# Does one ever need infinitely many cohomologies?

In a theory containing gauge fields or higher-form gauge fields, if the background spacetime is a complicated manifold, a nice way to represent the configuration of the gauge field mathematically is with a vector bundle. Often, important aspects of the configuration are captured in the cohomologies of the vector bundle, which are a small set of integers. (One context in which this is particularly true is compactification, where these integers typically count the massless fields seen in the uncompactified dimensions.)

Vector bundle cohomologies are usually very difficult to compute. In particular, this makes it impossible to obtain information for any significant set of vector bundles.

My question is whether this limitation prevents any particular calculations:

Q: Are there any contexts in physics where a calculation requires the computation of the cohomologies of a very large or even infinite number of vector bundles?

An example might be a partition function that requires a sum over cohomology values.

• The singular cohomology of a manifold vanishes in every degree higher than the dimension of the manifold. Hence for these cohomology theories the answer is negative or at least it forms no problem. Commented Oct 15, 2020 at 18:32
• @NDewolf Sorry, I haven't made myself clear. What I am asking about is whether one needs to know the cohomologies of an infinite number of vector bundles. I will edit the question - thanks for pointing this out. Commented Oct 15, 2020 at 19:10
• By cohomologies of a vector bundle, are you talking about its characteristic classes? Commented Oct 15, 2020 at 19:21
• @pianyon Cohomologies are abelian groups associated to the bundle. The simplest example is the zeroth cohomology, which is the group of sections of the bundle. Commented Oct 15, 2020 at 20:51
• So instead of singular cohomology and alternatives it seems you are interested in sheaf (hyper)cohomology? Commented Oct 15, 2020 at 21:25

I will try to say something about the string theory part of the answer.

Here there are some examples where an infinite number vector bundles/sheaves are important:

1) Holography: All known examples of quantum field theories with gravity duals have large $$N$$ limits. For large $$N$$ dualities to work, a gigantic number $$N$$ of color charges are needed. In the physical AdS/CFT correspondence that translates into the fact that there are allowed interactions in between an infinite number of "color bundles"(talking about the CFT side).

See Yin's talk on the computation of $$1/16$$ BPS states in $$N=4$$ SYM for an interesting example of a situation where an actual computation of an infinite number of representatives of Lie algebra cohomology classes is important Xi Yin - Comments on BPS states in N=4 SYM.

2) In topological strings and topological field theories the situation is rather similar. Large $$N$$ dualities in those contexts requiere an infinite amount of (quasi-coherent but not coherent) sheaves supported over a homology class. Then, to exactly compute the spectrum of open strings or gluons (and their interactions) we need to take into account an infinite number of sheaf cohomology classes at the same time. See On the Gauge Theory/Geometry Correspondence.

3) Quiver gauge theories: The possible fractional branes in a quiver are identified with the derived bounded category of quiver representations (or coherent sheaves in the associated quiver variety); see Topological branes from descent for a proof of the fact that any object in the derived category is a boundary condition for the B-model. The fact that every complex is bounded (all their nodes are zero, except for a finite number of them) does not rule out the possibility that an infinite number of complexes of sheaves could be relevant in some computation. See crystal melting and black holes for an interesting example.

4) Topological gravity: All the amplitudes in Kodaira-Spencer theory of gravity can be computed enforcing an infinite number of conditions coming from the symmetries of an infinite dimensional algebra. See Topological Strings and Integrable Hierarchies and Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions.