Vector Bundle Supermanifold

• Vector bundle supermanifolds are supermanifolds whose odd directions are organized by vector bundle data, typically realized as the exterior algebra of a locally free sheaf over a reduced manifold.
• Non-split structures arise as deformations of the split models, with cohomological obstructions and graded derivations governing the classification and holonomy phenomena.
• The theory integrates differential, holomorphic, and graded geometric methods, finding applications in homogeneous realizations, parallel transport frameworks, and BV geometry.

Searching arXiv for recent and foundational papers on vector bundle supermanifolds, split supermanifolds, and related bundle-based supergeometry. A vector bundle supermanifold is a supermanifold whose odd directions are organized by vector-bundle data, either exactly in the split sense or, more generally, through constructions that recover supergeometric structure from bundles, connections, higher vector bundles, or graded coverings. In the split case, the structure sheaf is the exterior algebra of a locally free sheaf over the reduced manifold, so the supermanifold is determined by a vector bundle of odd directions; in non-split settings, vector-bundle data continue to control deformation, obstruction, holonomy, mapping-space, and universal-classification phenomena (Kalus, 2014). The topic therefore occupies a central position between sheaf-theoretic supergeometry, bundle-valued differential geometry, homogeneous and holomorphic classification theory, and supergeometric models motivated by parallel transport and field theory (Groeger, 2013).

1. Split models and the basic bundle-theoretic notion

In the standard complex and smooth supergeometric framework, a split supermanifold is one whose structure sheaf is an exterior algebra over a locally free sheaf. One formulation is

$$\mathcal O_M \simeq \mathcal F_M \otimes \wedge \mathcal E_M,$$

where $\mathcal F_M$ is the sheaf of holomorphic functions on the reduced manifold $M$, and $\mathcal E_M$ is a locally free sheaf (Vishnyakova, 2010). Closely related is the split model

$$(M,\mathcal O_{\Lambda E}),\qquad \mathcal O_{\Lambda E}\cong \mathcal O_M\otimes \Lambda E,$$

where $E\to M$ is a holomorphic vector bundle of rank equal to the odd dimension (Kalus, 2014).

This formulation identifies the odd sector of the supermanifold with a vector bundle, so the supermanifold is literally built from bundle data on the reduced manifold. In this sense, $\Pi E := (M,\Gamma(\wedge^\bullet E^*))$ is the standard parity-shifted vector bundle supermanifold model, and it is used as the archetypal example of a supermanifold coming from a vector bundle (Zakharevich, 2017). The split condition is therefore the most direct realization of a vector bundle supermanifold: base manifold plus exterior algebra of odd fibers.

The retract or associated graded construction makes this bundle-theoretic core explicit even for a general supermanifold. If $\mathcal J_M$ is the odd ideal sheaf, then

$$\operatorname{gr}(M,\mathcal O_M)=(M,\operatorname{gr}\mathcal O_M),\qquad \operatorname{gr}\mathcal O_M^p=\mathcal J_M^p/\mathcal J_M^{p+1},$$

and this graded object is split (Vishnyakova, 2010). Thus every supermanifold possesses a canonical split approximation determined by vector-bundle data.

A further structural identification is that, for a split supermanifold $(M,\mathcal E_M)$, the tangent sheaf grading satisfies

$\mathcal F_M$0

so the odd bundle can be recovered from degree $\mathcal F_M$1 derivations of the tangent sheaf (Vishnyakova, 2010). This places the vector bundle not merely in the structure sheaf but also in the graded differential geometry of the supermanifold.

2. Non-split structures as deformations of vector-bundle models

Complex supermanifolds are described as deformations of split models associated with holomorphic vector bundles. In this setup, non-split supermanifold structures arise by deforming the multiplication in the sheaf $\mathcal F_M$2, and the resulting classification is governed by Green’s non-abelian cohomology (Kalus, 2014). The relevant subgroup is

$\mathcal F_M$3

consisting of automorphisms that raise $\mathcal F_M$4-degree by even positive amounts, and isomorphism classes of complex supermanifolds associated to $\mathcal F_M$5 are in bijection with

$\mathcal F_M$6

more precisely with the orbits of $\mathcal F_M$7 acting by conjugation on the pointed set $\mathcal F_M$8 (Kalus, 2014).

