Analytic geometry

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In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.

Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry.

Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometric shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom.

History

Ancient Greece

The Greek mathematician Menaechmus solved problems and proved theorems by using a method that had a strong resemblance to the use of coordinates and it has sometimes been maintained that he had introduced analytic geometry.

Apollonius of Perga, in On Determinate Section, dealt with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others. Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different from our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations (expressed in words) of curves. However, although Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case the coordinate system was superimposed upon a given curve a posteriori instead of a priori. That is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation.

Persia

The 11th-century Persian mathematician Omar Khayyam saw a strong relationship between geometry and algebra and was moving in the right direction when he helped close the gap between numerical and geometric algebra with his geometric solution of the general cubic equations, but the decisive step came later with Descartes. Omar Khayyam is credited with identifying the foundations of algebraic geometry, and his book Treatise on Demonstrations of Problems of Algebra (1070), which laid down the principles of analytic geometry, is part of the body of Persian mathematics that was eventually transmitted to Europe. Because of his thoroughgoing geometrical approach to algebraic equations, Khayyam can be considered a precursor to Descartes in the invention of analytic geometry.

Western Europe

Part of a series on
René Descartes

Philosophy

Works

People

Analytic geometry was independently invented by René Descartes and Pierre de Fermat, although Descartes is sometimes given sole credit. Cartesian geometry, the alternative term used for analytic geometry, is named after Descartes.

Descartes made significant progress with the methods in an essay titled La Géométrie (Geometry), one of the three accompanying essays (appendices) published in 1637 together with his Discourse on the Method for Rightly Directing One's Reason and Searching for Truth in the Sciences, commonly referred to as Discourse on Method. La Geometrie, written in his native French tongue, and its philosophical principles, provided a foundation for calculus in Europe. Initially the work was not well received, due, in part, to the many gaps in arguments and complicated equations. Only after the translation into Latin and the addition of commentary by van Schooten in 1649 (and further work thereafter) did Descartes's masterpiece receive due recognition.

Pierre de Fermat also pioneered the development of analytic geometry. Although not published in his lifetime, a manuscript form of Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci) was circulating in Paris in 1637, just prior to the publication of Descartes' Discourse. Clearly written and well received, the Introduction also laid the groundwork for analytical geometry. The key difference between Fermat's and Descartes' treatments is a matter of viewpoint: Fermat always started with an algebraic equation and then described the geometric curve that satisfied it, whereas Descartes started with geometric curves and produced their equations as one of several properties of the curves. As a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree. It was Leonhard Euler who first applied the coordinate method in a systematic study of space curves and surfaces.

Coordinates

Main article: Coordinate systems

 

Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.

In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates. Similarly, Euclidean space is given coordinates where every point has three coordinates. The value of the coordinates depends on the choice of the initial point of origin. There are a variety of coordinate systems used, but the most common are the following:

Cartesian coordinates (in a plane or space)

Main article: Cartesian coordinate system

The most common coordinate system to use is the Cartesian coordinate system, where each point has an x-coordinate representing its horizontal position, and a y-coordinate representing its vertical position. These are typically written as an ordered pair (x, y). This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates (x, y, z).

Polar coordinates (in a plane)

Main article: Polar coordinates

In polar coordinates, every point of the plane is represented by its distance r from the origin and its angle θ, with θ normally measured counterclockwise from the positive x-axis. Using this notation, points are typically written as an ordered pair (r, θ). One may transform back and forth between two-dimensional Cartesian and polar coordinates by using these formulae:

$${\displaystyle x=r\,\cos \theta ,\,y=r\,\sin \theta ;\,r={\sqrt {x^{2}+y^{2}}},\,\theta =\arctan(y/x).}$$

This system may be generalized to three-dimensional space through the use of cylindrical or spherical coordinates.

Cylindrical coordinates (in a space)

Main article: Cylindrical coordinates

In cylindrical coordinates, every point of space is represented by its height z, its radius r from the z-axis and the angle θ its projection on the xy-plane makes with respect to the horizontal axis.

Spherical coordinates (in a space)

Main article: Spherical coordinate system

In spherical coordinates, every point in space is represented by its distance ρ from the origin, the angle θ its projection on the xy-plane makes with respect to the horizontal axis, and the angle φ that it makes with respect to the z-axis. The names of the angles are often reversed in physics.

Equations and curves

Main articles: Solution set and Locus (mathematics)

In analytic geometry, any equation involving the coordinates specifies a subset of the plane, namely the solution set for the equation, or locus. For example, the equation y = x corresponds to the set of all the points on the plane whose x-coordinate and y-coordinate are equal. These points form a line, and y = x is said to be the equation for this line. In general, linear equations involving x and y specify lines, quadratic equations specify conic sections, and more complicated equations describe more complicated figures.

