Real versus complex

P. Deligne

We want to explain how to go back and forth between two modes of writing:

real: one considers real varieties, functions on them (possibly complex valued: F = f +ig), maps between them, …

complex: one considers complex varieties, and only holomorphic functions on them, complex analytic maps between them, …

A real analytic variety X can be complexified to a complex variety X_{ℂ}. In fact, only the germ of X_{ℂ} around X is defined. One can think of X as being given by X_{ℂ}, plus some auxiliary data. For some purposes, this auxiliary data is not needed, and computations are better done in “complex mode”, on X_{ℂ}.

To recover X from X_{ℂ}, one could give

(A) The antiholomorphic involution “complex conjugation” on X_{ℂ}, with fixed point X ⊂ X_{ℂ} — or

(B) the data of which (holomorphic) functions are real (on X).

There is not much difference between complex valued functions on X, and (holomorphic) functions on X_{ℂ}, and similarly for forms. A local coordinate system on X gives a (holomorphic) local coordinate system on X_{ℂ}.

If α is a form of maximal degree on X, viewed as an (holomorphic) form on X_{ℂ}, integration of α becomes, on X_{ℂ}, integration on a cycle, and we need to know this cycle only up to suitable homologies.

Example. To compute ∫ e^{−x^{2}/2+iax}dx, it is better to go to the holomorphic picture, where the integration cycle ℝ ⊂ ℂ is needed up to homologies given by 2-chains on which the integrand decays fast. In this setting, the completing the square change of variables y = (x − ia) makes good sense, giving the answer \sqrt{2π} e^{−a^{2}/2}.

Similarly, a (0|n) dimensional real varietyX can be complexified. Here, (holomorphic) functions on X_{ℂ} are the same as complex valued functions on X. The real X is recovered from X_{ℂ} by giving (A′) an antilinear involutuion σ of X_{ℂ}: \overline{X_{ℂ}} → X_{ℂ} or (B′) which functions are real. Here, to integrate a density is a purely algebraic story: only the density on X_{ℂ} matter, the real form X is irrelevant.

That integration is a purely algebraic story, making the real structure irrelevant, can happen also for ordinary varieties. Example: let V be a real vector space, Q a quadratic form, P a polynomial and dv a volume form. Then, up to an annoying square root, ∫ e^{−Q}P.dv depends only on e^{−Q}P dv on the complexification V_{ℂ}.

In dimension (n|m), Bernstein introduced the notion of cs-spaces. I like to think of a cs space as being a complex super space X_{ℂ}, plus the data of an “integration” cycle C on the reduced space. In many local computations only X_{ℂ} matters. If one wants to integrate densities (holomorphic sections of Ber(Ω^{1})), the cycle C matters only up to suitable homologies. Sometimes one wants to germify along C (which is the real locus for a real structure on the reduced space). This gives Bernstein definition, where the cs space is C, with the sheaf of rings 𝒪_{X_{ℂ}} restricted to C.

If X has complex structure J, X_{ℂ} is X × X̄, with X sitting in X × X̄ as the diagonal. As before, this is not much difference between a (complex valued) function on X and an (holomorphic) function on X_{ℂ} = X × X̄, defined near the diagonal. Holomorphic functions on X become functions f(x, y) on X_{ℂ} = X × X̄ depending only on x. In local coordinates: an holomorphic coordinate system on X, (z_{i}), gives one on X_{ℂ} = X×X̄: (z_{i}, z̄_{i}). A function on X is then viewed as an (holomorphic) function of z_{i} and z̄_{i}. It is an holomorphic function on X if it is an (holomorphic) function not depending on the z̄_{i}.

For functions spaces also it can be convenient to work in complexified spaces, with reality conditions coming as an afterthought, if needed. The complexification of \underline{Hom}(X, Y) is the space \underline{Hom}(X_{ℂ}, Y_{ℂ}) of (holomorphic) maps f from X_{ℂ} to Y_{ℂ}. One requires f to be defined only in a suitable neighborhood of X ⊂ X_{ℂ}, and that f(X) is close to Y ⊂ Y_{ℂ}.

