# The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions - Shing-Tung Yau, Steve Nadis (2010)

### GLOSSARY

*algebraic geometry:* a branch of mathematics that applies algebraic techniques—particularly those involving polynomial equations—to problems in geometry.

*anisotropy:* a property that varies in magnitude depending on the direction of measurement. Astronomers, for example, have detected variations in temperature—hot spots and cold spots—at different points in the sky, which are indications of temperature (and density) of anisotropies.

*anomaly:* a symmetry violation that is not apparent in classical theory but becomes evident when quantum effects are taken into consideration.

*anthropic principle:* the notion that the observed laws of nature must be consistent with the presence of intelligent life and, specifically, the presence of intelligent observers like us. Put in other terms, the universe looks the way it does because if conditions were even slightly different, life would not have formed and humans would not be around to observe it.

*Big Bang:* a theory that holds that our universe started from a state of extremely high temperature and density and has been expanding ever since.

*black hole:* an object so dense that nothing, not even light, can escape its intense gravitational field.

*boson:* one of two kinds of particles found in quantum theory, the other being fermions. Bosons include “messenger particles” that are carriers of the fundamental forces. (See *fermion*.)

*brane:* the basic object of string theory and M-theory, which can assume the form of a one-dimensional string and higher-dimensional objects, including a two-dimensional sheet or “membrane” (from which the term originated). When string theorists speak of branes, they’re generally referring to these higher-dimensional objects rather than to strings.

*bundle (or vector bundle or fiber bundle):* a topological space that is constructed from, or attached to, a manifold. To picture it, let’s assume the manifold is something familiar like a sphere. Then assume, to pick the simplest example, that a particular array of vectors (or a vector space) is attached to every point on the surface of the sphere. The bundle consists of the entire manifold—the sphere, in this case—plus all the arrays of vectors attached to it at every point. (See *tangent bundle*.)

*Calabi conjecture:* a mathematical hypothesis, put forth in the early 1950s by the geometer Eugenio Calabi, stating that spaces that satisfy certain topological requirements can also satisfy a stringent geometric (curvature) condition known as “Ricci flatness.” The conjecture also covered more general cases, where the Ricci curvature was not zero.

*Calabi-Yau manifold:* a broad class of geometric spaces with zero Ricci curvature that were shown to be mathematically possible in the proof of the Calabi conjecture. These spaces, or shapes, are “complex,” meaning that they must be of even dimension. The six-dimensional case is of special interest to string theory, where it serves as a candidate for the geometry of the theory’s six hidden, or “extra,” dimensions.

*calculus:* a set of tools—involving derivatives, integrals, limits, and infinite series—that were introduced to “modern” mathematics by Isaac Newton and Gottfried Leibniz.

*Cartesian product:* a way of combining two distinct geometric objects to create a new shape. The product of a circle and a line, for instance, is a cylinder. The product of two circles is a two-dimensional torus, or donut.

*Chern class:* a set of fixed properties, or invariants, that are used for characterizing the topology of complex manifolds. The number of Chern classes for a particular manifold equals the number of complex dimensions, with the last (or “top”) Chern class being equal to the Euler characteristic. Chern classes are named after the geometer S. S. Chern, who introduced the concept in the 1940s.

*classical physics:* physical laws, mostly developed before the twentieth century, that do not incorporate the principles of quantum mechanics.

*compact space:* a space that is bounded and finite in extent. A sphere is compact, whereas a plane is not.

*compactification:* the rolling up of a space so that it is “compact” or finite in extent. In string theory, different ways of wrapping, or compactifying, the extra dimensions lead to different physics.

*complex manifold:* a manifold that can be described in a mathematically consistent way by complex coordinates—its ordinary or real dimension being twice the complex dimension. All complex manifolds are also real manifolds of even dimension. However, not all real manifolds of even dimension are complex manifolds, because, in some cases, it is not possible to describe the entire manifold consistently with complex numbers. (See *manifold*.)

