Hodge Representations of Calabi-Yau 3-fold Type

Finally, to introduce the motivation of this thesis, we need to define Mumford-Tate groups and Mumford-Tate domains. To do this, we follow the construction in [GGK10].

In order to classify Mumford-Tate groups and Mumford-Tate domains, Griffiths, Green, and Kerr introduced the following notion:

Some results on the classification of Lie group Hodge representations can be seen in [GGK12] and [Zar83]. To avoid the complexities of working with $\mathbb{Q}$-algebraic groups, we "forget" the rational structure and work with the following notion:

Now we proceed to define real Lie algebra Hodge representations. To do this, we first cite the following fundamental result from Section IV of [GGK12]:

Let $G$ be a Hodge group. Then $G$ is a Lie group and we denote its Lie algebra by $\mathfrak{g}$. Fixing a Cartan subalgebra $\mathfrak{t}$, we know that $\mathfrak{g}$ has corresponding weight lattice $\Lambda_{w}$ and root lattice $\Lambda_{R}$. A grading element is an element $E$ of $\mathfrak{t}$ such that $\lambda(E)\in\mathbb{Z}$ for all $\lambda\in\Lambda_{R}$. Let $\tilde{G}$ be the simply connected simple Lie group with Lie algebra $\mathfrak{g}$. Then its center $Z(\tilde{G})=\frac{\Lambda_{w}}{\Lambda_{R}}$. We also know that all connected simple Lie groups with Lie algebra $\mathfrak{g}$ are of the form $\tilde{G}/C$ with $C$ a subgroup of $Z(\tilde{G})$, so the simple Lie groups with Lie algebra $\mathfrak{g}$ are in one to one correspondence with subgroups of $Z(\tilde{G})$, each of which is isomorphic to some $\frac{\Lambda}{\Lambda_{R}}$ where $\Lambda$ is a sublattice of $\Lambda_{w}$ containing $\Lambda_{R}$. Suppose that under this correspondence, $G$ is labeled by $\Lambda$. Let $\Lambda^{*}\mathrel{\mathop{\mathchar 58\relax}}=\mathrm{Hom}(\Lambda,2\pi i\mathbb{Z}).$ Then the maximal torus of $G$ is compact by Theorem 1.7, and is given by

$$T\simeq\mathfrak{t}/\Lambda^{*}.$$

In practice, real Lie algebra Hodge representations are easier to compute than real Lie group Hodge representations and yield important insights into the latter. In this paper, we will classify all the real Hodge Representations of $CY$ 3-fold type.

The goal of this thesis is to identify the real Lie algebra Hodge representations that can arise when $V_{\mathbb{R}}$ is the period domain parameterizing polarized Hodge structure of CY 3-fold type. These consist of the data $E\in i\mathfrak{g_{\mathbb{R}}}\subset\mathrm{End}(V_{\mathbb{C}},Q)$, where

1. 1.

   $V_{\mathbb{R}}$ is a real space.

2. 2.

   $Q\mathrel{\mathop{\mathchar 58\relax}}V_{\mathbb{R}}\times V_{\mathbb{R}}\to\mathbb{R}$ is a non-degenerate skew form. Thus $\mathrm{Aut}(V_{\mathbb{R}},Q)\simeq Sp_{2d}\mathbb{R}$, where $2d={\rm dim\,}V_{\mathbb{R}}$.

3. 3.

   $\mathfrak{g}_{\mathbb{R}}$ is a real reductive Lie subalgebra of $\mathrm{End}(V_{\mathbb{R}},Q)\simeq\mathfrak{sp}_{2d}\mathbb{R}$.

4. 4.

   $E$ is a semisimple element of $i\mathfrak{g}_{\mathbb{R}}\subset\mathfrak{g}_{\mathbb{C}}$ acting on $V_{\mathbb{C}}=V_{\mathbb{R}}\otimes\mathbb{C}$ with eigenspace decomposition $V_{\mathbb{C}}=V_{\frac{3}{2}}\oplus V_{\frac{1}{2}}\oplus V_{-\frac{1}{2}}\oplus V_{-\frac{3}{2}}$, and ${\rm dim\,}V_{\frac{3}{2}}$.

5. 5.

   The complex dimensions of $V_{\frac{3}{2}},V_{\frac{1}{2}},V_{-\frac{1}{2}},V_{-\frac{3}{2}}$ respectively are 1, $a$, $a$, 1 for some positive integer $a$ and $\overline{V_{l}}=V_{-l}$.

