Twisted Jacobian Ring

• Twisted Jacobian ring is a graded algebra that extends the classical Jacobian ring by incorporating finite group symmetries and fixed-point data.
• It is constructed from polynomial rings modulo derivatives, with each group element's fixed locus contributing to its braided super-commutative structure.
• The ring underpins key computations in homological mirror symmetry and orbifold Floer theory, linking algebraic models to symplectic geometry.

The twisted Jacobian ring is a graded algebraic structure associated to a polynomial $W$ with an isolated singularity, equipped with a finite group action that preserves $W$. It generalizes the classical Jacobian ring, incorporating equivariant data through the group symmetry. The twisted Jacobian ring plays a central role in the study of Landau–Ginzburg orbifolds and appears as the endomorphism algebra of the twisted-diagonal matrix factorization in the category of equivariant matrix factorizations. It underpins key computations in homological mirror symmetry and orbifold Lagrangian Floer theory (Lee, 2021).

1. Classical and Twisted Jacobian Rings

Given $W\in\C[x_1,\dots,x_N]$ with an isolated critical point at the origin, the classical Jacobian ring is

$J(W)=\Jac(W):=\frac{\C[x_1,\dots,x_N]}{(\partial_{x_1}W,\dots,\partial_{x_N}W)},$

endowed with a residue pairing making it a finite-dimensional Frobenius algebra.

Introducing a finite group $G$ acting diagonally on $\C[x_1,\dots,x_N]$ and fixing $W$, the twisted Jacobian ring $J^G(W)$ becomes a $G$-graded vector space reflecting the fixed-point geometry of each group element: $J^G(W)=\bigoplus_{g\in G}\frac{\Omega^{\dim\Fix(g)}_{\Fix(g)}}{(dW|_{\Fix(g)})}\cdot\xi_g,$ where $\Fix(g)$ is the linear subspace fixed by $g$, $\Omega^k_{\Fix(g)}$ is the space of algebraic $k$-forms on $\Fix(g)$, $dW|_{\Fix(g)}$ generates the contraction ideal, and $\xi_g$ is a formal symbol of degree $N-\dim\Fix(g)\bmod2$. Each summand has a residue pairing, and the algebra structure arises from a braided super-commutative product: $$\bullet~:~ J^G(W)_g\otimes J^G(W)_h\longrightarrow J^G(W)_{gh}, \qquad \alpha\bullet\beta = (-1)^{\deg\alpha\,\deg\beta}\beta\bullet (h^{-1}\cdot\alpha),$$ as introduced by Shklyarov via a discrete-torsion-twisted Hochschild cup product.

2. Construction of the Twisted-Jacobian Algebra

For each $g\in G$:

• The summand $J^G(W)_g$ is computed from the quotient of top-degree forms (on $\Fix(g)$) by the ideal generated by contractions with $dW|_{\Fix(g)}$.
• The grading of each sector is determined by the codimension of fixed loci: $\deg(\xi_g)=N-\dim\Fix(g)\bmod2$.
• The product structure intertwines group actions and form degrees, resulting in a nontrivial fusion of sectors.

The overall structure simultaneously generalizes the Milnor algebra of $W$ and encodes group-twisted sectors akin to those appearing in orbifold cohomology theories.

3. The Twisted-Diagonal Matrix Factorization

Let $R=\C[x_1,\dots,x_N]$ and $S=R\otimes_\C R=\C[x_1,\dots,x_N,y_1,\dots,y_N]$. In the category $MF_{G\times G}(W(y)-W(x))$, the twisted-diagonal matrix factorization $\Delta_W^{G\times G}$ is constructed as

$$\Delta_W = \left(S\otimes\wedge^\bullet(\theta_1,\dots,\theta_N),\,d \right),\qquad d = \sum_{i=1}^N \left[(y_i-x_i)\theta_i + \nabla_iW(x,y)\,\partial_{\theta_i}\right],$$

where $\nabla_iW$ is the divided-difference operator. The group $G\times G$ acts diagonally, and twisting by group elements yields

$$\Delta_W^{G\times G} = \bigoplus_{h\in G} \Delta_h,$$

where each $\Delta_h$ is a Koszul-type factorization with differential $d_h$ adapted via the group element $h$.