*complex numbers:* numbers of the form *a* + *bi*, where a and b are real numbers and *i* is the square root of -1. Complex numbers can be broken down into two components, with *a* called the real part and *b* the imaginary part.

*conformal field theory:* a quantum field theory that retains scale invariance and conformal invariance. Whereas in an ordinary quantum field theory, the strong force that binds quarks changes with distance, in conformal field theory, that force remains the same at any distance.

*conformal invariance:* transformations that preserve angles. The notion of conformal invariance includes scale invariance, as changes in scale—such as uniformly blowing up or shrinking a space—also leave angles intact. (See *scale invariance*.)

*conifold:* a cone-shaped singularity. Singularities of this sort are commonly found on a Calabi-Yau manifold.

*conifold transition:* a process during which space tears in the vicinity of a conifold singularity on a Calabi-Yau manifold and is then repaired in a way that changes the topology of the original manifold. Topologically distinct Calabi-Yau manifolds can thus be linked through a conifold transition.

*conjecture:* a mathematical hypothesis that is initially proposed without a complete proof.

*convex:* an object, such as a sphere, that curves or bulges “outward” such that a line segment connecting every two points within that object also lies within that object.

*coordinates:* numbers that specify the position of a point in space or in spacetime. Cartesian coordinates, for example, are the standard coordinates on a plane in which each point is specified by two numbers, one being the distance from the origin in the *x* direction and the other being the distance from the origin in the *y* direction. This coordinate system is named after the French mathematician (and philosopher) René Descartes. More coordinates are required, of course, to localize a point in a higher-dimensional space.

*cosmic microwave background:* electromagnetic (microwave) radiation that is left over from the Big Bang and that has since cooled and diffused and now permeates the universe.

*cosmic strings:* one-dimensional objects—which can assume the form of long, extremely thin, and extremely massive filaments—that are predicted by some quantum field theory models to have formed during a phase transition in the early universe. Cosmic strings also arise naturally in some versions of string theory, corresponding to the fundamental strings of those theories.

*cosmological constant:* a term that counters the effects of gravity in the famous Einstein equation; the constant corresponds to the energy locked up in spacetime itself. The cosmological constant is basically the vacuum energy—a form of energy thought to pervade all of spacetime, thereby offering a possible explanation for the phenomenon of dark energy. (See *dark energy* and *vacuum energy*.)

*coupling constant*: a number that determines the strength of a physical interaction. The string coupling constant, for instance, governs the interactions of strings, determining how likely it is for one string to split into two, or for two strings to come together to make one.

*cubic equation:* an equation whose highest term is third order, as in *ax*^{3} + *bxy*^{2} + *cy* + *d* = 0.

*curvature:* a quantitative way of measuring the extent to which a surface or space deviates from flatness. For example, the curvature of a circle is given by the inverse of its radius: The smaller the curvature of a circle, the larger its radius. In more than one dimension, curvature not only is given by a number but also takes into account the different directions along which a manifold can curve. While two-dimensional surfaces can be completely described by one kind of curvature, different kinds of curvature are possible in higher dimensions.

*dark energy:* a mysterious form of energy, constituting more than 70 percent of the universe’s total energy, according to recent measurements. Dark energy may be the measured value of the vacuum energy. Cosmologists believe it is causing the universe to expand in an accelerated fashion.

*dark matter:* nonluminous matter of unknown form whose presence has been inferred but not directly detected. Dark matter is thought to comprise the bulk of the universe’s matter, accounting for about 25 percent of the universe’s total energy.

*D-brane:* a brane or multidimensional surface in string theory upon which *open strings* (those that are not closed loops) can end.

*decompactification:* the process during which curled up, compact dimensions “unwind” and become infinitely large.

*derivative:* the measure of how a function, or quantity, changes with respect to a particular variable or variables. For a given input (or number), a function yields a specific output (or number). The derivative measures how the output changes as the input itself deviates slightly from the original value. If one were to graph a function on, say, the *x-y* plane, the derivative of that function at a particular point equals the slope of the tangent line at that point.

