ADE Bundles over Surfaces

ADE Bundles over Surfaces††This paper is a contribution to the Special Issue
on Enumerative and Gauge-Theoretic Invariants in honor of Lothar Göttsche on
the occasion of his 60th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Gottsche.html

Yunxia CHEN a and Naichung Conan LEUNG b

Y.X. Chen and N.C. Leung

a) School of Mathematics, East China University of Science and Technology,
a) Meilong Road 130, Shanghai, China<EMAIL_ADDRESS>
b) The Institute of Mathematical Sciences and Department of Mathematics,
b) The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
<EMAIL_ADDRESS>
Received March 07, 2022, in final form October 21, 2022; Published online
November 05, 2022

This is a review paper about ADE bundles over surfaces. Based on the deep
connections between the geometry of surfaces and ADE Lie theory, we construct
the corresponding ADE bundles over surfaces and study some related problems.

ADE bundles; surfaces; Cox ring

53C10; 14H60

##  1 Introduction

Studies of curves on surfaces is one of the oldest problems in mathematics,
starting from the intersection theory of configurations of lines on a plane.
It is also an important tool in understanding the geometry of surfaces, see
for example [1, 12]. Enumeration of curves on surfaces is an old subject which
dates back to the 19th century. It has received much renewed interest and had
become a very active research area, probably started from the ground breaking
paper by Yau and Zaslow [56] which gave a beautiful formula for the number of
curves on a K3 surface in terms of a modular form. Namely the number of
rational curves on a K3 surface of a given degree is given by the coefficients
of the generating function

$q/\Delta(q)=\prod\nolimits_{m=1}^{\infty}(1-q^{m})^{-24},$

where $\Delta$ is the well-known modular form of weight 12. Beauville [4] (see
also [23]) gave a mathematical approach to this formula by interpreting the
BPS count in terms of compactified Jacobians of curves. For primitive classes,
the Yau–Zaslow formula was proven by Bryan and the second author [9, 10, 11,
38, 39] and the full conjecture was proven later by Klemm, Maulik,
Pandharipande and Scheidegger [33] (see also [48]) via mirror symmetry for a
Calabi–Yau threefold with a K3 fibration.

In [26], Göttsche gave an intriguing generalization of the Yau–Zaslow formula
which applies to any surface $X$ and any genus $g$ as long as the curve class
$C$ is sufficiently ample. Concretely the number of genus $g$ curves in $|C|$
passing through an appropriate number $r$ of points is given as the
coefficient of $q^{C\cdot(C-K)/2}$ in the following power series in $q$

$B_{1}^{K^{2}}B_{2}^{C\cdot
K}\big{(}DG_{2}\big{)}^{r}\big{(}D^{2}G_{2}\big{)}/\big{(}\Delta\big{(}D^{2}G_{2}\big{)}\big{)}^{\chi(O_{X})/2},$

where $D=q\frac{\mathrm{d}}{{\mathrm{d}}q}$,
$G_{2}(q)=-\frac{1}{24}+\sum_{k>0}\big{(}\sum_{d|k}d\big{)}q^{k}$ is the
Eisenstein series and $B_{1}(q)$ and $B_{2}(q)$ are universal power series in
$q$. The universality of this amazing conjecture has been solved by
Kool–Shende–Thomas [37] and Tzeng [55] independently in 2011 and 2012.

There is also a refined curve counting defined by Block and Göttsche which is
related to tropical geometry [5] and real algebraic geometry [27] as well.

In this article, we study very different aspects of the geometry of curves on
surfaces, namely the intricate relationships between configurations of low
degree curves, for instance lines, and representation theory of simple Lie
algebras. The most famous example is probably the 27 lines on a cubic surface
discovered by Cayley and Salmon in 1849 and their relationships with the
exceptional Lie algebra $E_{6}$ (see, e.g., [19, 46]).

The organization of this paper is as follows. In Sections 2, 3, 4 and 5, we
introduce the famous examples that reflect the deep connections between the
geometry of surfaces and Lie theory. Specifically, we study ADE singularities
in Section 2, cubic surfaces in Section 3, del Pezzo surfaces in Section 4 and
ADE surfaces in Section 5. Based on Section 5 (ADE surfaces), Sections 6 and 7
are related to F/string theory duality. In Sections 8, 9, 10 and 11, we
introduce some results related to deformation of ADE bundles over surfaces
with ADE singularities, based on Section 2 (ADE singularities). We study Cox
rings of ADE surfaces in Section 12. And the final Section 13 is a summary.

##  2 ADE singularities vs ADE Lie theory

In this section, we will recall ADE surface singularities, Lie algebras of ADE
types and their fundamental represenations. Another intricate relationship
between geometry of surfaces and Lie theory is the McKay correspondence [50].
The simplest type of surface singularity $p\in X$ is called a simple
singularity (also called a rational double point, canonical singularity, Du
Val singularity or ADE singularity) [1]. Locally it is given by the quotient
$\mathbb{C}^{2}/\Gamma$ of $\mathbb{C}^{2}$ by a finite subgroup
$\Gamma\subset{\mathrm{S}L}(2,\mathbb{C})$. The exceptional curve
$C=\bigcup_{i=1}^{r}C_{i}$ of its minimal resolution $\tilde{X}\rightarrow X$
is a union of smooth rational curves $C_{i}$’s satisfying $C_{i}\cdot
C_{i}=-2$, i.e., $(-2)$-curves, and the configuration can be described by its
dual graph which is a simply-laced Dynkin diagram, i.e., of ADE type.

Recall that a simple Lie algebra $\mathfrak{g}$ is called simply-laced if all
roots have the same length and they are exactly those of ADE types in the
classification of simple Lie algebras [29], see Figure 1.

$A_{n}\colon$

$D_{n}\colon$

$E_{6}\colon$

$E_{7}\colon$

$E_{8}\colon$

Figure 1: Dynkin diagrams of ADE types.

Nodes in a Dynkin diagram label fundamental representations of $\mathfrak{g}$.

The Lie algebra $A_{n}=\mathfrak{sl}(n+1)$ is the algebra of symmetries of a
volume form on $V\simeq\mathbb{C}^{n+1}$. The fundamental representations of
$\mathfrak{sl}(n+1)$ consist of the standard representation
$V\simeq\mathbb{C}^{n+1}$ together with its exterior powers $\Lambda^{k}V$
with $k=2,3,\dots,n$.

The Lie algebra $D_{n}=\mathfrak{o}(2n)$ is the algebra of infinitesimal
symmetries of a nondegenerate quadratic form $q\in S^{2}V^{\ast}$ on
$V\simeq\mathbb{C}^{2n}$. The fundamental representations of
$\mathfrak{o}(2n)$ consist of the standard representation $V$ together with
its exterior powers $\Lambda^{k}V$ with $k=2,3,\dots,n-2$ and also the two
spinor representations $S^{+}$ and $S^{-}$.

