Transversal Shifted Poisson Unfoldings

• The paper establishes that transversal shifted Poisson unfoldings are equivalent to flat splittings of the Poisson transverse controller, unifying deformation and transport of shifted Poisson structures.
• It employs derived Lie algebroids and homotopy stabilizers to control the deformation complexes and manage quantization anomalies through coherent null-homotopies.
• The framework connects classical transport, Lagrangian thickenings, and AKSZ constructions, thereby extending applications to BV observables and star-products.

Transversal shifted Poisson unfoldings formalize the problem of transporting a family of shifted Poisson structures flatly over a parameter space, while allowing the Poisson Maurer–Cartan element to be preserved not strictly but up to coherent homotopy. In the controller-theoretic formulation, a relative derived stack $p:X\to S$ endowed with a relative $n$-shifted Poisson structure $\pi$ determines a Poisson transverse controller $\mathbb U_\pi$, and the central classification statement identifies transversal shifted Poisson unfoldings with flat splittings of this controller (Corrêa et al., 7 Jul 2026). Earlier work on shifted Lagrangian thickenings, derived foliations, and AKSZ constructions supplies the principal geometric background for this notion by showing that shifted Poisson data can be re-expressed as Lagrangian thickening data and propagated to mapping stacks with the expected degree shift (Tomić, 29 Jun 2025, Tomić, 7 Jan 2026).

1. Relative shifted Poisson data and the transport problem

The basic input is a family $p:X\to S$ together with a relative $n$-shifted Poisson structure. In a strict affine chart $X=\Spec A$, $S=\Spec B$, the shifted polyvectors are defined by

$\Pol(A/B,n) := \prod_{q\ge 0} \underline{Hom}_A\!\left( \Sym_A^q\bigl(\mathbb L_{A/B}[n+1]\bigr),A \right),$

and the shifted dg Lie algebra is

$\mathfrak{pol}_n(A/B):=\Pol(A/B,n)[n+1].$

A relative $n$0-shifted Poisson structure is a Maurer–Cartan element

$n$1

Two complexes are distinguished from the outset. The Poisson deformation complex

$n$2

controls genuine deformations of the Poisson tensor on the fixed family, while the extended Poisson complex

$n$3

also includes weight-one terms, namely relative vector fields or infinitesimal reparametrizations. Here $n$4.

Object	Formula	Role
Poisson deformation complex	$n$5	Genuine deformations of $n$6
Extended Poisson complex	$n$7	Includes weight-one reparametrizations
Poisson transverse controller	$n$8	Encodes transverse symmetries with coherent null-homotopy

The inclusion of the weight-one part is decisive: a transverse transport problem needs it because changing a transverse lift by a relative vector field is part of the geometry. The guiding idea is therefore to separate deforming a Poisson structure fiberwise from transporting it transversely over the parameter space (Corrêa et al., 7 Jul 2026).

2. The Poisson transverse controller and homotopy stabilizers

The Poisson transverse controller is defined as the homotopy stabilizer of the Poisson Maurer–Cartan element under the ordinary transverse foliation controller,

$n$9

Informally, $\pi$0 is the object of transverse symmetries together with a homotopy proving that their induced infinitesimal effect on $\pi$1 is trivial.

This construction is an instance of a general mechanism. If a derived Lie algebra $\pi$2 acts on a filtered dg Lie algebra $\pi$3, and $\pi$4, then the derivative of the action gives

$\pi$5

and the homotopy stabilizer is $\pi$6. In the strict cone model, the stabilizer has underlying complex $\pi$7 with differential

$\pi$8

A closed degree-zero element is therefore a pair $\pi$9 such that the induced variation $\mathbb U_\pi$0 is null-homotopic via $\mathbb U_\pi$1.

