Quantum Number Fractionization in Cuprates: U(1) RVB Gauge Theory and Senthil Fisher Constructions

We critically look at Senthil Fisher constructions of quantum number fractionization in cuprates. The first construction, while mathematically correct is of limited relevance to cuprates. The second one has a missing local U(1) symmetry. Once this aspect is repaired, it reveals itself as the U(1) RVB gauge theory construction of quantum number fractionization. An approximate local $Z_2$ symmetry arises in the spin sector for a reason very different from Senthil and Fisher's. The charge sector continues to have the local U(1) symmetry and the metallic spin gap phase of the 2d cuprates may, in principle, carry (irrational) fractional charge $e^* \neq e$ obeying Haldane's fractional exclusion statistics like the 1d models.