added pointer to:

- Maxim Grigoriev, Dmitry Rudinsky,
*Notes on the $L_\infty$-approach to local gauge field theories*, Journal of Geometry and Physics**190**(2023) 104863 [arXiv2303.08990, doi:10.1016/j.geomphys.2023.104863]

added pointer to today’s

- Marco Benini, Pavel Safronov, Alexander Schenkel,
*Classical BV formalism for group actions*(arXiv:2104.14886)

Grammar fixed.

Anonymous

]]>I have added today’s reference

- Alberto Cattaneo, Pavel Mnev, Nicolai Reshetikhin,
*Perturbative quantum gauge theories on manifolds with boundary*, arxiv/1507.01221

at *BV-BRST complex* there used to be a paragraph on how over a manifold with a volume form the BV complex of multivectorfields is dual to the deRham complex.

I have reorganized that and then expanded to sections now called

Even though I expanded a good bit the material that used to be there, this still needs further work. But it should be a start. For more see Owen Gwilliam's thesis.

]]>like several of the comments above, this evokes BFV more than BV formalism

e.g. BFV feature constraints prominently whereas I thought BV was all aboout Lagrangians with symms??

btw, it's hard to search for notation like $T^*[1]M$ or am I just illiterate at seaerchin?

a basic reference would be appreciated ]]>

Yes, I like it a lot. That’s his contribution to our book.

]]>Paugam’s article looks quite readable!

]]>added that reference also to Berezin integral and integration over supermanifolds

]]>I have added to BV-BRST formalism a link to Frederic Paugam’s latest: *Homotopical Poisson reduction of gauge theories* .

added to BRST-BV formalism the pointer to the reference by Costello-Gwilliam where they indicate how the BV-complex ought to be thought of as a derived critical locus, and added a pointer to the page where I am trying to work out some details.

]]>I have started a section Lagrangian BV – the standard construction meant to provide a review and lightweight formalization of the standard story in the literature.

So far I have made it to the point where the antifield-antighost Tate-like resolution has been defined. Running out of steam for the moment.

]]>In Kontsevich’s approach to renormalizaton via OPE’s the sheaf of local fields also forms a D-module and the sections of that sheaf tensored with the volume form (properly shifted in degree) form an $L_\infty$-algebra. In the case of trasnlationally invariant theory he gets an algebra over a little disk operad in the appropriate dimension.

]]>I redid much of the Idea-section at BRST-BV formalism, trying to reduce repetition and add more information. i think it is better now than it was before,

]]>I have tried to clean up BRST-BV formalism a little.

And then I finally added some genuine content: in the section on homotopical symplectic reduction I state the main definitions and theorems. But formatting still a bit rough. Can try to polish later.

]]>Sorry, I meant FQFT !!

I am not versatile in your notation though I spent much of my time and therefore risk my career following your approach last two years. I correct/edited.

]]>AQFT is a fantastic insight into topological theories

What do you mean by that? One thing that AQFT (in the sense of Haag-Kastler local nets) does not apply to is TFTs. It applies to QFT on Minkowski space or, with some modifications, on Lorentzian spacetimes. Such as the worldsheet of a string.

]]>Hm, a discussion which is a bit of detour from the title above (BV-BRST formalism). But in paralellel there is $n$lab activity on BRST stuff! Just to remind you that BRST reduction is another side of the coin of mathematician’s semiinfinite cohomology for which I created the entry today; Koszul duality being one of the building blocks both for this and currently changing entry Chevalley-Eilenberg algebra. I wish somebody will systematically relate these nlab entries soon.

As far as the discussion above is concerned, FQFT is a fantastic insight into topological theories but for general QFTs we need to go back to analysis and geometry, beyond the topological terms (say in semiclassical expansion) which are of higher categorical nature, as most of the topology is.

]]>The opening paragraph is pure rubbish.

Ahm, I’d hope you know what you are talking about when you say this.

]]>Urs said:

Just focus on Edward Witten

It is funny you mention Witten because Witten is exactly the person that inspired me to proclaim my unhappiness in comment #12 :)

Zoran modified Edward Witten. I followed the link to his web site and had a look at the prominently linked article

The opening paragraph is pure rubbish.

Now I try to think of myself as open minded and if you tell me Witten’s recent paper has the potential to “bring it home”, then I’m happy to give it the benefit of doubt, but your description did not sound very much like physics to me, which is probably just my ignorance speaking.

Chern-Simons theory seems interesting. It has popped up in my reading list repeatedly over the years, but is it physics? Is the connection (pun?) from Chern-Simons to physics via BF theory? BF theory always seems “close” to being about physics. I remember many late nights in grad school with my mind swirling thinking about “area flux”, i.e. the flux of some field imposing area on the surface it penetrates and “volume flux” etc. Regge calculus. All super fun stuff and all close to physics in my opinion.

