String theory

• 1. Introduction

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  String theory asks physicists to imagine that the smallest ingredients of nature are not points at all. Instead, each one is a tiny one-dimensional object called a string. In any given version of the theory there is only one kind of string, and it may look like a small loop or a short segment of ordinary string. What makes it interesting is that it can vibrate in many different ways. Each vibrational state appears to an observer as a different particle, with its mass, charge and other properties set by the way the string is moving. One of those states has a remarkable identity. It corresponds to the graviton, the quantum particle that is theorized to carry the force of gravity. For that reason, string theory is a theory of quantum gravity. Because it could also describe the forces and the matter we see, many physicists hope it can grow into a theory of everything, a self-contained mathematical model of all fundamental forces and forms of matter. This documentary asks why gravity needed a new kind of theory at all. It asks what such a theory demands of space itself, and how an idea born in the study of the atomic nucleus ended up somewhere so different. It also asks why a subject still arguing about its own foundations has reshaped both black hole physics and pure mathematics.

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• 2. The Gravity Puzzle

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  By the late 1970s, two frameworks from the twentieth century had proven sufficient to explain most observed features of the universe, from elementary particles to atoms to the evolution of stars. The first is Albert Einstein's general theory of relativity, which explains gravity and the structure of spacetime on the largest scales. The second is quantum mechanics, a completely different formulation that uses probability to describe physical phenomena on the smallest scales. Despite these successes, one of the deepest problems in modern physics remained. General relativity is built within classical physics, while every other fundamental force is described by quantum mechanics. Reconciling them requires a quantum theory of gravity, yet applying the usual recipes of quantum theory to gravity runs into serious difficulties.

  Richard Feynman and others, working in the first half of the twentieth century, developed the standard tool for quantum calculations. Their method uses Feynman diagrams, which depict the paths of point-like particles and their interactions. String theory changes the picture at its root. The one-dimensional line tracing a point particle's path is replaced by a two-dimensional surface tracing a string's motion. This is the natural way to carry over the perturbation techniques of ordinary quantum field theory.

  The characteristic length of a string is assumed to be on the order of the Planck length, the scale at which quantum gravity is believed to become significant. At the much larger scales of a physics laboratory, such an object would be indistinguishable from a point particle, and its vibrational state would simply decide which particle it is. That pleasing simplicity has a price, because the mathematics will not work in the three dimensions of everyday space.

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• 3. Hidden Dimensions

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  In bosonic string theory, spacetime must have 26 dimensions. In superstring theory it must have 10, and in M-theory it must have 11. These numbers come from the mathematical consistency of the theories rather than from any observation. To describe real phenomena, physicists must therefore imagine scenarios in which the extra dimensions escape every experiment.

  The most common idea is compactification, in which some of the extra dimensions close up on themselves into tiny circles. A garden hose offers the standard analogy. Seen from far away, the hose looks like a line with only length. Walk up to it and a second dimension appears, its circumference, which an ant crawling on the surface would be free to travel around. In the limit where the curled dimensions are very small, spacetime looks effectively lower-dimensional.

  Not every way of curling up produces a universe like ours. In a viable model of particle physics, the compact dimensions must take the shape of a Calabi-Yau manifold, a special space typically taken to be six-dimensional in string theory. It is named after the mathematicians Eugenio Calabi and Shing-Tung Yau.

  There is a second route called the brane-world scenario. Here the observable universe is a four-dimensional subspace sitting inside a higher-dimensional space. The force-carrying bosons of particle physics come from open strings whose endpoints are stuck to that subspace. Gravity comes from closed strings that roam through the larger space around it. This arrangement offers a natural explanation for why gravity is so much weaker than the other forces.

  The notion of a curled-up extra dimension predates string theory by decades. In 1914, Gunnar Nordstrom noted that gravity in five dimensions describes both gravity and electromagnetism in four. Theodor Kaluza later combined a fifth dimension with general relativity, and he is usually the one credited with the idea. In 1926, the Swedish physicist Oskar Klein gave the unobservable dimension a physical meaning by wrapping it into a small circle.

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• 4. From Hadrons to Gravity

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  In the 1960s, Geoffrey Chew and Steven Frautschi found that mesons fall into families called Regge trajectories, with masses linked to spins. Yoichiro Nambu, Holger Bech Nielsen and Leonard Susskind later understood that this pattern is exactly what rotating strings would produce. String theory began as a never completely successful attempt to explain hadrons, the particles such as the proton and neutron that feel the strong interaction.

