07.06.28 · representation-theory / lie-algebraic

Supermanifolds as ringed spaces, the functor of points, and super Lie groups

shipped3 tiersLean: none

Anchor (Master): Deligne-Morgan 1999 *Notes on Supersymmetry following Joseph Bernstein* (Quantum Fields and Strings Vol. 1, AMS) §§1-2; Manin *Gauge Field Theory and Complex Geometry* (Springer, 2nd ed. 1997) Ch. 4; Kostant 1977 *Graded manifolds, graded Lie theory, and prequantization* (LNM 570); Batchelor 1979 *The structure of supermanifolds* (Trans. AMS 253); Carmeli-Caston-Fioresi *Mathematical Foundations of Supersymmetry* (EMS, 2011) Ch. 4-7

Intuition Beginner

A physicist writing supersymmetry needs coordinates that anticommute: two of them swapped pick up a minus sign, and each one squared gives zero. You cannot point at a place on a map where such a coordinate has the value 7, because no ordinary number behaves this way. The fix is to stop asking what points are and instead ask what functions live on the space. A supermanifold is defined by its ring of functions, and that ring is allowed to contain these sign-flipping symbols.

So you keep an ordinary space underneath — a plain manifold you can draw — and over it you layer a richer collection of functions. Each function is a finite sum: an ordinary part, plus terms carrying one anticommuting symbol, plus terms carrying two, and so on. Because each symbol squares to zero, the sums stop after finitely many terms.

This is the same move that algebraic geometry makes when it builds a space from its ring of functions rather than its points, which is why the scheme picture from earlier is the right mental template here.

Visual Beginner

Picture an ordinary flat sheet of paper as the visible space. Floating just above each point is a small stack of layers: the bottom layer holds ordinary functions, the next layer holds functions multiplied by one ghost coordinate, the layer above by a pair of ghost coordinates, and the top layer by all of them at once. The stack is short because once you use up every ghost coordinate there is nowhere higher to go.

The drawing makes two things plain. First, the visible sheet is the ordinary manifold you started with, and nothing about it changed. Second, all the new structure sits in the tower of functions above it, and the height of that tower counts how many anticommuting directions you added. A space written this way is called $R^{p ∣ q}$ : $p$ ordinary directions in the sheet, $q$ ghost directions in the tower.

Worked example Beginner

Take the smallest interesting case: one ordinary direction and one ghost direction, written $R^{1∣1}$ . The visible space is a single line with coordinate $x$ . The function tower has just two layers, because there is only one ghost symbol, call it $a$ , and $a$ times $a$ is zero.

A general function here is $f (x) = g (x) + h (x) a$ , where $g$ and $h$ are ordinary smooth functions of $x$ . The first piece is what you would see if you forgot the ghost; the second piece is the genuinely new part. Multiply two such functions and collect terms: any product of $a$ with itself drops out, so the answer again has only an ordinary part and a one-ghost part. The tower is closed under multiplication and never grows taller.

Now swap the order of two ghosts in a bigger example with symbols $a$ and $b$ . The rule $a b = - b a$ means the order matters and a sign appears. Setting $b$ equal to $a$ forces $a a = - a a$ , so $a a = 0$ , which is exactly why each ghost squares to zero. These two facts — ordering flips a sign, repetition kills the term — are the entire arithmetic of the new coordinates.

Check your understanding Beginner

Formal definition Intermediate+

A superspace is a $Z_{2}$ -graded locally ringed space $(M, O_{M})$ : a topological space $M$ together with a sheaf $O_{M}$ of supercommutative $R$ -algebras whose stalks are local rings, where supercommutative means $f g = (- 1)^{∣ f ∣∣ g ∣} g f$ for homogeneous sections with parities $∣ f ∣, ∣ g ∣ \in {0, 1}$ . The locally-ringed-space template is the one from 04.02.01, now in the graded category. The model superdomain of dimension $p ∣ q$ is the open set $U \subseteq R^{p}$ carrying the sheaf $$ \mathcal{O}U ;=; C^\infty_U \otimes{\mathbb{R}} \Lambda^\bullet(\xi_1,\dots,\xi_q), $$ the smooth functions on $U$ tensored with the Grassmann (exterior) algebra on $q$ odd generators $ξ_{1}, \dots, ξ_{q}$ satisfying $ξ_{i} ξ_{j} = - ξ_{j} ξ_{i}$ and $ξ_{i}^{2} = 0$ . A section is a finite sum $\sum_{I} f_{I} (x) ξ^{I}$ over multi-indices $I \subseteq {1, \dots, q}$ , with $ξ^{I} = ξ_{i_{1}} \dots ξ_{i_{k}}$ and $f_{I} \in C^{\infty} (U)$ .

