# Physical interpretations/meanings of the notion of a sheaf?

I fairly understand the fiber bundles, both the mathematical concept of fiber bundles and the physics use of fiber bundles. Because the fiber bundles are tightly connected to the gauge field theory in (quantum) field theory, and the general relativity in gravity theory. Here the gauge field theory are very powerful framework to descibe three of fundamental forces in the Standard Model, including (i) the U(1) gauge theory for Maxwell electro-magnetism. (ii) SU(2)$_{weak}$ Yang-Mills gauge theory for the nuclear weak interactions with a parity-violating chiral coupling to the SU(2) doublet of left-moving leptons/quarks particles in 3-generations ($\nu_e, e$), ($u,d$),($\nu_\mu, \mu$), ($c,s$), ($\nu_\tau, \tau$), ($t,b$). (iii) SU(3)$_{color}$ Yang-Mills gauge theory for the nuclear strong interactions, where the SU(3) gauge fields only coupling to quarks which carrying the SU(3) fundamental-representation charges in 3 colors, e.g. commonly called the 3 colors as $r,g,b$ (red, green, blue) --- say the color-triplets for all quarks; $(u_r,u_g,u_b)$, $(d_r,d_g,d_b)$, $(c_r,c_g,c_b)$,$(s_r,s_g,s_b)$,$(t_r,t_g,t_b)$,$(b_r,b_g,b_b)$. And the general relativity using Riemannian geometry on the curved spacetime with the Minskowski signature describes the fourth fundamental forces, the gravity.

A remarkable summary between the physics (gauge fields) and the math (fiber bundles) is made by Concept of nonintegrable phase factors and global formulation of gauge fields by Tai Tsun Wu and Chen Ning Yang in Phys. Rev. D 12, 3845, and there is a schematic table in the paper: However, even if I encounter the sheaf several times in different math/physics context, I don't have a clear understanding how this can be tightly connected to certain theory of physics.

So my question is:

What are the physical interpretations / meanings of the sheaf?

Perhaps there can be more than one branch of physics that really requires the use of the concept of sheaf in fundamental and deep ways?

Could there be a correspondence table for the sheaf and something-else in physics or anything in our natural real-word, just analogous to the table above?

• Where have you encountered sheaves in a (mainstream) physics context? There could at least be uses of sheaves in math that was inspired by physics (ex: take instantons from physics, consider the moduli space of instantons and then study that object using language of sheaves). – Chris Gerig Nov 14 '16 at 0:49
• Robert Ghrist and collaborators have introduced sheaf theory in engineering. See "Applications of sheaf cohomology and exact sequences to network coding." – Piyush Grover Nov 14 '16 at 17:03
• What sheaf do you want? What do you want it on? Do you want anything that doesn't come from the examples you have already implied? – AHusain Nov 14 '16 at 17:45
• This question appears a little un-directed. You could just as easily ask for correspondence tables for functions, topos, large cardinals, or any mathematical structure. Sheaves come up in a wide variety of situations. It seems to me your question should be answered by just getting comfortable with the definition and seeing a few canonical examples. Fibre bundles provide examples of sheaves, for example. Functions do, too. – Ryan Budney Nov 15 '16 at 8:02
• Can you make this a bit more precise? What kind of sheaves are you interested in? Vector bundles "are" sheaves (though it depends on what the meaning of "is" is). In particular, functions, or vector fields, or spinor/tensor fields, etc. are sections of sheaves, so you have lots of them already in any classical theory. – Peter Dalakov Nov 15 '16 at 8:46

Sheaves in physics -- Twistor theory (R. Jozsa, 1979), see also: Applications of Sheaf Cohomology in Twistor Theory

The formalism of sheaf theory for the description of complex manifolds and holomorphic bundles was applied to physics by Roger Penrose, in his development of twistor theory. In twistor theory a spacetime point is represented by a certain kind of subspace, i.e. a non-local object. The process of quantization in twistor space leads to a description in which the points of spacetime become "fuzzy", but certain relations associated with the causal structure are preserved. A possible mathematical model of such a structure is given by a sheaf where the partial sections represent the fuzzy points, global sections giving precise points.

Penrose himself wrote last year (2015) about the application of sheaf cohomology to twistor theory in a paper with the memorable title: "Palatial twistor theory and the twistor googly problem".

