PHY197F notes

Superstring Theory and Duality

String as LEGO

When we learn about atoms and molecules in school, we are taught that the fundamental constituents of everything are elementary particles. The key feature of a particle is that it is pointlike -- structureless. But what experimental evidence do we have that electrons and quarks are actually pointlike? Even the LHC, humanity's most powerful machine yet, can only reach down to about $10^{-20}$m. (If you blew up that LHC scale to my size, I would be nearly as big as the Milky Way.)

String theory is a brilliantly simple idea: the LEGOs of the universe are tiny one-dimensional vibrating strands of energy known as fundamental strings. These versatile beasties, as we know from our last class, can represent all the known subatomic particles by vibrating in different patterns. At low-res, a string looks and behaves just like a particle would. It only differs at high-res.

[wiggling string graphic]

String theory contains two types of strings: open strings and closed strings. As we saw near the end of the last class, groundstates of the former represent spin-1 messenger bosons of the Standard Model (photons, gluons, W+,W-,Z bosons) while groundstates of the latter represent spin-2 gravitons. Unification of gravity with the other three forces was something Einstein thought about for years and dreamed of achieving. Unfortunately he did not live long enough to see string theory. I like to think he might have found it profoundly satisfying.

Prediction of gravity and extra dimensions

The groundstate of a string corresponds to the lowest possible frequency produced by vibration in an organ pipe or a cello string. Excited states correspond to higher harmonics of the musical instrument, and they represent interacting particles with higher mass and spin.

Amazing feature: when you work out the quantum theory of strings, you get gravity for free! This does not impress most modern humans, because gravity was discovered centuries before string theory was. Had string theory been discovered first, it would be famous for having correctly predicted the existence of gravity and its inventors would probably have Nobel Prizes.

String theory also predicts extra dimensions of space, tightly curled up and hidden from view. Strings, unlike particles, can wrap around them. This has profound consequences for spacetime.

Superstring theory, with all its complicated moving parts, is capable of describing all the discovered subatomic particles in the Standard Model of particle physics -- plus possible zoos of undiscovered ones as well. It unifies forces and matter. It even describes the emergence of spacetime itself.

Onion layers

Will there be an end to peeling off onion layers of the structure of matter and force? String theory gives hints that it might be where the buck stops.

[periodic table of chemistry, depicting the usual array of atomic elements, coloured by whether they are produced in the big bang, cosmic rays, large stars, small stars, supernovae, or in human laboratories.]


[`periodic table' of subatomic physics, depicting quarks, leptons, spin-1 force carriers, and the Higgs.]

Building Gravity 3.0

So, is everything you thought you knew from high school wrong? No! Just upgraded.

Newton built Gravity 1.0. For him, gravity was a force transmitted instantaneously across space. Its key feature is unification of the terrestrial and celestial effects of gravity. Newtonian gravity might seem low-tech by today's standards, but it taught us enough to send humans to the Moon!

Einstein built Gravity 2.0: General Relativity (GR). For him, gravity was the geometry of the smooth fabric of spacetime. GR's key feature is that causality is baked in. GR superseded Newtonian gravity: it could handle describing more experiments. It obeys the Correspondence Principle: when gravity is weak everywhere, and all speeds stay much slower than light, you get back Newton's theory. So Einstein's theory of gravity stands on the shoulders of Newton's.

Various groups are working to build Gravity 3.0, i.e., quantum gravity. Maybe our different pathways to quantum gravity will converge in the end? String theory says that gravity is the lowest vibration mode of closed strings and it also obeys the correspondence principle -- i.e. it stands firmly on the shoulders of Einstein's theory of gravity. The string toolbox has proven especially fruitful in the past two decades. [picture depicting Einstein standing on the shoulders of Newton, and string theorists standing on the shoulders of Einstein in seeking Gravity 3.0.]


Mathematics has proven itself as an unreasonably effective way of working with the laws of Nature. This is why it is the language of modern theoretical physics. Math influences our standards of beauty in physics, but it is not the final arbiter. Usefulness for describing the natural world is.

Theoretical physicists studying interesting phenomena prefer to be able to calculate rather than just daydream about a topic. We like to have a model, even if it is not yet fully accurate or realistic. Hence my LEGO analogy.

[picture depicting a blue jay made of LEGO]
[picture depicting a blue jay made of LEGO]

I use string theory in my research because it has best ratio of features to bugs. It is my LEGO set.

What is a Dimension?

Physicists think of a dimension as an independent (perpendicular) direction in which objects could move. There are other definitions of dimension that are interesting to mathematicians, like a fractal dimension, but those ideas are beyond the scope of this course.

[three orthogonal axes depicting x, y, and z]

You are already familiar with four dimensions: (1) left-right (2) forwards-backwards (3) up-down and (4) past-future. The first three of these are space dimensions. The fourth is the dimension of time. Einstein's formulation of relativity taught us to bundle space and time together into spacetime.

