Applications of Algebraic Topology to physics

Question

I have always wondered about applications of Algebraic Topology to Physics, seeing as am I studying algebraic topology and physics is cool and pretty. My initial thoughts would be that since most invariants and constructions in algebraic topology can not tell the difference between a line and a point and $\mathbb{R}^4$ so how could we get anything physically useful?

Of course we know this is wrong. Or at least I am told it is wrong since several people tell me that both are used. I would love to see some examples of applications of topology or algebraic topology to getting actual results or concepts clarified in physics. One example I always here is "K-theory is the proper receptacle for charge" and maybe someone could start by elaborating on that.

I am sure there are other common examples I am missing.

Answer 1

First a warning: I don't know much about either algebraic topology or its uses of physics but I know of some places so hopefully you'll find this useful.

Topological defects in space

The standard (but very nice) example is Aharonov-Bohm effect which considers a solenoid and a charged particle. Idealizing the situation let the solenoid be infinite so that you'll obtain ${\mathbb R}^3$ with a line removed.

Because the particle is charged it transforms under the $U(1)$ gauge theory. More precisely, its phase will be parallel-transported along its path. If the path encloses the solenoid then the phase will be nontrivial whereas if it doesn't enclose it, the phase will be zero. This is because $$\phi \propto \oint_{\partial S} {\mathbf A} \cdot d{\mathbf x} = \int_S \nabla \times {\mathbf A} \cdot d{\mathbf S} = \int_S {\mathbf B}\cdot d{\mathbf S}$$ and note that $\mathbf B$ vanishes outside the solenoid.

The punchline is that because of the above argument the phase factor is a topological invariant for paths that go between some two fixed points. So this will produce an interference between topologically distinguishable paths (which might have a different phase factor).

Instantons

One place where homotopy pops up are Instantons in gauge theories.

Specifically, if you consider a Yang-mills theory in ${\mathbb R}^4$ (so this means Euclidean time) and you want the solution (which is a connection) to have a finite energy then its curvature has to vanish at infinity. This allows you to restrict your attention to $S^3$ (this is where the term instanton comes from; it is localized) and this is where homotopy enters to tell you about topologically inequivalent ways the field can wrap around $S^3$. Things like these are really big in modern physics (both QCD and string theory) because instantons give you a way to talk about non-perturbative phenomena in QFT. But I am afraid I can't really tell you anything more than this. (I hope I'll get to study these things more myself).

TQFT

Last point (which I know nearly nothing about) concerns Topological Quantum Field Theory like Chern-Simons theory. These again arise in string theory (as does all of modern mathematics). And again, I am sorry I cannot tell you more than this yet.

Answer 2

The fermion operators obey $b^2~=~{b^\dagger}^2$ $=~0$. This is a form of the rule d^2 = 0. Supersymmetry permit a cohomology of states $\psi~\in~ker(Q)/im(Q)$, which is a cohomology. The $Q$ obeys $Q^2~=~0$, physical states obey $Q\psi~=~0$, but where $\psi~\ne~Q\chi$. This is the basis of BRST (Becchi, Rouet, Stora and Tyutin) quantization.

Answer 3

Sean,

sorry I am responding to an old question but I have a very nice example of an application of advanced algebraic topology in physics (it's physics seen in experiments, not arbitrary speculative theories). It's the newly discovered "topological insulators".

One can topologically classify free Hamiltonians (hermitian matrices/operators) as a function of different symmetry classes and spatial dimension. It turns out that this can be done using topological K-theory (see periodic table in http://arxiv.org/abs/1002.3895 ; table 1 on page 8). There is a two-fold periodicity in symmetry-classes and dimension, originating from the Bott-periodicity of complex K-theory (classification of complex vector bundles up to stable equivalence). And there is a eight-fold periodicity in the other symmetry-classes originating from the Bott-periodicity of real K-theory. See more information here: http://arxiv.org/abs/0901.2686 and http://iopscience.iop.org/1367-2630/12/6/065010 (free access for both).

