Superstrings, why superstrings? While the Standard Model has been very successful in describing most of the phenomemon that we can experimentally investigate with the current generation of particle acceleraters, it leaves many unanswered questions about the fundamental nature of the universe. The goal of modern theoretical physics has been to find a "unified" description of the universe. This has historically been a very fruitful approach. For example Einstein-Maxwell theory unifies the forces of electricity and magnetism into the electromagnetic force. The Nobel prize winning work of Glashow, Salam, and Weinberg successfully showed that the electromagnetic and weak forces can be unified into a single electroweak force. There is actually some pretty strong evidence that the forces of the Standard Model should all unify as well. When we examine how the relative strengths of the strong force and electroweak force behave as we go to higher and higher energies, we find that they become the same at an energy of about 1016 GeV. In addition the gravitational force should become equally important at an energy of about 1019 GeV.The goal of string theory is to explain the "?" in the above diagram. The characteristic energy scale for quantum gravity is called the Planck Mass, and is given in terms of Planck constant, the speed of light, and Newton's constant, Physics at this high energy scale describes the universe as it existed during the first moments of the Big Bang. These high energy scales are completely beyond the range which can be created in the particle accelerators we currently have (or will have in the foreseeable future.) Most of the physical theories that we use to understand the universe that we live in also break down at the Planck scale. However, string theory shows unique promise in being able to describe the physics of the Planck scale and the Big Bang. In its final form string theory should be able to provide answers to answer questions like: • Where do the four forces that we see come from? • Why do we see the various types of particles that we do? • Why do particles have the masses and charges that we see? • Why do we live in 4 spacetime dimensions? • What is the nature of spacetime and gravity? Superstrings, string basics We are used to thinking of fundamental particles (like electrons) as point-like 0-dimensional objects. A generalization of this is fundamental strings which are 1-dimensional objects. They have no thickness but do have a length, typically 10-33 cm [that's a decimal point followed by 32 zeros and a 1]. This is very small compared to the length scales that we can reasonably measure, so these strings are so small that they practically look like point particles. However their stringy nature has important implications as we will see. Strings can be open or closed. As they move through spacetime they sweep out an imaginary surface called a worldsheet. These strings have certain vibrational modes which can be characterized by various quantum numbers such as mass, spin, etc. The basic idea is that each mode carries a set of quantum numbers that correspond to a distinct type of fundamental particle. This is the ultimate unification: all the fundamental particles we know can be described by one object, a string! [A very loose analogy can be made with say, a violin string. The vibrational modes are like the harmonics or notes of the violin string, and each type of particle corresponds to one of these notes.] As an example let's consider a closed string mode which looks like: This mode is characteristic of a spin-2 massless graviton (the particle that mediates the force of gravity). This is one of the most attractive features of string theory. It naturally and inevitably includes gravity as one of the fundamental interactions. Strings interact by splitting and joining. For example the anihilation of two closed strings into a single closed string occurs with a interaction that looks like: Notice that the worldsheet of the interaction is a smooth surface. This essentially accounts for another nice property of string theory. It is not plagued by infinities in the way that point particle quantum field theories are. The analogous Feynman diagram in a point particle field theory is: Notice that the interaction point occurs at a topological singularity in the diagram (where the 3 world-lines intersect). This leads to a break down of the point particle theory at high energies. If we glue two of the basic closed string interactions together, we get a process by which two closed strings interact by joining into an intermediate closed string which splits apart into two closed strings again: This is the leading contribution to this process and is called a tree level interaction. To compute quantum mechanical amplitudes using perturbation theory we add contributions from higher order quantum processes. Perturbation theory provides good answers as long as the contributions get smaller and smaller as we go to higher and higher orders. Then we only need to compute the first few diagrams to get accurate results. In string theory, higher order diagrams correspond to the number of holes (or handles) in the world sheet. The nice thing about this is that at each order in perturbation theory there is only one diagram. [In point particle field theories the number of diagrams grows exponentially at higher orders.] The bad news is that extracting answers from diagrams with more than about two handles is very difficult due to the complexity of the mathematics involved in dealing with these surfaces. Perturbation theory is a very useful tool for studying the physics at weak coupling, and most of our current understanding of particle