Homotopy

The geometric idea of homotopy is an interpolation between two paths on a manifold which share the same start and end point.

Mathematically speaking, given a manifold \(S\) and two fixed points \(p\) and \(q\) on the manifold, two paths \(\gamma_0, \gamma_1 : [0,T] \to S\) are homotopic if there exists a smooth function \(\gamma\):

\[ \gamma: [0,1] \times [0,T] \to S \]

such that \(\gamma(s, \cdot)\) is a path from \(p\) to \(q\) for each \(s\) and

\[ \gamma(0,t) = \gamma_0(t) \qquad \gamma(1,t) = \gamma_1(t) \]

Such a function \(\gamma\) is a homotopy between \(\gamma_0\) and \(\gamma_1\).

Things start to get interesting if we have a manifold where there is an obstruction to homotopy. This can happen if the manifold has ‘holes’ in it.

(Co)Homology

Cohomology is a way to study “holes” by looking at differential forms on manifolds.

A differential \(p\) -form \(\omega\) is closed if \(\mathrm{d}\omega = 0\) A differential \(p\) -form \(\beta\) is exact if there exists some \(p-1\) -form \(\alpha\) such that \(\beta = \mathrm{d}\alpha\).

Note that all exact forms are closed:

\[ \mathrm{d}\beta = \mathrm{d}^{2}\alpha = 0 \]

But not all closed forms are exact. The reason this is true is that not every manifold is simply connected – in other words, there are obstructions to homotopy. If \(\Omega\) is not simply connected, then the integral of a closed form around a loop could be non-zero, which breaks Stokes theorem.

Holonomy

The idea of a holonomy is a measure of the curvature of a manifold by inspecting the change in a vector as it is transported along a closed loop. This geometric notion is best described through a picture:

The tangent vector \(u\) is pushed along the loop \(\gamma\) on the sphere \(M\). The progression is indicated by a change in color, from red to orange, yellow, green, to blue. As it completes the loop, it ends up pointing in a different direction only because the sphere is curved.

Mathematically speaking, if \(\gamma: [0, T] \to M\) is a smooth path from \(p\) to \(q\) in the manifold \(M\), and \(E\) is a vector bundle with connection \(D\), given \(u \in E_p\), then

\[ H(\gamma, D)u \]

is the result of parallel transporting \(u\) to \(q\) along path \(\gamma\). The function \(H(\gamma, D)\) is the holonomy.

Application: Aharonov-Bohm Phase

Let the manifold \(M\) be:

\[ M = \mathbb{R}^3 - \{(0,0,z) | z \in \mathbb{R} \} \]

This is the usual three dimensional space, except there is a ‘hole’ along the vertical axis.

There is an obstruction to homotopy of the paths:

\[ \gamma_0(t) = (\cos(\pi t), \sin(\pi t), 0) \qquad \gamma_1(t) = (\cos(\pi t), -\sin(\pi t), 0) \]

Which can be visualized below:

There’s no way to smoothly deform \(\gamma_1\) into \(\gamma_2\) while keeping \(p\) and \(q\) fixed – you can’t get around the void at the origin!

In other words, we can find a 1-form \(A\) such that:

\[ \oint_\gamma A \neq 0 \]

for some smooth closed path \(\gamma: [0,T] \to M\), \(\gamma(0) = \gamma(T)\)

If we form the \(G\) -bundle \(E = M \times V\) where \(V\) is the vector space of the representation of the group \(U(1)\), then the holonomy:

\[ H(\gamma, D)u = e^{i\oint_\gamma A} u \]

describes the resulting wavefunction \(u\) of a quantum particle after being parallel transported around a loop \(\gamma\) in the base space.

Choosing \(\gamma = (\cos(2\pi t), \sin(2\pi t), 0)\) as the path, and the one-form as:

\[ A = \frac{2\pi}{r}\mathrm{d}\theta \]

Then \(e^{i\oint_\gamma A}\) is the Aharonov-Bohm phase seen in the Aharonov-Bohm effect.