Canonical Actions on Bundles – Philosophizing Identity Over Gauge Transformations.

Untitled

In physical applications, fiber bundles often come with a preferred group of transformations (usually the symmetry group of the system). The modem attitude of physicists is to regard this group as a fundamental structure which should be implemented from the very beginning enriching bundles with a further structure and defining a new category.

A similar feature appears on manifolds as well: for example, on ℜ2 one can restrict to Cartesian coordinates when we regard it just as a vector space endowed with a differentiable structure, but one can allow also translations if the “bigger” affine structure is considered. Moreover, coordinates can be chosen in much bigger sets: for instance one can fix the symplectic form w = dx ∧ dy on ℜ2 so that ℜ2 is covered by an atlas of canonical coordinates (which include all Cartesian ones). But ℜ2 also happens to be identifiable with the cotangent bundle T*ℜ so that we can restrict the previous symplectic atlas to allow only natural fibered coordinates. Finally, ℜ2 can be considered as a bare manifold so that general curvilinear coordinates should be allowed accordingly; only if the full (i.e., unrestricted) manifold structure is considered one can use a full maximal atlas. Other choices define instead maximal atlases in suitably restricted sub-classes of allowed charts. As any manifold structure is associated with a maximal atlas, geometric bundles are associated to “maximal trivializations”. However, it may happen that one can restrict (or enlarge) the allowed local trivializations, so that the same geometrical bundle can be trivialized just using the appropriate smaller class of local trivializations. In geometrical terms this corresponds, of course, to impose a further structure on the bare bundle. Of course, this newly structured bundle is defined by the same basic ingredients, i.e. the same base manifold M, the same total space B, the same projection π and the same standard fiber F, but it is characterized by a new maximal trivialization where, however, maximal refers now to a smaller set of local trivializations.

Examples are: vector bundles are characterized by linear local trivializations, affine bundles are characterized by affine local trivializations, principal bundles are characterized by left translations on the fiber group. Further examples come from Physics: gauge transformations are used as transition functions for the configuration bundles of any gauge theory. For these reasons we give the following definition of a fiber bundle with structure group.

A fiber bundle with structure group G is given by a sextuple B = (E, M, π; F ;>.., G) such that:

  • (E, M, π; F) is a fiber bundle. The structure group G is a Lie group (possibly a discrete one) and λ : G —–> Diff(F) defines a left action of G on the standard fiber F .
  • There is a family of preferred trivializations {(Uα, t(α)}α∈I of B such that the following holds: let the transition functions be gˆ(αβ) : Uαβ —–> Diff(F) and let eG be the neutral element of G. ∃ a family of maps g(αβ) : Uαβ —–> G such

    that, for each x ∈ Uαβγ = Uα ∩ Uβ ∩ Uγ

    g(αα)(x) = eG

    g(αβ)(x) = [g(βα)(x)]-1

    g(αβ)(x) . g(βγ)(x) . g(γα)(x) = eG

    and

    (αβ)(x) = λ(g(αβ)(x)) ∈ Diff(F)

The maps g(αβ) : Uαβ —–> G, which depend on the trivialization, are said to form a cocycle with values in G. They are called the transition functions with values in G (or also shortly the transition functions). The preferred trivializations will be said to be compatible with the structure. Whenever dealing with fiber bundles with structure group the choice of a compatible trivialization will be implicitly assumed.

Fiber bundles with structure group provide the suitable framework to deal with bundles with a preferred group of transformations. To see this, let us begin by introducing the notion of structure bundle of a fiber bundle with structure group B = (B, M, π; F; x, G).

Let B = (B, M, π; F; x, G) be a bundle with a structure group; let us fix a trivialization {(Uα, t(α)}α∈I and denote by g(αβ) : Uαβ —–> G its transition functions. By using the canonical left action L : G —–> Diff(G) of G onto itself, let us define gˆ(αβ) : Uαβ —–> Diff(G) given by gˆ(αβ)(x) = L (g(αβ)(x)); they obviously satisfy the cocycle properties. Now by constructing a (unique modulo isomorphisms) principal bundle PB = P(B) having G as structure group and g(αβ) as transition functions acting on G by left translation Lg : G —> G.

The principal bundle P(B) = (P, M, p; G) constructed above is called the structure bundle of B = (B, M, π; F; λ, G).

Notice that there is no similar canonical way of associating a structure bundle to a geometric bundle B = (B, M, π; F), since in that case the structure group G is at least partially undetermined.

