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can we have in physic or can we speak about 1-d conformal theory in physics ??

for example in this one dimensional theory what would be the generators $ x \partial _{x} $ or $ \partial _{x} $ ??

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Yes, such theories exist. They're known as "conformal quantum mechanics". See

http://scholar.google.com/scholar?q=%22conformal+quantum+mechanics%22&hl=en&lr=&btnG=Search
http://scholar.google.com/scholar?q=ads2-cft1&hl=en&lr=&btnG=Search

There is an $SL(2,{\mathbb R})$ symmetry in them, or its (e.g. supersymmetric) extensions. The potential $1/r^2$ may occur in such QM models. Because there is only time dimension, there is no $\partial_x$, just $\partial_t$, and similarly there is only $t \partial_t$, formally speaking.

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  • $\begingroup$ Hi @Lubos, shouldn't $CFT_1$ have larger symmetry, I mean full reparametrization invariance analogous to Virasoro symmetry in 2D CFTs? Can you elaborate on this part? I have seen people usually talk about $SL(2,R)$ symmetry for $CFT_1$. Don't understand the reason. Is it too large symmetry to give non-trivial results? $\endgroup$ – Physics Moron Sep 6 '17 at 6:30
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    $\begingroup$ @PhysicsMoron - first, in the CFT2 case, the Virasoro symmetry contains all the conformal transformations that map the plane or sphere onto itself.. The infinite-dimensional group of holomorphic maps fails to be one-to-one, so these transformations aren't really reversible and can't be considered a proper group acting on the whole theory on the plane itself. This statement couldn't be made about CFT1 but the SL(2,R) still plays a preferred role at least because it's a group obtained by a continuation from a general dimension. $\endgroup$ – Luboš Motl Sep 8 '17 at 6:50
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    $\begingroup$ One must be careful about the precise statements etc. For example, Strominger here arxiv.org/pdf/hep-th/9809027.pdf did argue that the enhancement to the infinite-dimensional group should be made in both cases, in some setup. Search for "enlarged" in that paper. $\endgroup$ – Luboš Motl Sep 8 '17 at 6:51
  • $\begingroup$ Thanks for the clarification! Actually people sometimes consider CFT$_1$ as one copy of the Virasoro algebra, like the Strominger's paper you referred to. I was wondering if one can compute conformal blocks (say) analogous to Virasoro blocks in 2D CFT for this single copy. In literature I could only find papers (e.g, arxiv.org/abs/1205.0443) which consider conformal quantum mechanics i.e, $SL(2,R)$ symmetry. Will those 'half-Virasoro' blocks be "trivial" or "not-so-easy-to-define"? $\endgroup$ – Physics Moron Sep 8 '17 at 7:14
  • $\begingroup$ OK, I would have to restudy these things, no expert to answer on the top of my head... Sorry. $\endgroup$ – Luboš Motl Sep 8 '17 at 18:22

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