# Is it true that in abelian Chern-Simons theory diffeomorphisms differ from ordinary gauge transformations trivially?

In Henneaux's Lectures on the Antifield BRST Formalism for Gauge Theories, it is claimed in Exercise 1 that diffeomorphisms $$\delta_\xi A_\mu=\xi^\rho\partial_\rho A_\mu+\partial_\mu\xi^\rho A_\rho$$ differ from ordinary gauge transformations $$\delta_\Lambda A_\mu=\partial_\mu\Lambda$$ by a trivial gauge transformation $$\delta_\mu A=\int\text{d}^Dy\,\mu_{\mu\nu}(x,y)\frac{\delta S}{\delta A_\nu(y)}$$, for some $$\mu_{\mu\nu}(x,y)=-\mu_{\nu\mu}(y,x)$$. In here $$S=\int\text{d}x\epsilon^{\mu\nu\rho}F_{\mu\nu}A_\rho$$ is the action for Abelian Chern-Simons theory.

I tried to proof this. The equations of motion are $$\frac{\delta S}{\delta A_\mu(x)}=\epsilon^{\mu\alpha\beta}\partial_\alpha A_\beta(x)$$. Therefore, what we want to proof is that there is a $$\Lambda$$ and $$\mu_{\mu\nu}(x,y)$$ such that $$\delta_\xi A_\mu-\delta_\Lambda A_\mu=\int\text{d}^Dy\,\mu_{\mu\nu}(x,y)\epsilon^{\nu\alpha\beta}\partial_\alpha A_\beta(y).$$ Choosing $$\mu_{\mu\nu}(x,y)=\epsilon_{\mu\nu\alpha}\xi^\alpha(x)\delta(x-y)$$ yields $$\int\text{d}^Dy\,\mu_{\mu\nu}(x,y)\epsilon^{\nu\alpha\beta}\partial_\alpha A_\beta(y)=\epsilon_{\mu\nu\gamma}\xi^\gamma(x)\epsilon^{\nu\alpha\beta}\partial_\alpha A_\beta(x)=\xi^\alpha\partial_\alpha A_\mu+\partial_\mu \xi^\beta A_\beta-\partial_\mu (\xi^\beta A_\beta)=\delta_\xi A_\mu-\delta_{\xi^\nu A_\nu}A_\mu.$$ Thus, this choice almost works. However, the parameter $$\Lambda$$ shouldn't depend on $$A$$. Is there another choice of $$\mu$$ and $$\Lambda$$ so that $$\Lambda$$ doesn't depend on $$A$$?

Investigating the problem further , I noticed the following. If indeed there is such a $$\mu$$ and $$\Lambda$$, then, whenever the equations of motion are satisfied we have that $$\delta_\xi A_\mu-\delta_\Lambda A_\mu=0$$. The equations of motion imply that $$A$$ is pure gauge, i.e. $$A_\mu=\partial_\mu \Omega$$ for some $$\Omega$$. Then $$\delta_\xi A_\mu-\delta_\Lambda A_\mu=\xi^\rho\partial_\rho\partial_\mu\Omega+\partial_\mu\xi^\rho\partial_\rho\Omega-\partial_\mu\Lambda=\partial_\mu(\xi^\rho\partial_\rho\Omega-\Lambda).$$ There is no way that this vanishes identically unless we choose $$\Lambda=\xi^\rho\partial_\rho\Omega=\xi^\rho A_\rho$$ (up to a constant). We conclude that the $$\Lambda$$ must depend on $$A$$ as we found above. Then, isn't the statement in the exercise wrong? Or are we allowed to use a different gauge parameter for every field?

The paradox comes from thinking that the gauge-transform parameter $$\Lambda$$ is not allowed to depend on the gauge field $$A$$ that is being transformed, but that is incorrect. Within any contractible patch, we can take the quantities $$A_a$$ (and also $$\xi^a$$) to be ordinary smooth functions, and then $$\Lambda=\xi^a A_a$$ is also a smooth function. The transform $$A_a\to A_a+\partial_a\Lambda$$ is a legal gauge transform for any smooth function $$\Lambda$$, including one of the form $$\Lambda=\xi^a A_a$$.
On a general manifold, the gauge field $$A_a$$ is defined only patchwise, and that's fine. A gauge transform $$A_a\to A_a+\partial_a\Lambda$$ is also defined patchwise, and setting $$\Lambda=\xi^a A_a$$ (defined patchwise) still gives a legal gauge transform because $$A_a$$ and $$\partial_a\Lambda$$ are both affected the same way by the transition functions that relate the different patches to each other.
• Thank you very much for your response! I understand that $\Lambda=\xi^\mu A_\mu=A(\xi)$ is a well defined gauge-transform parameter. My problem is however that for the vector fields to be equivalent one needs that $\delta_\xi-\delta_\Lambda=\int\text{d}^Dy\,\frac{\delta S}{\delta A_\mu(x)}\mu_{\mu\nu}(x,y)\frac{\delta}{\delta A_\nu(y)}$ should hold as a vector field equation. May 21 '20 at 21:39
• @IvánMauricioBurbano You're welcome, and sorry if I misunderstood your question. On the right-hand side of the equation in your comment, the integrand depends on $A$ through $\delta S/\delta A_\mu\sim \epsilon^{\mu\nu\rho}\partial_\nu A_\rho$. Similarly, if $\Lambda=\xi^\mu A_\mu$, then the $\delta_\Lambda$ on the left-hand side can be written $\int dx\ \partial_\nu (\xi^\mu A_\mu)\delta/\delta A_\nu$, if I'm not mistaken. If the $A$-dependence is allowed on the right-hand side, then shouldn't it also be allowed on the left-hand side? Or am I still misunderstanding your question? May 22 '20 at 1:09
• I didn't make myself clear. It is fine for vector fields to depend on the field $A$. However the parameters associated to them shouldn't. This is more clearly seen in the case of a non abelian YM or CS theory. In there, the vector field corresponding to a gauge transformation is $\delta_\Lambda=\int\text{d}^Dy(f^a_{bc}A^b_\mu \Lambda^c+\partial_\mu\Lambda^a)\frac{\delta}{\delta A^a_\mu}$. One, wouldn't consider however this vector field a gauge transformation if $\Lambda$ depended on $A$. Well, as I write this now I've found myself thinking, why not? May 22 '20 at 2:27