  Gabriele Veneziano supplied the first working formula. Data analyzed by R. Dolen, D. Horn and C. Schmid showed that two different ways of exchanging virtual particles were dual to each other, and each one described the whole process. Murray Gell-Mann promoted the result widely. Veneziano then turned to the gamma function, whose poles are evenly spaced, and combined gamma functions into a scattering amplitude that obeyed this duality. It fit near-beam scattering data about as well as the other approaches of its day.

  Hundreds of physicists then worked to complete the model, and surprises kept coming. Veneziano found that the lightest particle in his theory had to be a tachyon. Miguel Virasoro and Joel Shapiro found a different amplitude, now understood to describe closed strings. Claud Lovelace calculated a loop amplitude and noticed that it was inconsistent unless the theory had 26 dimensions. In 1969-1970, Nambu, Nielsen and Susskind gave the theory a description in space and time in terms of strings. In 1971, Pierre Ramond added fermions, and in the resulting theories the critical dimension was 10.

  The turning point came in 1974, when Tamiaki Yoneya discovered that all the known string theories contained a massless spin-two particle that behaved as a graviton should. John Schwarz and Joel Scherk reached the same conclusion and took the bold step of proposing a theory of gravity instead of a theory of hadrons. By then, quantum chromodynamics had been recognized as the correct theory of hadrons. The old program looked finished, and for the following decade all work on string theory was completely ignored. A handful of devotees kept it moving. In 1977, Ferdinando Gliozzi, Joel Scherk and David Olive realized that two earlier string models were separately inconsistent and had to be combined. In 1984, John Schwarz and Michael Green proved that the combined theory has space-time supersymmetry.

  That same period brought what is sometimes called the first superstring revolution. In the early 1980s, Edward Witten discovered that most theories of quantum gravity could not accommodate chiral fermions like the neutrino. Working with Luis Alvarez-Gaume, he concluded that type I string theories were inconsistent. Green and Schwarz then found a contribution to the anomaly that the pair had missed, which pinned the gauge group of type I theory to SO(32). Convinced, Witten became a high-profile advocate. Between 1984 and 1986, hundreds of physicists began to work in the field.

  David Gross, Jeffrey Harvey, Emil Martinec and Ryan Rohm discovered heterotic strings, whose gauge group was two copies of E8. Philip Candelas, Gary Horowitz, Andrew Strominger and Witten found that Calabi-Yau manifolds are the compactifications that preserve a realistic amount of supersymmetry. Then David Gross and Vipul Periwal discovered that string perturbation theory diverges, and Stephen Shenker showed it diverges much faster than in field theory. That pointed to missing non-perturbative objects that nobody had yet found.

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• 5. Dualities and M-Theory

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  By the 1990s, physicists had argued that there were only five consistent supersymmetric versions of string theory: type I, type IIA, type IIB, and two heterotic theories. It remained a mystery why there was not just one. The answer began to appear when researchers found that the five were related in intricate ways called dualities. In general, two theories are dual when they are mathematically different descriptions of the same phenomena.

  Chris Hull and Paul Townsend studied S-duality in the type IIB theory, and Ashoke Sen studied it for heterotic strings in four dimensions. It says that a system of strongly interacting strings can sometimes be viewed as a system of weakly interacting strings in a different theory. Type I turns out to be equivalent by S-duality to heterotic string theory. T-duality, by contrast, concerns a string moving around a circular extra dimension, which can also wind around the circle one or more times. A circle of one radius is equivalent to a circle of a related radius, and type IIA becomes equivalent to type IIB. Dualities are not limited to strings. Montonen-Olive duality is an S-duality between quantum field theories.

  A parallel story concerned higher-dimensional objects. In string theory, a brane generalizes the point particle. A point is a brane of dimension zero, a string is a brane of dimension one, and in dimension p the objects are called p-branes. The word comes from membrane, which is a two-dimensional brane. Joseph Polchinski discovered in the 1990s that the theory requires D-branes, the surfaces on which the endpoints of open strings must lie. The D refers to the Dirichlet boundary condition. He identified them with black-hole solutions of supergravity.

  Supergravity had its own history. In 1978, Werner Nahm showed that eleven is the maximum number of dimensions for a consistent supersymmetric theory. In the same year, Eugene Cremmer, Bernard Julia and Joel Scherk showed that supergravity is most elegant in exactly eleven dimensions. Hopes of building a realistic world from it faded when physicists, including Witten, saw that it could not readily produce chirality, the distinction between clockwise and counterclockwise in the laws of physics.