Each automorphism of P(B) naturally acts over B.

Let, in fact, {σ(α)}α∈I be a trivialization of PB together with its transition functions g(αβ) : Uαβ —–> G defined by σ(β) = σ(α) . g(αβ). Then any principal morphism Φ = (Φ, φ) over PB is locally represented by local maps ψ(α) : Uα —> G such that

Φ : [x, h]α ↦ [φ(α)(x), ψ(α)(x).h](α)

Since Φ is a global automorphism of PB for the above local expression, the following property holds true in Uαβ.

φ(α)(x) = φ(β)(x) ≡ x’

ψ(α)(x) = g(αβ)(x’) . ψ(β)(x) . g(βα)(x)

By using the family of maps {(φ(α), ψ(α))} one can thence define a family of global automorphisms of B. In fact, using the trivialization {(Uα, t(α)}α∈I, one can define local automorphisms of B given by

Φ(α)B : (x, y) ↦ (φ(α)(x), [λ(ψ(α)(x))](y))

These local maps glue together to give a global automorphism ΦB of the bundle B, due to the fact that g(αβ) are also transition functions of B with respect to its trivialization {(Uα, t(α)}α∈I.

In this way B is endowed with a preferred group of transformations, namely the group Aut(PB) of automorphisms of the structure bundle PB, represented on B by means of the canonical action. These transformations are called (generalized) gauge transformations. Vertical gauge transformations, i.e. gauge transformations projecting over the identity, are also called pure gauge transformations.

Principal Bundles Preserve Structures…

Untitled

A bundle P = (P, M ,π; G) is a principal bundle if the standard fiber is a Lie group G and ∃ (at least) one trivialization the transition functions of which act on G by left translations Lg : G → G : h ↦ f  g . h (where . denotes here the group multiplication).

The principal bundles are slightly different from affine bundles and vector bundles. In fact, while in affine bundles the fibers π-1(x) have a canonical structure of affine spaces and in vector bundles the fibers π-1(x) have a canonical structure of vector spaces, in principal bundles the fibers have no canonical Lie group structure. This is due to the fact that, while in affine bundles transition functions act by means of affine transformations and in vector bundles transition functions act by means of linear transformations, in principal bundles transition functions act by means of left translations which are not group automorphisms. Thus the fibers of a principal bundle do not carry a canonical group structure, but rather many non-canonical (trivialization-depending) group structures. In the fibers of a vector bundle there exists a preferred element (the “zero”) the definition of which does not depend on the local trivialization. On the contrary, in the fibers of a principal bundle there is no preferred point which is fixed by transition functions to be selected as an identity. Thus, while in affine bundles affine morphisms are those which preserve the affine structure of the fibers and in vector bundles linear morphisms are the ones which preserve the linear structure of the fibers, in a principal bundle P = (P, M, π; G) principal morphisms preserve instead a structure, the right action of G on P.

Let P = (P, M, π; G) be a principal bundle and {(Uα, t(α)}α∈I a trivialization. We can locally consider the maps

R(α)g : π-1(Uα) → π-1(Uα) : [x, h](α) ↦ [x, h . g](α) —– (1)

∃ a (global) right action Rg of G on P which is free, vertical and transitive on fibers; the local expression in the given trivialization of this action is given by R(α)g .

Using the local trivialization, we set p = [x, h](α) = [x, g(βα)(x) . h]β following diagram commutes:

Untitled

which clearly shows that the local expressions agree on the overlaps Uαβ, to define a right action. This is obviously a vertical action; it is free because of the following:

Rgp = p => [x, h . g](α) = [x, h](α) => h · g = h => g = e —– (2)

Finally, if p = [x, h1](α) and q = [x, h2](α) are two points in the same fiber of p, one can choose g = h2-1 . h1 (where · denotes the group multiplication) so that p = Rgq. This shows that the right action is also transitive on the fibers.

On the contrary, that a global left action cannot be defined by using the local maps

L(α)g : π-1(Uα) → π-1(Uα) : [x, h](α) ↦ [x, g . h](α) —– (3)

since these local maps do not satisfy a compatibility condition analogous to the condition of the commuting diagram.

let P = (P, M, π; G) and P’ = (P’, M’, π’ ; G’ ) be two principal bundles and θ : G → G’ be a homomorphism of Lie groups. A bundle morphism Φ = (Φ, φ) : P → P’ is a principal morphism with respect to θ if the following diagram is commutative:

Untitled

When G = G’ and θ = idG we just say that Φ is a principal morphism.

