Deformation theory
Encyclopedia
In mathematics
, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or vector of small quantities. The infinitesimal conditions are therefore the result of applying the approach of differential calculus
to solving a problem with constraint
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 physics
and engineering
. For example, in the geometry of numbers
a class of results called isolation theorems was recognised, with the topological interpretation of an open orbit (of a group action
) around a given solution. Perturbation theory
also looks at deformations, in general of operators.
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 geometry
. One expects, intuitively, that deformation theory, of the first order, should equate to the Zariski tangent space
to a moduli space
. The phenomena turn out to be rather subtle, though, in the general case.
In the case of Riemann surface
s, one can explain that the complex structure on the Riemann sphere
is isolated (no moduli). For genus 1, an elliptic curve
has a one-parameter family of complex structures, as shown in elliptic function
theory. The general Kodaira-Spencer theory identifies as the key to the deformation theory the sheaf cohomology
group
where Θ is (the sheaf of germs of sections of) the holomorphic tangent bundle
. There is an obstruction in the H2 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 H1 vanishes, also. For genus 1 the dimension is the Hodge number
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 duality
to relate the H1 to
where Ω is the holomorphic cotangent bundle
and the notation Ω[2] means the tensor square (not the second exterior power). In other words, deformations are regulated by holomorphic quadratic differential
s on a Riemann surface, again something known classically. The dimension of the moduli space, called Teichmüller space
in this case, is computed as 3g − 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.
arising in the context of algebras (and Hochschild cohomology) stimulated much interest in deformation theory in relation to string theory
(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 Kontsevich
is among those who have offered a generally accepted proof of this.
Mathematics
Mathematics 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 Pε, where ε is a small number, or vector of small quantities. The infinitesimal conditions are therefore the result of applying the approach of differential calculus
Differential calculus
In 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 constraint
Constraint (mathematics)
In 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 physics
Physics
Physics 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 engineering
Engineering
Engineering 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 numbers
Geometry of numbers
In 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 action
Group action
In 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 theory
Perturbation theory
Perturbation 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 manifoldComplex manifold
In 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 geometry
Italian school of algebraic geometry
In 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 space
Zariski tangent space
In 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 space
Moduli space
In 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 surface
Riemann surface
In 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 sphere
Riemann sphere
In 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 curve
Elliptic curve
In 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 function
Elliptic function
In 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 cohomology
Sheaf cohomology
In 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
- H1(Θ)
where Θ is (the sheaf of germs of sections of) the holomorphic tangent bundle
Tangent bundle
In 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 H2 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 H1 vanishes, also. For genus 1 the dimension is the Hodge number
- h1,0
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 duality
Serre duality
In 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 H1 to
- H0(Ω[2])
where Ω is the holomorphic cotangent bundle
Cotangent bundle
In 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 differential
Quadratic differential
In 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 space
Teichmüller space
In 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 3g − 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 conjectureDeligne conjecture
In 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 theory
String theory
String 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 Kontsevich
Maxim Kontsevich
Maxim 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.