Furthermore, $E_{6}$ is the algebra of infinitesimal symmetries of a specific
cubic form $c\in S^{3}V^{\ast}$ on $V\simeq\mathbb{C}^{27}$ and $E_{7}$ is the
algebra of infinitesimal symmetries of a specific quartic form $t\in
S^{4}V^{\ast}$ on $V\simeq\mathbb{C}^{56}$. The explicit forms of these cubic
form and quartic form can be described in terms of the geometry of del Pezzo
surfaces of degree 3 and 2 respectively. The situation for $E_{8}$ is more
complicated as its smallest representation $V$ is not a miniscule
representation, instead it is the adjoint representation of $E_{8}$. We will
call the above $V$’s as the standard representations of $\mathfrak{g}$, see
Figure 2.

$A_{n}\colon$$V$${\wedge}^{2}V$${\wedge}^{n-2}V$${\wedge}^{n-1}V$${\wedge}^{n}V$

$D_{n}\colon$$V$${\wedge}^{2}V$${\wedge}^{n-3}V$${\wedge}^{n-2}V$$S^{+}$$S^{-}$

$E_{6}\colon$$V$$V^{*}$

$E_{7}\colon$$V$

$E_{8}\colon$$V$

Figure 2: Dynkin diagrams with fundamental representations labelled.

##  3 Cubic surfaces vs $\boldsymbol{E_{6}}$ Lie theory

Let us come back to explain the relationships between cubic surfaces, or more
generally del Pezzo surfaces, with the representation theory of exceptional
Lie algebras of type $E$.

In this section, we will recall various ways to realize the famous $27$ lines
on cubic surfaces in $\mathbb{P}^{3}$. Each such geometric setting corresponds
to the branching rule for the $27$-dimensional standard representation of
$E_{6}$ to a Lie subalgebra.

Geometric settings |  Lie subalgebra of $E_{6}$
---|---
Degenerate to 3 planes | $\mathfrak{sl}(3)\times\mathfrak{sl}(3)\times\mathfrak{sl}(3)$
Degenerate to plane + quadric surface | $\mathfrak{sl}(6)\times\mathfrak{sl}(2)$
Blowing down a line | $\mathfrak{o}(10)$

We will also explain how to realize the $E_{6}$ structure from the
configuration of these 27 lines.

It is a classical result that there are exactly 27 lines on a smooth cubic
surfaces $X=\\{f(z)\allowbreak=0\\}\subset\mathbb{P}^{3}$ and the symmetry of
their intersection pattern is the Weyl group $W_{E_{6}}$ of $E_{6}$ (see,
e.g., [46, 49]). Could we recover the Lie algebra $E_{6}$ itself, rather than
just its Weyl group? Indeed this can be done from the geometry of these 27
lines.

One way to locate these lines is to consider a family of cubic surfaces of the
form [51]

$X(t)=\left\\{z_{1}z_{2}z_{3}+tf\left(z_{0},z_{1},z_{2},z_{3}\right)=0\right\\}\subset\mathbb{P}^{3}$

for a homogeneous polynomial $f$ of degree 3. Suppose $f$ is generic, then
$X(t)$ is a smooth cubic surface for any generic $t\neq 0$. When $t=0$, $X(0)$
is the union of 3 coordinate planes, which certainly contain infinitely many
lines. But only a few of them could survive for nearby $t$. To explain this,
we note that the singular set of $X(0)$ is the union of 3 coordinate axes. In
the smoothing $X\left(t\neq 0\right)$, each such singular point $p$ becomes a
vanishing $S^{1}$ as in the standard model of smoothing from
$\left\\{xy=0\right\\}$ to $\left\\{xy=t\right\\}$, with the exception when
$f$ vanishes at $p$. As $\deg(f)=3$, $f$ vanishes at 3 points on each
coordinate axis. We call these unstable points as without them the total
family of $X(t)$’s would be a semi-stable degeneration for small $t$. A line
on a coordinate plane, say $\left\\{z_{1}=0\right\\}\subset X(0)$, survives on
nearby $X(t)$’s if and only if it meets one of these 3 points on the
$z_{2}$-axis, as well as on $z_{3}$-axis. Therefore the total number of lines
on $X(0)$ which survive on nearby $X(t)$’s is $3\times 3+3\times 3+3\times
3=27.$

We will see that these 27 lines can be used to construct the fundamental
representation $L=\underline{27}$ of $E_{6}$. Here $\underline{27}$ means a
particular representation of $E_{6}$ of dimension 27. In fact, the above
description of these lines corresponds to the branching rule from $E_{6}$ to
its subalgebra
$\mathfrak{su}(3)_{1}\times\mathfrak{su}(3)_{2}\times\mathfrak{su}(3)_{3}$:

$\underline{27}=\underline{3_{1}}\times\underline{3_{2}^{\ast}}+\underline{3_{2}}\times\underline{3_{3}^{\ast}}+\underline{3_{3}}\times\underline{3_{1}^{\ast}}.$

Here $\underline{3_{i}}$ refers to the standard representation of the
$i^{th}$-factor $\mathfrak{su}(3)_{i}$ in
$\mathfrak{su}(3)_{1}\times\mathfrak{su}(3)_{2}\times\mathfrak{su}(3)_{3}$ and
$\underline{3_{i}^{\ast}}$ is its dual representation.

Similarly, we could degenerate $X(t)$ to a union of a plane $H$ and a smooth
quadric surface $Q$ in $\mathbb{P}^{3}$, i.e., $X(0)=H\cup Q$ [42]. Then there
are $6$ points on the curve $C=H\cap Q$ which stay on $X(t)$ for $t$
infinitesimally close to $0$. They play the same role as the $3$ points on
each coordinate axis in the previous example, namely they are the unstable
points for the family $X(t)$’s. A line on the plane $H$ joining any $2$ of
these $6$ points will deform to a line on nearby $X(t)$’s. The total number of
such lines is $\binom{6}{2}=15$. On the other hand, the quadric $Q$ has $2$
rulings, each is a $\mathbb{P}^{1}$-family of lines. A line in $Q$ passing
through one the these $6$ points in $C$ will also deform to one in nearby
$X(t)$, and the total number of such lines is $6\times 2=12$. All together, we
have $\binom{6}{2}+6\times 2=27$ lines on $X(t)$. This corresponds to the
branching rule for the fundamental representation $L_{E_{6}}=\underline{27}$
of $E_{6}$ to its subalgebra $\mathfrak{sl}(6)\times\mathfrak{sl}(2)\subset
E_{6}$ as follows:

$L_{E_{6}}\simeq\Lambda^{2}V_{6}+V_{6}\otimes V_{2},$

where $V_{6}\simeq\mathbb{C}^{6}$ and $V_{2}\simeq\mathbb{C}^{2}$ are the
standard representations of $\mathfrak{sl}(6)$ and $\mathfrak{sl}(2)$
respectively.

There is another way to see these lines once a particular line $l\subset X$ is
given. Any hyperplane section containing $l$ must be of the form $l+C$ for
some conic curve $C\subset X$ [46, 49]. The pencil of such hyperplane sections
degenerates into sum of three lines $l+l_{i}^{\prime}+l_{i}^{\prime\prime}$
five times $i=1,\dots,5$. These 10 divisors $l_{i}^{\prime}$’s and
$l_{i}^{{}^{\prime\prime}}$’s determine the standard representation of
$O(5,5)=D_{5}$ (see $D_{n}$-surfaces). The remaining 16 lines, which do not
intersect $l$, form a spinor representation of $D_{5}$ and

$\underline{27}=\underline{1}+\underline{16}+\underline{10}$

is the branching rule from $E_{6}$ to $E_{5}=D_{5}$. From the surface
perspective, these three types of lines correspond to lines with intersection
numbers $-1$, $0$ and $1$ with the given line $l$.

