In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
and
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 ...
, supermanifolds are generalizations of the
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
concept based on ideas coming from
supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories ...
. Several definitions are in use, some of which are described below.
Informal definition
An informal definition is commonly used in physics textbooks and introductory lectures. It defines a supermanifold as a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
with both
boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
ic and
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
ic coordinates. Locally, it is composed of
coordinate charts
In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an ...
that make it look like a "flat", "Euclidean"
superspace
Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions ''x'', ''y'', ''z'', ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann numb ...
. These local coordinates are often denoted by
:
where ''x'' is the (
real-number-valued)
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
coordinate, and
and
are
Grassmann-valued spatial "directions".
The physical interpretation of the Grassmann-valued coordinates are the subject of debate; explicit experimental searches for
supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories ...
have not yielded any positive results. However, the use of Grassmann variables allow for the tremendous simplification of a number of important mathematical results. This includes, among other things a compact definition of
functional integrals, the proper treatment of ghosts in
BRST quantization
In theoretical physics, the BRST formalism, or BRST quantization (where the ''BRST'' refers to the last names of Carlo Becchi, , Raymond Stora and Igor Tyutin) denotes a relatively rigorous mathematical approach to quantizing a field theory with ...
, the cancellation of infinities in
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, Witten's work on the
Atiyah-Singer index theorem, and more recent applications to
mirror symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
In 2D ther ...
.
The use of Grassmann-valued coordinates has spawned the field of
supermathematics, wherein large portions of geometry can be generalized to super-equivalents, including much of
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
and most of the theory of
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
s and
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
s (such as
Lie superalgebras, ''etc.'') However, issues remain, including the proper extension of
de Rham cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adap ...
to supermanifolds.
Definition
Three different definitions of supermanifolds are in use. One definition is as a sheaf over a
ringed space
In mathematics, a ringed space is a family of ( commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...
; this is sometimes called the "
algebro-geometric approach".
[ Alice Rogers, ''Supermanifolds: Theory and Applications'', World Scientific, (2007) ''(Se]
Chapter 1
'' This approach has a mathematical elegance, but can be problematic in various calculations and intuitive understanding. A second approach can be called a "concrete approach",
as it is capable of simply and naturally generalizing a broad class of concepts from ordinary mathematics. It requires the use of an infinite number of supersymmetric generators in its definition; however, all but a finite number of these generators carry no content, as the concrete approach requires the use of a
coarse topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
that renders almost all of them equivalent. Surprisingly, these two definitions, one with a finite number of supersymmetric generators, and one with an infinite number of generators, are equivalent.
[Rogers, ''Op. Cit.'' ''(See Chapter 8.)'']
A third approach describes a supermanifold as a
base topos of a
superpoint. This approach remains the topic of active research.
Algebro-geometric: as a sheaf
Although supermanifolds are special cases of
noncommutative manifolds, their local structure makes them better suited to study with the tools of standard
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
and
locally ringed space
In mathematics, a ringed space is a family of ( commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...
s.
A supermanifold M of dimension (''p'',''q'') is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
''M'' with a
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* ''The Sheaf'', a student-run newspaper se ...
of
superalgebra
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.
T ...
s, usually denoted ''O
M'' or C
∞(M), that is locally isomorphic to
, where the latter is a Grassmann (Exterior) algebra on ''q'' generators.
A supermanifold M of dimension (1,1) is sometimes called a
super-Riemann surface.
Historically, this approach is associated with
Felix Berezin,
Dimitry Leites, and
Bertram Kostant
Bertram Kostant (May 24, 1928 – February 2, 2017) was an American mathematician who worked in representation theory, differential geometry, and mathematical physics.
Early life and education
Kostant grew up in New York City, where he gradua ...
.
Concrete: as a smooth manifold
A different definition describes a supermanifold in a fashion that is similar to that of a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, except that the model space
has been replaced by the ''model superspace''
.
To correctly define this, it is necessary to explain what
and
are. These are given as the even and odd real subspaces of the one-dimensional space of
Grassmann number
In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra over the complex numbers. The special case of a 1-dimensional algebra is known as ...
s, which, by convention, are generated by a
countably infinite
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
number of anti-commuting variables: i.e. the one-dimensional space is given by
where ''V'' is infinite-dimensional. An element ''z'' is termed ''real'' if
; real elements consisting of only an even number of Grassmann generators form the space
of ''c-numbers'', while real elements consisting of only an odd number of Grassmann generators form the space
of ''a-numbers''. Note that ''c''-numbers commute, while ''a''-numbers anti-commute. The spaces
and
are then defined as the ''p''-fold and ''q''-fold Cartesian products of
and
.
