Supermanifold¶

In mathematics and mathematical physics, supermanifolds are generalizations of manifolds in which the algebra of functions includes both commuting and anticommuting variables.

Version: v1 · 2026-09-28 · History
Domain-specific #: 12376
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Supergeometry, Differential Geometry → Mathematics

Core Idea¶

Supermanifold is treated here as the recurring formal models and representations identity summarized by this source-grounded definition: In mathematics and mathematical physics, supermanifolds are generalizations of manifolds in which the algebra of functions includes both commuting and anticommuting variables.

In mathematics and mathematical physics, supermanifolds are generalizations of manifolds in which the algebra of functions includes both commuting and anticommuting variables. In the standard mathematical formulation, a smooth supermanifold is a locally ringed space whose structure sheaf is locally isomorphic to the tensor product of the ring of ordinary smooth functions C^\infty(\mathbb R^p) and a Grassmann algebra \Lambda(\xi_1,\dots,\xi_q) of the anticommuting variables. In complex-analytic and algebraic settings, smooth functions are replaced with holomorphic functions or algebraic functions, respectively.

An ordinary manifold is recovered from a supermanifold as the corresponding reduced space of commuting variables, sometimes called the body or reduced manifold. Supermanifolds provide the basic objects of supergeometry. They were introduced in connection with supersymmetry and are used in areas of mathematical physics such as quantum field theory and string theory, as well as in purely mathematical subjects including the theory of Lie superalgebras and supergroups.

For Supermanifold, the abstraction is narrower than the article's general subject matter: a positive case must preserve In mathematics and mathematical physics, supermanifolds are generalizations of manifolds in which the algebra of functions includes both commuting and anticommuting variables. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in formal models and representations, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — A more concrete coordinate-based formalism, used especially in parts of the physics literature, describes supermanifolds by charts with values in the even and odd parts of a Grassmann algebra.
Constitutive relation — The relationship between this formalism and the locally ringed-space definition requires some care, and is often clarified using the functor of points.
Operating condition — The underlying ordinary manifold |M| , sometimes called the reduced manifold or body, is obtained by quotienting \mathcal O_M by the sheaf of ideals generated by its odd elements.
Recognition evidence — A different, more concrete approach describes a supermanifold in a way analogous to a smooth manifold, except that the model space \mathbb{R}^p is replaced by a model superspace built from the even and odd parts of a Grassmann algebra.
Admissible variation — Its even and odd parts are denoted by \Lambda(V){\bar 0} and \Lambda(V) .
Characteristic consequence — In the terminology used by DeWitt and Rogers, the even elements are called c-numbers and the odd elements a-numbers.
Failure boundary — As in the case of an ordinary manifold, a supermanifold is then described by an atlas of charts with suitably smooth transition functions.

What It Is Not¶

Not the whole field of formal models and representations. The node requires the specific identity stated by In mathematics and mathematical physics, supermanifolds are generalizations of manifolds in which the algebra of functions includes both commuting and anticommuting variables.
Not an over-broad reading. A different, more concrete approach describes a supermanifold in a way analogous to a smooth manifold, except that the model space \mathbb{R}^p is replaced by a model superspace built from the even and odd parts of a Grassmann algebra.
Not an over-broad reading. Unlike a regular manifold, a supermanifold is not entirely composed of a set of points.
Not an over-broad reading. The odd elements of the structure sheaf are nilpotent and anti-commute, so they do not define additional points of the underlying topological space.
Not automatically Stratifold. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Supermanifold applies literally inside formal models and representations wherever the source-defined carrier and relation can be established. Its documented habitats include:

Definition. A more concrete coordinate-based formalism, used especially in parts of the physics literature, describes supermanifolds by charts with values in the even and odd parts of a Grassmann algebra.
Definition. Variants of this definition are used in the smooth, complex-analytic, and algebraic categories.
Definition. In the terminology used by DeWitt and Rogers, the even elements are called c-numbers and the odd elements a-numbers.
Definition. As in the case of an ordinary manifold, a supermanifold is then described by an atlas of charts with suitably smooth transition functions.
Definition. With suitable restrictions on transition functions, the two approaches are equivalent; more generally, the comparison is best understood using the functor of points.
Properties. Instead, one takes the dual point of view that the structure of a supermanifold M is contained in its sheaf O M of "smooth functions".