The $(G\times G)$-action is specified on generators by

$$(h_1\times h_2): \Delta_g\longrightarrow \Delta_{h_1\,g\,h_2^{-1}},\qquad r(x,y)\theta_I \mapsto r(h_1x,h_2y)\,\rho(h_2)(\theta_I).$$

4. Algebra Isomorphism: Twisted Jacobian and Endomorphisms

A key result ((Lee, 2021), Theorem A) establishes an isomorphism of graded algebras when $(W,G)$ is a Landau–Ginzburg orbifold and $G$ is finite abelian: $\End_{MF_{1\times G}}(\Delta_W^{G\times G}) \cong J^G(W).$ This correspondence matches each generator $\xi_g\in J^G(W)_g$ with an explicit closed endomorphism in the matrix factorization category,

$\Phi(\xi_g) = \exp(\eta_g)(\theta_{I_g}) \in \Hom(\Delta_1,\Delta_g),$

where $\eta_g$ encodes divided-difference expressions intrinsic to $W$. The proof leverages Koszul duality, spectral sequences, and explicit computations of composition products, showing that endomorphism algebra operations reconstruct the twisted Jacobian product formula.

5. Explicit Example: $A_{n-1}$-Singularity with $\mathbb{Z}/n$-Action

For $W(x)=x^n$ and $G=\mathbb{Z}/n$ acting via $g(x)=\zeta x$ ($\zeta=e^{2\pi i/n}$), the fixed-point subspaces are trivial except at the identity. The structure simplifies to: $J^G(W) = \underbrace{\C[x]/(x^{n-1})}_{g^0\text{ sector}}\oplus\bigoplus_{k=1}^{n-1}\C\,\xi_{g^k},$ with nontrivial sectors one-dimensional and graded by $2k/n \bmod 2$. The algebra product satisfies:

• $\xi_{g^k}\bullet\xi_{g^\ell}=0$ unless $k+\ell\equiv0\mod n$,
• in which case the result lands in the identity sector, represented by the class of $x^{n-1}$. On the matrix factorization side, $(1\times G)$-equivariant endomorphisms of $\bigoplus_{k=0}^{n-1}\Delta_{g^k}$ precisely realize this algebraic structure.

6. Applications to Floer Theory and Homological Mirror Symmetry

For symplectic manifolds $(X,\omega)$ with Landau–Ginzburg mirrors $(\check{X},W)$, a single Lagrangian brane $\scr{L}\subset X$ with bounding cochains is mirrored by $W$, and the Floer cohomology algebra $HF^*(\scr{L},b)$ is often isomorphic to the Jacobian ring of $W$.

If $X$ admits a finite group symmetry $G$, the orbifold quotient $X/G$ introduces additional structure. The Seidel Lagrangian $\bar{\scr{L}}\subset X/G$ is mirrored by the Landau–Ginzburg orbifold $(\check{X},W,G)$. Cho and Lee show under transversality conditions that

$HF^*(\pi^{-1}(\bar{\scr{L}}),\pi^{-1}(\bar{\scr{L}})) \cong J^G(W)$

as twisted algebras. This identifies $J^G(W)$ as a purely algebraic model for the orbifold Lagrangian–Floer algebra predicted by homological mirror symmetry. Group-fixed sectors of Floer cohomology correspond precisely to the $\Delta_h$ summands, and the Floer product structure mirrors the composition of endomorphisms in the equivariant matrix factorization category (Lee, 2021).

7. Significance and Broader Context

The twisted Jacobian ring amalgamates singularity theory, equivariant algebra, and symplectic geometry, providing a calculable invariant for Landau–Ginzburg orbifolds. Its depiction as an equivariant endomorphism algebra enables direct algebraic modeling of orbifold Floer theory, thus bridging algebraic geometry, representation theory, and mathematical physics in the context of homological mirror symmetry. The explicit description and isomorphism are instrumental in rendering the computations of orbifold Fukaya categories and establishing bridges between different flavors of mirror symmetry constructions (Lee, 2021).