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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.

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Maybe also this question at MO and this one here might be of relevance. –  Marek Dec 6 '10 at 18:23

10 Answers 10

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.

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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).

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The Hasan-Kane article you cite does not contain the word K-theory. Can you explain how table 1 on page 8 is supposed to involve topological K-theory? –  Rasmus Jan 30 '11 at 20:43

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.

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Your answer could use a little elaboration for the uninitiated, still there is absolutely no reason to down-vote it. –  user346 Jan 20 '11 at 6:04
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thanks for this, i don't understand the reason for the downvote at all. –  Sean Tilson Jan 20 '11 at 6:19

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 .

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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.

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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.

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

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While it has the root topology in it, I don't think this is what I am looking for. –  Sean Tilson Dec 6 '10 at 18:32
    
@Sean: actually, I think it might. I don't know much about this stuff but it seems awfully similar to TQFT in that it encodes degrees of freedom of the system in its global properties. And indeed reading a wikipedia article on topological order we can see it mentions TQFT (references [6]--[9] therein). E.g. [6] talks about Chern-Simons theory for 2+1 space-times (so presumably this encodes evolution of some properties of surfaces). I have to say I am intrigued :-) –  Marek Dec 21 '10 at 9:48

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.

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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).

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The calculation of extended charges moving in time, requires finite difference calculations that only a computer can perform. This is one of the principle reasons why 20th century Physicists chose to use the point charge model. Quantum Mechanics (QM) was invented, as a probability theory, to incorporate the point property and attempt to explain the wave like phenomena.

In QM, the probability density of the charged entity is of course the square magnitude of the wave function, much like the energy density of the electromagnetic (EM) field is the squared magnitude of the field densities. No matter how small the region is, there is a chance of finding the charge entity, which implies the charged entity is a point charge. As any physics student knows, this implies infinite self energy, and no physical explanation for angular momentum, magnetic spin or radiation (energy exchange).

Dirac "solved" this by equating the QM wave function to the EM fields or the QM probability density to the EM energy density. He also used the bimorphic mapping of the 2x2 group of complex unitary transformations to the proper Lorentz group, which the EM field equations are invariant to. The Lorentz transform and Poincare space were invented, in order to accommodate the illusion of a point charge, because spatially extended charge entities were just too complicated to deal with at the time.

What is most amusing is Special Relativity (SR) assumes as a hypothesis that Space has no properties (no Aether), and General Relativity (GR) states that Space is curved, yet the second order GR uses the first order SR as its foundation. As everyone knows, QM and GR are completely incompatible, so they decided to change the point charge to a Planck scale String. This String theory already has millions of Calabi-Yau manifolds that could potentially be used as the "fundamental" String, no field equations, no experimental results (Planck scale), but it has become the alter of Modern Physics.

There are powerful computers today, that can deal with spatially extended charges evolving in time, so as to truly understand Physics. We live in a 3-dim(sensional) world, so using a model of 0-dim entities as the constituents of this 3-dim Universe we live in is rather absurd. There are those who are using the Parson ring model of the electron and proton, that have avoided most all contradictions, have explained away the weak and strong forces, have described a physical mechanism for inertia, have calculated gravity as a fourth order effect of EM (anti gravity Space travel), and it is all based upon classical Physics.

If you wish to study Modern Physics, then please visit www.commonsensescience.org.

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