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Representations of Cohomological Hall Algebras and Donaldson–Thomas Theory with Classical Structure Groups

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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 \(\sigma \) 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 in which the quiver representations have orthogonal or symplectic structure groups. The associated invariants are called orientifold Donaldson–Thomas invariants. We prove the orientifold analogue of the integrality conjecture for \(\sigma \)-symmetric quivers. We also formulate precise conjectures regarding the geometric meaning of orientifold Donaldson–Thomas 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 construct explicit Poincaré–Birkhoff–Witt-type bases of the CoHM of finite type quivers.

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Notes

  1. Because signs are included in the definition of \(x^{\prime }\), additional sign substitutions (as occur in Theorem 3.3) are not needed in these equations.

  2. Analogously to Conjecture 3.7, we should really restrict to a subalgebra of \({\mathcal {H}}_{Q,W,\mu =0}^{\theta \text {-} {\text {ss}}}\).

  3. Such a choice cannot always be made in type \(E_8\).

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Acknowledgements

The author would like to thank Ben Davison and Sven Meinhardt for discussions. The author is also grateful for the comments and suggestions of anonymous referees. Parts of this work were completed at National Taiwan University and the KIAS Winter School on Derived Categories and Wall-Crossing. The author would like to thank Wu-yen Chuang, Michel van Garrel and Bumsig Kim for the invitations. During the preparation of this work the author was supported by the Research Grants Council of the Hong Kong SAR, China (GRF HKU 703712) and by the Max Planck Institute for Mathematics.

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  1. Department of Mathematics and Statistics, Utah State University, Logan, Utah, 84322, USA

    Matthew B. Young

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Young, M.B. Representations of Cohomological Hall Algebras and Donaldson–Thomas Theory with Classical Structure Groups. Commun. Math. Phys. 380, 273–322 (2020). https://doi.org/10.1007/s00220-020-03877-z

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