January String Theory. University of Cambridge Part III Mathematical Tripos. David Tong. Department of Applied Mathematics and Theoretical Physics, . Abstract: This set of notes is based on the course “Introduction to String Theory” which was taught by Prof. Kostas Skenderis in the spring of. INTRODUCTION TO STRING THEORY∗ version Gerard 't Hooft. Institute for Theoretical Physics. Utrecht University, Leuvenlaan 4. CC Utrecht.

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String theory is not, in contrast to general relativity and quantum field the- String theory is a proposal for a unifying framework of high energy physics. 10, Relativistic strings: Nambu-Goto action, equations of motion and boundary conditions, (PDF). Static gauge, transverse velocity, and string action. In these lecture notes, an introduction to superstring theory is presented. Classi- BRST in string theory and the physical spectrum.

Each of these operations is called a symmetry , and the collection of these symmetries satisfies certain technical properties making it into what mathematicians call a group. In this particular example, the group is known as the dihedral group of order 6 because it has six elements. A general group may describe finitely many or infinitely many symmetries; if there are only finitely many symmetries, it is called a finite group.

Mathematicians often strive for a classification or list of all mathematical objects of a given type. It is generally believed that finite groups are too diverse to admit a useful classification. A more modest but still challenging problem is to classify all finite simple groups. These are finite groups which may be used as building blocks for constructing arbitrary finite groups in the same way that prime numbers can be used to construct arbitrary whole numbers by taking products.

This classification theorem identifies several infinite families of groups as well as 26 additional groups which do not fit into any family. The latter groups are called the "sporadic" groups, and each one owes its existence to a remarkable combination of circumstances. The largest sporadic group, the so-called monster group , has over 10 53 elements, more than a thousand times the number of atoms in the Earth. A seemingly unrelated construction is the j -function of number theory.

This object belongs to a special class of functions called modular functions , whose graphs form a certain kind of repeating pattern.

In the late s, mathematicians John McKay and John Thompson noticed that certain numbers arising in the analysis of the monster group namely, the dimensions of its irreducible representations are related to numbers that appear in a formula for the j -function namely, the coefficients of its Fourier series.

In , Richard Borcherds constructed a bridge between the theory of modular functions and finite groups and, in the process, explained the observations of McKay and Thompson.

STRINGS '95: Proceedings

Since the s, the connection between string theory and moonshine has led to further results in mathematics and physics. Harvey proposed a generalization of this moonshine phenomenon called umbral moonshine , [] and their conjecture was proved mathematically by Duncan, Michael Griffin, and Ken Ono.

Some of the structures reintroduced by string theory arose for the first time much earlier as part of the program of classical unification started by Albert Einstein. Thereafter, German mathematician Theodor Kaluza combined the fifth dimension with general relativity , and only Kaluza is usually credited with the idea.

In , the Swedish physicist Oskar Klein gave a physical interpretation of the unobservable extra dimension—it is wrapped into a small circle. Einstein introduced a non-symmetric metric tensor , while much later Brans and Dicke added a scalar component to gravity. These ideas would be revived within string theory, where they are demanded by consistency conditions.

String theory was originally developed during the late s and early s as a never completely successful theory of hadrons , the subatomic particles like the proton and neutron that feel the strong interaction.

In the s, Geoffrey Chew and Steven Frautschi discovered that the mesons make families called Regge trajectories with masses related to spins in a way that was later understood by Yoichiro Nambu , Holger Bech Nielsen and Leonard Susskind to be the relationship expected from rotating strings. Chew advocated making a theory for the interactions of these trajectories that did not presume that they were composed of any fundamental particles, but would construct their interactions from self-consistency conditions on the S-matrix.

String Theory For Dummies Cheat Sheet

The S-matrix approach was started by Werner Heisenberg in the s as a way of constructing a theory that did not rely on the local notions of space and time, which Heisenberg believed break down at the nuclear scale. While the scale was off by many orders of magnitude, the approach he advocated was ideally suited for a theory of quantum gravity. Working with experimental data, R. Dolen, D. Horn and C. Schmid developed some sum rules for hadron exchange. When a particle and antiparticle scatter, virtual particles can be exchanged in two qualitatively different ways.

In the s-channel, the two particles annihilate to make temporary intermediate states that fall apart into the final state particles. In the t-channel, the particles exchange intermediate states by emission and absorption. In field theory, the two contributions add together, one giving a continuous background contribution, the other giving peaks at certain energies. In the data, it was clear that the peaks were stealing from the background—the authors interpreted this as saying that the t-channel contribution was dual to the s-channel one, meaning both described the whole amplitude and included the other.