Definition (supermanifold). A smooth supermanifold of dimension $p ∣ q$ is a superspace $(M, O_{M})$ in which $M$ is a second-countable Hausdorff space and every point has an open neighbourhood $V$ with an isomorphism of graded-ringed spaces $(V, O_{M} ∣_{V}) ≅ (U, O_{U})$ onto a model superdomain. The reduced manifold $M_{red}$ is the ordinary $p$ -manifold $M$ equipped with $O_{M} / N$ , where $N$ is the nilpotent ideal generated by odd elements; quotienting by $N$ is the body map $O_{M} ↠ C_{M_{red}}^{\infty}$ that forgets the ghosts.

Morphisms. A morphism $Φ : (M, O_{M}) \to (N, O_{N})$ is a continuous map $ϕ : M \to N$ together with a sheaf homomorphism $ϕ^{♯} : O_{N} \to ϕ_{*} O_{M}$ of supercommutative algebras (even, hence parity-preserving), compatible with the local-ring structure. The chart theorem of Leĭtes makes this concrete: a morphism into $R^{m ∣ n}$ is determined by the pullbacks of the $m$ even and $n$ odd coordinate functions, subject only to the body of each even pullback landing where the underlying $ϕ$ sends it. Unlike for ordinary manifolds, a morphism is not determined by its effect on points — the odd coordinates have no point-values to read off, which is the gap the functor of points (below) repairs.

Counterexamples to common slips

A supermanifold is not a set with extra "anticommuting points." There are no such points; $M$ is an honest topological space and all odd data sit in $O_{M}$ . Treating $ξ_{i}$ as a numerical coordinate is the error the ringed-space definition exists to prevent.
The body map $O_{M} \to C_{M_{red}}^{\infty}$ is not an isomorphism: it kills the nilpotent ideal $N$ . A supermanifold of dimension $p ∣ q$ with $q \geq 1$ carries strictly more functions than its reduced manifold.
Two supermanifolds with the same reduced manifold $M$ and the same odd dimension $q$ need not be isomorphic in the holomorphic or algebraic category, even though Batchelor's theorem (next section) makes them isomorphic in the smooth category. The structure sheaf can be a non-split extension.

Key theorem with proof Intermediate+

Theorem (Batchelor 1979). Every smooth supermanifold $(M, O_{M})$ of dimension $p ∣ q$ is isomorphic, as a graded-ringed space, to the split supermanifold $(M, Γ_{M} (Λ^{∙} E^{*}))$ of a rank- $q$ vector bundle $E \to M$ , where $Λ^{∙} E^{*}$ is the exterior bundle of the dual and $Γ_{M}$ its sheaf of smooth sections. The bundle $E$ is determined up to isomorphism by $O_{M}$ .

Proof. Let $N \subset O_{M}$ be the ideal generated by odd sections; its powers give a finite filtration $O_{M} \supset N \supset N^{2} \supset \dots \supset N^{q + 1} = 0$ . The associated graded sheaf $gr O_{M} = ⨁_{k} N^{k} / N^{k + 1}$ is the split model: $N / N^{2}$ is locally free of rank $q$ over $C_{M}^{\infty} = O_{M} / N$ , defining a rank- $q$ bundle $E^{*} \to M$ , and $N^{k} / N^{k + 1} ≅ Λ^{k} E^{*}$ . So $gr O_{M} ≅ Γ_{M} (Λ^{∙} E^{*})$ identically.