*differentiable:* a term that applies to “smooth” functions whose derivative can be taken at every point. A function is called infinitely differentiable if a limitless number of derivatives can be taken.

*differential equation:* an equation involving derivatives that shows how something changes with respect to one or more variables. Ordinary differential equations involve just one variable, whereas partial differential equations involve two or more independent variables. When processes in the physical and natural world are described mathematically, it is usually through differential equations.

*differential geometry:* the branch of geometry involving calculus (as opposed to algebra) that studies how the property of a space, such as its curvature, changes as you move about the space.

*dimension:* an independent direction, or “degree of freedom,” in which one can move in space or time. We can also think of the dimensionality of a space as the minimum number of coordinates needed to specify the position of a point in space. We call a plane “two-dimensional” because just two numbers—an *x* and a *y* coordinate—are needed to specify a position. Our everyday world has three spatial dimensions (left-right, forward-backward, up-down), whereas the spacetime we’re thought to inhabit has four dimensions—three of space and one of time. In addition, string theory (among other theories) holds that spacetime has additional spatial dimensions that are small, curled up, and concealed from view.

*Dirac equation:* a set of four interconnected equations, formulated by the British physicist Paul Dirac, that describe the behavior and dynamics of freely moving (and hence noninteracting) “spin-½” particles such as electrons.

*duality:* two theories that, at least superficially, appear to be different yet give rise to identical physics.

*Einstein equations:* the equations of general relativity that describe gravity, taking into account the theory of special relativity. Expressed in other terms, these equations can be used to determine the curvature of spacetime due to the presence of mass and energy.

*electromagnetic force:* one of four known forces in nature, this force combines electricity and magnetism.

*elementary particle:* a particle that is not known to have any substructure. Quarks, leptons, and gauge bosons are the elementary particles of the Standard Model—particles that we believe to be indivisible and fundamental.

*entropy:* a measure of the disorder of a physical system, with disordered systems having large entropy and ordered systems having low entropy. The entropy can also be thought of as the number of ways of arranging a system’s ingredients without changing the system’s overall properties (such as its volume, temperature, or pressure).

*Euclidean geometry:* the mathematical system attributed to the Greek mathematician Euclid in which the Pythagorean theorem always holds; the angles of a triangle always add up to 180 degrees; and for a straight line and a point not on that straight line, only one line can be drawn through that point (in the same plane) that does not intersect the given line. (This is the so-called parallel postulate.) Other versions of geometry have subsequently been developed, falling under the rubric of “non-Euclidean,” where these principles do not always hold.

*Euler characteristic (or Euler number):* an integer that helps characterize a topological space in a very general sense. The Euler characteristic, the simplest and the oldest known “topological invariant” of a space, was first introduced by Leonhard Euler for polyhedra and has since been generalized to other spaces. The Euler characteristic of a polyhedron, for example, is given by the number of vertices minus the number of edges plus the number of faces.

*event horizon:* the surface surrounding a black hole beyond which nothing, not even light, can escape.

*family (of particles):* See *generation.*

*fermion:* a particle of half-integer spin. This class of particles includes quarks and leptons, the so-called matter particles of the Standard Model.

*field:* a physical concept, introduced by the nineteenth-century physicist Michael Faraday, that assigns a specific value, such as a number or vector, to each point in spacetime. While a field can describe the force exerted on a particle at a given point in space, it can also describe the particle itself.

*field theory:* a theory in which both particles and forces are described by fields.

*flux:* lines of force, like the familiar electric and magnetic fields, that correspond to the special fields of string theory.

*function:* a mathematical expression of the form, for example, of *f*(*x*) = 3*x*^{2}, where every input value of *x* leads to a single output value for the function *f*(*x*).

*fundamental group:* a way of classifying spaces in topology. In spaces with a *trivial* fundamental group, every loop you can draw in that space can be shrunk down to a point without tearing a hole in the space. Spaces with a *nontrivial* fundamental group have noncontractible loops—that is, loops that cannot be shrunk down to a point owing to the presence of some obstruction (such as a hole).

