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I have started a category:reference page
such as to be able to point to it for reference, e.g. from Kontsevich 15 etc.
Damien Calaque’s contribution has just appeared, so I’ve added a link.
It says there about dcct and other articles:
Observe that these references also deal with variants of shifted pre-symplectic structures on stacks, but the non-degeneracy condition is almost never satisfied as everything takes place in the realm of underived stacks.
I was reminded of our conversation here.
By the way, the talk slides link at Prequantum field theories from Shifted symplectic structures doesn’t work.
Anel’s paper added too.
Good final flourish to the paper!
Thanks for the alerts. I have fixed that link.
It says there about dcct and other articles:
Observe that these references also deal with variants of shifted pre-symplectic structures on stacks, but the non-degeneracy condition is almost never satisfied as everything takes place in the realm of underived stacks.
It is noteworthy that non-degenerate symplectic structure in gauge field theory is all about recitifying a homotopy-theoretic structure: the gauge-fixing of the BV-BRST complex which ensures the non-degenerate graded symplectic structure is a means to quantize homotopy-theoretically by doing it naively but degreewise, respecting a differential (chapter 11). This is in direct analogy to how a naive Lie algebra considered degreewise and respecting a differential (hence a dg-Lie algebra) is a rigidified model for a strong homotopy Lie algebra.
While it is true that presently this homotopy-rigidified trick is the only known way to quantize gauge field theory, in general, it is clear from the point of view of homotopy theory that this must be but a tool and a convenience, not a fundamental necessity. It ought to be true that there is a homotpy-quantization procedure which reads in the non-gauge fixed and hence degenerate presymplectic current (aka shifted pre-symplectic structure) and quantizes it right away.
This is ultimately what Marco Benini and Alexander Schenkel are headed for in their homotopical AQFT. Their toy example of free electromagnetism sort of works this way already, but there is a long way to go until this will be understood generally. Meanwhile, it is good to have a clear picture of the role that symplectic rather than pre-symplectic structure plays in QFT.
So a way needs to be found to realise
quantization is the result of forming the homotopy quotient of the space of Lagrangian data by these duality relations?
That sounds possibly circular, since how would one know about duality relations without first having an independent construction of quantization in the first place. But who knows what the future will bring.
But the point under discussion above is a different one: The Lagrangian densities that define Lagrangian field theory (up to renormalization choices) a priori yield higher pre-symplectic structure, not higher symplectic structure. The latter is obtained only after auxiliary fields are adjoined and a choice of BV-gauge fixing is made, which is directly analogous to choosing a differential graded algebra over an operad as a rectified model for an $\infty$-algebra over an $\infty$ -operad. It works and is often convenient, sometimes it may even be the only tool under control, but it is not part of the definition of the homotopy theoretic concept.
That’s why I always thought it pays to have a thorough look at higher pre-quantum geometry first, before hastening to make assumptions about what higher quantum geometry should be like. First things first. In any case, it is not a fault or omission of higher pre-quantum geometry not to feature derived geometry and non-degenerate shifted symplectic form, rather this is the nature of the subject of Lagrangian field theory. Or so I think.
And the extension to look at differential graded algebroids belongs to same approach? I see from this abstract that your former student is developing this:
we explain how to apply this machinery to the case of non-split formal moduli problems under a given derived affine scheme; this situation has been dealt with recently by Joost Nuiten, and requires to replace differential graded Lie algebras with differential graded Lie algebroids.
Now by “the same approach” you are referring to the general topic of rectification, right? Yes, the concept of dg-Lie algebroids is (or should be) the rectification of the concept of general $\infty$-Lie algebroids. To be more precise I would have called Joost’s def 2.1 in arXiv:1712.03442 that of dg-Lie-Rinehart pairs, but of course the difference is negligible in a context of dg-geometry.
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