Cocycles

Prerequisites: Group Theory, Cohomology. Full path (9 pages)

• Why is it interesting?
• Intuitive
• Concrete
• Abstract

Why is it interesting?

"In field theory and in modern elementary particle theory, cocycles are used to describe anomalies." (Source)

1-cocycles are used to understand anomalies in the Lagrangian framework. The 1-cocycles show up in the non-conservation of classically conserved currents.

2-cocycles are used to understand anomalies in the Hamiltonian framework. They have the effect that it becomes impossible to satisfy Gauss' law. (Source)

Intuitive

A cocycle is a bookkeeping device for a consistency condition. It appears whenever a rule is "almost" compatible with how operations combine, and we want to record the mismatch and check that the mismatch is itself consistent.

The standard example from quantum mechanics: a symmetry acts on states only up to a phase, because multiplying a state by a phase does not change the physics. Take two symmetry operations. Applying their operators one after the other can differ by a phase from applying the operator of the combined operation. That phase, one number for each pair of operations, is a 2-cocycle. It cannot be arbitrary: for three operations, the two ways of grouping them must give the same answer, and this forces a relation among the phases (the cocycle condition). Some phases are fake, because we could have removed them by choosing different phase conventions for the individual operators from the start. Those are called coboundaries. Only the phases that cannot be removed this way carry physics. A well-known case is the spin-1/2 representation of rotations, where a full turn multiplies the state by minus one.

The same pattern explains anomalies. A classical symmetry may fail to be a symmetry of the quantum theory, and the failure, the change of the effective action under a symmetry transformation, has to satisfy a consistency condition because doing two transformations in a row must agree with doing their product. This makes the failure a 1-cocycle. It is a genuine anomaly only if it cannot be removed by adding a local counterterm to the action, which is the statement that it is not a coboundary.

Concrete

For a great introduction see https://inspirehep.net/record/213998/files/v15-n3-p99.pdf and chapter III in Analysis On Infinite Dimensional Manifolds: Schrodinger Representation For Quantized Fields by R. Jackiw

Abstract

Group cohomology. Let $G$ be a group and $A$ an abelian group on which $G$ acts by automorphisms ($A$ is a $G$-module). An $n$-cochain is a function $f : G^n \to A$, and the coboundary operator $\delta : C^n(G, A) \to C^{n+1}(G, A)$ is

$$(\delta f)(g_1, \dots, g_{n+1}) = g_1 \cdot f(g_2, \dots, g_{n+1}) + \sum_{i=1}^{n} (-1)^i f(g_1, \dots, g_i g_{i+1}, \dots, g_{n+1}) + (-1)^{n+1} f(g_1, \dots, g_n) .$$

One checks $\delta^2 = 0$. The $n$-cocycles are $Z^n = \ker \delta$, the $n$-coboundaries are $B^n = \operatorname{im} \delta$, and $H^n(G, A) = Z^n / B^n$ is the $n$-th cohomology group. In low degrees:

• A 1-cocycle satisfies $f(gh) = f(g) + g \cdot f(h)$ (a crossed homomorphism). It is a coboundary if $f(g) = g \cdot a - a$ for some $a \in A$. For trivial action, $H^1(G, A) = \mathrm{Hom}(G, A)$.
• A 2-cocycle satisfies $g_1 \cdot f(g_2, g_3) - f(g_1 g_2, g_3) + f(g_1, g_2 g_3) - f(g_1, g_2) = 0$.

Projective representations. Let $U$ be a projective unitary representation, $U(g) U(h) = \omega(g, h)\, U(gh)$ with $\omega(g,h) \in U(1)$. Associativity of operator multiplication, $(U(g_1) U(g_2)) U(g_3) = U(g_1) (U(g_2) U(g_3))$, gives

$$\omega(g_1, g_2)\, \omega(g_1 g_2, g_3) = \omega(g_2, g_3)\, \omega(g_1, g_2 g_3) ,$$

which is the 2-cocycle condition for $U(1)$ with trivial action, written multiplicatively. Rephasing $U'(g) = \lambda(g) U(g)$ changes $\omega'(g,h) = \omega(g,h)\, \lambda(g) \lambda(h) / \lambda(gh)$, that is, by a coboundary. Projective representations up to rephasing are therefore classified by $H^2(G, U(1))$, and each class defines a central extension of $G$ by $U(1)$ on which the representation becomes an ordinary one. For Lie algebras the analogous statement uses Lie algebra cohomology $H^2(\mathfrak{g}, \mathbb{R})$. Two standard physical cases are the mass as a central charge in the projective representations of the Galilei group, and the Virasoro algebra, the central extension of the Witt algebra by the cocycle $\frac{c}{12}(m^3 - m)\, \delta_{m+n, 0}$.

Anomalies. Let the gauge group $\mathcal{G}$ act on connections from the right, $A \mapsto A^g$, and let $W[A]$ be the effective action. The variation $\alpha(g; A) = W[A^g] - W[A]$ satisfies

$$\alpha(gh; A) = \alpha(g; A) + \alpha(h; A^g) ,$$

the 1-cocycle condition for $\mathcal{G}$ with values in functionals of $A$. Adding a local counterterm $W \to W + L[A]$ shifts $\alpha$ by $L[A^g] - L[A]$, a coboundary. An anomaly is a cocycle that is not the coboundary of a local functional, so it is classified by a cohomology group of local functionals. The infinitesimal version of the cocycle condition is the Wess-Zumino consistency condition. In the Hamiltonian picture the anomaly shows up as a 2-cocycle, the Schwinger term in the commutators of the Gauss law generators.

Čech cocycles. For an open cover $\{U_i\}$ of a manifold, the transition functions $g_{ij} : U_i \cap U_j \to G$ of a principal $G$-bundle satisfy $g_{ij} g_{jk} = g_{ik}$ on triple overlaps. This is a Čech 1-cocycle (non-abelian when $G$ is). Changing the local trivializations changes it by a coboundary, $g_{ij} \to h_i\, g_{ij}\, h_j^{-1}$, and isomorphism classes of bundles correspond to $\check{H}^1(M; G)$.

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