As far as we know, quantum mechanics doesn't make sense with two or more time dimensions, because we can't define energy unambiguously. On the other hand, extra space dimensions are fine. There is no theoretical reason to believe that the idea of extra space dimensions is inherently evil. In fact, theoretical physicists have done a lot of thinking about whether extra dimensions of space might exist and -- if they do -- how we would measure their existence. I am one of the people who enjoys this kind of theoretical exploration. :-)

Trying to think in extra dimensions tends to make our brains hurt. 🤔 What tends to be much easier is imagining a lower-dimensional world than ours by projecting our world onto a surface.

Living in Flatland

Imagine that all people and objects in our lives are two-dimensional (2D), living in a 2D world -- say on a tabletop. What's different about their lives than for our lives? Take some time while you're reading this to try to draw on pieces of paper what life would be like in only 2D. How do flatlanders eat? Can they have a digestive tract? Try to also think about how would you explain the idea of a 3rd dimension to a flatlander. How would you explain a soccer ball?

[2D amoeba with mouth and anus]

Note: if a Flatlander circle inside a square tried to grow, it would bump into the square. But if I first lifted the circle up into the third dimension a little bit, then let it grow, then set it down again, it could reach its goal. Yay! Similarly, in four or more space dimensions, you can smoothly undo a knot in a rope without having to cut the rope. There is enough extra space to make the move.

[a blue circle inside a red square, then outside it]

Higher-D to us is like 3D is to the flatlanders. It's hard for our brains to comprehend naturally. The really neat thing is that theoretical physicists have developed ways of making extra-dimensional physics easier to grasp. Essentially, what we do is work with projections, which is like looking at a 3D object from different 2D perspectives and then knitting the 2D perspectives together into a 3D whole. This is a bit like assembling a CT scan: what the CT machine does is takes lots of 2D pictures of you and then uses software to join the pictures together into a 3D image that your doctor can interpret.

Ant on Telephone Wire

Theories incorporating extra dimensions go back over 80 years, but string theory is new in requiring more space dimensions. How would we visualize having extra dimensions that are too small for us to see?

Consider an ant on a telephone wire:-

[an ant on a telephone wire]

If I look at this from far away without my glasses on, I might think the wire was 1D. If I look at it very close up, I can see the curled-up dimensions as well. The wire is 3D. At every point along the extended dimension, there are curled-up extra dimensions. If we want to discover small curled-up extra dimensions, we need to be capable of resolving distances that are very small. This is why hunting for extra dimensions isn't easy.

Experimental Constraints on Extra Dimensions

Extra dimensions have not yet been seen. Physicists are looking hard for evidence of extra dimensions, in a variety of very ingenious tabletop and collider experiments as well as astrophysical observations. The experimental constraints differ for gravity-only extra dimensions (GOEDs) versus universal extra dimensions (UEDs). UEDs are egalitarian: everyone can play in them. GOEDs are picky: only gravity can play in them. State-of-the-art experiment says that UEDs must be smaller than a few billionths of a billionth of a metre. Pretty small, eh! For GOEDs, the constraints are much weaker because gravity is the weakest force, making precise experiments difficult. The limit that experimentalists have obtained so far is that any GOEDs -- which are dimensions into which we cannot move and into which we cannot shine light (photons) -- must be smaller than the width of a human hair. 👀

The Kaluza-Klein Energy Toll

Quantum physics teaches us that entering a curled-up extra dimension costs money (energy). There is a bare minimum cover charge to pay. 💰

Why is there a toll booth on curled-up extra dimensions? All microscopic objects (like electrons, or photons which make up light) have both particle-(/string-)like and wave-like behaviours. Waves are characterized by their wavelength. Having a shorter wavelength costs more energy because the wave is more frenetic. To explore a curled-up extra dimension, a particle/string must fit an integer number of its wavelengths around the circumference, or else the waves would end up cancelling each other out, telling you that the object is not allowed to play in there. The smaller the curled-up extra dimension, the more expensive the cover charge. Rich observers may be able to observe more space dimensions than poor observers: they have enough money to open doors to playing around in extra dimensions. 💩

The two most extreme energy contexts we know of in the cosmos are the big bang explosion 💥 that created our universe in the beginning, and the singularity at the heart of a black hole. Extra dimensions may play an important role in the physics of big bangs and black holes.

Duality as Dictionary

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.

An example is a phenomenon in string theory called T-duality. String theorists discovered that if you study string theory on a spacetime with a curled-up extra dimension that is large, it is physically equivalent to string theory on a spacetime with a curled-up extra dimension that is small. The ability of a string to wind its body around the curled-up extra dimension is crucial to making this fly.

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]

(Note: topics beyond this point constitute pretty advanced material, and will not be on the final exam. They are presented mostly for students who want the enrichment. :D )

An Earlier Duality in Maxwell's Equations

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.

T-duality and Other Superstring Dualities

The first well-understood duality in string theory was called T-duality. This symmetry 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. 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.

M Theory

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.

Testability of String Theory

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 this in class, depending on student demand.

