The backreacted Calabi–Yau category conjecture

Let $\mathcal{P}_{\mathcal{C},\lambda,h,s}=\{\mathcal{C},\lambda,h,s\}$ be a holography package consisting of a smooth cyclic $A_\infty$-category $\mathcal{C}$, an object $\lambda\in\mathcal{C}$, a trivialization $(d_{hoch}+uB)h=ch(\lambda)$, and a splitting $s$. For $N\in\mathbb{N}$, let $Z^{c,br}(\mathcal{P}_{\mathcal{C},\lambda,h,s})$ denote the backreacted closed string field theory partition function, and let $\mathcal{C}^{br}_N$ and $s^{br}$ be the conjectural backreacted Calabi–Yau category and splitting. Then $HH_*(\mathcal{C})\cong HH_*(\mathcal{C}^{br}_N)$ and

$$Z_{\mathcal{C}^{br}_N,s^{br}}^c=Z^{c,br}(\mathcal{P}_{\mathcal{C},\lambda,h,s})|_{\nu=N}\in Sym(HH_*(\mathcal{C})[u^{-1}]).$$

Backreacted Calabi–Yau category conjecture. Given a holography package $\mathcal{P}_{\mathcal{C},\lambda,h,s}$ as above, there is a family of Calabi–Yau categories $\mathcal{C}^{br}_N$, depending on $N\in\mathbb{N}$, together with a splitting $s^{br}$ such that

$$HH_*(\mathcal{C})\cong HH_*(\mathcal{C}^{br}_N)$$

and

$$Z_{\mathcal{C}^{br}_N,s^{br}}^c=Z^{c,br}(\mathcal{P}_{\mathcal{C},\lambda,h,s})|_{\nu=N}\in Sym(HH_*(\mathcal{C})[u^{-1}]).$$

The left-hand side uses the closed partition function associated with the conjectural category $\mathcal{C}^{br}_N$ and splitting $s^{br}$, while the right-hand side is the backreacted closed partition function evaluated at $\nu=N$; the additional assumption referred to in the source is required for the relevant expression to be defined. The conjecture is motivated by categorical versions of backreaction phenomena such as the deformed conifold construction, but the supplied text gives no resolution.