Twistor theory was proposed by Roger Penrose
in 1967 as a possible path to quantum gravity
and has evolved into a branch of theoretical
and mathematical physics. Penrose proposed
that twistor space should be the basic arena
for physics from which space-time itself should
emerge. It leads to a powerful set of mathematical
tools that have applications to differential
and integral geometry, nonlinear differential
equations and representation theory and in
physics to relativity and quantum field theory,
in particular to scattering amplitudes.
== Overview ==
Mathematically, projective twistor space
P
T
{\displaystyle \mathbb {PT} }
is a three-dimensional complex manifold, complex
projective 3-space
C
P
3
{\displaystyle \mathbb {CP} ^{3}}
. Physically it has the interpretation as
the space of massless particles with spin.
It is the projectivisation of a 4-dimensional
complex vector space, non-projective twistor
space
T
{\displaystyle \mathbb {T} }
with a Hermitian form of signature (2,2) and
a holomorphic volume form. This can be most
naturally understood as the space of chiral
(Weyl) spinors for the conformal group
S
O
(
4
,
2
)
/
Z
2
{\displaystyle SO(4,2)/\mathbb {Z} _{2}}
of Minkowski space; it is the fundamental
representation of the spin group
S
U
(
2
,
2
)
{\displaystyle SU(2,2)}
of the conformal group. This definition can
be extended to arbitrary dimensions except
that beyond dimension four, one defines projective
twistor space to be the space of projective
pure spinors for the conformal group.In its
original form, twistor theory encodes physical
fields on Minkowski space into complex analytic
objects on twistor space via the Penrose transform.
This is especially natural for massless fields
of arbitrary spin. In the first instance these
are obtained via contour integral formulae
in terms of free holomorphic functions on
regions in twistor space. The holomorphic
twistor functions that give rise to solutions
to the massless field equations are more correctly
understood as Cech representatives of analytic
cohomology classes on regions in
P
T
{\displaystyle \mathbb {PT} }
. These correspondences have been extended
to certain nonlinear fields, including self-dual
gravity in Penrose's non-linear graviton construction
and self-dual Yang-Mills in Ward's construction;
the former gives rise to deformations of the
underlying complex structure of regions in
P
T
{\displaystyle \mathbb {PT} }
, and the latter to certain holomorphic vector
bundles over regions in
P
T
{\displaystyle \mathbb {PT} }
. These constructions have had wide applications.The
self-duality condition is a major limitation
for incorporating the full nonlinearities
of physical theories, although it does suffice
for Yang-Mills-Higgs monopoles and instantons.
An early attempt to overcome this restriction
was the introduction of ambitwistors by Witten
and by Isenberg, Yasskin & Green. Ambitwistor
space is the space of complexified light rays
or massless particles and can be regarded
as a complexification or cotangent bundle
of the original twistor description. These
apply to general fields but the field equations
are no longer so simply expressed.
Twistorial formulae for interactions beyond
the self-dual sector first arose from Witten's
twistor-string theory. This is a quantum theory
of holomorphic maps of a Riemann surface into
twistor space. It gave rise to the remarkably
compact RSV (Roiban, Spradlin & Volovich)
formulae for tree-level S-matrices of Yang-Mills
theories, but its gravity degrees of freedom
gave rise to a version of conformal supergravity
limiting its applicability; conformal gravity
is an unphysical theory containing ghosts,
but its interactions are combined with those
of Yang-Mills in loops amplitudes calculated
via twistor-string theory.Despite its shortcomings,
twistor-string theory led to rapid developments
in the study of scattering amplitudes. One
was the so-called MHV formalism loosely based
on disconnected strings, but was given a more
basic foundation in terms of a twistor action
for full Yang-Mills theory in twistor space.
Another key development was the introduction
of BCFW recursion. This has a natural formulation
in twistor space that in turn led to remarkable
formulations of scattering amplitudes in terms
of grassmannian integral formulae and polytopes.
These ideas have evolved more recently into
the positive grassmannian and amplituhedron.
Twistor string theory was extended first by
generalising the RSV Yang-Mills amplitude
formula, and then by finding the underlying
string theory. The extension to gravity was
given by Cachazo & Skinner, and formulated
as a twistor-string theory for maximal supergravity
by David Skinner. Analogous formulae were
then found in all dimensions by Cachazo, He
& Yuan for Yang-Mills and gravity and subsequently
for a variety of other theories. They were
then understood as string theories in ambitwistor
space by Mason & Skinner in a general framework
that includes the original twistor-string
and extends to give a number of new models
and formulae. As string theories they have
the same critical dimensions as conventional
string theory; for example the type II supersymmetric
versions are critical in 10 dimensions and
are equivalent to the full field theory of
type II supergravities in 10 dimensions (this
is distinct from conventional string theories
that also have a further infinite hierarchy
of massive higher spin states that provide
an ultraviolet completion). They extend to
give formulae for loop amplitudes and can
be defined on curved backgrounds.
