Primitive Form of K. Saito

• Primitive Form of K. Saito is a distinguished holomorphic relative volume form defined on the miniversal deformation of isolated, quasi-homogeneous hypersurface singularities.
• It uniquely encodes Frobenius manifold structures by establishing flat metrics and special Kähler geometries through vanishing cohomology and Gauss–Manin connections.
• Its recursive construction via BV-algebra methods and the Gelfand–Leray framework connects singularity theory with mirror symmetry and quantum cohomology applications.

The primitive form of K. Saito is a distinguished relative holomorphic volume form defined for any miniversal deformation of an isolated, quasi-homogeneous hypersurface singularity. The primitive form encodes semisimple Frobenius manifold structures on parameter spaces of singularities, realizes canonical special geometries via periods on vanishing cohomology, and, through the Gelfand–Leray construction, provides a universal Seiberg–Witten differential for the associated singular geometry. Beyond its original algebro-geometric and singularity-theoretic context, the primitive form has become central in the study of mirror symmetry, quantum singularity theory, and categorical approaches to noncommutative Hodge theory. The construction and unique characterization of the primitive form draws on Hodge-theoretic, polyvector, and Batalin–Vilkovisky algebra perspectives, with deep ties to Gromov–Witten theory and deformation quantization.

1. Quasi-homogeneous Singularity and Mini-versal Deformation

Let $X\cong\mathbb{C}^{n+1}$, with coordinates $x=(x_1,\ldots,x_{n+1})$. A holomorphic function $f:X\to\mathbb{C}$ has an isolated critical point at $0$ and is quasi-homogeneous of weights $q_i>0$ if

$$f(\lambda^{q_1}x_1,\ldots,\lambda^{q_{n+1}}x_{n+1}) = \lambda\, f(x_1,\ldots,x_{n+1}), \quad \lambda\in\mathbb{C}^*.$$

The Jacobian (Milnor) algebra $$R_f=\mathbb{C}[x_1,\ldots,x_{n+1}]/(\partial_1 f,\ldots,\partial_{n+1} f)$$ is finite-dimensional ($\mu<\infty$). The miniversal unfolding is

$F(x,u) = f(x) + \sum_{i=1}^\mu u^i\, \phi_i(x),$

where $\{\phi_i\}$ is a $\mathbb{C}$-basis of $R_f$ and $u\in M\cong\mathbb{C}^\mu$. The hypersurface $F(\cdot,u)=0$ generically has only Morse-type singularities; $M$ parameterizes all versal deformations. For such $F$, the primitive form is constructed as a section of the Brieskorn lattice and encodes the intersection structure of vanishing cohomology through the variation of semi-infinite Hodge filtration (Li et al., 2018, Li et al., 2013, Tu, 2019, Tu, 2019, Basalaev, 31 Dec 2025).

2. Characterization and Properties of the Primitive Form

K. Saito’s existence/uniqueness theorem asserts that, up to an overall constant, there is a unique family of relative volume forms $\zeta(u,x)\in H^{(0)}_F$ called the primitive form, characterized by:

• Flatness (primitivity) of the residue pairing: The pairing $$g^\zeta(V,W)=\operatorname{Res}_F\big(\nabla_{V}\zeta^{(-1)},\nabla_{W}\zeta^{(-1)}\big)$$ is constant in suitable coordinates, where $\nabla$ is the Gauss–Manin connection, and $\zeta^{(-1)}$ is the Gelfand–Leray descendant.
• Normalization: At $u=0$, the top-degree part of the higher residue pairing is normalized to $1$, i.e., for the top basis element $\phi_\mu$, $\operatorname{Res}_f(dx,\phi_\mu dx)=1$.
• Homogeneity: $\zeta$ is an eigenvector for $(\nabla_E + t\partial_t)$, where $E$ is the Euler vector field on $M$ and $t$ the Brieskorn lattice variable, i.e., $(\nabla_E + t\partial_t)\zeta = r\,\zeta$ for some rational $r$.
• Holonomicity: Triple derivatives of $\zeta$ under $\nabla$ satisfy pole conditions that enforce associativity of the induced product.

These properties guarantee that the Kodaira–Spencer map furnishes an isomorphism between tangent vectors to deformation space and vanishing cohomology classes, with the residue pairing inducing a nondegenerate, flat metric (the Frobenius metric) in flat coordinates.

3. Constructive Approaches and Recursion

A primitive form $\zeta(u,x)$ can be explicitly constructed. One writes

$$\zeta(u,x) = \sum_{i=1}^\mu \zeta_i(u)\, \phi_i(x)\, dx$$

and solves for the $\zeta_i(u)$ recursively to satisfy the conditions above. In polyvector notation, following the gauge-theoretic approach, the construction produces a unique flat section via the semi-infinite period map and a suitable choice of “opposite filtration” $L$, ensuring compatibility with homogeneity and Lagrangian complementarity (Li et al., 2013).