A more recent development comes from string theory, reviewed in Geometry and Physics (M. Atiyah, R. Dijkgraaf, N. Hitchin, 2010). The Donaldson–Thomas invariant of certain sheaves can be used to evaluate the string partition function.

I expected someone to give an answer of the following form; since that still has not happened let me give it a shot. I should mention that I am by no means an expert on this topic, and many others here could probably write up a much nicer version of this answer; I welcome any comments and corrections.

To start with, the dictionary in the paper by Yang and Wu quoted in the OP is a summary of the way to rephrase gauge theory in terms of differential geometry. Here the latter provides a coordinate-independent (global), and therefore natural, way of thinking about the former. Fiber bundles are the proper way of attaching geometric structures to spaces in order to account for concepts from QFT like gauge groups, spin, and so on. Reversely, gauge theory seems to be "the way" to interpret or understand (or: "the example" of) the geometric notions in the context of theoretical physics. (Edit: One could either add general relativity as a second important example, or argue that it is a gauge theory with gauge group the Poincaré group, albeit the action for the gauge fields has a different form than for Yang-Mills theory.)

To me the broadness of the OP's question, asking for such a dictionary for the case of sheaves in general, sounds rather like the question: what are sheaves, really? Let's first try to get this question out of the way. In short I would say that a sheaf enables one to attach more general structures (i.e. than for fiber bundles) to a topological space, again in a nice and global manner. Indeed, it allows one to think of the structure locally (via restriction: this is the presheaf condition) if one wants to understand the global situation (equality can be tested locally, and compatible local data gives rise to global data: these are the sheaf conditions). I'll leave detailed answers to others: see for example

Next one might ask for examples of sheaves in theoretical physics to understand the definition better. Different examples are related to different structures that sheaves may account for, and give rise to different dictionaries summarizing it. The crux here is that the notion of a sheaf really is quite general. Even if gauge theory might be the example giving fiber bundles and so on a meaning in theoretical physics, I don't think that we can expect there to be a canonical dictionary for sheaves in theoretical physics as well. This is reflected in the comments and other answers to the question, each pointing to different examples. Let me add two more examples:

• First of all, back to the table in the OP: many standard notions from differential geometry (and thus gauge theory) can be rephrased in terms of sheaves. For examples of this type I like the last two references above (see also Section II.3 of Mac Lane and Moerdijk's book mentioned above).
• An important example in theoretical physics where sheaves (as far as I know) have to be used are provided by supermanifolds. You can try Wikipedia and the references there; another classic is the notes written up by Deligne and Morgan of Bernstein's lectures on supersymmetry, of which a slightly more complete version can be found in the first volume of Quantum Fields and Strings: A Course for Mathematicians.

Edit: For me personally the latter is the context in physics where I first encountered sheaves. At the time my MSc thesis advisor, André Henriques, told me to learn what a sheaf is by talking with mathematicians. Since the abstract definition of a sheaf is perhaps a bit intransparent when one first sees it, at least as a physicist, this might really be the way to go!

In mathematics, sheaves can often be resolved by vector bundles. That is, given a sheaf $\mathcal{F}$, one can find vector bundles $E_i$ and an exact sequence:

$$\cdots \to E_2 \to E_1 \to E_0 \to \mathcal{F} \to 0$$

In physics, one finds gauge theories as the low energy theories on branes in string theory. Having these branes end on each other in certain ways can be interpreted in terms of exact sequences like the above, and the corresponding object can be interpreted as a sheaf; see various papers of Michael Douglas.

• Thanks very much, +1, but it seems to me that anything beyond the gauge theoretic fiber bundle may not be necessarily essential to describe fundamental forces or something else in physics. Could we see why the sheaf is necessary for physics, more than the principle bundle is for physics? – wonderich Nov 14 '16 at 2:42
• @wonderich Why do you believe the sheaf is necessary for physics? – Steven Stadnicki Nov 15 '16 at 3:48
• @Steven I dont know, maybe it is not necessary for physics at all. So that is why I asked. For example, C N Yang thought that only part of math is relevant to the real world physics. Although in opposite, Dirac thinks every beautiful math shall be used in the nature physics in some way. – wonderich Nov 15 '16 at 3:53