  In 1987, Eric Bergshoeff, Ergin Sezgin and Paul Townsend showed that eleven-dimensional supergravity contains two-dimensional branes that look like sheets. Soon afterward, Michael Duff, Paul Howe, Takeo Inami and Kellogg Stelle curled one dimension into a circle. A membrane wrapped around it looks like a string in ten dimensions, and it reproduces exactly the strings of type IIA theory.

  Then, at the annual string theory conference held at the University of Southern California in 1995, Witten suggested something surprising. All five superstring theories, he said, are limiting cases of one theory in eleven dimensions. His proposal drew on the work of Sen, Hull, Townsend and Duff. In the months that followed, hundreds of new papers appeared online confirming different parts of it, a flurry now called the second superstring revolution.

  The name M-theory was deliberately noncommittal. In their 1996 paper, Petr Horava and Witten wrote that since the theory had been proposed to be a supermembrane theory, but there were reasons to doubt that interpretation, "we will non-committally call it the M-theory, leaving to the future the relation of M to membranes." Witten has suggested the M could stand for "magic", "mystery", or "membrane" according to taste.

  In 1997, Tom Banks, Willy Fischler, Stephen Shenker and Leonard Susskind proposed the BFSS matrix model, which describes the behavior of a set of nine large matrices. They showed that its low energy limit is eleven-dimensional supergravity, and proposed that the model is exactly equivalent to M-theory. A 1998 paper by Alain Connes, Michael R. Douglas and Albert Schwarz then tied matrix models and M-theory to noncommutative geometry.

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• 6. Black Holes and Holography

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  In the 1870s, the Austrian physicist Ludwig Boltzmann defined entropy as the natural logarithm of the number of microstates that produce the same large-scale features of a gas. In the 1970s, Jacob Bekenstein suggested that a black hole's entropy is proportional to the area of its event horizon, not to its volume. Combined with ideas from Stephen Hawking, this yielded a precise result called the Bekenstein-Hawking formula. Yet by the 1990s, no one could derive it by counting microstates in a theory of quantum gravity, and finding such a derivation was seen as an important test of any candidate theory.

  In 1996, Andrew Strominger and Cumrun Vafa passed that test for certain black holes in string theory. Their key observation was that D-branes, which look like fluctuating membranes when weakly interacting, become dense and massive with event horizons when the interactions are strong. They counted the ways of arranging D-branes to match a given mass and charge, and the count reproduced the formula exactly. Their black holes were unlike real astrophysical ones. They were extremal, meaning they had the lowest possible mass for a given charge, and they lived in five-dimensional spacetime with unphysical supersymmetry. In 1998, Strominger argued that the result could be generalized to any consistent theory of quantum gravity. In 2010, he and collaborators extended some results to non-extremal astrophysical black holes.

  In late 1997, Juan Maldacena proposed the anti-de Sitter/conformal field theory correspondence, or AdS/CFT. Steven Gubser, Igor Klebanov and Alexander Markovich Polyakov elaborated important aspects of it, as did Witten. By 2010, Maldacena's article had over 7000 citations, making it the most highly cited article in high energy physics.

  The correspondence starts from anti-de Sitter space, a model of spacetime closely related to hyperbolic space. Picture a hyperbolic disk tiled with equal-sized triangles and squares, where the circular edge is infinitely far from every interior point. Stack such disks to represent the universe at successive moments, and you get a solid cylinder, with time running up its length. Its surface looks locally like the flat spacetime of non-gravitational physics. The claim is that a quantum field theory living on that boundary is equivalent to a gravitational theory such as string theory in the interior. A dictionary translates between them, and predictions agree exactly. If two particles have a 40 percent chance of colliding in one theory, the matching collections have a 40 percent chance in the other.

  In 1975, Hawking published a calculation suggesting that black holes emit a dim radiation. This seemed to mean that black holes destroy information, in conflict with the Schrodinger equation of quantum mechanics. The clash became known as the black hole information paradox. Through AdS/CFT, a black hole corresponds to particles on the boundary that obey ordinary quantum rules, so it must also evolve consistently. In 2005, Hawking announced that the paradox had been settled in favor of information conservation.