*gauge theory:* a field theory, such as the Standard Model, in which symmetries are “gauged.” If a particular symmetry is gauged (which we then refer to as a gauge symmetry), that symmetry can be applied differently to a field at different points in spacetime, and yet the physics doesn’t change. Special fields called gauge fields must be added to the theory so that the physics remains invariant when symmetries are gauged.

*Gaussian:* a random probability distribution that is sometimes called a bell curve. This probability distribution is named after the geometer Carl Friedrich Gauss, who used it in his astronomical analyses, among other applications.

*general relativity:* Albert Einstein’s theory that unites Newtonian gravity with his own theory of special relativity. General relativity describes the gravitational potential as a metric and the gravitational force as the curvature of four-dimensional spacetime.

*generation (or family):* the organization of matter particles into three groups, each consisting of two quarks and two leptons. The particles in these groups would be identical, except that the masses increase with each generation.

*genus:* simply put, the number of holes in a two-dimensional surface or space. An ordinary donut, for instance, is of genus 1, whereas a sphere, which lacks a hole, is of genus 0.

*geodesic:* a path that is generally the shortest distance between two points on a given surface. On a two-dimensional plane, this path is a line segment. On a two-dimensional sphere, the geodesic lies along a so-called great circle that passes through the two points and has its center in the middle of the sphere. Depending on which way one travels on this great circle, the geodesic can be either the shortest path between those two points or the shortest path between those points compared with any path nearby.

*geometric analysis:* a mathematical approach that applies the techniques of differential calculus to geometric problems.

*geometry:* the branch of mathematics that concerns the size, shape, and curvature of a given space.

*gravitational waves:* disturbances or fluctuations of the gravitational field due to the presence of massive objects or localized sources of energy. These waves travel at the speed of light, as predicted by Einstein’s theory of gravity. Although there have been no direct detections of gravitational waves, there has been indirect evidence that they exist.

*gravity:* the weakest of the four forces of nature on the basis of current measurements. Newton viewed gravity as the mutual attraction of two massive objects, whereas Einstein showed that the force can be thought of in terms of the curvature of spacetime.

*Heisenberg uncertainty principle:* See *uncertainty principle*.

*heterotic string theory:* a class that includes two of the five string theories—the E_{8} X E_{8} and the SO(32) theories—which differ in terms of their choice of (gauge) symmetry groups. Both heterotic string theories involve only “closed” strings or loops rather than open strings.

*Higgs field:* a hypothetical field—a component of which consists of the Higgs boson or Higgs particle—that is responsible for endowing particles in the Standard Model with mass. The Higgs field is expected to be observed for the first time at the Large Hadron Collider.

*Hodge diamond:* a matrix or an array of Hodge numbers that provides detailed topological information about a Kähler manifold from which one can determine the Euler characteristic and other topological features. The Hodge diamond for a six-dimensional Calabi-Yau manifold consists of a four-by-four array. (Arrays of different sizes correspond to spaces of other even dimensions.) Hodge numbers, which are named after the Scottish geometer W. V. D. Hodge, offer clues into a space’s internal structure.

*holonomy:* a concept in differential geometry, related to curvature, that involves moving vectors around a closed loop in a parallel manner. The holonomy of a surface (or manifold), loosely speaking, is a measure of the degree to which tangent vectors are transformed as one moves around a loop on that surface.

*inflation:* a postulated exponential growth spurt during the universe’s first fraction of a second. The idea, first suggested by the physicist Alan Guth in 1979, simultaneously solves many cosmological puzzles, while also helping to explain the origin of matter and our expanding universe. Inflation is consistent with observations in astronomy and cosmology but has not been proven.

*integral:* one of the principal tools of calculus, taking an integral (or integration) offers a way of finding the area bounded by a curve. The calculation breaks up the bounded region into infinitesimally thin rectangles and adds up the areas of all the rectangles contained therein.

