The example I am aware of so far is quintic 3-fold equipped with $SU(3)$ holonomy. Why it is more natural to talk about $CY_3$ folds or Calibi-Yau 3 folds without talking about algebraic curves in string theory? Why it is not more natural to talk about curves first? Or why it is more intuitive to movtivate 3-fold first rather than motivate algebraic curves? Reminder: This is not a popular science question. I need mathematical and theoretical physics justification. Please give me the reference if possible.
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Possible duplicates: http://physics.stackexchange.com/q/4972/2451 , http://physics.stackexchange.com/q/13945/2451 , http://physics.stackexchange.com/q/10495/2451 and links therein. – Qmechanic May 02 '15 at 06:57
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I do not think those posts do answer my question from technical view points at all. My question is why we do not want to prefer algebraic surfaces to algebraic curves. Shouldn't curves come more naturall first then the surfaces? The quintic 3 fold is the example I have seen with SU(3) holonomy. I need a better math explaination telling me why we can exclude algebraic curves. – user45765 May 02 '15 at 17:10