In particular, the constraint conditions that the representations are level 3 (see Subsection 1.3 for specific definition of level) and have first Hodge number $h^{3,0}=1$ ensure that we get Hodge representations associated to CY 3-fold type. For more background information, the reader might consult Section 1 and 2 of [HR20] as well as Appendix B of [Rob14]. In order to phrase our results, we assume a fixed torus and a fixed Weyl Chamber. To enumerate all such tuples of $(\mathfrak{g}_{\mathbb{R}},E,V_{\mathbb{R}})$, we take the following steps:

Suppose $V_{\mathbb{R}}=V^{1}_{\mathbb{R}}\oplus V^{2}_{\mathbb{R}}$ as $\mathfrak{g}_{\mathbb{R}}$ representations. Then $V_{\mathbb{C}}=V^{1}_{\mathbb{C}}\oplus V^{2}_{\mathbb{C}}$. Hence,

$$V_{\mathbb{C}}=V_{\frac{3}{2}}\oplus V_{\frac{1}{2}}\oplus V_{-\frac{1}{2}}\oplus V_{-\frac{3}{2}}$$

if and only if

$$V^{1}_{\mathbb{C}}=V^{1}_{\frac{3}{2}}\oplus V^{1}_{\frac{1}{2}}\oplus V^{1}_{-\frac{1}{2}}\oplus V^{1}_{-\frac{3}{2}}\qquad\text{and }\qquad V^{2}_{\mathbb{C}}=V^{2}_{\frac{3}{2}}\oplus V^{2}_{\frac{1}{2}}\oplus V^{2}_{-\frac{1}{2}}\oplus V^{2}_{-\frac{3}{2}}.$$

Moreover, ${\rm dim\,}V_{l}={\rm dim\,}V^{1}_{l}+{\rm dim\,}V^{2}_{l}$ so condition 5 above translates to

$$1={\rm dim\,}V_{\pm\frac{3}{2}}={\rm dim\,}V^{1}_{\pm\frac{3}{2}}+{\rm dim\,}V^{2}_{\pm\frac{3}{2}}.$$

Thus, without loss of generality, $V^{1}_{\mathbb{C}}=V^{1}_{\frac{1}{2}}\oplus V^{1}_{-\frac{1}{2}}$ and $V^{2}_{\mathbb{C}}=V^{2}_{\frac{3}{2}}\oplus V^{2}_{\frac{1}{2}}\oplus V^{2}_{-\frac{1}{2}}\oplus V^{2}_{-\frac{3}{2}}$, where ${\rm dim\,}V^{2}_{\frac{3}{2}}={\rm dim\,}V^{2}_{-\frac{3}{2}}=1$. And we say that $V^{1}_{\mathbb{R}}$ is a level 1 Hodge representation and $V^{2}_{\mathbb{R}}$ is a level 3 Hodge representation. These two types of representations are classified respectively in Sections 2 and 3.

To simplify the computations, instead of directly computing $\mathfrak{g}_{\mathbb{R}}$ and $V_{\mathbb{R}}$, we will first work with the complexifications $\mathfrak{g}_{\mathbb{C}}=\mathfrak{g}_{\mathbb{R}}\otimes\mathbb{C}$ and $V_{\mathbb{C}}=V_{\mathbb{R}}\otimes\mathbb{C}$, which will be carried out in the following subsections. Then we can give explicit formulae of $\mathfrak{g}_{\mathbb{R}}$ from $\mathfrak{g}_{\mathbb{C}}$ in most cases, namely when $E=A^{i}$, the unique Cartan element that is dual to simple root $\alpha_{i}$ for some $i$, although recovering $V_{\mathbb{R}}$ is more complicated. In order to recover $\mathfrak{g}_{\mathbb{R}}$, one should first observe that the grading element $E$ corresponds to a noncompact root as explained in Chapter IV of [GGK12]. Then by consulting [Kna02], we enumerate all the underlying real forms $\mathfrak{g}_{\mathbb{R}}$ in Sections 2 and 3.