Usually, a single equation corresponds to a curve on the plane. This is not always the case: the trivial equation x = x specifies the entire plane, and the equation x² + y² = 0 specifies only the single point (0, 0). In three dimensions, a single equation usually gives a surface, and a curve must be specified as the intersection of two surfaces (see below), or as a system of parametric equations. The equation x² + y² = r² is the equation for any circle centered at the origin (0, 0) with a radius of r.

Lines and planes

Main articles: Line (geometry) and Plane (geometry)

Lines in a Cartesian plane, or more generally, in affine coordinates, can be described algebraically by linear equations. In two dimensions, the equation for non-vertical lines is often given in the slope-intercept form:

$${\displaystyle y=mx+b}$$

where:

• m is the slope or gradient of the line.
• b is the y-intercept of the line.
• x is the independent variable of the function y = f(x).

In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the normal vector) to indicate its "inclination".

Specifically, let ${\displaystyle \mathbf {r} _{0}}$ be the position vector of some point ${\displaystyle P_{0}=(x_{0},y_{0},z_{0})}$, and let ${\displaystyle \mathbf {n} =(a,b,c)}$ be a nonzero vector. The plane determined by this point and vector consists of those points ${\displaystyle P}$, with position vector ${\displaystyle \mathbf {r} }$, such that the vector drawn from ${\displaystyle P_{0}}$ to ${\displaystyle P}$ is perpendicular to ${\displaystyle \mathbf {n} }$. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points ${\displaystyle \mathbf {r} }$ such that

$${\displaystyle \mathbf {n} \cdot (\mathbf {r} -\mathbf {r} _{0})=0.}$$

(The dot here means a dot product, not scalar multiplication.) Expanded this becomes

$${\displaystyle a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0,}$$

which is the point-normal form of the equation of a plane.^{[citation needed]} This is just a linear equation:

$${\displaystyle ax+by+cz+d=0,{\text{ where }}d=-(ax_{0}+by_{0}+cz_{0}).}$$

Conversely, it is easily shown that if a, b, c and d are constants and a, b, and c are not all zero, then the graph of the equation

$${\displaystyle ax+by+cz+d=0,}$$

is a plane having the vector ${\displaystyle \mathbf {n} =(a,b,c)}$ as a normal. This familiar equation for a plane is called the general form of the equation of the plane.

In three dimensions, lines can not be described by a single linear equation, so they are frequently described by parametric equations:

$${\displaystyle x=x_{0}+at}$$

$${\displaystyle y=y_{0}+bt}$$

$${\displaystyle z=z_{0}+ct}$$

where:

• x, y, and z are all functions of the independent variable t which ranges over the real numbers.
• (x₀, y₀, z₀) is any point on the line.
• a, b, and c are related to the slope of the line, such that the vector (a, b, c) is parallel to the line.

Conic sections

Main article: Conic section

In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section – though it may be degenerate, and all conic sections arise in this way. The equation will be of the form

$${\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0{\text{ with }}A,B,C{\text{ not all zero.}}}$$

As scaling all six constants yields the same locus of zeros, one can consider conics as points in the five-dimensional projective space ${\displaystyle \mathbf {P} ^{5}.}$

The conic sections described by this equation can be classified using the discriminant

$${\displaystyle B^{2}-4AC.}$$

If the conic is non-degenerate, then:

• if ${\displaystyle B^{2}-4AC<0}$, the equation represents an ellipse;
  • if ${\displaystyle A=C}$ and ${\displaystyle B=0}$, the equation represents a circle, which is a special case of an ellipse;
• if ${\displaystyle B^{2}-4AC=0}$, the equation represents a parabola;
• if ${\displaystyle B^{2}-4AC>0}$, the equation represents a hyperbola;
  • if we also have ${\displaystyle A+C=0}$, the equation represents a rectangular hyperbola.

Quadric surfaces

Main article: Quadric surface

A quadric, or quadric surface, is a 2-dimensional surface in 3-dimensional space defined as the locus of zeros of a quadratic polynomial. In coordinates x₁, x₂,x₃, the general quadric is defined by the algebraic equation

$${\displaystyle \sum _{i,j=1}^{3}x_{i}Q_{ij}x_{j}+\sum _{i=1}^{3}P_{i}x_{i}+R=0.}$$

Quadric surfaces include ellipsoids (including the sphere), paraboloids, hyperboloids, cylinders, cones, and planes.

Distance and angle

Main articles: Distance and Angle

 

The distance formula on the plane follows from the Pythagorean theorem.