The result was widely advertised by Murray Gell-Mann , leading Gabriele Veneziano to construct a scattering amplitude that had the property of Dolen—Horn—Schmid duality, later renamed world-sheet duality. The amplitude needed poles where the particles appear, on straight line trajectories, and there is a special mathematical function whose poles are evenly spaced on half the real line—the gamma function — which was widely used in Regge theory.

By manipulating combinations of gamma functions, Veneziano was able to find a consistent scattering amplitude with poles on straight lines, with mostly positive residues, which obeyed duality and had the appropriate Regge scaling at high energy. The amplitude could fit near-beam scattering data as well as other Regge type fits, and had a suggestive integral representation that could be used for generalization.

Over the next years, hundreds of physicists worked to complete the bootstrap program for this model, with many surprises. Veneziano himself discovered that for the scattering amplitude to describe the scattering of a particle that appears in the theory, an obvious self-consistency condition, the lightest particle must be a tachyon.

Miguel Virasoro and Joel Shapiro found a different amplitude now understood to be that of closed strings, while Ziro Koba and Holger Nielsen generalized Veneziano's integral representation to multiparticle scattering. Veneziano and Sergio Fubini introduced an operator formalism for computing the scattering amplitudes that was a forerunner of world-sheet conformal theory , while Virasoro understood how to remove the poles with wrong-sign residues using a constraint on the states.

Claud Lovelace calculated a loop amplitude, and noted that there is an inconsistency unless the dimension of the theory is Charles Thorn , Peter Goddard and Richard Brower went on to prove that there are no wrong-sign propagating states in dimensions less than or equal to In —70, Yoichiro Nambu , Holger Bech Nielsen , and Leonard Susskind recognized that the theory could be given a description in space and time in terms of strings.

The scattering amplitudes were derived systematically from the action principle by Peter Goddard , Jeffrey Goldstone , Claudio Rebbi , and Charles Thorn , giving a space-time picture to the vertex operators introduced by Veneziano and Fubini and a geometrical interpretation to the Virasoro conditions. In , Pierre Ramond added fermions to the model, which led him to formulate a two-dimensional supersymmetry to cancel the wrong-sign states.

In the fermion theories, the critical dimension was Stanley Mandelstam formulated a world sheet conformal theory for both the bose and fermi case, giving a two-dimensional field theoretic path-integral to generate the operator formalism. Michio Kaku and Keiji Kikkawa gave a different formulation of the bosonic string, as a string field theory , with infinitely many particle types and with fields taking values not on points, but on loops and curves.

In , Tamiaki Yoneya discovered that all the known string theories included a massless spin-two particle that obeyed the correct Ward identities to be a graviton. John Schwarz and Joel Scherk came to the same conclusion and made the bold leap to suggest that string theory was a theory of gravity, not a theory of hadrons.

String Theory For Dummies Cheat Sheet

They reintroduced Kaluza—Klein theory as a way of making sense of the extra dimensions. At the same time, quantum chromodynamics was recognized as the correct theory of hadrons, shifting the attention of physicists and apparently leaving the bootstrap program in the dustbin of history.

String theory eventually made it out of the dustbin, but for the following decade all work on the theory was completely ignored. Still, the theory continued to develop at a steady pace thanks to the work of a handful of devotees. Ferdinando Gliozzi , Joel Scherk, and David Olive realized in that the original Ramond and Neveu Schwarz-strings were separately inconsistent and needed to be combined. The resulting theory did not have a tachyon, and was proven to have space-time supersymmetry by John Schwarz and Michael Green in The same year, Alexander Polyakov gave the theory a modern path integral formulation, and went on to develop conformal field theory extensively.

In , Daniel Friedan showed that the equations of motions of string theory, which are generalizations of the Einstein equations of general relativity , emerge from the renormalization group equations for the two-dimensional field theory. The consistency conditions had been so strong, that the entire theory was nearly uniquely determined, with only a few discrete choices. In the early s, Edward Witten discovered that most theories of quantum gravity could not accommodate chiral fermions like the neutrino.

In coming to understand this calculation, Edward Witten became convinced that string theory was truly a consistent theory of gravity, and he became a high-profile advocate. Following Witten's lead, between and , hundreds of physicists started to work in this field, and this is sometimes called the first superstring revolution.

