First, I'm not a physicist so I have just a little background in physics. I have been reading some noncommutative geometry books and papers (Connes, Rosenberg, Kontsevich etc) and a lot of high machinery from algebraic geometry such as étale cohomology and motives appears in such books, however I could not guess where these structures arise in physical situations. How algebraic geometry and motives appears in physics? Why do physicists needs to use a projective scheme? When this scheme (or other structure) needs a noncommutative analog?
[Physics] How algebraic geometry and motives appears in physics
algebraic-geometrycategory-theorymathematical physicsnon-commutative-geometrystring-theory
Related Solutions
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.
Griffiths and Harris' "Principles of Algebraic Geometry" (Wiley) is the best for your purposes (read only the parts on Kahler geometry). The sections on algebraic geometry in "Mirror Symmetry" (Clay/AMS) are essentially a Crib Notes version of that paper and some of the classic CY and special geometry papers referred to above.
What you should keep in mind going in is the following:
Kahler manifolds are complex manifolds with a hermitian inner product on tangent vectors which have a metric that is determined (locally) by a single function. It is the geometry in which the metric and the complex structure "get along very nicely." This simplifies lots of calculations and adds new symmetries. That's why we know so much about them.
Best Answer
Algebraic geometry as such appears because it happens to capture important aspects of the geometry of strings.
For instance the partition functions of superstrings are elliptic genera and the best way to understand this is to regard a torus-shaped string worlsheet as an elliptic curve, regard the moduli space of possible worldsheet tori as the moduli stack of elliptic curves or actually as the derived moduli stack of derived elliptic curves in derived algebraic geometry to finally understand that the Witten genus (superstring partition function) is but the shadow of the string orientation of tmf.
Similarly the target space Calabi-Yau geometries of interest due to the relation between supersymmetry and Calabi-Yau manifolds is best understood with tools from algebraic geometry. Similar statements apply to a bunch of other compactification geometries.
Now motives is another story. Motivic structure enters quantum physics in two dual guises, related to on the one hand algebraic deformation quantization and on the other hand to geometric quantization.
In the first case one observes that formal deformation quantization of $n$-dimensional field theory amounts to choosing an inverse equivalence to the formality map from $E_n$-algebras to $P_n$-algebras, this is explained here. The automorphism infinity-group of either side therefore naturally acts on the space of quantization choices and one shows (conjectured by Kontsevich, recently proven by Dolgushev) that the connected component group of this is the Grothendieck-Teichmüller group, a quotient of the motivic Galois group. Related to this in some way is Connes "cosmic Galois group" acting on the space of renormalizations of perturbative quantum field theory. According to Kontsevich, this explains the role of motivc structures in correlation functions in perturbative field theory, see at Motivic Galois group action on the space of quantizations.
On the other hand, in full non-perturbative geometric quantization in its modern cohomological form as geometric quantization by push-forward one finds a "cohesive" form of actual motivic cohomology exhibited by actual pure motives. In effect, a local ("extended") action functional on a space of histories is exhibited by a correspondence with the action itself exhibited by a twisted bivariant cocycle on the correspondence space, and the motivic path integral quantization of this corresponds to the induced pull-push index transform.
This is explained in the last section of arXiv:1310.7930 "differential cohomology in a cohesive topos" with more details in the thesis "Cohomological quantization of local prequantum boundary field theory".