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We are finalizing a contribution for a book on mathematical aspects of quantum field theory:
Domenico Fiorenza, Hisham Sati, Urs Schreiber,
Abstract This text is a gentle exposition of some basic constructions and results in the extended prequantum theory of Chern-Simons-type gauge field theories. We explain in some detail how the action functional of ordinary 3d Chern-Simons theory is naturally localized (“extended”, “mutli-tiered”) to a map on the universal moduli stack of principal connections, a map that itself modulates a circle-principal 3-connection on that moduli stack, and how the iterated transgressions of this extended Lagrangian unify the action functional with its prequantum bundle and with the WZW-functional. At the end we provide a brief review and outlook of the higher prequantum field theory of which this is a first example. This includes a higher geometric description of the supersymmetric case.
Comments are welcome!
“mutli-tiered” in abstract.
With all this systematic treatment of field theories, do you think you are almost glimpsing what you called
the theory-as-in-logic of theories-as-in-physics?
Hi David,
thanks for catching that typo!
With all this systematic treatment of field theories, do you think you are almost glimpsing what you called
the theory-as-in-logic of theories-as-in-physics?
Maybe glimpsing some central aspects of this.
In physics one routineley speaks of a Lagrangian/action functional as being “a theory”, “a quantum field theory”. This is somewhat loosely speaking, but it is quite common practice. In this sense: what we discuss is what it means for a Lagrangian to be “extended” in the sense that its prequantum theory matches the notion of “extended” in “extended topological quantum field theory”, and that such an extended Lagrangian is nothing but a universal characteristic map, but refined to smooth cohomology/smooth stacks and then to differential cohomology.
While we are not quite there yet, the next step should be a proof that going beyond the geometric prequantum theory of such extended Lagrangians to their genuine geometric quantization indeed yields the corresponding extended quantum field theory.
We talked elsewhere about how a plain monoidal $n$-functor on $n$-dimensional cobordisms may be less than what qualifies as a “physical theory”, and that one may need to know which Lagrangian it comes from under quantization in order to interpret it as actual physics. To the extent that this is correct, I suppose one could regard the extended prequantum theory that we discuss as a step that gets closer to a full formalization of what “a (fundamental) physical theory” is. But before all dust has settled, this remains a bit speculative.
Suppose that works out as hoped for. In summary it would be a formalization and refinement of the traditional idea that a “quantum field theory” is a Lagrangian and its quantization. To the extent that we are happy to call this a physical theory, what we provide would give a good bit of its formalization as a theory-as-in-logic. For we show that this is accurately axiomatized in cohesive homotopy type theory.
On the other hand, the quantum field theories obtained this way directly are “topological” only. Non-topological field theories sit in this story in two ways:
as holographic duals in one dimension lower. The extented prequantum theory that we discuss already sees parts of this, but certainly not yet the whole story, currently. There is more work needed here.
as fragments of a topological theory. For instance pure Yang-Mills theory is not topological, but Einstein-Yang-Mills theory is. An example for this that fits squarely into our disucussion is string field theory which (as inidicated there) is a higher Chern-Simons type field theory the way we discuss, and which does contain things theories like Yang-Mills theory etc. as fragments.
So, I’d think (or hope) that we are on a good way towards providing what you are rightly asking for. But at the moment a bit more still has to be worked out, I’d say. Maybe as at the end of the interview that I gave you one can conclude: for the moment it is remarkable that with these minimal axioms we get this far at all.
Let’s come back to this question in a year from now, next Christmas, and reflect on how much the completeness of the picture has improved by then.
What is fragment ? A sector ?
What is fragment ? A sector ?
Keep some of the fields of Einstein-Yang-Mills theory fixed (namely the field of gravity) then in the remaining fields it is Yang-Mills theory on the given curved background.
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