Why is (1,1) worldsheet SUSY enhanced to (2,2) SUSY after compactification of type II string theory on a CY threefold? I have a quick question. In case of compactification on an Calabi Yau cutting off 3/4th SUSY from the type-II string in 10D leads to an enhancement of worldsheet SUSY. It has been claimed since time of Gepner that (1,1) SUSY enhances to (2,2) SUSY on worldsheet upon compactification and it's known to be accidental rather than some special moduli enhancement of usual sort. Does anyone know the precise reason?
 A: The idea is that an $\mathcal N=2$ worldsheet supersymmetry requires two supercharges and a U(1) current that rotates the supercharges into each other. In the compactified theory, though this current turns out to be unphysical, you can use it to explicitly construct the supercharges in the left and right-moving sectors. Roughly, one supercharge is a relic of the gauge symmetry in the uncompactified theory, while the other comes from the global symmetry of the vacuum (and so is "accidental", in a sense).
Here I sketch the derivation of this fact. Focus on the holomorphic sector for now. The spacetime supersymmetry current is $V^\alpha_{-1/2}=e^{-\phi/2}S_\alpha\Sigma$ (and likewise for the dotted index $\dot\alpha$), where $e^{-\phi/2}$ is the spin field for the $\beta\gamma$ superghosts, $S_\alpha$ and $S_\dot\alpha$ are the spin fields for the worldsheet fermions in the compactified (4D) theory, and $\Sigma$ and $\bar\Sigma$ are the Ramond-sector ground states ($\Sigma$ has no further indices since left and right movers each have only $\mathcal N=1$). Since during Calabi-Yau compactification the internal SCFT in the holomorphic sector has $\hat{c}=6$ and the currents are of weight $(0,1)$, it is straightforward to show that $\Sigma$ is of weight $(0,\frac38)$.
Using that the (spacetime) supercharges are simply contour integrals of the SUSY currents $V^\alpha_{-1/2}$ and $V^\dot\alpha_{-1/2}$, followed by the definition of the supersymmetry algebra, the following OPEs can be derived for the 4D spin fields:
$$
\Sigma(z)\bar\Sigma(w)\sim(z-w)^{-3/4}+\frac12(z-w)^{1/4}J(w)
\\\Sigma(z)\Sigma(w)\sim(z-w)^{-3/4}\mathcal O(w)
$$
where $J$ is some current and $\mathcal O$ is just an operator with dimension 3/2. Our goal is to show that $J(z)$ completes the N=2 superconformal algebra.
Using these OPEs, we can easily construct the four-point function $\langle\Sigma(z_1)\bar\Sigma(z_2)\Sigma(z_3)\bar\Sigma(z_4)\rangle$. Since the current $J$ appears in the $\Sigma\bar\Sigma$ OPE, by taking appropriate limits of $z_i-z_j$, we can obtain the following OPEs:
$$
J(z)\Sigma(w)\sim\frac32\frac{\Sigma(w)}{z-w}
\\J(z)\bar\Sigma(w)\sim -\frac32\frac{\bar\Sigma(w)}{z-w}
\\J(z)J(w)\sim 3(z-w)^{-2}
$$
Uplifting news: $J$ is clearly a dimension 1, U(1)-current (and although, as mentioned, it does not commute with the BRST charge, this is immaterial). We can now bosonise the current into $J=i\sqrt3\partial H$, whereupon $\Sigma$ must be $\exp\left(i\frac{\sqrt3}2 H\right)$, from the $J\Sigma$ OPE and dimensional analysis (i.e. that $\Sigma$ has $h=3/8$). Here $H(z)$ is just a run-of-the-mill free scalar field, used to compute the splitting the supercurrent into definite $J$-charge components.
The OPE of the ($\mathcal N=1$) supercurrent of the internal SCFT $G_\text{int}$ with $\Sigma$ is determined using the BRST invariance of the gravitino vertex operator constructed from $\Sigma$ to eliminate any singular terms other than $\sim (z-w)^{-1/2}$.
Writing $G_\text{int}=\sum_i \exp\left(i\sqrt3 H\right) G_i$ and using the $\Sigma G$ OPE, it is seen that $G_\text{int}$ splits into $\frac{1}{2\sqrt2}(G_++G_-)$, where $G_\pm$ has definite charge $\pm1$ under the current $J$. With a little bit more fiddling with known $TT$ and $TG$ OPEs and using that the modes of $T_\text{int}$ and $G_\text{int}$ obey the $\mathcal N=1$ superconformal algebra already, we get the final result:
The modes of $T_\text{int}$, $G_\pm$ and $J$ satisfy the $\mathcal N=2$  superconformal algebra with central charge 9.
In the case of the type-II string, this is repeated for the antiholomorphic sector, demonstrating that the previous (1,1) worldsheet SUSY is enlarged to (2,2) SUSY due to the target spacetime SUSY. Moreover, using that the U(1) charge in each sector is quantized in half-integers (not proven here), the implication also runs the other way: (2,2) worldsheet SUSY and U(1) charge quantization in the compactified theory ensure that the target spacetime has $\mathcal N=2$ spacetime SUSY, as we expect for CY-threefold compactification.

[1] Polchinski, String Theory. Volume II, Superstring Theory and Beyond, Chapter 18.5.