In analytic geometry, geometric notions such as distance and angle measure are defined using formulas. These definitions are designed to be consistent with the underlying Euclidean geometry. For example, using Cartesian coordinates on the plane, the distance between two points (x₁, y₁) and (x₂, y₂) is defined by the formula

$${\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}},}$$

which can be viewed as a version of the Pythagorean theorem. Similarly, the angle that a line makes with the horizontal can be defined by the formula

$${\displaystyle \theta =\arctan(m),}$$

where m is the slope of the line.

In three dimensions, distance is given by the generalization of the Pythagorean theorem:

$${\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}},}$$

while the angle between two vectors is given by the dot product. The dot product of two Euclidean vectors A and B is defined by

$${\displaystyle \mathbf {A} \cdot \mathbf {B} {\stackrel {\mathrm {def} }{=}}\left\|\mathbf {A} \right\|\left\|\mathbf {B} \right\|\cos \theta ,}$$

where θ is the angle between A and B.

Transformations

a) y = f(x) = |x|       b) y = f(x+3)       c) y = f(x)-3       d) y = 1/2 f(x)

Transformations are applied to a parent function to turn it into a new function with similar characteristics.

The graph of ${\displaystyle R(x,y)}$ is changed by standard transformations as follows:

• Changing ${\displaystyle x}$ to ${\displaystyle x-h}$ moves the graph to the right ${\displaystyle h}$ units.
• Changing ${\displaystyle y}$ to ${\displaystyle y-k}$ moves the graph up ${\displaystyle k}$ units.
• Changing ${\displaystyle x}$ to ${\displaystyle x/b}$ stretches the graph horizontally by a factor of ${\displaystyle b}$. (think of the ${\displaystyle x}$ as being dilated)
• Changing ${\displaystyle y}$ to ${\displaystyle y/a}$ stretches the graph vertically.
• Changing ${\displaystyle x}$ to ${\displaystyle x\cos A+y\sin A}$ and changing ${\displaystyle y}$ to ${\displaystyle -x\sin A+y\cos A}$ rotates the graph by an angle ${\displaystyle A}$.

There are other standard transformation not typically studied in elementary analytic geometry because the transformations change the shape of objects in ways not usually considered. Skewing is an example of a transformation not usually considered. For more information, consult the Wikipedia article on affine transformations.

For example, the parent function ${\displaystyle y=1/x}$ has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote, and occupies either the 1st and 3rd or 2nd and 4th quadrant. In general, if ${\displaystyle y=f(x)}$, then it can be transformed into ${\displaystyle y=af(b(x-k))+h}$. In the new transformed function, ${\displaystyle a}$ is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative ${\displaystyle a}$ values, the function is reflected in the ${\displaystyle x}$-axis. The ${\displaystyle b}$ value compresses the graph of the function horizontally if greater than 1 and stretches the function horizontally if less than 1, and like ${\displaystyle a}$, reflects the function in the ${\displaystyle y}$-axis when it is negative. The ${\displaystyle k}$ and ${\displaystyle h}$ values introduce translations, ${\displaystyle h}$, vertical, and ${\displaystyle k}$ horizontal. Positive ${\displaystyle h}$ and ${\displaystyle k}$ values mean the function is translated to the positive end of its axis and negative meaning translation towards the negative end.

Transformations can be applied to any geometric equation whether or not the equation represents a function. Transformations can be considered as individual transactions or in combinations.

Suppose that ${\displaystyle R(x,y)}$ is a relation in the ${\displaystyle xy}$ plane. For example,

$${\displaystyle x^{2}+y^{2}-1=0}$$

is the relation that describes the unit circle.

Finding intersections of geometric objects

Main article: Intersection (geometry)

For two geometric objects P and Q represented by the relations ${\displaystyle P(x,y)}$ and ${\displaystyle Q(x,y)}$ the intersection is the collection of all points ${\displaystyle (x,y)}$ which are in both relations.

For example, ${\displaystyle P}$ might be the circle with radius 1 and center ${\displaystyle (0,0)}$: ${\displaystyle P=\{(x,y)|x^{2}+y^{2}=1\}}$ and ${\displaystyle Q}$ might be the circle with radius 1 and center ${\displaystyle (1,0):Q=\{(x,y)|(x-1)^{2}+y^{2}=1\}}$. The intersection of these two circles is the collection of points which make both equations true. Does the point ${\displaystyle (0,0)}$ make both equations true? Using ${\displaystyle (0,0)}$ for ${\displaystyle (x,y)}$, the equation for ${\displaystyle Q}$ becomes ${\displaystyle (0-1)^{2}+0^{2}=1}$ or ${\displaystyle (-1)^{2}=1}$ which is true, so ${\displaystyle (0,0)}$ is in the relation ${\displaystyle Q}$. On the other hand, still using ${\displaystyle (0,0)}$ for ${\displaystyle (x,y)}$ the equation for ${\displaystyle P}$ becomes ${\displaystyle 0^{2}+0^{2}=1}$ or ${\displaystyle 0=1}$ which is false. ${\displaystyle (0,0)}$ is not in ${\displaystyle P}$ so it is not in the intersection.

