Torsors Made Easy

John Baez

December 27, 2009

When you first read about "torsors", they're bound to seem a bit too darn sophisticated for their own good. Mostly this is because nobody explains them very well. So, I want to convince you that they're actually very simple and natural. I'll do this by giving you three examples from physics:

• In Newtonian mechanics, we can only measure energy differences, not energies themselves. The reason is that we can add any real number to our definition of energy without changing any of the physics. This means it doesn't make much sense to ask what the energy of a system is - we can answer this question only after picking an arbitrary convention about what counts as "zero energy". What makes more sense is to talk about the difference between the energy of a system in one state and the energy of that system in some other state.

  We can express this in terms of torsors as follows: energy differences lie in the group of real numbers R, but energies themselves do not: they lie in an "R-torsor".

• In electromagnetism, we can only measure voltage differences, not voltages themselves. The reason is that we can add any real number to our definition of the electromagnetic potential without changing any of the physics. In practical work with electrical circuits, we usually say the electromagnetic potential of the ground is zero. But this is an arbitrary choice.

  We can express this in terms of torsors as follows: voltage differences lie in the group of real numbers R, but voltages themselves do not: they lie in an "R-torsor".

• In quantum mechanics, we can only measure relative phases, not phases themselves. The reason is that we can multiply the phase of a quantum state by any unit complex number without changing any of the physics. So, it doesn't make much sense to ask what the phase of a quantum state is - we can answer this question only after picking an arbitrary convention. What makes sense more sense is to talk about the relative phase between two states that differ only by a phase.

  We can express this in terms of torsors as follows: relative phases lie in the group of unit complex numbers, which is called U(1), but phases themselves do not: they lie in a "U(1)-torsor".

These three examples are closely related. First of all, the energy of a charged particle at rest equals its charge times its voltage - so the fact that we can add any real number to our definition of voltage without changing any physics is closely connected to the similar fact for energy. Secondly, in quantum mechanics two states whose energy differs by E will experience a change of exp(-iEt) in their relative phase after a time t elapses.

Sometimes people think something is a torsor but later realize it's not! As Philip Dorrell pointed out to me, people once thought only temperature differences could be measured - but then they discovered absolute zero. As soon as we pick units of temperature, temperatures are elements of an R-torsor. When absolute zero was discovered, this R-torsor was revealed to be R itself.

But by now, you should be burning with curiosity about what a torsor actually is!

For any group G we can define a concept of "G-torsor". For starters, a G-torsor is a set X equipped with an "action" of G. Suppose we write the group operation as multiplication and write the identity element as 1. Then an action of G on X lets us multiply any element g of G and any element x of X to get an element gx of X, in such a way that the following axioms hold:

1x = x

and

(g₁ g₂) x = g₁ (g₂ x).

Group actions are a dime a dozen. The special thing that makes a group action be a torsor is this: for any two elements x₁ and x₂ of our torsor there exists a unique group element g with

g x₁ = x₂.

This means that for any two elements of a torsor we can talk about their "ratio". Their ratio, x₂/x₁, is the unique group element g for which the above equation holds. In other words:

(x₂/x₁) x₁ = x₂.

(Writing the group operation as multiplication is nice for the example of phases in quantum mechanics: a relative phase is indeed a ratio of phases. But we can also write the group operation as addition. This notation is more appropriate for energies in classical mechanics, since an energy difference is computed using subtraction.)

So here's the difference between a group and a torsor. I'll say it in additive language, just for a change of pace. In a group G you can can add elements and also subtract them. But you can't add elements of a G-torsor X. Instead, you can add an element of G to an element of X and get another element of X. You can also subtract two elements of X and get an element of G.

Here are two examples of torsors from everyday life, contributed by Phineas Porter:

• Dates — in the calendrical sense — are elements of a Z-torsor where Z is the group of integers. That is, it makes sense to subtract two dates and conclude that your dentist appointment is 6 days before Christmas, but there's no good reason you'd want to add the day of your dentist appointment and Christmas day. (Here I'm talking about a calendar that goes off to infinity in both directions, so I'm ignoring the Big Bang.)
• Notes — in the musical sense — are also elements of a Z-torsor. That is, it makes sense to ask how many half-steps one note is above another, but there's no good reason you'd want to add notes. (Here I'm talking about an equal-tempered chromatic scale, with equally spaced notes that go off to infinity in both directions.)

