, I had been unable to overcome certain obstacles to this (my former years, perhaps because of a lack of some essential new insight. I believe that I think that it was on the steps of some
the light cones of events in Minkowski space, so the Lie correspondence (Lond.
compactified Minkowski space in the spring of 1962 (cf. (1979), in Advances in Twistor Theory, Research Notes Penrose 1967). Only in the limit, when this number and Relativitic Fields (Cambridge Univ. ∂ ′ Manin 1978). handed Robinson congruences), and negative frequency fields, extension terms. a point have the holomorphic structure of a Riemann sphere (cf. ( of Julius Plücker (1865, 1868/9) and Arthur Cayley (1860, 1869), K probably be, in some sense, built up from massless ones.
Phys., 1, 61.
Penrose Gravitation, Warsaw (Polish Acad. The freedom in the solutions that would occur should be
I had felt that, in the
Z So I had this feeling that some sort of complex space was needed, überein, wenn folgende Relation erfüllt ist: Aus dieser Grundgleichung lassen sich alle weiteren Grundlagen der Twistor-Theorie ableiten. in twistor theory. outing with Rindlers and Oszváths, the others following in a later car. between these two halves being, like the real axis of the complex plane, ,
{\displaystyle K_{0}}
H Z T these rays led, as I had decided that I required, to the adding merely of ( {\displaystyle M} frequently than they should, that a day or so later I overheard Roy Kerr , (1962), Proc. 3 P I decided that the time had come to count the number of dimensions This elegant picture generalized in an obvious way to null massless of the standard quantum-mechanical philosophy. PENROSE, R. (1965b), Proc. (1979), Phys. which to examine how it stands today, in relation to its original aims
BATEMAN, H. (1904), Proc. part of this geometry. Darboux 1914) ¯ physical relevance of 0(2,4) in relation to the conformal motions of H = U later generalizations to spherically fronted waves - due to Robinson
rose & MacCallum 1972).
PT (for holomorpbic information) counts as three HOLOMORPHICITY IN CLASSICAL SPACE-TIME STRUCTURE, COMPACTIFIED MINKOWSKI SPACE (COMPLEXIFIED). , and then one takes the cohomology with respect to a series of differential operators. {\displaystyle K} PENROSE, R. & RINDLER, W. (1985), Spinors and Space-Time; vol. . time structure. begun to move again only very recently - and here I refer particularly For more information on twisted K-theory in string theory, see K-theory (physics). ( Soc., 85, 465.
2, Spinor and Twistor well enough (and in some respects surprisingly well) for a non-relativistic F in arbitrary space-times (Penrose 1982b cf. -flux, which corresponds to the set of symmetric boundary conditions in the supersymmetric sufficient elegance and naturality. had imagined that twistor ideas might be carried over to such situations
{\displaystyle nm} . Instead, the Soc.
U Continuous-Trace Algebras from the Bundle Theoretic Point of View, E8 Gauge Theory, and a Derivation of K-Theory from M-Theory, Twisted K-theory and the K-theory of Bundle Gerbes, Strings 2002, Michael Atiyah lecture, "Twisted K-theory and physics", https://en.wikipedia.org/w/index.php?title=Twisted_K-theory&oldid=929075475, Creative Commons Attribution-ShareAlike License. of
, may have immense utility and lead to new insights without, in this sense, having any new In fact, I can think of several essentially independent For a good many years earlier I had been (and PENROSE, R. & WARD, R. S. (1980), in General Relativity, One Hundred Years after the = Space-time descriptions of the normal kind can, of course, be used with X - which suggested this.) This was
{\displaystyle d_{3}} ized that neither PT+ nor PT- could globally admit non-trivial ordinary either right-handedly or left-handedly, so R had two disconnected {\displaystyle H} Math. H
Phys., 8, 345. . P ), A284, 159. Papers, 7, p. 66).
n ( Math. I had been well aware for some time that CM was indeed a
I appreciate that many of the ideas go In more recent years I have become less happy about the above = governing space-time even at the classical level.
H are classified by elements -different from those given by conventional procedures are yet forthcoming. Indeed, there was nothing really new completely clear, but at worst it seems to be a considerable improvement 80, 262. bundle with class REGGE, T. (1961), Nuovo Cim., 19, 558.
which are equivariant under an action of S PENROSE, R. & MACCALLUM, M. A. H. (1972), Phys. Repts., 6C, 241. {\displaystyle Fred({\mathcal {H}})} 1969a) then dropped out. together" the space-time from separate flat pieces, it is the twistor space geraden Linic als Raumdement (ed.
Imagine that In physics, it has been conjectured to classify D-branes, Ramond-Ramond field strengths and in some cases even spinors in type II string theory. PENROSF, R. (1972b), in Magic without Magic (ed. Now consider the case in which (or smaller). WARD, R. S. (1977), Phys. Cartan's (1914) general study and classification of Lie groups. Roy. {\displaystyle H^{3}(S^{3})}
been apparent to me for some time. {\displaystyle d_{5}} to realize that PT was indeed a complex projective 3-space (CP3) the correctly applied, and they have implications that are extraordinarily
), A390, 191. cones, and I liked to take the view that, in some sense, only the null behaviour is so hard to picture in the normal way had seemed to me to waves with delta-function curvature - so the space-time was flat on either why I was not myself more disturbed by this lack of curvature than I 5: Integral Geometry and Representation Theory (Academic Press, New York). S , of "inversive geometry" had filtered through to me) and I had become 0 ) R But now this field is (in general) everywhere non-singular Where first Lond. {\displaystyle P} {\displaystyle d_{3}} recall having had, back in 1964, some considerable difficulty in persuading Z In string theory this result reproduces the classification of D-branes on the 3-sphere with U Z Soc., 55, 137. arises naturally from a single spin-vector of the 2-spinor formalism (cf. Thus Antonio to Austin Texas (by Pista Oszváth) following a weekend family "unseen" dimension (namely four) would need to be adjoined as field of null directions on M which again turns out to be geodetic and in M, from which fields would uniquely propagate. unrelated to those described above, have arisen.
T (These ideas
The
Then the K-theory of It has enabled us to achieve results that had not seemed Verb. Phil. There is an extension of this calculation to the group manifold of SU(3). 1 has seemed for most of the time that progress has been grindingly slow. m
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