We could realize the $E_{6}$ structure more concretely. We recall that the
standard representation $\underline{27}$ of $E_{6}$ admits a cubic form

$c\colon\
\underline{27}\otimes\underline{27}\otimes\underline{27}\rightarrow\mathbb{C}$

such that $E_{6}\simeq\operatorname{Aut}(\underline{27},c)$, similar to
$D_{n}\simeq\operatorname{Aut}(\underline{2n},q)$ as the symmetry group of a
quadratic form on its standard representation. We consider the direct sum of
line bundles from lines on $X$:

$\mathcal{L}^{E_{6}}=\mathop{\textstyle\bigoplus}\limits_{i=1}^{27}O(l_{i}).$

If 3 of them $l_{1}$, $l_{2}$, $l_{3}$ form a triangle, i.e., $l_{i}\cdot
l_{j}=1$ whenever $i\neq j$, then $l_{1}+l_{2}+l_{3}$ is a hyperplane section.
Therefore

$O(l_{1})\otimes O(l_{2})\otimes O(l_{3})\simeq O(1)\simeq K^{-1},$

where $K$ is the canonical line bundle of $X$. With suitable choices of these
isomorphisms [42] and using all triangles in $X$, we obtain

$c_{\mathcal{L}}\colon\
\mathcal{L}^{E_{6}}\otimes\mathcal{L}^{E_{6}}\otimes\mathcal{L}^{E_{6}}\rightarrow
K^{-1}$

so that the fiberwise cubic structures give a conformal $E_{6}$-bundle
$\mathcal{E}^{E_{6}}$ over the cubic surface $X$.

The root lattice $\Lambda_{E_{6}}$ of $E_{6}$ can be identified as the
orthogonal complement $\langle K\rangle^{\bot}$ of $K$ in
$\operatorname{Pic}(X)\simeq H^{2}(X,\mathbb{Z})\simeq\mathbb{Z}^{1,6}$. Here
$\mathbb{Z}^{1,6}$ denotes the lattice $\mathbb{Z}^{7}$ with the quadratic
form $(1)\oplus(-1)^{\oplus 6}$, which is isomorphic to $H^{2}(X,\mathbb{Z})$
equipped with the intersection form. This is because any smooth cubic surface
is a blowup of $\mathbb{P}^{2}$ at 6 points. Note that $\alpha\in
H^{2}(X,\mathbb{Z})$ is a root in $\Lambda_{E_{6}}$ if and only if
$\alpha\cdot\alpha=-2$ and $\alpha\cdot K=0$, i.e., $\alpha\in\langle
K\rangle^{\bot}\simeq\Lambda_{E_{6}}$. We denote the collection of all roots
as $\Phi$. If $\alpha\in\Phi$ can be represented by an effective divisor $C$,
then $C\simeq\mathbb{P}^{1}$ is called an $(-2)$-curve. Furthermore, (1)
$l\in\operatorname{Pic}(X)\simeq H^{2}(X,\mathbb{Z})$ satisfying $l\cdot l=-1$
and $l\cdot K=-1$ is always represented by a unique line on $X$; (2)
$f\in\operatorname{Pic}(X)\simeq H^{2}(X,\mathbb{Z})$ satisfying $f\cdot f=0$
and $f\cdot K=-2$ will be called a ruling, or conic bundle, as it defines a
$\mathbb{P}^{1}$-bundle $\Phi_{f}\colon X\rightarrow|f|\simeq\mathbb{P}^{1}$
on $X$ with fiber degree 2; (3) $h\in\operatorname{Pic}(X)\simeq
H^{2}(X,\mathbb{Z})$ satisfying $h\cdot h=1$ and $h\cdot K=-3$ gives
$\Phi_{h}\colon X\rightarrow|h|\simeq\mathbb{P}^{2}$ which realizes $X$ as a
blowup of $\mathbb{P}^{2}$ at 6 points. Similar structures hold for all del
Pezzo surfaces and they are closely related to the fundamental representations
$L$, $R$ and $H$ of $E_{n}$ corresponding the left node, the right node and
the top node in the Dynkin diagram of $E_{n}$’s, see Figure 3.

$L$$R$$H$ Figure 3: Dynkin diagram of $E_{n}$ with $L$, $R$ and $H$ labelled.

In fact, the Lie algebra $E_{6}$-bundle of fiberwise infinitesimal
automorphisms of $\big{(}\mathcal{L}^{E_{6}},c_{\mathcal{L}}\big{)}$ is
naturally

$\mathcal{E}^{E_{6}}=O_{X}^{\oplus
6}\mathop{\textstyle\bigoplus}\limits_{\begin{subarray}{c}\alpha\cdot K=0\\\
\alpha\cdot\alpha=-2\end{subarray}}O_{X}(\alpha),$

similar to the root space decomposition of Lie algebra
$\operatorname{Lie}(E_{6})=\mathfrak{h}\mathop{\textstyle\bigoplus}\limits_{\alpha\in\Phi}\mathfrak{g}_{\alpha}$.

Given any smooth anti-canonical curve $\Sigma\in|-K|$, then it is always an
elliptic curve by the adjunction formula. For any $\alpha\in\Phi$,
$O(\alpha)|_{\Sigma}$ has degree 0 and therefore defines a flat line bundle
over $\Sigma$. Indeed $\mathcal{E}^{E_{6}}|_{\Sigma}$ is a flat $E_{6}$-bundle
over $\Sigma$ and every flat $E_{6}$-bundle over $\Sigma$ always arises this
way for an embedding of $\Sigma$ into some cubic surface. Such a
correspondence works for any $E_{n}$-bundle, or even any ADE bundle, as we
will describe next.

##  4 Del Pezzo surfaces vs $\boldsymbol{E_{n}}$ Lie theory

Cubic surfaces are degree three del Pezzo surfaces. In this section, we
explain the analogous results between the configuration of lines in degree
$9-n$ del Pezzo surfaces and representation theory for the Lie algebra $E_{n}$
for any $n$ between $1$ and $8$, as given in [42].

A smooth surface $X$ is called a del Pezzo surface if $K^{-1}$ is ample. Its
degree $\deg X=K\cdot K$ is always between $1$ and $9$. A del Pezzo surface of
degree $d$ is always a blowup of $\mathbb{P}^{2}$ at $n=9-d$ points in general
position, with the exception of $\mathbb{P}^{1}\times\mathbb{P}^{1}$. We will
denote them as $X_{n}$ and $X_{1}^{\prime}=\mathbb{P}^{1}\times\mathbb{P}^{1}$
respectively. For example $X_{0}=\mathbb{P}^{2}$, $X_{5}$ is the complete
intersection of two quadric hypersurfaces in $\mathbb{P}^{4}$, $X_{6}$ is a
cubic surface in $\mathbb{P}^{3}$, $X_{7}$ is a branched cover of
$\mathbb{P}^{2}$ branched along a quartic curve and $X_{8}$ has a pencil of
elliptic curves, i.e., $|-K|\simeq\mathbb{P}^{1}$. If we blowup the unique
base point of this pencil, we obtain a surface $X_{9}$ which is a rational
elliptic surface. We will see in later sections that the geometry of this
surface is related to affine Kac–Moody Lie algebras of type $\hat{E}_{8}$’s.