The intersection of ${\displaystyle P}$ and ${\displaystyle Q}$ can be found by solving the simultaneous equations:

$${\displaystyle x^{2}+y^{2}=1}$$

$${\displaystyle (x-1)^{2}+y^{2}=1.}$$

Traditional methods for finding intersections include substitution and elimination.

Substitution: Solve the first equation for ${\displaystyle y}$ in terms of ${\displaystyle x}$ and then substitute the expression for ${\displaystyle y}$ into the second equation:

$${\displaystyle x^{2}+y^{2}=1}$$

$${\displaystyle y^{2}=1-x^{2}.}$$

We then substitute this value for ${\displaystyle y^{2}}$ into the other equation and proceed to solve for ${\displaystyle x}$:

$${\displaystyle (x-1)^{2}+(1-x^{2})=1}$$

$${\displaystyle x^{2}-2x+1+1-x^{2}=1}$$

$${\displaystyle -2x=-1}$$

$${\displaystyle x=1/2.}$$

Next, we place this value of ${\displaystyle x}$ in either of the original equations and solve for ${\displaystyle y}$:

$${\displaystyle (1/2)^{2}+y^{2}=1}$$

$${\displaystyle y^{2}=3/4}$$

$${\displaystyle y={\frac {\pm {\sqrt {3}}}{2}}.}$$

So our intersection has two points:

$${\displaystyle \left(1/2,{\frac {+{\sqrt {3}}}{2}}\right)\;\;{\text{and}}\;\;\left(1/2,{\frac {-{\sqrt {3}}}{2}}\right).}$$

Elimination: Add (or subtract) a multiple of one equation to the other equation so that one of the variables is eliminated. For our current example, if we subtract the first equation from the second we get ${\displaystyle (x-1)^{2}-x^{2}=0}$. The ${\displaystyle y^{2}}$ in the first equation is subtracted from the ${\displaystyle y^{2}}$ in the second equation leaving no ${\displaystyle y}$ term. The variable ${\displaystyle y}$ has been eliminated. We then solve the remaining equation for ${\displaystyle x}$, in the same way as in the substitution method:

$${\displaystyle x^{2}-2x+1+1-x^{2}=1}$$

$${\displaystyle -2x=-1}$$

$${\displaystyle x=1/2.}$$

We then place this value of ${\displaystyle x}$ in either of the original equations and solve for ${\displaystyle y}$:

$${\displaystyle (1/2)^{2}+y^{2}=1}$$

$${\displaystyle y^{2}=3/4}$$

$${\displaystyle y={\frac {\pm {\sqrt {3}}}{2}}.}$$

So our intersection has two points:

$${\displaystyle \left(1/2,{\frac {+{\sqrt {3}}}{2}}\right)\;\;{\text{and}}\;\;\left(1/2,{\frac {-{\sqrt {3}}}{2}}\right).}$$

For conic sections, as many as 4 points might be in the intersection.

Finding intercepts

Main articles: x-intercept and y-intercept

One type of intersection which is widely studied is the intersection of a geometric object with the ${\displaystyle x}$ and ${\displaystyle y}$ coordinate axes.

The intersection of a geometric object and the ${\displaystyle y}$-axis is called the ${\displaystyle y}$-intercept of the object. The intersection of a geometric object and the ${\displaystyle x}$-axis is called the ${\displaystyle x}$-intercept of the object.

For the line ${\displaystyle y=mx+b}$, the parameter ${\displaystyle b}$ specifies the point where the line crosses the ${\displaystyle y}$ axis. Depending on the context, either ${\displaystyle b}$ or the point ${\displaystyle (0,b)}$ is called the ${\displaystyle y}$-intercept.

Tangents and normals

Tangent lines and planes

Main article: Tangent

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Informally, it is a line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c on the curve if the line passes through the point (c, f(c)) on the curve and has slope f'(c) where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.

As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.

Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space.

Normal line and vector

Main article: Normal (geometry)

In geometry, a normal is an object such as a line or vector that is perpendicular to a given object. For example, in the two-dimensional case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point.

In the three-dimensional case a surface normal, or simply normal, to a surface at a point P is a vector that is perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality.

Topological quantum field theory

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Topological_quantum_field_theory

In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants.

Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for mathematical work related to topological field theory.

In condensed matter physics, topological quantum field theories are the low-energy effective theories of topologically ordered states, such as fractional quantum Hall states, string-net condensed states, and other strongly correlated quantum liquid states.

