Author: Prof. A.W. Peet, University of Toronto
![[my avatar]](../images/avatars/an32sq.png)
Creative Commons licence: BY-NC-SA 4.0 Int'l
Last updated: 2014-06-14
Duality is when two different physical systems turn out to have the same basic physics. This means that they secretly possess the same underlying dynamics. This state of affairs is very uncommon: hardly any physical systems in the universe are dual to one another. But string theorists have found amazing riches of dualities since the earliest days of the First Superstring Revolution which began in 1984 with Michael Green and John Schwarz’s discovery of how to cancel anomalies in superstring theory.
As we discussed last week, having a duality is like being in the region of the Venn diagram below where English and French words correspond one-to-one. Everything that can be said in this intersection region has a description in the other language with the same meaning, even if it may be more cumbersome to express it.
![[The idea of duality as a dictionary]](images/dualitydictionary.jpg)
Suppose for a moment that there are no electric charges present, and that there are no electric currents either. Then the dynamical equations for electromagnetism -- encoded in Maxwell’s equations -- actually possess a surprising symmetry! For art’s sake, we display the equations here \begin{align} {\vec{\nabla}}\cdot{\vec{E}} &= 0 & {\vec{\nabla}} \times {\vec{E}} - {\frac{1}{c}}{\frac{\partial}{\partial t}}{\vec{B}} &= {\vec{0}} \cr {\vec{\nabla}}\cdot{\vec{B}} &= 0 & {\vec{\nabla}}\times {\vec{B}} + {\frac{1}{c}}{\frac{\partial}{\partial t}}{\vec{E}} &= {\vec{0}} \end{align}
It is a pretty neat mathematical fact that Maxwell’s equations are completely unchanged if you replace the electric field $E$ by the magnetic field $B$ and replace $B$ with $-E$. This is a baby version of duality. You can probably see it just by eye in the above equations.
Special types of gauge theories which are significantly more complicated than Maxwell’s theory of electromagnetism are able to support dualities, even with sources of charges and currents turned on. This is pretty amazing.
Our signature duality from last week was T-duality. This is a symmetry of superstring theories. It exchanges momentum (Kaluza-Klein) modes of strings running around a compact extra dimension of space with winding modes of strings wrapped around that same dimension. At the same time, it exchanges small radius with large radius for that dimension. (The only radius that does not get altered under T-duality is that which is exactly 1 in units of the string length $\ell_s$.) T-duality relies on the extended nature of the body of the string. Particle theory does not support T-duality: it is not a rich enough theory to be able to do so.
T-duality relates Type IIA superstring theory to Type IIB. It also relates Heterotic SO(32) to Heterotic E8xE8 string theory. For Type I, the fifth kind of superstring theory living in ten dimensions, does something a wee bit more complicated, which is well understood. If you are interested in more mathematical details, you can delve into the quantum geometry parts of Brian Greene's book The Elegant Universe.
As we already know, superstring theory contains both closed strings and open strings. Closed strings do not have endpoints, but open strings do. You can imagine a closed string as looking like a teeny tiny version of a wiggly rubber band, and an open strings as looking like a teeny tiny version of a wiggly jump rope. 
![[a cartoon of a closed string (like a wiggly rubber band) and an open string (like a wiggly jump rope) ]](images/openclosed.jpg)
Starting from the basic dynamics of strings, it is possible to figure out the allowed behaviour of string endpoints. For our case of open strings there are only two physically consistent choices:
D-branes have tension (a bit like strings, but different) and force charges. D-brane tension refers to the restoring force that you feel pushing you back upwards if you deform a trampoline surface downwards and vice versa. It is analogous to the force you feel on your fingertip from a guitar string when you are in the act of plucking it.
Open strings can start and end on the same D-brane, or they can start on one D-brane and end on a different D-brane. The body of the string can do anything consistent with the basic principles of string theory while its ends are on D-branes; as you might expect, the wigglier configurations cost more energy. The picture illustrates what the picture might look like for the case of three D-branes. (The figure is labelled by $N=3$, where the $N$ here refers to the number of D-branes.)
![[A cartoon of three parallel D2-branes with open strings starting and ending on all the different options]](images/keeptrack.jpg)
Superstring theory lives in ten spacetime dimensions, one time and nine space. D$p$-branes are D-branes with spatial dimension $p$, which is an integer somewhere between 0 and 9. In this terminology, a 0-brane is physicist-speak for a particle (a point particle has no spatial extent in any direction!). Similarly, a 1-brane is a string, because it has spatial extent in one direction. A 2-brane is called a membrane, which is where the word brane
comes from. After 2-branes, physicists no longer have a specific English word for the object, so we use $p$-brane
in general.
Which types of D-branes you can have depends on which superstring theory you are talking about. Neither of the heterotic theories has any D-branes of any kind. The Type I theory only has stable D1-branes, D5-branes, and D9-branes. Type IIA only has stable even-dimensional D-branes, while Type IIB only has stable odd-dimensional D-branes. This sort of level of detail cannot be understood without investing in some heavy math and physics, which is not taught until grad school.
Consider a stack of $N$ D-branes. They can be coincident or separated. How can we figure out what effect they have on the motion of other objects? We can start by asking how much they warp the spacetime they live in, and what force-charges they carry. This is something that can be calculated using modern string theory technology, and was first discovered in full by Joseph Polchinski in 1995.
D-branes are heavy when the string coupling $g_s$ is weak and light when it is strong. They also warp spacetime. The more D-branes there are, the bigger the warping of spacetime, and the larger are their higher-dimensional versions of electrical charges. They can also carry angular momenta as well.
Whenever dimensions of spacetime are rolled up into a compact manifold (like Brian Greene’s precious Calabi-Yaus), D-branes can become more important than previously thought. The reason is that D-branes are able to wrap dimensions of space that are aligned with their worldvolume, and can become massless when the radius (or other geometrical info about the compact manifold) is tuned to a special value. Because D-brane tension scales differently with the coupling strength of string theory than it does for fundamental strings, they have different effects on the physics in the remaining spatial dimensions.
Relying on earlier work by British physicists Chris Hull and Paul Townsend, in 1995 American superstar Edward Witten announced a very powerful web of new dualities involving all five of the superstring theories living in ten spacetime dimensions and an eleven-dimensional theory known as M Theory. He proposed that the reason we had thought of Type I, IIA, IIB, HE and HO superstring theories as physically distinct is that we had not noticed that they were all connected (in their different ways) to a unifying eleven dimensional theory. He absolutely rocked the string theory community with his amazing announcement. It would later turn out to be relevant to a much broader set of subfields of physics researchers than just string theorists.
For the case of Type IIB superstring theory, there is a very simple relationship between the size of the eleventh dimension $R_{11}$ and the coupling constant $g_s$ of string theory, which characterizes the strength of interactions between strings: $$ R_{11}=g_s\ell_s \,, $$ where $\ell_s$ denotes the string length. What was amazing about Witten’s relationship above was that it explained why we never knew about the 11th dimension before 1995: we had not turned the string coupling up large enough in our minds to see it! When $g_s$ is small, $R$ is negligible compared to the string length and we would not know to look for it; all we would see is the physics of the string length. It was not until string theorists started asking questions about strong coupling physics, which is inaccessible using the standard techniques of perturbation theory, that we discovered any evidence of M theory.
Initially the properties of M Theory seemed quite opaque. We still have a great deal to learn about it and much active research work is in progress in this context. Back in 1995, M stood for “Mystery”, “Mother”, “Matrix”, or whatever other words beginning with M felt fashionable for string theorists of the day. 
There are extended objects that live in eleven dimensional M theory (which has one time dimension and ten space dimensions), called M2-branes and M5-branes. These are stable 2- and 5-dimensional friends of D-branes which only live up in eleven dimensions within the confines of M Theory. The example which is worth showing you is how the M2-brane of M Theory is connected with two objects in Type IIA superstring theory: the D2-brane and the fundamental string. Let us see how this works.
Consider an M2-brane living up in D=11. Suppose that we wrap one of its braney dimensions around the curled up 11th dimension of M theory. Then what we would get down in D=10 in Type IIA superstring theory is a stable object with only one remaining dimension. The only thing it can be is the fundamental string, and that is exactly what it is! Neato! Suppose on the other hand that we do NOT wrap either of the dimensions of the M2-brane around the curled up 11th dimension: then we would get a stable 2-dimensional object down in D=10. This is precisely the D2-brane of Type IIA superstring theory. Things are similar for the M5-brane. Overall, string theorists say that M Theory has the ability to unify disparate objects into a more coherent whole. It does this by having an extra dimension of space at its disposal.
In fact, M Theory is even more powerful than our single M/IIA duality example mentioned above. It is able to unify all five superstring theories together in various limits! The take-home message is that different superstring theories in ten dimensions are connected via eleven dimensions. This is one of the central and powerful ideas of modern superstring theory.
Once string theorists had put these linkages together, Polchinski made a visual representation in the form of the deerskin diagram
depicted below. It gives you a cartoony way of seeing how M Theory, living in eleven dimensions, is connected with all five flavours of superstring theory living in ten dimensions.
![[the deerskin diagram of superstring duality]](images/deerskin.jpg)
This topic is political, in the sense that even a lot of professional physicists with PhDs are quite ignorant about it. What I would like to warn you of is that there are articles and books out there for laypeople by folks outside the field of string theory (eg people with chips on their shoulder like Jon Woit and Lee Smolin) who like to shout to anyone listening that string theory is inherently untestable and therefore not real physics. Despite their opinions, string theory actually is testable. There are even publications in highly-respected journals to prove it, like this one from 2006. I can say more about the sociology of this in class, depending on student demand.
IMPORTANT NOTE
This material we just discussed today is the most technical of the whole course. It was quite advanced, and you will be responsible only for the general conceptual aspects on the Test, not the technical details.