Applied to a shifted Poisson element $\mathbb U_\pi$2, the condition becomes

$\mathbb U_\pi$3

Thus a transverse symmetry $\mathbb U_\pi$4 of the Hamiltonian foliation is permitted only together with a homotopy $\mathbb U_\pi$5 certifying that the variation of $\mathbb U_\pi$6 is $\mathbb U_\pi$7-exact. This is the precise sense in which the Poisson Maurer–Cartan element is preserved up to coherent homotopy. In strict affine models, $\mathbb U_\pi$8 becomes a crossed dg-Lie algebroid

$\mathbb U_\pi$9

where $p:X\to S$0 is the homotopy stabilizer of the action of basic derivations on $p:X\to S$1 (Corrêa et al., 7 Jul 2026).

3. Flat splittings and the classification of unfoldings

For a derived Lie algebroid $p:X\to S$2, the derived space of flat splittings is denoted $p:X\to S$3. A strict point is a section $p:X\to S$4 satisfying

$p:X\to S$5

A flat splitting is therefore an anchor splitting with zero curvature.

The geometric unfolding space is defined as the homotopy fiber of the Poisson variation map,

$p:X\to S$6

An unfolding consists of two pieces of data: a transverse unfolding of the Hamiltonian foliation, and a coherent null-homotopy of the induced variation of $p:X\to S$7. Under controller-admissibility, the principal theorem identifies the two descriptions: $p:X\to S$8 Equivalently, transversal shifted Poisson unfoldings are exactly flat splittings of the Poisson transverse controller.

The local strict models refine this statement. In a strict affine chart, a flat splitting is represented by a section $p:X\to S$9 with zero curvature. In a crossed or non-effective model, one may instead have a lift $n$0 whose curvature lies in the image of the inner map: $n$1 with Bianchi identity

$n$2

The effective quotient controller is

$n$3

and in the effective case unfoldings reduce to ordinary flat sections of $n$4 (Corrêa et al., 7 Jul 2026).

4. Classical transport, vertical symmetries, and local deformation theory

A flat splitting transports more than the Poisson tensor itself. If $n$5, then the induced connection on twisted polyvectors is

$n$6

This corrected operator is necessary because the naive derivation $n$7 alone does not commute with $n$8; the homotopy correction $n$9 cancels the anomaly.

The same splitting acts on the vertical kernel $X=\Spec A$0 by the adjoint action,

$X=\Spec A$1

As a consequence, flat transport is induced on the vertical symmetry sheaf, the full twisted polyvector complex, the deformation complex, and the Poisson cohomology sheaves. The formal deformation theory of a flat splitting $X=\Spec A$2 is controlled by

$X=\Spec A$3

or, in a smooth de Rham model,

$X=\Spec A$4

Its cohomology gives the standard tangent-obstruction hierarchy: $X=\Spec A$5

This establishes that a transversal shifted Poisson unfolding is not merely a method for selecting compatible fibers. It acts on vertical symmetries, Poisson cohomology, and local deformation theory, and in the smooth proper classical case it recovers the Gauss–Manin connection (Corrêa et al., 7 Jul 2026).

5. $X=\Spec A$6-adic lifting and transport anomalies

The quantized problem is formulated as a lifting problem for transport rather than as fiberwise quantization alone. The quantum question is not merely to quantize each fiber, but to lift a flat classical unfolding to a flat quantum unfolding. One assumes an $X=\Spec A$7-adically complete quantized object with a filtered quantum symmetry extension whose classical limit is the classical controller. A quantized unfolding is then a flat lift

$X=\Spec A$8

of the classical splitting $X=\Spec A$9.

For a partial lift modulo $S=\Spec B$0, the next obstruction lies in the associated graded vertical kernel: $S=\Spec B$1 and the main obstruction class is

$S=\Spec B$2

This degree-two class is the transport anomaly. It vanishes if and only if the lift extends one more order; if it vanishes, extensions form a torsor under $S=\Spec B$3; infinitesimal automorphisms are governed by $S=\Spec B$4.

