This entry is supposed to be a survey of and motivation for the role of fiber bundles in physics.
physics, mathematical physics, philosophy of physics
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(see also Chern-Weil theory, parameterized homotopy theory)
All of physics has two aspects: a local or even infinitesimal aspect, and a global aspect. Much of traditional lore deals just with the local and infinitesimal aspects – the perturbative aspects – and fiber bundles play little role there. But they are the all-important structure that govern the global – the non-perturbative – aspects. Bundles are the global structure of physical fields and they are irrelevant only for the crude local and perturbative description of reality.
One way to think of fiber bundles is that they are the data to globally twist functions (on spacetime, say) where “global twist” is much in the sense of “global anomaly” and the like, namely an effect visible on topologically nontrivial spaces when moving around non-contractible cycles. The concept of monodromy – which may be more familiar to physicists – is closely related: monodromy is something exhibited by a connection on a bundle and specifically by a flat bundle. For a discrete structure group (gauge group) every bundle is flat, and in this case non-trivial bundles and non-trivial monodromy come down to essentially the same thing (see also at local system).
More explicitly, suppose $X$ denotes spacetime and $F$ denotes some space that one wants to map into. For instance $F$ might be the complex numbers and a free scalar field would be a function $X\to F$. For the following it is useful to talk about functions a bit more indirectly: observe that the projection $pr_2 \colon F \times X \to X$ from the product of $F$ with $X$ down to $X$ is such that a section of this map, namely a $\Phi \colon X \to X\times F$ such that
is just the same as data a function $X \to F$. We think of the projection $X \times F \overset{pr_2}{\to} X$ as encoding the fact that there is one copy of $F$ associated with each point of $X$, and think of a function with values in $F$ as something that, of course, takes values in $F$ over each point of $X$. One says that this projection, suggestively shown vertically,
is the trivial $F$-fiber bundle over $X$.
If $F$ is equipped with the structure of a vector space then one calls this a trivial vector bundle, etc.
The point being that more generally we may add a global “twist” to the $F$-valued functions by making the space $F$ vary to some degree as we move along $X$. For a fiber bundle one requires that it doesn’t change much: in fact the word “fiber” in “fiber bundle” refers to the fact that all fibers (over all points of $X$) are equivalent. But the point is that any $F$ may be equivalent to itself in more than one way (it may have “automorphisms”), and this allows non-trivial global structure even though all fibers look alike.
In this sense, a general $F$-fiber bundle on some $X$ is defined to be a space $P$ equipped with a map
to the base space $X$ (e.g. to spacetime), such that locally it looks like the trivial $F$-fiber bundle, up to equivalence.
To say this more technically: $P \overset{fb}{\to} X$ is called an $F$-fiber bundle if there exists a cover (open cover) of $X$ by patches (e.g. coordinate charts!) $U_i \to X$ for some index set $I$, such that for each patch $U_i$ (with $i \in I$) there exists a fiberwise equivalence between the restriction $P|_{U_i}$ of $P$ to $U_i$, and the trivial $F$-fiber bundle $F\times U_i \to U_i$ over the patch $U_i$.
To say this again in terms of sections: this means that a section $\Phi$ of $P \overset{fb}{\to} X$
is locally on each (coordinate) patch $U_i$ simply an $F$-valued function, but when we change patches (change coordinates) then there may be a non-trivial identifications (notably: gauge transformations) that relates the values of the function on one patch to that on another patch, where they overlap.
Even if this may seem a bit roundabout on first sight, this is actually something at the very heart of modern physics, in that it embodies the two central principles of modern physics, namely
the principle of locality;
the gauge principle.
The first roughly says that every global phenomenon in physics must come from local data. In the above discussion this means that any “globally $F$-valued thing on spacetime $X$” must come from just $F$-valued functions on local (coordinate) charts $U_i \hookrightarrow X$ of spacetime. BUT – and this is key now –, second, the gauge principle says that we may never strictly identify any two phenomena in physics (neither locally nor globally) but we must always ask instead for gauge transformations connecting two maybe seemingly different phenomena. Hence combining the gauge principle with the locality principle means that if an $F$-valued something on spacetime is locally given by plain $F$-valued functions, then it should be globally given by gluing these $F$-valued functions together not by identification but by gauge equivalence. The result may be a structure that has global twists, and the nature of these global twists is precisely what an $F$-fiber bundle embodies.
For instance the gauge fields in Yang-Mills theory, hence in electromagnetism, in QED and in QCD, hence in the standard model of the known universe, are not really just the local differential 1-forms “$A_\mu^a$” known from so many textbooks, but are globally really connections on principal bundles (or their associated bundles) and this is all-important once one passes to non-perturbative Yang-Mills theory, hence to the full story, instead of its infinitesimal or local approximation.
