Charged-representation anomaly formula for elliptic Calabi–Yau threefolds

Let $X\to B$ be an elliptic Calabi–Yau threefold with $\mathbb{Q}$-factorial terminal singularities $\{P\}$, the relative minimal model of a Weierstrass model $W\to B$. Assume that $\operatorname{rk}(\operatorname{MW}(X))=r$, that

$$\Sigma=\Sigma_0\cup\Sigma_1\cup\cdots\cup\Sigma_N$$

is the discriminant, with simple algebra $\mathfrak{g}_i$ associated to $\Sigma_i$, and that the Weierstrass model is otherwise general. Define

$$\mathcal R'=30K_B^2+\frac12\left(\chi_{top}(X)-\sum_Pm(P)+2\sum_P\tau(P)\right).$$

Here $g_i=g(\Sigma_i)$, $g'_i=g(\Sigma'_i)$, and $Q$ and $C_r$ are codimension-two strata. Charged-representation anomaly conjecture.

$$\mathcal R'=\sum_i(g_i-1)(\dim\operatorname{adj}_i)_{ch}+(g'_i-g_i)(\dim\rho_{0,i})_{ch}+\sum_Q(\dim\rho_Q)_{ch}+\sum_{C_r}1+\sum_P\tau(P).$$

If $Q\in\Sigma_i\cap\Sigma_j$, the associated representation $\rho_Q$ is a tensor product representation for $\mathfrak{g}_i\oplus\mathfrak{g}_j$. This extends the anomaly-counting formula to Mordell–Weil rank and terminal-singularity contributions; the supplied text gives no resolution status.