In fact, the $X_{n}$’s and $X_{1}^{\prime}$ correspond to simple Lie algebras
of type $E_{n}$, $1\leq n\leq 9$ and $\mathbb{C}$ if the Abelian Lie algebra
$\mathbb{C}$ is also included as a rank one simple Lie algebra. We recall that
$E_{n}$’s with $n\leq 5$ coincide with certain classical Lie algebras, as can
be seen directly from Dynkin diagrams. Concretely, we have $E_{5}\simeq
D_{5}=\mathfrak{o}(10)$, $E_{4}\simeq A_{4}=\mathfrak{sl}(5)$, $E_{3}\simeq
A_{2}\times A_{1}=\mathfrak{sl}(3)\times\mathfrak{sl}(2)$,
$E_{2}\simeq\mathfrak{sl}(2)\times\mathbb{C}$, $E_{1}\simeq\mathbb{C}$,
$E_{1}^{\prime}\simeq\mathfrak{sl}(2)$ and $E_{0}=0$. We remark that there is
a physical explanation of having two $E_{n}$’s when $n=1$ [31] via
supergravity in eleven dimensions where $E_{n}$ arises as one compactifies the
11-dimensional space-time along a $n$-dimensional torus.

The first relationship between the geometry of a del Pezzo surface $X_{n}$ and
the Lie algebra $E_{n}$ is the following: the orthogonal complement $\langle
K\rangle^{\bot}\subset H^{2}(X_{n},\mathbb{Z})\simeq\mathbb{Z}^{n,1}$ is a
root system $\Lambda_{E_{n}}$ for $E_{n}$.

Similar to the cubic surface case, a line $l\subset X_{n}$ means a curve in
$X_{n}$ of degree one with respect to the anti-canonical class, which is
equivalent to $l\in H^{2}(X_{n},\mathbb{Z})$ satisfying $l\cdot l=-1$ and
$l\cdot K=-1$. It is because each such cohomology class can be represented as
the Poincare dual of a unique line in $X_{n}$. The number of lines in $X_{n}$
is equal to the dimension of the standard representation $L$ of $E_{n}$,
unless $n=8$ in which case it is $\dim L-8=240$ because $L$ is no longer a
minuscule representation. Explicitly these numbers are 1, 3, 6, 10, 16, 27, 56
and 240. Similarly $f\in\operatorname{Pic}(X)\simeq H^{2}(X,\mathbb{Z})$
satisfying $f\cdot f=0$ and $f\cdot K=-2$ will be called a ruling, or conic
bundle, as it defines a $\mathbb{P}^{1}$-bundle $\Phi_{f}\colon
X\rightarrow|f|\simeq\mathbb{P}^{1}$ on $X$ with fiber degree 2. The number of
rulings in $X_{n}$ is equal to the dimension of the fundamental representation
$R$ of $E_{n}$ for $n<7$.

We define

$\displaystyle\mathcal{E}^{E_{n}}=O^{\oplus
n}\mathop{\textstyle\bigoplus}\limits_{\begin{subarray}{c}\alpha\cdot K=0\\\
\alpha\cdot\alpha=-2\end{subarray}}O(\alpha),\qquad\mathcal{L}^{E_{n}}=\mathop{\textstyle\bigoplus}\limits_{\begin{subarray}{c}l\cdot
K=-1\\\ l\cdot
l=-1\end{subarray}}O(l),\qquad\mathcal{R}^{E_{n}}=\mathop{\textstyle\bigoplus}\limits_{\begin{subarray}{c}f\cdot
K=-2\\\ f\cdot f=0\end{subarray}}O(f).$

Then (1) $\mathcal{E}^{E_{n}}$ is always an $E_{n}$-bundle over $X_{n}$; (2)
$\mathcal{L}^{E_{n}}$ is an $\mathcal{E}^{E_{n}}$-representation bundle over
$X_{n}$ corresponding to the $E_{n}$ representation $L$, for $n<8$ and (3)
$\mathcal{R}^{E_{n}}$ is an $\mathcal{E}^{E_{n}}$-representation bundle over
$X_{n}$ corresponding to the $E_{n}$-representation $R$, for $n<7$. The
restriction on $n$ is related to the fact that $L$ for $E_{8}$ and $R$ for
$E_{7}$ are the adjoint representations.

If $l_{1}$ and $l_{2}$ are two lines in $X_{n}$ satisfying $l_{1}\cdot
l_{2}=1$, then $l_{1}+l_{2}$ is a ruling. This is reflected by the fact that
$R$ is an irreducible component of the tensor product $E_{n}$-representation
$L\otimes L$ and in fact we have a natural bundle homomorphism

$\mathcal{L}^{E_{n}}\otimes\mathcal{L}^{E_{n}}\rightarrow\mathcal{R}^{E_{n}}$

over any $X_{n}$.

For example $X_{1}$ is the blowup of $\mathbb{P}^{2}$ and the exceptional
curve is its unique line. The two exceptional curves in $X_{2}$ together with
the strict transform of the line joining two blow up points in
$\mathbb{P}^{2}$ give the 3 lines in $X_{2}$. Having two different types of
lines in $X_{2}$ reflects that $L_{E_{2}}$ is a reducible representation of
$E_{2}$ and it is also responsible for the fact that there are two different
degree 8 del Pezzo surfaces, namely $X_{1}$ and
$X_{1}^{\prime}=\mathbb{P}^{1}\times\mathbb{P}^{1}$. The $6=3\times 2$ lines
in $X_{3}$ is reflecting the fact that $X_{3}$ is the blow up of
$\mathbb{P}^{2}$ in 2 different ways, which in turns is the origin of the
Cremona transformation. In terms of the representation $L_{E_{3}}$ of
$E_{3}\simeq\mathfrak{sl}(3)\times\mathfrak{sl}(2)$ we have

$L_{E_{3}}\simeq V_{3}\otimes V_{2},$

where $V_{3}$ and $V_{2}$ are the standard representations of
$\mathfrak{sl}(3)$ and $\mathfrak{sl}(2)$ respectively. On $X_{4}$, there are
10 lines and 5 rulings. Under the identification of $E_{4}$ with
$\mathfrak{sl}(5)$, we have $R_{E_{4}}\simeq V_{5}$ the standard
representation of $\mathfrak{sl}(5)$ and $L_{E_{4}}\simeq\Lambda^{3}V_{5}$. We
could also see the relationship $L_{E_{4}}\simeq\Lambda^{3}R_{E_{4}}$ from the
fact that every 3 distinct rulings on $X_{4}$ determines a unique line in
$X_{4}$ which is a bisection to each of these 3 rulings. In fact, we have a
natural bundle isomorphism

