# Applications of topological chiral homology and factorization algebras (aka higher Hochschild cohomology)

I recently heard a talk about these topics and found them very interesting. The talk was centered on the formal structure and didn't really focus on examples.

So my question is: what is your favorite application of topological chiral homology? (or its other variants and specialisations)

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Here's a good start: math.northwestern.edu/~jnkf/writ/cotangentcomplex.pdf –  Dylan Wilson Mar 16 '13 at 4:00
(and John's other papers... like this one, joint with Hiro Tanaka and David Ayala: math.northwestern.edu/~jnkf/writ/singular-factorization.pdf ) –  Dylan Wilson Mar 16 '13 at 4:01
thank dylan for the links, although they truly fly way above my head. but they aren't too big on examples, now are they? :P –  Jacob Bell Mar 16 '13 at 10:08
I know what you mean... Part of it is that this is so general that many computable examples are actually just dressed up versions of classical calculations (e.g. integrating over $S^1$ is the same as some version of Hochschild cohomology). But I think this is exciting: it means there's a lot to do! –  Dylan Wilson Mar 16 '13 at 14:30
Also here is a really cool talk by Jacob Lurie that hints at a use of factorization homology towards the end to obtain something really neat in number theory: cornell.edu/video/?videoID=2099 –  Dylan Wilson Mar 16 '13 at 14:31

It might be worth pointing out that topological chiral homology is a specialization to the topological (or locally constant) setting of a construction that originated (AFAIK) in conformal field theory, namely the functor of conformal blocks for a vertex algebra and its derived formulation as chiral homology, due to Beilinson and Drinfeld. As such there are loads of concrete applications and calculations.

Perhaps the best known is the proof of the Verlinde formula, calculating the dimension of spaces of nonabelian theta functions on a Riemann surface by realizing them as conformal blocks and using the gluing properties of conformal field theory to reduce to a calculation in the representation theory of loop groups. (Recently Gaitsgory gave a derived enhancement of this calculation, showing that the full chiral homology of the integrable level k vertex algebra of a loop group is still the space of nonabelian theta functions).

Perhaps my favorite application still is the one that motivated Beilinson and Drinfeld to introduce chiral homology in the first place. Namely, they wished to construct special $D$-modules on the moduli of $G$-bundles on a Riemann surface which were eigen-objects for Hecke functors, with eigenvalues particular $G^\vee$-local systems on the surface. (This is the central goal of the geometric Langlands program). They achieved this goal when the local system is a so-called "oper" by showing first how to use the Feigin-Frenkel theorem (a loop algebra version of the Harish Chandra isomorphism for the center of the enveloping algebra) to solve a local version of the problem. In other words, they showed that Feigin-Frenkel provide representations of the loop algebra which are eigenobjects for Hecke functors. But then the mechanism of conformal blocks (or chiral homology) allowed them to globalize this construction on a Riemann surface, resulting in the desired global objects. The in-depth examination that resulted of what vertex algebras really mean geometrically led to the notion of factorization algebras and chiral homology which are now becoming so influential, thanks to work of (alphabetically) Costello, Francis, Gaitsgory, Gwilliam, Lurie, Rozenblyum and others.

The Beilinson-Drinfeld idea of chiral homology captures in a profound geometric way the adelic structure of moduli spaces of bundles on curves. The recent work of Gaitsgory and Lurie (as well as Rozenblyum, Barlev, Zhu and others) greatly furthers the Beilinson-Drinfeld dream of realizing the entire geometric Langlands correspondence on a Riemann surface as the "integral" (chiral homology) over the surface of a simpler local statement.

In short, chiral homology is providing a formula for the "path integral" over a Riemann surface, telling us how to integrate more categorical but simpler local invariants to get global invariants. For an introduction to more recent developments (and in particular the migration of factorization algerbas outside of the confines of CFT or TFT), I recommend Costello's ICM address (about recovering the Witten genus through factorization algebras). Also in a sense made precise I think by Ayala and Rozenblyum, all the values of any extended TFT (in the sense of the cobordism hypothesis) are given by calculating chiral homology of an appropriate local quantity (i.e., a higher category is a kind of "partially defined E_n algebra"), so in principle the examples are everywhere!

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I often heard the fact that topological chiral homology comes from chiral homology, but also that it's really a completely different thing. Also, I don't understand all details but I find the bit in Lurie's survey where he defines TCH quite readable. So what's the purpose of this comment? It's the following question: is there a readable introduction to chiral homology (which possibly makes clear why TCH is analogous to it)? –  Jacob Bell Mar 16 '13 at 20:47
Jacob -- I'm not sure I understand the distinction between the two, but maybe someone else can clarify. I thought the Beilinson-Drinfeld notion corresponds to the holomorphic case, and TCH to the locally constant case, of one notion that Costello and Gwilliam consider in a general $C^\infty$ setting, but I might be missing some subtleties. As for references I would look at Francis-Gaitsgory, which is very nicely written. –  David Ben-Zvi Mar 16 '13 at 22:27

Topological chiral homology allows one to write down explicit formulas for the value on a given manifold M with corners of the framed n-dimensional TFT that sends the point to the given En-algebra and takes values in the (∞,n)-category of En-algebras, En−1-algebras in bimodules between En-algebras, and so on all the way up to n-morphisms, leaving only equivalences as (n+1)-morphisms.

For example, for n=1 we recover the 1-TFT that sends the point to the given ∞-algebra and the framed circle to its (topological) Hochschild homology.

See Theorem 4.1.24 in Jacob Lurie's survey on TFTs.

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Jacob himself is very forthright about the limitations of his Theorem 4.1.24 in his illuminating Remark 4.1.27, which starts out "Theorem 4.1.24 is usually not very satisfying, because it describes a functor ... whose values on closed framed n-manifolds are objects of an (\infty,1) category S, rather than concrete invariants like numbers". –  Peter May Mar 16 '13 at 17:45
While that might seem like a limitation, objects of a category are usually better invariants than numbers. For instance, if the target category is chain complexes, you get a number of invariants: Euler characteristic and (more generally) Betti numbers or cohomology groups of the chain complex. If your target is spaces, all of a space's topological invariants become an invariant... –  Hiro Lee Tanaka Mar 10 '14 at 22:37
So I think the limitation isn't in the versatility of the invariants, but in the computability. Lifting Donaldson invariants to the spectrum level (Manolescu) or Casson invariants to a Floer cochain complex (Atiyah-Floer conjecture) are good examples. –  Hiro Lee Tanaka Mar 10 '14 at 22:38