graded structures derived from vanishing-cycle cohomology on quiver moduli that encode wall-crossing phenomena and DT invariants.
• Their construction employs vanishing-cycle functors and Hall-type correspondences to yield associative bialgebra and localized Hopf algebra frameworks.
• They connect representation theory, quantum groups, and categorified invariants through functorial operations like quiver mutation, edge contraction, and dimensional reduction.

A critical Cohomological Hall algebra (critical CoHA) is a mathematical structure associated to a quiver $Q$ with potential $W$, constructed via vanishing-cycle cohomology on moduli spaces of quiver representations. The critical CoHA, originating from work by Kontsevich–Soibelman, forms a graded algebra encoding wall-crossing phenomena, categorified Donaldson–Thomas invariants, and deep links to representation theory of Yangians, $W$-algebras, and geometric representation theory. Its construction fundamentally utilizes the vanishing-cycles functor with support on the critical locus of a trace-of-potential function, leading to a bialgebra or, in localized settings, a Hopf algebra structure. Critical CoHAs have powerful functorial properties under quiver mutation, edge contraction, and dimensional reduction, connecting $3$-Calabi–Yau (CY3) and $2$-Calabi–Yau (CY2) Hall algebras and elucidating connections between noncommutative geometry and quantum groups.

1. Definition and Structural Construction

Given a finite quiver $Q=(Q_0, Q_1)$ and a potential $W\in\mathbb{C} Q/[\mathbb{C} Q, \mathbb{C} Q]$ (a finite $\mathbb{C}$-linear combination of cyclic paths), one considers for dimension vector $\gamma\in\mathbb{N}^{Q_0}$ the affine space of representations

$$M_\gamma(Q) = \prod_{a\in Q_1}\mathrm{Hom}\bigl(\mathbb{C}^{\gamma_{s(a)}}, \mathbb{C}^{\gamma_{t(a)}}\bigr)$$

with an action of $G_\gamma = \prod_{i\in Q_0}GL(\gamma_i)$. The trace-of-potential function $$\operatorname{Tr} W_\gamma: M_\gamma(Q)\to\mathbb{C}$$ is induced by evaluating $W$ on matrix data. The critical CoHA is constructed as

$$\mathcal{H}^{\rm crit}_Q = \bigoplus_{\gamma} H^*_{c,G_\gamma}\bigl(M_\gamma(Q), \phi_{\operatorname{Tr} W_\gamma}\bigr)\otimes T^{-\chi_Q(\gamma,\gamma)/2}$$

where $\phi_{\operatorname{Tr} W_\gamma}$ is the vanishing-cycle functor applied to the constant sheaf and $\chi_Q(\gamma,\gamma')$ is the Ringel–Euler form.

Multiplication is defined by a Hall-type correspondence: given $\gamma = \gamma^1+\gamma^2$, the representation space $M_{\gamma^1, \gamma^2}\subset M_\gamma$ parameterizes representations admitting $G_\gamma$-stable subspaces. Through a sequence of restriction, pushforward, Thom–Sebastiani isomorphism, and degree shifts, the multiplication map

$$m: \mathcal{H}^{\rm crit}_Q \otimes \mathcal{H}^{\rm crit}_Q \rightarrow \mathcal{H}^{\rm crit}_Q$$

results in an associative, graded algebra structure. The product is expressed by explicit pull–push operations and perverse sheaf manipulations, with cohomological degree appropriately adjusted by the Euler form (Yang et al., 2016, Davison, 2013, 2401.04839).

2. Vanishing Cycles and Dimension Reduction

The critical CoHA is fundamentally built from vanishing-cycle cohomology, linking representation-theoretic moduli and the geometry of critical loci. The vanishing-cycle functor $\phi_f$ on a regular function $f:X\to\mathbb{C}$ captures cohomology supported on $\mathrm{Crit}(f)$. For quivers with a “cut” (i.e., presentations where $W$ is linear in certain loops), the dimension-reduction theorem applies: $$H^*_{c,G_v}\left(\mathrm{Rep}(Q,v), \phi_{\mathrm{Tr}_W}\right) \simeq H^{\rm BM}_{G_v}\left(\mathrm{Rep}(\mathrm{Jac}(Q,W)/C, v) \times \mathrm{Lie}(G_v), \mathbb{Q}\right)$$ This reduction yields isomorphisms between vanishing-cycle cohomology (CY3) and Borel–Moore homology of representation spaces for Jacobian/preprojective algebras (CY2), with the Hall products related by Euler class twists (Yang et al., 2016, Davison, 2013).

3. Hopf Algebra, Drinfeld Double, and Functoriality

The critical CoHA admits a localized coproduct, making it a (localized) graded Hopf algebra: $$\Delta: \mathcal{H}^{\rm crit}_Q \longrightarrow (\mathcal{H}^{\rm crit}_Q \otimes \mathcal{H}^{\rm crit}_Q)[L^{-1}]$$ The Drinfeld double $D(\mathcal{H}^{\rm crit}_Q)$ is constructed using the canonical skew-pairing, yielding a Hopf algebra that encodes quantum group symmetries and wall-crossing structures.