It remains to lift the isomorphism from the associated graded to $O_{M}$ itself, that is, to split the filtration. Cover $M$ by superdomains $V_{α}$ on which $O_{M} ∣_{V_{α}} ≅ Γ (Λ^{∙} E^{*}) ∣_{V_{α}}$ is split, with chosen splitting maps $s_{α} : Γ (Λ^{∙} E^{*}) ∣_{V_{α}} \to O_{M} ∣_{V_{α}}$ . On overlaps the $s_{α}$ differ by automorphisms inducing the identity on $gr$ ; such automorphisms form a sheaf of groups that is unipotent (each acts as identity plus a strictly $N$ -raising part, nilpotent of order $\leq q$ ), hence a sheaf of $C_{M}^{\infty}$ -modules after taking logarithms. A smooth partition of unity ${ρ_{α}}$ subordinate to ${V_{α}}$ averages the local splittings: the obstruction to a global splitting lives in $H^{1} (M, S)$ for a sheaf $S$ of $C_{M}^{\infty}$ -modules, and such sheaves are fine, so $H^{1}$ vanishes. The averaged splitting is a global graded-ringed-space isomorphism $O_{M} ≅ Γ_{M} (Λ^{∙} E^{*})$ . Uniqueness of $E$ follows from $E^{*} ≅ N / N^{2}$ , an invariant of $O_{M}$ . $□$

Bridge. Batchelor's theorem builds toward every smooth computation in supergeometry: it licenses replacing an abstract structure sheaf by sections of an exterior bundle, so that Berezin integration and the Berezinian of 12.19.02 can be set up on the concrete split model. The foundational reason it holds is the fineness of $C_{M}^{\infty}$ -module sheaves — partitions of unity kill the splitting obstruction — and this is exactly the same partition-of-unity mechanism that builds connections and Riemannian metrics on ordinary manifolds. The result generalises the observation that a smooth vector bundle is, up to isomorphism, recoverable from its sheaf of sections. The central insight is that smoothness is what makes supergeometry split: putting these together, in the holomorphic category fine resolutions are unavailable, the $H^{1}$ obstruction is genuine, and non-split supermanifolds (super Riemann surfaces, the moduli of which feed superstring perturbation theory) appear again in complex supergeometry as the objects Batchelor's theorem cannot reach.

Exercises Intermediate+

Exercise 4 (hard, short-answer).

Show that the tangent sheaf $T_{M} = D er (O_{M})$ of derivations of the structure sheaf is a locally free $O_{M}$ -module of rank $p ∣ q$ , and exhibit a local basis on the model superdomain.

Hint

On $R^{p ∣ q}$ a derivation is determined by its values on the coordinates $x^{i}$ and $ξ^{j}$ ; build the partial derivatives.

Answer

On a model superdomain with coordinates $(x^{1}, \dots, x^{p}; ξ^{1}, \dots, ξ^{q})$ , define the even derivations $\partial / \partial x^{i}$ (ordinary partials, extended to annihilate the $ξ$ ) and the odd derivations $\partial / \partial ξ^{j}$ characterised by $\partial ξ^{k} / \partial ξ^{j} = δ_{j}^{k}$ , the Leibniz rule with Koszul signs, and annihilating the $x^{i}$ . Any derivation $D$ is then $D = \sum_{i} D (x^{i}) \partial_{x^{i}} + \sum_{j} D (ξ^{j}) \partial_{ξ^{j}}$ , since these determine $D$ on generators and a derivation is fixed by its values on algebra generators. The coefficients $D (x^{i})$ and $D (ξ^{j})$ are arbitrary sections of the appropriate parity, so ${\partial_{x^{i}}, \partial_{ξ^{j}}}$ is a free $O_{M}$ -basis of rank $p$ even plus $q$ odd. Locally free of rank $p ∣ q$ follows. Rubric: full credit for constructing both families of partials, the spanning argument, and the freeness conclusion.

Exercise 5 (hard, short-answer).

Using Batchelor's theorem, classify the smooth supermanifolds of dimension $p ∣1$ up to isomorphism in terms of ordinary geometric data over the reduced manifold $M$ .

Hint

With $q = 1$ the bundle $E$ has rank $1$ ; ask what a smooth real line bundle over $M$ is up to isomorphism.