Notably what is called a Yang-Mills instanton in general and the QCD instanton in particular is nothing but the underlying nontrivial class of the principal bundle underlying the Yang-Mills gauge field. Specifically, what physicists call the instanton number for SU(2)-gauge field theory in 4-dimensions is precisely what mathematically is called the second Chern-class, a “characteristic class” of these gauge bundles.
YM Instanton = class of principal bundle underlying the non-perturbative gauge field
To appreciate the utmost relevance of this, observe that the non-perturbative vacuum of the observable world is a “sea of instantons” with about one YM instanton per femtometer to the 4th. See for instance the first sections of (Schaefer-Shuryak 98) for a review of this fact. So the very substance of the physical world, the very vacuum that we inhabit, is all controled by non-trivial fiber bundles and is inexplicable without these.
Also monopole solutions in physics are mathematically nontrivial principal bundles. For instance the Dirac monopole (that appears in Dirac charge quantization) or the Yang monopole.
Similarly fiber bundles control all other topologically non-trivial aspects of physics. For instance most quantum anomalies are the statement that what looks like an action function to feed into the path integral, is globally really the section of a non-trivial bundle – notably a Pfaffian line bundle resulting from the fermionic path integrals. Moreover all classical anomalies are statements of nontrivializability of certain fiber bundles.
Indeed, as the discussion there shows, quantization as such, if done non-perturbatively, is all about lifting differential form data to line bundle data, this is called the prequantum line bundle which exists over any globally quantizable phase spaces and controls all of its quantum theory. It reflects itself in many central extensions that govern quantum physics, such as the Heisenberg group central extension of the Hamiltonian translation and generally and crucially the quantomorphism group central extension of the Hamiltonian diffeomorphisms of phase space. All these central extensions are non-trivial fiber bundles, and the “quantum” in “quantization” to a large extent a reference to the discrete (quantized) characteristic classes of these bundles. One can indeed understand quantization as such as the lift of infinitesimal classical differential form data to global bundle data. This is described in detail at quantization – Motivation from classical mechanics and Lie theory.
But actually the role of fiber bundles reaches a good bit deeper still. Quantization is just a certain extension step in the general story, but already classical field theory cannot be understood globally without a notion of bundle. Notably the very formalization of what a classical field really is says: a section of a field bundle. For instance the global nature of spinors, hence spin structures and their subtle effect on fermion physics are all encoded by the corresponding spinor bundles.
Two aspects of bundles in physics come together in the theory of gauge fields and combine to produce higher fiber bundles: namely we saw above that a gauge field is itself already a bundle (with a connection), and hence the bundle of which a gauge field is a section has to be a “second-order bundle”. This is called gerbe or 2-bundle: the only way to realize the Yang-Mills field both locally and globally accurately is to consider it as a section of a bundle whose typical fiber is $\mathbf{B}G$, the universal moduli stack of $G$-principal bundles. For more on this see on the nLab at The traditional idea of field bundles and its problems.
All of this becomes even more pronounced as one digs deeper into local quantum field theory, with locality formalized as in the cobordism theorem that classifies local topological field theories. Then already the local Lagrangians and local action functionals themselves are higher connections on higher bundles over the higher moduli stack of fields. For instance the fully local formulation of Chern-Simons theory exhibits the Chern-Simons action functional — with all its global gauge invariance correctly realized – as a universal Chern-Simons circle 3-bundle. This is such that by transgression to lower codimension it reproduces all the global gauge structure of this field theory, such as in codimension 2 the WZW gerbe (itself a fiber 2-bundle: the background B-field of the WZW model!), in codimension 1 the prequantum line bundle on the moduli space of connections whose sections in turn yield the Hitchin bundle of conformal blocks on the moduli space of Riemann surfaces.
And so on and so forth. In short: all global structure in field theory is controled by fiber bundles, and all the more the more the field theory is quantum and gauge. The only reason why this can be ignored to some extent is because field theory is a complex subject and maybe the majority of discussions about it concerns really only a small little perturbative local aspect of it. But this is not the reality. The QCD vacuum that we inhabit is filled with a sea of non-trivial bundles and the whole quantum structure of the laws of nature are bundle-theoretic at its very heart.
Expositions include
L. Mangiarotti, Gennadi Sardanashvily, Connections in Classical and Quantum Field Theory, World Scientific, 2000
Luciano Boi, Geometrical and topological foundations of theoretical physics: From gauge theories to string program, 2003 (pdf)
A.P. Balachandran, G. Marmo, B.-S. Skagerstam, A. Stern, Gauge Theories and Fibre Bundles - Applications to Particle Dynamics, Springer “Lecture Notes in Physics”, 188 (1983) (arXiv:1702.08910)
P. Balachandran, G. Marmo, B.-S. Skagerstam, A. Stern, Classical Topology and Quantum States (World Scientific, Singapore, 1991).
See also