Here are two important examples of torsors from calculus and geometry:

• When you calculate the indefinite integral of some function f you get something like F + C, where the constant C can any real number. So, there's a whole set of choices: we call any one of these an "antiderivative" of f. And this set is an R-torsor! If I have an antiderivative of f, I can add any number to it and get another antiderivative of f. If I have two antiderivatives of f, I can take their difference and get a real number. But I can't add two antiderivatives of f and get another antiderivative of f.

  So, if we want to show off, we can define the indefinite integral of f to be the whole set of all antiderivatives of f. In other words: the set of all functions whose derivative is f. This set is an R-torsor. So, we can say "The indefinite integral of f is not a function: it's a torsor".

  In short, that mysterious "+C" we learn about in calculus is mysterious because we need torsors to really understand it! So call up your old calculus teacher and tell them this: "You should have told us about torsors!"

  This stuff shows up a lot in physics. We calculate potential energy by doing an integral of force over distance. This is why potential energies are defined only up to an additive constant... which is why energies are elements of an R-torsor.

• When you learn about vectors in the plane, sometimes you pick a point in the plane and call it the "origin". Then you can take any point in the plane and draw an arrow going from the origin to that point. And this lets you add points in the plane by adding their arrows:

  
      /^
  v+w/ | 
    /  | 
   /   |w
  /    |
  ----->
    v
  

  This makes the plane into a group.

  But sometimes you don't want to pick an origin. The world doesn't come with one, after all. Without an origin, you can't draw points as arrows. All you can do is draw them as "dots". You can't add these dots... but you can still subtract two of them and get an arrow! So, now you are treating the plane as a torsor!

  Whenever I teach students about vectors they find this stuff very confusing. The reason is that we're using vectors in two different ways. If your teacher didn't explain this stuff well, call them up and tell them "You should have told us about torsors!"

  This stuff also shows up a lot in physics. For example, it makes sense to add velocities, but not positions. Which position do you get by adding the position of Mt. Gibraltar and the the position of Mt. Everest? It makes no physical sense! But it does make sense to subtract positions - and in fact we do this as part of computing velocities. But when we subtract one position from another, we don't get a position: we get an arrow with a length and direction.

  The reason is that velocities really live in a group, but positions live in a torsor! Of course, velocities don't live in just any old group - they live in a special sort of group called a "vector space". Similarly for positions: we say they live in an "affine space". An affine space is a torsor for a vector space.

Again, these two examples are closely related, since the the position of a particle is an antiderivative of its velocity.

The second example has an important moral:

and more generally:

In fact, to turn a G-torsor into the group G, all we need to do is pick a point in our G-torsor and call it the "identity". I'll explain this in more detail later. First, a little philosophy.

The main reason torsors seem subtle is that they correct a bad habit that's deeply ingrained in all of us. Whenever we see a G-torsor, we are tempted to say "Oh look - it's G!"

When we see energies, we say "Oh look - real numbers!" When we see voltages, we say "Oh look - real numbers!" And when we see phases, we say "Oh look - unit complex numbers!" But it's not quite true - it's only energy differences and voltage differences that are real numbers, and relative phases that are unit complex numbers.

You will argue that I'm being nitpicky. If you're doing Newtonian mechanics, as soon as you pick one state of your system and declare it to have energy zero, then you can think of energies as real numbers. What could possibly be wrong with that?

Well, it's not too bad. I'll admit it: if you've survived this far without torsors, you can probably continue to survive without them. You can always pretend a torsor is a group. But, it involves an arbitrary choice! This means that when you do this, you're imposing some structure on the situation which is not really there already. Through long experience in mathematics, I've learned that making arbitrary choices causes problems. I could list specific technical problems, but the basic problem is that it distorts your thinking. So, the idea of torsors is to avoid pretending something is a group when it doesn't come naturally equipped with an identity element.

You can think about it this way. The group G is itself a G-torsor in an obvious way. And if you hand me any other G-torsor X, I can pretend it's G as soon as I pick one element x₁ in X and declare it to be the identity element of G. More precisely, I get a map from X to G which sends any element x₂ to the unique g such that

g x₁ = x₂.

