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Faithful readers of Vivastgasse 7 may have noticed that we haven’t posted anything in a while now- the reasons are numerous, but it mostly boils down to the fact that two of us (Anton and I) are in the middle of resolving several quasi-bureaucratic/ quasi-academic things: Anton is right now in Ukraine, finishing up his thesis and busy planning his move to Paris in a couple of months for his first postdoc. I’m busy planning a move as well- off to Durham next month for five months. So both of us are sort of ‘out-of-commission’ right now. However, I do intend to post something soon (maybe this weekend?) either on schemes over the mysterious “field with one element” or better yet, an unpacking of our guest-blogger Sniggy Mahanta’s post on conformal field theories. (Thanks goes to AJ Tolland for pointing out some gross inaccuracies in that post!)

But here is the main reason for this (non-mathematical) post- to vent!

Readers should be warned that the author is not an expert of CFT and, in fact, not even a novice in physics. What follows should be taken with a hefty pinch of salt.

Given any Riemann surface $X$ (as a target manifold) one is able to associate to it an $SCFT(X)$ or a super conformal field theory. The word super can just be construed as a $\mathbb{Z}_2$-grading of the theory. This is a simplistic version as normally one should also take into account several other parameters like a B-field and so on. Within an SCFT there is a topological sector called a TQFT (topological quantum field theory) which is insensitive to the metric on the target space. There are axiomatic descriptions of this theory due to Atiyah and Segal and in its latest version possibly due to Costello. Roughly an $n$-dimensional TQFT is a functor satisfying a lot of axioms from $n$-manifolds with labelled boundaries (incoming and outgoing) to symmetric monoidal DG (differential graded) categories with some twisting.

(Via Anton (my fellow-blogger) and his wife Masha- their seven year old daughter Ivanka’s notes from Hirzebruch’s opening lecture at the 50th Arbeitstagung here a couple of weeks ago)