A BV-algebra-based recursion for the primitive form is available: fix a good basis $\{\omega_\alpha\}$ and set

$$\zeta(s,z) = \sum_{\alpha=1}^\mu d_\alpha(s,z)\, \omega_\alpha,$$

expanded in $s$-degree as $\zeta = \zeta_{(0)} + \zeta_{(1)} + \dots$. Basalaev’s recursive formula (Basalaev, 31 Dec 2025):

• $\zeta_{(0)}$ is the identity in the good basis,
• For $p\geq 1$,

$$\zeta_{(p)} = - \left[\, \pi_{>0} \big( (\Phi_f^\omega)^{-1}( \sum_{a=1}^p \frac{(F-f)^a}{z^a a!}\, \zeta_{(p-a)} ) \big) \right]_f,$$

where $\Phi_f^\omega$ is the BV trivialization, $\pi_{>0}$ projects to positive $z$-powers, and $[\,\cdot\,]_f$ projects to Jac$(f)$ in the good basis.

4. Variation of Semi-infinite Hodge Structure and Frobenius Manifolds

The primitive form endows the base $M$ of the miniversal deformation with a Frobenius manifold structure, making $M$ into an analytic variety equipped with:

• a flat metric,
• a compatible, commutative, associative product on tangent spaces (encoded via multiplication on vanishing cohomology),
• a prepotential $\mathcal{F}_0$ determined by the three-point functions (third derivatives of $\mathcal{F}_0$ given by the pairing of triple derivatives of $\zeta$).

The Brieskorn lattice, together with the higher residue pairing and primitive form, realizes the axioms of a polarized variation of semi-infinite Hodge structure (VSHS). For weighted-homogeneous $f$, the space of primitive forms (modulo scaling) is isomorphic to the moduli of good opposite filtrations; in particular, for ADE and 14 exceptional unimodular singularities, the primitive form is unique up to scale (Li et al., 2013, Tu, 2019, Tu, 2019).

The period integrals of the Gelfand–Leray residue of $\zeta$ over vanishing $n$-cycles yield flat and dual special coordinates and encode special Kähler geometry as realized in four-dimensional $\mathcal{N}=2$ SCFTs (Li et al., 2018).

5. Deformation Theory, Mirror Symmetry, and Irrelevant Couplings

The construction accommodates parameters $u^i$ of negative scaling weight (“irrelevant” in RG terminology), and the primitive form depends nontrivially on them. This allows extension of the Seiberg–Witten geometry and corresponding special Kähler structure to include these non-marginal directions (Li et al., 2018).

Primitive forms underpin Landau–Ginzburg B-models, providing the Frobenius manifold data for the full genus expansion. LG/LG mirror symmetry equates Givental-type B-model correlators constructed from Saito–primitive-form data $(\zeta,E)$ with Fan–Jarvis–Ruan–Witten (FJRW) A-model invariants. Explicit computations for exceptional unimodular singularities validate this correspondence (Li et al., 2013).

The primitive form also appears categorically: via the negative cyclic homology of the category of matrix factorizations MF$(W)$ (and its $G$-equivariant generalizations), the VSHS structure and Frobenius manifold potential are reconstructed, and the categorical primitive form coincides canonically with Saito’s geometric primitive form (Tu, 2019, Tu, 2019).

6. BV-algebras, Givental Formalism, and Recursive Structures

The Batalin–Vilkovisky (BV) algebra approach realizes the Brieskorn lattice cohomologically and identifies primitive forms as generating elements with prescribed period and flatness properties. Choices of BV trivialization, and particularly the explicit recursion in the good basis, parallel and generalize Saito’s linear algebraic constructions.

The Givental $R$-matrix—which governs higher genus effects and quantization of Dubrovin–Frobenius structures—is constructed explicitly in the BV setup. The primitive form changes under the action of such $R$-matrices, reflecting different trivializations and underlying semi-infinite variations. This approach unifies deformation-theoretic, cohomological, and operadic aspects of primitive form theory (Basalaev, 31 Dec 2025).

7. Summary Table: Key Data for Saito Primitive Form

Data	Mathematical Object / Operation	Role for Primitive Form
Singular function	$f(x)$, quasi-homogeneous with isolated critical point	Base singularity
Deformation space	Miniversal unfolding $F(x,u)=f(x)+\sum u^i\phi_i(x)$	Parameter domain
Brieskorn lattice	$H^{(0)}_F$	Ambient for $\zeta$
Primitive form	$\zeta(u,x)\in H^{(0)}_F$	Flat generator
Gauss–Manin conn.	$\nabla_V[g\,dx]=[Vg\,dx+(VF)/z\,g\,dx]$	Flatness/Hodge theory
Residue pairing	$\operatorname{Res}_F$	Frobenius metric
Gelfand–Leray form	$$\omega_{\text{SW}}=\operatorname{Res}_{W=0}\frac{\zeta}{dW}$$	Periodic/SW geometry

The above summary table encapsulates the central algebraic–analytic data and their geometric roles. This infrastructure supports a range of applications, including singularity theory, mirror symmetry, and quantum cohomology. The categorical, deformation-theoretic, and BV-algebraic reconstructions strongly reinforce the fundamental and universal nature of the primitive form in modern mathematics and physics (Li et al., 2018, Li et al., 2013, Tu, 2019, Tu, 2019, Basalaev, 31 Dec 2025).