Given an irreducible faithful representation $U$ of $\mathfrak{g}_{\mathbb{C}}$ with highest weight $\mu$, we have $\mathfrak{g}_{\mathbb{C}}\subset\mathrm{Aut}(U)$. Under the adjoint representation, $\mathfrak{g}_{\mathbb{C}}$ decomposes as $\mathfrak{g}^{ss}_{\mathbb{C}}\oplus\mathfrak{z}$, where $\mathfrak{g}^{ss}_{\mathbb{C}}$ denotes the semisimple part and $\mathfrak{z}$ denotes the center. By Schur’s lemma (see Appendix C), if the semisimple part of $\mathfrak{g}_{\mathbb{C}}$ is simple, then $\mathfrak{z}$ is 0 or 1 dimensional. In general, the dimension of $\mathfrak{z}$ does not exceed the number of simple direct summands in $\mathfrak{g}^{ss}_{\mathbb{C}}$. Given a semisimple element $E$, $E=E_{ss}+E^{\prime}$ for unique choices of $E_{ss}\in\mathfrak{g}_{\mathbb{C}}^{ss}$ and $E^{\prime}\in\mathfrak{z}$. By virtue of infinitesimal period relation(IPR) on the dual of the Mumford-Tate domain, we without loss of generality assume that $\alpha_{i}(E_{ss})\in\{0,1\}$ for each simple root $\alpha_{i}$ throughout this paper [Rob14]. On $U$, $E_{ss}$ acts via weights and $E^{\prime}$ acts as $c\mathds{1}$ for some $c\in\mathbb{Q}$. Thus, $\mu(E)=\mu(E_{ss})+c$ on $U$ and $\mu^{*}(E)=\mu^{*}(E_{ss})-c$ on $U^{*}$, where $\mu^{*}$ is the highest weight of $\mathfrak{g}_{\mathbb{C}}$ on $U^{*}$. In order to recover $\mathfrak{g}_{\mathbb{R}}$, we need the tuples $(\mathfrak{g}_{\mathbb{C}},E_{ss},c)$, which we enumerate in the following Sections 2 and 3. More specifically, the grading element $E$ determines the maximal compact subalgebra $\mathfrak{k}$ of $\mathfrak{g}_{\mathbb{R}}^{ss}$, which is just the direct sum of the even $E$-eigenspaces in the adjoint representation. This is enough to determine the real form $\mathfrak{g}_{\mathbb{R}}^{ss}$ using Vogan diagram classification, and [Kna02] is a good source of information. Then if $c\neq 0$, we have $\mathfrak{g}_{\mathbb{C}}=\mathfrak{g}_{\mathbb{C}}^{ss}\oplus\mathbb{C}$ and therefore $\mathfrak{g}_{\mathbb{R}}=\mathfrak{g}_{\mathbb{R}}^{ss}\oplus\mathbb{R}.$

When $V_{\mathbb{R}}$ is an irreducible representation of $\mathfrak{g}_{\mathbb{R}}$, there are three possible cases:

$$V_{\mathbb{R}}\otimes\mathbb{C}=\left\{\begin{array}[]{ll}U\text{ and }U=U^{*}\text{ as representations of $\mathfrak{g}_{\mathbb{C}}$}\\ \hfill\quad(U\text{ is a {real} representation of $\mathfrak{g}_{\mathbb{R}}$})\\ U\oplus U^{*}\text{ and }U\neq U^{*}\text{ as representations of $\mathfrak{g}_{\mathbb{C}}$}\\ \hfill\quad(U\text{ is a {complex} representation of $\mathfrak{g}_{\mathbb{R}}$})\\ U\oplus U^{*}\text{ and }U=U^{*}\text{ as representations of $\mathfrak{g}_{\mathbb{C}}$}\\ \hfill\quad(U\text{ is a {quaternionic} representation of $\mathfrak{g}_{\mathbb{R}}$})\end{array}\right.$$

where $U$ is an irreducible representation of $\mathfrak{g}_{\mathbb{C}}$ with highest weight $\mu$.

To distinguish the three possible cases of $U$, one could immediately observe that once we have $U\neq U^{*}$ (or equivalently $\mu\neq\mu^{*}$), $U$ is complex. When $U=U^{*}$ (or equivalently $\mu=\mu^{*}$), we determine whether $U$ is real or quaternionic using the following test: Define

$$H_{\phi}\mathrel{\mathop{\mathchar 58\relax}}=2\sum_{\alpha_{j}(E_{ss})=0}A^{j}.$$

Then $U$ is quaternionic if and only if $\mu(H_{\phi})$ is odd, and real if and only if $\mu(H_{\phi})$ is even.
In general, when $U$ is real or quaternionic, we necessarily have $c=0$. When $U$ is complex, $c=n/2-\mu(E_{ss})$ where $n$ is the level of a Hodge representation.

