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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 15th 2013
    • (edited Sep 15th 2013)

    Started an entry in “category:motivation” on fiber bundles in physics.

    (prompted by this Physics.SE question)

    • CommentRowNumber2.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 16th 2013
    Imho, I think it is a disservice to say a bundle IS a field.
    Bundles are the global structure in whihc physical fields live.

    Yang-Mills hence E&M? Dirac's monopole as one of the earliest examples existing in a bundle, i.e. the Hopf fibration, seems to get short shrift.

    Anomalies from non-trivializability might well have a reference to the Gribov ambiguity.
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 16th 2013
    • (edited Sep 16th 2013)

    Imho, I think it is a disservice to say a bundle IS a field.

    It is a basic fact that a gauge field is a bundle equipped with a connection. For discrete gauge theory the connection disappears and then a gauge field is precisely a bundle. This fact was very much amplified back then by Freed-Quinn 93 (if it takes appeal to third party auhority) Since it is a true fact, I can’t see why it would be a disservice in a scientific context to state it.

    Bundles are the global structure in whihc physical fields live.

    Not gauge fields. Gauge fields are sections of gerbes. I know that this is not widely advertized fact (due to the focus on perturbation theory), but it is easily seen to be true.

    • CommentRowNumber4.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 16th 2013
    It is a basic fact that a gauge field is a bundle equipped with a connection.

    THEN no distinction between different sections of the same bundle with connection?
    that is, they correspond to just one gauge field?
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 16th 2013
    • (edited Sep 16th 2013)

    Hey Jim,

    let’s get on the same page here, there are two levels of bundles here:

    1. a GG-gauge field on some space XX is itself a GG-principal bundle over XX equipped with GG-principal connection;

      A gauge transformation is an isomorphism of such GG-principal bundles with GG-principal connections over XX.

    2. Now what is it that a gauge field over XX is itself a section of? This must be some “second order” bundle over XX, namely something such that a section of this second order bundle is itself an ordinary bundle (with connection).

      Such “second order bundles” which are such that its sections are themselves ordinary GG-principal bundles are called GG-gerbes . Or BG\mathbf{B}G-fiber 2-bundles, to have a more conceptual term.

    So if we forget the connections for a moment then:

    1. the “field 2-bundle” for GG-gauge fields on XX is the 2-bundle (BG)×X⟶X(\mathbf{B}G) \times X \longrightarrow X, hence the trivial GG-gerbe on XX;

    2. a section of the “field 2-bundle” is equivalently a map X⟶BGX \longrightarrow \mathbf{B}G and that is equivalently an ordinary GG-principal bundle on XX.

    • CommentRowNumber6.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 16th 2013
    a G-gauge field on some space X is itself a G-principal bundle over X equipped with G-principal connection

    I'm lost already - granted that, I would go on without protest

    how do you write a gauge field in an action?
    the simple example of the `gauge field' for the Dirac monopole might could help
    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 16th 2013

    a G-gauge field on some space X is itself a G-principal bundle over X equipped with G-principal connection

    I’m lost already - granted that, I would go on without protest

    But yes, that’s what it is. What physicists call GG-gauge fields are mathematically GG-principal connections. What they call the “instanton sector” of the gauge field is mathematically the class of the underlying GG-principal bundle.

    how do you write a gauge field in an action?

    For ∇\nabla a GG-gauge field for GG a semisimple Lie group with Killing form ⟨−,−⟩\langle -,-\rangle, the canonical action functional on it (the “Yang-Mills action functional”) is the integral over Riemannian base space of the curvature 2-form wedged with its Hodge dual

    ∇↦exp(iℏ∫ X⟨F ∇∧*F ∇⟩). \nabla \mapsto \exp\left( \frac{i}{\hbar} \int_X \left\langle F_\nabla \wedge \ast F_\nabla\right\rangle \right) \,.

    Moreover, for γ:S 1⟶X\gamma \colon \S^1 \longrightarrow X a path in XX, thought of as the trajectory of a charged particle, then the interaction functional is the holonomy of the connection along this path

    (γ,∇)↦hol ∇(γ). (\gamma, \nabla) \mapsto hol_\nabla(\gamma) \,.
    • CommentRowNumber8.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 17th 2013
    OH ``What physicists call G-gauge fields are mathematically G-principal connections. What they call the “instanton sector” of the gauge field is mathematically the class of the underlying G-principal bundle.''

    Perhaps my problem is linguistic - a connection is a field - fine
    but a connection IS a bundle with a connection??

    even the “Yang-Mills action functional” is defined on the space of connections
    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 17th 2013
    • (edited Sep 17th 2013)

    Not sure, Jim, where we are miscommunicating. A GG-gauge field is a GG-principal connection and every GG-principal connection has an underlying GG-principal bundle. That’s all. It’s a standard fact. (You have co-authored various articles involving this fact, if I may say so.)

    • CommentRowNumber10.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 17th 2013
    As a theologian once said, the problem is with the word IS.

    I agree:
    A G-gauge field is a G-principal connection and every G-principal connection has an underlying G-principal bundle.

    so I'm happy with
    A G-gauge field is a G-principal connection ON a G-principal bundle.
    but to say

    A G-gauge field is G-principal bundle with a G-principal connection.

    distorts my perspective.

    Anyone else see my point?
    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 17th 2013

    Jim, if you allow then I’ll drop out of this discussion. If you go back through our exchange I think it is hard (or else unfair) to argue that I stated anything which is not standard.

    • CommentRowNumber12.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 18th 2013
    Yes,let us drop the discussion since what is standard in one culture may not be in another.
    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMay 12th 2014
    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMar 3rd 2019

    added pointer to the exposition

    • Adam Marsh, Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results (arXiv:1607.03089)

    (updated on the arXiv this week) here and elsewhere

    diff, v13, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2020

    added some doi-s to the references and brought them into chronological order

    diff, v17, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeOct 19th 2020

    added pointer to

    Will give this book a page of its own…

    diff, v18, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeOct 19th 2020

    added pointer to

    diff, v19, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeOct 19th 2020

    added publication data to:

    diff, v20, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeDec 27th 2023

    added pointer to the early discussion of

    and a scan of their Table I (here)

    diff, v25, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeApr 10th 2024

    added pointer to:

    diff, v29, current

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