In dynamics, all continuous time dynamical systems, with and without noise, are Witten-type TQFTs and the phenomenon of spontaneous breakdown of the corresponding topological supersymmetry encompasses such well-established concepts as chaos, turbulence, 1/f and crackling noises, self-organized criticality etc.

Overview

In a topological field theory, correlation functions do not depend on the metric of spacetime. This means that the theory is not sensitive to changes in the shape of spacetime; if spacetime warps or contracts, the correlation functions do not change. Consequently, they are topological invariants.

Topological field theories are not very interesting on flat Minkowski spacetime used in particle physics. Minkowski space can be contracted to a point, so a TQFT applied to Minkowski space results in trivial topological invariants. Consequently, TQFTs are usually applied to curved spacetimes, such as, for example, Riemann surfaces. Most of the known topological field theories are defined on spacetimes of dimension less than five. It seems that a few higher-dimensional theories exist, but they are not very well understood.

Quantum gravity is believed to be background-independent (in some suitable sense), and TQFTs provide examples of background independent quantum field theories. This has prompted ongoing theoretical investigations into this class of models.

(Caveat: It is often said that TQFTs have only finitely many degrees of freedom. This is not a fundamental property. It happens to be true in most of the examples that physicists and mathematicians study, but it is not necessary. A topological sigma model targets infinite-dimensional projective space, and if such a thing could be defined it would have countably infinitely many degrees of freedom.)

Specific models

The known topological field theories fall into two general classes: Schwarz-type TQFTs and Witten-type TQFTs. Witten TQFTs are also sometimes referred to as cohomological field theories. See (Schwarz 2000).

Schwarz-type TQFTs

In Schwarz-type TQFTs, the correlation functions or partition functions of the system are computed by the path integral of metric-independent action functionals. For instance, in the BF model, the spacetime is a two-dimensional manifold M, the observables are constructed from a two-form F, an auxiliary scalar B, and their derivatives. The action (which determines the path integral) is

${\displaystyle S=\int \limits _{M}BF}$

The spacetime metric does not appear anywhere in the theory, so the theory is explicitly topologically invariant. The first example appeared in 1977 and is due to A. Schwarz; its action functional is:

${\displaystyle S=\int \limits _{M}A\wedge dA.}$

Another more famous example is Chern–Simons theory, which can be applied to knot invariants. In general, partition functions depend on a metric but the above examples are metric-independent.

Witten-type TQFTs

The first example of Witten-type TQFTs appeared in Witten's paper in 1988 (Witten 1988a), i.e. topological Yang–Mills theory in four dimensions. Though its action functional contains the spacetime metric g_αβ, after a topological twist it turns out to be metric independent. The independence of the stress-energy tensor T^αβ of the system from the metric depends on whether the BRST-operator is closed. Following Witten's example many other examples can be found in string theory.

Witten-type TQFTs arise if the following conditions are satisfied:

1. The action ${\displaystyle S}$ of the TQFT has a symmetry, i.e. if ${\displaystyle \delta }$ denotes a symmetry transformation (e.g. a Lie derivative) then ${\displaystyle \delta S=0}$ holds.
2. The symmetry transformation is exact, i.e. ${\displaystyle \delta ^{2}=0}$
3. There are existing observables ${\displaystyle O_{1},\dots ,O_{n}}$ which satisfy ${\displaystyle \delta O_{i}=0}$ for all ${\displaystyle i\in \{1,\dots ,n\}}$.
4. The stress-energy-tensor (or similar physical quantities) is of the form ${\displaystyle T^{\alpha \beta }=\delta G^{\alpha \beta }}$ for an arbitrary tensor ${\displaystyle G^{\alpha \beta }}$.

As an example (Linker 2015): Given a 2-form field ${\displaystyle B}$ with the differential operator ${\displaystyle \delta }$ which satisfies ${\displaystyle \delta ^{2}=0}$, then the action ${\displaystyle S=\int \limits _{M}B\wedge \delta B}$ has a symmetry if ${\displaystyle \delta B\wedge \delta B=0}$ since

${\displaystyle \delta S=\int \limits _{M}\delta (B\wedge \delta B)=\int \limits _{M}\delta B\wedge \delta B+\int \limits _{M}B\wedge \delta ^{2}B=0}$.

Further, the following holds (under the condition that ${\displaystyle \delta }$ is independent on ${\displaystyle B}$ and acts similarly to a functional derivative):

${\displaystyle {\frac {\delta }{\delta B^{\alpha \beta }}}S=\int \limits _{M}{\frac {\delta }{\delta B^{\alpha \beta }}}B\wedge \delta B+\int \limits _{M}B\wedge \delta {\frac {\delta }{\delta B^{\alpha \beta }}}B=\int \limits _{M}{\frac {\delta }{\delta B^{\alpha \beta }}}B\wedge \delta B-\int \limits _{M}\delta B\wedge {\frac {\delta }{\delta B^{\alpha \beta }}}B=-2\int \limits _{M}\delta B\wedge {\frac {\delta }{\delta B^{\alpha \beta }}}B}$.