*invariant (or topological invariant):* a number or another fixed property of a space that does not change under the allowed transformations of a given mathematical theory. A topological invariant, for example, does not change under the continuous deformation (such as stretching, shrinking, or bending) of the original space from one shape to another. In Euclidean geometry, an invariant does not change under translations and rotations. In a conformal theory, an invariant does not change under conformal transformations that preserve angles.

*K3 surface:* a Calabi-Yau manifold of four real dimensions—or, equivalently, of two complex dimensions—named after the geometers Ernst Kummer, Erich Kähler, and Kunihiko Kodaira, the three K’s of K3. The name of these surfaces, or manifolds, also refers to the famous Himalayan mountain K2.

*Kähler manifold:* a complex manifold named after the geometer Erich Kähler and endowed with a special kind of holonomy that preserves the manifold’s complex structure under the operation of parallel transport.

*Kaluza-Klein theory:* Originally an attempt to unify general relativity and electromagnetism by introducing an extra (fifth) dimension, *Kaluza-Klein* is sometimes used as shorthand for the general approach of unifying the forces of nature by postulating the existence of an additional, unseen dimension (or dimensions).

*landscape:* in string theory, this is the range of possible shapes, or geometries, that the unseen dimensions could assume, which also depends on the number of ways that fluxes can be placed in that internal space. Put in other terms, the landscape consists of the range of possible vacuum states, or vacua, that are allowed by string theory.

*lemma:* a proven statement in mathematics that, rather than being considered an endpoint in itself, is normally regarded as a stepping-stone toward the proof of a broader, more powerful statement. But lemmas can be useful in themselves, too—sometimes more than initially realized.

*lepton:* the class of elementary particles that includes electrons and neutrinos. Unlike quarks, which are the other fermions, leptons do not experience the strong force and hence do not get trapped in atomic nuclei.

*linear equation:* an equation (in the case of two variables) of the general form *ax* + *by* + *c* = 0. An equation of this sort has no higher-order terms (such as *x*^{2}, *y*^{2}, or *xy*) and maps out a straight line. Another key feature of a linear equation is that a change in one variable, *x*, leads to a proportional change in the other variable, *y*, and vice versa. However, linear equations need not have only two variables, *x* and *y*, and can instead have any number of variables.

*manifold:* a topological space that is locally Euclidean, meaning that every point lies in a neighborhood that resembles flat space.

*matrix:* a two-dimensional (rectangular or square) array of numbers or more complicated algebraic entities. Two matrices can be added, subtracted, multiplied, and divided according to a relatively simple set of rules. A matrix can be expressed in the abbreviated form, a* _{ij}* where

*i*is the row number and

*j*the column number.

*metric:* a mathematical object (technically called a tensor) used to measure distances on a space or manifold. On a curved space, the metric determines the extent to which the actual distance deviates from the number given by the Pythagorean formula. Knowing a space’s metric is equivalent to knowing the geometry of that space.

*minimal surface:* a surface whose area is “locally minimized,” meaning that the surface area cannot be reduced by replacing small patches with any other possible nearby surface in the same ambient (or background) space.

*mirror symmetry:* a correspondence between two topologically distinct Calabi-Yau manifolds that give rise to the exact same physical theory.

*moduli space:* For a given topological object such as a Calabi-Yau manifold, the moduli space consists of the set of all possible *geometric* structures—the continuous set of manifolds encompassing all possible shape and size settings.

*M-theory:* a theory that unites the five separate string theories into a single, all-embracing theory with eleven spacetime dimensions. The principal ingredients of M-theory are branes, especially the two-dimensional (M2) and five-dimensional (M5) branes. In M-theory, strings are considered one-dimensional manifestations of branes. M-theory was introduced by Edward Witten—and largely conceived by him as well—during the “second string revolution” of 1995.

*neutron star:* a dense star, composed almost entirely of neutrons, that forms as a remnant following the gravitational collapse of a massive star that has exhausted its nuclear fuel.

*Newton’s constant:* the coefficient G, which determines the strength of gravity according to Newton’s law. Although Newton’s law has, of course, been supplanted by Einstein’s general relativity, it still remains a good approximation in many cases.

