Superpotential
Every family on this site is Koszul — quadratic, with the trivial module \(k\) resolved as economically as possible. For such an algebra the six defining relations are not independent data: they are all encoded in a single tensor, the superpotential.
From relations to a potential
Let \(A = T(V)/(R)\) be a quadratic algebra on \(V = A_1\) (here \(\dim V = 4\)), with relation space \(R \subseteq V \otimes V\) (here \(\dim R = 6\)). When \(A\) is Artin–Schelter regular of global dimension \(d\) it is \(m\)-Koszul and its Koszul dual \(A^{!}\) is a Frobenius algebra concentrated in degrees \(0, \dots, d\). The one-dimensional top piece \(A^{!}_d\) is a nondegenerate functional on \(A^{!}_d\)’s predecessors, and dualising it produces a distinguished element
\[ w \;\in\; V^{\otimes d}, \]the superpotential of \(A\). For a quadratic algebra of global dimension \(d = 4\) — a noncommutative \(\mathbb{P}^3\) — the superpotential lives in \(V^{\otimes 4}\): it is a degree-\(4\) “noncommutative polynomial” in \(x_1, x_2, x_3, x_4\).
The algebra is recovered from \(w\) by differentiation. Writing \(\partial_\xi\) for contraction of the left-most tensor slot against \(\xi \in V^{*}\), the relation space is
\[ R \;=\; \partial^{\,d-2}(w) \;=\; \operatorname{span}\bigl\{\, \partial_\xi \partial_\eta\, w : \xi,\eta \in V^{*} \,\bigr\} \quad (d = 4), \]so that \(A = T(V)/(\partial^{d-2} w)\) is the derivation-quotient algebra of \(w\). This is the point of view of Dubois-Violette and Bocklandt: the relations are exactly the “second partial derivatives” of one quartic potential. The superpotential is unique up to a nonzero scalar.
The cleanest example is the commutative polynomial ring \(k[x_1,x_2,x_3,x_4]\), whose superpotential is the fully antisymmetric tensor
\[ w \;=\; \sum_{\sigma \in S_4} \operatorname{sgn}(\sigma)\, x_{\sigma(1)} x_{\sigma(2)} x_{\sigma(3)} x_{\sigma(4)}, \]the noncommutative determinant; its derivatives are the commutators \(x_i x_j - x_j x_i\).
Twisted superpotentials
In general \(w\) is not invariant under cyclically rotating its four tensor slots; it is only invariant up to a twist by a graded automorphism \(\nu\) of \(A\):
\[ w \;=\; (\nu \otimes \mathrm{id} \otimes \mathrm{id} \otimes \mathrm{id})\,(\tau\, w), \]where \(\tau\) is the cyclic permutation of the slots. This \(\nu\) is precisely the Nakayama automorphism, and \(w\) is an honest (untwisted, cyclically invariant) superpotential exactly when \(\nu\) is trivial — that is, exactly when \(A\) is Calabi–Yau.
Because the superpotential is only defined up to scalar and its coefficients are just a repackaging of the relations, it is not itself a useful “invariant” of a family — it carries the parameters in the obvious way. The invariant worth recording is its twist, the Nakayama automorphism.
Computation
The superpotential and its twist are computed with the
AssociativeAlgebras
package in Macaulay2 (superpotential and
nakayamaAut),
which work through the Frobenius structure on the Koszul dual; see
the Nakayama automorphism for the results
and reproduction details.
Per-invariant overview tables: point schemes, line schemes, \(\mathrm{HH}^i_0(A)\), \(\mathrm{HH}^i(\operatorname{qgr} A)\), the centre, normal elements, the Kodaira–Spencer map, and the Nakayama automorphism.