physics and string theory is based on it. However it is far from complete. The answers to many of the deepest questions will only be found once we have a complete non-perturbative description of the theory. Superstrings, D-branes Strings can have various kinds of boundary conditions. For example closed strings have periodic boundary conditions (the string comes back onto itself). Open strings can have two different kinds of boundary conditions called Neumann and Dirichlet boundary conditions. With Neumann boundary conditions the endpoint is free to move about but no momentum flows out. With Dirichlet boundary conditions the endpoint is fixed to move only on some manifold. This manifold is called a D-brane or Dp-brane ('p' is an integer which is the number of spatial dimensions of the manifold). For example we see open strings with one or both endpoints fixed on a 2-dimensional D-brane or D2-brane: D-branes can have dimensions ranging from -1 to the number of spatial dimensions in our spacetime. For example superstrings live in a 10-dimensional spacetime which has 9 spatial dimensions and one time dimension. Therefore the D9-brane is the upper limit in superstring theory. Notice that in this case the endpoints are fixed on a manifold that fills all of space so it is really free to move anywhere and this is just a Neumann boundary condition! The case p= -1 is when all the space and time coordinates are fixed, this is called an instanton or D-instanton. When p=0 all the spatial coordinates are fixed so the endpoint must live at a single point in space, therefore the D0-brane is also called a D-particle. Likewise the D1-brane is also called a D-string. Incidently the suffix 'brane' is borrowed from the word 'membrane' which is reserved for 2-dimensional manifolds or 2-branes! D-branes are actually dynamical objects which have fluctuations and can move around. This was first shown by physicist Joseph Polchinski. For example they interact with gravity. In the diagram below we see one way in which an closed string (graviton) can interact with a D2-brane. Notice how the closed string becomes an open string with endpoints on the D-brane at the intermediate point in the interaction. We now see that string theory is more than just a theory of strings! Superstrings, supersymmetric strings There are two types of particles in nature - fermions and bosons. A fundamental theory of nature must contain both of these types. When we include fermions in the worldsheet theory of the string, we automatically get a new type of symmetry called supersymetry which relates bosons and fermions. Fermions and bosons are grouped together into supermultiplets which are related under the symmetry. This is the reason for the "super" in “supersymmetry”. A consistent quantum field theory of superstrings exists only in 10 spacetime dimensions! Otherwise there are quantum effects which render the theory inconsistent or 'anomalous'. In 10 spacetime dimensions the effects can precisely cancel leaving the theory anomaly free. It may seem to be a problem to have 10 spacetime dimensions instead of the 4 spacetime dimensions that we observe, but we will see that in getting from 10 to 4 we actually find some interesting physics. In terms of weak coupling perturbation theory there appear to be only five different consistent superstring theories known as Type I SO(32), Type IIA, Type IIB, SO(32) Heterotic and E8 x E8 Heterotic. Type IIB Type IIA E8 x E8 Heterotic SO(32) Heterotic Type I SO(32) String Type Closed Closed Closed Closed Open (& closed) 10d Supersymmetry N=2(chiral) N=2(non-chiral) N=1 N=1 N=1 10d Gauge groups none none E8 x E8 SO(32) SO(32) D-branes -1,1,3,5,7 0,2,4,6,8 none none 1,5,9 • Type I SO(32): This is a theory which contains open superstrings. It has one (N=1) supersymmetry in 10 dimensions. Open strings can carry gauge degrees of freedom at their endpoints, and cancellation of anomalies uniquely constrains the guage group to be SO(32). It contains D-branes with 1, 5, and 9 spatial dimensions. • Type IIA: This is a theory of closed superstrings which has two (N=2) supersymmetries in ten dimensions. The two gravitini (superpartners to the graviton) move in opposite directions on the closed string world sheet and have opposite chiralities under the 10 dimensional Lorentz group, so this is a non-chiral theory. There is no gauge group. It contains D-branes with 0, 2, 4, 6, and 8 spatial dimensions. • Type IIB: This is also a closed superstring theory with N=2 supersymmetry. However in this case the two gravitini have the same chiralities under the 10 dimensional Lorentz group, so this is a chiral theory. Again there is no gauge group, but it contains D-branes with -1, 1, 3, 5, and 7 spatial dimensions. • SO(32) Heterotic: This is a closed string theory with worldsheet fields moving in one direction on the world sheet which have a supersymmetry and fields moving in the opposite direction which have no supersymmetry. The result is N=1 supersymmetry in 10 dimensions. The non-supersymmetric fields contribute massless vector bosons to the spectrum which by anomaly cancellation are required to have an SO(32) gauge symmetry. • E8 x E8 Heterotic: This theory is identical to the SO(32) Heterotic string, except that the gauge group is E8 X E8 which is the only other gauge group allowed by anomaly cancellation. We see that the Heterotic theories don't contain D-branes. They do however contain a fivebrane soliton which is not a D-brane. The IIA and IIB theories also contain this fivebrane soliton in addition to the D-branes. This fivebrane is usually called the "Neveu-Schwarz fivebrane" or "NS fivebrane". It is worthwhile to note that the E8 x E8 Heterotic string