Cohomology

Prerequisites: Differential Forms, Topology. Full path (7 pages)

Intuitive
Concrete
Abstract
Why is it interesting?

Intuitive

Cohomology is a way to detect holes and other global features of a space by doing local calculus on it.

Take the magnetic field around a long straight wire carrying a current. Outside the wire the field has zero curl everywhere. In a region without holes, a curl-free field is always the gradient of some function, a potential. Here, if we walk once around the wire and add up the field along the path, we get a nonzero answer proportional to the current. A gradient would give zero for any closed path. So the field is locally a gradient but there is no single potential that works all the way around. The obstruction exists because the region outside the wire has a hole in it.

Cohomology turns this into a systematic count. It looks at all the objects that pass the local test (zero curl, or more generally "closed") and removes the ones that pass it for a trivial reason (gradients, or more generally "exact"). What is left measures how the space is connected: how many separate pieces it has, how many independent loops around holes, how many enclosed cavities.

The same logic applies in many places in physics. Whenever we have an operation that gives zero when applied twice, everything that is the result of applying it once passes the test "applying it gives zero" automatically, and the interesting objects are those that pass the test without being of that form. BRST cohomology, described below, is the version of this used to find the physical states of a gauge theory.

Concrete

The most direct version is de Rham cohomology, built from differential forms on a manifold $M$ . The exterior derivative $d$ maps $p$ -forms to $(p+1)$ -forms and satisfies $d^2 = 0$ . In $\mathbb{R}^3$ this contains the identities of vector calculus: $d$ on functions is the gradient, on 1-forms the curl, on 2-forms the divergence, and $d^2 = 0$ says $\nabla \times \nabla f = 0$ and $\nabla \cdot (\nabla \times \vec{A}) = 0$ .

A form $\omega$ is closed if $d\omega = 0$ and exact if $\omega = d\eta$ for some form $\eta$ . Every exact form is closed. The $p$ -th de Rham cohomology group is

$H^p_{dR}(M) = \frac{\{ \omega \in \Omega^p(M) : d\omega = 0 \}}{\{ d\eta : \eta \in \Omega^{p-1}(M) \}} ,$

so two closed forms define the same class if they differ by an exact form. Its dimension $b_p$ is the $p$ -th Betti number.

Examples.

$H^0_{dR}(M)$ consists of the functions with $df = 0$ , the locally constant functions. So $b_0$ is the number of connected components.
On $\mathbb{R}^n$ (or any star-shaped region) every closed form of degree $p \ge 1$ is exact (Poincaré lemma). So $H^p_{dR}(\mathbb{R}^n) = 0$ for $p \ge 1$ .
On the punctured plane $\mathbb{R}^2 \setminus \{0\}$ take

$\omega = \frac{x\, dy - y\, dx}{x^2 + y^2} .$

A short calculation gives $d\omega = 0$ . Integrating over the unit circle, with $x = \cos\phi$ , $y = \sin\phi$ , gives $\oint \omega = \int_0^{2\pi} d\phi = 2\pi$ . If $\omega$ were $df$ for a function $f$ on the punctured plane, the integral over any closed curve would vanish. So $\omega$ is closed but not exact, and in fact $H^1_{dR}(\mathbb{R}^2 \setminus \{0\}) \cong \mathbb{R}$ , spanned by $[\omega]$ . Locally $\omega = d\phi$ with the polar angle $\phi$ , which cannot be defined continuously all the way around the origin. - For the 2-sphere, $b_0 = 1$ , $b_1 = 0$ , $b_2 = 1$ . The Euler characteristic $\chi = \sum_p (-1)^p b_p = 2$ .

In electrodynamics. The field strength is a closed 2-form, $dF = 0$ (the homogeneous Maxwell equations). On a region where $H^2_{dR} = 0$ we can write $F = dA$ globally. On $\mathbb{R}^3 \setminus \{0\}$ , which has $H^2_{dR} \cong \mathbb{R}$ , the magnetic field of a monopole gives a closed 2-form with nonzero flux through a sphere around the origin. Such a form is not exact, so no vector potential exists on the whole region, and one has to work with potentials on patches. In the Aharonov-Bohm setup the region outside the solenoid has $H^1_{dR} \cong \mathbb{R}$ . There the potential satisfies $dA = 0$ , but $\oint A$ equals the enclosed flux, so $A$ is closed and not exact, and the phase it produces is observable.

The BRST construction described below follows the same pattern, with $Q_{BRST}$ playing the role of $d$ : physical states are the closed states modulo exact ones.

Abstract

A cochain complex is a sequence of abelian groups (or vector spaces, or modules) with maps

$\cdots \to C^{n-1} \xrightarrow{d_{n-1}} C^n \xrightarrow{d_n} C^{n+1} \to \cdots, \qquad d_n \circ d_{n-1} = 0 .$

Its cohomology is $H^n(C) = \ker d_n / \operatorname{im} d_{n-1}$ . A cochain map $f : C \to D$ (with $f d = d f$ ) induces maps $H^n(C) \to H^n(D)$ , and chain-homotopic maps induce the same map. A short exact sequence of complexes $0 \to A \to B \to C \to 0$ gives a long exact sequence $\cdots \to H^n(A) \to H^n(B) \to H^n(C) \to H^{n+1}(A) \to \cdots$ . Different cohomology theories differ in which complex is assigned to a given object.