A trivial principal bundle (M x G, M, π; G) naturally admits the global unity section I ∈ Γ(M x G), defined with respect to a global trivialization, I : x ↦ (x, e), e being the unit element of G. Also, principal bundles allow global sections iff they are trivial. In fact, on principal bundles there is a canonical correspondence between local sections and local trivializations, due to the presence of the global right action.

Interleaves

Untitled

Many important spaces in topology and algebraic geometry have no odd-dimensional homology. For such spaces, functorial spatial homology truncation simplifies considerably. On the theory side, the simplification arises as follows: To define general spatial homology truncation, we used intermediate auxiliary structures, the n-truncation structures. For spaces that lack odd-dimensional homology, these structures can be replaced by a much simpler structure. Again every such space can be embedded in such a structure, which is the analogon of the general theory. On the application side, the crucial simplification is that the truncation functor t<n will not require that in truncating a given continuous map, the map preserve additional structure on the domain and codomain of the map. In general, t<n is defined on the category CWn⊃∂, meaning that a map must preserve chosen subgroups “Y ”. Such a condition is generally necessary on maps, for otherwise no truncation exists. So arbitrary continuous maps between spaces with trivial odd-dimensional homology can be functorially truncated. In particular the compression rigidity obstructions arising in the general theory will not arise for maps between such spaces.

Let ICW be the full subcategory of CW whose objects are simply connected CW-complexes K with finitely generated even-dimensional homology and vanishing odd-dimensional homology for any coefficient group. We call ICW the interleaf category.

For example, the space K = S22 e3 is simply connected and has vanishing integral homology in odd dimensions. However, H3(K;Z/2) = Z/2 ≠ 0.

Let X be a space whose odd-dimensional homology vanishes for any coefficient group. Then the even-dimensional integral homology of X is torsion-free.

Taking the coefficient group Q/Z, we have

Tor(H2k(X),Q/Z) = H2k+1(X) ⊗ Q/Z ⊕ Tor(H2k(X),Q/Z) = H2k+1(X;Q/Z) = 0.

Thus H2k(X) is torsion-free, since the group Tor(H2k(X),Q/Z) is isomorphic to the torsion subgroup of H2k(X).

Any simply connected closed 4-manifold is in ICW. Indeed, such a manifold is homotopy equivalent to a CW-complex of the form

Vi=1kSi2ƒe4

where the homotopy class of the attaching map ƒ : S3 → Vi=1k Si2 may be viewed as a symmetric k × k matrix with integer entries, as π3(Vi=1kSi2) ≅ M(k), with M(k) the additive group of such matrices.

Any simply connected closed 6-manifold with vanishing integral middle homology group is in ICW. If G is any coefficient group, then H1(M;G) ≅ H1(M) ⊗ G ⊕ Tor(H0M,G) = 0, since H0(M) = Z. By Poincaré duality,

0 = H3(M) ≅ H3(M) ≅ Hom(H3M,Z) ⊕ Ext(H2M,Z),

so that H2(M) is free. This implies that Tor(H2M,G) = 0 and hence H3(M;G) ≅ H3(M) ⊗ G ⊕ Tor(H2M,G) = 0. Finally, by G-coefficient Poincaré duality,

H5(M;G) ≅ H1(M;G) ≅ Hom(H1M,G) ⊕ Ext(H0M,G) = Ext(Z,G) = 0

Any smooth, compact toric variety X is in ICW: Danilov’s Theorem implies that H(X;Z) is torsion-free and the map A(X) → H(X;Z) given by composing the canonical map from Chow groups to homology, Ak(X) = An−k(X) → H2n−2k(X;Z), where n is the complex dimension of X, with Poincaré duality H2n−2k(X;Z) ≅ H2k(X;Z), is an isomorphism. Since the odd-dimensional cohomology of X is not in the image of this map, this asserts in particular that Hodd(X;Z) = 0. By Poincaré duality, Heven(X;Z) is free and Hodd(X;Z) = 0. These two statements allow us to deduce from the universal coefficient theorem that Hodd(X;G) = 0 for any coefficient group G. If we only wanted to establish Hodd(X;Z) = 0, then it would of course have been enough to know that the canonical, degree-doubling map A(X) → H(X;Z) is onto. One may then immediately reduce to the case of projective toric varieties because every complete fan Δ has a projective subdivision Δ, the corresponding proper birational morphism X(Δ) → X(Δ) induces a surjection H(X(Δ);Z) → H(X(Δ);Z) and the diagram

Untitled

commutes.