The Bianchi identity ensures that the curvature defect is closed, and changing the partial lift shifts it by a coboundary. Accordingly, the obstruction theory is the standard filtered Maurer–Cartan obstruction theory interpreted as anomaly theory for transport. This makes anomaly questions intrinsic to the unfolding formalism rather than external corrections appended after quantization (Corrêa et al., 7 Jul 2026).

6. Geometric antecedents: Lagrangian thickenings, foliations, and AKSZ

The controller-theoretic notion emerged against a background in which shifted Poisson structures were progressively identified with geometric thickening data. For derived schemes locally of finite presentation over a field $S=\Spec B$5 of characteristic $S=\Spec B$6, the main theorem of "Shifted Lagrangian thickenings of shifted Poisson derived schemes" establishes the equivalence

$S=\Spec B$7

In that framework, a shifted Poisson structure produces a derived foliation

$S=\Spec B$8

whose formal integration yields a formal thickening

$S=\Spec B$9

The key intermediate theorem identifies isotropic, and under perfectness assumptions Lagrangian, structures on the foliation side with those on the formal leaves prestack side. As a corollary, if $\Pol(A/B,n) := \prod_{q\ge 0} \underline{Hom}_A\!\left( \Sym_A^q\bigl(\mathbb L_{A/B}[n+1]\bigr),A \right),$0 is a compact oriented $\Pol(A/B,n) := \prod_{q\ge 0} \underline{Hom}_A\!\left( \Sym_A^q\bigl(\mathbb L_{A/B}[n+1]\bigr),A \right),$1-dimensional manifold and $\Pol(A/B,n) := \prod_{q\ge 0} \underline{Hom}_A\!\left( \Sym_A^q\bigl(\mathbb L_{A/B}[n+1]\bigr),A \right),$2 carries an $\Pol(A/B,n) := \prod_{q\ge 0} \underline{Hom}_A\!\left( \Sym_A^q\bigl(\mathbb L_{A/B}[n+1]\bigr),A \right),$3-shifted Poisson structure, then $\Pol(A/B,n) := \prod_{q\ge 0} \underline{Hom}_A\!\left( \Sym_A^q\bigl(\mathbb L_{A/B}[n+1]\bigr),A \right),$4 has an $\Pol(A/B,n) := \prod_{q\ge 0} \underline{Hom}_A\!\left( \Sym_A^q\bigl(\mathbb L_{A/B}[n+1]\bigr),A \right),$5-shifted Poisson structure (Tomić, 29 Jun 2025).

The paper "AKSZ construction for shifted Poisson structures" extends this picture to derived prestacks having a deformation theory and proves the Poisson analogue of the AKSZ theorem: if $\Pol(A/B,n) := \prod_{q\ge 0} \underline{Hom}_A\!\left( \Sym_A^q\bigl(\mathbb L_{A/B}[n+1]\bigr),A \right),$6 is an $\Pol(A/B,n) := \prod_{q\ge 0} \underline{Hom}_A\!\left( \Sym_A^q\bigl(\mathbb L_{A/B}[n+1]\bigr),A \right),$7-shifted derived Poisson formal prestack, $\Pol(A/B,n) := \prod_{q\ge 0} \underline{Hom}_A\!\left( \Sym_A^q\bigl(\mathbb L_{A/B}[n+1]\bigr),A \right),$8 is a $\Pol(A/B,n) := \prod_{q\ge 0} \underline{Hom}_A\!\left( \Sym_A^q\bigl(\mathbb L_{A/B}[n+1]\bigr),A \right),$9-oriented prestack, and the mapping prestack $\mathfrak{pol}_n(A/B):=\Pol(A/B,n)[n+1].$0 is locally of finite presentation, then $\mathfrak{pol}_n(A/B):=\Pol(A/B,n)[n+1].$1 carries a natural $\mathfrak{pol}_n(A/B):=\Pol(A/B,n)[n+1].$2-shifted Poisson structure. The technical bridge is the equivalence

