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In this paper on page 125, the authors claim that one can twist the $N=2$ theory by introducing a term in the action

$\frac{1}{2}\int R \phi$,

where $\phi$ is the bosonized version of the field we want to shift the spin of, and $R$ is the scalar curvature.

I don't see how this could work at all. Further, which twist is being used here? Presumably the $A$-twist from the context the claim appears in in the paper, but where in this term is a choice of $U(1)$ action to couple to the worldsheet holonomy?

As an additional question, is there any way to think about twisting this way in 4 or other dimensions?

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up vote 5 down vote accepted

This is really one of the numerous standard identities or "tricks" one may exploit in two-dimensional conformal field theories, a part of the "bosonization" techniques.

I am not aware of generalizations to higher dimensions and I don't really believe they exist because it is only $d=2$ in which fields have the same minimum number of components (one), regardless of the spin, so one may relate fields of different spins in various ways and continuously change the spin. But look at equation 3.16 and below it in this paper to see why the minimum coupling to the curvature scalar changes the mass of fields in the AdS space (there are probably better references).

To see why it's valid (up to boundary terms) in $d=2$, imagine for example (there are other methods based on equations of motion, action, holonomies etc.) that you study the stress-energy tensor for the fields like $b,c$ which have spins $\lambda,1-\lambda$ (equation 2.5.5 in Polchinski's String Theory, Volume I). These fields are given as $:\exp(\pm\phi):$ in terms of your bosonic field. Well, $i$ is usually needed in the exponent, I don't want to go into that.

The stress-energy tensor is given by 2.5.11a $$T(z) = :(\partial b)c: - \lambda \partial(:bc:) $$ However, $-:bc:=j$, the current (2.5.14) which may also be written as a multiple of $\partial\phi$. So the stress-energy tensor has an extra term of the form $\lambda \partial^2 \phi$ in it – and it's the same addition you get by varying $\lambda R^{(2)}$ with respect to the world sheet metric (the usual derivation of the stress-energy tensor). Here, $\lambda$ plays the role of the desired spin of $b$ and you will have to work out the factors of two and the signs etc. which I was sloppy about.

Note that the extra term doesn't matter on a flat world sheet. Alternatively, you may see the emergence of the curvature from the comparison of $\partial^2 b$ and $\nabla^2 b$ forms of the equation of motion, and in other ways.

A similar construction actually works for the re-bosonization of bosons, too. I find this case messier because one needs two distinct bosons etc. I was never trained in bosonization and similar things so the proof above is not necessarily the standard one, the simplest one, or the most correctly formulated or most complete one.

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