tag:blogger.com,1999:blog-10614348.post5194272524318239517..comments2024-01-22T11:26:37.599-08:00Comments on TGD diary: The only game in the town: again!Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-10614348.post-90645300377640282432015-03-01T06:07:13.361-08:002015-03-01T06:07:13.361-08:00
I want to emphasise that I have nothing against s...<br />I want to emphasise that I have nothing against string models. Strings in 4-D space-time are part of TGD and it is now rather clear that gravitational constant appears as fundamental constant besides Kahler coupling strength and CP_2 radius defining universal scale. The ratio hbarG/R^2 is dictated by quantum criticality.<br /><br />The especially interesting aspect is quaternion conformal symmetry generalising complex analyticity to quaternion conformal analyticity: the generalisation of Cauchy-Riemann equations is carried out decades ago but I found it only now.Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-89388368303611397532015-02-22T04:21:03.196-08:002015-02-22T04:21:03.196-08:00Thank you for clarifying that point about conforma...Thank you for clarifying that point about conformal invariance!<br />So, in two dimensions, infinitesimal conformal transformations are functions which obey the Cauchy-Riemann equations.<br />Conformal vector fields can be considered as a natural generalization of Killing vector fields. <br />For example, in the case of the black hole, there are metrics which allow the existence of Killing horizons and these Killing horizons coincide with their event horizons. <br />In terms of the dual CFT, CFT observers cannot exchange information faster than light. <br />The motivation behind their new work (the meaning of "the only game in the town") seems to be that string-like theory in flat space is, broadly defined, the simplest example of a weakly coupled theory of gravity.<br />Thanks again for your very interesting post and your kind answer!Giuliohttps://www.blogger.com/profile/03054492374012236142noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-28063482284413343702015-02-21T21:02:20.498-08:002015-02-21T21:02:20.498-08:00
Thank you for asking since this is extremely impo...<br />Thank you for asking since this is extremely important point that I have tried to get through for more that two decades!<br /><br />2-D conformal invariance is due to 2-dimensionality of the basic geometric objects, Riemann surfaces. One can generalise 2-D conformal symmetry in non-trivial manner to light-like 3-surfaces (having one-light-like direction), which are metrically 2-dimensional. <br /><br />Light-cone boundary with topology S^2xR_+ is the simplest example. Denote by r the radial light-like coordinate.<br /><br />*There is no contribution from the light-like radial direction to the induced metric and it is just ds^2= r^2dOmega^2, where dOmega^2 is the metric of S^2. The metric is effectively 2-D and allows conformal transformations of S^2 depending parametrically on r as generalised conformal transformations. By a suitable choice of radial dependence the conformal transformations act even as isometries so that ordinary 2-D conformal group acts as isometries. <br /><br />Another huge conformal symmetry emerges in M^4xCP_2. Now one can consider symplectic transformations of S^2xCP_2 depending parametrically on r. This group has structure of conformal group with r taking the role of z so that finite-D group defining Kac-Moody ext rends to infinite-D symplectic group of M^4xCP_2. <br /> <br />These two kinds of symmetries mean gigantic extension of the conformal symmetries of string models. These symmetries act at imbedding space level. This of course also replaces AdS/CFT correspondence with the correspondence realised in TGD framework. <br /><br />A further generalisation of the conformal symmetries of light-cone boundary are to the <br />conformal symmetries of light-like partonic orbits<br />since the effective metric 2-dimensionality is enough. These conformal symmetries correspond to ordinary Kac-Moody type symmetries acting at space-time level. <br /><br />Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-53379731598061747712015-02-21T12:29:35.804-08:002015-02-21T12:29:35.804-08:00"When one speaks of AdS/CFT, one should not f..."When one speaks of AdS/CFT, one should not forget that it is based on one particular definition of super-conformal invariance: the conformal invariance associated with 2-D surfaces"<br />Sorry, I can't understand what do you mean. Yangian structure? T-duality in the AdS sigma model ? two dimensional Ising model with CFT?<br />Could you please elaborate more? Thank you very much.Giuliohttps://www.blogger.com/profile/03054492374012236142noreply@blogger.com