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In their proposed compactification of superstrings [4], Candelas, Horowitz, Strominger and Witten took the matric product of a maximal symmetric four dimensional spacetime M with a six dimensional CalabiYau vacua X as the ten dimensional spacetime; they identified the YangMills connection with the SU (3) connection of the CalabiYau metric and set the dilaton to be a constant. To make this theory compatible with the standard grand unified field theory, Witten [28] and HoravaWitten [20] proposed to use higher rank bundles for strong coupled heterotic string theory so that the gauge groups can be SU (4) or SU (5). Mathematically, this approach relies on UhlenbeckYaus theorem on constructing HermitianYangMills connections on stable bundles [27]. Many authors, including Friedman, Morgan and Witten [18]; Donagi, Ovrat, Pantev and Reinbacher [12]; Andreas [1], Kachru [21] and others, have worked

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