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I'm studying TQFT and I'm stucked on this part of the paper of my teacher:

My teacher didn't explain a lot about it and I've never followed an advanced course on differential geometry or algebraic topology, so I'm spending days in studying cohomology and trying to understand what is a de Rham current and all the stuff written there. But there are too many things I have no idea about, and I just would like to have an idea of what is going here, so I'm here to ask you if is it possible to have a sketch of at least some of the following question without going too much detail:

  1. What is a de Rham current? Could I imagine it like a delta distribution which is non-zero only in a submanifold (in this case the knot)?
  2. What is the relationship between forms and currents? Are currents in some sense a generalization of forms that involves distributions?
  3. Why is the linking number the one written there?
  4. Why does he say "without the introduction of any regularization"?
  5. Is $\int_C \omega=\int_{M_0}\omega\wedge J_C$ a sort of generalization of Stokes theorem?
  6. Is the existence of the 1-current $\alpha_{\Sigma}$ such that $J_C=d\alpha_{\Sigma}$ a generalization of Poincaré lemma for currents?
  7. Since $B$ is a 1-form e $\alpha_L$ is a 1-current, does the limit $B\rightarrow\alpha_L$ have a physical/mathematical meaning or is it just a "symbolic" limit?
  8. Is the main idea of what is happening here something like: "we have written the CS theory on a homology sphere by introducing a 1-form $B$, such that the coefficients of the Taylor expansion of the new generating functional $G[B]$ in powers of $B$ coincide with the correlation functions of the curvature and it turns out that we don't need gauge fixing to compute it. Now we want this theory to be defined on a knot, so we give some properties to the 1-form $B$: we decide that $B$ is a de Rham current made of sum of de Rham currents defined on all the knots of the link weighed with the charge (which gives the representation of the knot), and for the properties of de Rham currents the integral is now defined on the link (on which the currents are defined) by means of Stokes theorem, so we obtain the linking number and we can compute the expectation value"?

This is the paper: https://arxiv.org/pdf/1402.3140.pdf

Thank you very much if you could answer to some of these confused questions.

polology
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1 Answers1

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I can answer some of the questions, but I'm more familiar with forms than currents

  1. Currents are linear forms on differential $k$ forms, so they are dual to $k$ forms (similar to distributions being dual to smooth functions, I think). This forms a nice framework to combine differential forms and homology since integrating along a $k$ dim cycle (submanifold) or wedging with an $n-k$ form and then integrating are both distributions.
  2. I think distribution valued n-k forms are examples of currents.
  3. The linking number measures the number of links, so it is the intersection of one knot with a surface whose boundary is the other knot (I'll relate this to the formula in a bit). If this is unclear, I think some simple examples would make it clear.
  4. idk
  5. The key word is either Thom class or Euler class I think. The idea is if $X$ is a dim $k$ submanifold of dim $n$ $Y$, we can take a tubular neighborhood of $X$ in $Y$ and this is isomorphic to the normal bundle which is an $n-k$ dimensional vector bundle over $X$. We can take a degree $n-k$ "bump form" $\omega_X$ on $Y$ which vanishes outside the normal bundle and such that if we integrate it along any $n-k$ dimensional fiber of the tubular neighborhood (considered as a vector bundle over $X$), we get 1. Now, this satisfies for any $k$ form $\alpha$, $\int_Y \alpha\wedge \omega_X=\int_X i^*\alpha$ ($i$ is the inclusion map for $X$). $J_C$ is this $\omega_X$ for $X=C$, as is $a_\Sigma$ for $X=\Sigma$. Your interpretation of a delta function nonvanishing only on the manifold is basically right, just keep in mind it's an $n-k$ form.

Now, $\int_{M_0} a_\Sigma \wedge J_C=\int_\Sigma i_\Sigma^* J_C$ i.e. the integral of the pullback of $J_C$ to $\Sigma$, but $J_C$ vanishes outside a tubular neighborhood of $C$ and integrates to 1 along any cross section, thus the pullback is a bunch of bumps on $\Sigma$ and integrates to the number of times $C$ intersects $\Sigma$. Try mimicking the argument but using $C$ instead to see if you get it.

  1. Funnily enough, it's this one which is Stoke's (I will ignore signs here and lazily omit pullback maps): For any 1 form $\omega$ on $M_0$, $$ \int_{M_0} \omega\wedge J_C = \int_C \omega =\int_\Sigma d\omega $$$$=\int_{M_0} d\omega \wedge a_\Sigma =\int_{M_0} \omega \wedge da_\Sigma $$ where in the last line we used that $M_0$ is closed. Perhaps it should've (more accurately?) said $J_C$ and $da_\Sigma$ are in the same cohomology class, but since you're just using them for integrals, it's only the cohomology class that really matters.

7,8. I don't know much QFT, nor do I know what B is, so I cannot comment

Toyesh Jayaswal
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