Furthermore I must mention that the 10-symmetry classes mathematically originate from Cartans classification of symmetric spaces and the labels in the above-mentioned table comes from this classification.

(I just saw that topological insulators were mentioned above, but not these aspects).

Answer 4

Marek and Eric have given good answers. I think many particle physicists first encountered homotopy theory in the context of magnetic monopoles. Take the Standard Model gauge group $H=SU(3) \times SU(2) \times U(1)$ and embed it into a GUT gauge group $G$ such as $SU(5)$ or $SO(10)$. Assuming there is no accidental degeneracy, the space of minima of the symmetry breaking potential is the coset $G/H$. Static, finite energy configurations must approach a point in $G/H$ at spatial infinity and so are classified by $\pi_2(G/H)$ which equals $\pi_1(H)$ provided that $\pi_1(G)= 0$. Since $\pi_1(H)=\mathbb{Z}$ there is an integer valued topological charge for these configurations which turns out to be magnetic monopole charge. Sidney Colemans lectures ( http://ccdb4fs.kek.jp/cgi-bin/img/allpdf?198211084 ) explain this in much more detail. Condensed matter systems have a much wider range of "Higgs" fields (i.e. order parameters) and so have much more interesting and complicated symmetry breaking patterns and a much richer classification of topological defects by homotopy groups. Mermin has a very nice review here: http://rmp.aps.org/abstract/RMP/v51/i3/p591_1 .

Answer 5

Another fun example of topology applied to physics is Witten's cohomological field theory trick. Mathematicians usually regard this as a way of making novel conjectures about the topology of moduli spaces. Physicists see it as a way of using the topology of moduli spaces to make limited checks of the validity of physics conjectures.

The idea is that, in some supersymmetric theories, you can match the correlation functions of some observables in the physical theory with the correlation functions of observables in a topologically twisted version of the physical theory. The relevant path integrals in the topological theory 'localize' to integrals of differential forms over moduli spaces of solutions to instanton equations embedded in the space of fields. This is remarkable: a measure defined on a infinite-dimensional space of distributions ends up being supported on a finite-dimensional moduli space lying with in it, a miraculous cancellation. These integrals are 'just' intersection numbers, so 'hooray! we can make an exact computation in the original physical theory!'. Most of the time these correlation functions are 'protected', like BPS invariants; some symmetry makes their behavior very regular.

A lot of interesting mathematics shows up in the details of localization. For example, index theorems show up when you're counting fermion zero modes to figure out the dimension of the space localized to.

Answer 6

All of Marek's examples are good. (I had to write a new answer due to space limitations on comments.) Instantons are probably the single best place to explore this relationship. Maxwell's equations in vacuo read dF = 0 and d*F = 0, where F is the field strength tensor. The charge of a particle (by Gauss' law) can be obtained by calculating the integral of *F, on a surrounding sphere, while the magnetic charge (always zero, as we have not yet observed magnetic monopoles reliably) is the integral of F. Now if you're studying algebraic topology, F is the Chern form of the connection defined by the gauge field (vector potential), namely it represents the first Chern class of this bundle. This is the prime example of how a characteristic class -- which measures the topological type of the bundle -- appears in physics as a quantum number, magnetic charge.

Indeed, in quantum field theory we are instructed to integrate over ALL connections, including those for different topological types of bundles -- so the configuration space has different components. The minimum (Euclidean) energy configurations in these different components are called "instantons."

There are many other examples which involve mildly exotic theories of physics.

Answer 7

I'm looking for similar examples. The given examples are good -- I just want to add one more, that I recently discovered: Topological insulators

Answer 8

The following three physics.stackexchange articles:

(1) Reconciling topological insulators and topological order

(2) Group Cohomology and Topological Field Theories

(3) Do Category Theory and/or Quantum Logic add value in physics?

contain examples of Applications of Algebraic Topology to physics.

Answer 9

Twistor theory uses sheaf cohomology, see e.g. Gentle introduction to twistors.

And twistor theory itself has applications to perturbative quantum field theory (MHV amplitudes).