Given a quiver $Q$, a potential for $Q$ is a cyclically invariant element in its path algebra or its completion. Cyclic invariance means that it is a linear combination of cyclic paths; cyclic invariance can be in a ${\mathbf Z}_2$-graded sense in which case it is called a superpotential. This is in agreement with another notion of a superpotential in Donaldson-Thomas theory. The space of superpotentials has a necklace Lie algebra structure.
It is introduced in
(What is a relation to Paul S. Aspinwall, Lukasz M. Fidkowski, Superpotentials for quiver gauge theories, hep-th/0506041 ?)
Relation to generalized cluster categories is in
The proof that an important class of “exact” $d$-Calabi-Yau algebras are derived from superpotentials is in
An important counterexample of a Calabi-Yau algebras which are not of this form (but are related to topology) is exhibited in
B. Davison, Superpotential algebras and manifolds, Adv. Math. 231(2), 879–912 (2012)
B. Davison, The critical CoHA of a quiver with potential, The Quarterly Journal of Mathematics 68(2):635–703, 2017
Ben Davison, Sven Meinhardt, Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras, arXiv:1601.02479
Last revised on September 20, 2022 at 15:06:23. See the history of this page for a list of all contributions to it.