Answer

By Batchelor every smooth $p ∣1$ supermanifold is split, $O_{M} ≅ Γ (Λ^{∙} E^{*})$ with $E \to M$ a rank- $1$ real vector bundle (a real line bundle), and $Λ^{∙} E^{*} = R \oplus E^{*}$ since $E^{*}$ has rank $1$ . Hence the isomorphism classes correspond to the isomorphism classes of smooth real line bundles over $M$ , classified by $H^{1} (M, Z /2)$ via the first Stiefel-Whitney class. Over a contractible or simply connected $M$ there is only the split product. Rubric: full credit for reducing to line bundles and naming the $H^{1} (M, Z /2)$ classification.

Advanced results Master

The morphism theory is awkward until it is reorganised by the functor of points. By the Yoneda principle of 01.02.09, a supermanifold $M$ is determined by the functor $\underline{M} : S \mapsto Hom (S, M)$ from the (opposite) category of supermanifolds to sets; an element of $\underline{M} (S)$ is an $S$ -point of $M$ . For the model superdomain $R^{p ∣ q}$ this functor is computed explicitly: an $S$ -point is a tuple of $p$ even and $q$ odd global sections of $O_{S}$ whose even bodies satisfy the open-set constraint. Thus odd coordinates acquire genuine "values" — but only against an auxiliary odd parameter space $S$ , never as numbers. This is the precise content of the physicists' practice of treating $θ$ 's as anticommuting numbers: those numbers are the odd sections of an unnamed parameter superalgebra $O_{S}$ .

The functorial viewpoint comes with the even-rules principle (Deligne-Morgan §2). A construction natural in $S$ and defined using only the even part $O_{S}^{ev}$ — where everything commutes — automatically extends to a construction in the super category, with all Koszul signs forced by naturality. Concretely: to define a super structure (a bracket, a group law, a tensor) it suffices to give the rule on $S$ -points using a large auxiliary Grassmann algebra and ordinary commutative algebra, then invoke naturality to descend. The sign ambiguities that plague coordinate computations are resolved once and for all, because there is exactly one extension compatible with all base changes $S^{'} \to S$ . This is the formal engine behind the manipulations of 12.19.02.

Super Lie groups are group objects in the category of supermanifolds: a supermanifold $G$ with morphisms $m : G \times G \to G$ , $i : G \to G$ , $e : pt \to G$ satisfying the group axioms as commutative diagrams, equivalently a functor $S \mapsto \underline{G} (S)$ valued in ordinary groups. The reduced manifold $G_{0} = G_{red}$ is an ordinary Lie group, and the tangent superspace at the identity is a Lie superalgebra $g$ in the sense of 07.06.27, with even part $g_{\overset{ˉ}{0}} = Lie (G_{0})$ .

Theorem (super Harish-Chandra equivalence; Kostant-Koszul). The category of super Lie groups is equivalent to the category of super Harish-Chandra pairs $(G_{0}, g)$ : an ordinary Lie group $G_{0}$ , a Lie superalgebra $g = g_{\overset{ˉ}{0}} \oplus g_{\overset{ˉ}{1}}$ with an isomorphism $g_{\overset{ˉ}{0}} ≅ Lie (G_{0})$ , and an action of $G_{0}$ on $g$ by automorphisms extending the adjoint action of $G_{0}$ on $Lie (G_{0})$ and differentiating to the bracket $[\cdot, \cdot] : g_{\overset{ˉ}{0}} \otimes g \to g$ . The pair packages the discrete/topological content ( $G_{0}$ ) and the infinitesimal content ( $g$ , including the odd directions $g_{\overset{ˉ}{1}}$ that no ordinary group sees) into purely classical data, trading the structure sheaf for a Lie group plus a Lie superalgebra.

The worked example is the super-Poincaré group. Its reduced group is the ordinary Poincaré group $G_{0} = R^{1, 3} ⋊ Spin (1, 3)$ , and its Lie superalgebra is the super-Poincaré algebra of 12.19.01: $g_{\overset{ˉ}{0}}$ the Poincaré algebra, $g_{\overset{ˉ}{1}}$ the supercharges $Q_{α}, \overset{ˉ}{Q}_{\overset{α}{˙}}$ transforming in the spinor representation supplied by the spinor bundle of 03.09.05, with ${Q_{α}, \overset{ˉ}{Q}_{\dot{β}}} = 2 σ_{α \dot{β}}^{μ} P_{μ}$ . The associated homogeneous superspace is super-Minkowski space $R^{4∣4}$ , the supermanifold whose ring of functions is the superfields of 12.19.02. The Berezinian of 12.19.02 reappears here as the transition cocycle of the Berezin (orientation) line bundle $Ber (M) = det (Λ^{p} T_{\overset{ˉ}{0}}^{*}) \otimes det (T_{\overset{ˉ}{1}}^{*})^{- 1}$ , the super-analogue of the determinant line whose sections are what one integrates.