  The correspondence has also reached into laboratory physics. Heavy ions such as gold or lead, collided at high energies, briefly form a quark-gluon plasma at about two trillion kelvin. In 2005, Dam Thanh Son and collaborators described it using black holes in five-dimensional spacetime and predicted a universal value for the ratio of its shear viscosity to its entropy density. In 2008, the Relativistic Heavy Ion Collider at Brookhaven National Laboratory confirmed the predicted value. Subir Sachdev and others hope for similar help with superconductors and superfluids, and string methods have already described the transition of a superfluid to an insulator in clouds of cold atoms held in criss-crossing lasers.

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• 7. Mathematical Dividends

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  In May 1991, the physicists Philip Candelas, Xenia de la Ossa, Paul Green and Linda Parkes announced that a six-dimensional Calabi-Yau manifold can contain exactly 317,206,375 curves of degree three. They found the number by using mirror symmetry, and the result startled geometers.

  Mirror symmetry grew out of the late 1980s, when several physicists noticed that a compactification of string theory does not determine its Calabi-Yau manifold uniquely. Type IIA and type IIB theories can be compactified on two entirely different manifolds and yield the same physics. Such manifolds are called mirror manifolds. The relationship lets mathematicians turn hard questions about one manifold into easier questions about its mirror, which is especially useful in enumerative geometry, the branch concerned with counting solutions to geometric problems.

  The classical counts have a long pedigree. The nineteenth-century mathematicians Arthur Cayley and George Salmon showed that exactly 27 straight lines lie on a cubic surface such as the Clebsch cubic. Hermann Schubert found that a quintic Calabi-Yau manifold holds exactly 2,875 lines. In 1986, Sheldon Katz proved that the number of conics on the quintic is 609,250. By 1991, most classical problems of the field had been solved and interest was fading.

  The physicists' method went far beyond what mathematicians had obtained, but it relied on physical intuition rather than proof. Mathematicians went on to construct rigorous arguments of their own. Today, the homological mirror symmetry program of Maxim Kontsevich and the SYZ conjecture of Strominger, Yau and Eric Zaslow are major approaches.

  A second bridge concerns group theory, the mathematics of symmetry. The classification of finite simple groups identifies several infinite families plus 26 additional sporadic groups. The largest, the monster group, has more elements than a thousand times the number of atoms in the Earth. In the late 1970s, John McKay and John Thompson noticed that numbers from the monster group matched numbers in a formula from number theory. John Horton Conway and Simon Norton named the link monstrous moonshine because it seemed so far-fetched.

  In 1992, Richard Borcherds explained the observations, using ideas from string theory in an essential way. He built on earlier work by Igor Frenkel, James Lepowsky and Arne Meurman, who had realized the monster group as the symmetries of a particular string theory. Borcherds received the Fields medal in 1998. In 2010, Tohru Eguchi, Hirosi Ooguri and Yuji Tachikawa found a similar link between the Mathieu group and string theory. Miranda Cheng, John Duncan and Jeffrey Harvey proposed a generalization called umbral moonshine, and Duncan, Michael Griffin and Ken Ono proved their conjecture.

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• 8. Testing the Theory

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  The Standard Model of particle physics unifies three of nature's fundamental forces: electromagnetism and the strong and weak nuclear forces. Yet it leaves out gravity and cannot explain the hierarchy problem, the structure of fermion masses or dark matter. String theorists respond by choosing a shape for the extra dimensions. One popular recipe starts with the ten-dimensional heterotic theory and curls six dimensions into a Calabi-Yau manifold. The result is a model roughly similar to the Standard Model, plus additional particles that no one has found.

  That last detail is the heart of the difficulty. Partly because of theoretical complications and partly because of the extremely high energies required, no experimental evidence yet points unambiguously to any model as a correct fundamental description. Some in the community therefore criticize these unification efforts and question whether continued research is worthwhile. Two further problems deepen the doubt. The full theory has no satisfactory definition in all circumstances, and it is not known in general how to define it outside perturbation theory. And the theory is thought to describe an enormous landscape of possible universes, with no clear principle for selecting the vacuum state that fixes the properties of ours.

  Cosmology offers another possible testing ground. The Big Bang model leaves open why the universe looks the same in all directions, why it appears flat, and why magnetic monopoles are not observed. Alan Guth and others developed cosmic inflation in the 1980s, a period of extremely rapid expansion driven by a hypothetical particle called the inflaton. Its properties should ultimately come from a more fundamental theory such as string theory, and researchers have tried to find it among the particles the theory describes. Observations of the cosmic microwave background, the radiation filling the sky since around 380,000 years after the Big Bang, might eventually support or reject such proposals.

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