$\mathcal{L}^{E_{4}}\simeq\Lambda^{3}\mathcal{R}^{E_{4}}(K).$

On $X_{5}$ there are 16 lines and 10 rulings, which are related to a spinor
representation $S_{10}^{+}$ and the standard representation $V_{10}$ of
$E_{5}\simeq D_{5}=\mathfrak{o}(10)$. The defining quadratic form of
$\mathfrak{o}(10)$ on $V_{10}\simeq\mathbb{C}^{10}$ is reflected by the
geometric fact that given any two rulings $f_{1}$ and $f_{2}$, we have
$f_{1}\cdot f_{2}\leq 2$ and the equality sign holds if and only if
$f_{1}+f_{2}=-K$. In fact, we have a fiberwise quadratic form $q$ over
$X_{5}$,

$q\colon\ \mathcal{R}^{E_{5}}\otimes\mathcal{R}^{E_{5}}\rightarrow O(-K),$

so that the Lie algebra $E_{5}$-bundle over $X_{5}$ is the bundle of
infinitesimal symmetries of $q$ on $\mathcal{R}^{E_{5}}$ [40]. A smooth
$X_{5}$ is the complete intersection of two quadric hypersurfaces $Q_{0}$ and
$Q$ in $\mathbb{P}^{4}$. If $Q_{0}$ varies in a one parameter family
$Q_{0}(t)$ and degenerates into a union of two hyperplanes
$Q_{0}(0)=H^{\prime}\cup H^{\prime\prime}$, thus $X_{5}(0)=X_{5}^{\prime}\cup
X_{5}^{\prime\prime}$, then its geometry is governed by the reduction of the
Lie algebra $E_{5}\simeq D_{5}=\mathfrak{o}(10)$ to $A_{3}\times A_{1}\times
A_{1}\simeq D_{3}\times D_{2}=\mathfrak{o}(6)\times\mathfrak{o}(4)$. For
instance the branching rule

$S_{10}^{+}\simeq S_{6}^{+}\otimes S_{4}^{+}+S_{6}^{-}\otimes S_{4}^{-}$

is reflected by the following geometric statements: $X_{5}^{\prime}$ and
$X_{5}^{\prime\prime}$ are quadric surfaces and therefore each admits two
rulings. Lines on each ruling passing through one of the four points in the
curve $Q\cap H^{\prime}\cap H^{\prime\prime}$, which are the unstable points
for the total family $X_{5}(t)$, are exactly those lines on $X_{5}(0)$ which
will survive for nearby $X_{5}(t)$’s. Thus the $16=4\times 2+4\times 2$ lines
on $X_{5}(t)$ is reflecting the above branching rule of $S_{10}^{+}$. The
relationship between the branching rule $V_{10}\simeq V_{6}+V_{4}$ and the
geometry of rulings on $X_{5}(t)$ can be described in a similar fashion.

The relationship between the geometry of cubic surfaces $X_{6}$ and
representation theory of $E_{6}$ has been discussed in the previous section.
Recall that a smooth degree 2 del Pezzo surface $X_{7}$ is a double cover of
$\mathbb{P}^{2}$, branched along a quartic curve $Q\subset\mathbb{P}^{2}$. A
bitangent line to $Q$ in $\mathbb{P}^{2}$ determines a pair of lines in
$X_{7}$ as its double cover, thus the 28 bitangents to $Q$ gives 56 lines in
$X_{7}$. If $l$ and $l^{\prime}$ is any such pair of lines in $X_{7}$, then
$l+l^{\prime}=-K$ and we have a fiberwise non-degenerate quadratic form $q$ on
$\mathcal{L}^{E_{7}}$,

$q\colon\ \mathcal{L}^{E_{7}}\otimes\mathcal{L}^{E_{7}}\rightarrow O(-K).$

Furthermore, there is a fiberwise quartic form $f$ on $\mathcal{L}^{E_{7}}$,

$f\colon\
\mathcal{L}^{E_{7}}\otimes\mathcal{L}^{E_{7}}\otimes\mathcal{L}^{E_{7}}\otimes\mathcal{L}^{E_{7}}\rightarrow
O(-2K),$

so that its bundle of infinitesimal symmetries is the Lie algebra
$E_{7}$-bundle $\mathcal{E}^{E_{7}}$ over $X_{7}$. We also have

$\mathcal{E}^{E_{7}}\simeq\bigl{(}O^{\oplus
7}+\mathcal{R}^{E_{7}}\bigr{)}(K).$

If we degenerate the smooth quartic branch curve $Q\subset\mathbb{P}^{2}$ to a
double conic $2C$, then we obtain a family of smooth del Pezzo surface
$X_{7}(t)$ degenerating to a nonnormal surface $X_{7}(0)=X_{7}^{\prime}\cup
X_{7}^{\prime\prime}$ with $X_{7}^{\prime}\simeq X_{7}^{\prime\prime}$
$\simeq\mathbb{P}^{2}\supset C$. There are 8 unstable points on $C\subset
X_{7}(0)$. Each line in $X_{7}^{\prime}$ or $X_{7}^{\prime\prime}$ passing
through 2 of these 8 unstable points will survive on nearby $X_{7}(t)$’s, thus
giving $\binom{8}{2}+\binom{8}{2}=56$ lines on $X_{7}$. This is reflected by
the branching rule of $L_{E_{7}}$ of $E_{7}$ to its subalgebra
$A_{7}=\mathfrak{sl}(8)$,

$L_{E_{7}}\simeq\Lambda^{2}V_{8}+\Lambda^{2}V_{8}^{\ast},$

where $V_{8}\simeq\mathbb{C}^{8}$ is the standard representation of
$\mathfrak{sl}(8)$. For degree 1 del Pezzo surfaces $X_{8}$, the relationships
between their geometry with $E_{8}$ are discussed in [40].

##  5 ADE surfaces vs ADE Lie theory

In this section, the above relationships between the representation theory of
$E_{n}$ and the geometry of del Pezzo surfaces will be generalized to other
simply-laced Lie algebras, i.e., Lie algebras of type ADE [42]. We call the
corresponding surfaces ADE surfaces. In this unified description, an
$E_{n}$-surface is an $X_{n+1}$ together with a choice of a line, or
equivalently $X_{n}$ with a point in it.