Synthesis. The ringed-space definition is the foundational reason supergeometry is geometry and not formal symbol-pushing: a supermanifold is a space whose ring of functions carries odd elements, exactly as a scheme of 04.02.01 is a space whose ring of functions carries nilpotents, and this is the central insight that lets the same locally-ringed-space machinery serve both. Batchelor's theorem and the functor of points are dual halves of a single working method — the first concretises the smooth structure sheaf as an exterior bundle, the second makes the odd coordinates operational against auxiliary parameters — and putting these together yields the even-rules principle that fixes every Koszul sign by naturality. The super Harish-Chandra equivalence generalises the Lie-group/Lie-algebra dictionary to the graded world and is exactly what reduces the super-Poincaré group to the classical Poincaré group plus the Lie superalgebra of 07.06.27; this is the precise sense in which the algebraic side 07.06.27 and the physics coordinate side 12.19.02 are the two faces of one geometric object, and the seam between them — which this unit fills — is the supermanifold itself. The pattern recurs in complex supergeometry, where the failure of Batchelor splitting makes super Riemann surfaces and their moduli the genuinely new objects.

Full proof set Master

Batchelor's theorem is proved in full in the Key theorem section. The remaining Master claims are recorded here.

Proposition (Schur-style determinacy: morphisms by coordinates). Let $N$ have a global chart $≅ R^{m ∣ n}$ with even coordinates $y^{1}, \dots, y^{m}$ and odd coordinates $η^{1}, \dots, η^{n}$ . Then for any supermanifold $M$ , a morphism $Φ : M \to N$ is uniquely determined by the data of $m$ even sections $Φ^{♯} (y^{a}) \in O_{M} (M)^{ev}$ and $n$ odd sections $Φ^{♯} (η^{b}) \in O_{M} (M)^{odd}$ , subject only to the constraint that the body of $(Φ^{♯} (y^{a}))$ lands in the image domain. Conversely any such data define a morphism.

Proof. A morphism is a pair $(ϕ, ϕ^{♯})$ with $ϕ^{♯} : O_{N} \to ϕ_{*} O_{M}$ an even algebra-sheaf map. Since $O_{N} = C^{\infty} \otimes Λ^{∙} (η)$ is generated as a supercommutative $R$ -algebra by the $y^{a}$ (whose smooth-function completions are generated under composition with smooth functions, by Hadamard's lemma and the Leĭtes chart theorem) together with the $η^{b}$ , an algebra homomorphism is determined by its values on these generators. Parity-preservation forces $ϕ^{♯} (y^{a})$ even and $ϕ^{♯} (η^{b})$ odd. The body of $ϕ^{♯} (y^{a})$ recovers $ϕ$ on $M_{red}$ , fixing the continuous map; the open-domain constraint is exactly that $ϕ$ maps into the correct chart. Conversely, given such sections, the assignment $f (y) \mapsto f (Φ^{♯} (y))$ (interpreting $f (Φ^{♯} (y))$ by Taylor expansion in the nilpotent parts, which terminates because nilpotents are nilpotent of order $\leq n$ ) plus $η^{b} \mapsto Φ^{♯} (η^{b})$ extends uniquely to an even algebra map, hence a morphism. $□$

Proposition (the functor of points is full and faithful). The assignment $M \mapsto \underline{M} = Hom (-, M)$ is a fully faithful functor from supermanifolds to functors on the category of supermanifolds; a natural transformation $\underline{M} \to \underline{N}$ is induced by a unique morphism $M \to N$ .

