M Theory Lesson 89

In an interesting exchange on the Cafe, Schreiber made the remark, "we are understanding that all these differential graded algebras are really just the Koszul dual incarnation of the Lie version of n-groupoids of physical configurations and/or states." In physical M theory we like to avoid phrasing everything in terms of complicated algebras, because the physics does not require this and the algebras arise naturally from operads anyway. The point is that Koszul duality is a fundamental process for operads themselves, playing the role of 2-ality for the n-logos Machian correspondence.

In a short paper on operads, Kapranov (who introduced modular operads with Getzler) discusses Koszul duals. Examples are the duals of the operads for associative, commutative and Lie algebras, which are respectively the associative, Lie and commutative operads. That is, associativity is self-dual but commutativity and Lie structures are interchanged. These operads are linear. More generally there is a functor $K$ on the category of operads (this is just about 1-operads) which has the property that $K^{2}$ is a kind of isomorphism. Sounds like a monad, doesn't it? Indeed, Kapranov says that one should think of $F$ as a type of cohomology theory on the category of operads. It's a bit like the power set monad for the topos Set, which is why the comment of the enigmatic David Corfield regarding duality based on the number 2 is particularly relevant here.

Another example of Koszul dual operads are the homologies of (a) uncompactified moduli spaces (actually varieties) of $n$ punctured Riemann surfaces of genus $g$ and (b) the compactified spaces.

So it would seem that the absolutely brilliant Schreiber is fast converging on the right mathematics, and he's doing it the hard way, without really thinking about the physics at all. Somebody (respectable) needs to tell him that there is no Dark Energy or no SUSY partners or no KK modes.