*non-Euclidean geometry:* the geometry that applies to spaces that are not flat, such as a sphere, where parallel lines can intersect, contrary to Euclid’s fifth postulate. In a non-Euclidean space, the sum of the angles of a triangle may be more or less than 180 degrees.

*non-Kähler manifold:* a class of complex manifolds that includes Kähler manifolds but also includes manifolds that cannot support a Kähler metric.

*nonlinear equation:* an equation that is not linear, meaning that changing one variable can lead to a disproportionate change in another variable.

*orthogonal:* perpendicular.

*parallel transport:* a way of moving vectors along a path on a surface or manifold that keeps the lengths of those vectors constant while keeping the angles between any two vectors constant as well. Parallel transport is easy to visualize on a flat, two-dimensional plane, but in more complicated, curved spaces, we may need to solve differential equations to determine the precise way of moving vectors around.

*phase transition:* the sudden change of matter, or a system, from one state to another. Boiling, freezing, and melting are familiar examples of phase transitions.

*Planck scale:* a scale of length (about 10^{-33} centimeters), time (about 10^{-43} seconds), energy (about 10^{28} electron-volts), and mass (about 10^{-8} kilograms) at which the quantum mechanical effects on gravity must be taken into account.

*Platonic solids:* the five “regular” (convex) polyhedrons—the tetrahedron, hexahedron (cube), octahedron, dodecahedron, and icosahedron—that satisfy the following properties: Their faces are made up of congruent polygons with every edge being the same length, and the same number of faces meet at every vertex. The Greek philosopher Plato theorized that the elements of the universe were composed of these solids, which were subsequently named after him.

*Poincaré conjecture (in three dimensions):* a famous conjecture posed by Henri Poincaré more than a century ago, which holds that if any loop drawn in a three-dimensional space can be shrunk to a point, without ripping the space or the loop itself, then that space is equivalent, topologically speaking, to a sphere.

*polygon:* a closed path in geometry composed of line segments such as a triangle, square, and pentagon.

*polyhedron:* a geometric object consisting of flat faces that meet at straight edges. Three-dimensional polyhedrons consist of polygonal faces that meet at edges, and the edges, in turn, meet at vertices. The tetrahedron and cube are familiar examples.

*polynomial:* functions that involve addition, subtraction, and multiplication, and non-negative, whole-number exponents. Although polynomial equations may look simple at first glance, they are often very difficult (and sometimes impossible) to solve.

*positive mass theorem (or positive energy theorem):* the statement that in any isolated physical system, the total mass or energy must be positive.

*product:* the result of multiplying two or more numbers (or other quantities).

*Pythagorean theorem:* a formula dictating that the sum of the squares of the two sides of a right triangle equals the square of the hypotenuse (*a*^{2} + *b*^{2} = *c*^{2}).

*quadratic equation:* an equation with a second-order (or “squared”) term, as in *ax*^{2} + *bx* + *c* = 0.

*quantum field theory:* a mathematical framework that combines quantum mechanics and field theory. Quantum field theories serve as the principal formalism underlying particle physics today.

*quantum fluctuations:* random variations on submicroscopic scales due to quantum effects such as the uncertainty principle.

*quantum geometry:* a form of geometry thought to be necessary to provide realistic physical descriptions on ultramicroscopic scales, where quantum effects become important.

*quantum gravity:* a sought-after theory that would unite quantum mechanics and general relativity and provide a microscopic or quantum description of gravity that’s comparable to the descriptions we have for the three other forces. String theory offers one attempt at creating a theory of quantum gravity.

*quantum mechanics:* a set of laws dictating the behavior of the universe on atomic scales. Quantum mechanics holds, among other things, that a particle can be equivalently expressed as a wave, and vice versa. Another central notion is that in some situations, physical quantities like energy, momentum, and charge only come in discrete amounts called *quanta*, rather than assuming any possible value.