Proof. This is the Yoneda lemma of 01.02.09 applied in the category $SMan$ : for any category $C$ and objects $M, N$ , the map $Hom_{C} (M, N) \to Nat (\underline{M}, \underline{N})$ sending $f$ to post-composition $f \circ (-)$ is a bijection, with inverse evaluating a natural transformation at the identity $id_{M} \in \underline{M} (M)$ . No special feature of supermanifolds is used beyond their forming a category; the content is that the representable-functor embedding loses nothing, so computing with $S$ -points for all $S$ is equivalent to working with the supermanifolds themselves. $□$

Proposition (super Harish-Chandra reconstruction). Given a super Harish-Chandra pair $(G_{0}, g)$ , the structure sheaf $O_{G} = H o m_{U (g_{\overset{ˉ}{0}})} (U (g), C_{G_{0}}^{\infty})$ defines a super Lie group $G$ with $G_{red} = G_{0}$ and $Lie (G) = g$ , and every super Lie group arises this way; the construction is inverse to $G \mapsto (G_{red}, Lie (G))$ .

Proof sketch. The enveloping superalgebra $U (g)$ is a free right $U (g_{\overset{ˉ}{0}})$ -module on $Λ^{∙} g_{\overset{ˉ}{1}}$ by the super-PBW theorem of 07.06.27. The $C_{G_{0}}^{\infty}$ -module of $U (g_{\overset{ˉ}{0}})$ -linear maps from $U (g)$ is therefore locally $C_{G_{0}}^{\infty} \otimes Λ^{∙} g_{\overset{ˉ}{1}}^{*}$ , the split model of dimension $dim G_{0} ∣ dim g_{\overset{ˉ}{1}}$ ; the $G_{0}$ -action and the bracket assemble it into a sheaf of supercommutative Hopf algebras, whose co-operations are the group morphisms $m, i, e$ . The left-invariant derivations recover $g$ , and reducing modulo nilpotents recovers $G_{0}$ . Functoriality of $U (-)$ and of $C^{\infty} (-)$ makes the two passages mutually inverse on morphisms. $□$

Connections Master

Lie superalgebras 07.06.27 are the infinitesimal object this unit globalises. The graded bracket and super-Jacobi identity defined there are exactly the structure on $g = Lie (G)$ for a super Lie group $G$ ; the super-PBW theorem proved there is the engine of the Harish-Chandra reconstruction, supplying the free-module decomposition $U (g) ≅ U (g_{\overset{ˉ}{0}}) \otimes Λ^{∙} g_{\overset{ˉ}{1}}$ that produces the split structure sheaf. This unit is the geometric successor: it is where the odd part $g_{\overset{ˉ}{1}}$ , which has no ordinary group integrating it, finds its integrated home as the odd tangent directions of a supermanifold.

Schemes and locally ringed spaces 04.02.01 supply the template. A supermanifold is a locally ringed space whose structure sheaf is supercommutative, the $Z_{2}$ -graded refinement of the commutative structure sheaf of a scheme; the body map $O_{M} \to C_{M_{red}}^{\infty}$ is the super-analogue of quotienting a scheme by its nilradical. The same "space = its functions" philosophy that lets a scheme carry nilpotents lets a supermanifold carry odd elements, and the functor-of-points formalism is imported directly from the scheme-theoretic one, where it was first developed.

The spinor bundle 03.09.05 is what makes the worked example geometric rather than formal. The odd directions $g_{\overset{ˉ}{1}}$ of the super-Poincaré algebra are the supercharges $Q_{α}, \overset{ˉ}{Q}_{\overset{α}{˙}}$ , which transform in the spinor representation of $Spin (1, 3)$ ; the spinor bundle provides exactly this representation as the fibre of the odd tangent bundle of super-Minkowski space $R^{4∣4}$ . Without the spin geometry there, the odd coordinates would be a bare Grassmann algebra with no Lorentz content.

Category, functor, Yoneda, and adjunction 01.02.09 underwrite the functor-of-points statement. The full faithfulness of $M \mapsto \underline{M}$ is the Yoneda lemma in $SMan$ , and the even-rules principle is a naturality argument: a construction defined functorially on the even-points subfunctor extends uniquely, because there is exactly one extension compatible with all base changes. The categorical machinery is not decoration here; it is the only rigorous way to give odd coordinates "values."

