I am studying Chris Elliott's notes on Line and Surface Operators in Gauge Theories (available here).

In the notes, there's a mention of the fact that (for $G = U(1)$),

$$W_{\gamma, n}(A) = e^{in\oint_{\gamma}A}.$$

The gauge field $A$ is not actually a 1-form, but upon choosing a principal $U(1)$-bundle the connections on that bundle become a torsor for $\Omega^1(X)$.

Is there an intuitive way to understand the idea of a torsor in this context?

EDIT: I found a nice post by John Baez from 2009: http://math.ucr.edu/home/baez/torsors.html, which explains a few things.


Oversimplified & in a nutshell:

  1. Recall the slogan

    A $G$-torsor is like the group $G$ that has forgotten its neutral element.

  2. Example: An affine space $A$ is a torsor for a vector space $V$.

  3. The space of $U(1)$ gauge fields is an affine space, while $\Omega^1(X)$ is a vector space.

  • $\begingroup$ Thanks for the succinct answer, @Qmechanic. Is there a relation between the notion of "framing" and a torsor? $\endgroup$ – leastaction Mar 27 '18 at 20:33
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    $\begingroup$ @leastaction don't think so $\endgroup$ – Ryan Thorngren Mar 27 '18 at 22:34

A torsor T is just an algebra equipped with a ternary operation a,b,c ∈ T ↦ abc ∈ T such that abb = a = bba and (abc)de = ab(cde).

The archetypical example of a torsor is a group G whose product a,b ∈ G ↦ a·b ∈ G and inverse a ∈ G ↦ a⁻¹ ∈ G yield the torsor operation abc ≡ a·b⁻¹·c. This operation is "affine" in the sense that it is covariant with respect to multiplication on left and right - (g·a·h)(g·b·h)(g·c·h) = g·abc·h; so the structure removes the "first class" standing of the group identity.

The group operations themselves can be recovered from the ternary operation, once a group identity e ∈ G has been specified, by defining a·b ≡ aeb, a⁻¹ ≡ eae. Then it is a routine matter to show that the group axioms follow from the torsor axioms.

Relations suggested by this example, such as these (ade)(bde)(cde) = (abc)de, (abc)(abd)(abe) = ab(cde), (abe)(ace)(ade) = a(dcb)e, (abc)de = a(dcb)e = ab(cde), can all be proven from the torsor axioms.

The torsors that are associated with Abelian groups are precisely those for which the identity abc = cba also holds. Consequently, these may be termed Abelian torsors.

A torsor T contains within it a natural group structure in two ways. First, each e ∈ T can be taken as the identity of a group T_e whose operations are defined as just indicated. This, you might call the "tangent group" at e. That they are all groups confirms the equal standing that all points have in the torsor: that any of them can be taken as the group identity.

Second, one can also define a formal quotient a,b ∈ T ↦ a\b ≡ ρ[(a,b)] with the equivalence classes taken with respect to the equivalence relation ρ ∈ T×T generated from the relations (cba,d) ρ (a,bcd). This implements the relation cba\d = a\bcd. The resulting algebra δT ≡ (T×T)/ρ is a group with product (a\b)·(c\d) = cba\d = a\bcd and (thus) a group identity equal to a\a for all a ∈ T, and an inverse (a\b)⁻¹ = b\a.

This group acts on the torsor on the right by the action a(b\c) = abc. You may confirm this by noting that a(b\c)(d\e) = (abc)de = ab(cde) = a(b\cde); while the well-definedness of the operation follows since a(dcb\e) = a(dcb)e = ab(cde).

The tangent groups T_e, for each e ∈ T are isomorphic to each other and to δT; with the isomorphism given by the correspondences π_e: a ∈ T_e ↦ e\a ∈ δT, π_e⁻¹: a\b ∈ δT ↦ eab ∈ T_e, which are inverses of one another. The composition π_f⁻¹ ∘ π_e: a ∈ T_e ↦ fea ∈ T_f yields the isomorphisms between the respective tangent groups.

From this, you can also talk about a Lie torsor T as a manifold whose torsor operation satisfies suitable smoothness properties. The tangent groups and group action then provide the structure of a principal bundle on the torsor T with fibre δT. In fact, the very definition of principal bundles (and associated bundles) can themselves be rendered in a more transparent and direct fashion in an analogous way to what has been done here.

Also related are affine geometries, which bear the same relation to vector spaces as torsors do to groups. In addition to the Abelian torsor operation a,b,c ∈ A ↦ abc = a - b + c ∈ A of an affine geometry A, you also have the "barycenter" operation a,c ∈ A, λ ∈ F ↦ [a,λ,c] = (1-λ)a + λc ∈ A, where F is the underlying field.

To recover the torsor properties, and to recover the structure of a vector space over the field F for the fibres A_o, for each o ∈ A, it is enough to postulate the following identities [a,0,c] = a, [a,1,c] = c and [a,λν(1-ν),[b,μ,c]] = [[a,λν(1-μ),b],ν,[a,λμ(1-ν),c]]; and to define the torsor operation by abc = [b,1/(1-λ),a],λ,[b,1/λ,c]] for any λ ∈ F - {0,1} ... the independence from λ following as a consequence of the other properties. This suffices to characterize Affine geometries over all fields except the 2 and 3 element fields; which must be handled specially.

Other, more needlessly complex and obfuscated, definitions have been posed for torsors and are in common use (the same for affine geometries, as well as principal and associated bundles). It is a routine matter to establish the equivalence of them each to what has been described here. The alternative constructions are mostly unnecessary, except to attach some names to ideas that can be more transparently developed and expressed, as just done here. You may want to try the exercise of drawing the relevant correspondences, just to see how they relate to this and to make the ideas described by them accessible.

It bears repeating an observation I've previously made elsewhere on this matter. I don't know why mathematicians obfuscate their formalism by going out of their way to avoid writing things in a natural, obvious and transparent way as done here. But I think a large part of it is that what we call "job security". Like the Egyptian scribes who kept a monopoly on orthography by intentionally using an overwrought system even when much better (and more easily teachable) alternatives were eventually available (the Sinai script and Phoenician), they may do it to over-intellectualize the subject and put an intellectual firewall around the subject so as to erect a barrier to entry.

This is, of course, just the tip of the iceberg. Most other fields of mathematics have a firewall of obfuscation erected around them, similar to what you've just seen here. This problem is deeply ingrained in human nature and is, needless to say, why I think the time has come to simply automate the field and remove the human from the picture.

Just like when Logic Theorist ( https://history-computer.com/Library/Logic%20Theorist%20memorandum.pdf ) found the dramatically simpler proof of Pons Asinorum; the kind of simplification you just saw here is what you will see arise as a result. Or perhaps a framework for automation is already operational at this location, and I'm speaking in the past tense with this very description being a product of that automation. :)

Higher Order Logic Theorem Proving
https://www.google.com/search?q=Theorem+Provers+in+Higher+Order+Logic ;
Fully Automated Theorem Provers
https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20050184118.pdf ;
Automated Theorem Discovery


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