nLab complex supermanifold

Idea

There are two different notions that one might mean by “complex supermanifolds”, and both are in use (the terminology is a mess!):

1. In the first sense, complex supermanifolds (alias cs manifolds) generalize the notion of smooth manifolds with their sheaf of smooth complex-valued functions, just as ordinary supermanifolds are a generalization of ordinary manifold with their sheaf of smooth real-valued functions.

   However, considering ordinary smooth manifolds as ringed spaces with either their sheaves of real or complex smooth functions gives two equivalent categories, whereas this is not true in the case of real and complex supermanifolds; the corresponding functor is neither essentially surjective nor fully faithful.

   (For $X$ a complex supermanifold in this sense, the underlying reduced manifold $X_{red}$ is not a complex manifold but just a smooth manifold regarded as a ringed space with structure sheaf taken to be the sheaf of $\mathbb{C}$-valued smooth functions on the ordinary real manifold.)

2. In the second sense, complex supermanifolds are super(complex manifolds), a supergeometric version of complex manifolds.

Smooth complex supermanifolds

A smooth complex supermanifold (alias: cs manifold; c for “complex”, s for “super”) is a ringed space $X = (|X|, O_X)$ such that

• the structure sheaf $O_X$ a sheaf of commutative complex super algebras

• locally $O_X$ is isomorphic to $C^\infty(\mathbb{R}^d) \otimes_{\mathbb{C}} \wedge^\bullet \mathbb{C}^\delta$

The pair $d|\delta$ is known as the superdimension of $X$.

Write cSDiff for the category of complex supermanifolds.

Example

The functor

$$\Pi : \text{real vector bundles} \to SDiff$$

has a complex analogue

$$\Pi : \text{complex vector bundles} \to cSDiff.$$

Let $E \to X$ be a complex vector bundle of rank $\delta$. This gives rise to the complex supermanifold $\Pi E$, in the same way as a real vector bundle gives rise to a real supermanifold: the structure sheaf is given by sections of the exterior algebra of the dual of $E$.

Remark

The algebra $C^\infty(X) := O_X(X)$ does not in general have a $\mathbb{C}$-antilinear involution $\bar{-} : C^\infty(X) \to C^\infty(X)$ but there does exist a canonical complex conjugation on the quotient $C^\infty(X)$ by the ideal of nilpotent sections, which is $C^\infty(X_{red}; \mathbb{C})$. So on a complex supermanifold we have complex conjugation only on the reduced manifold.

As for ordinary supermanifolds (and with same proof as in the real case), we have the following two statements.

Theorem

1. Every complex supermanifold is isomorphic to one of the form $\Pi E$.

2. The functor $C^\infty$ is fully faithful:

   $$cSDiff(X,Y) \simeq ComplexSuperAlg(C^\infty(Y), C^\infty(X)).$$

Remark

It turns out that a $\mathbb{C}$-super algebra homomorphism

$$\phi : C^\infty(Y) \to C^\infty(X)$$

automatically satisfies

$$\phi_{red}(\overline{f_{red}}) = \overline{\phi_{red}(f_{red})}.$$

Example

Define the complex supermanifold $\mathbb{R}_{cs}^{d|\delta}$ as $\mathbb{R}^d$ with structure sheaf $U\mapsto C^\infty(U) \otimes_{\mathbb{C}} \wedge^\bullet \mathbb{C}^\delta$.

Then for $S$ an arbitrary complex supermanifold we have

$$\mathbb{R}_{cs}^{d|\delta}(S) = cSDiff(S, \mathbb{R}_{cs}^{d|\delta}) = \{ (x_1, \cdots, x_d, \theta_1, \cdots, \theta_{\delta})\},$$

where $x_i \in C^\infty(S)^{ev}$, $\theta_j \in C^\infty(S)^{odd}$, and $(x_i)_{red}$ is real: $\overline{(x_i)_{red}} = (x_i)_{red} \}$.