Outside formal models and representations, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Theory or should be marked as analogy.

Clarity¶

A clear use of Supermanifold names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics and mathematical physics, supermanifolds are generalizations of manifolds in which the algebra of functions includes both commuting and anticommuting variables. The strongest recognition evidence in the frozen account is: A different, more concrete approach describes a supermanifold in a way analogous to a smooth manifold, except that the model space \mathbb{R}^p is replaced by a model superspace built from the even and odd parts of a Grassmann algebra. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification A different, more concrete approach describes a supermanifold in a way analogous to a smooth manifold, except that the model space \mathbb{R}^p is replaced by a model superspace built from the even and odd parts of a Grassmann algebra. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Supermanifold compresses multiple formal models and representations details into a stable diagnostic relation. The source shows both the central mechanism—the relationship between this formalism and the locally ringed-space definition requires some care, and is often clarified using the functor of points.—and the practical consequence—in the terminology used by DeWitt and Rogers, the even elements are called c-numbers and the odd elements a-numbers. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the formal models and representations entities to which the claim applies.
State the relation. Use the source-grounded identity: In mathematics and mathematical physics, supermanifolds are generalizations of manifolds in which the algebra of functions includes both commuting and anticommuting variables.
Check operation and conditions. The underlying ordinary manifold |M| , sometimes called the reduced manifold or body, is obtained by quotienting \mathcal O_M by the sheaf of ideals generated by its odd elements.
Demand recognition evidence. A different, more concrete approach describes a supermanifold in a way analogous to a smooth manifold, except that the model space \mathbb{R}^p is replaced by a model superspace built from the even and odd parts of a Grassmann algebra.
Test variation. Change an implementation or setting while preserving its even and odd parts are denoted by \Lambda(V){\bar 0} and \Lambda(V) .
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Theory.

Knowledge Transfer¶

Within the home domain. Knowledge about Supermanifold transfers literally when a new case preserves the same carrier type, relation, and recognition test. A more concrete coordinate-based formalism, used especially in parts of the physics literature, describes supermanifolds by charts with values in the even and odd parts of a Grassmann algebra. Variants of this definition are used in the smooth, complex-analytic, and algebraic categories.

Beyond the home domain. No canonical parent is asserted for Supermanifold. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

They were introduced in connection with supersymmetry and are used in areas of mathematical physics such as quantum field theory and string theory, as well as in purely mathematical subjects including the theory of Lie superalgebras and supergroups. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In mathematics and mathematical physics, supermanifolds are generalizations of manifolds in which the algebra of functions includes both commuting and anticommuting variables; recognition evidence → A different, more concrete approach describes a supermanifold in a way analogous to a smooth manifold, except that the model space \mathbb{R}^p is replaced by a model superspace built from the even and odd parts of a Grassmann algebra

Applied / In Practice¶

As in the case of an ordinary manifold, a supermanifold is then described by an atlas of charts with suitably smooth transition functions. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Definition; invariant → In mathematics and mathematical physics, supermanifolds are generalizations of manifolds in which the algebra of functions includes both commuting and anticommuting variables; boundary → the case exits the class when a different, more concrete approach describes a supermanifold in a way analogous to a smooth manifold, except that the model space \mathbb{R}^p is replaced by a model superspace built from the even and odd parts of a Grassmann algebra

Structural Tensions¶

T1 — Stable identity versus admissible variation. A different, more concrete approach describes a supermanifold in a way analogous to a smooth manifold, except that the model space \mathbb{R}^p is replaced by a model superspace built from the even and odd parts of a Grassmann algebra. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. Unlike a regular manifold, a supermanifold is not entirely composed of a set of points. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. The odd elements of the structure sheaf are nilpotent and anti-commute, so they do not define additional points of the underlying topological space. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. If M is a supermanifold of dimension (p,q), then the underlying space M inherits the structure of a differentiable manifold whose sheaf of smooth functions is O_M/I , where I is the ideal generated by all odd functions. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. A more concrete coordinate-based formalism, used especially in parts of the physics literature, describes supermanifolds by charts with values in the even and odd parts of a Grassmann algebra. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Supermanifold literally, co-instantiate Theory, or only resemble it?