De Rham and singular cohomology. For a smooth manifold $M$ , the de Rham complex $(\Omega^\bullet(M), d)$ gives $H^\bullet_{dR}(M)$ . Pullback of forms makes this a contravariant functor, and it is invariant under smooth homotopy. The wedge product descends to cohomology and makes $H^\bullet_{dR}(M)$ a graded commutative ring. De Rham's theorem states that integration of forms over smooth singular chains, $\omega \mapsto \big(\sigma \mapsto \int_\sigma \omega\big)$ , induces an isomorphism of rings $H^\bullet_{dR}(M) \cong H^\bullet(M; \mathbb{R})$ with singular cohomology with real coefficients. The integrality information in $H^\bullet(M; \mathbb{Z})$ , including torsion, is lost in this passage.

Characteristic classes. For a connection with curvature $F$ on a principal $G$ -bundle, every $\mathrm{Ad}$ -invariant polynomial on the Lie algebra evaluated on $F$ gives a closed form whose de Rham class does not depend on the connection (Chern-Weil). For a $U(1)$ connection written as a real 1-form $a$ with curvature $f = da$ (covariant derivative $d - i a$ ), the class $[f/2\pi]$ is the image of the first Chern class, and $\int_\Sigma f/2\pi \in \mathbb{Z}$ for every closed oriented surface $\Sigma$ . This is the topological content of Dirac's charge quantization.

Other cohomologies used in physics.

Čech and sheaf cohomology. For example, principal $U(1)$ -bundles up to isomorphism correspond to $H^1(M; \underline{U(1)}) \cong H^2(M; \mathbb{Z})$ , with the sheaf of smooth $U(1)$ -valued functions.
Group cohomology and Lie algebra (Chevalley-Eilenberg) cohomology, which classify central extensions and projective representations, and enter the description of anomalies (see cocycles).
BRST cohomology. For a gauge theory with first-class constraints the BRST differential $s$ satisfies $s^2 = 0$ , and under regularity conditions its cohomology in ghost number zero is the algebra of gauge-invariant observables, the functions on the reduced phase space. In the quantum theory the operator $Q_{BRST}$ plays the role of $s$ on the state space, as described below.

Why is it interesting?

In a theory with a given gauge symmetry (redundancy) we are interested in what our actual non-redundant physical states are. These can be found with the help of a special kind of cohomology, called BRST cohomology.

The BRST operator is used to identify physical states. These are to be contrasted with unphysical states that only appear as convenient computational tools.

A physical state satisfies

$Q_{BRST} |\Psi_{phys} \rangle = 0.$

However, there are also unphysical states that satisfy this condition trivially, because the BRST charge satisfies $Q_{BRST}^2=0$ . Every state that can be written in the form $|\xi\rangle = Q_{BRST} |\Psi \rangle$ yields zero when we act on it with $Q_{BRST}$ because $Q_{BRST}^2=0$ .

These states are non-physical, because they have norm zero (for a hermitian $Q_{BRST}$ ): $\langle \xi |\xi\rangle=0$ .

A state that can be written as $|\xi\rangle = Q_{BRST} |\Psi \rangle$ is called BRST exact. A state that satisfies $Q_{BRST} |\Psi_{phys} \rangle = 0$ is called BRST closed. From the discussion above, we know that the BRST exact states are a subset of all BRST closed states.

We can add to any physical state an BRST exact state without changing anything:

$|\tilde{\Psi}_{phys} \rangle = |\Psi_{phys} \rangle + Q_{BRST} |\Psi \rangle .$

The additional term drops out of every BRST invariant observable ( $[Q_{BRST},\hat O]=0$ ), because $Q_{BRST}$ is hermitian and annihilates $|\Psi_{phys}\rangle$ :

$\langle \tilde{\Psi}_{phys} | \hat O |\tilde{\Psi}_{phys} \rangle = \langle {\Psi}_{phys} | \hat O |{\Psi}_{phys} \rangle .$

After this observation we are in the position to identify the true, non-redundant physical states.

A physical state is an equivalence class, i.e. the set of all states that we can get from a given state by adding BRST exact states to it. It makes sense to identify all such states with each other, because as we have seen above, they all describe exactly the same physics.

The set of all such equivalence classes is called the BRST cohomology.

This process of "factoring out redundant things" by defining equivalence classes is, of course, the usual procedure of building a quotient space.

The truly physical Hilbert space, i.e. the set of all non-redundant truly physical states can then be defined as

$\mathcal{H}_{phys} = \frac{\text{Ker}Q_{BRST}}{\text{Im}Q_{BRST}} .$

See section 5.3 in this paper.

Another important role is played be deRham cohomology in electrodynamics. (See Gauge Fields, Knots and Gravity by John Baez, Javier P Muniain)

Path to this page

Leads to: Cocycles, Gerbes.