# Sign error in the (Chern-Simon induced) Bose-Fermi transmutation literature?

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In Bose-Fermi transmutation [(started with Polyakov 1988)][1] one considers a Chern-Simon action
\begin{align}
S_{CS}= \int d^3 x \frac{\epsilon^{\mu\nu\rho}}{4\pi} a_\mu \partial_\nu a_\rho
\end{align}
(my $\epsilon^{012}=+1$) and computes the quantum expectation of, say, two, Wilson loops
\begin{align}
& \left\langle \exp\left(i\oint_{C_1} dx^\mu a_\lambda(x) + i\oint_{C_2} dy^\nu a_\nu(y)\right) \right\rangle_{CS} \nonumber \\[.2cm]
=& \ \exp\left(-\frac{1}{2} \left(\oint_{C_1} dx^\mu \oint_{C_1} dy^\nu + \oint_{C_2} dx^\mu \oint_{C_2} dy^\nu + 2\oint_{C_1} dx^\mu \oint_{C_2} dy^\nu \right) \left\langle a_\mu(x) a_\nu(y) \right\rangle_{CS}\right).
\end{align}
There are three terms in the exponent. The point is one can show the last term would give rise to an exchange phase of $\pi$ while the first two terms would give rise to the Berry phase of a Euclidean $SO(3)$ spinor, hence the name "Bose-Fermi transmutation". My problem is the sign of the phase computed in the literature does not seem right to me. (It is known that Polyakov's original paper has another numerical error besides my problem, so I will point to, say, [GHKL 1989][2], as "literature"). Here I wish someone could help me check it, or clarify if I had any confusion.

I will demonstrate the problem in the last term -- the braiding term -- because its computation involves less steps.

From the above, the last term is equal to (I call it $i\Theta_{br}$ as it will later be equal to the braiding phase)
\begin{align}
i\Theta_{br} =& -\oint_{C_1} dx^\mu \oint_{C_2} dy^\nu \ \langle a_\mu(x) a_\nu(y)\rangle_{CS} \\
=& -\oint_{C_1} dx^\mu \int_\Sigma (d^2 r)^{\nu\rho} \ 2\partial_{[r^\nu|} \langle a_\mu(x) a_{|\rho]}(r) \rangle_{CS}
\end{align}
where in the second equality I used the Stoke's Theorem and $\Sigma$ is a surface bounded by $C_2$. The quantum expectation can be easily computed via
\begin{align}
0 =& \int \mathcal{D}a \ \frac{\delta}{\delta a_\sigma(r)} \left( a_\mu(x) \exp(iS_{CS}+iS_{g.f.})\right) \\
=& \int \mathcal{D}a \ \left(\delta^\sigma_\mu \delta^3(r-x) + a_\mu(x) \frac{i\epsilon^{\sigma\lambda\kappa}}{2\pi} \partial_\lambda a_\kappa(r) + \mbox{(g.f. term)}\right) \exp(iS_{CS}+iS_{g.f.})
\end{align}
("g.f." stands for gauge fixing) which leads to
\begin{align}
2\partial_{[r^\nu|} \langle a_\mu(x) a_{|\rho]}(r) \rangle_{CS} = 2\pi i \epsilon_{\mu\nu\rho} \delta^3(r-x) + \mbox{(g.f. term)}
\end{align}
where my $\epsilon_{012}=+1$ so that $\epsilon_{\sigma\nu\rho} \epsilon^{\sigma\lambda\kappa} = 2\delta^\lambda_{[\nu} \delta^\kappa_{\rho]}$; the gauge fixing term can be omitted since its contribution must vanish when integrated around the Wilson loop which is gauge independent. Therefore
\begin{align}
i\Theta_{br}=-2\pi i \ \oint_{C_1} dx^\mu \int_{\Sigma} (d^2 r)^{\nu\rho} \ \epsilon_{\mu\nu\rho} \ \delta^3(r-x).
\end{align}
The integral just computes the linking number between $C_1$ and $C_2$ (note that everything so far is purely topological and does not involve any geometry). In particular, if $C_2$ forms a circular loop in the $01$-plane, say $\{(\cos(s), \sin(s), 0)| 0\leq s < 2\pi\}$, and $C_1$ threads through it in the $+2$-direction (counter-clockwise braiding), say $\{(0, \cos(s)-1, \sin(s))| 0\leq s < 2\pi\}$, then the integral yields $+1$ and hence the braiding phase is $\Theta_{br}=-2\pi$. (The exchange phase, being half the braiding phase, would be $\Theta_{ex}=-\pi$, hence "Bose-Fermi transmutation".)

I am pretty sure this sign is correct because in the condensed matter "flux attachment" point of view, if a particle has charge $-1$ (or $+1$) under $a$, then $S_{CS}$ attaches $+2\pi$ (or $-2\pi$) $a$-flux to the particle, so another particle going counter-clockwise around it will pick up a phase of $-2\pi$ which equals the $\Theta_{br}$ I computed above.

However, in the literature, the quantum expectation is presented as
\begin{align}
\langle a_\mu(x) a_\nu(y)\rangle_{CS} = \frac{-i}{2} \epsilon_{\mu\nu\rho} \frac{(x-y)^\rho}{|x-y|^3}
\end{align}
(this is not topological but since the literature focused on flat Euclidean space, it is fine). Here my problem comes. From it one finds
\begin{align}
& 2\partial_{[y^{\, \nu}|}\langle a_\mu(x) a_{|\rho]}(y)\rangle_{CS} \\
=& \ \epsilon_{\sigma\nu\rho}\epsilon^{\sigma\lambda\kappa} \partial_{y^{\, \lambda}} \left( \frac{-i}{2} \epsilon_{\mu\kappa\alpha} \frac{(x-y)^\alpha}{|x-y|^3} \right) \\
=& \ \frac{+i}{2}\epsilon_{\sigma\nu\rho} \: \left(\delta^\lambda_\mu \delta^\sigma_\alpha - \delta^\lambda_\alpha \delta^\sigma_\mu\right) \partial_{x^\lambda} \frac{(x-y)^\alpha}{|x-y|^3} \\
=& \ \frac{i}{2} \epsilon_{\mu\nu\rho} (-4\pi) \delta^3(x-y) \ + \ \mbox{($\partial_{x^\mu}$ term)}.
\end{align}
The $\partial_{x^\mu}$ term is a gauge fixing term that vanishes under $\oint dx^\mu$. The first term is OPPOSITE to what I found. Thus, the quantum expectation in the literature leads to a braiding phase of $+2\pi$ when braided once counter-clockwise, which is OPPOSITE to what I computed and argued based on "flux attachment" reasoning.

One might say, these are phases being integer multiples of $2\pi$, so why care about the sign? Because the same sign problem is carried over into the computation of the Berry phase terms too, which depend on the geometry around the loops (due to regularization) and can take arbitrary value. (This Berry phase is the crucial point in the literature.) Related to this, it seems this sign error is also carried over to the more recent literature about the spin connection contribution to flux attachment (and the way they then interpret the fractional quantum Hall "shift"). Moreover, in the anyon context one can also consider $S_{CS}$ whose denominator is not $4\pi$, then the braiding phase can also take arbitrary value.

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