Polarized Variations of Semi-Infinite Hodge Structures

• PVSIHS is a refined Hodge-theoretic framework that unifies classical de Rham methods with noncommutative invariants in quantum cohomology and Fukaya categories.
• It canonically encodes deformation, connection, and polarization data using structures like the Getzler–Gauss–Manin connection and higher residue pairing.
• Its application in mirror symmetry bridges categorical and enumerative invariants, offering tools to verify open–closed map isomorphisms and compute Gromov–Witten invariants.

A polarized variation of semi-infinite Hodge structures (PVSIHS) is a refinement of Hodge theoretic apparatus suited to both classical and noncommutative geometry, encapsulating rich structures arising in quantum cohomology, Fukaya categories, and their interplay through mirror symmetry. The PVSIHS formalism—first systematically developed for noncommutative settings by Barannikov, Kontsevich, Ganatra, Sheridan, and others—enables a canonical encoding of deformation, connection, and polarization data in both commutative (e.g., de Rham or quantum cohomology) and noncommutative (e.g., Fukaya, $A_\infty$, or derived) contexts. Its centrality in relating homological and enumerative aspects of mirror symmetry has been established through explicit constructions and Morita invariance.

1. Formal Definition and Structural Data

Let $S$ be a complex base or parameter space, and $u$ a formal parameter of degree 2, with coefficients in a suitable field extension $\BbK$. A polarized $\mathbb{Z}$-graded variation of semi-infinite Hodge structure (PVSIHS) over $S$ consists of the following data:

• A locally free $\mathcal{O}_S$-module (the “Hodge bundle”) $H = \bigoplus_{p \in \mathbb{Z}} H^p$.
• A flat $u$-connection,

$u \nabla: H \longrightarrow \Omega^1_S \otimes H$

• A decreasing filtration $F^\bullet H$ by $\mathcal{O}_S$-submodules.
• A nondegenerate sesquilinear pairing,

$$Q: H \otimes_{\mathcal{O}_S} H \rightarrow \mathcal{O}_S[[u]],$$

with the following properties for all vector fields $X$ and sections $s, t$: - Flatness and compatibility with $Q$: $\nabla \circ d = d \circ \nabla$,

$$Q(\nabla_X s, t) + Q(s, \nabla_X t) = X \cdot Q(s, t)$$

• Griffiths transversality:

  $\nabla(F^p H) \subset \Omega^1_S \otimes F^{p-1} H$

• u-symmetry and nondegeneracy:

  $Q(s, t) = (-1)^{|s||t| + n} Q(t, s)^*$

  where $*$ inverts $u$, $n = \dim X$ (in geometric cases), and $|\cdot|$ denotes degree.

In the commutative (classical) case, one recovers:

• $S$: formal deformation space of a Calabi–Yau manifold,
• $H$: de Rham cohomology,
• $\nabla$: Gauss–Manin connection,
• $F^\bullet$: Hodge filtration,
• $Q$: twisted Poincaré pairing.

In the noncommutative case (following (Ganatra et al., 6 Nov 2025, Sheridan, 2015)), the construction is intrinsically in terms of categories and their homological invariants:

• $H = HC^-_*(A)$, negative cyclic homology of an $A_\infty$-category $A$,
• $\nabla$: Getzler–Gauss–Manin connection on $HC^-_*(A)$,
• $F^p H$: $u$-adic filtration / Hodge filtration from the Connes $B$-operator,
• $Q$: higher residue pairing.

PVSIHS in this generality capture the variation of algebraic, analytic, and categorical structures under deformation, integral to both the formulation of mirror symmetry and to noncommutative Hodge theory.

2. PVSIHS in Noncommutative and Geometric Contexts

The passage from classical to noncommutative PVSIHS involves several precise replacements: | Component | Classical Setting | Noncommutative/A_\infty Setting | |--------------------------|-------------------------------------------------|----------------------------------------------| | Module/Hodge bundle | De Rham cohomology | Negative cyclic homology $HC^-$ | | Connection | Gauss–Manin connection | Getzler–Gauss–Manin connection | | Filtration | Hodge filtration | $u$-adic/Connes $B$-filtration | | Pairing | Twisted Poincaré pairing | Higher residue pairing |

For a (possibly bulk-deformed) relative Fukaya category $F^{(X, D; \Lambda)}$ over the universal Novikov field $\Lambda$, $HC^-_*$ is constructed from the filtered negative cyclic complex $fCC^-_*(F) = (CC_*^{nu}(F)[[u]], b + uB)$, where $CC_*^{nu}(F)$ is the non-unital Hochschild chain complex, $b$ the Hochschild differential, and $B$ the Connes operator. The Getzler–Gauss–Manin connection is defined by explicit cochain-level formulae, e.g.,

$$\nabla_v(\alpha) = v(\alpha) - u^{-1} b^{1|1}(v(\mu^*) | \alpha) - B^{1|1}(v(\mu^*) | \alpha)$$

where $v$ is a derivation on coefficients, and $v(\mu^*)$ is the Kodaira–Spencer class (cf. (Sheridan, 2015)).

The higher residue pairing is induced via a trace construction using the diagonal $(A,A)$-bimodule and the Feigin–Losev–Shoikhet trace, yielding a sesquilinear map of the form

$$\langle \alpha, \beta \rangle_{res} = -\widetilde{\int} \left( \widetilde{\wedge}_{A_\Delta}(\alpha, \beta^\vee) \right)$$

where $\alpha, \beta$ are negative cyclic chains.

