nLab real space

Idea

In the context of real-oriented cohomology theory, notably in KR-theory, by “real space” (“real manifold”) one means a topological space (a manifold) equipped with an action of the cyclic group of order 2 $\mathbb{Z}/2$ (a “non-linear real structure”).

In the context of string theory real spaces appear as orientifold target spacetimes. The involution fixed points here are known as O-planes.

Examples

Real circles

There are three non-equivalent real structures on the circle $S^1$, usually denoted

• $S^{2,0}$ for the trivial involution;

• $S^{1,1}$ for the reflection involution (identifying the two semi-circles);

• $S^{0,2}$ for the antipodal involution (rotation by $\pi$).

Accordingly real-oriented cohomology theory is bigraded in a way modeled on this bigrading.

(This is standard notation, but maybe $S^{1,0}$, $S^{\tfrac{1}{2}, \tfrac{1}{2}}, S^{0,1}$ would be more suggestive. Indeed the quotients in the first and the last case are actually circles, while in the second case it is the semi-circle.)

Complexified cartesian spaces

The complex $n$-dimensional complexified cartesian space ${\mathbb{C}}^n$ equipped with its conjugative involution is a real space. Explicitly, this involution sends $(z^1,\ldots, z^n)$ to $(\bar{z^1},\ldots, \bar{z^n})$.

Complex projective spaces.

The complex $n$-dimensional complex projective space ${\mathbb{P}}^n_{\mathbb{C}}$ equipped with a conjugation involution is a real space. For each choice of affine chart, the conjugation involution of this chart (which is biholomorphic to ${\mathbb{C}}^n$) extends to a conjugation involution on ${\mathbb{P}}^n_{\mathbb{C}}$. Any two conjugation involutions are ${\mathbb{Z}}/2$-equivariantly diffeomorphic.

For example, the Riemann surface ${\mathbb{P}}^1_{\mathbb{C}}$ is diffeomorphic to the 2-sphere $S^2$ and its conjugation involution is the antipodal action.

Complex general and special linear groups.

For $A$ a square matrix, the determinant of its conjugation transpose equals the conjugate of its determinant. In symbols, ${\mathrm{det}} A^* = \overline{{\mathrm{det}} A}$. Hence sending a square matrix to its conjugate transpose is an involution on the complex general $GL(n,{\mathbb{C}})$ and special $SL(n,{\mathbb{C}})$.

In particular, the real space $SL(1,{\mathbb{C}})$ equipped with this conjugate transpose involution is equivariantly diffeomorphic to $S^{0,2}$, the circle equipped with its antipodal action (dicussed above).

Properties

Relation to real cobordism

There is a “real” analog of complex cobordism cohomology theory $MU$, the MR cohomology theory $M \mathbb{R}$.

While $\pi_\bullet(M \mathbb{R})$ is not the cobordism ring of real manifolds, still every real manifold does give a class in $M \mathbb{R}$ (Kriz 01, p. 13). For details see here: pdf.

Related concepts

• equivariant cohomology

• real structure

• real oriented cohomology theory

  • KR-theory

References

• Michael Atiyah, K-theory and reality, The Quarterly Journal of Mathematics. Oxford. Second Series 17 1 (1966) 367-386 [doi:10.1093/qmath/17.1.367, pdf, ISSN:0033-5606]

• Igor Kriz, Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence, Topology 40 (2001) 317-399 (pdf)