Triply Dyonic

I thought I’d mentioned this already, but I could not find anything after a search on the blog so somehow I think I must have forgotten to. It is a cute thing about a certain favourite solution (or class of solutions) of Einstein’s equations that I’ve talked about here before. I’m talking about the Taub-NUT solution (and its cousin, Taub-Bolt). Taub-NUT is sort of interesting for lots of reasons. Many, in fact. One of them concerns it having both mass [tex]M[/tex] and another parameter called “nut charge”, [tex]N[/tex]. There are several ways to think about what nut charge is, but one curious way is that is is sort of a “magnetic” counterpart to the ordinary mass, which can be thought of as an “electric” quantity.

The language is based on analogy with electromagnetism, where, in the usual spherical polar coordinates [tex](r,t,\theta,\phi)[/tex], the electric charge [tex]Q[/tex] is associated with [tex](r,t)[/tex] components of the Maxwell tensor, and magnetic charge [tex]P[/tex] associated with [tex](\theta,\phi)[/tex] parts. (Think [tex]F_{rt}\sim Q/r^2[/tex] and [tex]F_{\theta\phi}\sim P\sin\theta[/tex], following from [tex]A_t\sim -Q/r[/tex], [tex]A_\phi\sim P\cos\theta[/tex], etc.) Similarly, as you might know from the Schwarzschild solution, you have a [tex]g_{tt}[/tex] component of the metric with a leading [tex]M/r[/tex] piece that gives what we think of as mass, as measured asymptotically, which I can call “electric” as compared to a [tex]g_{t\phi}[/tex] part that has a leading [tex] N \cos\theta[/tex] part, which gives the “magnetic” mass.

So in some sense, the Taub-NUT solution is “dyon”, for mass. (“Dyon” is usually a term used when a particle has both magnetic and electric charge.) (It is this dyonic property that allows Euclidean Taub-NUT to play a nice role in the subject of gravitational instantons (see the classic paper of Hawking), where it ends up being the self–dual solution for the [tex]R\wedge R[/tex] sector that’s analogous to [tex]F\wedge F[/tex] for ordinary instanton solutions in Yang–Mills.)

Well, the cool thing is that if you embed Taub–NUT into Einstein-Maxwell theory (so, add a minimal electromagnetic sector to the theory), you can find (as Brill did a long time ago) a charged Taub–NUT solution. But guess what? What comes for free with that electric solution is a magnetic part. You get not just an [tex]A_t[/tex] but an [tex]A_\phi[/tex], for consistency.

So Taub–NUT is doubly a dyon in Einstein-Maxwell!

But it does not stop there. Many years ago, I discovered the first non–trivial embedding of Taub–NUT into string theory. Typically, you’d have some electromagnetic sector turned on, as usual, but in closed string theory, you also naturally have a scalar field called the dilaton [tex]\Phi[/tex] turned on, as well as the antisymmetric tensor [tex]B_{\mu\nu}[/tex]. Think of them as generically naturally present since one sets the strength of the string coupling while the other counts winding of the strings. Well, in four dimensions, the tensor is equivalent under hodge duality to a scalar which we can call the axion, [tex]a[/tex]. The axion and dilation are natural partners in duality and its extensions, again sort of like electric and magnetic sectors are. So something that has both axion and dilation charge is dyonic in that sense too. Guess what? Taub–NUT comes out (in those embeddings) as naturally dyonic in that sense.

So Taub–NUT is triply a dyon in string theory!

That’s all.

-cvj

P.S. Oh, the (click to enlarge) figures at the top of the post are from a paper I wrote last month about the phase structure of Taub-NUT and Taub-Bolt, along with a dyonic deformation, in the extended thermodynamics I’ve blogged about a few times recently (see list of related posts below). They are the properly labelled versions of what was calling the “swoosh” here. You can find references to the work of Hawking, Brill, and myself, that I mentioned above in that paper, along with the original Taub-NUT references too.