Let G be a complex, simply connected, semisimple Lie group and P ⊂ G a connected parabolic subgroup. Then the homogeneous space G/P is in ICW. It is simply connected, since the fibration P → G → G/P induces an exact sequence

1 = π1(G) → π1(G/P) → π0(P) → π0(G) = 0,

which shows that π1(G/P) → π0(P) is a bijection. Accordingly, ∃ elements sw(P) ∈ H2l(w)(G/P;Z) (“Schubert classes,” given geometrically by Schubert cells), indexed by w ranging over a certain subset of the Weyl group of G, that form a basis for H(G/P;Z). (For w in the Weyl group, l(w) denotes the length of w when written as a reduced word in certain specified generators of the Weyl group.) In particular Heven(G/P;Z) is free and Hodd(G/P;Z) = 0. Thus Hodd(G/P;G) = 0 for any coefficient group G.

The linear groups SL(n, C), n ≥ 2, and the subgroups S p(2n, C) ⊂ SL(2n, C) of transformations preserving the alternating bilinear form

x1yn+1 +···+ xny2n −xn+1y1 −···−x2nyn

on C2n × C2n are examples of complex, simply connected, semisimple Lie groups. A parabolic subgroup is a closed subgroup that contains a Borel group B. For G = SL(n,C), B is the group of all upper-triangular matrices in SL(n,C). In this case, G/B is the complete flag manifold

G/B = {0 ⊂ V1 ⊂···⊂ Vn−1 ⊂ Cn}

of flags of subspaces Vi with dimVi = i. For G = Sp(2n,C), the Borel subgroups B are the subgroups preserving a half-flag of isotropic subspaces and the quotient G/B is the variety of all such flags. Any parabolic subgroup P may be described as the subgroup that preserves some partial flag. Thus (partial) flag manifolds are in ICW. A special case is that of a maximal parabolic subgroup, preserving a single subspace V. The corresponding quotient SL(n, C)/P is a Grassmannian G(k, n) of k-dimensional subspaces of Cn. For G = Sp(2n,C), one obtains Lagrangian Grassmannians of isotropic k-dimensional subspaces, 1 ≤ k ≤ n. So Grassmannians are objects in ICW. The interleaf category is closed under forming fibrations.

Grothendieck’s Abstract Homotopy Theory

HS21

Let E be a Grothendieck topos (think of E as the category, Sh(X), of set valued sheaves on a space X). Within E, we can pick out a subcategory, C, of locally finite, locally constant objects in E. (If X is a space with E = Sh(X), C corresponds to those sheaves whose espace étale is a finite covering space of X.) Picking a base point in X generalises to picking a ‘fibre functor’ F : C → Setsfin, a functor satisfying various conditions implying that it is pro-representable. (If x0 ∈ X is a base point {x0} → X induces a ‘fibre functor’ Sh(X) → Sh{x0} ≅ Sets, by pullback.)

If F is pro-representable by P, then π1(E, F) is defined to be Aut(P), which is a profinite group. Grothendieck proves there is an equivalence of categories C ≃ π1(E) − Setsfin, the category of finite π1(E)-sets. If X is a locally nicely behaved space such as a CW-complex and E = Sh(X), then π1(E) is the profinite completion of π1(X). This profinite completion occurs only because Grothendieck considers locally finite objects. Without this restriction, a covering space Y of X would correspond to a π1(X) – set, Y′, but if Y is a finite covering of X then the homomorphism from π1(X) to the finite group of transformations of Y factors through the profinite completion of π1(X). This is defined by : if G is a group, Gˆ = lim(G/H : H ◅ G, H of finite index) is its profinite completion. This idea of using covering spaces or their analogue in E raises several important points:

a) These are homotopy theoretic results, but no paths are used. The argument involving sheaf theory, the theory of (pro)representable functors, etc., is of a purely categorical nature. This means it is applicable to spaces where the use of paths, and other homotopies is impossible because of bad (or unknown) local properties. Such spaces have been studied within Shape Theory and Strong Shape Theory, although not by using Grothendieck’s fundamental group, nor using sheaf theory.

b) As no paths are used, these methods can also be applied to non-spaces, e.g. locales and possibly to their non-commutative analogues, quantales. For instance, classically one could consider a field k and an algebraic closure K of k and then choose C to be a category of étale algebras over k, in such a way that π1(E) ≅ Gal(K/k), the Galois group of k. It, in fact, leads to a classification theorem for Grothendieck toposes. From this viewpoint, low dimensional homotopy theory is ssen as being part of Galois theory, or vice versa.

c) This underlines the fact that π1(X) classifies covering spaces – but for i > 1, πi(X) does not seem to classify anything other than maps from Si into X!

