In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Hitchin integrable system is an
integrable system
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
depending on the choice of a complex
reductive group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...
and a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
, introduced by
Nigel Hitchin
Nigel James Hitchin FRS (born 2 August 1946) is a British mathematician working in the fields of differential geometry, gauge theory, algebraic geometry, and mathematical physics. He is a Professor Emeritus of Mathematics at the University of O ...
in 1987. It lies on the crossroads of
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, the theory of
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s and
integrable system
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
theory. It also plays an important role in the
geometric Langlands correspondence In mathematics, the geometric Langlands correspondence is a reformulation of the Langlands correspondence obtained by replacing the number fields appearing in the original number theoretic version by function fields and applying techniques from al ...
over the field of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s through
conformal field theory
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes ...
. A
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
zero analogue of the Hitchin system, the
Garnier system, was discovered by René Garnier somewhat earlier as a certain limit of the
Schlesinger equations, and Garnier solved his system by defining spectral curves. (The Garnier system is the classical limit of the
Gaudin model
In physics, the Gaudin model, sometimes known as the ''quantum'' Gaudin model, is a model, or a large class of models, in statistical mechanics first described in its simplest case by Michel Gaudin. They are exactly solvable models, and are also ...
. In turn, the Schlesinger equations are the classical limit of the
Knizhnik–Zamolodchikov equations
In mathematical physics the Knizhnik–Zamolodchikov equations, or KZ equations, are linear differential equations satisfied by the correlation functions (on the Riemann sphere) of two-dimensional conformal field theories associated with an affin ...
). Almost all integrable systems of
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
can be obtained as particular cases of the Hitchin system or their common generalization defined by Bottacin and Markman in 1994.
Description
Using the language of algebraic geometry, the
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
of the system is a partial
compactification
Compactification may refer to:
* Compactification (mathematics), making a topological space compact
* Compactification (physics), the "curling up" of extra dimensions in string theory
See also
* Compaction (disambiguation)
Compaction may refer t ...
of the
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. It may be described also as the dual bundle to the tangent bundle. This may ...
to the
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
of
stable ''G''-bundles for some
reductive group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...
''G'', on some compact
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
. This space is endowed with a
canonical symplectic form
In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T^Q of a manifold Q. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus p ...
. Suppose for simplicity that ''G''=GL(''n'', ℂ), the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
; then the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
s can be described as follows: the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
to the moduli space of ''G''-bundles at the bundle ''F'' is
:
which by
Serre duality In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Alexa ...
is dual to
:
where
is the
canonical bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''.
Over the complex numbers, it ...
, so a pair
:
called a Hitchin pair or
Higgs bundle
In mathematics, a Higgs bundle is a pair (E,\varphi) consisting of a holomorphic vector bundle ''E'' and a Higgs field \varphi, a holomorphic 1-form taking values in the bundle of endomorphisms of ''E'' such that \varphi \wedge \varphi=0. Such pai ...
, defines a point in the cotangent bundle. Taking
:
one obtains elements in
:
which is a vector space which does not depend on
. So taking any basis in these vector spaces we obtain functions ''H
i'', which are Hitchin's hamiltonians. The construction for general reductive group is similar and uses
invariant polynomial In mathematics, an invariant polynomial is a polynomial P that is invariant under a group \Gamma acting on a vector space V. Therefore, P is a \Gamma-invariant polynomial if
:P(\gamma x) = P(x)
for all \gamma \in \Gamma and x \in V.
Cases of p ...
s on the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
of ''G''.
For trivial reasons these functions are algebraically independent, and some calculations show that their number is exactly half of the dimension of the phase space. The nontrivial part is a proof of
Poisson commutativity
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. ...
of these functions. They therefore define an integrable system in the symplectic or
Arnol'd–Liouville sense.
Hitchin fibration
The Hitchin fibration is the map from the moduli space of Hitchin pairs to
characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...
s, a higher genus analogue of the map Garnier used to define the spectral curves. used Hitchin fibrations over
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s in his proof of the
fundamental lemma.
See also
*
Yang–Mills equations
In physics and mathematics, and especially differential geometry and gauge theory, the Yang–Mills equations are a system of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the Eu ...
*
Higgs bundle
In mathematics, a Higgs bundle is a pair (E,\varphi) consisting of a holomorphic vector bundle ''E'' and a Higgs field \varphi, a holomorphic 1-form taking values in the bundle of endomorphisms of ''E'' such that \varphi \wedge \varphi=0. Such pai ...
*
Nonabelian Hodge correspondence
In algebraic geometry and differential geometry, the nonabelian Hodge correspondence or Corlette–Simpson correspondence (named after Kevin Corlette and Carlos Simpson) is a correspondence between Higgs bundles and representations of the fundame ...
*
Character variety
In the mathematics of moduli theory, given an algebraic, reductive, Lie group G and a finitely generated group \pi, the G-''character variety of'' \pi is a space of equivalence classes of group homomorphisms from \pi to G:
:\mathfrak(\pi,G)=\ ...
*
Hitchin's equations In mathematics, and in particular differential geometry and gauge theory, Hitchin's equations are a system of partial differential equations for a connection and Higgs field on a vector bundle or principal bundle over a Riemann surface, written ...
References
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{{Integrable systems
Algebraic geometry
Dynamical systems
Hamiltonian mechanics
Integrable systems
Lie groups
Differential geometry