*quark:* the class of elementary, subatomic particles—of which there are six varieties in all—that make up protons and neutrons. Quarks experience the strong force, unlike the other fermions, leptons, which do not.

*relativistic:* a term that applies to particles or other objects traveling at velocities that approach the speed of light.

*Ricci curvature:* a kind of curvature that is related to the flow of matter in spacetime, according to Einstein’s equations of general relativity.

*Riemann surface:* a one-dimensional complex manifold or, equivalently, a two-dimensional real surface. In string theory, the surface swept out by a string moving through spacetime is considered a Riemann surface.

*Riemannian geometry:* a mathematical framework for studying the curvature of spaces of arbitrary dimension. This form of geometry, introduced by Georg Friedrich Bernhard Riemann, lies at the heart of general relativity.

*scalar field:* a field that can be completely described by a single number at each point in space. A number corresponding to the temperature at each point in space is one example of a scalar field.

*scale invariance:* a phenomenon that is true, regardless of physical scale. In a scale-invariant system, the physics remains unchanged if the size of a system (or the notion of distance in the system) is uniformly expanded or shrunk.

*singularity:* a point in spacetime where the curvature and other physical quantities such as density become infinite, and conventional laws of physics break down. The center of a black hole and the moment of the Big Bang are both thought to be singularities.

*slope:* a term denoting the steepness or gradient of a curve—a measure of how much the steepness changes vertically compared with changes in the horizontal direction.

*smooth:* infinitely differentiable. A *smooth manifold* is a manifold that is everywhere differentiable, arbitrarily often, which means that a derivative can be taken at any point on the manifold, as many times as one cares to.

*spacetime:* In the four-dimensional version, spacetime is the union of the three dimensions of space with the single dimension of time to create a single, combined entity. This notion was introduced at the turn of the twentieth century by Albert Einstein and Hermann Minkowski. The concept of spacetime, however, is not restricted to four dimensions. String theory is based on a ten-dimensional spacetime, and M-theory, to which it is related, is based on an eleven-dimensional spacetime.

*special relativity:* a theory devised by Einstein that unifies space and time, stating that the laws of physics should be the same for all observers moving at constant velocity, regardless of their speed. The speed of light (*c*) is the same for all observers, according to special relativity. Einstein also showed that for a particle at rest, energy (*E*) and mass (*m*) are related by the formula *E = mc** ^{2}*.

*sphere:* as employed by geometers, the term typically refers to the two-dimensional surface of a ball, rather than the three-dimensional object itself. The concept of the sphere, however, is not limited to two dimensions and can instead apply to objects of any dimension, from zero on up.

*Standard Model:* a theory of particle physics that describes the known elementary particles and the interactions (strong, weak, and electromagnetic) between them. Gravity is not included in the Standard Model.

*string theory:* a physical theory, incorporating both quantum mechanics and general relativity, that is widely regarded as the leading candidate for a theory of quantum gravity. String theory posits that the fundamental building blocks of nature are not pointlike particles but are instead tiny, one-dimensional strands called strings, which come in either open or closed (loop) forms. There are five varieties of string theories—Type I, Type IIA, Type IIB, Heterotic E_{8}╳ E_{8}, Heterotic SO(32)—all of which are related to each other. The term *superstring theory* is sometimes used in place of *string theory* to explicitly show that the theory incorporates supersymmetry.

*strong force:* the force responsible for binding quarks inside protons and neutrons and for keeping protons and neutrons together to form atomic nuclei.

*submanifold:* a space of lower dimension sitting inside a higher-dimensional space. One can think of a donut, for example, as a continuous ring of circles, with each of those circles being a submanifold within the bigger structure or manifold—that being the donut itself.

*superpartners:* pairs of particles, consisting of a fermion and a boson, that are related to each other through supersymmetry.

*supersymmetry:* a mathematical symmetry that relates fermions to bosons. It is important to note that the bosons inferred by supersymmetry, which would be “superpartners” to the known quarks and leptons, have yet to be observed. Although supersymmetry is an important feature of most string theories, finding firm evidence of it would not necessarily prove that string theory is “right.”