has historically been considered to be the most promising string theory for describing the physics beyond the Standard Model. It was discovered in 1987 by Gross, Harvey, Martinec, and Rohm and for a long time it was thought to be the only string theory relevant for describing our universe. This is because the SU(3) x SU(2) x U(1) gauge group of the standard model can fit quite nicely within one of the E8 gauge groups. The matter under the other E8 would not interact except through gravity, and might provide a answer to the Dark Matter problem in astrophysics. Due to our lack of a full understanding of string theory, answers to questions such as how is supersymmetry broken and why are there only 3 generations of particles in the Standard Model have remained unanswered. Most of these questions are related to the issue of compactification (discussed on the next page). What we have learned is that string theory contains all the essential elements to be a successful unified theory of particle interactions, and it is virtually the only candidate which does so. However, we don't yet know how these elements specifically come together to describe the physics that we currently observe. Superstrings, extra dimensions Superstrings live in a 10-dimensional spacetime, but we observe a 4-dimensional spacetime. Somehow we need to link the two if superstrings are to describe our universe. To do this we curl up the extra 6 dimensions into a small compact space. If the size of the compact space is of order the string scale (10-33 cm) we wouldn't be able to detect the presence of these extra dimensions directly - they're just too small. The end result is that we get back to our familiar (3+1)-dimensional world, but there is a tiny "ball" of 6-dimensional space associated with every point in our 4-dimensional universe. This is shown in an extremely schematic way in the following illustration: This is actually a very old idea dating back to the 1920's and the work of Kaluza and Klein. This mechanism is often called Kaluza-Klein theory or compactification. In the original work of Kaluza it was shown that if we start with a theory of general relativity in 5-spacetime dimensions and then curl up one of the dimensions into a circle we end up with a 4-dimensional theory of general relativity plus electromagnetism! The reason why this works is that electromagnetism is a U(1) gauge theory, and U(1) is just the group of rotations around a circle. If we assume that the electron has a degree of freedom corresponding to point on a circle, and that this point is free to vary on the circle as we move around in spacetime, we find that the theory must contain the photon and that the electron obeys the equations of motion of electromagnetism (namely Maxwell's equations). The Kaluza-Klein mechanism simply gives a geometrical explanation for this circle: it comes from an actual fifth dimension that has been curled up. In this simple example we see that even though the compact dimensions maybe too small to detect directly, they still can have profound physical implications. [Incidentally the work of Kaluza and Klein leaked over into the popular culture launching all kinds of fantasies about the "Fifth dimension"!] How would we ever really know if there were extra dimensions and how could we detect them if we had particle accelerators with high enough energies? From quantum mechanics we know that if a spatial dimension is periodic the momentum in that dimension is quantized, p = n / R (n=0,1,2,3,....), whereas if a spatial dimension is unconstrained the momentum can take on a continuum of values. As the radius of the compact dimension decreases (the circle becomes very small) then the gap between the allowed momentum values becomes very wide. Thus we have a Kaluza Klein tower of momentum states. If we take the radius of the circle to be very large (the dimension is de-compactifying) then the allowed values of the momentum become very closely spaced and begin to form a continuum. These Kaluza-Klein momentum states will show up in the mass spectrum of the uncompactifed world. In particular, a massless state in the higher dimensional theory will show up in the lower dimensional theory as a tower of equally spaced massive states just as in the picture shown above. A particle accelerator would then observe a set of particles with masses equally spaced from each other. Unfortunately, we'd need a very high energy accelerator to see even the lightest massive particle. Strings have a fascinating extra property when compactified: they can wind around a compact dimension which leads to winding modes in the mass spectrum. A closed string can wind around a periodic dimension an integral number of times. Similar to the Kaluza-Klein case they contribute a momentum which goes as p = w R (w=0,1,2,...). The crucial difference here is that this goes the other way with respect to the radius of the compact dimension, R. So now as the compact dimension becomes very small these winding modes are becoming very light! Now to make contact with our 4-dimensional world we need to compactify the 10-dimensional superstring theory on a 6-dimensional compact manifold. Needless to say, the Kaluza Klein picture described above becomes a bit more complicated. One way could simply be to put the extra 6 dimensions on 6 circles, which is just a 6-dimensional Torus. As it turns out this would preserve too much supersymmetry. It is believed that some supersymmetry exists in our 4-dimensional world at an energy scale above 1 TeV (this is the focus of much of the current and future research at the highest energy accelerators around the word!). To preserve the minimal amount of supersymmetry, N=1 in 4 