Popular Science Presentation
In theoretical physics we use mathematics to describe Nature, often creating new mathematical structures during our research, just like Newton did when he invented calculus to formulate his "Laws of Motion" describing the motion of planets, stars or even swinging weights.
Later on, in the 19th century calculus evolved into classical field theory and physicists were able to successfully describe electromagnetism. But with the discovery of the electron, there was suddenly some new physics to understand and particle physics was born. Using the new framework of quantum mechanics and experimental observation, the scientific community gained an understanding of the fundamental particles and their division into two classes: bosons and fermions, particles that transmit forces and make up matter, respectively.
At the same time, increasing observational evidence showed that light travelled at one fixed speed according to any observer. This fact, together with new mathematics that Einstein developed to describe it gave birth to the special theory of Relativity. During the 20th century physicists worked hard and were able to merge this together with quantum mechanics, which gave origin to the relativistic quantum field theory. This theory is the foundation of our understanding of subatomic particle physics in the form of the Standard Model, which has so successfully described the phenomena occuring at the LHC.
However, Einstein continued his work and improved Newton's theory of gravitation. Bringing the new concepts of differential geometry into physics, he created the General Theory of Relativity. This theory has been proven to be a great description of Nature. Nevertheless, some of its features that have been observed, such as black holes and an expanding universe, have led the scientific community to believe that our knowledge is still somehow incomplete.
On one hand, quantum field theory describes elementary particles extremely well, but it neglects gravitational effects, and on the other hand we have general relativity which gives great insight into gravitational effects but ignores quantum effects. This is where String Theory comes, since we believe that it is the theory that closes this gap.
It is well known that a good quantum theory of gravity must include a massless particle that carries the gravitational force and has two units of spin, the graviton. In string theory, particles arise from excitations modes of the string (just like how the string of an instrument can vibrate to produce several harmonics), and the graviton is one of those excitations.
This is however not enough for us to say that we have a good theory of quantum gravity, since the graviton could also have been added to quantum field theory by hand, but that was shown to produce disastrous behaviour at small distances because the particles were point-like. In string theory this problem is solved since the fundamental objects have a non-zero length such which leads to a nice behaviour at small distances.
If we want our string theory to include fermions, then we have to add a special kind of symmetry called supersymmetry, which means that for every boson there must be a correponding fermion. Supersymmetric partners to known particles have not yet been found, but theorists believe this is because those particles would be too heavy to be detected with the technology we have nowadays. There is however a chance the new improvements on LHC might be enough to find supersymmetry which would be compelling evidence that string theory is the theory of quantum gravity.
Another surprising revelation was that superstring theories include more structures than the fundamental strings. There are also higher dimensional objects called D-branes, which are membranes where the ends of open strings are localized and correspond to a collective excitation of strings.
Finally, one has also to remember that superstring theories are only consistent if the number of space-time dimensions is ten. This means that to give good descriptions of Nature one has to compactify the unwanted spacetime dimensions, which makes the number of possible string theories grow quite a lot since there are many possible ways to make six dimensions much smaller than the other four. This compactification procedure is very interesting on its own and gives origin to very nice mathematical structures.
Depending on how supersymmetry works and whether or not the strings are required to be closed loops, there are five different string theories in ten dimensions. In spite of that, one can also say that that number is shrinking, since string theorists have discovered that what they thought were five totally distinct seem in fact to be related to each other through dualities and all special limits of one more fundamental theory, which they call M theory.
Before string theory won the full attention of the theoretical physics community, the most popular unified theory was an eleven dimensional theory of supergravity, which is supersymmetry combined with gravity. Technically speaking, M theory is is the unknown eleven-dimensional theory whose low energy limit is that supergravity theory in eleven dimensions and from which the known superstring theories emerge as special limits.
We still do not know much about the fundamental M theory, but a lot has been learned about how it relates to superstrings in ten spacetime dimensions and judging from all these relationships, it is certainly a very interesting and rich theory.
Cosmological observations have shown that most of the mass in the universe occurs in the form of dark matter. One leading candidate for the composition of dark matter is the WIMP (Weakly Interacting Massive Particle), and many people believe the WIMP to be a spin 1/2 fermion called neutralino which are the supersymmetric partner of gauge bosons and Higgs particles. They are very heavy, interact very weakly with other particels which makes them excellent candidates to explain dark matter.
Supersymmetry needs to be broken in some way because there is no exact match between fermions and bosons, but one can break supersymmetry without losing all the advantages of having it. One example of such mechanism is how the Higgs boson gives mass to the other particles. One of the reason originally string theorists liked supersymmetry was that the vacuum energy of bosons and fermions canceled each other. However, when supersymmetry in string theory is broken that cancellation does not occur anymore, which might explain the cosmological constant behind the expansion of the Universe. String theory also provides us with new scalar fields that could in principle have driven the inflationary phase of our Universe, even though this is stil an active area of research.
String theory also provides an excellent framework to study black holes. When it was discovered that black holes can decay by quantum processes, it was also discovered that black holes seem to have the thermodynamic properties of temperature and entropy. Entropy is a quantity that measures the number of allowed quantum states. One of the great successes of string theory was to show precisely how the counting of quantum states for a system of D-branes relates with the entropy as the area of the black hole horizon which appears in the D-brane setup.
Finally another great contribution from string theory has come in the form of the so called AdS/CFT duality, a powerful realization of the holographic principle which states that the description of gravity in some space is encoded in its boundary. AdS/CFT relates some string theories with highly symmetric quantum field theories and the most studied case corresponds to a string theory in five dimensions which is dual to a quantum field theory in four dimensions. This correspondence has the striking property that when the gravitational theory is difficult to solve then the computations in the corresponding quantum field theory become trivial, and vice-versa. This duality then gives great insights into quantum gravity, as it provides a non-perturbative formulation of string theory, and at the same time it provides a powerful toolkit for studying strongly coupled quantum field theories, as is the case of QCD, the theory describing the strong force which keeps atoms together.