String theory in the context of T-duality

Physics beyond the Standard Model (BSM) refers to the theoretical developments needed to explain the deficiencies of the Standard Model, such as the inability to explain the fundamental parameters of the Standard Model, the strong CP problem, neutrino oscillations, matter–antimatter asymmetry, and the nature of dark matter and dark energy. Another problem lies within the mathematical framework of the Standard Model itself: the Standard Model is inconsistent with that of general relativity, and one or both theories break down under certain conditions, such as spacetime singularities like the Big Bang and black hole event horizons.

Theories that lie beyond the Standard Model include various extensions of the standard model through supersymmetry, such as the Minimal Supersymmetric Standard Model (MSSM) and Next-to-Minimal Supersymmetric Standard Model (NMSSM), and entirely novel explanations, such as string theory, M-theory, and extra dimensions. As these theories tend to reproduce the entirety of current phenomena, the question of which theory is the right one, or at least the "best step" towards a Theory of Everything, can only be settled via experiments, and is one of the most active areas of research in both theoretical and experimental physics.

String is a long flexible tool made from fibers twisted together into a single strand, or from multiple such strands which are in turn twisted together. String is used to tie, bind, or hang other objects. It is also used as a material to make things, such as textiles, and in arts and crafts. String is a simple tool, and its use by humans is known to have been developed tens of thousands of years ago. String may also be a component in other tools, and in devices as diverse as weapons, musical instruments, and toys. The ubiquity of string as a tool has led to conceptual and scientific uses of the term, including strings in computer science, cosmic strings, and string theory.

Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings.

'Superstring theory' is a shorthand for supersymmetric string theory because unlike bosonic string theory, it is the version of string theory that accounts for both fermions and bosons and incorporates supersymmetry to model gravity.

In physics, M-theory is a theory that unifies all consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1995. Witten's announcement initiated a flurry of research activity known as the second superstring revolution. Prior to Witten's announcement, string theorists had identified five versions of superstring theory. Although these theories initially appeared to be very different, work by many physicists showed that the theories were related in intricate and nontrivial ways. Physicists found that apparently distinct theories could be unified by mathematical transformations called S-duality and T-duality. Witten's conjecture was based in part on the existence of these dualities and in part on the relationship of the string theories to a field theory called eleven-dimensional supergravity.

Although a complete formulation of M-theory is not known, such a formulation should describe two- and five-dimensional objects called branes and should be approximated by eleven-dimensional supergravity at low energies. Modern attempts to formulate M-theory are typically based on matrix theory or the AdS/CFT correspondence. According to Witten, the M should stand for "magic", "mystery" or "membrane" (according to one's taste), and the true meaning of the title should be decided when a more fundamental formulation of the theory is known.

In particle physics, the baryon number (B) is an additive quantum number of a system. It is defined as$${\displaystyle B={\frac {1}{3}}(n_{\text{q}}-n_{\rm {\overline {q}}}),}$$where ⁠${\displaystyle n_{\rm {q}}}$⁠ is the number of quarks, and ⁠${\displaystyle n_{\rm {\overline {q}}}}$⁠ is the number of antiquarks. Baryons (three quarks) have B = +1, mesons (one quark, one antiquark) have B = 0, and antibaryons (three antiquarks) have B = −1. Exotic hadrons like pentaquarks (four quarks, one antiquark) and tetraquarks (two quarks, two antiquarks) are also classified as baryons and mesons depending on their baryon number. In the Standard Model B conservation is an accidental symmetry which means that it appears in the Standard Model but is often violated when going beyond it. Physics beyond the Standard Model theories that contain baryon number violation are, for example, Standard Model with extra dimensions, Supersymmetry, Grand Unified Theory and String theory.

In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that the curve travels counterclockwise around the point, i.e., the curve's number of turns. For certain open plane curves, the number of turns may be a non-integer. The winding number depends on the orientation of the curve, and it is negative if the curve travels around the point clockwise.

Winding numbers are fundamental objects of study in algebraic topology, and they play an important role in vector calculus, complex analysis, geometric topology, differential geometry, and physics (such as in string theory).

In mathematics, a modular form is a holomorphic function on the complex upper half-plane, ${\displaystyle {\mathcal {H}}}$, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modular forms has origins in complex analysis, with important connections with number theory. Modular forms also appear in other areas, such as algebraic topology, sphere packing, and string theory.

Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group ${\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )\subset \mathrm {SL} _{2}(\mathbb {R} )}$. Every modular form is attached to a Galois representation.

In mathematics, and especially differential geometry and mathematical physics, gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theory that admits gauge symmetry. In mathematics theory means a mathematical theory, encapsulating the general study of a collection of concepts or phenomena, whereas in the physical sense a gauge theory is a mathematical model of some natural phenomenon.