T6 — Autonomy versus reduction. The relationship between this formalism and the locally ringed-space definition requires some care, and is often clarified using the functor of points. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Supermanifold distinguish that the broader parent Theory leaves together?

Structural–Framed Character¶

Supermanifold is mixed or framed-leaning. Its structural side is the repeatable organization summarized by In mathematics and mathematical physics, supermanifolds are generalizations of manifolds in which the algebra of functions includes both commuting and anticommuting variables. Its framed side is the formal models and representations vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: The underlying ordinary manifold |M| , sometimes called the reduced manifold or body, is obtained by quotienting \mathcal O_M by the sheaf of ideals generated by its odd elements. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Theory. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In mathematics and mathematical physics, supermanifolds are generalizations of manifolds in which the algebra of functions includes both commuting and anticommuting variables. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: A more concrete coordinate-based formalism, used especially in parts of the physics literature, describes supermanifolds by charts with values in the even and odd parts of a Grassmann algebra. The relationship between this formalism and the locally ringed-space definition requires some care, and is often clarified using the functor of points. It further constrains recognition and variation through: The underlying ordinary manifold |M| , sometimes called the reduced manifold or body, is obtained by quotienting \mathcal OM by the sheaf of ideals generated by its odd elements. A different, more concrete approach describes a supermanifold in a way analogous to a smooth manifold, except that the model space \mathbb{R}^p is replaced by a model superspace built from the even and odd parts of a Grassmann algebra.

What is domain-bound. formal models and representations supplies the operative entities, technical vocabulary, warrants, and exceptions that make Supermanifold literal. Its documented scope includes the condition that A more concrete coordinate-based formalism, used especially in parts of the physics literature, describes supermanifolds by charts with values in the even and odd parts of a Grassmann algebra. Another bounded application condition is that Variants of this definition are used in the smooth, complex-analytic, and algebraic categories. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—Its even and odd parts are denoted by \Lambda(V){\bar 0} and \Lambda(V){\bar 1} .—and future graph densification may discover a defensible relation only if it preserves that boundary.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Supermanifold. The reviewed identity is: In mathematics and mathematical physics, supermanifolds are generalizations of manifolds in which the algebra of functions includes both commuting and anticommuting variables. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Neighborhood in Abstraction Space¶

Supermanifold sits in a moderately populated region (47^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Algebraic Structures, Groups & Operators (42 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Theory. The parent omits the specialist differentia. Tell: Can the case establish In mathematics and mathematical physics, supermanifolds are generalizations of manifolds in which the algebra of functions includes both commuting and anticommuting variables?
Stratifold. A stratified topological space equipped with a sheaf of smooth functions and manifold strata satisfying controlled local conditions, used as a geometric model for homology theories. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Exotic R4. A smooth four-manifold homeomorphic but not diffeomorphic to ordinary Euclidean four-space, exposing the unique dimension-four split between topological and smooth equivalence. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Solvmanifold. A homogeneous manifold of a connected solvable Lie group, with compact lattice quotients forming the special convention central to topology and Lie theory. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Supermanifold remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside formal models and representations lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Theory?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Supermanifold (revision 1350705926).
Preserved source candidate: http://www.worldscientific.com/doi/suppl/10.1142/1878/suppl_file/1878_chap01.pdf
Preserved source candidate: https://arxiv.org/abs/hep-th/9205088
Preserved source candidate: http://www.math.ias.edu/QFT/fall/
Preserved source candidate: http://www.map.mpim-bonn.mpg.de/Super_manifolds:_an_incomplete_survey

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.