Calabi-Yau infinity algebras and Topological Quantum Field Theories

A Calabi-Yau ($CY$) structure of dimension $d$ on a compact $A_\infty$ algebra $A$ is
a degree $d$ non-degenerate, cyclically invariant pairing on $A$. It is well known that the
complex of Hochschild chains of a compact $A_\infty$, $CY$ algebra has the structure of
a TQFT, namely it is an algebra over a PROP of chains in the moduli space
$\mathcal{M}_{g,\vec{m},\vec{n}}$ of Riemann surface of genus $g$ with $m$ incoming
and $n$ outgoing marked points with real tangent directions, where $m,n \geq 1$. It is
necessary though for several applications to go beyond compact algebras.
In this talk we will introduce a new notion called a pre-$CY$ algebra which is a
Non-Commutative analogue of a solution to the Maurer-Cartan equation for the Schouten
bracket on polyvector fields. Hochschild chains of a pre-$CY$ algebra have the structure
of a TQFT. A special case of a pre-$CY$ structure on a homologically smooth algebra is
a $CY_\infty$ algebra. We will show that $CY_\infty$ structures on an $A_\infty$ algebra
are classified by its negative cyclic homology. An example of a $CY_\infty$ algebra is the
dg-algebra of homotopy classes of familes of paths in a compact oriented manifold.
Moreover the bounded derived category of coherent sheaves of a Fano manifold, along
with the choice of a section of its anticanonical bundle, has a pre-$CY$ structure. All
these are the result of joint work with Maxim Kontsevich.