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I am changing the page title – this used to be “A first idea of quantum field theory”, which of course still redirects. The “A first idea…” seemed a good title for when this was an ongoing lecture that was being posted to PhysicsForums. I enjoyed the double meaning one could read into it, but it’s a bad idea to carve such jokes into stone. And now that the material takes its place among the other chapters of geometry of physics, with the web of cross-links becoming thicker, the canonical page name clearly is “perturbative quantum field theory”.
added to the list of references (here) pointer to today’s
where the approach that I had taken in these lecture notes is now developed much more extensively.
Re #2: Very nice!
Equation (7) claims that the on-shell field space is the pullback of Γ_M(F) and Γ_M(F) over sections of the variational cotangent bundle.
Should it not be the pullback of Γ_M(F) and a point over the variational cotangent bundle instead?
Yes, it’s great that this writeup now exists.
Regarding your question:
Let me highlight that a big issue in these formalizations is that the most naive idea of taking the critical locus, namely as the pullback of the point $0 \in \Omega^1_{dR}(-)$ along $\mathrm{d} S \,\colon\, \mathcal{F} \to \Omega^1_{dR}(-)$, comes out wrong: It contains plots that annihiliate $\mathrm{d}S$ simply by not being étale onto their domain.
One way to rectify this is to take that pullback in the space of sections. The 0-section that is being pulled back serves as the “point” in the space of sections.
I’ll check if Gregory would enjoy to say more about this, here in the Forum…
Re #4: I see, but what is even more confusing is this: since the map 0: Γ_M(F)→Γ(∧^d T*M⊗V*F) factors through the point (i.e., the terminal object in smooth sets), the resulting pullback is necessarily the categorical product of Γ_M(F) and the (categorical) fiber of the map EL over 0. In other words, unless I severely misunderstood the notation and/or the context, such a pullback cannot resolve the indicated problem.
Now I understand what you were after in the original question. I see, right, this must be a typo.
I should better shut up and let the authors reply when they are back from the holidays. But just to briefly relay Gregory’s reaction smuggled out of his almost offline vacation:
The point to notice is that $\wedge^d T^\ast M \otimes V^\ast F$ is canonically a vector bundle only over $F$ but just a fiber bundle over $M$, so that the “0-map” in (7) is indeed a non-constant map. It is instructive to see this in the very special case of the pullback (7) for the degenerate case of 0d field theory discussed in Example 5.33.
Hello Urs and Dmitri. Thank you for the interest in the paper, let me expand slightly on the above.
The crucial part is that the “0-map” from Eq. (7) is actually the canonical 0-section $0_\mathcal{F} : \mathcal{F} \longrightarrow T^*_\mathrm{var} \mathcal{F}$ of the (smooth set) vector bundle $T^*_{\mathrm{var}} \mathcal{F} \longrightarrow \mathcal{F}$, where $\mathcal{F}= \Gamma_M(F)$ and $T^*_{\mathrm{var}} \mathcal{F} = \Gamma_M( V^*F \otimes \wedge^d T^* M )$. In particular, the $0$-section is not the constant map, i.e. does not factor through the point. In fact, there does not exist such ”constant 0-map” since the variational cotangent bundle is not a vector space ( $V^*F \otimes \wedge^d T^* M$ is not a vector bundle over $M$, and so its sections do not form a vector space).
For the sake of completeness, let me make this fully explicit (which we should probably do in the paper itself, in retrospect). The (dual) vertical tangent bundle $V^*F\rightarrow F \rightarrow M$ of an arbitrary fiber bundle $\pi_M: F\rightarrow M$ is only a vector bundle over the total space $F$. It follows, the tensor product bundle $V^*F \otimes \wedge^d T^* M$ can only be taken over $F$, i.e., it really stands for $V^*F \otimes_F \pi^*_M(\wedge^d T^*M)$. Since the latter is a vector bundle $\pi_F : V^*F \otimes \wedge^d T^* M \rightarrow F$ (over $F$), there is a canonical $0$-section $0_F : F \rightarrow V^*F \otimes \wedge^d T^* M$. Taking sections over $M$ yields the field space $\Gamma_M(F)$ and the variational cotangent bundle $\Gamma_M(V^*F \otimes \wedge^d T^* M)$. Postcomposition of sections of $V^*F \otimes \wedge^d T^* M$ with the projection $\pi_F$ yields the (smooth set) bundle projection $\pi_{\mathcal{F}}: T^*_\mathrm{var} \mathcal{F} \rightarrow \mathcal{F}$, while postcomposition of sections of $F$ with the (manifold) zero section $0_F$ yields the (smooth set) zero section $0_{\mathcal{F}}: \mathcal{F}\rightarrow T^*_\mathrm{var} \mathcal{F}$.
Let me know if this clears up the notation used.
Happy new year!
By the way, I assigned this paper for my student seminar: https://dmitripavlov.org/homotopy
Looks like we will have several talks on it, at least 6 people signed up. We’ve already found some typos, which might be posted here, if it’s okay.
Hi, I had a minor question regarding what I think might be a typo in the paper. In Lemma 2.17 on page 15 of the paper Field Theory via Higher Geometry I: Smooth Sets of Fields, I am unclear on what $\Phi=\pi_F\circ X_\Phi$ is meant to be. In particular, I am unsure of what $X_\Phi$ is. If I assume this to be a typo, I could replace this with $Z_\Phi$, but this also does not make sense in context of $Z_\Phi$ covering $\Phi$. If there is any clarification the authors could make, that would be fantastic!
Hi Dmitri, Jhoo. I’m glad to hear you are finding the paper useful.
On Jhoo’s question: This is a typo, thanks for spotting that. Indeed, it should say $\phi = \pi_F \circ Z_\phi$, i.e., as the diagram below and the proof does. However, I am not sure what you mean by this not making sense in the context of $Z_\phi$ covering $\phi$, since this is (by definition) precisely what it means to cover a map $\phi: M \rightarrow F$ via a map $Z_\phi:M \rightarrow VF$, with respect to the canonical projection $VF\rightarrow F$. Do let me know if there is something I am missing here.
On that note, we are currently finishing up the second version of the preprint which should be uploaded by the end of the week. This will definitely be better for your seminar. It will have plenty of typos fixed, and furthermore some minor additions / corrections (e.g. a couple of Lemmata on “forms via the classifying space $\Omega^\bullet_\mathrm{dR}$ on $\mathcal{F}\times M$” vs “local forms via $J^\infinity_M F$ on $\mathcal{F}\times M$”, and the introduction of the “prolonged shell” where needed).
Please feel free to post questions here and I’ll do my best to clarify. It will be good to have these recorded ‘in public’. Regarding pure typos, which you deem need no clarification, perhaps send these via email to keep this post free of clutter, as it is likely there will be too many. Thanks!
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