This map is 1-1 and onto, so it makes X isomorphic to G. However, it depends on an arbitrary choice choice - so we say this isomorphism isn't "canonical".

So if you want to impress your friends, tell them: Any group G is a G-torsor, and every other G-torsor is isomorphic to G - but not canonically!

If this is enough for you, fine. But if you know some fancy physics, you may know that this stuff about energy differences, the electromagnetic potential, and relative phases is the tip of a big iceberg called "gauge theory". Gauge theory is all about how you can only tell what something is relative to something else. This is where torsors become important. Furthermore, to compare two things in different locations, you have to carry one to the other along a path through space - and your answers will depend on this path.

The mathematics of gauge theory involves fiber bundles. Thus, you shouldn't be surprised that torsors also show up in the theory of fiber bundles! What's the fiber of a principal G-bundle? It's a bit like the group G... but what it actually is, is a G-torsor.

Here's a famous example: the set of orthonormal frames at some point of a n-dimensional Riemannian manifold is not the group O(n), but it's an O(n)-torsor. You can take any frame and rotate it by an element of O(n); you can take two frames and work out their "difference", which is an element of O(n) - but the frames don't form a group. We can pretend the frames are the group O(n) - but only after we arbitrarily choose one frame and decree it to be the identity. Then every other frame is a rotated version of this one, so we can pretend it is a rotation!

Here we see a specific technical problem that hits us if we try to pretend torsors are groups. Yes, you can pretend the frames at some point of a Riemannian manifold are the same as elements of O(n), but doing so involves an arbitrary choice of frame. And if you try to make such a choice at every point of your manifold, it's usually impossible to do it in a smoothly varying way! So, you can't do this trick and still play games involving calculus. It's better not to even try.

Here's an even fancier example, involving spinors. To define spinors on a manifold, we need to give that manifold a "spin structure".

Suppose you have a manifold that admits a spin structure. How many spin structures does it have?

This question has a nice answer: if a manifold M admits a spin structure, its set of spin structures is isomorphic to H¹(M,Z/2) - the first cohomology with coefficients in Z/2.

For example, since the two-sphere has a spin structure, it has just one, since H¹(S²,Z/2) = {0}. The torus has a spin structure, so it has four, since H¹(T²,Z/2) = Z/2 x Z/2.

I don't know if you know enough algebraic topology to see how pleasant this answer is. Among other things, an element of H¹(M,Z/2) assigns an element of Z/2 to any loop in M. Crudely speaking, this tells you how the spin structure "twists" as you go around that loop - i.e., whether a fermion switches sign when you carry it around that loop.

But actually, the last sentence was a lie! You can't really measure in any invariant way how much a spin structure twists as you go around a circle; you'd need a connection on your spin bundle to do that. All you can really do is compare how two different spin structures twist as you go around a loop, and get an element of Z/2 from that.

In other words, there is not a god-given map

[spin structures] → H¹(M,Z/2)

but if you fix any one spin structure and take it as the one to compare others to, you get a map

[spin structures] → H¹(M,Z/2)

which is 1-1 and onto.

Another way to put this is that starting with any spin structure, you can get all the rest by "inserting twists" given by elements of H¹(M,Z/2). But there's in general no god-given "untwisted" spin structure.

We can summarize this by saying that spin structures form a torsor for the group H¹(M,Z/2).

Torsors are everywhere. You have only to open your eyes and look!

Or open your ears and listen - they also show up in music theory.

Finally, one more remark for people who want to go further. Near the beginning of this essay, I said "as soon as we pick units of temperature, temperatures are elements of an R-torsor". We need to pick units of temperature to know what it means to "add 1" to a temperature. So, where should we think of temperatures a living before we pick units? We should think of them as living on a line whose symmetries include not just translations but also dilations - in other words, the "stretchings" or "squashings" that result from a change of units. Picking an origin reduces the symmetry group to just dilations - and indeed, there's a distinguished choice of origin, namely absolutely zero. Picking units reduces the symmetry group to just translations, giving us an R-torsor - and indeed, there's a distinguished choice of units, namely Planck units. Picking both lets us think of temperatures as real numbers. This combination of translations and dilations arises because R is not just a group, but a ring. So, there's a more sophisticated concept than that of "torsor" allowing both translations and dilations whenever you start with a ring.

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