Cohomological Hall algebras replace finite-field extension counts by cohomology. A quiver representation category supplies moduli stacks, short exact sequences supply correspondences, and pull-push operations turn cohomology classes into a multiplication. With potentials and vanishing cycles, the same machinery organizes Donaldson–Thomas and BPS invariants.
This chapter extends Hall Algebras, Quiver Moduli, Hall–Littlewood Geometry and KLR Algebras.
Classical Hall multiplication
Over a finite field, Hall multiplication counts extensions. If [M] and [N] are basis elements, their product sums over [E] weighted by the number of short exact sequences 0→N→E→M→0.
A cohomological Hall algebra, or CoHA, replaces that count by the topology of the moduli correspondence of extensions.
Representation spaces and quotient stacks
For dimension vector d, the affine space Rep(Q,d) carries the base-change action G_d=∏_iGL_{d_i}. The quotient stack [Rep(Q,d)/G_d] remembers both representations and their automorphisms.
The CoHA vector space is built from equivariant cohomology or Borel–Moore homology of these spaces, summed over all dimension vectors, with grading shifts depending on the chosen convention and Euler form.
Extension correspondence
Fix d=e+f. Consider representations E together with a subrepresentation N of dimension f. The quotient M=E/N has dimension e.
This defines a correspondence between the product of moduli for M and N and the moduli for E. Pullback from the product, cup product on the correspondence and proper pushforward toward E define Hall multiplication.
Associativity follows from comparing flags of two nested subrepresentations, the geometric counterpart of associativity of composing extensions.
The one-vertex no-arrow example
For one vertex and no arrows, Rep(Q,n) is a point and the stack is BGL_n. The CoHA is built from H^*(BGL_n) across all n.
Multiplication is induced by block-upper-triangular correspondences encoding subspaces C^m⊂C^{m+n}. Even this trivial quiver therefore produces nontrivial cohomological multiplication through classifying-space geometry.
Shuffle formulas
For many quivers, localization identifies CoHA multiplication with shuffle products in symmetric variables. One multiplies two symmetric functions and sums over shuffles, inserting kernel factors determined by arrows and equivariant weights.
Shuffle descriptions make the algebra computational while retaining its geometric origin.
Quivers with potential
A potential W is a linear combination of cyclic paths. Taking traces on representations produces a function Tr(W) on Rep(Q,d).
Critical points of Tr(W) correspond to representations of the Jacobian algebra determined by cyclic derivatives of W. Vanishing-cycle cohomology detects the singularity of this critical locus.
Critical CoHA
Kontsevich and Soibelman’s critical CoHA uses vanishing cycles associated with Tr(W). The Hall correspondence respects the potential sufficiently to define multiplication on the critical cohomology.
This is the natural home for cohomological Donaldson–Thomas invariants of quivers with potential.
Donaldson–Thomas invariants
Donaldson–Thomas theory counts stable objects with signs or motivic/cohomological refinements. In quiver settings, stability parameters select semistable representations while vanishing cycles encode the local critical geometry.
CoHA packages all dimension vectors together, so wall-crossing identities become factorization statements inside one algebraic structure.
BPS spaces
Primitive or indecomposable pieces of the CoHA are interpreted as BPS cohomology in appropriate settings. Integrality theorems explain why generating functions built from rational-looking stack counts reorganize into integer-dimensional BPS spaces.
This is categorified positivity again: integers arise because there are actual graded vector spaces behind the generating series.
Wall crossing
Changing a stability condition changes which representations are stable. Harder–Narasimhan filtrations factor an arbitrary object into semistable pieces of ordered slope.
Hall multiplication records the recombination of those pieces. Wall-crossing formulas express equality of two different ordered factorizations of the same total Hall element.
Yangians and quantum groups
CoHAs of quivers and related moduli spaces produce actions of Yangians, quantum loop algebras and other quantum groups on cohomology of quiver varieties and moduli spaces.
This places CoHA between KLR categorification and BFN Coulomb-branch algebras: all three convert geometric correspondences into quantum representation operators.
Dimensional reduction
For special potentials, vanishing-cycle calculations can reduce to ordinary cohomology of lower-dimensional representation varieties. This technique connects critical CoHAs to preprojective algebras and quiver varieties.
It provides the bridge to the next chapter.
Verification workflow
- Fix ordinary cohomology, equivariant cohomology or Borel–Moore homology conventions.
- Track degree shifts from the Euler form.
- Distinguish ordinary CoHA from critical CoHA.
- State the quiver potential before using vanishing cycles.
- Separate stability-dependent DT invariants from the underlying Hall algebra.
- Check properness conditions for pushforwards.
- Do not treat BPS interpretations as identical across every CoHA framework.
Practice
1. What replaces finite-field extension counts? 2. Why use quotient stacks? 3. What correspondence defines multiplication? 4. What does a potential produce? 5. Why do vanishing cycles enter? 6. What does wall crossing express?
Answers. Cohomology of extension moduli; stacks retain automorphisms; short exact sequences/subrepresentations; a trace function whose critical locus is the Jacobian representation space; they measure critical singularity; equality of different Hall factorizations into semistable sectors.
Representation Mathematics
Begin with Canonical Bases, continue to Preprojective Algebras and q-Schur Algebras, then return to the BTT Mathematics Learning Hub.