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What's a concrete example where groupoids are better suited to describe conventional gauge theory?

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I've recently started reading about groupoids. I understand that they (higher-groupoids) are useful in the context of speculative theories, like string theory, where we are dealing with branes etc. However groupoids are also valuable in conventional gauge theories like, for example, the standard model. To quote from the book What is a Mathematical Concept?

 From […] to the objects of field values for gauge field theory, which need to keep track of gauge equivalences, we find we are treating geometric forms of groupoid. If in the latter case instead we take the simple quotient, the plain set of equivalences classes, which amount to taking gauge symmetries to be redundant, the physics goes wrong in some sense, in that we cannot retain a local quantum field theory. Furthermore, in the case of gauge equivalence, we do not just have one level of arrows between points or objects, but arrows between arrows and so on, representing equivalences between gauge equivalences. 

What's a concrete example to demonstrate this? So far, I've only found very abstract examples that talk about some gauge transformations $g$ and $h$ abstractly

So what I'm looking for, is an example that is really explicit. For example, let's take a concrete structure group like $U(1)$. Then let's specify some gauge transformations $g(x)= ...$ and $h(x)= ...$, where the dots should be replaced by concrete functions of $x$. 

  • So how, can we demonstrate concretely that in gauge theories, "arrows between arrows", i.e. gauge transformations of gauge transformations are also important?
  • And how can we see that something goes wrong if we naively factor out local gauge transformations from the description?
asked 5 days ago in Theoretical Physics by JakobS (70 points) [ no revision ]

Hi JakobS. I believe all this is saying is that when you take a quotient of a QFT by a symmetry group G it is naturally understood as a "homotopy" or "(higher) groupoid" quotient, obtained by "adding arrows" to the configuration space of the theory, ie. we introduce a new degree of freedom, the G gauge field, rather than take degrees of freedom away. Then we have arrows between arrows which are the gauge transformations and things can get even hairier if you start with something like a 2-group action instead of just a group action.

1 Answer

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The core of the easy argument of how groupoids are necessary to retain both locality as well as instanton sectors in gauge theory is in my talk notes "Higher field bundles for gauge fields". For more elementary technical background see the master thesis "Stacks in gauge theory"  that I supervised. For the latest developments see the references at "homotopical AQFT".

But the quickest route to see that groupoids already are all over the place in gauge theory is to notice that the BRST-complex is nothing but the dual incarnation (the Chevalley-Eilenberg algebra) of the infinitesimal approximation of the gauge groupoid: Just like every Lie group is infinitesimally approximated by (and under good conditions reconstructible from) a Lie algebra, so every Lie groupoid is infinitesimally approximated (and in good situations reconstructible) by a Lie algebroid . This Lie algebroid for gauge theory is embodied by the BRST complex.

Hence when you open books on "Quantization of Gauge Theories", such as that by Henneaux-Teitelboim, which tell you that such quantization proceeds by the BRST complex, they are secretly saying that it proceeds by Lie algebroids, hence by Lie groupoids. Only that they don't say these words, but that's exactly what it is.

This is discussed in geometry of physics -- A first idea of quantum  field theory in the sections 10. Gauge symmetries  11. Reduced phase space and 12. Gauge fixing .

answered 7 hours ago by Urs Schreiber (5,745 points) [ revision history ]

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