Existence and properties of sigma-model full VOAs for Calabi–Yau manifolds
Construct, for every compact Calabi–Yau manifold X of complex dimension d, a unitary N=(2,2) full vertex operator algebra F_X with central charge (3d,3d) such that (i) F_X is of Calabi–Yau type as defined in the paper, (ii) the A-twist Hodge numbers of F_X satisfy h^A_{p,q}(F_X)=h_{p,q}(X), (iii) the cohomology ring H(F_X,d_A) is isomorphic to the quantum cohomology ring QH(X), and (iv) the Witten index Ind_{F_X}(q,y) equals the elliptic genus Ell(X;q,y).
References
Conjecture \ref{conj_sigma} Let $X$ be a compact Calabi-Yau manifold of complex dimension $d$. Then, there is a unitary $N=(2,2)$ full vertex operator algebra $F_X$ of central charge $({3d},{3d})$ such that: \begin{enumerate} \item $F_X$ is of CY type; \item The $A$-Hodge numbers of $F_X$ coincide with the Hodge numbers in Kähler geometry: $h_{p,q}A(F_X) = h_{p,q}(X)$; \item The cohomology ring $H(F_X,d_A)$ is isomorphic to the quantum cohomology ring $QH(X)$; \item The Witten index coincides with the elliptic genus \begin{align*} \mathrm{Ind}_{F_X}(q,y) = Ell(X;q,y). \end{align*} \end{enumerate}