Archive for April, 2010

III.6 Calabi-Yau Manifolds by Eric Zaslow: Basic Definitions, Complex Manifolds and Hermitian Structure, Holonomy, and Calabi-Yau Manifolds in Riemannian Geometry: Page 163.

Wednesday, April 7th, 2010

Calabi-Yau manifolds play a prominent role in string theory and mirror symmetry. 

A manifold is called orientable if you can choose coordinate systems, defined on overlapping sets, at each point so that the Jacobian between the two sets of points is positive, det(yi/xj) > 0.  The Calabi-Yau manifold is the natural extension of this to complex numbers.  In this case for each coordinate system z there is a holomorphic function f(z) that is nonvanishing and satisfies the compatibility condition.  That condition is if there is another corresponding coordinate system w(z) then the corresponding function g is related to f by the equation: f = g det(wa/zb) .  A Calabi-Yau manifold may be thought of as a complex manifold with complex orientation.

The notion  of a holomorphic function on a complex manifold does not depend on the coordinates used to express the function.   The local geometry of a complex manifold looks like an open set in Cn and the tangent space at a point looks like Cn.

One may utilize Hermitian inner products represented by Hermitian matrices,gab’, with respect to a basis, ea, on a complex vector space.

On a Riemannian manifold you can move a vector of constant length and pointing in the same direction along a path and return to the same point.  Curvature reflects the fact that the end vector depends upon the path you take.  When traversing a closed loop the end vector can also be expressed by a matrix operator or holonomy matrix that sends the starting vector to the ending vector.  The group generated by these vectors is called the holonomy group of the manifold.  The holonomy group lies in the orthogonal group of length preserving matrices, O(m).  If the manifold is oriented then the holonomy group must lie in SO(m), as exhibited by transporting an oriented vector basis around the loop.

Complex manifolds of dimension n are also real manifolds of dimension 2n.  Real manifolds obtained in this way have additional structure.  One can multiply complex coordinate directions by i so there must be an operator on the real tangent space that squares to -1.   This operator has eigenvalues ±1.  One can think of these as holomorphic and anti-holomorphic directions.  The Hermitian property states that these directions are orthogonal.  If the directions stay orthogonal after transport around a closed loop then the manifold is called Kähler.

If Kähler then the holonomy group is a subgroup of U(n) which itself is a subgroup of SO(2m).  So, complex manifolds always have real orientations.  A characterization of the Kähler property is if gab’are components of the Hermitian metric in some coordinate patch then there exists a function φ such that gab’ = 2φ/zab.   

Given the nonmetric definition of a Calabi-Yau manifold a compatible Kähler structure leads to a metric definition of the Calabi-Yau manifold.  That the Kähler structure leads to a holonomy that lies in SU(n) Ì U(n) the natural analogue of the case of real orientation.

III.5 Buildings by Mark Ronan, Apartments and Chambers: Page 162.

Saturday, April 3rd, 2010

Invertible linear transformations on a vector space form a group called the general linear group, GLn(K), where n is the dimension of the vector space and K is the field of scalars.  Given a basis for the vector space then each group element can be written as an n x n matrix whose determinant is non-zero.

The subgroups of this group is studied.  The subgroups such as all lines, or two dimensional subspaces, are studied as projective spaces.  The study of the subspaces removes the special role of zero in the original space.

If one restricts the space GLn(K) to those linear transformations with determinant equal to one then we have the space SLn(K).

Another example is the group O(n) which consists of all linear transformations α of an n-dimensional real inner product space such that <αv,αw> = <v,w>, for any two vectors v and w (or in matrix terms for all real matrices A such that AAT = I).

Many groups may be created from GLn(K) and these subgroups are referred top as the classical groups.  The classical groups are either simple or close to simple.  Close to simple may mean for example that we can make them simple by quotienting out by the subgroup of scalar matrices.  When K is the field of real or complex numbers then the classical groups become the Lie groups. 

The simple Lie groups fall into one of four families called: An, Bn, Cn and Dn.  Then there are other types such as: E6, E7, E8, F4 and G2.  The subscripts are related to the dimensions of the groups.  For example, the groups of type An are the groups of invertible linear transformations of n+1 dimensions.

Lie groups have analogues over any field K where they are called of Lie type.  K can be a finite group in which case the the groups are finite.  Almost all finite simple groups are of Lie type.

Jacques Tits found a geometric theory that would embrace groups called the theory of buildings.

The building associated with the groups GLn(K) and SLn(K) are of type An-1.  The building is an abstract simplical complex.   It is a high dimensional analogue of a graph.  It has vertices, some pairs of vertices form edges, triples of vertices form edges and k vertices form k-1 dimensional simplexes.

A simplex is a convex hull of a finite set of points.  A three dimensional simplex is a tetrahedron. 

To form a building of type An-1 we take all 1-spaces, in projective space and treat them as vertices.   The simplexes are formed as nested sequences of proper subspaces.  For example, a 2-space inside a 3-space inside a 7-space form a triangle whose vertices are these three subspaces.

A chamber is formed as a simplex of maximal dimension n-1.  This would be a 1-space inside a 2-space inside a 3-space and so on. 

