Worldsheet Instanton Corrections to the Kaluza-Klein Monopole

The paper itself:

[Published: JHEP10(2005)028] [arXiv.org: hep-th/0507204] [local PDF (285 Kb)]

Presentation at Great Lakes Strings 2006:

First, the briefest explanation I've come up with:

My advisor Jeff Harvey and I studied a type of object in string theory called a "Kaluza-Klein monopole". These objects were first described over twenty years ago by people studying how gravity works when there is an "extra dimension". Because string theory includes both gravity and extra dimensions, string theorists soon adopted these objects in their own work.

In our paper, we show that the proper description of Kaluza-Klein monopoles in string theory is subtly different than their description in ordinary gravity. The details are complicated, but the reason is essentially that string theory isn't just gravity with extra dimensions. Strange string configurations called "worldsheet instantons" change the shape of the object and give rise to other features not present in the gravity-only solution. Knowing this might lead to corrections or new insights in previous string theory research; we are now beginning to look for such applications.

And now for a bit more detail:

If you really want all the details, our paper itself is available from the links above. I suspect, though, that it may be a wee bit too technical for most of my friends and family to understand. At all. So for the sake of those who want a glimpse of our actual results, I've decided to attempt to give an explanation that a non-scientist might be able to understand. Admittedly, it would have to be a fairly interested and motivated non-scientist, but my friends and family are generally pretty clever folks.

I'm not going to start from scratch, of course. I'll assume that readers understand some background about string theory, most importantly the ideas of "extra dimensions" and "T-duality". You can get a feel for those ideas from my Introduction to String Theory presentation (at least skim the slides and the parts of the explanatory text in black). If you're in a hurry, you can jump straight to my slides about extra dimensions and their effects, and then to my explanation of T-duality (which you won't understand in detail without reading the previous slide and the following one, at least). You don't need to understand everything on those slides (especially not the colored, indented text), but they'll at least get your oriented to what the pictures below mean.

Once you're up to speed on those basics, you can dive into the "background" section below, which will introduce the "Kaluza-Klein monopole" and other objects that are relevant to our work. And finally, you'll be able to read a brief summary of what we actually did. I won't be able to explain much about how we carried out our calculations, but I'll at least try to explain the results.

Background

The first question I should answer is probably "What is a Kaluza-Klein monopole?" And to answer that, I'd better explain what those words mean. (Yes, I'm aware that practically every word in the paper's title is pure jargon. Sorry!)

Kaluza-Klein refers to the two people whose work in the 1920s first showed that (Einstein's) gravity with an extra dimension too small to see would look like it described both gravity and electromagnetism. I won't subject you to the details, but I wish I could! That attempt to "unify" gravity and E&M didn't work on its own, but the idea may still be important.
Monopole refers to a source of "magnetic charge" in electromagnetism. All of the magnets that physicists have found in experiments so far come from "magnetic dipoles", which have both a north pole and a south pole. A magnetic monopole would (e.g.) have only a south pole, much like an electron only has negative electric charge.

[A KK-monopole shown as a band around a vertical cylinder.
It can slide up and down along the cylinder's axis.]

So a Kaluza-Klein monopole is a solution to the equations of gravity with an extra dimension that looks like a magnetic monopole for the Kaluza-Klein electromagnetic field.

To picture that solution, consider the illustration at right. The surface of the "pinched" brown cylinder represents just two of the dimensions: the vertical direction is one of the three that we're familiar with, and the circle around the cylinder is the "extra dimension". The Kaluza-Klein monopole corresponds to a special kind of "pinch" that sits at a specific point in the vertical direction, so it could move up and down if it wanted to. But doesn't have any particular position around the extra dimension circle, so trying to rotate it around the circle does nothing at all.

You should imagine that it also sits at a specific point in the familiar "front-back" and "right-left" directions, but they aren't shown here. Now, string theory has a total of nine space dimensions, and we've only mentioned four of them so far. The Kaluza-Klein monopole is "stretched" around the other five "extra" dimensions, with no particular position to keep track of.

The important point here is that a Kaluza-Klein monopole has three "degrees of freedom": its position in each of the familiar three dimensions of space.

[An H-monopole shown as a point on a vertical cylinder. It
can slide up and down or around the cylinder.]

In string theory, T-duality is known to relate the Kaluza-Klein monopole to another object, the "NS5-brane". This object is "stretched" around five spatial dimensions (that's the "5" in "5-brane"), but sits at a specific point around the circle (as shown at right). It also has specific positions in the three familiar dimensions (including the vertical one shown in the picture).