dimensions, we need to compactify on a special kind of 6-manifold called a Calabi-Yau manifold. The properties of the Calabi-Yau manifold can have important implications for low energy physics such as the types of particles observed, their masses and quantum numbers, and the number of generations. One of the outstanding problems in the field has been the fact that there are many many Calabi-Yau manifolds (thousands upon thousands?) and we have no way of knowing which one to use. In a sense we started with a virtually unique 10-dimensional string theory and have found that possibilities for 4-dimensional physics are far from unique, at least at the level of our current (and incomplete) understanding. The long-standing hope of string theorists is that a detailed knowledge of the full non-perturbative structure of the theory, will lead us to an explanation of how and why our universe flowed from the 10-dimensional physics that probably existed during the high energy phase of the Big Bang, down to the low energy 4-dimensional physics that we observe today. [Possibly we will find a unique Calabi-Yau manifold that does the trick.] Some important work of Andrew Strominger has shown that Calabi-Yau manifolds can be continuously connected to one another through conifold transitions and that we can move between different Calabi-Yau manifolds by varying parameters in the theory. This suggests the possibility that the various 4-dimensional theories arising from different Calabi-Yau manifolds might actually be different phases of an single underlying theory. Superstrings string duality The five superstring theories appear to be very different when viewed in terms of their descriptions in weakly coupled perturbation theory. In fact they are all related to each other by various string dualities. We say two theories are dual when they both describe the same physics. The first kind of duality that we will discuss is called T-duality. This duality relates a theory which is compactified on a circle with radius R, to another theory compactified on a circle with radius 1/R. Therefore when one theory has a dimension curled up into a small circle, the other theory has a dimension which is on a very large circle (it is barely compactified at all) but they both describe the same physics! The Type IIA and Type IIB superstring theories are related by T-duality and the SO(32) Heterotic and E8 x E8 Heterotic theories are also related by T-duality. The next duality that we will consider is called S-duality. Simply put, this duality relates the strong coupling limit of one theory to the weak coupling limit of another theory. (Note that the weak coupling descriptions of both theories can be quite different though.) For example the SO(32) Heterotic string and the Type I string theories are S-dual in 10 dimensions. These means that the strong coupling limit of the SO(32) Heterotic string is the weakly coupled Type I string and visa versa. One way to find evidence for a duality between strong and weak coupling is to compare the spectrum of light states in each picture and see if they agree. For example the Type I string theory has a D-string state that is heavy at weak coupling, but light at strong coupling. This D-string carries the same light fields as the worldsheet of the SO(32) Heterotic string, so when the Type I theory is very strongly coupled this D-string is becomes very light and we see the weakly coupled Heterotic string description emerging. The other S-duality in 10 dimensions is the self duality of the IIB string: the strong coupling limit of the IIB string is another weakly coupled IIB string theory. The IIB theory also has a D-string (with more supersymmetry than the Type I D-string and hence different physics) which becomes a light state at strong coupling, but this D-string looks like another fundamental Type IIB string. In 1995, physicist and mathematician Edward Witten pioneered the idea that Type IIA and E8 x E8 string theories are related to each other through a new 11-dimensional theory which he called "M-theory". This revelation provided the missing link that related all of the superstring theories through a chain of dualities. The dualities between the various string theories provide strong evidence that they are simply different descriptions of the same underlying theory. Each description has its own regime of validity, and in certain limits another description takes over just when the original one is breaks down. What is this "M-theory" shown above? Superstrings, M-theory M-theory is described at low energies by an effective theory called 11-dimensional supergravity. This theory has a membrane and 5-branes as solitons, but no strings. How can we get the strings that we've come to know and love from this theory? We can compactify the 11-dimensional M-theory on a small circle to get a 10-dimensional theory. If we take a membrane with the topology of a torus and wrap one of its dimensions on this compact circle, the membrane will become a closed string! In the limit where the circle becomes very small we recover the Type IIA superstring. How do we know that M-theory on a circle gives the IIA superstring, and not the IIB or Heterotic superstrings? The answer to this question comes from a careful analysis of the massless fields that we get upon compactification of 11-dimensional supergravity on a circle. Another easy check is that we can find an M-theory origin for the D-brane states unique to the IIA theory. Recall that the IIA theory contains D0,D2,D4,D6,D8-branes as well as the NS fivebrane. The following table summarizes the situation: M-theory on circle ----------- IIA in 10 dimensions Wrap membrane on circle ------ IIA superstring Shrink