nLab quiver with potential

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

• H. Derksen, J. Weyman, A. Zelevinsky, Quivers with potentials and their representations I: Mutations, Sel. math., New ser. 14, 59–119 (2008) doi Quivers with potentials and their representations II: Applications to cluster algebras, J. Amer. Math. Soc. 23 (2010) 749-790 doi

(What is a relation to Paul S. Aspinwall, Lukasz M. Fidkowski, Superpotentials for quiver gauge theories, hep-th/0506041 ?)

• Raf Bocklandt, Travis Schedler, Michael Wemyss, Superpotentials and higher order derivations, J. Pure and Appl. Alg. 214:9 (2010) 1501-1522 doi

Relation to generalized cluster categories is in

• Claire Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential, Ann. Inst. Fourier 59:6 (2009) 2525-2590 link

An analytic version of the theory is in

• Zheng Hua, Bernhard Keller, Quivers with analytic potentials, arxiv:1909.13517
• Okke van Garderen, Cyclic A-infinity algebras and Calabi-Yau structures in the analytic setting, arxiv:2306.00771

The proof that an important class of “exact” $d$-Calabi-Yau algebras are derived from superpotentials is in

• Michel Van den Bergh, Calabi-Yau algebras and superpotentials, Sel. Math. New Ser. 21, 555–603 (2015). doi

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