*symmetry:* an action on an object, physical system, or equation in a way that leaves it unchanged. A circle, for example, remains unchanged under rotations about its center. A square and an equilateral triangle, similarly, remain unchanged under rotations about the center of 90 and 120 degrees, respectively. A square is not symmetric, however, under rotations of 45 degrees, as its appearance would change to what is sometimes called a square diamond that’s tipped on one corner.

*symmetry breaking:* a process that reduces the amount of observed symmetry in a system. Bear in mind that after symmetry is “broken” in this manner, it can still exist, though remaining hidden rather than visible.

*symmetry group:* a specific set of operations—such as rotations, reflections, and translations—that leaves an object invariant.

*tangent:* the best linear approximation to a curve at a particular point on that curve. (The same definition holds for higher-dimensional curves and their tangents.)

*tangent bundle:* a particular type of bundle made by attaching a tangent space to every point on the manifold. The tangent space encompasses all the vectors that are tangent to the manifold at that point. If the manifold is a two-dimensional sphere, for instance, then the tangent space is a two-dimensional plane that contains all the tangent vectors. If the manifold is a three-dimensional object, then the tangent space will be three-dimensional as well. (See *bundle*.)

*tension:* a quantity that measures a string’s resistance to being stretched or vibrated. A string’s tension is similar to its linear energy density.

*theorem:* a statement or proposition proven through formal, mathematical reasoning.

*topology:* a general way of characterizing a geometrical space. Topology concerns itself only with the gross features of that space rather than its exact size or shape. In topology, shapes are classified into groups that can be deformed into each other by stretching or compressing, without tearing their structure or changing the number of interior holes.

*torus* (plural, *tori*): a class of topological shapes that include the two-dimensional surface of a donut, plus higher-dimensional generalizations thereof.

*tunneling (or quantum tunneling):* a phenomenon, such as a particle crossing through a barrier into a different region, that is forbidden according to classical physics but is allowed (or has a nonzero probability) in quantum physics.

*uncertainty principle (also known as the Heisenberg uncertainty principle):* a tenet of quantum mechanics that holds that both the position and the momentum of an object cannot be known with absolute precision. The more precisely one of those variables is known, the greater the uncertainty that is attached to the other.

*unified field theory:* an attempt to account for all the forces of nature within a single, overarching framework. Albert Einstein devoted the last thirty years of his life to this goal, which has not yet been fully realized.

*vacuum:* a state, essentially devoid of matter, that represents the lowest possible energy density, or *ground state*, of a given system.

*vacuum energy:* the energy associated with “empty” space. The energy carried by the vacuum is not zero, however, because in quantum theory, space is never quite empty. Particles are continually popping into existence for a fleeting moment and then disappearing into nothingness. (See *cosmological constant*.)

*vector:* a geometric object (a line segment in one dimension) that has both length (or magnitude) and direction.

*vertex:* the point at which two or more edges of a shape meet.

*warping (related to “warp factor” and “warped product”):* the idea that the geometry of the four-dimensional spacetime we inhabit is not independent of the hidden extra dimensions but is instead influenced by the internal dimensions of string theory.

*weak force:* this force, one of the four known forces of nature, is responsible for radioactive decay, among other processes.

*world sheet:* the surface traced out by a string moving in spacetime.

*Yang-Mills equations:* generalizations of Maxwell’s equations that describe electromagnetism. The Yang-Mills equations are now used by physicists to describe the strong and weak forces, as well as the electroweak force that combines the electromagnetic and weak interactions. The equations are part of Yang-Mills theory or *gauge theory*, which was developed in the 1950s by the physicists Chen Ning Yang and Robert Mills.

*Yukawa coupling constant:* a number that determines the strength of coupling, or interaction, between a scalar field and a fermion—a noted example being the interactions of quarks or leptons with the Higgs field. Since the mass of particles stems from their interactions with the Higgs field, the Yukawa coupling constant is closely related to particle mass.