can anyone tell me how many boson or fermion g2 algebra has?
This is a good question.
A very good introduction to Lie Algebra G2 is given in
D. Baraglia, G$_2$ Geometry and Integrable Systems, Ph.D. thesis, Hertford College, Oxford University, 2009:
http://people.maths.ox.ac.uk/hitchin/hitchinstudents/baraglia.pdf
See Chapter 2, staring on page 87, on conformal G2 holonomy. See, also, Chapter 6, starting on page 105, on G2 manifolds.
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