These structures are Morita invariant and, when the Hodge-to-de Rham spectral sequence degenerates at $E_1$, yield a genuine polarized VSHS.

3. The Cyclic Open–Closed Map and Morphisms of PVSIHS

The cyclic open–closed map $OC^{cyc}$ plays a pivotal role in comparing the noncommutative and classical PVSIHS: $$OC^{cyc}: \sigma(n) fCC^-_*(F^{(X, D; \Lambda)}) \to QH^*(X; \Lambda)[[u]]$$ is defined by counting pseudoholomorphic discs with one outgoing interior puncture and several marked points, with explicit decomposition into “no-unit” and “unit-inserted” classes. It acts at the chain level and, after verifying compatibility with differentials, induces a homological map

$$OC^{cyc}: HC^-_*(F^{(X, D; \Lambda)})[-n] \to QH^*(X; \Lambda)[[u]].$$

This map is a morphism of PVSIHS, as it strictly preserves the connection (Theorem 6.4), compatible pairings (cyclic Cardy identity, Theorem 6.3), and $u$-adic filtrations:

• Connection: $$(id \otimes OC^{cyc}) \circ \nabla^{GGM} = \nabla^{DG} \circ OC^{cyc}$$
• Pairing: $$\langle OC^{cyc}(\alpha), OC^{cyc}(\beta) \rangle_{QH} = (-1)^n (n+1)/2 \cdot \langle \alpha, \beta \rangle_{res}$$

This provides the bridge for translating noncommutative Hodge-theoretic data into classically meaningful (quantum cohomological) invariants.

4. Isomorphism Criteria and Mirror Symmetry

A split-generation and smoothness criterion (from Sanda–Ganatra [Sanda], [Ganatra2016]) ensures isomorphism of the open–closed and closed–open maps (and thus their cyclic analogues) under suitable hypotheses:

• $X$ is connected, grading group is $\mathbb{Z}$.
• $A \subset F^{(X, D; \Lambda)}$ is a homologically smooth full subcategory.

If $A$ split-generates $F^{(X, D; \Lambda)}$, then both

$$CO: QH^*(X; \Lambda) \to HH^*(A),\quad OC: HH_*(A)[-n] \to QH^*(X; \Lambda)$$

are isomorphisms, as are their extensions

$$HC^- (A) \xleftrightarrow{\sim} QH^*(X;\Lambda)[[u]]$$

at the level of PVSIHS. In particular, in the case of a Batyrev mirror pair, homological mirror symmetry equivalence ($$D^\pi F^{(X, D; \Lambda)} \simeq D^b \mathrm{Coh}(\check{X})$$) implies, functorially, an isomorphism of the respective PVSIHS.

The proof fundamentally relies on the adjointness of $OC$ and $CO$ under nondegenerate pairings, split-generation, and compatibility of the cyclic and quantum structures through spectral-sequence arguments.

5. Extraction of Genus-Zero Gromov–Witten Invariants

In the Calabi–Yau case, the PVSIHS machinery achieves a direct connection between Fukaya categorical data and Gromov–Witten invariants. The big A-model VSHS $QH^*(X;\Lambda)[[u]]$ is miniversal, enabling the deformation theory of $X$ to be encoded in the quantum cohomology algebra. Selecting a canonical opposite filtration and the dilaton shift, one constructs a Frobenius manifold with genus-zero potential

$$F_0(t) = \frac{1}{2} \sum_{i,j} \langle \varphi_i, \varphi_j \rangle_Q t^i t^j + \sum_{k \ge 3} \frac{1}{k!} \langle \varphi_{i_1} \star \cdots \star \varphi_{i_k}, 1 \rangle_Q t^{i_1} \cdots t^{i_k}$$

where the coefficients correspond precisely to the genus-zero Gromov–Witten invariants.

Since the isomorphism $OC^{cyc}$ of PVSIHS identifies all such structures, these invariants can be entirely computed in terms of the relative Fukaya category and its cyclic Hodge theory, completing the topological–categorical bridge. In cases where a Batyrev mirror pair is present, the equivalence furnishes equality between A-model invariants and B-model Yukawa couplings, reducing enumerative mirror symmetry to homological mirror symmetry, as established in these frameworks.

6. Examples and Consequences in Mirror Symmetry

Explicit examples where PVSIHS structures manifest include:

• Fukaya categories $\mathcal{F}(X)$ of Calabi–Yau (and Fano) symplectic manifolds, yielding the A-model VSHS, and derived categories of coherent sheaves on Calabi–Yau varieties for the B-model.
• Mirror symmetry equivalences supply isomorphisms of PVSIHS between these models, through negative cyclic homology (for the Fukaya side) and periodic cyclic homology (for the derived category side) (Sheridan, 2015).
• In the $\mathbb{Z}/2$-graded (Fano, LG) case, the semi-infinite nature is essential: the filtration is not exhaustive, and the standard Hodge theoretical structures do not reduce to the classical case.

The PVSIHS formalism, together with Morita invariance and concrete pairing/connection constructions, underpins the passage from classical deformation theory to noncommutative and categorical Hodge theory. The entire apparatus serves as the foundation for relating categorical and enumerative mirror symmetry in both commutative and noncommutative geometry.