The gauge group of these closed strings was two copies of E8 , and either copy could easily and naturally include the standard model. Philip Candelas , Gary Horowitz , Andrew Strominger and Edward Witten found that the Calabi—Yau manifolds are the compactifications that preserve a realistic amount of supersymmetry, while Lance Dixon and others worked out the physical properties of orbifolds , distinctive geometrical singularities allowed in string theory.

Cumrun Vafa generalized T-duality from circles to arbitrary manifolds, creating the mathematical field of mirror symmetry. Daniel Friedan , Emil Martinec and Stephen Shenker further developed the covariant quantization of the superstring using conformal field theory techniques. David Gross and Vipul Periwal discovered that string perturbation theory was divergent. Stephen Shenker showed it diverged much faster than in field theory suggesting that new non-perturbative objects were missing.

In the s, Joseph Polchinski discovered that the theory requires higher-dimensional objects, called D-branes and identified these with the black-hole solutions of supergravity. These were understood to be the new objects suggested by the perturbative divergences, and they opened up a new field with rich mathematical structure.

It quickly became clear that D-branes and other p-branes, not just strings, formed the matter content of the string theories, and the physical interpretation of the strings and branes was revealed—they are a type of black hole.

Leonard Susskind had incorporated the holographic principle of Gerardus 't Hooft into string theory, identifying the long highly excited string states with ordinary thermal black hole states. As suggested by 't Hooft, the fluctuations of the black hole horizon, the world-sheet or world-volume theory, describes not only the degrees of freedom of the black hole, but all nearby objects too. In , at the annual conference of string theorists at the University of Southern California USC , Edward Witten gave a speech on string theory that in essence united the five string theories that existed at the time, and giving birth to a new dimensional theory called M-theory.

M-theory was also foreshadowed in the work of Paul Townsend at approximately the same time. The flurry of activity that began at this time is sometimes called the second superstring revolution. Andrew Strominger and Cumrun Vafa calculated the entropy of certain configurations of D-branes and found agreement with the semi-classical answer for extreme charged black holes.

Witten noted that the effective description of the physics of D-branes at low energies is by a supersymmetric gauge theory, and found geometrical interpretations of mathematical structures in gauge theory that he and Nathan Seiberg had earlier discovered in terms of the location of the branes.

In , Juan Maldacena noted that the low energy excitations of a theory near a black hole consist of objects close to the horizon, which for extreme charged black holes looks like an anti-de Sitter space. It is a concrete realization of the holographic principle , which has far-reaching implications for black holes , locality and information in physics, as well as the nature of the gravitational interaction.

To construct models of particle physics based on string theory, physicists typically begin by specifying a shape for the extra dimensions of spacetime.

Each of these different shapes corresponds to a different possible universe, or "vacuum state", with a different collection of particles and forces. String theory as it is currently understood has an enormous number of vacuum states, typically estimated to be around 10 , and these might be sufficiently diverse to accommodate almost any phenomena that might be observed at low energies.

Many critics of string theory have expressed concerns about the large number of possible universes described by string theory.

In his book Not Even Wrong , Peter Woit , a lecturer in the mathematics department at Columbia University , has argued that the large number of different physical scenarios renders string theory vacuous as a framework for constructing models of particle physics. According to Woit,. The possible existence of, say, 10 consistent different vacuum states for superstring theory probably destroys the hope of using the theory to predict anything.

If one picks among this large set just those states whose properties agree with present experimental observations, it is likely there still will be such a large number of these that one can get just about whatever value one wants for the results of any new observation. Some physicists believe this large number of solutions is actually a virtue because it may allow a natural anthropic explanation of the observed values of physical constants , in particular the small value of the cosmological constant.

In , Steven Weinberg published an article in which he argued that the cosmological constant could not have been too large, or else galaxies and intelligent life would not have been able to develop. String theorist Leonard Susskind has argued that string theory provides a natural anthropic explanation of the small value of the cosmological constant.

The fact that the observed universe has a small cosmological constant is just a tautological consequence of the fact that a small value is required for life to exist. Speculative scientific ideas fail not just when they make incorrect predictions, but also when they turn out to be vacuous and incapable of predicting anything. One of the fundamental properties of Einstein's general theory of relativity is that it is background independent , meaning that the formulation of the theory does not in any way privilege a particular spacetime geometry.

One of the main criticisms of string theory from early on is that it is not manifestly background independent.

In string theory, one must typically specify a fixed reference geometry for spacetime, and all other possible geometries are described as perturbations of this fixed one.

