I am trying to check the Weyl transformation of the massless vertex operator in Polchinski closed bosonic string in the Polyakov approach (p105, Eq 3.6.16).To do that one needs to calculate something of the form $$ \int d^2\sigma d^2\sigma'd^2\sigma'' f(\sigma',\sigma'') g(\sigma) \frac{\delta}{\delta X^\lambda(\sigma')} \frac{\delta}{\delta X_\lambda(\sigma'')} \Big[ \partial_a X^\mu \partial_b X^\nu e^{ik.X (\sigma)} \Big]_r $$ Here $[F]_r$ is the regularised form of $F$, but my understanding is that you can just pass the functional derivatives into the square brackets. Here $f(\sigma',\sigma'')$ and $g(\sigma)$ are some functions whose explicit form does not matter for my question.

I have been able to work out the case where one or two of the functional derivatives acts on the $e^{ik\cdot X}$ and my result seems to be in line with Polchinski's result. However I am stuck when both functional derivatives act on the $ \partial_a X^\mu \partial_b X^\nu (\sigma)$. This generates a factor $\big[\partial_a \delta^2 (\sigma'-\sigma) \big] \times \big[\partial_b \delta^2 (\sigma''-\sigma) \big]$. Partial integration to free up a delta function so it can be integrated gives \begin{align*} & -\int d^2\sigma d^2\sigma'd^2\sigma'' \delta(\sigma'-\sigma) f(\sigma',\sigma'') \partial_a \Big\{ g(\sigma) \partial_b \delta^2 (\sigma''-\sigma) \Big[ e^{ik.X (\sigma)} \Big]_r \Big\} \\ =& -\int d^2\sigma d^2\sigma'' f(\sigma,\sigma'') \partial_a \Big\{ g(\sigma) \partial_b \delta^2 (\sigma''-\sigma) \Big[ e^{ik.X (\sigma)} \Big]_r \Big\}\\ =& +\int d^2\sigma d^2\sigma'' \partial_a f(\sigma,\sigma'') g(\sigma) \partial_b \delta^2 (\sigma''-\sigma) \Big[ e^{ik.X (\sigma)} \Big]_r \\ =& -\int d^2\sigma d^2\sigma'' \delta^2 (\sigma''-\sigma) \partial_b\Big[\partial_a f(\sigma,\sigma'') g(\sigma) \Big[ e^{ik.X (\sigma)} \Big]_r \Big]\\ = & -\int d^2\sigma \partial_b\Big[\partial_a f(\sigma,\sigma'') g(\sigma) \Big[ e^{ik.X (\sigma)} \Big]_r \Big]_{\sigma''=\sigma} \end{align*} Are these manipulations correct? It doesn't look like it when I compare the detailed outcome with Polchinski's result. In particular I don't expect a contribution containing $\partial_b g(\sigma)$. What is the correct way to do this?

PS. I realise that this is a pure mathematics question, but as more than one string theorist must have done this caclulation before I feel it is better suited here and not on the math SE.


1 Answer 1


A detailed and closely related discussion is given in this post

Briefly, the specific term you are looking at gives the 4th term in (3.6.17a) and the 3rd term in (3.6.17c) when contracted with $g^{ab}$. it’s a little subtle, you also need (3.6.18), (3.6.15b), replace normal by covariant derivatives (since they act on scalars this is trivial) and use that the worldsheet metric is covariantly constant. consider primarily $\delta_{\rm W}[\partial_a X^\mu \partial_b X^\nu e^{ik\cdot X}]_r$ rather than $\delta_{\rm W}V_1$, and integrate by parts in $\sigma'$, $\sigma''$ after using the chain rule on derivatives of the delta functions to obtain derivatives w.r.t. primed coordinates first. All terms/factors in (3.6.14) - (3.6.18) in Polchinski are correct.

  • $\begingroup$ Thanks. I agree with all your points, but even if you do that you get derivatives on $\delta \omega$. Using again partial derivatives gives a contribution $\partial_a [\sqrt{g} (g^{ab} ....)]$. How to get rid of that? $\endgroup$ Feb 11, 2020 at 12:35
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    $\begingroup$ if you indeed work with covariant derivatives you get $\nabla_a[\sqrt{g}g^{ab}\dots]=\sqrt{g}g^{ab}\nabla_a[\dots]$ because the metric is covariantly constant. you then differentiate $[\dots]$ and get two terms, one from the exponential and one containing $\nabla^2X$: you use (3.6.18) for the latter, whereas the former will give you something you will recognise when you compare to Polchinski's result. I could add further details if you want, but it might take a while to find time to write it. $\endgroup$ Feb 11, 2020 at 12:53
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    $\begingroup$ incidentally, you might notice that nice things happen if we had a scheme for which $\gamma=-1$. this is not discussed in Polchinski, but it is precisely the prescription developed in arxiv.org/abs/1912.01055, you could take a look if you're feeling adventurous! The relevant sections are 2.4.4, 5.5, and 5.8. The formalism used there is in precise agreement with (3.6.16) in Polchinski when $\gamma=-1$, and it automatically yields (3.6.18), and its generalisation to arbitrary operators. $\endgroup$ Feb 11, 2020 at 12:56
  • $\begingroup$ ok. let me have a further look then. $\endgroup$ Feb 11, 2020 at 12:57
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    $\begingroup$ i am indeed. by the way, this $\gamma=-1$ renormalisation prescription is not new, it was first developed by Polchinski back in 88. the advantage is that is globally well defined even for offshell vertex operators that appear in loops. It didn’t receive much attention until now (largely because it is fairly complicated, but it is well worth the effort if one is interested in higher genus and perhaps even non-perturbative string theory). Polchinski called it Weyl normal ordering. $\endgroup$ Feb 11, 2020 at 16:45

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