Gauge theory in mathematics is typically concerned with the study of gauge-theoretic equations. These are differential equations involving connections on vector bundles or principal bundles, or involving sections of vector bundles, and so there are strong links between gauge theory and geometric analysis. These equations are often physically meaningful, corresponding to important concepts in quantum field theory or string theory, but also have important mathematical significance. For example, the Yang–Mills equations are a system of partial differential equations for a connection on a principal bundle, and in physics solutions to these equations correspond to vacuum solutions to the equations of motion for a classical field theory, particles known as instantons.

The Minimal Supersymmetric Standard Model (MSSM) is an extension to the Standard Model that realizes supersymmetry. MSSM is the minimal supersymmetrical model as it considers only "the [minimum] number of new particle states and new interactions consistent with "Reality". Supersymmetry pairs bosons with fermions, so every Standard Model particle has a (yet undiscovered) superpartner. If discovered, such superparticles could be candidates for dark matter, and could provide evidence for grand unification or the viability of string theory. The failure to find evidence for MSSM using the Large Hadron Collider has strengthened an inclination to abandon it.

In physics, Ginzburg–Landau theory, often called Landau–Ginzburg theory, named after Vitaly Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenological model which could describe type-I superconductors without examining their microscopic properties. One GL-type superconductor is the famous YBCO, and generally all cuprates.

Later, a version of Ginzburg–Landau theory was derived from the Bardeen–Cooper–Schrieffer microscopic theory by Lev Gor'kov, thus showing that it also appears in some limit of microscopic theory and giving microscopic interpretation of all its parameters. The theory can also be given a general geometric setting, placing it in the context of Riemannian geometry, where in many cases exact solutions can be given. This general setting then extends to quantum field theory and string theory, again owing to its solvability, and its close relation to other, similar systems.

In physics, a string is a physical entity postulated in string theory and related subjects. Unlike elementary particles, which are zero-dimensional or point-like by definition, strings are one-dimensional extended entities. Researchers often have an interest in string theories because theories in which the fundamental entities are strings rather than point particles automatically have many properties that some physicists expect to hold in a fundamental theory of physics. Most notably, a theory of strings that evolve and interact according to the rules of quantum mechanics will automatically describe quantum gravity.

Bosonic string theory is the original version of string theory, developed in the late 1960s. It is so called because it contains only bosons in the spectrum.

In the 1980s, supersymmetry was discovered in the context of string theory, and a new version of string theory called superstring theory (supersymmetric string theory) became the real focus. Nevertheless, bosonic string theory remains a very useful model to understand many general features of perturbative string theory, and many theoretical difficulties of superstrings can actually already be found in the context of bosonic strings.

Edward Witten (born August 26, 1951) is an American theoretical physicist known for his contributions to string theory, topological quantum field theory, and various areas of mathematics. He is a professor emeritus in the school of natural sciences at the Institute for Advanced Study in Princeton. Witten is a researcher in string theory, quantum gravity, supersymmetric quantum field theories, and other areas of mathematical physics. Witten's work has also significantly impacted pure mathematics. In 1990, he became the first physicist to be awarded a Fields Medal by the International Mathematical Union, for his mathematical insights in physics, such as his 1981 proof of the positive energy theorem in general relativity, and his interpretation of the Jones invariants of knots as Feynman integrals. He is considered the practical founder of M-theory.

The history of string theory spans several decades of intense research including two superstring revolutions. Through the combined efforts of many researchers, string theory has developed into a broad and varied subject with connections to quantum gravity, particle and condensed matter physics, cosmology, and pure mathematics.

In theoretical physics, S-duality (short for strong–weak duality, or Sen duality) is an equivalence of two physical theories, which may be either quantum field theories or string theories. S-duality is useful for doing calculations in theoretical physics because it relates a theory in which calculations are difficult to a theory in which they are easier.

In quantum field theory, S-duality generalizes a well established fact from classical electrodynamics, namely the invariance of Maxwell's equations under the interchange of electric and magnetic fields. One of the earliest known examples of S-duality in quantum field theory is Montonen–Olive duality which relates two versions of a quantum field theory called N = 4 supersymmetric Yang–Mills theory. Recent work of Anton Kapustin and Edward Witten suggests that Montonen–Olive duality is closely related to a research program in mathematics called the geometric Langlands program. Another realization of S-duality in quantum field theory is Seiberg duality, which relates two versions of a theory called N=1 supersymmetric Yang–Mills theory.

In supersymmetry, eleven-dimensional supergravity is the theory of supergravity in the highest number of dimensions allowed for a supersymmetric theory. It contains a graviton, a gravitino, and a 3-form gauge field, with their interactions uniquely fixed by supersymmetry. Discovered in 1978 by Eugène Cremmer, Bernard Julia, and Joël Scherk, it quickly became a popular candidate for a theory of everything during the 1980s. However, interest in it soon faded due to numerous difficulties that arise when trying to construct physically realistic models. It came back to prominence in the mid-1990s when it was found to be the low energy limit of M-theory, making it crucial for understanding various aspects of string theory.