First we describe $D_{n}$-surfaces. Given any ruling $f$ on $X_{n+1}$, its
linear system defines a $\mathbb{P}^{1}$-bundle on $X_{n+1}$

$\Phi_{f}\colon\ X_{n+1}\rightarrow|f|\simeq\mathbb{P}^{1}.$

For a generic $X_{n+1}$, there are $n$ singular fibers and each is a union of
two lines intersecting at a point. To see this, we note the Euler
characteristic of $X_{n+1}$ equals

$\chi(X_{n+1})=\chi\big{(}\mathbb{P}^{2}\big{)}+n+1=n+4.$

Smooth (resp. singular) fibers have Euler characteristic $2$ (resp. $3$). We
have

$\chi(X_{n+1})=\chi\big{(}\mathbb{P}^{1}\times\mathbb{P}^{1}\big{)}+\text{\\#}\
(\text{singular fibers})$

and therefore there are $n$ singular fibers [3]. Using these fiberwise lines,
i.e., $l\cdot f=0$, we construct a rank $2n$ vector bundle
$\mathcal{L}^{D_{n}}$ over $X_{n+1}$,

$\mathcal{L}^{D_{n}}=\mathop{\textstyle\bigoplus}\limits_{\begin{subarray}{c}l\cdot
K=-1\\\ l\cdot l=-1\\\ l\cdot f=0\end{subarray}}O(l).$

Notice that any two such lines $l$ and $l^{\prime}$ intersect if and only if
$l+l^{\prime}$ is a singular fiber of $\Phi_{f}$. When this happens, we have

$O(l)\otimes O(l^{\prime})\simeq O(f).$

Putting these isomorphisms together, we obtain a fiberwise quadratic form
$q_{\mathcal{L}}$ on $\mathcal{L}^{D_{n}}$ over $X_{n+1}$,

$q_{\mathcal{L}}\colon\
\mathcal{L}^{D_{n}}\otimes\mathcal{L}^{D_{n}}\rightarrow O(f).$

The bundle of infinitesimal symmetries of
$\big{(}\mathcal{L}^{D_{n}},q_{\mathcal{L}}\big{)}$ is naturally

$\mathcal{E}^{D_{n}}=O^{\oplus
n}\mathop{\textstyle\bigoplus}\limits_{\begin{subarray}{c}\alpha\cdot K=0\\\
\alpha\cdot\alpha=-2\\\ \alpha\cdot f=0\end{subarray}}O(\alpha),$

which is a $D_{n}$-Lie algebra bundle over $X_{n+1}$ so that
$\mathcal{L}^{D_{n}}$ is its representation bundle corresponding to the
standard representation of $D_{n}=\mathfrak{o}(2n)$. We call $(X_{n+1},f)$ a
$D_{n}$-surface, denoted as $X_{D_{n}}$.

In fact, the orthogonal complement of $K$ and $f$ in
$H^{2}(X_{n+1},\mathbb{Z})$ is the root lattice of type $D_{n}$

$\langle K,f\rangle^{\bot}\simeq\Lambda_{D_{n}}.$

Furthermore, this holds true for any $n$ without the restriction $n\leq 8$.

In a similar fashion, an $A_{n}$-surface $X_{A_{n}}$ is a pair $(X_{n+1},h)$
with $h\cdot K=-3$ and $h\cdot h=1$. The rank $n+1$ vector bundle

$\mathcal{L}^{A_{n}}=\mathop{\textstyle\bigoplus}\limits_{\begin{subarray}{c}l\cdot
K=-1\\\ l\cdot l=-1\\\ l\cdot h=0\end{subarray}}O(l)$

over $X_{A_{n}}$ admits a fiberwise determinant morphism

$\det\colon\
\Lambda^{n+1}\mathcal{L}^{A_{n}}\overset{\sim}{\rightarrow}O(K+3h).$

This is because

$\Phi_{h}\colon\ X_{n+1}\rightarrow|h|\simeq\mathbb{P}^{2},$

which realizes $X_{n+1}$ as a blowup of $\mathbb{P}^{2}$ at $n+1$ points and
exceptional curves for $\Phi_{h}$ are precisely those lines used to construct
$\mathcal{L}^{A_{n}}$. The bundle of infinitesimal symmetries of
$\big{(}\mathcal{L}^{A_{n}},\det\big{)}$ is naturally

$\mathcal{E}^{A_{n}}=O^{\oplus
n}\mathop{\textstyle\bigoplus}\limits_{\begin{subarray}{c}\alpha\cdot K=0\\\
\alpha\cdot\alpha=-2\\\ \alpha\cdot h=0\end{subarray}}O(\alpha),$

which is an $A_{n}$-Lie algebra bundle over $X_{A_{n}}$ so that
$\mathcal{L}^{A_{n}}$ is its representation bundle corresponding to the
standard representation of $A_{n}=\mathfrak{sl}(n+1)$. Again the orthogonal
complement of $K$ and $h$ in $H^{2}(X_{n+1},\mathbb{Z})$ is the root lattice
of type $A_{n}$

$\langle K,h\rangle^{\bot}\simeq\Lambda_{A_{n}}.$

We remark that an $E_{n}$-surface can be interpreted as $(X_{n+1},l)$, as in
the $D_{n}$ and $A_{n}$ cases. But since $l$ is an exceptional curve and
therefore it can be blown down to $X_{n}$. We also remark that a choice of
$h$, $f$ or $l$ in $X_{n+1}$ defines an $A_{n}$-surface, $D_{n}$-surface or
$E_{n}$-surface accordingly. From the Lie theoretical perspective, they
correspond to the fundamental representations $H$, $R$ and $L$ for $E_{n+1}$.
By removing the corresponding nodes in the $E_{n+1}$ Dynkin diagram, we also
obtain Dynkin diagrams of type $A$, $D$ and $E$ respectively.

An ADE surface is a surface $X_{n+1}$, $\mathbb{P}^{2}$ blown up at $n+1$
points in general position, together with a divisor $h$, $r$ or $l$ as below:

$A_{n}$-surface | $(X_{n+1},h)\text{ with }h\cdot K=-3\text{ and }h\cdot h=1$
---|---
$D_{n}$-surface | $(X_{n+1},r)\text{ with }r\cdot K=-2\text{ and }r\cdot r=0$
$E_{n}$-surface | $(X_{n+1},l)\text{ with }l\cdot K=-1\text{ and }l\cdot l=-1$

##  6 F/String theory duality

In this section, we recall a physical motivation for the construction of
$E_{n}$-bundles over del Pezzo surfaces of degree $9-n$ given in [17, 20, 21,
24]. Physically, if $G$ is a simple subgroup of $E_{8}\times E_{8}$, then
$G$-bundles are related to the duality between F-theory and heterotic string
theory. Among other things, this duality predicts that the moduli of flat
$E_{n}$-bundles over a fixed elliptic curve $\Sigma$ can be identified with
the moduli of del Pezzo surfaces with a fixed anti-canonical curve $\Sigma$.

Given any smooth elliptic curve $\Sigma$ and any ADE Lie group $G$ of adjoint
type, if $\Sigma$ is an anti-canonical curve in a $G$-surface $X$, then the
natural $\mathfrak{g}$ Lie algebra bundle $\mathcal{E}^{\mathfrak{g}}$
restricts to a flat $G$-bundle over $\Sigma$. In fact, every flat $G$-bundle
on $\Sigma$ arises this way. To state the result, we denote
$\mathcal{M}_{\Sigma}^{G}$ to be the moduli space of flat $G$-bundles over
$\Sigma$ and $\mathcal{S}_{\Sigma}^{G}$ to be the moduli space of pairs
$(X,\Sigma\in|-K_{X}|)$ with $X$ being an ADE surface of type $G$ and $\Sigma$
is an anti-canonical curve in $X$.

######  Theorem 6.1 ([42]).

Given $\Sigma$ and $G$ as above, there is an open dense embedding

$\Phi\colon\ \mathcal{S}_{\Sigma}^{G}\rightarrow\mathcal{M}_{\Sigma}^{G}$

given by the restriction of the natural $\mathfrak{g}$-bundle
$\mathcal{E}^{\mathfrak{g}}$ over $X$ to $\Sigma$. Furthermore, there is a
natural compactification $\mathcal{\bar{S}}_{\Sigma}^{G}$ of
$\mathcal{S}_{\Sigma}^{G}$ given by those surfaces $X$ equipped with
$G$-configurations and $\Phi$ extends to an isomorphism
$\Phi\colon\mathcal{\bar{S}}_{\Sigma}^{G}\overset{\sim}{\rightarrow}\mathcal{M}_{\Sigma}^{G}$.

This particular form is given in [42] and its generalization for non-simply
laced $G$ is given in [43, 44]. Such a correspondence was originally motivated
from the duality between F-theory and heterotic string theory in physics by
the work of Friedman–Morgan–Witten [24] and Donagi [20] where different proofs
of this correspondence are also given.

Let us very briefly describe how such a correspondence arises from the
physical duality between F-theory and heterotic string theory. The space-time
in F-theory is a Calabi–Yau fourfold $Z$ equipped with an elliptic K3
fibration over a complex surface $B$ and the space-time in heterotic string
theory is a Calabi–Yau threefold $Y$ equipped with an elliptic fibration over
the same complex surface $B$ and coupled with an $E_{8}\times E_{8}$ Hermitian
Yang–Mills bundle over $Y$. When the two theories are dual to each other, in a
certain adiabatic limit, the duality becomes a fiberwise duality. Namely an
elliptic K3 fiber $X$ over $b\in B$ in $Z$ is dual to an elliptic curve fiber
$\Sigma$ over $b\in B$ in $Y$ coupled with a flat $E_{8}\times E_{8}$-bundle
over $\Sigma$. To obtain a geometric correspondence, the dilaton field in the
string theory should vanish, which corresponds to a type II degeneration of
the elliptic K3 surface, that is $X$ is a fiber sum $X_{1}\\#_{\Sigma}X_{2}$,
each $X_{i}$ is a rational elliptic surface with section with $\Sigma$ being a
fiber. In particular, $X_{i}$ is a blowup of $\mathbb{P}^{2}$ at $9$ points
with one exceptional curve identified with the section, namely $X_{i}$ is an
$E_{8}$-surface as defined previously. Hence each $(X_{i},\Sigma)$ is an
$E_{8}$-surface and therefore gives an $E_{8}$-bundle over $\Sigma$ by our
above discussions. Together we obtain the flat $E_{8}\times E_{8}$-bundle over
$\Sigma$.

##  7 Generalization to Kac–Moody cases

In this section, we generalize the above results for $E_{n}$-bundles in
Theorem 6.1 to Kac–Moody $\hat{E}_{n}$-bundles in [41, Theorems 2 and 3]. If
we blowup $\mathbb{P}^{2}$ at 9 points, then the resulting surface $X_{9}$ is
no longer a del Pezzo surface. As $K\cdot K=0$, $\langle
K_{X}\rangle^{\perp}\subset H^{2}(X_{9},\mathbb{Z})$ is only non-positive
definite and there are infinitely many roots. The latter is because for any
root $\alpha$, namely $\alpha\cdot\alpha=-2$ and $\alpha\cdot K=0$,
$\alpha+nK$ is also a root for any integer $n$ [15, 46].

In fact, $\Lambda_{E_{9}}\triangleq\langle K_{X}\rangle^{\perp}$ is the root
system of the affine Kac–Moody Lie algebra $\hat{E}_{8}$, or the loop algebra
$LE_{8}$ of $E_{8}$, up to central extension. The following is the list of
Dynkin diagrams of simply-laced affine Kac–Moody Lie algebras, which coincides
with extended Dynkin diagrams of simply-laced simple Lie algebras, namely ADE
types (see [32] for details):

$\hat{A}_{n}\colon$

$\hat{D}_{n}\colon$

$\hat{E}_{6}\colon$

$\hat{E}_{7}\colon$

$\hat{E}_{8}\colon$

Figure 4: Dynkin diagrams of affine ADE types.

We could also realize other $\hat{E}_{n}$’s with $n\leq 8$ from rational
surfaces as follows: Let $C_{1},\dots,C_{8-n}$ be an $A_{8-n}$-chain of
$\left(-2\right)$-curves in $X_{9}$, i.e., $C_{i}\simeq\mathbb{P}^{1}$,
$C_{i}\cdot C_{i}=-2$, $C_{i}\cdot C_{i+1}=1$ and other $C_{i}\cdot C_{j}$’s
are zero. Then we could blow down this chain of $\left(-2\right)$-curves in
$X_{9}$ to obtain a rational surface, denoted $X_{\hat{E}_{n}}$, with a
rational singular point of type $A_{8-n}$. Equivalently $X_{\hat{E}_{n}}$ is a
singular del Pezzo surface of degree $8-n$ with a canonical singularity of
type $A_{8-n}$, for $n\leq 7$. When $n=8$, further care is needed as the
representation $L_{n}$ is no longer a miniscule representation in this case.
We call these $X_{\hat{E}_{n}}$’s $\hat{E}_{n}$-surfaces.

Then we have the following results [41] generalizing the construction of
$E_{n}$-bundles over del Pezzo surfaces and their relationships with flat
$E_{n}$-bundles over elliptic curves.

######  Theorem 7.1 ([41]).

For any $\hat{E}_{n}$-surface $X_{\hat{E}_{n}}$, the orthogonal complement of
$K$ in $H^{2}\bigl{(}X_{\hat{E}_{n}},\mathbb{Z}\bigr{)}$ is isomorphic to the
root system of the affine Kac–Moody Lie algebra $\hat{E}_{n}$, i.e.,

$\langle K\rangle^{\bot}\simeq\Lambda_{\hat{E}_{n}}.$

Furthermore, the real root system of $\hat{E}_{n}$ is
$\Delta^{\mathrm{r}e}\bigl{(}\hat{E}_{n}\bigr{)}\simeq\langle
K\rangle^{\bot}\cap\\{\alpha\cdot\alpha=-2\\}$, the imaginary root system of
$\hat{E}_{n}$ is $\Delta^{\mathrm{i}m}\bigl{(}\hat{E}_{n}\bigr{)}\simeq
Z_{\neq 0}\langle K\rangle$ and the null root is $-K$.

There is also a canonically defined $\hat{E}_{n}$-bundle
$\mathcal{E}^{\hat{E}_{n}}$ over $X_{\hat{E}_{n}}$,

$\displaystyle\mathcal{E}^{\hat{E}_{n}}$ $\displaystyle=\mathcal{O}^{\oplus
n+1}\oplus\mathop{\textstyle\bigoplus}_{\alpha\in\Delta^{\mathrm{r}e}(\hat{E}_{n})}O(\alpha)\mathop{\textstyle\bigoplus}_{\beta\in\Delta^{\mathrm{i}m}(\hat{E}_{n})}O(\beta)$
$\displaystyle\simeq\mathcal{E}^{E_{n}}\otimes\Bigl{(}\mathop{\textstyle\bigoplus}_{n\in\mathbb{Z}}O(nK)\Bigr{)}\oplus
O.$

Note that the last isomorphism is a realization of the Kac–Moody bundle as a
central extension of a loop algebra bundle over $X$, which depends on a choice
of a fixed line in $X$.