membrane to zero size - D0-brane Unwrapped membrane ----------- D2-brane Wrap fivebrane on circle ----- D4-brane Unwrapped fivebrane ---------- NS fivebrane The two that have been left out are the D6 and D8-branes. The D6-brane can be interpreted as a "Kaluza Klein Monopole" which is a special kind of solution to 11-dimensional supergravity when it's compactified on a circle. The D8-brane doesn't really have clear interpretation in terms of M-theory at this point in time; that's a topic for current research! We can also get a consistent 10-dimensional theory if we compactify M-theory on a small line segment. That is, take one dimension (the 11-th dimension) to have a finite length. The endpoints of the line segment define oundaries with 9 spatial dimensions. An open membrane can end on these boundaries. Since the intersection of the membrane and a boundary is a string, we see that the 9+1 dimensional worldvolume of the each boundary can contain strings which come from the ends of membranes. As it turns out, in order for anomalies to cancel in the supergravity theory, we also need each boundary to carry an E8 gauge group. Therefore as we take the space between the boundaries to be very small we're left with a 10-dimensional theory with strings and an E8 x E8 gauge group. This is the E8 x E8 heterotic string! So given this new phase 11-dimensional phase of string theory, and the various dualities between string theories, we're led to the very exciting prospect that there is only a single fundamental underlying theory -- M-theory. The five superstring theories and 11-D Supergravity can be thought of as classical limits. Previously, we've tried to deduce their quantum theories by expanding around these classical limits using perturbation theory. Perturbation has its limits, so by studying non-perturbative aspects of these theories using dualities, supersymmetry, etc. we've come to the conclusion that there only seems to be one unique quantum theory behind it all. This uniqueness is very appealing, and much of the work in this field will be directed toward formulating the full quantum M-theory. Superstrings, black holes The classical description of gravity known as General Relativity, contains solutions which are called "black holes". There are many different kinds of black hole solutions but they share some common characteristics. The event horizon is a surface in spacetime which, loosely speaking, divides the inside of the black hole from the outside. The gravitational attraction of a black hole is so strong that any object that crosses the event horizon, including light, can never escape out of the black hole. Classical black holes are therefore relatively featureless, but they can be described by a set of observable parameters such as mass, charge, and angular momentum. Black holes turn out to be important "laboratories" in which to test string theory, because the effects of quantum gravity turn out to be important even for large macroscopic holes. Black holes aren't really "black" since they radiate! Using semi-classical reasoning, Stephen Hawking showed black holes emit a thermal spectrum of radiation at their event horizon. Since string theory is, among other things, a theory of quantum gravity, it should be able to describe black holes in a consistent way. In fact there are black hole solutions which satisfy the string equations of motion. These equations of motion resemble the equations of general relativity with some extra matter fields coming from string theory. Superstring theories also have some special black hole solutions which are themselves supersymmetric, in that they preserve some supersymmetry. One of the most dramatic recent results in string theory is the derivation of the Bekenstein-Hawking entropy formula for black holes obtained by counting the microscopic string states which form a black hole. Bekenstein noted that black holes obey an "area law", dM = K dA, where 'A' is the area of the event horizon and 'K' is a constant of proportionality. Since the total mass 'M' of a black hole is just its rest energy, Bekenstein realized that this is similar to the thermodynamic law for entropy, dE = T dS. Hawking later performed a semiclassical calculation to show that the temperature of a black hole is given by T = 4 k [where k is a constant called the "surface gravity"]. Therefore the entropy of a black hole should be written as S = A/4. Physicists Andrew Strominger and Cumrin Vafa, showed that this exact entropy formula can be derived microscopically (including the factor of 1/4) by counting the degeneracy of quantum states of configurations of strings and D-branes which correspond to black holes in string theory. This is compelling evidence that D-branes can provide a short distance weak coupling description of certain black holes! For example, the class of black holes studied by Strominger and Vafa are described by 5-branes, 1-branes and open strings traveling down the 1-brane all wrapped on a 5-dimensional torus, which gives an effective one dimensional object -- a black hole. Hawking radiation can also be understood in terms of the same configuration, but with open strings traveling in both directions. The open strings interact, and radiation is emitted in the form of closed strings. The system decays into the configuration shown above. Explicit calculations show that for certain types of supersymmetric black holes, the string theory answer agrees with the semi-classical supergravity answer including non-trivial frequency dependent corrections called greybody factors. This is more evidence that string theory is a consistent and accurate fundamental theory of quantum gravity. Home page. Physics home page.
