Based on the Ramanujam's modular functions, somehow these magic numbers 10 and 26 spacetime dimensions appear in string theory. The dimensions can be viewed as 8 + 2 and 24 + 2. The number 2 is added due to relativity. But what are these two dimensions? Or this 2 represents something different?
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$\begingroup$ These aren't two dimensions; these are two possible numbers of dimensions. In other words, they are two possibilities for the number of dimensions. $\endgroup$– HDE 226868Commented Nov 16, 2014 at 20:56
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$\begingroup$ @HDE226868 Could you kindly explain a little further? What are the two Possibilities of dimensions? Is it 0 or some n number of dimensions? $\endgroup$– GworldCommented Nov 16, 2014 at 20:59
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$\begingroup$ I think that this may help you better than I can. $\endgroup$– HDE 226868Commented Nov 16, 2014 at 21:04
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2$\begingroup$ What does this bit mean: "They can be generalized as 8 ( 8 + 2). 2 is added due to relativity."? $\endgroup$– innisfreeCommented Nov 16, 2014 at 21:08
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1$\begingroup$ 26 is not an option, since that's the choice for bosonic string theory, which doesn't incorporate fermions. Also, one shouldn't really think of the dimension having to be fixed for the theory to be consistent - it's not very enlightening. What we want is the central charge to vanish to get rid of the trace anomaly, which eventually leads to $D=26$. $\endgroup$– JamalSCommented Nov 16, 2014 at 21:44
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1 Answer
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In a nutshell, the $1+1=2$ extra dimensions of spacetime can (in an appropriate gauge) be viewed as a longitudinal spatial direction and a temporal direction. This makes sense because of Minkowski signature of spacetime.
The point is that the physical modes of the string can be identified with the 24 (8) transversal directions of the critical bosonic (super) string, respectively. This splitting of spacetime is the starting point of light-cone quantization of string theory. The specific spacetime dimension is needed to cancel the conformal anomaly.
answered Nov 16, 2014 at 21:36
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$\begingroup$ This is similar to d.o.f. of a photon. A photon has 4 d.o.f in 4-d space-time, but only 2 physical after 2 removed by gauge symmetry, Ramanujam's modular functions give the physical, vibrating d.o.f. of the string, then we add 2 for the gauge symmetry to get them all. $\endgroup$ Commented Nov 16, 2014 at 22:25
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