The deformation problem can be recast in terms of even derivations of positive $\mathcal F_M$9-degree,

$M$0

and the exponential isomorphism

$M$1

connects the non-abelian and abelian descriptions (Kalus, 2014). For odd dimension $M$2 there are only split structures; for odd dimension $M$3 the classification is essentially abelian; and for odd dimension $M$4 genuinely non-abelian phenomena appear (Kalus, 2014).

For ranks $M$5 and $M$6, and under the assumption

$M$7

the non-abelian cohomology $M$8 can be embedded into a computable subset of the abelian cohomology $M$9 (Kalus, 2014). This does not remove the bundle from the theory; rather, it shows that the entire non-split structure remains organized by the original bundle $\mathcal E_M$0, its derivations, and cohomological obstructions attached to the split model.

A related but more geometric reformulation is given by the graded-covering approach to non-split supermanifolds. For a supermanifold of odd dimension $\mathcal E_M$1, the functor

$\mathcal E_M$2

encodes the data of a non-split supermanifold in an $\mathcal E_M$3-fold vector bundle with extra symmetries, and then recovers it as a graded manifold of degree $\mathcal E_M$4 (Rotkiewicz et al., 2021). This means that non-split supergeometry can be represented by a collection of vector bundles and morphisms between them. In odd dimension $\mathcal E_M$5, the first obstruction class

$\mathcal E_M$6

is identified with the Atiyah class of a skew-symmetric double vector bundle, so the obstruction to splitting becomes geometric vector-bundle data rather than merely an abstract cohomology class (Rotkiewicz et al., 2021).

This suggests that “vector bundle supermanifold” has both a strict meaning and a broader one. Strictly, it denotes a split supermanifold determined by an odd bundle. More broadly, it includes non-split supermanifolds that admit faithful representation by higher vector bundles, graded manifolds, or deformation data attached to a split bundle model.

3. Homogeneous and holomorphic realizations

Homogeneous supergeometry makes the vector-bundle content of split supermanifolds especially explicit. If $\mathcal E_M$7 is a complex Lie supergroup and $\mathcal E_M$8 is a homogeneous supermanifold, then the retract

$\mathcal E_M$9

is split, and its associated homogeneous vector bundle $$(M,\mathcal O_{\Lambda E}),\qquad \mathcal O_{\Lambda E}\cong \mathcal O_M\otimes \Lambda E,$$0 is determined by the $$(M,\mathcal O_{\Lambda E}),\qquad \mathcal O_{\Lambda E}\cong \mathcal O_M\otimes \Lambda E,$$1-module

$$(M,\mathcal O_{\Lambda E}),\qquad \mathcal O_{\Lambda E}\cong \mathcal O_M\otimes \Lambda E,$$2

This identifies the odd bundle directly from the odd isotropy quotient (Vishnyakova, 2010).

For compact homogeneous spaces $$(M,\mathcal O_{\Lambda E}),\qquad \mathcal O_{\Lambda E}\cong \mathcal O_M\otimes \Lambda E,$$3, the associated vector bundle of a split homogeneous supermanifold is a homogeneous subbundle of a trivial homogeneous bundle $$(M,\mathcal O_{\Lambda E}),\qquad \mathcal O_{\Lambda E}\cong \mathcal O_M\otimes \Lambda E,$$4 determined by a finite-dimensional $$(M,\mathcal O_{\Lambda E}),\qquad \mathcal O_{\Lambda E}\cong \mathcal O_M\otimes \Lambda E,$$5-module $$(M,\mathcal O_{\Lambda E}),\qquad \mathcal O_{\Lambda E}\cong \mathcal O_M\otimes \Lambda E,$$6, and conversely any such homogeneous subbundle comes from a split homogeneous supermanifold (Vishnyakova, 2010). Thus the vector bundle supermanifold is not an abstract locally ringed space alone; it can be realized concretely as homogeneous bundle data inside a trivial bundle.

The relation between holomorphic functions and vector-bundle structure is equally direct. Since

$$(M,\mathcal O_{\Lambda E}),\qquad \mathcal O_{\Lambda E}\cong \mathcal O_M\otimes \Lambda E,$$7

global holomorphic functions decompose as

$$(M,\mathcal O_{\Lambda E}),\qquad \mathcal O_{\Lambda E}\cong \mathcal O_M\otimes \Lambda E,$$8

Therefore the existence of nonconstant holomorphic functions is controlled by the exterior powers of the odd bundle (Vishnyakova, 2010). For homogeneous split supermanifolds, the criterion is representation-theoretic: if there are no non-trivial $$(M,\mathcal O_{\Lambda E}),\qquad \mathcal O_{\Lambda E}\cong \mathcal O_M\otimes \Lambda E,$$9-submodules in the relevant odd isotropy quotient, then $E\to M$0 (Vishnyakova, 2010).