The physics superspace 12.19.02 is the coordinate face of this same object. Its $R^{4∣4}$ with coordinates $(x^{μ}, θ^{α}, \overset{ˉ}{θ}^{\overset{α}{˙}})$ , its superfields, its Berezin integral, and its Berezinian are, respectively, the worked-example supermanifold of this unit, its structure-sheaf sections, the integration against the Berezin line bundle, and the transition cocycle of that bundle. This unit supplies the ringed-space foundation that the coordinate calculus there tacitly assumes, closing the seam between the algebraic 07.06.27 and physical 12.19.02 developments.

Historical & philosophical context Master

The systematic theory of supermanifolds emerged in the mid-1970s from two directions at once. Berezin and Leĭtes gave the structure-sheaf definition in 1975 (Supermanifolds, Soviet Math. Dokl. 16, 1218-1222) ^{[Berezin-Leĭtes 1975]}, building on Berezin's earlier calculus of anticommuting variables; Kostant arrived independently at "graded manifolds" with a Lie-theoretic emphasis in his 1977 lectures (Graded manifolds, graded Lie theory, and prequantization, LNM 570) ^{[Kostant 1977]}, where the equivalence with what are now called super Harish-Chandra pairs is essentially established. Batchelor's 1979 splitting theorem (The structure of supermanifolds, Trans. AMS 253) ^{[Batchelor 1979]} settled the smooth classification by reducing it to vector bundles, and Manin's Gauge Field Theory and Complex Geometry ^{[Manin 1997]} made the ringed-space and complex-analytic theory canonical, introducing the Berezinian line bundle as the supergeometric determinant. The text that turned this apparatus toward a mathematician's account of supersymmetry is Deligne and Morgan's Notes on Supersymmetry following Joseph Bernstein ^{[Deligne-Morgan 1999]}, written precisely to install the foundations the physics literature presumed.

The philosophical content is a decisive commitment to the functor-of-points stance. Physicists had long manipulated anticommuting coordinates $θ$ as though they were numbers, and the manipulations gave correct answers, yet there is no set on which a $θ$ takes numerical values. The resolution — that the space is its sheaf of functions, and odd coordinates are detected only against auxiliary odd parameters — is the same conceptual move Grothendieck made for schemes, here imported wholesale: meaning is relational, carried by maps into the object from all probe objects $S$ , not by an intrinsic set of points. The even-rules principle is the payoff: it certifies that the physicists' sign-laden computations are not heuristics but theorems, because naturality leaves no freedom. Supergeometry thus stands as a case study in how a rigorous foundation can ratify, rather than overturn, a working physical formalism, by relocating its objects from a category of sets to a category of functors.

Bibliography Master

@incollection{delignemorgan1999susy,
  author    = {Deligne, Pierre and Morgan, John W.},
  title     = {Notes on Supersymmetry (following Joseph Bernstein)},
  booktitle = {Quantum Fields and Strings: A Course for Mathematicians, Vol. 1},
  editor    = {Deligne, P. and Etingof, P. and Freed, D. S. and others},
  publisher = {American Mathematical Society},
  year      = {1999},
  pages     = {41--97}
}

@article{batchelor1979,
  author  = {Batchelor, Marjorie},
  title   = {The structure of supermanifolds},
  journal = {Transactions of the American Mathematical Society},
  volume  = {253},
  pages   = {329--338},
  year    = {1979}
}

@incollection{kostant1977,
  author    = {Kostant, Bertram},
  title     = {Graded manifolds, graded {L}ie theory, and prequantization},
  booktitle = {Differential Geometrical Methods in Mathematical Physics},
  series    = {Lecture Notes in Mathematics},
  volume    = {570},
  publisher = {Springer},
  year      = {1977},
  pages     = {177--306}
}

@book{manin1997gauge,
  author    = {Manin, Yuri I.},
  title     = {Gauge Field Theory and Complex Geometry},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {289},
  edition   = {2nd},
  publisher = {Springer},
  year      = {1997}
}

@article{berezinleites1975,
  author  = {Berezin, Felix A. and Le\u{\i}tes, Dimitry A.},
  title   = {Supermanifolds},
  journal = {Soviet Mathematics Doklady},
  volume  = {16},
  pages   = {1218--1222},
  year    = {1975}
}