Functoriality is manifest under quiver edge contraction: if $Q'$ is obtained by contracting an arrow $a$ in $Q$, there is a natural algebra and coalgebra homomorphism

$$f_a: \mathcal{H}^{\rm crit}_Q \longrightarrow \mathcal{H}^{\rm crit}_{Q'}$$

which descends compatibly to the Drinfeld double and preserves the full Hopf algebra structure. Such morphisms commute with mutation operations (via explicit Coxeter element actions) and are compatible with dimensional reduction, ensuring that CY3 and CY2 CoHAs are linked under contraction and reduction (2401.04839).

4. Comparison with Preprojective CoHAs and Yangian Connections

For the Ginzburg quiver $Q^{\rm Gin}$ with potential $W^{\rm Gin}$, the critical CoHA becomes isomorphic (as an algebra) to the preprojective CoHA, generalizing the Schiffmann–Vasserot K-theoretic construction. Explicitly,

$$\mathcal{H}^{\rm crit}(Q^{\rm Gin}, W^{\rm Gin}) \simeq \mathcal{P}(Q)$$

via an explicit sign twist on the product. This identification transports the standard Yangian action (for ADE-type Dynkin quivers) to the critical CoHA side: $$\rho: Y^+_\hbar(\mathfrak{g}_Q) \to \mathcal{H}^{\rm crit}(Q^{\rm Gin}, W^{\rm Gin})$$ sending Drinfeld generators to classes of tautological representations, realizing the Maulik–Okounkov Yangian as a Hall algebra of vanishing cycles (Yang et al., 2016).

The induced coproduct and PBW-basis structure carry over to the critical CoHA. All known Yangian module representations, including Verma and finite-dimensional modules, lift compatibly, enabling the realization of geometric representation-theoretic phenomena in the critical CoHA formalism (Yang et al., 2016, Davison, 2013).

5. Applications: Wall-Crossing, DT Invariants, and BPS Algebras

Critical CoHAs encode wall-crossing automorphisms and scattering diagrams. The scattering diagram $D(Q,W)$ is a wall-and-chamber structure in the weight space $\mathrm{Hom}(\mathbb{N}^I, \mathbb{R})$, with wall-crossing governed by the vanishing-cycle cohomology in each chamber. Edge contractions induce embeddings of scattering diagrams and thus sub-diagram inclusions at the level of wall-crossing automorphisms.

Donaldson–Thomas partition series for $(Q,W)$ are constructed from critical CoHA cohomology classes and are functorially mapped under edge contractions, preserving both cohomological and numerical invariants. BPS Lie algebras are extracted as perverse filtrations of critical CoHAs, with their universal enveloping algebra isomorphic to the CoHA, but with a nontrivial induced product structure (2401.04839, Davison, 2022).

Explicitly, in the case of the triple-loop Jordan quiver,

$$\mathcal{H}^{\rm crit}_{Q^{(3)}, W} \cong \mathcal{H}_{\mathbb{A}^2} = \bigoplus_{d\geq 0} H^{\rm BM} \bigl( \mathrm{Hilb}_d(\mathbb{A}^2) \bigr)$$

identifying the CoHA of zero-dimensional sheaves on $\mathbb{A}^2$ with $W_{1+\infty}^+$ and, after equivariant deformation, with the positive half of the affine Yangian of $\widehat{\mathfrak{gl}}(1)$ (Davison, 2022).

6. Key Examples and Explicit Computations

A fundamental example is the Jordan quiver (single vertex, single loop), where the critical CoHA recovers the Schiffmann–Vasserot Hall algebra of the commuting variety, with the Hall multiplication coinciding with the usual commuting variety Hall algebra up to sign, and realizes the action of Heisenberg–Virasoro subalgebra of $Y^+(\mathfrak{sl}_1)$ on Hilbert scheme cohomology (Yang et al., 2016).

For quivers arising from surface fundamental groups, suitable critical CoHAs recover the (shifted) cohomology of character varieties, giving a bialgebra structure intertwining 2D and 3D moduli-theoretic pictures (Davison, 2013).

7. Categorification, Integrality, and PBW Filtrations

Critical CoHAs categorify Donaldson–Thomas theory, supporting PBW-type filtrations whose associated graded is (super)commutative on a primitive subspace. For symmetric quivers with $W=0$, this realizes Lusztig's construction of the universal enveloping algebra $U(\mathfrak{n}_+)$ from perverse sheaves on representation varieties.

The integrality conjecture asserts that weight-polynomials of the Hodge structures on the critical CoHA are Laurent polynomials, reflecting categorified integrality for motivic DT invariants. This structure is encoded by a Yangian PBW-theorem for the critical CoHA bialgebra, as established by Davison–Meinhardt (Davison, 2013).

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References

• "On two cohomological Hall algebras" (Yang et al., 2016)
• "The critical CoHA of a quiver with potential" (Davison, 2013)
• "Critical Cohomological Hall Algebra and Edge Contraction" (2401.04839)
• "Affine BPS algebras, W algebras, and the cohomological Hall algebra of $\mathbb{A}^2$" (Davison, 2022)