On $E\to M$1, where every vector bundle splits as a sum of line bundles

$E\to M$2

the classification of supermanifold structures becomes explicit in terms of cohomology of line bundles and compact-group orbits (Kalus, 2014). This makes $E\to M$3 a model case in which the bundle-theoretic nature of low-dimensional supermanifolds is completely computable.

4. Bundle-based differential geometry and parallel transport

Vector-bundle supermanifolds are not only classificatory objects; they also support differential-geometric constructions analogous to those on ordinary vector bundles. In the sheaf-theoretic supermanifold setting, a super vector bundle $E\to M$4 is a sheaf of locally free $E\to M$5-modules with $E\to M$6-grading

$E\to M$7

and the super tangent bundle is the sheaf of superderivations

$E\to M$8

(Groeger, 2013). An $E\to M$9-connection is then an even map on the pulled-back bundle $\Pi E := (M,\Gamma(\wedge^\bullet E^*))$0 satisfying the Leibniz rule, and it provides the basis for supergeometric parallel transport along $\Pi E := (M,\Gamma(\wedge^\bullet E^*))$1-paths (Groeger, 2013).

For a superpath $\Pi E := (M,\Gamma(\wedge^\bullet E^*))$2, parallel sections satisfy

$\Pi E := (M,\Gamma(\wedge^\bullet E^*))$3

which becomes a linear ODE

$\Pi E := (M,\Gamma(\wedge^\bullet E^*))$4

with solution given by a path-ordered exponential (Groeger, 2013). Parallel transport composes under concatenation,

$\Pi E := (M,\Gamma(\wedge^\bullet E^*))$5

and inverts under path reversal (Groeger, 2013). On closed $\Pi E := (M,\Gamma(\wedge^\bullet E^*))$6-loops, the gauge-invariant trace

$\Pi E := (M,\Gamma(\wedge^\bullet E^*))$7

is the super Wilson loop (Groeger, 2013).

Holonomy is then defined at an $\Pi E := (M,\Gamma(\wedge^\bullet E^*))$8-point $\Pi E := (M,\Gamma(\wedge^\bullet E^*))$9 by

$\mathcal J_M$0

and the theory includes a supergeometric Ambrose–Singer theorem and a holonomy principle (Groeger, 2013). In particular, the Lie algebra is generated by curvature transported back to the base point, and parallel sections are characterized by invariance under the holonomy functor (Groeger, 2013). This situates vector bundles over supermanifolds within a fully developed transport-and-curvature formalism.

A distinct but related construction concerns $\mathcal J_M$1-parallel transport. For a $\mathcal J_M$2-graded vector bundle $\mathcal J_M$3 over a manifold $\mathcal J_M$4, there is a natural bijection between $\mathcal J_M$5 parallel transport on $\mathcal J_M$6 and even connections on $\mathcal J_M$7 (Dumitrescu, 2010). The proof passes through the odd tangent bundle $\mathcal J_M$8, whose functions are differential forms,

$\mathcal J_M$9

and uses odd-trivial connections on the pullback bundle $$\operatorname{gr}(M,\mathcal O_M)=(M,\operatorname{gr}\mathcal O_M),\qquad \operatorname{gr}\mathcal O_M^p=\mathcal J_M^p/\mathcal J_M^{p+1},$$0 (Dumitrescu, 2010). Although the base is an ordinary manifold in that theorem, the argument is explicitly supergeometric and shows how bundle data over $$\operatorname{gr}(M,\mathcal O_M)=(M,\operatorname{gr}\mathcal O_M),\qquad \operatorname{gr}\mathcal O_M^p=\mathcal J_M^p/\mathcal J_M^{p+1},$$1 encode superpath transport.