@book{carmeli2011,
  author    = {Carmeli, Claudio and Caston, Lauren and Fioresi, Rita},
  title     = {Mathematical Foundations of Supersymmetry},
  series    = {EMS Series of Lectures in Mathematics},
  publisher = {European Mathematical Society},
  year      = {2011}
}

@book{varadarajan2004,
  author    = {Varadarajan, V. S.},
  title     = {Supersymmetry for Mathematicians: An Introduction},
  series    = {Courant Lecture Notes},
  volume    = {11},
  publisher = {American Mathematical Society},
  year      = {2004}
}

Prerequisites

07.06.27
04.02.01
03.09.05
01.02.09

Tier anchors

beginner: Deligne-Morgan 1999 *Notes on Supersymmetry following Joseph Bernstein* (in *Quantum Fields and Strings*, AMS) §1.1-1.2 (the informal superdomain $\mathbb{R}^{p|q}$ and its functions); Varadarajan *Supersymmetry for Mathematicians* (Courant Lecture Notes 11) Ch. 4 introduction
intermediate: Deligne-Morgan 1999 §§1-2 (the ringed-space definition, morphisms, the functor of points); Leĭtes 1980 *Introduction to the theory of supermanifolds* (Russ. Math. Surveys 35) §1-2; Varadarajan *Supersymmetry for Mathematicians* (CLN 11) Ch. 4
master: Deligne-Morgan 1999 *Notes on Supersymmetry following Joseph Bernstein* (Quantum Fields and Strings Vol. 1, AMS) §§1-2; Manin *Gauge Field Theory and Complex Geometry* (Springer, 2nd ed. 1997) Ch. 4; Kostant 1977 *Graded manifolds, graded Lie theory, and prequantization* (LNM 570); Batchelor 1979 *The structure of supermanifolds* (Trans. AMS 253); Carmeli-Caston-Fioresi *Mathematical Foundations of Supersymmetry* (EMS, 2011) Ch. 4-7

References

Deligne-Morgan — Notes on Supersymmetry following Joseph Bernstein · P. Deligne and J. W. Morgan, in *Quantum Fields and Strings: A Course for Mathematicians*, Vol. 1 (AMS, 1999), §1 (the ringed-space definition of a supermanifold, the model $\mathbb{R}^{p|q}$, morphisms and the chart theorem), §2 (the functor of points / even-rules principle, super Lie groups, the super Harish-Chandra pair $(G_0,\mathfrak{g})$, the super-Poincaré group and super-Minkowski space $\mathbb{R}^{4|4}$)
Batchelor — The structure of supermanifolds · M. Batchelor, Transactions of the AMS 253 (1979) 329-338: every smooth ($C^\infty$) supermanifold is isomorphic to the split form $(M,\Gamma(\Lambda^\bullet E^\ast))$ associated with a vector bundle $E\to M$; the proof patches local splittings with a smooth partition of unity, and the result fails in the holomorphic and algebraic categories
Kostant — Graded manifolds, graded Lie theory, and prequantization · B. Kostant, in *Differential Geometrical Methods in Mathematical Physics*, Springer Lecture Notes in Mathematics 570 (1977) 177-306: graded (super) manifolds via structure sheaves, graded Lie groups, and the equivalence with super Harish-Chandra pairs $(G_0,\mathfrak{g})$
Manin — Gauge Field Theory and Complex Geometry · Yu. I. Manin, Grundlehren 289, Springer (2nd ed. 1997), Ch. 4 (supermanifolds as ringed spaces, the Berezinian and the orientation/Berezin line bundle, super Grassmannians); the originating systematic treatment alongside Berezin-Leĭtes 1975
Berezin-Leĭtes — Supermanifolds · F. A. Berezin and D. A. Leĭtes, Soviet Math. Dokl. 16 (1975) 1218-1222: the original definition of a supermanifold (graded manifold) via a sheaf of $\mathbb{Z}_2$-graded commutative algebras locally $C^\infty\otimes\Lambda^\bullet$

Estimated time

beginner: 18m
intermediate: 45m
master: 85m