Anyway, if physics is about to make a come back as you suggest, I’ll be happy to watch from here on the sidelines.

]]>That’s a pretty wide tar-brush there, Eric. Care to explain what you mean more specifically?

First, let me repeat I was talking about physics and not mathematics. Historically, the two have progressed hand in hand, but I do not see that happening anymore. I am hopeful that things will change, but I’m not very happy with the direction physics has gone. I think and have always thought that (higher) category theory will provide a language to “bring things back home” for physics, but I haven’t seen anything in a long time (except maybe a few glimmers with Kantization) that tells me we’re any closer.

Back when I was a physics student, I was fascinated by the Einstein-Bohr debates, but I always interpreted the outcome differently than most. I think most physicists today would argue that among the two, Bohr was “most right”. I guess I never gave up on Einstein. I still expect him to be vindicated one day. With the “triumph” of Bohr, physicists stopped asking questions. Giants like Feynman warned:

“I think it is safe to say that no one understands quantum mechanics. Do not keep saying to yourself, if you can possibly avoid it, ’ But how can it be like that?’ because you will go ’down the drain’ into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.”

Consequently, many people (sheep?) decided to stop asking those questions: “How can it be like that?”

But in my opinion, if you are not asking that question, you are not truly a physicist as Einstein probably would have defined it because that is the only question that should matter to a physicist: “How can it be like that?”

]]>Concering physics (by which I take it is meant fundamental physics):

I see us entering in a valuable consolidation phase. In the 80s and 90s there was an explosion of ideas that overstretched what the community could cope with, to the extent that many now think it was all wasted.

But the opposite is true: we are seeing now these ideas finally be worked out and fixed.

Just focus on Edward Witten, arguably the epicenter of that explosion of ideas. Did you see his last article? This is somewhat remarkable: is a followup of

E. Witten,

*Some computations in background independent Open-String Field Theory*, Phys. Rev. D47 (1993) 3405-3410.E. Witten,

*Quantum background independence in String Theory*

These had been visionary articles. Back then in the 90s this was taken as the way physics should be done. But of course these articles had gaps. The new one now observes that using AQFT techniques, these gaps can be filled.

I think everybody who has been following the scene in the last two decades or so will appreciate that something noteworthy is going on when finally AQFT methods are being applied to demonstrate the infinitesimal background independence of the string perturbation series.

As another example, consider the development of TCFT. That’s quite amazing. Witten has been talking about the A-model and the B-model for ages, Kontsevich has been making famous conjectures about it. Now it suddenly exists.

Witten had argued in 1992 that the string perturbation series for the A- and the B-model – i.e. the thing that computes the physics in the emergent spacetime that these models describe strings propagating in – is Chern-Simons theory. That has now suddenly become a theorem (due to Costello).

I think these developments are most remarkable. I think eventually the whole development $\sim 1985$ to $\sim 1999$ or so (before the effective collaps of the system) of fundamental physics will be redone now, with the appropriate tools that were missing back then.

I believe this to the extent that I am co-editing a book based on this statement: Mathematical Foundations of Quantum Field and Perturbative String Theory (schreiber)

]]>The “awe” is disappearing. Especially when it comes to physics. I don’t think anyone really understands anything better today than we did 75 years ago. I see people using a bunch of fancy tools to tackle questions whose foundations are questionable to begin with.

That’s a pretty wide tar-brush there, Eric. Care to explain what you mean more specifically?

]]>Eric wrote:

As an aside, the more I learn, the more disappointed I become. The “awe” is disappearing. Especially when it comes to physics.

As far as math goes, my awe level keeps going up, especially when I relax, stop trying to keep up with what other people are doing (math as a competitive *business*), and simply think about what whatever I like. There are really deep mysteries which we are getting closer to understanding, yet grow and expand the more we learn about them. Like the Riemann hypothesis, to take a sort of random example. At first it seems like a mildly entertaining question about whether a bunch of zeroes of some function lie on a line, with a cute application to the asymptotic density of primes. But then you see it’s related to a mysterious dynamical system, and the geometry of a mysterious space such that functions on this space are integers, and “the mysteries of the real prime”, and the equally mysterious “field with one element”, and “the search for a deeper base”… and you start to suspect that it’s not a puzzle we’ll solve by simply blundering around and working harder using known techniques, but something that could - if we let it! - force a revolution in basic concepts.

In short, math has *layers of depth* that one can probe only by sustained thought, which lead one to ever more staggeringly cosmic insights… and there’s no sign I see that this process is anywhere close to being *done*.

Which is why it’s a pity civilization may not last long enough for humankind to get to the *really* cool stuff.

How do you know you are even walking in the right direction?

In math, it’s surprisingly easy to tell when you’re walking in *a* right direction, and Urs is certainly walking in one.

When it comes to physics, there’s the element of trying to guess what Nature likes best, and that’s much more tricky - there’s an extra element of chance or luck or fate involved…

]]>