Buildings have important subgeometries called apartments.  In the An-1 case one takes a basis for the vector space and then generates subspaces formed by subsets of the basis.  Certain subgeometries, as mentioned above, of apartments are called chambers. 

In the A3 case the vector space is 4 dimensional.  A basis has four elements.  There are 4 subspaces formed by of one basis vector,  six formed by two basis vectors and four formed by three basis vectors.  One may view the four one spaces as vertices of a tetrahedron, the six two spaces as the midpoints of the edges and the four 3 spaces as the midpoints of the faces.  The apartment has 24 chambers.  There are six chambers for each face.  The triangular chambers form a triangular tiling of the tetrahedron.  The tailing surface is topologically equivalent to s sphere.  All apartments of a building that are topologically equivalent to a sphere are called spherical buildings. 

The buildings for groups of Lie type are all spherical.  The apartments of other Lie types are related to the regular and semi regular polyhedra in n dimensions.

Any two chambers of a building lie in a common apartment.  All apartments of a building are isomorphic.  Apartments have a nice intersection property. 

The theory of spherical buildings gives a geometric basis for the Lie groups and the other non Lie groups mentioned above.  A theorem of Tits on the existence of automorphisms shows that the group’s themselves must also exist.

If the group K is a p-adic field then one may construct an affine, as well as an associated spherical, building.  The affine building carries more information and yields the spherical one as a structure at infinity.  The apartments are tilings of Euclidean space.

One may further generalize to hyperbolic buildings, whose apartments are tilings of hyperbolic space, that arise in the study of hyperbolic Kac-Moody groups.

III.4 Braid Groups by F. E. A. Johnson: Page 161.

Friday, April 2nd, 2010

A braid determines a permutation by the rule the i th hole on the first plane z  label of the corresponding i th hole on the second plane.  This is a surjective homomorphism Bn z Sn that maps si to the transposition (i, i+1).  This is not an isomorphism because Bn is infinite.  si has infinite order and the transposition (i, i+1) squares to the identity (and is finite).

In his 1925 paper “Theorie der Zöpfe ” Artin showed that multiplication in Bn is completely described by the relations:

sisj = sjsi  (|i-j| 2),

sisi+1si = si+1sisi+1.

In physics the Artin relations are called the Yang-Baxter equations.  They are important in  statistical physics.

Artin also devised the method of “Combing the Braid” which allowed for the identification of the identity element from an arbitrary word in the generators.  An alternative algebraic method by Garside (1967) also decides when two elements are conjugate.

In 2001 a proof by Bigelow and independently by Kramer proved that braid groups are linear.

The braid groups described here are braid groups of the plane, because we started with the two parallel planes being punctured.  This method may be generalized to more general curves than the plane.

III.3 Bayesian Analysis and III.4 Braid Groups by F. E. A. Johnson: Page 160.

Thursday, April 1st, 2010

A conditional probability example is:  suppose n unbiased coins are tossed and three heads are observed.  If the number of coins tossed is between one and ten then we can compute the probability of seeing three heads given that there are n tosses, P(3 Heads | n Tosses), for each value of n from one to ten.   We may instead want to know P( n Tosses | 3 Heads).  By Bayes theorem we have P( n Tosses | 3 Heads) = P( 3 Heads | n Tosses) P( n Tosses) / P( 3 Heads).  A prior distribution is used to guess P( n Tosses) and this is used in the equality.  Using this prior distribution and the equality one obtains the posterior distribution.  If we assume the prior distribution is equal likelihood for each of the number of possible coins from 1 to 10 ( P( n Tosses) = 1/10) then we get the posterior distribution proportional to (1/10) P( 3 Heads | n Tosses).  This example seems very contrived to me.  It makes several assumptions without support, these are: the number of tosses is between 1 and 10, the prior distribution of equally likely tosses and it does not address how to account for the fact that if one has 2 tosses then one cannot observe 3 heads.

A physical model of a mathematical n-braid is that of two parallel planes each with n holes,  a string is run from each hole in the first plane to a hole in the second plane so that no two strings go to the same hole.  The result a physical model of an n-braid.  One can not double back or create knots but can stretch, contract, bend, and otherwise move the strings about in 3-dimensions and still end up with the “same” n-braid.  This notion of “same” is an equivalence relation called a braid isotropy.

Braid composition may proceed as follows.  Consider three parallel planes with with one n-braid between an end plane and the middle plane and a second n-braid between the middle plane and the other end plane.  With the same n-holes in the middle plane.  One may join the two strings, one from each end plane, at the middle plane and remove the middle plane.  The result is the composition of the two n-braids, X and Y, to form the n-braid XY.

With this notion of composition, n-braids, form a group Bn.  One may create an n-braid that acts as X-1.  Then the composition XX-1 = I, the trivial braid where the holes are the same and the strings simply connect to the same hole in the second plane.

Take the trivial n-braid and switch the i th and i+1 st string attachments in the second plane, call this si.  As a group BN, the braid group, is generated by the elements (si)1≤i≤n-1.  There is a similarity between si and the adjacent transpositions that generate the group Sn of permutations of {1,2, … , n}.