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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 11th 2018

    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”.

    diff, v187, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2023
    • (edited Dec 29th 2023)

    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.

    diff, v199, current

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 29th 2023
    • (edited Dec 29th 2023)

    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?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2023

    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 0Ω dR 1()0 \in \Omega^1_{dR}(-) along dS:Ω dR 1()\mathrm{d} S \,\colon\, \mathcal{F} \to \Omega^1_{dR}(-), comes out wrong: It contains plots that annihiliate dS\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…

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 29th 2023
    • (edited Dec 29th 2023)

    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.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2023

    Now I understand what you were after in the original question. I see, right, this must be a typo.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeDec 30th 2023
    • (edited Dec 30th 2023)

    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 dT *MV *F\wedge^d T^\ast M \otimes V^\ast F is canonically a vector bundle only over FF but just a fiber bundle over MM, 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.

    • CommentRowNumber8.
    • CommentAuthorGrigorios
    • CommentTimeJan 3rd 2024

    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 :T var *0_\mathcal{F} : \mathcal{F} \longrightarrow T^*_\mathrm{var} \mathcal{F} of the (smooth set) vector bundle T var *T^*_{\mathrm{var}} \mathcal{F} \longrightarrow \mathcal{F}, where =Γ M(F)\mathcal{F}= \Gamma_M(F) and T var *=Γ M(V *F dT *M)T^*_{\mathrm{var}} \mathcal{F} = \Gamma_M( V^*F \otimes \wedge^d T^* M ) . In particular, the 00-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 dT *MV^*F \otimes \wedge^d T^* M is not a vector bundle over MM, 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 *FFMV^*F\rightarrow F \rightarrow M of an arbitrary fiber bundle π M:FM\pi_M: F\rightarrow M is only a vector bundle over the total space FF. It follows, the tensor product bundle V *F dT *MV^*F \otimes \wedge^d T^* M can only be taken over FF, i.e., it really stands for V *F Fπ M *( dT *M)V^*F \otimes_F \pi^*_M(\wedge^d T^*M). Since the latter is a vector bundle π F:V *F dT *MF\pi_F : V^*F \otimes \wedge^d T^* M \rightarrow F (over FF), there is a canonical 00-section 0 F:FV *F dT *M0_F : F \rightarrow V^*F \otimes \wedge^d T^* M. Taking sections over MM yields the field space Γ M(F)\Gamma_M(F) and the variational cotangent bundle Γ M(V *F dT *M)\Gamma_M(V^*F \otimes \wedge^d T^* M). Postcomposition of sections of V *F dT *MV^*F \otimes \wedge^d T^* M with the projection π F\pi_F yields the (smooth set) bundle projection π :T var *\pi_{\mathcal{F}}: T^*_\mathrm{var} \mathcal{F} \rightarrow \mathcal{F}, while postcomposition of sections of FF with the (manifold) zero section 0 F0_F yields the (smooth set) zero section 0 :T var *0_{\mathcal{F}}: \mathcal{F}\rightarrow T^*_\mathrm{var} \mathcal{F}.

    Let me know if this clears up the notation used.

    Happy new year!

    • CommentRowNumber9.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 31st 2024
    • (edited Jan 31st 2024)

    By the way, I assigned this paper for my student seminar:

    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.

    • CommentRowNumber10.
    • CommentAuthorjhoo
    • CommentTimeJan 31st 2024

    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 Φ=π FX Φ\Phi=\pi_F\circ X_\Phi is meant to be. In particular, I am unsure of what X ΦX_\Phi is. If I assume this to be a typo, I could replace this with Z ΦZ_\Phi, but this also does not make sense in context of Z ΦZ_\Phi covering Φ\Phi. If there is any clarification the authors could make, that would be fantastic!

    • CommentRowNumber11.
    • CommentAuthorGrigorios
    • CommentTimeJan 31st 2024
    • (edited Jan 31st 2024)

    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 ϕ=π FZ ϕ \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 ϕZ_\phi covering ϕ\phi, since this is (by definition) precisely what it means to cover a map ϕ:MF\phi: M \rightarrow F via a map Z ϕ:MVFZ_\phi:M \rightarrow VF, with respect to the canonical projection VFFVF\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 Ω dR \Omega^\bullet_\mathrm{dR} on ×M\mathcal{F}\times M” vs “local forms via J M FJ^\infinity_M F on ×M\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!