The expression ${\displaystyle {\frac {\delta }{\delta B^{\alpha \beta }}}S}$ is proportional to ${\displaystyle \delta G}$ with another 2-form ${\displaystyle G}$ .

Now any averages of observables ${\displaystyle \left\langle O_{i}\right\rangle :=\int d\mu O_{i}e^{iS}}$ for the corresponding Haar measure ${\displaystyle \mu }$ are independent on the "geometric" field ${\displaystyle B}$ and are therefore topological:

${\displaystyle {\frac {\delta }{\delta B}}\left\langle O_{i}\right\rangle =\int d\mu O_{i}i{\frac {\delta }{\delta B}}Se^{iS}\propto \int d\mu O_{i}\delta Ge^{iS}=\delta \left(\int d\mu O_{i}Ge^{iS}\right)=0}$.

The third equality uses the fact that ${\displaystyle \delta O_{i}=\delta S=0}$ and the invariance of the Haar measure under symmetry transformations. Since ${\displaystyle \int d\mu O_{i}Ge^{iS}}$ is only a number, its Lie derivative vanishes.

Mathematical formulations

The original Atiyah–Segal axioms

Atiyah suggested a set of axioms for topological quantum field theory, inspired by Segal's proposed axioms for conformal field theory (subsequently, Segal's idea was summarized in Segal (2001)), and Witten's geometric meaning of supersymmetry in Witten (1982). Atiyah's axioms are constructed by gluing the boundary with a differentiable (topological or continuous) transformation, while Segal's axioms are for conformal transformations. These axioms have been relatively useful for mathematical treatments of Schwarz-type QFTs, although it isn't clear that they capture the whole structure of Witten-type QFTs. The basic idea is that a TQFT is a functor from a certain category of cobordisms to the category of vector spaces.

There are in fact two different sets of axioms which could reasonably be called the Atiyah axioms. These axioms differ basically in whether or not they apply to a TQFT defined on a single fixed n-dimensional Riemannian / Lorentzian spacetime M or a TQFT defined on all n-dimensional spacetimes at once.

Let Λ be a commutative ring with 1 (for almost all real-world purposes we will have Λ = Z, R or C). Atiyah originally proposed the axioms of a topological quantum field theory (TQFT) in dimension d defined over a ground ring Λ as following:

• A finitely generated Λ-module Z(Σ) associated to each oriented closed smooth d-dimensional manifold Σ (corresponding to the homotopy axiom),
• An element Z(M) ∈ Z(∂M) associated to each oriented smooth (d + 1)-dimensional manifold (with boundary) M (corresponding to an additive axiom).

These data are subject to the following axioms (4 and 5 were added by Atiyah):

1. Z is functorial with respect to orientation preserving diffeomorphisms of Σ and M,
2. Z is involutory, i.e. Z(Σ*) = Z(Σ)* where Σ* is Σ with opposite orientation and Z(Σ)* denotes the dual module,
3. Z is multiplicative.
4. Z(${\displaystyle \emptyset }$) = Λ for the d-dimensional empty manifold and Z(${\displaystyle \emptyset }$) = 1 for the (d + 1)-dimensional empty manifold.
5. Z(M*) = Z(M) (the hermitian axiom). If ${\displaystyle \partial M=\Sigma _{0}^{*}\cup \Sigma _{1}}$ so that Z(M) can be viewed as a linear transformation between hermitian vector spaces, then this is equivalent to Z(M*) being the adjoint of Z(M).

Remark. If for a closed manifold M we view Z(M) as a numerical invariant, then for a manifold with a boundary we should think of Z(M) ∈ Z(∂M) as a "relative" invariant. Let f : Σ → Σ be an orientation-preserving diffeomorphism, and identify opposite ends of Σ × I by f. This gives a manifold Σ_f and our axioms imply

${\displaystyle Z(\Sigma _{f})=\operatorname {Trace} \ \Sigma (f)}$

where Σ(f) is the induced automorphism of Z(Σ).

Remark. For a manifold M with boundary Σ we can always form the double ${\displaystyle M\cup _{\Sigma }M^{*}}$ which is a closed manifold. The fifth axiom shows that

${\displaystyle Z\left(M\cup _{\Sigma }M^{*}\right)=|Z(M)|^{2}}$

where on the right we compute the norm in the hermitian (possibly indefinite) metric.

The relation to physics

Physically (2) + (4) are related to relativistic invariance while (3) + (5) are indicative of the quantum nature of the theory.