Bryce DeWitt
Bryce Seligman DeWitt (January 8, 1923 – September 23, 2004), was an American theoretical physicist noted for his work in gravitation and quantum field theory.
Life
He was born Carl Bryce Seligman, but he and his three brothers, including th ...
, ''Supermanifolds'', (1984) Cambridge University Press ''(See chapter 2.)''
Just as in the case of an ordinary manifold, the supermanifold is then defined as a collection of
charts
A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent tabu ...
glued together with differentiable transition functions.
This definition in terms of charts requires that the transition functions have a
smooth structure In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold.
Definition
A smooth structure on a manifold M is ...
and a non-vanishing
Jacobian
In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to:
* Jacobian matrix and determinant
* Jacobian elliptic functions
* Jacobian variety
*Intermediate Jacobian
In mathematics, the intermediate Jacobian of a compact Kähle ...
. This can only be accomplished if the individual charts use a topology that is
considerably coarser than the vector-space topology on the Grassmann algebra. This topology is obtained by projecting
down to
and then using the natural topology on that. The resulting topology is ''not''
Hausdorff, but may be termed "projectively Hausdorff".
That this definition is equivalent to the first one is not at all obvious; however, it is the use of the coarse topology that makes it so, by rendering most of the "points" identical. That is,
with the coarse topology is essentially isomorphic
to
Properties
Unlike a regular manifold, a supermanifold is not entirely composed of a set of points. Instead, one takes the dual point of view that the structure of a supermanifold M is contained in its sheaf ''O
M'' of "smooth functions". In the dual point of view, an injective map corresponds to a surjection of sheaves, and a surjective map corresponds to an injection of sheaves.
An alternative approach to the dual point of view is to use the
functor of points In algebraic geometry, a functor represented by a scheme ''X'' is a set-valued contravariant functor on the category of schemes such that the value of the functor at each scheme ''S'' is (up to natural bijections) the set of all morphisms S \to X. T ...
.
If M is a supermanifold of dimension (''p'',''q''), then the underlying space ''M'' inherits the structure of a
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
whose sheaf of smooth functions is ''O
M/I'', where ''I'' is the
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
generated by all odd functions. Thus ''M'' is called the underlying space, or the body, of M. The quotient map ''O
M'' → ''O
M/I'' corresponds to an injective map ''M'' → M; thus ''M'' is a submanifold of M.
Examples
* Let ''M'' be a manifold. The ''odd tangent bundle'' ΠT''M'' is a supermanifold given by the sheaf Ω(''M'') of differential forms on ''M''.
* More generally, let ''E'' → ''M'' be a
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
. Then Π''E'' is a supermanifold given by the sheaf Γ(ΛE
*). In fact, Π is a
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
from the
category of vector bundles
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
*C ...
to the
category of supermanifolds.
*
Lie supergroups are examples of supermanifolds.
Batchelor's theorem
Batchelor's theorem states that every supermanifold is noncanonically isomorphic to a supermanifold of the form Π''E''. The word "noncanonically" prevents one from concluding that supermanifolds are simply glorified vector bundles; although the functor Π maps surjectively onto the isomorphism classes of supermanifolds, it is not an
equivalence of categories
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences fr ...
. It was published by
Marjorie Batchelor
Marjorie Blake (Marj) Batchelor-Winter is an American mathematician known for her work on coalgebras and supermanifolds. She is an emeritus staff member in the department of pure mathematics and mathematical statistics at the University of Cambr ...
in 1979.
The
proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a c ...
of Batchelor's theorem relies in an essential way on the existence of a
partition of unity
In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X:
* there is a neighbourhood of where all but a finite number of the functions of are 0 ...
, so it does not hold for complex or real-analytic supermanifolds.
Odd symplectic structures
Odd symplectic form
In many physical and geometric applications, a supermanifold comes equipped with an Grassmann-odd
symplectic structure. All natural geometric objects on a supermanifold are graded. In particular, the bundle of two-forms is equipped with a grading. An odd symplectic form ω on a supermanifold is a closed, odd form, inducing a non-degenerate pairing on ''TM''. Such a supermanifold is called a
P-manifold. Its graded dimension is necessarily (''n'',''n''), because the odd symplectic form induces a pairing of odd and even variables. There is a version of the Darboux theorem for P-manifolds, which allows one
to equip a P-manifold locally with a set of coordinates where the odd symplectic form ω is written as
:
where
are even coordinates, and
odd coordinates. (An odd symplectic form should not be confused with a Grassmann-even
symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping that is
; Bilinear: Linear in each argument ...
on a supermanifold. In contrast, the Darboux version of an even symplectic form is
:
where
are even coordinates,
odd coordinates and
are either +1 or −1.)