There also exists a more elementary bundle-based approximation to supergeometry in which one fixes an ordinary smooth manifold $$\operatorname{gr}(M,\mathcal O_M)=(M,\operatorname{gr}\mathcal O_M),\qquad \operatorname{gr}\mathcal O_M^p=\mathcal J_M^p/\mathcal J_M^{p+1},$$2 and an auxiliary vector bundle $$\operatorname{gr}(M,\mathcal O_M)=(M,\operatorname{gr}\mathcal O_M),\qquad \operatorname{gr}\mathcal O_M^p=\mathcal J_M^p/\mathcal J_M^{p+1},$$3, and defines the super tangent bundle

$$\operatorname{gr}(M,\mathcal O_M)=(M,\operatorname{gr}\mathcal O_M),\qquad \operatorname{gr}\mathcal O_M^p=\mathcal J_M^p/\mathcal J_M^{p+1},$$4

The even part is $$\operatorname{gr}(M,\mathcal O_M)=(M,\operatorname{gr}\mathcal O_M),\qquad \operatorname{gr}\mathcal O_M^p=\mathcal J_M^p/\mathcal J_M^{p+1},$$5, the odd part is $$\operatorname{gr}(M,\mathcal O_M)=(M,\operatorname{gr}\mathcal O_M),\qquad \operatorname{gr}\mathcal O_M^p=\mathcal J_M^p/\mathcal J_M^{p+1},$$6, and one imposes

$$\operatorname{gr}(M,\mathcal O_M)=(M,\operatorname{gr}\mathcal O_M),\qquad \operatorname{gr}\mathcal O_M^p=\mathcal J_M^p/\mathcal J_M^{p+1},$$7

This framework develops super differential forms, super connections, curvature, super metrics, and a Levi-Civita super connection, but it does not construct a supermanifold in the usual Berezin–Leites sense (Boroojerdian, 2020). It is therefore best understood as vector-bundle-based graded differential geometry rather than a full vector bundle supermanifold.

5. Mapping supermanifolds and intrinsic vector-bundle-type structure

Mapping supermanifolds provide a particularly direct example of a space that is “vector-bundle-type” in the supergeometric sense. For a smooth manifold $$\operatorname{gr}(M,\mathcal O_M)=(M,\operatorname{gr}\mathcal O_M),\qquad \operatorname{gr}\mathcal O_M^p=\mathcal J_M^p/\mathcal J_M^{p+1},$$8, the supermanifold of maps from $$\operatorname{gr}(M,\mathcal O_M)=(M,\operatorname{gr}\mathcal O_M),\qquad \operatorname{gr}\mathcal O_M^p=\mathcal J_M^p/\mathcal J_M^{p+1},$$9 to $(M,\mathcal E_M)$0 is described intrinsically by the isomorphism

$(M,\mathcal E_M)$1

where $(M,\mathcal E_M)$2 is the base point map and $(M,\mathcal E_M)$3 is parity reversal (Yan, 29 Jun 2025).

A morphism $(M,\mathcal E_M)$4 has pullback

$(M,\mathcal E_M)$5

with $(M,\mathcal E_M)$6, $(M,\mathcal E_M)$7, and $(M,\mathcal E_M)$8 linear (Yan, 29 Jun 2025). The homomorphism condition forces

1. $(M,\mathcal E_M)$9 and $\mathcal F_M$00 to be linearly dependent in $\mathcal F_M$01,
2. $\mathcal F_M$02 for all $\mathcal F_M$03, so the only surviving odd data are dependent tangent vectors (Yan, 29 Jun 2025).

These dependent odd directions are encoded by the decomposable bivector bundle. The fiber over $\mathcal F_M$04 consists of bivectors $\mathcal F_M$05 where $\mathcal F_M$06 are linearly dependent, including the zero bivector (Yan, 29 Jun 2025). Because parity reversal makes the fiber purely odd, the resulting supermanifold is of vector-bundle type (Yan, 29 Jun 2025).

The reduced fiber

$\mathcal F_M$07

has dimension

$\mathcal F_M$08

so the reduced manifold has total dimension $\mathcal F_M$09 (Yan, 29 Jun 2025). The paper emphasizes that this parameterization is intrinsic and does not require a connection, unlike connection-dependent approaches involving Hessians and auxiliary fields (Yan, 29 Jun 2025). This gives an explicit modern example of a supermanifold canonically built from bundle data over an ordinary manifold.

6. Universal and higher bundle classifications

The classification of super vector bundles themselves motivates generalized classifying spaces and higher-categorical analogues. A super vector bundle of rank $\mathcal F_M$10 over a supermanifold $\mathcal F_M$11 is a sheaf $\mathcal F_M$12 of $\mathcal F_M$13-graded $\mathcal F_M$14-modules that is locally free of rank $\mathcal F_M$15 (Afshari et al., 2018). To classify such bundles, the paper on universal super vector bundles introduces the $\mathcal F_M$16-grassmannian, a new classifying space designed to retain odd information that ordinary supergrassmannians do not capture (Afshari et al., 2018).