Σ is meant to indicate the physical space (usually, d = 3 for standard physics) and the extra dimension in Σ × I is "imaginary" time. The space Z(Σ) is the Hilbert space of the quantum theory and a physical theory, with a Hamiltonian H, will have a time evolution operator e^itH or an "imaginary time" operator e^−tH. The main feature of topological QFTs is that H = 0, which implies that there is no real dynamics or propagation, along the cylinder Σ × I. However, there can be non-trivial "propagation" (or tunneling amplitudes) from Σ₀ to Σ₁ through an intervening manifold M with ${\displaystyle \partial M=\Sigma _{0}^{*}\cup \Sigma _{1}}$; this reflects the topology of M.

If ∂M = Σ, then the distinguished vector Z(M) in the Hilbert space Z(Σ) is thought of as the vacuum state defined by M. For a closed manifold M the number Z(M) is the vacuum expectation value. In analogy with statistical mechanics it is also called the partition function.

The reason why a theory with a zero Hamiltonian can be sensibly formulated resides in the Feynman path integral approach to QFT. This incorporates relativistic invariance (which applies to general (d + 1)-dimensional "spacetimes") and the theory is formally defined by a suitable Lagrangian—a functional of the classical fields of the theory. A Lagrangian which involves only first derivatives in time formally leads to a zero Hamiltonian, but the Lagrangian itself may have non-trivial features which relate to the topology of M.

Atiyah's examples

In 1988, M. Atiyah published a paper in which he described many new examples of topological quantum field theory that were considered at that time (Atiyah 1988). It contains some new topological invariants along with some new ideas: Casson invariant, Donaldson invariant, Gromov's theory, Floer homology and Jones–Witten theory.

d = 0

In this case Σ consists of finitely many points. To a single point we associate a vector space V = Z(point) and to n-points the n-fold tensor product: V^⊗n = V ⊗ … ⊗ V. The symmetric group S_n acts on V^⊗n. A standard way to get the quantum Hilbert space is to start with a classical symplectic manifold (or phase space) and then quantize it. Let us extend S_n to a compact Lie group G and consider "integrable" orbits for which the symplectic structure comes from a line bundle, then quantization leads to the irreducible representations V of G. This is the physical interpretation of the Borel–Weil theorem or the Borel–Weil–Bott theorem. The Lagrangian of these theories is the classical action (holonomy of the line bundle). Thus topological QFT's with d = 0 relate naturally to the classical representation theory of Lie groups and the Symmetry group.

d = 1

We should consider periodic boundary conditions given by closed loops in a compact symplectic manifold X. Along with Witten (1982) holonomy such loops as used in the case of d = 0 as a Lagrangian are then used to modify the Hamiltonian. For a closed surface M the invariant Z(M) of the theory is the number of pseudo holomorphic maps f : M → X in the sense of Gromov (they are ordinary holomorphic maps if X is a Kähler manifold). If this number becomes infinite i.e. if there are "moduli", then we must fix further data on M. This can be done by picking some points P_i and then looking at holomorphic maps f : M → X with f(P_i) constrained to lie on a fixed hyperplane. Witten (1988b) has written down the relevant Lagrangian for this theory. Floer has given a rigorous treatment, i.e. Floer homology, based on Witten's Morse theory ideas; for the case when the boundary conditions are over the interval instead of being periodic, the path initial and end-points lie on two fixed Lagrangian submanifolds. This theory has been developed as Gromov–Witten invariant theory.

Another example is Holomorphic Conformal Field Theory. This might not have been considered strictly topological quantum field theory at the time because Hilbert spaces are infinite dimensional. The conformal field theories are also related to the compact Lie group G in which the classical phase consists of a central extension of the loop group (LG). Quantizing these produces the Hilbert spaces of the theory of irreducible (projective) representations of LG. The group Diff₊(S¹) now substitutes for the symmetric group and plays an important role. As a result, the partition function in such theories depends on complex structure, thus it is not purely topological.

d = 2

Jones–Witten theory is the most important theory in this case. Here the classical phase space, associated with a closed surface Σ is the moduli space of a flat G-bundle over Σ. The Lagrangian is an integer multiple of the Chern–Simons function of a G-connection on a 3-manifold (which has to be "framed"). The integer multiple k, called the level, is a parameter of the theory and k → ∞ gives the classical limit. This theory can be naturally coupled with the d = 0 theory to produce a "relative" theory. The details have been described by Witten who shows that the partition function for a (framed) link in the 3-sphere is just the value of the Jones polynomial for a suitable root of unity. The theory can be defined over the relevant cyclotomic field, see Atiyah (1988). By considering a Riemann surface with boundary, we can couple it to the d = 1 conformal theory instead of coupling d = 2 theory to d = 0. This has developed into Jones–Witten theory and has led to the discovery of deep connections between knot theory and quantum field theory.

