Complex Structures| Complex Structures |
Complex Vector Space > s.a. Polar
Decomposition; Complex Numbers; i.
* Idea: A real vector space
V with a tensor J which effectively converts it into a
complex one, by allowing us to define the product of x ∈ V
and λ = a + ib ∈ \(\mathbb C\) by
λx: = ax + bJx; In other words, it is
as if we defined the imaginary unit by ix:= Jx, etc.
$ Def: A complex structure on a real
even-dimensional vector space V is a map J: V →
V, with J2 = −I.
Almost Complex Manifold
$ Weakly almost complex: An n-manifold M
such that [TM]−[M × \(\mathbb R\)n]
is the image of the map KC(M)
→ KR(M) defined by taking
the underlying real vector bundle.
$ Almost complex: An n-manifold M with a (smooth assignment of a)
complex structure on the tangent space at each point, or, equivalently, such that its frame bundle
is reducible to a GL(n/2, \(\mathbb C\))-bundle.
* Necessary conditions:
The dimension n is even, M is orientable.
* Necessary and sufficient condition: There is a lift
r: BU(n) → BO(n) of f : M → BO(n)
to g: M → BU(n) with r \(\circ\) g = f.
Complex Manifold
$ Complex structure:
An integrable almost complex structure on a manifold.
* Idea: The integrability
condition allows to introduce local complex coordinates on M such
that the transition functions between different patches are holomorphic.
* Isomorphism: A map which preserves
the complex structure of a manifold is a biholomorphic map (f and
f −1 both holomorphic).
* Necessary conditions:
The dimension n is even, M is orientable.
@ References: Fernández de Córdoba & Isidro a1505 [generalised complex geometry, and thermodynamical fluctuation theory].
And Other Structures
> s.a. symplectic structure [Kähler structure].
$ And Hilbert space: On a real
\({\cal H}\) with (weakly non-degenerate) symplectic form Ω, define
(i) A complex structure J; (ii) A new real inner product
s(x, y):= −Ω(Jx, y),
or s(Jx, y) = B(x, y);
(iii) A Hermitian inner product h(x, y):=
s(x, y) + i Ω(x, y),
i.e., Ω is the imaginary part of a Hermitian inner product.
@ Generalizations:
Varsaie JMP(99) [on supermanifolds].
Mathematical Examples
* On spheres:
Of all S2n, only
S2 and S6 have
an almost complex structure, and only S2 has a
complex structure [@ in Steenrod 51].
* On projective spaces:
\(\mathbb R\)P2n does not have one, it is not
even orientable; \(\mathbb R\)P2n−1 is
weakly almost complex; \(\mathbb H\)Pn is not even
weakly almost complex for n ≥ 2; \(\mathbb H\)P1
is w.a.c.; \(\tilde G_{n,k}\): & P Sankaran.
Examples and Applications
> s.a. extensions of general relativity;
phase space; {& P Sankaran: Calgary seminar 4.10.1990}.
* Idea: For a field theory, a choice of complex structure
on the phase space is equivalent to a choice of decomposition into positive and negative frequency modes.
* On the phase space of a linear field theory: If the
background spacetime has a timelike Killing vector field ta,
then a complex structure which is compatible with the symplectic one is J:=
\(-(-{\cal L}_t{}^2)^{-1/2}{\cal L}_t\), acting on solutions of the field equations.
* On the Klein-Gordon phase space:
The general prescription translates into J(φ, π):=
(|Δ|−1/2π,
−|Δ|1/2 φ).
* On the Maxwell phase space:
The general prescription translates into
J(Ai, Ei):=
(|Δ|−1/2Ei,
−|Δ|1/2Ai).
@ General references: Esposito in(93)gq/95 [complex spacetime with torsion];
Marshakov & Niemi MPLA(05)ht [examples, gauge theory].
@ And quantum mechanics: Isidro IJGMP(05)ht/04 [complex structure on phase space];
Marmo et al IJGMP(05)ht [and classical limit];
Isidro IJMPA(06) [complex geometry and Planck cone];
Gibbons a1111-conf
[quantum mechanics as a real theory with a multitude of complex structures];
Aste a1905;
> s.a. Complex Numbers; geometric quantization.
@ For field theory: Gibbons & Pohle NPB(93)gq
[on space of solutions in curved background, and quantization];
Much & Oeckl a1812
[for Klein-Gordon theory in globally hyperbolic spacetimes].
References > s.a. Riemann Surface.
@ General: Yano 65;
in Chern 79;
Flaherty 76;
in Griffiths & Harris 78;
Kodaira 86;
in Willmore 93.
@ Complex manifolds: Newman ln;
Wells 80 [analysis].
@ Generalizations: Bandelloni & Lazzarini JMP(98) [Kodaira-Spencer deformation theory];
Sobczyk in(13)-a0710 [geometry of moving planes].
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 31 may 2019