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This Concept Map, created with IHMC CmapTools, has information related to: Symmetries of WCW.cmap, SYMMETRIES OF WCW 2. Super-conformal symmetries related to isometries of WCW. a) Guess: isometries of WCW are symmet- ries of Kähler action for vacuum extremals Symplectic transformations of dCDxCP_2 satisfy this condition. They have structure of conformal group with light-like radial co- ordinate r of δCDxCP_2 taking the role of complex coordinate z. r is not invariant un- der Lorentz boosts. Hamiltonians can be chosen to be proportional to power r^n of of r and n has interpretation as radial con- formal weight. b) Symplectic algebra can be extended to super-conformal algebra. The contractions of complexified gamma matrices of WCW with Killing vector fields of isometry gene- rators are good candidates for fermionic super-algebra generators anti-commuting to matrix elements of WCW Kähler metric. The outcome is geometrization of fermionic statistic in terms of WCW geometry. c) Physics intuition suggests that WCW gamma matrices are expressible as linear combinations of second quantized oscilla- ctor operators for induced leptonic spinor fields contracted with covariantly constant right-handed neutrino spinor mode and in- tegrated over partonic two-surface and in- volving contraction with symplectic isomet- ry generator for δCD. Flux integral is in qu- estio. It does not involve induced metric. This is essential for conformal invariance. d) Besides this one has algebra defined in terms of isometry generators of δCDx CP_2 localized with respect to r. Now con- served stringy conformal charges associa- ted with string ends at boundaries of CD are in question and therefore 1-D integrals. These generators are labelled by conformal weight associated with spinor modes at string world sheet. This algebra is also a good candidate for isometries of WCW acting on string curves inside 3-surface. It corresponds to sub-algebra of symplectic isometries of δCDxCP_2. Both fermionic and quark type generators are possible., SYMMETRIES OF WCW 4. Yangian symmetries might be also present. Strings and partonic 2-surface correspond to two conformal weight like quantum num- bers and 3-dimensionality of basic objects suggest a further quantum number. a) Yangian symmetries are extensions of symmetries defined by finite-D Lie-al- gebras. The essential feature is that gene- rators are multilocal and can be graded by non-negative integer n telling that ge- nerator is n+1-local. n is somewhat analo- gous to conformal weight. Yangian algeb- ra has co-algebra structure: co-algebra operation is analogous to time reversal of algebra operation and the construction of n-local generators relies on it. The con- struction is recursive and starts from n=0, and n=1 generators. b) Yangian symmetries for 4-D conformal algebra appear in N=4 SUSYs and the successes of twistor Grassman ap- proach relate closely to underlying Yangian symmetry. Scattering amplitudes are con- structed from Yangian invariants. c) In TGD 3-surfaces replace point like par- ticles and strong form of holography redu- ces the situation to the level of partonic 2- surfaces and string ends as it seems. The natural guess suggested also by the non- locality of zero energy states in the scale of CD is that the conformal algebras gene- ralizes to Yangians multilocal with respect to loci defined by partonic 2-surfaces at same or even both boundary of CD. d) The n-local generators might play a role in the mathematical description of bound states, say hadrons, involving several parto- nic 2-surfaces. Also elememntary particles correspond to pairs of wormhole contacts with throats connected by Kähler magnetic flux tubes., SYMMETRIES OF WCW 1. Maximization of symmetries. a) The fundamental symmetries if the theory are isometries of the Kähler geometry of WCW. Kähler function de- termines the line element and corre- sponds to Kähler action for preferred extremals having as its 3-D ends given 3-surfaces at boundaries of CD. Iso- metries cannot be symmetries of Käh- ler action since they would not con- tribute to the Kähler metric. b) Already in string models the Kähler geometry for loop spaces is essenti- ally unique. Riemann connection does not exist in mathematically respectable manner in infinite-dimensional context unless there exists maximal group of isometries. c) Conjecture: Physics is fixed comple- tely by the Kähler geometric existence of WCW. Infinite-D geometric existence unique and thus also physics. d) Note: basic geometric objects in ZEO are pairs formed by 3-surfaces residing are at different light-like boun- daries of causal diamond CD. CDs are characterized by moduli space and WCW decomposes to sub-WCWs associated with CDs., SYMMETRIES OF WCW 3. The special role of light-like 3-surfaces. a) 3-D light-like boundaries of CD are metrically 2-dimensional and therefore has infinite-dimensio- nal conformal group extending the conformal sym- metries of 2-D manifolds as conformal symmetries. Also the isometries of light-cone boundary form an infinite-D group essentially identical with conformal symmetres of sphere S^2 since radial scaling de- pending on S^2 coordinates can compensate the conformal scaling. b) Arbitrary light-like 3-surface and therefore the light-like 3-surfaces defined by the regions at which the induced metric of space-time surface changes its signature from Minkowskian to Euclidian, possess this extended conformal symmetry. These exceptio- nal conformal symmetries raise 4-D Minkowski spa- ce to unique position. Also space-time dimension D=4 is unique. M^4 and CP_2 are also the only spaces allowing twistor space with Kähler structure. c) Natural guess: these symmetries are symmetries of the theory - maybe even isometries of WCW. How these symmetries extend to the level of imbedding space? Could one use slicing of CD by surfaces pa- rallel to its either boundary to extend symplectic isometries or do the isometries co-incide with sym- plectic transformations only at partonic 2-surface and string world sheets or their ends. d) Holography states that 3-surfaces code for phy- and is implied by General Coordinate Invariance. Strong form of GCI implies strong form of hologra- phy stating that partonic 2-surfaces and their 4-D tangent pace data code for physics. Does this imply effective reduction ofconformal invariance to that assigned with WCW isometry generators defined in terms of data from partonic 2-surfaces and stringy charges for induced spinor fields: note that vanish- ing of classical W boson fields guaranteeing well- defined em charge for spinor modes implies the reduction. e) Do the conformal symmetries associated with light-like partonic orbits correspond to gauge sym- metries? Is this the case when these transformations are trivial at boundaries of CD and do not therefore affect 3-surface. Or does one obtain a hierarchy of symmetry breakings characterized by sub-algebras of conformal algebras isomorphic with conformal al- lgebra itself. This hierarchy could correspond to the hierarchy of Planck constants assignable to n-furca- tions of 3-surfaces at boundaries of CD due to failure of strict determinism for Kähler action.
