M-theory on Elliptic Calabi-Yau Threefolds and 6d Anomalies

Abstract

We consider the 8-supercharge 5d su(N) gauge theories from M-theory compactified on elliptic Calabi-Yau threefolds. By matching the triple intersection numbers in the elliptic Calabi-Yau with the 5d Chern-Simons levels, we determine the charged matter contents for these theories. We show that all these 5d theories can be lifted to 6d N = (1, 0) theories while satisfying the anomaly cancellation equations. This suggests that the 5d theories obtained from M-theory compactified on elliptic Calabi-Yau threefolds have a natural 12d description, which as we know is F-theory. Furthermore, we compute the Euler characteristics of the elliptic Calabi-Yau threefolds.

Figures (6)

• Figure 1: Singular fibers of the I$_N^s$ elliptic Calabi-Yau threefolds. The base $B$ is an algebraic (compact or not) surface, and $E_0$ is the curve that supports the singular fiber. The singular fiber consists of $N$ copies of $\mathbb{P}^1$, denoted by $\varepsilon_i$, intersecting with each other as an affine $su(N)$ Dynkin diagram (in the figure we ignore the affine node). $D_i$ is the surface swept out by $\varepsilon_i$ in the singular fiber along the curve $E_0$. The triple intersection numbers $D_i\cdot D_j\cdot D_k$ are the main quantities of interest in this paper.
• Figure 2: M-theory and F-theory on elliptic Calabi-Yau threefolds. We determine the 5$d$ charged matter contents by matching the triple intersection numbers in the internal Calabi-Yau threefold with the 5$d$ Chern-Simons levels \ref{['match']}. On the other hand, the 6$d$ matter contents can be determined from anomaly cancellation equations. We show that the above diagram commutes for the I$_N^s$ elliptic Calabi-Yau threefolds with any algebraic base $B$.
• Figure 3: The network of resolutions for the I$_2^s$ model. Each letter stands for a (partial) resolution and each arrow represents a blowup. Starting from $\mathscr{E}_0$, there is a unique (small) resolution $\mathscr{T}$.
• Figure 5: (a) The singular fiber for the resolved I$_{2n}^s$ model $\mathscr{T}$\ref{['I2n']}. (b) The singular fiber for the resolved I$_{2n+1}^s$ model $\mathscr{T}$\ref{['I2n1']}. Here $\varepsilon_i$ is the fiber class of the surface $D_i$, each of which is a copy of $\mathbb{P}^1$. The classes for $D_i$ are labeled next to the nodes (see \ref{['Cclass']}). $E_i$ is the exceptional divisor for the $i$-th blowup, while the subscript of $\varepsilon_i$ labels the position of the node in the affine $su(N)$ Dynkin diagram.
• Figure 6: The singular fibers of the I$_N^s$ Weierstrass model. Over the codimension one locus $e_0=0$ in the base, the singular fiber is of the type I$_N^s$. Over the codimension two singular loci $e_0=a_1=0$ and $e_0=P_N=0$, the singular fiber enhances to I$_{N-4}^*$ and I$_{N+1}^s$, respectively.