Choose a grading element $E\in\mathfrak{g}_{\mathbb{C}}$. According to the discussion in Subsection 1.5, we can write $\mathfrak{g}_{\mathbb{C}}=\mathfrak{g}_{\mathbb{C}}^{ss}\oplus\mathfrak{z}$, where $\mathfrak{z}$ is a zero or one dimensional center, and $\mathfrak{g}_{\mathbb{C}}^{ss}=[\mathfrak{g}_{\mathbb{C}},\mathfrak{g}_{\mathbb{C}}]$ is the semisimple part. Then we can write grading element $E=E_{ss}+E^{\prime}$, where $E_{ss}\in\mathfrak{g}_{\mathbb{C}}^{ss}$ and $E^{\prime}\in\mathfrak{z}$. Assuming $V_{\mathbb{R}}$ is irreducible, by Subsection 1.6, there are three possible cases: $U$ is a real, complex or quaternionic representation of $\mathfrak{g}_{\mathbb{R}}^{ss}$. In case of level 1 Hodge representations, we want

$$V_{\mathbb{R}}\otimes\mathbb{C}=V_{1/2}\oplus V_{-1/2}.$$

By discussion in Subsections 1.4-1.6, it suffices to locate tuples
$(\mathfrak{g}^{ss}_{\mathbb{C}},E_{ss},\mu,c)$, where $\mathfrak{g}_{\mathbb{C}}$ is a complex semisimple Lie algebra, $E_{ss}$ is a grading element of $\mathfrak{g}^{ss}_{\mathbb{C}}$, and $\mu$ is the highest weight on $\mathfrak{g}_{\mathbb{C}}^{ss}$’s complex irreducible representation $U$. We first claim that $\mathfrak{g}^{ss}_{\mathbb{C}}$ must be simple. To see this, let $\mathfrak{g}_{\mathbb{C}}^{ss}=\oplus_{i=1}^{n}\mathfrak{g}_{i}$ where each $\mathfrak{g}_{i}$ is simple. Then each irreducible representation of $\mathfrak{g}_{\mathbb{C}}^{ss}$ decomposes as $U=\otimes_{i=1}^{n}U_{i}$ where $U_{i}$ is an irreducible representation of $\mathfrak{g}_{i}$. Moreover, suppose that $E$ is a grading element of $\mathfrak{g}_{\mathbb{C}}^{ss}$, then there exists a unique grading element $E_{i}$ from each $\mathfrak{g}_{i}$ such that $E=\sum_{i=1}^{n}E_{i}$, and $\mu(E)+\mu^{*}(E)=\sum_{i=1}^{n}\mu_{i}(E_{i})+\mu^{*}(E_{i})$. Note that for each $i$, $\mu_{i}(E_{i})+\mu^{*}_{i}(E_{i})\geq 1$, and the tuples we search for must satisfy $\mu(E)+\mu^{*}(E)=1$. Hence, $\mathfrak{g}_{\mathbb{C}}^{ss}$ must be simple as claimed.

Now $U$ decomposes into $E_{ss}$’s eigenspaces $U_{\mu(E_{ss})}$ and $U_{\mu(E_{ss})-1}$, and correspondingly $U^{*}$ to decompose into $E_{ss}$’s eigenspaces $U^{*}_{1-\mu(E_{ss})}$ and $U^{*}_{-\mu(E_{ss})}$. Recall that $E^{\prime}$ acts on $U$ via $c\mathds{1}$ and on $U^{*}$ via $-c\mathds{1}$ for some $c\in\mathbb{Q}$. Then as $E=E_{ss}+E^{\prime}$’s eigenspaces, $U$ and $U^{*}$ respectively decompose as:

$$U=U_{\mu(E_{ss})+c}\oplus U_{\mu(E_{ss})+c-1}$$

$$U^{*}=U_{1-\mu(E_{ss})-c}\oplus U^{*}_{-\mu(E_{ss})-c}$$

When $U$ is real, we must have $U=U_{1/2}\oplus U_{-1/2}$ as eigenspaces of $E_{ss}$, and ${\rm dim\,}U_{1/2}={\rm dim\,}U_{-{1}/{2}}$. In this case, $c=0$. When $U$ is complex or quaternionic, with $c={1}/{2}-\mu(E_{ss})$, we get $U=U_{{1}/{2}}\oplus U_{-{1}/{2}}$ and $U^{*}=U^{*}_{{1}/{2}}\oplus U^{*}_{-{1}/{2}}$ as desired. Moreover, since ${\rm dim\,}U_{{1}/{2}}={\rm dim\,}U^{*}_{-{1}/{2}}$ and ${\rm dim\,}U_{-{1}/{2}}={\rm dim\,}U^{*}_{{1}/{2}}$, we have ${\rm dim\,}V_{1/2}={\rm dim\,}V_{-1/2}$ as desired.

In this section, we assume that $\mathfrak{g}_{\mathbb{C}}^{ss}$ is simple. We treat the situation when $\mathfrak{g}_{\mathbb{C}}^{ss}$ is semisimple in Section 3.2.