d = 3

Donaldson has defined the integer invariant of smooth 4-manifolds by using moduli spaces of SU(2)-instantons. These invariants are polynomials on the second homology. Thus 4-manifolds should have extra data consisting of the symmetric algebra of H₂. Witten (1988a) has produced a super-symmetric Lagrangian which formally reproduces the Donaldson theory. Witten's formula might be understood as an infinite-dimensional analogue of the Gauss–Bonnet theorem. At a later date, this theory was further developed and became the Seiberg–Witten gauge theory which reduces SU(2) to U(1) in N = 2, d = 4 gauge theory. The Hamiltonian version of the theory has been developed by Floer in terms of the space of connections on a 3-manifold. Floer uses the Chern–Simons function, which is the Lagrangian of Jones–Witten theory to modify the Hamiltonian. For details, see Atiyah (1988). Witten (1988a) has also shown how one can couple the d = 3 and d = 1 theories together: this is quite analogous to the coupling between d = 2 and d = 0 in Jones–Witten theory.

Now, topological field theory is viewed as a functor, not on a fixed dimension but on all dimensions at the same time.

The case of a fixed spacetime

Let Bord_M be the category whose morphisms are n-dimensional submanifolds of M and whose objects are connected components of the boundaries of such submanifolds. Regard two morphisms as equivalent if they are homotopic via submanifolds of M, and so form the quotient category hBord_M: The objects in hBord_M are the objects of Bord_M, and the morphisms of hBord_M are homotopy equivalence classes of morphisms in Bord_M. A TQFT on M is a symmetric monoidal functor from hBord_M to the category of vector spaces.

Note that cobordisms can, if their boundaries match, be sewn together to form a new bordism. This is the composition law for morphisms in the cobordism category. Since functors are required to preserve composition, this says that the linear map corresponding to a sewn together morphism is just the composition of the linear map for each piece.

There is an equivalence of categories between the category of 2-dimensional topological quantum field theories and the category of commutative Frobenius algebras.

All n-dimensional spacetimes at once

The pair of pants is a (1+1)-dimensional bordism, which corresponds to a product or coproduct in a 2-dimensional TQFT.

To consider all spacetimes at once, it is necessary to replace hBord_M by a larger category. So let Bord_n be the category of bordisms, i.e. the category whose morphisms are n-dimensional manifolds with boundary, and whose objects are the connected components of the boundaries of n-dimensional manifolds. (Note that any (n−1)-dimensional manifold may appear as an object in Bord_n.) As above, regard two morphisms in Bord_n as equivalent if they are homotopic, and form the quotient category hBord_n. Bord_n is a monoidal category under the operation which maps two bordisms to the bordism made from their disjoint union. A TQFT on n-dimensional manifolds is then a functor from hBord_n to the category of vector spaces, which maps disjoint unions of bordisms to their tensor product.

For example, for (1 + 1)-dimensional bordisms (2-dimensional bordisms between 1-dimensional manifolds), the map associated with a pair of pants gives a product or coproduct, depending on how the boundary components are grouped – which is commutative or cocommutative, while the map associated with a disk gives a counit (trace) or unit (scalars), depending on the grouping of boundary components, and thus (1+1)-dimension TQFTs correspond to Frobenius algebras.

Furthermore, we can consider simultaneously 4-dimensional, 3-dimensional and 2-dimensional manifolds related by the above bordisms, and from them we can obtain ample and important examples.

Development at a later time

Looking at the development of topological quantum field theory, we should consider its many applications to Seiberg–Witten gauge theory, topological string theory, the relationship between knot theory and quantum field theory, and quantum knot invariants. Furthermore, it has generated topics of great interest in both mathematics and physics. Also of important recent interest are non-local operators in TQFT (Gukov & Kapustin (2013)). If string theory is viewed as the fundamental, then non-local TQFTs can be viewed as non-physical models that provide a computationally efficient approximation to local string theory.

Witten-type TQFTs and dynamical systems

Main article: Supersymmetric theory of stochastic dynamics

Stochastic (partial) differential equations (SDEs) are the foundation for models of everything in nature above the scale of quantum degeneracy and coherence and are essentially Witten-type TQFTs. All SDEs possess topological or BRST supersymmetry, ${\displaystyle \delta }$, and in the operator representation of stochastic dynamics is the exterior derivative, which is commutative with the stochastic evolution operator. This supersymmetry preserves the continuity of phase space by continuous flows, and the phenomenon of supersymmetric spontaneous breakdown by a global non-supersymmetric ground state encompasses such well-established physical concepts as chaos, turbulence, 1/f and crackling noises, self-organized criticality etc. The topological sector of the theory for any SDE can be recognized as a Witten-type TQFT.