19 June 2011

My work in this paper builds directly on a previous paper I wrote with my advisor Jeff Harvey a few years ago. We showed that "Kaluza-Klein monopoles" (a type of object in string theory) have a subtly different "shape" than people had previously thought. But our new description was puzzling, because the equations we found weren't quite compatible with the usual equations of string theory.

In my recent work, I showed that this monopole system makes perfect sense as an example of a new mathematical formalism for string theory called "doubled geometry". The doubled description of this system simultaneously blends the KK-monopole with a related string theory object called the NS5-brane. By combining the two, it makes their underlying structure and relationship clear. The real value of this work is that it is the first explicit application of the doubled formalism to a broad new class of systems that we previously had no clear way to understand.

As before, you can get the *full* details from my
paper itself. (The slides from my Great Lakes Strings talk that
are linked above might also be of interest to the experts out
there, though I suspect they won't make a whole lot of sense
divorced from my actual presentation. Too bad they didn't have
audio recording going my day.) But that's highly technical; my
hope here is to hit that happy intermediate point where
interested laypeople can actually get the gist of what I'm up
to.

Given that this new work builds on a previous paper that I've
described in this way, I'm going to treat that earlier explanation as required
background for this one. Focus on the parts *before* the
"our results" section: that will give you a basic feel for what
the "Kaluza-Klein monopole" and "NS5-brane" are. The main
connection between that earlier work and my new paper is the
"new degree of freedom" (something akin to a new direction of
motion) for the KK-monopole to make it match the NS5-brane
properly.

The important part of those earlier conclusions is that the
KK-monopole turns out to be localized at a "point" in that new
degree of freedom (that I called "torsion strength" on the other
page). But the new degree of freedom emphatically *isn't*
a direction in space (not even an "extra dimension"). So what
exactly does it mean? The usual equations of string theory
aren't clear. The new degree of freedom is clearly related to
the behavior of a sort of magnetic field (the "torsion") that
exists all around the KK-monopole: as shown in the figure at
right, that field has a "direction" that corresponds to the new
"position" of the monopole. But this can't be the full story,
because that field has been known for a long time and its
direction doesn't affect the "shape" of the monopole the way our
earlier results said that the new degree of freedom should.

Another way to try to understand what this new degree of freedom means is to consider how strings behave near the center of the monopole. Because the extra dimension becomes small there, it is easy for strings to stretch around the circle and in fact one would expect a clump of winding strings to settle there automatically (as suggested in the figure at left). The exact state of this clump of strings is determined by the monopole's "position" in the new degree of freedom, and the clump's effect on other objects nearby is in principle expected to give the same results as the altered "shape" that our results described. (As a rough analogy, if you're driving a car and it keeps pulling left, that could be either because the road is slanted [altered shape] or because there's wind blowing [clump of strings].) But nobody really knows how to write down equations that describe exactly what that clump of strings looks like or how they affect other objects: this description is correct, but not very useful.

This is where the new "doubled geometry" framework comes to the rescue. Doubled geometry is a mathematical technique for describing a string theory system and its T-dual simultaneously. The basic idea may be subtle unless you've studied quantum mechanics, but I'll give it a go.

In everyday life, when we want to describe an object's
motion, we need to specify two things: its *position* and
its *velocity*. One of the most remarkable insights of
quantum mechanics is that these two types of information aren't
independent after all. If you know everything there is to know
about an object's velocity (in a complicated quantum sense),
that automatically also tells you everything you can know about
its position, too. (And vice versa.) In other words, there is a
mathematical way to convert knowledge about the "velocity state"
into knowledge about the "position state" (for the math people
out there, this is a Fourier transform).

A related feature of quantum mechanics is that every moving
object can also be thought of as a *wave*. The faster it
moves, the more rapidly the wave oscillates. This is illustrated
in the first and last images below: the first image shows a
slow-moving object with a long wavelength, and the last image
shows a fast-moving object with a short wavelength. But when the
motion is around an extra dimension as shown here, this has a
surprising effect: speeds whose wavelengths don't fit "just
right" around the extra dimension aren't allowed! (The middle
picture below is an example of this: the wave doesn't smoothly
reconnect with itself in front.) To be precise, the object's
speed must produce a wave that fits exactly an integer number of
times around the extra dimension:
*n* = 0, 1, 2, 3....

Putting all this together, it means that if we know all about
the object's *n* = 0, 1, 2, 3...
velocity, there's a straightforward mathematical way to figure
everything possible about its position around the extra
dimension. But now we bring string theory into the story!
Strings can move around the extra dimension like any other
object (as shown at the bottom of the picture at right). But
they can also *stretch* around the circle as shown. The
number of times they wind around the circle is obviously an
integer: a string has a "winding number"
*w* = 0, 1, 2, 3....

Remarkably, the mathematics used to describe a string's
winding number (an integer, *w*) is nearly identical to the
mathematics used to describe its velocity (also an integer,
*n*). (The central idea of T-duality boils down to swapping
*n* and *w*.) So it is natural to ask: if we know all
about the string's
*w* = 0, 1, 2, 3... winding, what
would happen if we used it in that same mathematical formula
(the Fourier transform) that changed velocity into position? The
answer, it turns out, is that we get something that acts
mathematically just like a position should act, but it doesn't
correspond to any of the actual directions that describe the
shape and location of objects in the familiar world. It's almost
as if the extra dimension gives the equations twice as many
"directions" as they should have!

And that is where "doubled geometry" comes in. The usual equations describing string theory can only really handle one of these "directions" at a time, because they only describe the shape of the extra dimension (its geometry) with the familiar type of position. Anything interesting about the other "direction" gets sort of brushed under the rug. "Doubled geometry" is a new way of setting up those fundamental string equations that treats these two very different sorts of "directions" on an even footing. It's almost as if you're holding both descriptions in your head at the same time (while keeping track of the velocity/winding relationship between them).

If all of that is reminding you of the new degree of freedom that my advisor and I studied for the Kaluza-Klein monopole, you're absolutely right. Just as the NS5-brane sits at a specific "real" position around the extra dimension, the KK-monopole sits at a specific position in the "winding" direction that we can't really visualize in a normal way. But with the help of the doubled geometry formalism for the equations of string theory, we can keep track of both types of position at once in a completely equal way. That allows us to picture the KK-monopole and the NS5-brane almost as a single type of object: it's our choice whether we think of the KK-monopole as "real" (with the NS5-brane hiding away to represent the monopole's "position" in the mysterious other "direction") or whether we think of the NS5-brane with its specific position as "real" (with the KK-monopole hiding in the background). Doubled geometry allows us to draw insight from both descriptions at once.

In showing how to work with the Kaluza-Klein monopole and NS5-brane in the doubled geometry formalism, my paper goes a long way toward answering the questions at the end of my description of our earlier work. My hope is that this system will prove to be a useful example of how pairs of objects like this can be described in the doubled language. That's important, because it seems likely that objects with funny paired behavior and "extra degrees of freedom" like this are much more common in string theory than people have often thought. Having an explicit example of this unfamiliar behavior that is nevertheless related to a familiar system may be exactly what the field needs to get a good handle on them. With luck, studying these objects and the doubled geometry equations will help us to eventually find a complete mathematical description of string theory itself.

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Any questions or comments? Write to me: jensens@alma.edu

Copyright © 2011 by Steuard Jensen.