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 $\sigma$ 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 $\sigma$-symmetric quivers. We also formulate precise conjectures regarding the geometric meaning of these invariants and the freeness of the CoHM of a $\sigma$-symmetric quiver. We prove the freeness conjecture for disjoint union quivers, loop quivers and the affine Dynkin quiver of type $\widetilde{A}_1$. We also verify the geometric conjecture in a number of examples. Finally, we describe the CoHM of finite type quivers by constructing explicit Poincar\'{e}-Birkhoff-Witt type bases of these representations.

中文翻译：

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具有经典结构群的上同调霍尔代数和 Donaldson-Thomas 理论的表示

我们介绍了 Kontsevich 和 Soibelman 的上同调霍尔代数的一类新表示，我们将其称为上同调霍尔模块，或简称 CoHM。这些表示是由具有逆变对合 $\sigma$ 的箭袋的自对偶表示构成的，并为三折弦理论中的 BPS 状态空间提供了数学模型。我们使用 CoHM 来定义箭袋的上同调 Donaldson-Thomas 理论的推广，它允许箭袋表示具有正交和辛结构群。相关的不变量称为 orientifold Donaldson-Thomas 不变量。我们证明了 $\sigma$-对称箭袋的 orientifold Donaldson-Thomas 不变量的完整性猜想。我们还就这些不变量的几何意义和 $\sigma$ 对称箭袋的 CoHM 的自由度制定了精确的猜想。我们证明了不相交联合箭袋、循环箭袋和 $\widetilde{A}_1$ 类型的仿射 Dynkin 箭袋的自由度猜想。我们还在一些例子中验证了几何猜想。最后，我们通过构造这些表示的显式 Poincar\'{e}-Birkhoff-Witt 类型基来描述有限类型颤动的 CoHM。