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I heard that there are relations between elliptic curves and topological quantum field theory (TQFT).

I googled and found that something called "elliptic genus" might be the key word to relate these two fields. (It might be wrong, though.)

However, I still lack knowledge to search good references.

Could you give me some references that explain the relations between elliptic curves and TQFT for people who have some basic knowledge of these two fields separately?

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You might want to look up "Witten genus" and "topological modular form." Peter Teichner has some good survey articles, although I'm not sure any specifically address what you're looking for. For example, you could try "What is an elliptic object?" – Matt Mar 30 '13 at 21:56
Right; the thrust of the Stolz--Teichner program is to use TQFTs to describe cocycles for certain "chromatic cohomology theories". The first two (rational cohomology at height 0 and complex K-theory at height 1) have been worked out, and so the next task is to give cocycles for elliptic cohomology at height 2. – Aaron Mazel-Gee Apr 2 '13 at 1:28

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