In

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

,

**deformation theory** is the study of

*infinitesimal conditions* associated with varying a solution

*P* of a problem to slightly different solutions

*P*_{ε}, where ε is a small number, or vector of small quantities. The infinitesimal conditions are therefore the result of applying the approach of

differential calculusIn mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....

to solving a problem with

constraintIn mathematics, a constraint is a condition that a solution to an optimization problem must satisfy. There are two types of constraints: equality constraints and inequality constraints...

s. One can think of a structure that is not completely rigid, and that deforms slightly to accommodate forces applied from outside; this explains the name.

Some characteristic phenomena are: the derivation of first-order equations by treating the ε quantities as having negligible squares; the possibility of

*isolated solutions*, in that varying a solution may not be possible,

*or* does not bring anything new; and the question of whether the infinitesimal constraints actually 'integrate', so that their solution does provide small variations. In some form these considerations have a history of centuries in mathematics, but also in

physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

and

engineeringEngineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

. For example, in the

geometry of numbersIn number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by ....

a class of results called

*isolation theorems* was recognised, with the topological interpretation of an

*open orbit* (of a

group actionIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

) around a given solution.

Perturbation theoryPerturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...

also looks at deformations, in general of operators.

## Deformations of complex manifolds

The most salient deformation theory in mathematics has been that of

complex manifoldIn differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

s and algebraic varieties. This was put on a firm basis by foundational work of

Kunihiko Kodaira and D. C. Spencer, after deformation techniques had received a great deal of more tentative application in the

Italian school of algebraic geometryIn relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more done internationally in birational geometry, particularly on algebraic surfaces. There were in the region of 30 to 40 leading mathematicians who made major...

. One expects, intuitively, that deformation theory, of the first order, should equate to the

Zariski tangent spaceIn algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point P on an algebraic variety V...

to a

moduli spaceIn algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...

. The phenomena turn out to be rather subtle, though, in the general case.

In the case of

Riemann surfaceIn mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

s, one can explain that the complex structure on the

Riemann sphereIn mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...

is isolated (no moduli). For genus 1, an

elliptic curveIn mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...

has a one-parameter family of complex structures, as shown in

elliptic functionIn complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...

theory. The general Kodaira-Spencer theory identifies as the key to the deformation theory the

sheaf cohomologyIn mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...

group

*H*^{1}(Θ)

where Θ is (the sheaf of germs of sections of) the holomorphic

tangent bundleIn differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...

. There is an obstruction in the

*H*^{2} of the same sheaf; which is always zero in case of a curve, for general reasons of dimension. In the case of genus 0 the

*H*^{1} vanishes, also. For genus 1 the dimension is the Hodge number

*h*^{1,0}

which is therefore 1.

It is known that all curves of genus one have equations of form y^2 = x^3 + ax + b. These obviously depend on two parameters, a and b, whereas the isomorphism classes of such curves have only one parameter. Hence there must be an equation relating those a and b which describe isomorphic elliptic curves. It turns out that curves for which b^2/a^3 has the same value, describe isomorphic curves. I.e. varying a and b is one way to deform the structure of the curve y^2 = x^3 + ax + b, but not all variations of a,b actually change the isomorphism class of the curve.

One can go further with the case of genus

*g* > 1, using

Serre dualityIn algebraic geometry, a branch of mathematics, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n . It shows that a cohomology group Hi is the dual space of another one, Hn−i...

to relate the

*H*^{1} to

*H*^{0}(Ω^{[2]})

where Ω is the holomorphic

cotangent bundleIn mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...

and the notation Ω

^{[2]} means the

*tensor square* (

*not* the second exterior power). In other words, deformations are regulated by holomorphic

quadratic differentialIn mathematics, a quadratic differential on a Riemann surface is a section of the symmetric square of the holomorphic cotangent bundle.If the section is holomorphic, then the quadratic differentialis said to be holomorphic...

s on a Riemann surface, again something known classically. The dimension of the moduli space, called

Teichmüller spaceIn mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism...

in this case, is computed as 3

*g* − 3, by the Riemann-Roch theorem.

These examples are the beginning of a theory applying to holomorphic families of complex manifolds, of any dimension. Further developments included: the extension by Spencer of the techniques to other structures of differential geometry; the assimilation of the Kodaira-Spencer theory into the abstract algebraic geometry of Grothendieck, with a consequent substantive clarification of earlier work; and deformation theory of other structures, such as algebras.

## Relationship to string theory

The so-called

Deligne conjectureIn mathematics, there are a number of so-called Deligne conjectures, provided by Pierre Deligne. These are independent conjectures in various fields of mathematics....

arising in the context of algebras (and Hochschild cohomology) stimulated much interest in deformation theory in relation to

string theoryString theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...

(roughly speaking, to formalise the idea that a string theory can be regarded as a deformation of a point-particle theory). This is now accepted as proved, after some hitches with early announcements.

Maxim KontsevichMaxim Lvovich Kontsevich is a Russian mathematician. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami...

is among those who have offered a generally accepted proof of this.