Example

For

$$\mathbb{R}^{2|1}(S) = \{ (x,y,\theta) | x,y \in C^\infty(S)^{ev}, \theta \in C^\infty(S)^{odd}; x,y real \}$$

we shall write

$$\simeq \{ (z,\bar z, \theta) | z, \bar z \in C^\infty(S)^{ev}; \overline{z_{red}} = (\overline z )_{red} \}.$$

The special case of odd dimension 1

The following observation will be used in examples constructed below.

Example

Consider real or complex smooth supermanifolds of superdimension $n|1$. A morphism of such supermanifolds (real or complex) induces a morphism of their algebras of functions, in particular, a morphism of their odd degrees as modules over even degrees. Since the odd dimension is 1, the odd degrees contain precisely the module sections $\Gamma(L^*)$ of the dual of the line bundle $L$ provided by Batchelor's theorem. By the smooth Serre–Swan theorem, such morphisms are induced by morphisms of the corresponding line bundles. Thus, the category of smooth supermanifolds of odd dimension 1 is equivalent to the category of line bundles over smooth manifolds.

Comparison of real and complex smooth supermanifolds

Using , we work with the corresponding commutative superalgebras over real or complex numbers, respectively.

The complexification functor yields a faithful functor from real to complex smooth supermanifolds.

The complexification functor is not full.

Example

Maps of smooth real supermanifolds $\mathbf{R}^{0|1}\to\mathbf{R}^{0|1}$ can be identified with homomorphisms of real algebras $\Lambda_{\mathbf{R}}\mathbf{R}\to\Lambda_{\mathbf{R}}\mathbf{R}$, which themselves can be identified with elements of $\mathbf{R}$ by restricting to the odd part.

Maps of smooth complex supermanifolds $\mathbf{R}^{0|1}\to\mathbf{R}^{0|1}$ can be identified with homomorphisms of complex algebras $\Lambda_{\mathbf{C}}\mathbf{C}\to\Lambda_{\mathbf{C}}\mathbf{C}$, which themselves can be identified with elements of $\mathbf{C}$ by restricting to the odd part.

The complexification functor maps $\mathbf{R}\to\mathbf{C}$ via the canonical inclusion. Accordingly, a morphism of smooth complex supermanifolds $\mathbf{R}^{0|1}\to\mathbf{R}^{0|1}$ given by an element $a\in\mathbf{C}\setminus\mathbf{R}$ does not come from a morphism of smooth real supermanifolds.

Example

Consider real or complex smooth supermanifolds of superdimension $1|1$ whose reduced part is a circle. By , there are exactly two such real supermanifolds, given by the two nonisomorphic real line bundles over $S^1$: the trivial line bundle (which gives $S^1\times\mathbf{R}^{0|1}$) and the Möbius line bundle. There is exactly one such complex supermanifold, given by the trivial complex line bundle over $S^1$. The complexification functor makes the two real supermanifolds isomorphic. Indeed, such isomorphisms are in bijection with smooth paths from $1$ to $-1$ in $\mathbf{C}^\times$. No such path factors through $\mathbf{R}^\times$, so none of these isomorphisms is in the image of the complexification functor.

The complexification functor is not essentially surjective.

Example

Consider real or complex smooth supermanifolds of superdimension $2|1$ whose reduced part is a 2-sphere. By , there is exactly one such real supermanifold, given by the trivial real line bundle over $S^2$, which yields $S^2\times\mathbf{R}^{0|1}$. The complex supermanifolds are classified by complex line bundles over $S^2$, which themselves are classified by $H^2(S^2,\mathbf{Z})\cong\mathbf{Z}$. The complexification functor sends the real supermanifold $S^2\times\mathbf{R}^{0|1}$ to its complex counterpart $S^2\times\mathbf{R}^{0|1}$. Thus, every complex supermanifold classified by $n\in\mathbf{Z}\setminus\{0\}$ is not in the essential image of the complexification functor.

Related concepts

• supermanifold, complex manifold

• super Riemann surface

References

• Pierre Deligne, John Morgan, Section 4.8 in: Notes on supersymmetry, in Quantum Fields and Strings, A course for mathematicians, vol 1, Amer. Math. Soc. (1999) 41-98 [web version, pdf, pdf]