Supersymmetry and Unification

One question you might have had about unification and string theory is whether it manages to unify matter and force together. I mean, string theory is impressive and all, unifying spin-one messenger bosons (like photons) with spin-two messenger bosons (gravitons), but can it unify bosons and fermions together? This is something the Standard Model cannot do.

It turns out that, in order to unify force and matter, physicists had to expand the concept of symmetries to include ones with fermionic parameters, not just bosonic parameters like rotation angles. The resulting symmetry was dubbed supersymmetry. I gather that the origin of this word is in the French, à la supermarché. It refers to a market that has more items than just a regular marché. By analogy, supersymmetry has more moving parts to it than regular symmetry. The neat thing about supersymmetry is that is is the unique exension of Poincaré symmetry that provides a 1-1 pairing between bosons and fermions.

SUSY Nomenclature

Physicists are lazy. All other things being equal, we don't go for a baroque solution over a simple one. We also love to abbreviate stuff to make it quicker to write - which is why we make such heavy use of math. Physicists working on supersymmetry got quickly bored of writing out such a long word and decided to abbreviate it as SUSY. This is pronounced like the girl's name Suzy.

Supersymmetry does not change the mass of a particle. So every particle's superpartner has the same mass as it does. The superpartner always has spin-half less than the original, unless the original had spin zero in which case its superpartner gets spin-half (not minus-half, because spin always has to be a non-negative number for it to make sense as a physics concept).

Superpartners of fermions get an s- prefix, for example the superpartner of the electron is the selectron. Superpartners of bosons, on the other hand, have a different nomenclature: they are designated by an -ino suffix.

W+,W-,Z1Winos, Zino1/2

While SUSY does change the spin by half, it does not change the mass. But... where are the stops, the Winos, etc? Shouldn't we have seen them already? In particular, shouldn't there be a particle called the selectron which has a spin of zero and also has a mass of 511keV? Well, if SUSY were a preserved symmetry, then yes, these particles should have been measurable already. But we certainly have not seen any kind of supersymmetric particle evidence yet, although physicists smarter than I have been looking for years. The conclusion is that SUSY must be a broken symmetry today, if it was ever a symmetry. It is not known whether SUSY is even a high-energy symmetry of nature. We hope to be lucky enough to find out by using LHC and cosmological data, but it is not clear if this wish will be realized. There is no compelling theoretical or experimental reason why SUSY has to be true. SUSY may be helpful for understanding the nature of dark matter and maybe even dark energy. It might end up being crucial to define a consistent theory of quantum gravity at the shortest imaginable length scales. We just do not know yet.

The Super in Superstring

Superstrings have an absolutely beautiful property that bosonic strings do not: they automatically take care of cancelling out the dangerous tachyon that might destabilize the whole edifice. SUSY for strings ensures that the mass of all string states, including the groundstate, is non-negative. Superstrings therefore come in bosonic and fermionic types. You can imagine electrons and its other lepton friends as coming from fermionic open string modes, while photons could be bosonic open string modes. And so forth.

Superstrings can be shown to be anomaly-free. This was a gigantic discovery by Green and Schwarz in 1984 that kicked off the First Superstring Revolution, in which theoretical particle physicists learned how to do quantum mechanics with superstrings. What does being anomaly-free mean? Basically, it means that your theory is free of nasty mathematical inconsistencies which are so problematic that they render a theory useless for practical applications. So getting your theory to be anomaly-free is a big bonus. It gives you street cred with other physicists.

Superstring anomaly cancellation, a highly technical topic I can only teach graduate students once they have had two semesters of technical Masters-level coursework, is a very complicated beast. But the essence of it distills down to one very simple equation: $$ D_{\rm superstring}-2=8 $$ This tells us that the dimension of spacetime must be $D_{\rm superstring}=10$. In other words, one time dimension and nine space dimensions. For bosonic strings, the equation is $$ D_{\rm bosonic\ string}-2=24 $$ which means $D_{\rm bosonic\ string}=26$.

The Type I superstring is an open superstring, and it has eight fermionic modes balancing the eight bosonic ones describing the spacetime coordinates. All of the other four superstring theories are closed superstring theories. The Type IIA superstring has eight left-handed fermionic modes and eight right-handed ones. It is nonchiral (has no handedness). The Type IIB superstring has sixteen left-handed fermionic modes and is chiral (has a handedness). The heterotic superstring theories are hybrids, which have left-moving bosonic string hybridized with right-moving superstring. The difficulty with making hybrids is that you have 26 left-moving bits of the spacetime coordinates but 10 right-moving bits. This seems silly because we know that there can only be 10 sensible dimensions built out of these bits. What the heterotic string theory discoverers found was that they could take the 16 unbalanced bits and form gauge symmetries out of them. Anomaly cancellation demands that the gauge group you build this way has to have 496 generators. This is a rather large number, and there are only two solutions: $SO(32)$ and $E8\times E8$ for the gauge symmetry. These two hybrids fill out the total of five known superstring theories.