Example. (i) If v is a section of the complexified tangent bundle of X, with X compact and X and v real analytic, exp(tv) (t small) belongs to Diff(X)_{ℂ}. (ii) (After G. Segal), Let S^{1} be the unit circle in ℂ. A point of Diff(S^{1})_{ℂ} is an holomorphic map from an annulus 1 − η < r < 1 + η to an annulus 1 − ε < r < 1 + ε. It should have a restriction to S^{1} close to a diffeomorphism of S^{1}.

Composition is only partially defined. If f maps S^{1} inside S^{1}, one can attach to f the “annulus” [f] between f(S^{1}) and S^{1} in ℂ. Both boundary components of [f] are parametrized by S^{1}, using the identity, resp. f. To composition corresponds gluing of annuli, and this is how a good substitute to Diff(S^{1})_{ℂ} is the monoid of isomorphism classes of annuli with parametrized boundaries. In conformal field theory, one meets complex (projective) representations of real semi-group underlying the complex semi-group Diff(S^{1})_{ℂ}. At the Lie algebra level, if W is the Lie algebra of vector fields on S^{1}, with complexification W_{ℂ}, a complex (projective) representation of the real Lie algebra r(W_{ℂ}) underlying W_{ℂ} is the same as a complex (projective) representation of r(W_{ℂ}) ⊗ ℂ = W_{ℂ} ⊗_{ℝ} ℂ = W_{ℂ} × W_{ℂ}. (iii) Take X = ℝ^{0|1}, hence X_{ℂ} = ℂ^{0|1}. Here Hom(X, Y) = Y, with the sheaf of ring Ω^{∗}, while Hom(X_{ℂ}, Y_{ℂ}), taken in the holomorphic world is Y_{ℂ}, with the holomorphic Ω^{∗}, and (Y, Ω^{∗})_{ℂ} = (Y_{ℂ}, Ω^{∗}).

In the complex setting, if X is of dimension (0|1), and Y of dimension (n|0), a real structure on Y induces a cs structure on \underline{Hom}(X, Y). Indeed, \underline{Hom}(X, Y) has a reduced space Y.

The same is not true for X of dimension (0, 2). Indeed, take Y = ℂ and coordinates θ_{1}, θ_{2} on X. A map to Y, i.e. an (holomorphic) function, is written

a + α_{1}θ_{1} + α_{2}θ_{2} + bθ_{1}θ_{2} :

the space \underline{Hom}(X, Y) is a ℂ^{2|2} with coordinates (a, b, α_{1}, α_{2}). If we give a real structure on Y, we know what it means for a to be real. To know what it means for b to be real, we need in addition to give a real structure on X.

In practice, this will not matter, for two reasons:

(A) The spaces of externals 𝒮 ⊂ Hom(X, Y) we will consider will be such that 𝒮_{red} embeds into Hom(X_{red}, Y) (and is a space of extremal there), so that a real structure on Y gives one on 𝒮_{red}: we get a cs structure on 𝒮. (B) In path integrals, integrals on fields playing the role of “c” above can be handled algebraically.

In the real setting, if we are to consider Hom(X, Y) with Y having a complex structure, it is usually convenient to work on Hom(X, Y)_{ℂ} = Hom(X_{ℂ}, Y_{ℂ}).

The fact that Y was complex gives us two transversal foliations on Y_{ℂ} = Y × Ȳ. In local oordinates: for (z_{i}) local coordinates on Y, we get local coordinates z_{i}, z̄_{i} on Y_{ℂ}. It follows that f ∈ Hom(X_{ℂ}, Y_{ℂ}) is given by the f_{i} = z_{i}(f) and f̄_{i} = z̄_{i}(f), where the f_{i} and f̄_{i} are (independent) holomorphic functions on X_{ℂ} (or complex valued functions on X). If on X_{ℂ} we have a real structure X, we get the real structure Hom(X, Y) on Hom(X_{ℂ}, Y_{ℂ}) by telling that f is real in f_{i} and f̄_{i} are complex conjugate (on X).