Canonical Actions on Bundles – Philosophizing Identity Over Gauge Transformations.

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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.

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Deleuzo-Foucauldian Ontological Overview From the Machine to the Archive. Thought of the Day 26.0

In his book on Foucault first published in 1986, Deleuze drew a diagram in the last chapter, Foldings, that depicts in overview the Outside as abstract machine, defined by the line of the outside (1), which separates the unformed interplay of forces and resistance from the strategies and strata that filter the affects of power relations to become “the world of knowledge”.

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The central Fold of subjectification, of ‘Life’ is “hollowed out” and ignored by the forces of the outside as they are realized in the strata fulfilling the obligation of the diagram to “come to fruition in the archive.” This is dual process of integration and differentiation. The residual dust of the affective relations produced by force upon force, integrate into the strata even as they differentiate to forms of realization – visible or articulable. The ‘empty’ fissure/fold attracts and repels these moving curvilinear strategies as they differentiate and ”hop over” it. Ostensibly, the Fold of subjectification effectuates change as both continuously topological, and as discontinuously catastrophic (as in leaping over). So, the process of crystallization from informal to formal paradoxically integrates as it differentiates. Deleuze’s somewhat paradoxical description follows:

The informal relations between forces differentiate from one another by creating heterogeneous curves which pass through the neighborhood of particular features (statements) and that of the scenes which distribute them into figures of light (visibilities). And at the same time the relations between forces became integrated, precisely in the formal relations between the two, from one side to the other of differentiation. This is because the relations between forces ignored the fissure within the strata, which begins only below them. They are apt to hollow out the fissure by being actualized in the strata, but also to hop over it in both senses of the term by becoming differentiated even as they become integrated. Gilles Deleuze, Sean Hand-Foucault

So this “pineal gland” figure of the Fold is the “center of the cyclone”, where life is lived “par excellence” as a “slow Being”.

As clarifying as Deleuze’s diagram is in summarizing the layered dimensionality of the Foucauldian/Deleuzian hybrid, some modifications will be drawn off to alternatively express the realizations of the play of informal forces as this diagram takes on the particular features of a Research Creation praxis. True to the originating wax tablet diagramma, the relations are drawn and redrawn, in recognition, after Bergson’s notion of recognition as the intensive point where memory meets action of the contemporary social field that situates it. The shifts from the 19C to 20C disciplinary diagram of Foucault’s focus modulates with the late 20C society of control diagram formulated by Deleuze. The shorthand for the force field relevant to the research creation diagram of practice-led arts research today is a transdisciplinary diagram, the gamespace of just-in-time capitalism, which necessarily elicits mutations in the Foucault/Deleuze model. Generating the power-resistance relations in this outside qua gamespace are, among others, the revitalized forces of the military-academic-entertainment complex that fuel economic models such as the Creative Industries that pervade the conditions of play in artistic research. McKenzie Wark concludes his book GAMER THEORY, with prescient comments on the black hole quality of a topology of the outside qua contemporary “gamespace” from Deleuze and Guattari (ATP) and Guy Debord. “Only by going further and further into gamespace might one come out the other side of it, to realize a topology beyond the limiting forms of the game. Deleuze and Guattari: “… one can never go far enough in the direction of [topology]: you haven’t seen anything yet — an irreversible process. And when we consider what there is of a profoundly artificial nature […] we cry out, ‘More perversion! More artifice!’ — to a point where the earth becomes so artificial that the movement of [topology] creates of necessity and by itself a new earth.”

Math Conundrum in Thomas Pynchon

vqylv

Her idea of banter
Likely isn’t Cantor
Nor is she apt to murmur low Axioms of Zermelo,
She’s been kissed by geniuses, Amateur Frobeniuses
One by one in swank array, Bright as any Poincaré…

and so on in that vein.

It was when I came upon the word “automorphic”…Earth making its automorphic way round the sun again and yet again…periodic functions, and their generalized form, automorphic functions as a prelude to a scholarly discussion of time travel:

Time no longer ‘passes,’ with a linear velocity, but ‘returns,’ with an angular one. All is ruled by the Automorphic Dispensation. We are returned to ourselves eternally, or, if you like, timelessly.

You find an awful lot of hyperbolas in Against the Day. For example: the hyperbolic geometry in connection with automorphic functions; the “Automorphic Dispensation” which seems to be a “function… by which, almost as a by-product, ordinary Euclidean space is transformed to Lobachevskian”; and that “perfect hyper-hyperboloid” that “only Miles” Blundell, the one character to have comprehended the meaning of space-time, “can see in its entirety.” There are (hyperbolic) wave equations (and a whole family of Vibes) and the “noted Quaternionist V. Ganesh Rao of Calcutta University” who by rotating himself in an imaginary direction performs something “like reincarnation on a budget, without the element of karma to worry about.”

Or as the NewYorker puts it,

The readers will encounter many references to, and, frequently, extended disquisitions on, such matters as Hamilton’s Quaternions, Gibbsian vector analysis, Riemann spheres, Prandtl’s discovery of the boundary layer, the Hilbert Pólya Conjecture, the Minkowskian space-time track, and Zermelo’s Axiom of Choice. Inserting this stuff into novelistic situations produces passages like this one, describing a meeting of an outfit known as the Transnoctial Discussion Group.

“Time moves on but one axis,” advised Dr. Blope, “past to future—the only turnings possible being turns of a hundred and eighty degrees. In the Quaternions, a ninety-degree direction would correspond to an additional axiswhose unit is √-1. A turn through any other angle would require for its unit a complex number.”

“Yet mappings in which a linear axis becomes curvilinear—functions of a complex variable such as w=ez, where a straight line in the z-plane maps to a circle in the w-plane,” said Dr. Rao, “do suggest the possibility of linear time becoming circular, and so achieving eternal return as simply, or should I say complexly, as that.”. . . As if the hour itself in growing later had exposed some obscure fatality, the discussion moved to the subject of the luminiferous Æther, as to which exchanges of opinion—relying, like Quaternions, largely on faith—often failed to avoid a certain vehemence……..

Still coming to grips with this ?????????