Over the $\mathcal F_M$17-grassmannian $\mathcal F_M$18, the canonical super vector bundle

$\mathcal F_M$19

is obtained by gluing local trivial bundles

$\mathcal F_M$20

via explicit transition morphisms (Afshari et al., 2018). For any finite-type super vector bundle $\mathcal F_M$21, a Gauss supermap produces a classifying morphism

$\mathcal F_M$22

and the pullback theorem states

$\mathcal F_M$23

This is the universal-bundle statement in supergeometry (Afshari et al., 2018).

A higher-categorical extension is provided by super $\mathcal F_M$24-vector bundles. These are modeled by super algebras, graded bimodules, and parity-preserving intertwiners, assembled by bicategorical descent into a $\mathcal F_M$25-stack (Kristel et al., 2021). An object is a quadruple

$\mathcal F_M$26

where $\mathcal F_M$27 is a super algebra bundle over $\mathcal F_M$28, $\mathcal F_M$29 is an invertible bimodule bundle over $\mathcal F_M$30, and $\mathcal F_M$31 is an invertible even intertwiner over $\mathcal F_M$32 satisfying coherence over $\mathcal F_M$33 (Kristel et al., 2021). These $\mathcal F_M$34-vector bundles contain bundle gerbes and ordinary algebra bundles as full sub-bicategories, thereby unifying structures previously treated separately (Kristel et al., 2021).

A plausible implication is that the phrase “vector bundle supermanifold” now spans a hierarchy. At the lowest level it denotes a split supermanifold modeled on one vector bundle. At higher levels it extends to graded manifolds represented by multiple compatible vector bundles, and to categorified bundle theories whose fibers are super $\mathcal F_M$35-vector spaces rather than ordinary super vector spaces.

7. Cotangent-type total spaces, BV geometry, and localization

The total space of a vector bundle over a supermanifold can itself carry important supergeometric structures. A prominent example is the BV supermanifold

$\mathcal F_M$36

whose structure sheaf is

$\mathcal F_M$37

(Noja, 2022). In local coordinates $\mathcal F_M$38, the fiber coordinates satisfy

$\mathcal F_M$39

and the transition functions display the standard odd symplectic cotangent-type geometry (Noja, 2022).

Globally, the cotangent sheaf of $\mathcal F_M$40 sits in the short exact sequence

$\mathcal F_M$41

Thus the $\mathcal F_M$42-forms on the BV supermanifold are an extension of the pullback cotangent bundle by the pullback tangent bundle (Noja, 2022). In the holomorphic category, this sequence splits if and only if the super Atiyah class $\mathcal F_M$43 vanishes, and this is equivalent to the existence of a holomorphic affine connection (Noja, 2022). The result links the geometry of a cotangent-type vector bundle supermanifold to the characteristic non-split geometry of the base.

Localization theory on supermanifolds also uses vector-bundle supermanifolds as basic examples. The standard model $\mathcal F_M$44 appears explicitly, while the general localization formula is expressed in terms of the normal bundle

$\mathcal F_M$45

of a vanishing subsupermanifold $\mathcal F_M$46 (Zakharevich, 2017). If $\mathcal F_M$47 is an odd vector field with non-degenerate vanishing subsupermanifold $\mathcal F_M$48, then the integral of a $\mathcal F_M$49-invariant compactly supported density localizes to $\mathcal F_M$50 and is expressed using the Berezinian of the induced odd automorphism of $\mathcal F_M$51 (Zakharevich, 2017). The local normal form of an even function near a non-degenerate critical subsupermanifold is controlled by the quadratic structure on the normal bundle, including the odd symplectic part (Zakharevich, 2017). This shows that even when a theory is not primarily about vector bundle supermanifolds, its coordinate-free formulation is often bundle-theoretic.

Overall, the theory of vector bundle supermanifolds is the theory of how odd directions, and often the full supergeometry, are encoded by vector bundles or systems of vector bundles. In split geometry this encoding is literal; in deformation theory it is cohomological; in homogeneous and holomorphic settings it is representation-theoretic; in parallel transport it is differential-geometric; in mapping-space constructions it yields intrinsic vector-bundle-type supermanifolds; and in higher and BV settings it extends to $\mathcal F_M$52-fold vector bundles, cotangent-type extensions, and categorified bundles (Kalus, 2014).