Representations of cohomological Hall algebras and Donaldson-Thomas theory with classical structure groups

Abstract

We introduce a new class of representations of the cohomological Hall algebras of Kontsevich and Soibelman, which we call cohomological Hall modules, or CoHM for short. These representations are constructed from self-dual representations of a quiver with contravariant involution σ and provide a mathematical model for the space of BPS states in orientifold string theory. We use the CoHM to define a generalization of the cohomological Donaldson-Thomas theory of quivers which allows the quiver representations to have orthogonal and symplectic structure groups. The associated invariants are called orientifold Donaldson-Thomas invariants. We prove the integrality conjecture for orientifold Donaldson-Thomas invariants of σ-symmetric quivers. We also formulate precise conjectures regarding the geometric meaning of these invariants and the freeness of the CoHM of a σ-symmetric quiver. We prove the freeness conjecture for disjoint union quivers, loop quivers and the affine Dynkin quiver of type A 1 . We also verify the geometric conjecture in a number of examples. Finally, we describe the CoHM of finite type quivers by constructing explicit Poincaré-Birkhoff-Witt type bases of these representations.

Tautological bases of CoHA modules

“…For any M ∈ U S , the condition m v ∈ M S<v for all v ∈ C(S) i is a closed condition: when writing m v = u∈S i c u,v m u as in the proof of Lemma 2.7, it is given by the vanishing of c u,v for all u ∈ S i with u > v. This also shows that Z S is a linear subspace of U S of the required dimension. Lemma 2.8 implies that Z S ⊂ N d has dimension (13) d(S) = v∈C(S)…”

“…The next natural question is a study of modules over the CoHA H [12,2,3,4,13]. A structure of an H-module can be introduced on the cohomology of the moduli spaces of stable framed representations of Q, similar to the action of quantum affine algebras on (equivariant) cohomology of Nakajima quiver varieties [8,9].…”

Quiver Schur algebras and cohomological Hall algebras

“…The idea of replacing ordinary quiver representations by self-dual representations has also been exploited in the representation theory of Hall algebras (in the finite field setting) [51] and cohomological Hall algebras [52] by Young. In the finite field case, Young defined a "Hall module" over the Hall algebra of Q, and showed that it carries a natural action of the aforementioned Enomoto-Kashiwara algebra B θ (g Q ).…”

“…As an application of Theorem D, we obtain an explicit description of the faithful polynomial representation of a mixed quiver Schur algebra (Theorem 6.15). Moreover, we reinterpret the description of the CoHM as a shuffle module [52,Theorem 3.3] in terms of Demazure operators of types A-D (Corollary 6. 19).…”

Degenerate versions of Green's theorem for Hall modules

“…While such representations do not form an abelian category in any natural way, there is a modification of the Hall algebra construction which produces a module over the Hall algebra, called the Hall module [30]. In various settings, Hall modules have been shown to be related to canonical bases [5], [27], representations of quantum groups [28] and Donaldson-Thomas theory with classical structure groups [29], [7]. While the Hall module M Q,Fq of Rep Fq (Q) has a natural comodule structure, the most naive analogue of Green's theorem does not hold: M Q,Fq is not a Hopf module over H Q,Fq .…”