Antibracket
Given an odd symplectic 2-form ω one may define a
Poisson bracket
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. T ...
known as the antibracket of any two functions ''F'' and ''G'' on a supermanifold by
::
Here
and
are the right and left
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s respectively and ''z'' are the coordinates of the supermanifold. Equipped with this bracket, the algebra of functions on a supermanifold becomes an
antibracket algebra
In mathematics and theoretical physics, a Gerstenhaber algebra (sometimes called an antibracket algebra or braid algebra) is an algebraic structure discovered by Murray Gerstenhaber (1963) that combines the structures of a supercommutative ring an ...
.
A
coordinate transformation
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
that preserves the antibracket is called a
P-transformation. If the
Berezinian of a P-transformation is equal to one then it is called an
SP-transformation.
P and SP-manifolds
Using the
Darboux theorem
Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among ...
for odd symplectic forms one can show that P-manifolds are constructed from open sets of superspaces
glued together by P-transformations. A manifold is said to be an
SP-manifold if these transition functions can be chosen to be SP-transformations. Equivalently one may define an SP-manifold as a supermanifold with a nondegenerate odd 2-form ω and a
density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
ρ such that on each
coordinate patch there exist
Darboux coordinates in which ρ is identically equal to one.
Laplacian
One may define a
Laplacian operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
Δ on an SP-manifold as the operator which takes a function ''H'' to one half of the
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
of the corresponding
Hamiltonian vector field In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is ...
. Explicitly one defines
:::
.
In Darboux coordinates this definition reduces to
::::
where ''x''
a and θ
a are even and odd coordinates such that
::::
.
The Laplacian is odd and nilpotent
::::
.
One may define the
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be view ...
of functions ''H'' with respect to the Laplacian. I
Geometry of Batalin-Vilkovisky quantization Albert Schwarz
Albert Solomonovich Schwarz (; russian: А. С. Шварц; born June 24, 1934) is a Soviet and American mathematician and a theoretical physicist educated in the Soviet Union and now a professor at the University of California, Davis.
Early lif ...
has proven that the integral of a function ''H'' over a
Lagrangian submanifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sympl ...
''L'' depends only on the cohomology class of ''H'' and on the
homology class of the body of ''L'' in the body of the ambient supermanifold.
SUSY
A pre-SUSY-structure on a supermanifold of dimension
(''n'',''m'') is an odd ''m''-dimensional
distribution
.
With such a distribution one associates
its Frobenius tensor
(since ''P'' is odd, the skew-symmetric Frobenius
tensor is a symmetric operation).
If this tensor is non-degenerate,
e.g. lies in an open orbit of
,
''M'' is called ''a SUSY-manifold''.
SUSY-structure in dimension (1, ''k'')
is the same as odd
contact structure
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution (differential geometry), distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. ...
.
See also
*
Superspace
Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions ''x'', ''y'', ''z'', ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann numb ...
*
Supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories ...
*
Supergeometry
Supergeometry is differential geometry of modules over graded commutative algebras, supermanifolds and graded manifolds. Supergeometry is part and parcel of many classical and quantum field theories involving odd fields, e.g., SUSY field theory, ...
*
Graded manifold
*
Batalin–Vilkovisky formalism
In theoretical physics, the Batalin–Vilkovisky (BV) formalism (named for Igor Batalin and Grigori Vilkovisky) was developed as a method for determining the ghost structure for Lagrangian gauge theories, such as gravity and supergravity, whose ...
References
* Joseph Bernstein,
Lectures on Supersymmetry (notes by Dennis Gaitsgory), ''Quantum Field Theory program at IAS: Fall Term''
* A. Schwarz,
Geometry of Batalin-Vilkovisky quantization, ArXiv hep-th/9205088
* C. Bartocci, U. Bruzzo, D. Hernandez Ruiperez, ''The Geometry of Supermanifolds'' (Kluwer, 1991)
* L. Mangiarotti,
G. Sardanashvily, ''Connections in Classical and Quantum Field Theory'' (World Scientific, 2000) ()
External links
Super manifolds: an incomplete surveyat the Manifold Atlas.
{{Supersymmetry topics
Supersymmetry
Generalized manifolds
Structures on manifolds
Mathematical physics