== The twistor correspondence ==
Denote Minkowski space by
M
{\displaystyle \mathbb {M} }
, with coordinates
x
a
=
(
t
,
x
,
y
,
z
)
{\displaystyle x^{a}=(t,x,y,z)}
and Lorentzian metric
η
a
b
{\displaystyle \eta _{ab}}
signature
(
1
,
3
)
{\displaystyle (1,3)}
. Introduce 2-component spinor indices
A
=
0
,
1
,
A
′
=
0
′
,
1
′
{\displaystyle A=0,1,\,A'=0',1'}
, and setNon-projective twistor space
T
{\displaystyle \mathbb {T} }
is a four-dimensional complex vector space
with coordinates are denoted by
Z
α
=
(
ω
A
,
π
A
′
)
{\displaystyle Z^{\alpha }=(\omega ^{A},\pi
_{A'})}
where
ω
A
{\displaystyle \omega ^{A}}
and
π
A
′
{\displaystyle \pi _{A'}}
are two constant Weyl spinors. The hermitian
form can be expressed by defining a complex
conjugation from
T
{\displaystyle \mathbb {T} }
to its dual
T
∗
{\displaystyle \mathbb {T} ^{*}}
by
Z
¯
α
=
(
π
¯
A
,
ω
¯
A
′
)
{\displaystyle {\bar {Z}}_{\alpha }=({\bar
{\pi }}_{A},{\bar {\omega }}^{A'})}
so that the Hermitian form can be expressed
as
Z
α
Z
¯
α
=
ω
A
π
¯
A
+
ω
¯
A
′
π
A
′
.
{\displaystyle Z^{\alpha }{\bar {Z}}_{\alpha
}=\omega ^{A}{\bar {\pi }}_{A}+{\bar {\omega
}}^{A'}\pi _{A'}\,.}
This together with the holomorphic volume
form,
ε
α
β
γ
δ
Z
α
d
Z
β
∧
d
Z
γ
∧
d
Z
δ
{\displaystyle \varepsilon _{\alpha \beta
\gamma \delta }Z^{\alpha }dZ^{\beta }\wedge
dZ^{\gamma }\wedge dZ^{\delta }}
is invariant under the group SU(2,2), a quadruple
cover of the conformal group C(1,3) of compactified
Minkowski spacetime.
Points in Minkowski space are related to subspaces
of twistor space through the incidence relation
ω
A
=
i
x
A
A
′
π
A
′
.
{\displaystyle \omega ^{A}=ix^{AA'}\pi _{A'}.}
The incidence relation is preserved under
an overall re-scaling of the twistor, so usually
one works in projective twistor space
P
T
{\displaystyle \mathbb {PT} }
, which is isomorphic as a complex manifold
to
C
P
3
{\displaystyle \mathbb {CP} ^{3}}
. A point
x
∈
M
{\displaystyle x\in M}
thereby determines a line
C
P
1
{\displaystyle \mathbb {CP} ^{1}}
in
P
T
{\displaystyle \mathbb {PT} }
parametrised by
π
A
′
{\displaystyle \pi _{A'}}
. A twistor
Z
α
{\displaystyle Z^{\alpha }}
is easiest understood in space-time for complex
values of the coordinates where it defines
a totally null two-plane that is self-dual.
If
x
{\displaystyle x}
is taken to be real, then if
Z
α
Z
¯
α
{\displaystyle Z^{\alpha }{\bar {Z}}_{\alpha
}}
vanishes, then
x
{\displaystyle x}
lies on a light ray, whereas if i
Z
α
Z
¯
α
{\displaystyle Z^{\alpha }{\bar {Z}}_{\alpha
}}
is non-vanishing, there are no solutions,
and indeed then
Z
α
{\displaystyle Z^{\alpha }}
corresponds to a massless particle with spin
that are not localised in real space-time.
== Variations ==
=== Supertwistors ===
Supertwistors are a supersymmetric extension
of twistors introduced by Alan Ferber in 1978.
Non-projective twistor space is extended by
fermionic coordinates where
N
{\displaystyle {\mathcal {N}}}
is the number of supersymmetries so that a
twistor is now given by
(
ω
A
,
π
A
′
,
η
i
)
,
i
=
1
,
…
,
N
{\displaystyle (\omega ^{A},\pi _{A'},\eta
^{i}),i=1,\ldots ,{\mathcal {N}}}
with
η
i
{\displaystyle \eta ^{i}}
anticommuting. The super conformal group
S
U
(
2
,
2
|
N
)
{\displaystyle SU(2,2|{\mathcal {N}})}
naturally acts on this space and a supersymmetric
version of the Penrose transform takes cohomology
classes on supertwistor space to massless
supersymmetric multiplets on super Minkowski
space. The
N
=
4
{\displaystyle {\mathcal {N}}=4}
case provides the target for Penrose's original
twistor-string and 
the
N
=
8
{\displaystyle {\mathcal {N}}=8}
case is that for Skinner's supergravity generalisation.
=== Hyper-Kähler manifolds ===
Hyperkähler manifolds of dimension
4
k
{\displaystyle 4k}
also admit a twistor correspondence with a
twistor space of complex dimension
2
k
+
1
{\displaystyle 2k+1}
.
=== Palatial twistor theory ===
The nonlinear graviton construction encodes
only anti-self-dual, i.e., left-handed fields.
A first step towards the problem of modifying
twistor space so as to encode a general gravitational
field is to ask to encode right handed fields.
Infinitesimally, these are encoded in twistor
functions or cohomology classes of homogeneity
–6. The task of using such twistor functions
in a fully nonlinear way so as to obtain a
right-handed non-linear graviton has been
referred to as the (gravitational) googly
problem (the word "googly" is a term used
in the game of cricket for a ball bowled with
right-handed helicity using the apparent action
that would normally give rise to left-handed
helicity). The most recent proposal in this
direction by Penrose in 2015 was based on
noncommutative geometry on twistor space and
referred to as palatial twistor theory (named
after Buckingham Palace, the place in which
Michael Atiyah suggested to Penrose the use
of a type of "noncommutative algebra", an
important component of the theory).
== See also ==
Background independence
Complex spacetime
History of loop quantum gravity
Penrose transform
Spin network
Twistor strings
== Notes