This is abstract homotopy theory par excellence.

Einstein Algebra and General Theory of Relativity Preserve the Empirical Structure of the Theories

8ZQfK

In general relativity, we represent possible universes using relativistic spacetimes, which are Lorentzian manifolds (M, g), where M is a smooth four dimensional manifold, and g is a smooth Lorentzian metric. An isometry between spacetimes (M,g) and (M,g′) is a smooth map φ : M → M′ such that φ ∗ (g′) = g, where φ∗ is the pullback along φ. We do not require isometries to be diffeomorphisms, so these are not necessarily isomorphisms, i.e., they may not be invertible. Two spacetimes (M,g), (M′,g′) are isomorphic, if there is an invertible isometry between them, i.e., if there exists a diffeomorphism φ : M → M′ that is also an isometry. We then say the spacetimes are isometric.

The use of category theoretic tools to examine relationships between theories is motivated by a simple observation: The class of models of a physical theory often has the structure of a category. In what follows, we will represent general relativity with the category GR, whose objects are relativistic spacetimes (M,g) and whose arrows are isometries between spacetimes.

According to the criterion for theoretical equivalence that we will consider, two theories are equivalent if their categories of models are “isomorphic” in an appropriate sense. In order to describe this sense, we need some basic notions from category theory. Two (covariant) functors F : C → D and G : C → D are naturally isomorphic if there is a family ηc : Fc → Gc of isomorphisms of D indexed by the objects c of C that satisfies ηc ◦ Ff = Gf ◦ ηc for every arrow f : c → c′ in C. The family of maps η is called a natural isomorphism and denoted η : F ⇒ G. The existence of a natural isomorphism between two functors captures a sense in which the functors are themselves “isomorphic” to one another as maps between categories. Categories C and D are dual if there are contravariant functors F : C → D and G : D → C such that GF is naturally isomorphic to the identity functor 1C and FG is naturally isomorphic to the identity functor 1D. Roughly speaking, F and G give a duality, or contravariant equivalence, between two categories if they are contravariant isomorphisms in the category of categories up to isomorphism in the category of functors. One can think of dual categories as “mirror images” of one another, in the sense that the two categories differ only in that the directions of their arrows are systematically reversed.

For the purposes of capturing the relationship between general relativity and the theory of Einstein algebras, we will appeal to the following standard of equivalence.

Theories T1 and T2 are equivalent if the category of models of T1 is dual to the category of models of T2.

Equivalence differs from duality only in that the two functors realizing an equivalence are covariant, rather than contravariant. When T1 and T2 are equivalent in either sense, there is a way to “translate” (or perhaps better, “transform”) models of T1 into models of T2, and vice versa. These transformations take objects of one category – models of one theory—to objects of the other in a way that preserves all of the structure of the arrows between objects, including, for instance, the group structure of the automorphisms of each object, the inclusion relations of “sub-objects”, and so on. These transformations are guaranteed to be inverses to one another “up to isomorphism,” in the sense that if one begins with an object of one category, maps using a functor realizing (half) an equivalence or duality to the corresponding object of the other category, and then maps back with the appropriate corresponding functor, the object one ends up with is isomorphic to the object with which one began. In the case of the theory of Einstein algebras and general relativity, there is also a precise sense in which they preserve the empirical structure of the theories.

Category-Theoretic Sinks

The concept dual to that of source is called sink. Whereas the concepts of sources and sinks are dual to each other, frequently sources occur more naturally than sinks.

A sink is a pair ((fi)i∈I, A), sometimes denoted by (fi,A)I or (Aifi A)I consisting of an object A (the codomain of the sink) and a family of morphisms fi : Ai → A indexed by some class I. The family (Ai)i∈I is called the domain of the sink. Composition of sinks is defined in the (obvious) way dual to that of composition of sources.

Untitled

In Set, a sink (Aifi A)I is an epi-sink if and only if it is jointly surjective, i.e., iff A = ∪i∈I fi[Ai]. In every construct, all jointly surjective sinks are epi-sinks. The converse implication holds, e.g., in Vec, Pos, Top, and Σ-Seq. A category A is thin if and only if every sink in A is an epi-sink.