$\mathfrak{pol}_n(A/B):=\Pol(A/B,n)[n+1].$3

now proved for formal prestacks, together with the passage from a Poisson structure on $\mathfrak{pol}_n(A/B):=\Pol(A/B,n)[n+1].$4 to an $\mathfrak{pol}_n(A/B):=\Pol(A/B,n)[n+1].$5-shifted Lagrangian thickening and back after applying the mapping-stack functor. The same work also extends the nondegenerate equivalence

$\mathfrak{pol}_n(A/B):=\Pol(A/B,n)[n+1].$6

to prestacks with deformation theory and local finite presentation, and gives applications to mapping stacks with non-proper source and to the BV formalism (Tomić, 7 Jan 2026).

The deeper foundations go back to the identification of non-degenerate $\mathfrak{pol}_n(A/B):=\Pol(A/B,n)[n+1].$7-shifted Poisson and $\mathfrak{pol}_n(A/B):=\Pol(A/B,n)[n+1].$8-shifted symplectic structures on derived Artin $\mathfrak{pol}_n(A/B):=\Pol(A/B,n)[n+1].$9-stacks, together with the compatible-pairs and obstruction-tower formalism of "Shifted Poisson and symplectic structures on derived $n$00-stacks" (Pridham, 2015). Safronov’s "Lectures on shifted Poisson geometry" further organizes the subject around polyvector Maurer–Cartan models, additivity, relative/coisotropic structures, and intersection theorems producing lower-shifted Poisson structures on derived intersections (Safronov, 2017). These works do not define transversal shifted Poisson unfoldings as a standalone notion. This suggests, however, that the later unfolding formalism can be read as a transport-theoretic refinement of an existing equivalence between Poisson data, foliation data, coisotropic data, and Lagrangian thickening data.

7. Applications, scope, and terminological boundaries

The controller framework is realized in several concrete settings. For star-products, the controller becomes the crossed Lie algebra

$n$01

and a flat splitting gives flat transport of star-products, with induced flat connections on Hochschild and cyclic complexes. For BV observables, a flat unfolding gives the corrected transport operator

$n$02

which commutes with the BV differential and transports classical BV cohomology, quantum BV observables, and factorization algebra structures. If observables assemble into a factorization algebra, the transport preserves factorization products. AKSZ transgression is formal and functorial, so a target-side stabilizer $n$03 transgresses to $n$04, and a target flat unfolding transgresses to a flat unfolding of the mapping-stack theory. For the Poisson sigma model, the AKSZ target is $n$05 with Hamiltonian

$n$06

and if the transport anomalies vanish then the Cattaneo–Felder/Kontsevich boundary product is horizontal over the parameter space: $n$07 A particularly clean anomaly-free example is provided by a flat bundle of Lie algebras $n$08, whose dual bundle carries a fiberwise linear Poisson structure and whose Rees enveloping algebra $n$09 gives a flat quantization (Corrêa et al., 7 Jul 2026).

Several misconceptions are thereby excluded. A transversal shifted Poisson unfolding is not simply the deformation theory of $n$10; it is a theory of transport of $n$11 along the base. It is also not identical with nondegeneracy, coisotropicity, or AKSZ transgression taken separately, although it interacts with all of them. Conversely, earlier foundational papers in shifted Poisson geometry provide the necessary Maurer–Cartan, foliation, and Lagrangian apparatus, but they do not themselves define the controller-theoretic notion of transversal unfolding (Pridham, 2015, Safronov, 2017).

A further terminological boundary is supplied by an unrelated statistical literature on Poisson processes with shifted trajectories. In that setting, the problem is the adaptive estimation of a non-homogeneous Poisson intensity from independent trajectories with random shifts, and the resulting inverse problem is a deconvolution or unfolding problem controlled by the shift density and analyzed over Besov balls by Meyer-wavelet thresholding. That use of “shifted” and “unfolding” concerns random misalignment of counting-process intensities, not shifted Poisson structures in derived geometry (Bigot et al., 2011).