As before, the most important point is the number of degrees of freedom here. The NS5-brane has four degrees of freedom: its position in familiar three-dimensional space and its position around the circle.

But that is already a puzzle! Two objects related by T-duality are supposed to just be different descriptions of the same physics. So how could one of them possibly be missing a whole degree of freedom? That's already a sign that something interesting is going on.

Years ago, my advisor and two collaborators suggested for this reason that the Kaluza-Klein monopole must "sit at a specific point" in some other degree of freedom. And they explained that rather than that being a direction in space, the appropriate degree of freedom would probably be the intensity of the "torsion" (which you can think of as being a sort of generalized magnetic field that shows up in string theory). But they weren't able to work out the details of that configuration, and it looked like changing the torsion from one intensity to another didn't actually change the physical behavior of the object at all (just like rotating around the circle didn't change the Kaluza-Klein monopole). So the puzzle remained unresolved.

[The same NS5-brane, but now 'smeared'
around the cylinder to look like a band, much like the
KK-monopole.]

But there's a second puzzle as well. When you go through the math and work out the T-duality by hand, you don't get the NS5-brane shown above. The trouble is, the math used to find the T-dual of a solution in gravity can only relate solutions in which rotating the circle doesn't change anything (like the Kaluza-Klein monopole). So rather than getting an NS5-brane with a specific position around the circle, you get a "smeared" version as pictured here. It's as if you've averaged over every possible position it could have around the circle.

This second puzzle was resolved not long ago by David Tong. Rather than finding the T-dual of the gravity solution directly, he began from a description of it in string theory called a "gauged linear sigma model". Even in that model, naively applying the string theory T-duality rules leads to a "smeared" NS5-brane. But a more careful treatment reveals that the effects of "worldsheet instantons" (explained below) cause the NS5-brane to "pick" a specific point around the circle.

Our results

Happily, Tong's solution to the second puzzle gave us the tools we needed to solve the first. Jeff and I wrote down the "gauged linear sigma model" in string theory that described the Kaluza-Klein monopole, and went looking for "worldsheet instanton" effects. This is probably a good point to define those words, too.

Worldsheet refers to the path that a string sweeps out as is passes through time. You can see from this slide of my "Intro. to String Theory" talk that a particle passing through time sweeps out a line as it goes (picture it like an airplane's contrail). That is called the particle's "worldline". So as you can see on this slide (or maybe this one), a string sweeps out a two-dimensional surface, its "worldsheet". String theory, at its core, is the theory of how waves propagate along that sheet.
Instantons might require a little more explaining. In the modern perspective on quantum mechanics, calculations involve a sum over every possible history, every possible configuration that could show up in the theory. (I describe that in the context of field theory on this slide.) Because there are infinitely many configurations that are very similar to each other, we usually write this as an integral, which is basically just the continuous version of a sum.

But sometimes assuming everything is continuous is a mistake! You will miss any configurations that have a different topology than the one you started with. (For example, if you start with a ball of Silly Putty and only squish it continuously, you won't ever get a doughnut shape, because poking your finger through the middle is a discontinuous change. A ball and a doughnut have different topologies.)

When there are configurations in your quantum mechanical theory that are topologically different in a way that depends on time, those are called different "instanton sectors". So the sum over histories has to include both a continuous integral (over configurations with the same topology) and a discrete sum (over the different possible topologies). Including the discrete sum can lead to important changes in the answer.

In our case, the "gauged linear sigma model" means that there is something like an electromagnetic field; describing that field means putting a bunch of little "vector arrows" on the string worldsheet. When those vectors are arranged to swirl around in vortices, configurations with different numbers of vortices turn out to have different topologies. (I should really have a picture to illustrate that, but I haven't made one yet.) And because one of the directions on the string worldsheet is time, these configurations are called "worldsheet instantons".

Once we included the sum over those instanton configurations, we found that the "shape" of the solution was changed. In particular, we discovered that the shape was different depending on the intensity of the "torsion"! And we were able to find at least an approximation to the explicit form of that corrected solution. It turns out to show features that do indicate that it is "sitting at a specific point", but at a point in this "torsion strength" degree of freedom rather than at a point around the circle.

I have always been interested in how "T-dual coordinates" like the torsion strength work, and I think our results help to give some insight into them. Hopefully, we and others will find that having the correct solution for string theory will answer other questions as well.