In his book The Trouble With Physics , physicist Lee Smolin of the Perimeter Institute for Theoretical Physics claims that this is the principal weakness of string theory as a theory of quantum gravity, saying that string theory has failed to incorporate this important insight from general relativity.

Others have disagreed with Smolin's characterization of string theory. In a review of Smolin's book, string theorist Joseph Polchinski writes. New physical theories are often discovered using a mathematical language that is not the most suitable for them… In string theory it has always been clear that the physics is background-independent even if the language being used is not, and the search for more suitable language continues.

Polchinski notes that an important open problem in quantum gravity is to develop holographic descriptions of gravity which do not require the gravitational field to be asymptotically anti-de Sitter. Since the superstring revolutions of the s and s, string theory has become the dominant paradigm of high energy theoretical physics.

In an interview from , Nobel laureate David Gross made the following controversial comments about the reasons for the popularity of string theory:. The most important [reason] is that there are no other good ideas around. That's what gets most people into it. When people started to get interested in string theory they didn't know anything about it. In fact, the first reaction of most people is that the theory is extremely ugly and unpleasant, at least that was the case a few years ago when the understanding of string theory was much less developed.

It was difficult for people to learn about it and to be turned on. So I think the real reason why people have got attracted by it is because there is no other game in town.

All other approaches of constructing grand unified theories, which were more conservative to begin with, and only gradually became more and more radical, have failed, and this game hasn't failed yet. Several other high-profile theorists and commentators have expressed similar views, suggesting that there are no viable alternatives to string theory.

Many critics of string theory have commented on this state of affairs. In his book criticizing string theory, Peter Woit views the status of string theory research as unhealthy and detrimental to the future of fundamental physics. He argues that the extreme popularity of string theory among theoretical physicists is partly a consequence of the financial structure of academia and the fierce competition for scarce resources. According to Smolin,.

String theory is a powerful, well-motivated idea and deserves much of the work that has been devoted to it. If it has so far failed, the principal reason is that its intrinsic flaws are closely tied to its strengths—and, of course, the story is unfinished, since string theory may well turn out to be part of the truth.

The real question is not why we have expended so much energy on string theory but why we haven't expended nearly enough on alternative approaches. Smolin goes on to offer a number of prescriptions for how scientists might encourage a greater diversity of approaches to quantum gravity research. From Wikipedia, the free encyclopedia. This article is about physics. For string algorithms, see String computer science. For other uses, see String disambiguation. For a more accessible and less technical introduction to this topic, see Introduction to M-theory.

Related concepts. Verlinde H. Verlinde Witten Yau Zaslow. Main article: String physics. Main articles: S-duality and T-duality. Matrix theory physics. String phenomenology. String cosmology. Mirror symmetry string theory. Monstrous moonshine. History of string theory.

String theory landscape. Background independence. Retrieved 25 July Archived from the original on November 5, Retrieved December 31, CS1 maint: Dirichlet Branes and Mirror Symmetry. Clay Mathematics Monographs. American Mathematical Society.

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Kaluza—Klein theory Compactification Why 10 dimensions? Supergravity Superspace Lie superalgebra Lie supergroup. Matrix theory Introduction to M-theory. Authority control BNF: Knowledge is your reward. Use OCW to guide your own life-long learning, or to teach others. We don't offer credit or certification for using OCW. Made for sharing. Download files for later. Send to friends and colleagues.

Modify, remix, and reuse just remember to cite OCW as the source. Lecture Notes. Professor Zwiebach has not thoroughly proofread these notes but believes they will be useful to people that want to know what goes on during lecture and what material gets covered Lecture note files. Need help getting started?

Don't show me this again Welcome! Announcements, introduction Lorentz transformations Light-cone coordinates.The most important [reason] is that there are no other good ideas around. Open strings are photons and non gravitational interactions. Originally, these results of Candelas were justified on physical grounds. The notion of parallel universes given by string theory would be tackled to understand consciousness and make an attempt to clarify the notion of other world or universe and parallel universes, the problems of time travel, soul and death in the universe or universes of strings which gets twisted and compactified.

Cumrun Vafa generalized T-duality from circles to arbitrary manifolds, creating the mathematical field of mirror symmetry. Branes are frequently studied from a purely mathematical point of view, and they are described as objects of certain categories , such as the derived category of coherent sheaves on a complex algebraic variety , or the Fukaya category of a symplectic manifold. String theory and M-theory: