## Abstract

We propose a new method for computing motivic Donaldson–Thomas invariants of a symmetric quiver which relies on Koszul duality between supercommutative algebras and Lie superalgebras and completely bypasses cohomological Hall algebras. Specifically, we define, for a given symmetric quiver Q, a supercommutative quadratic algebra A_{Q}, and study the Lie superalgebra g_{Q} that corresponds to A_{Q} under Koszul duality. We introduce an action of the first Weyl algebra on g_{Q} and prove that the motivic Donaldson–Thomas invariants of Q may be computed via the Poincaré series of the kernel of the operator ∂_{t}. This gives a new proof of positivity for motivic Donaldson–Thomas invariants. Along the way, we prove that the algebra A_{Q} is numerically Koszul for every symmetric quiver Q and conjecture that it is in fact Koszul; we show that this conjecture holds for a certain class of quivers.

Original language | English |
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Article number | 106 |

Journal | Letters in Mathematical Physics |

Volume | 112 |

Issue number | 5 |

DOIs | |

State | Published - Oct 2022 |

Externally published | Yes |

## Keywords

- Donaldson-Thomas invariants
- Koszul duality
- Lie superalgebras
- quadratic algebras
- symmetric quivers