Differential Operators in Polyakov Action What do the differential operators in the Polyakov action mean? How does one derive the Polyakov action and treat the differential operators?
 A: The differential operators in the action linked to in the question are $\partial_a,\partial_b$ which simply mean
$$ \partial_a \equiv \frac{\partial}{\partial \sigma^a} $$
i.e. a partial derivative with respect to a world sheet coordinate. Similarly for $b$. Note that $a,b=0,1$ parameterize the two coordinates on the world sheet, $\sigma^0=\tau$ (time) and $\sigma^1=\sigma$ (space along the string).
The action is a relatively "fundamental" starting point of any theory, so one may perhaps answer that it can't be derived from anything more fundamental.
However, we may also say that it may be "derived" from a more heuristically comprehensible Nambu-Goto action, as is explained in the 1st chapter of every string theory textbook (Polchinski; Green, Schwarz, Witten etc.). The Nambu-Goto action is $-T$ times the proper "area" of the world sheet, a generalization of the proper time along the particles' world lines (which are multiplied by $-m$, the negative mass; here, $T$ is the string tension).
The Nambu-Goto action is highly nonlinear but by introducing an auxiliary world sheet metric whose equations of motion say that it is proportional to the induced metric, we may convert the nonlinear action to the Polyakov action above - and the latter is quadratic in the derivatives of the bosonic fields, so it is a free Klein-Gordon action in 2 dimensions, a much better starting point for calculations and for well-definedness of the theory.
