# Inner product on Hochschild homology in 2d TCFTs

This should be an easy question for some people. Take a compact $A(\infty)$ algebra with a cyclically symmetric non-degenerate inner product. In Kontsevich and Soibelman's article "Notes on $A(\infty)$ algebras, $A(\infty)$ categories and non-commutative geometry", they describe a way that this induces an inner product structure on the Hochschild homology of A. I was wondering whether there was a clean closed formula for this inner product so that if I wanted, say, to compute the inner product on loop homology for some simple spaces like S^n, I could just follow the formula. If someone is willing to put in the extra work, I do want the formula for all $A(\infty)$ algebras (not just graded algebras or dg algebras), though, and with correct signs. Hopefully this doesn't try anyone's patience too much :).

In theory, one gets the answer from a certain graph and a "Feynman diagram" algorithm described in KS, but I am not sure from reading them how to turn their algorithm into a formula in this seemingly simple case. I spent some time just trying to guess a formula but I couldn't come up with anything that looked "right" so I thought I'd save some time by asking.

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