Wednesday, January 2, 2013

Gauge theory and the variational bicomplex

Hi! Former Notre Dame math student here, posting at Liviu's suggestion to expand on a conversation we were having on Facebook.

Suppose you have a fiber bundle $E \rightarrow M$. Interpret $M$ as "spacetime;" then sections of $E$ are "fields." (Particle dynamics can be recovered by taking $M = \mathbb{R}$.) To set up a classical dynamics on these fields, one writes down a Lagrangian $L$ and associated action functional $S = \int_M L$, then obtains field equations by requiring $\delta S = 0$. When I first read the derivation of these Euler-Lagrange equations in a physics book, I felt like a trick had been played. It wasn't clear to me what the Lagrangian really "was," in a formal mathematical sense, and the formula $\frac{d}{dt}(\delta q) = \delta \dot{q}$ seemed a bit magic.

As usual, the nlab came to my rescue and told me about the "variational bicomplex." ( This is a doubly-graded complex of differential forms on the infinite jet bundle $j_{\infty}(E)$. In particular, any differential form on a finite jet bundle $j_k(E)$ gives you an element of the variational bicomplex via pullback. And both fields and Lagrangians look like forms on finite jet bundles of $E$. A field is a $0$-form on the $0$-jet bundle. A Lagrangian is a bit more complicated- since the action functional is the integral of $L$ over $M$, $L$ must be an $n$-form on $M$... but since its values depend on the 1-jet of the field you're at, it's more like an $n$-form on the $1$-jet bundle.

Write $D$ for the exterior derivative on forms on $j_{\infty}(E)$ (or, for that matter, on any finite jet bundle). We'd like to be able to split $D$ into a sum $d + \delta$, where $d$ is the derivative "along M" and $\delta$ is the derivative along the fiber. What this requires is a splitting of the tangent space at any $\infty$-jet $\varphi$ into horizontal directions and vertical directions. The vertical directions are already there, since we have a fiber bundle, so we just need the horizontal ones. Local coordinates on $j_{\infty}(E)$ are $\{x_1, \ldots, x_n, q_1, \ldots, q_k, \partial_i q_j, \partial_i \partial_j q_k, \ldots \}$.

Trickily, the vectors $\frac{\partial}{\partial x_i}$ are NOT an appropriate choice of horizontal vectors, even if the bundle $E$ happens to be trivial! (As is always the case when $M = \mathbb{R}$.) This is precisely because we want an equation like $\frac{d}{dt}(\delta q) = \delta \dot{q}$. In other words, if we're at the jet $\varphi$, then when we go out from $\varphi$ in a horizontal direction (say the $x_1$ direction), the coordinates $q_1, \ldots, q_k$ of $\varphi$ should change in a manner specified by the coordinates $\partial_1 q_1, \ldots, \partial_1 q_k$ of $\varphi$.

But how should the coordinates $\partial_i q_j$ of $\varphi$ change themselves? Now we need to look at the $2$-jet component of $\varphi$, and it's the same all the way up. Now we see why the full $\infty$-jet bundle was needed- we'd be stuck if we cut ourselves off at a finite jet bundle.

Given this horizontal/vertical splitting of tangent spaces to the $\infty$-jet bundle, we're within our rights to talk about a bicomplex of differential forms on $j_{\infty}(E)$. Sections of $E$ (i.e. fields) yield $(0,0)$-forms, and Lagrangians yield $(n,0)$ forms. We may now happily take a variational derivative of $L$: it's just $\delta L$, the derivative in the vertical direction. This is an $(n,1)$-form, and the usual Euler-Lagrange argument beefs up to show that any $(n,1)$-form splits uniquely as $E + d\Theta$, where $\Theta$ is an $(n-1,1)$-form and $E$ is a "source form," i.e. an $(n,1)$-form such that, when contracted with a vertical vector (represented by a path of germs $\varphi_s$), the result only depends on the values of $\varphi_s$ at the spacetime point in question and not on the higher jet components. So $\delta(L) = E(L) + d\Theta$; the field equations are $E(L) = 0$.

Lots of other nice stuff falls out of this framework: i.e. an infinitesimal symmetry of the system is a vertical vector field $v$ such that $\iota_v \delta L = d \sigma$ for some $(n-1,0)$-form $\sigma$. (Such terms $d \sigma$ affect the action only by a boundary contribution, which can be assumed to be zero in your favorite way). Then you can immediately consider the $(n-1,0)$-form $\sigma - \iota_v \Theta$; being an $(n-1)$-form on spacetime, it represents a Noether current. To see that it's conserved, compute $d(\sigma - \iota_v \Theta) = \iota_v \delta L - \iota_v d\Theta = \iota_v E(L)$. But, at a solution to the field equations, $E(L) = 0$, so Noether's theorem is just a bit of playing around with differential forms.


So, what if you want to study gauge theories? Suppose you have a Lie group $G$ and a $G$-principal bundle $P \rightarrow M$. Then "fields" should be $G$-connections on $P$. These aren't naturally sections of a bundle-- rather, they're an affine space for sections of the bundle $\Omega^1(M; \mathfrak{g})$ where $\mathfrak{g}$ denotes the bundle associated to $P$ via the adjoint representation of $G$. So "variations" in a gauge field look just like the variations above, where $E = \Omega^1(M; \mathfrak{g})$. But somehow this seems unnatural, and I'd like a more convincing way of saying this stuff in a gauge-theory setting.

Furthermore, the principal bundle $P$ shouldn't need to be fixed. Fields should be "bundle plus connection" rather than just "connection on a fixed bundle." Apparently this is where differential cocycles come in. A differential cocycle is supposed to (roughly?) capture the notion of a differential form AND an integer cocycle representing the same real cohomology class. The form gives us the connection, and the integer cohomology class gives us the bundle. (?) Unfortunately, I don't know much about these beasts. What I'd like to know is:

(1) is there a way to set up a variational bicomplex for gauge theory, where the "sections of $E$" are replaced by differential cocycles?

(2) when $G$ is trivial, you don't recover the non-gauge theory. Is there a more general framework which subsumes both gauge fields and non-gauge fields?

(0) what do I need to know about differential cohomology, cocycles, etc., to understand these things? The paper with the right definition, I think, is "Quadratic functions in geometry, topology, and M-theory" (Hopkins, Singer), but it's a bit formidable.

Comments, questions, answers, oracular enlightenment all appreciated!
-Andy Manion
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