T-duality and Isometries of Spacetime

I’ve just been to an excellent seminar on Double Field Theory by its co-creator, Chris Hull. You may know that string theory exhibits a meta-symmetry called T-duality. More precisely, it’s equivalent to put closed strings on circles of radius $R$ and $1/R$.

This is the simplest version of T-duality, when spacetime has no background fields. Now suppose we turn on the Kalb-Ramond field $B$. This is just an excitation of the string which generalizes electromagnetic potential.

This has the effect of making T-duality more complicated. In fact it promotes the $R\to 1/R$ symmetry to $O(d,d;\mathbb{Z})$ where $d$ is the dimension of your torus. Importantly for this to work, we must choose a $B$ field which is constant in the compact directions, otherwise we lose the isometries that gave us T-duality in the first place.

Under this T-duality, the $B$ field and $G$ metric get mixed up. This can have dramatic consequences for the underlying geometry! In particular our new metric may not patch together by diffeomorphisms on our spacetime. Similarly our new Kalb-Ramond field $B$ may not patch together via diffeomorphisms and gauge transformations. We call such strange backgrounds non-geometric.

To express this more succintly, let’s package diffeomorphisms and gauge transformations together under the name generalized diffeomorphisms. We can now say that T-duality does not respect the patching conditions of generalized diffeomorphisms. Put another way, the $O(d,d)$ group does not embed within the group of generalized diffeomorphisms of our spacetime!

This lack of geometry is rather irritating. We physicists tend to like to picture things, and T-duality has just ruined our intuition! But here’s where Double Field Theory comes in. The idea is to double the coordinates of your compact space, so that $O(d,d)$ transformations just become rotations! Now T-duality clearly embeds within generalized diffeomorphisms and geometry has returned.

All this complexity got me thinking about an easier problem – what do we mean by an isometry in a theory with background fields? In vacuum isometries are defined as diffeomorphisms which preserve the metric. Infinitesimally these are generated by Killing vector fields, defined to obey the equation

$\displaystyle \mathcal{L}_K g=0$

Now suppose you add in background fields, in the form of an energy-momentum tensor $T$. If we want a Killing vector $K$ to generate an overall symmetry then we’d better have

$\displaystyle \mathcal{L}_K T=0$

In fact this equation follows from the last one through Einstein’s equations. If your metric solves gravity with background fields, then any isometry of the metric automatically preserves the energy momentum tensor. This is known as the matter collineation theorem.

But hang on, the energy momentum tensor doesn’t capture all the dynamics of a background field. Working with a Kalb-Ramond field for instance, it’s the potential $B$ which is the important quantity. So if we want our Killing vector field to be a symmetry of the full system we must also have

$\displaystyle \mathcal{L}_K B=0$

at least up to a gauge transformation of $B$. Visually if we have a magnetic field pointing upwards everywhere then our symmetry diffeomorphism had better not twist it round!

So from a physical perspective, we should really view background fields as an integral part of spacetime geometry. It’s then natural to combine fields with the metric to create a generalized metric. A cute observation perhaps, but it’s not immediately useful!

Here’s where T-duality joins the party. The extended objects of string theory (and their low energy descriptions in supergravity) possess duality symmetries which exchange pieces of the generalized metric. So in a stringy world it’s simplest to work with the generalized metric as a whole.

And that brings us full circle. Double Field Theory exactly manifests the duality symmetries of the generalized metric! Not only is this mathematically helpful, it’s also an important conceptual step on the road to unification via strings. If that road exists.

Integrating Differentials in Thermodynamics

I’ve just realised I made a mistake when teaching my statistical physics course last term. Fortunately it was a minor and careless maths mistake, rather than any lack of physics clarity. But it’s time to set the record straight!

Often in thermodynamics you will derive equations in differential form. For example, you might be given some equations of state and asked to derive the entropy of a system using the first law

$\displaystyle dE = TdS - pdV$

My error pertained to exactly such a situation. My students had derived the equation

$\displaystyle dS = (V/E)^{1/2}dE+(E/V)^{1/2}dV$

and were asked to integrate this up to find $S$. Naively you simply integrate each separately and add the answers. But of course this is wrong! Or more precisely this is only correct if you get the limits of integration exactly right.

Let’s return to my cryptic comment about limits of integration later, and for now I’ll recap the correct way to go about the problem. There are four steps.

1. Rewrite it as a system of partial DEs

This is easy – we just have

$\displaystyle \partial S/\partial E = (V/E)^{1/2} \textrm{ and } \partial S / \partial V = (E/V)^{1/2}$

2. Integrate w.r.t. E adding an integration function $g(V)$

Again we do what it says on the tin, namely

$\displaystyle S(E,V) = 2 (EV)^{1/2} + g(V)$

3. Substitute in the $\partial S/\partial V$ equation to derive an ODE for $g$

We get $dG/dV = 0$ in this case, easy as.

4. Solve this ODE and write down the full answer

Immediately we know that $g$ is just a constant function, so we can write

$\displaystyle S(E,V) = 2 (EV)^{1/2} + \textrm{const}$

Contrast this with the wrong answer from naively integrating up and adding each term. This would have produced $4(EV)^{1/2}$, a factor of $2$ out!

So what of my mysterious comment about limits above. Well, because $dS$ is an exact differential, we know that we can integrate it over any path and will get the same answer. This path independence is an important reason that the entropy is a genuine physical quantity, whereas there’s no absolute notion of heat. In particular we can find $S$ by integrating along the $x'$ axis to $x' = x$ then in the $y'$ direction from $(x',y')=(x,0)$ to $(x',y')=(x,y)$.

Mathematically this looks like

$\displaystyle S(E,V) = \int_{(0,0)}^{(E,V)} dS = \int_{(0,0)}^{(E,0)}(V'/E')^{1/2}dE' + \int_{(E,0)}^{(E,V)}(E'/V')^{1/2}dV'$

The first integral now gives $0$ since $V=0$ along the $E$ axis. The second term gives the correct answer $S(E,V) = 2(EV)^{1/2}$ as required.

In case you want a third way to solve this problem correctly, check out this webpage which proves another means of integrating differentials correctly!

So there you have it – your health warning for today. Take care when integrating your differentials!

Bibliographies and The arXiv

I’m currently writing up my first paper! The hope is that my collaborators and I will release the paper in the next couple of months. When we do, it’ll go on the arXiv – a publically accessible preprint server.

This open-access policy is adopted pretty much universally throughout mathematics and theoretical physics. I think it’s extremely good for science to be freely accessible to all. There’s still a place for journals, allowing research to be ranked by quality and rigorously peer reviewed. But the arXiv is vital in maintaining the pace of research, particularly in hot topic areas.

Every piece of work on the arXiv gets its own unique identifier. I rather like codes, so I tend to remember these numbers for my favourite papers. Just typing the number into Google search immediately takes you to the relevant document.

My current paper draft is peppered with arXiv numbers referring to important papers we need to cite. When we come to making a bibliography I’ll need to convert these into a standard form. Technically this involves making a BibTex file, and referring to it in my typesetting program.

I thought this would take ages, but it turns out that there’s an online Easter Egg solving the problem in a flash. Inspire HEP is a database of physics papers, providing all the metadata you could ever need including ready formatted BibTex. And it even has a feature which automatically generates a bibliography for you – check it out!

If you’re writing up your first paper and this tip helped you out, do drop me a line in the comments! And to the curators of arXiv and Inspire HEP – a huge thank you from me and the whole physics community.