Given any fixed smooth elliptic curve $\Sigma$, let
$\mathcal{S}_{\Sigma}^{\hat{E}_{n}}$ be the moduli space of
$\hat{E}_{n}$-surfaces $X_{\hat{E}_{n}}$ containing $\Sigma$ as an anti-
canonical curve and $\mathcal{M}_{\Sigma}^{\hat{E}_{n}}$ be the moduli space
of holomorphic $\hat{E}_{n}$-bundles over $\Sigma$, in particular

$\mathcal{M}_{\Sigma}^{\hat{E}_{n}}\simeq\operatorname{Hom}\bigl{(}\Lambda_{\hat{E}_{n}},\Sigma\bigr{)}/W_{\hat{E}_{n}\times\mathbb{Z}_{2}}.$

######  Theorem 7.2 ([41]).

Given any smooth elliptic curve $\Sigma,$ there is an open embedding

$\Phi\colon\
\mathcal{S}_{\Sigma}^{\hat{E}_{n}}\rightarrow\mathcal{M}_{\Sigma}^{\hat{E}_{n}}$

given by the restriction of the canonically defined $\hat{E}_{n}$-bundle
$\mathcal{E}^{\hat{E}_{n}}$ over $X_{\hat{E}_{n}}$ to $\Sigma\subset
X_{\hat{E}_{n}}$. Furthermore, there is a natural extension
$\Phi\colon\mathcal{\bar{S}}_{\Sigma}^{\hat{E}_{n}}\overset{\sim}{\rightarrow}\mathcal{M}_{\Sigma}^{\hat{E}_{n}}$.

In above discussions, we contract an $A_{d}$-chain of $(-2)$-curves on $X_{9}$
to realize affine $E_{9-d}$ structures on rational surfaces and elliptic
curves. However, in Section 4, $E_{n}$ structures on rational surfaces and
elliptic curves also appear for considerations of $\mathbb{P}^{2}$ blown up at
$n$ points. In general we could consider $A_{d}$-chains of $(-2)$-curves on
$\mathbb{P}^{2}$ blown up at $n+1$ points. Corresponding Lie algebras are
listed in the following magic triangle of Julia [31], see Table 1.

Table 1: Julia’s magic triangle. | $n=8$ | $n=7$ | $n=6$ | $n=5$ | $n=4$ | $n=3$ | $n=2$ | $n=1$ | $n=0$
---|---|---|---|---|---|---|---|---|---
$d=11$ | + |  |  |  |  |  |  |  |
$d=10$ |  $\mathbb{R}$ or $A_{1}$ | + |  |  |  |  |  |  |
$d=9$ | $\mathbb{R}\times A_{1}$ | $\mathbb{R}$ |  |  |  |  |  |  |
$d=8$ | $A_{1}\times A_{2}$ |  $\mathbb{R}\times A_{1}$ or $A_{2}$ | $A_{1}$ |  |  |  |  |  |
$d=7$ | $E_{4}$ | $\mathbb{R}\times A_{2}$ | $\mathbb{R}\times A_{1}$ | $\mathbb{R}$ | + |  |  |  |
$d=6$ | $E_{5}$ | $A_{1}\times A_{3}$ | $\mathbb{R}\times A_{1}^{2}$ |  $\mathbb{R}^{2}$ or $A_{1}^{2}$ | $\mathbb{R}$ |  |  |  |
$d=5$ | $E_{6}$ | $A_{5}$ | $A_{2}^{2}$ | $\mathbb{R}\times A_{1}^{2}$ | $\mathbb{R}\times A_{1}$ | $A_{1}$ |  |  |
$d=4$ | $E_{7}$ | $D_{6}$ | $A_{5}$ | $A_{1}\times A_{3}$ | $\mathbb{R}\times A_{2}$ |  $\mathbb{R}\times A_{1}$ or $A_{2}$ | $\mathbb{R}$ | + |
$d=3$ | $E_{8}$ | $E_{7}$ | $E_{6}$ | $E_{5}$ | $E_{4}$ | $A_{1}\times A_{2}$ | $\mathbb{R}\times A_{1}$ |  $\mathbb{R}$ or $A_{1}$ | +

Notice that there is a symmetry between $d$ and $n$ in this magic triangle,
extending our earlier descriptions on $E_{n}$ bundles and affine $E_{n}$
bundles. The motivation of Julia is from the studies of 11-dimensional
supergravity in which the bosonic fields are pairs $(g,C)$, where $g$ is a
metric tensor on $\mathbb{R}^{1,10}$ and $C$ is a three form field on
$\mathbb{R}^{1,10}$, called the $C$-field. Consider a physical toroidal
compactification, namely one replaces $\mathbb{R}^{1,10}$ by
$\mathbb{R}^{1,10-n}\times T^{n}$ and requires the volume of $T^{n}$ to shrink
to zero size. By viewing $g$ as a family of metrics on $T^{n}$ parametrized by
$\mathbb{R}^{1,10-n}$, we might expect to obtain an effective theory which is
a sigma model for maps from $\mathbb{R}^{1,10-n}$ to
${\mathrm{S}L}(n,\mathbb{Z})\backslash{\mathrm{S}L}(n,\mathbb{R})/{\mathrm{S}O}(n)$,
as the latter space parametrizes Einstein metrics on $T^{n}$. However, the
$C$-field will also decompose and regroup with components of $g$ to enhance
the sigma model to $E_{n,\mathbb{Z}}\backslash E_{n}^{\mathrm{s}plit}/K_{n}$
where $K_{n}$ is the maximal compact subgroup of the split Lie group
$E_{n}^{\mathrm{s}plit}$ of type $E_{n}$. Various structures that we mentioned
on ADE structures would have their counterparts in this supergravity theory,
which is a fascinating connection between algebraic geometry and physics.

##  8 ADE bundles over surfaces with ADE singularities

In this section, we explain that ADE singularities on a surface $X$ with
$q=p_{g}=0$ leads to ADE bundles over $X$, as stated in Theorem 8.1. Suppose
$X^{\prime}$ is a singular surface with $q(X^{\prime})=0$ and with a simple
singularity $p$ and $X$ is its minimal resolution with exceptional divisor
$C=\bigcup_{i=1}^{r}C_{i}$. As explained in Section 2, each irreducible
component $C_{i}$ is an $(-2)$-curve, i.e., $C_{i}\simeq\mathbb{P}^{1}$ with
$C_{i}\cdot C_{i}=-2$. In particular, $C_{i}\cdot K=0$ by the adjunction
formula. The dual graph for the configuration of these $C_{i}$’s is a Dynkin
diagram of ADE type, thus there is a corresponding simple Lie algebra
$\mathfrak{g}$ of ADE type, and we also call the corresponding singularity $p$
an ADE singularity. The $\mathbb{Z}$-span of the $C_{i}$’s is a root lattice
$\Lambda_{\mathfrak{g}}$ of type $\mathfrak{g}$ inside $H^{2}(X,\mathbb{Z})$,

$\mathbb{Z}\langle C_{1},\dots,C_{r}\rangle=\Lambda_{\mathfrak{g}}\subset
H^{2}(X,\mathbb{Z}).$

We denote the set of roots in $\Lambda_{\mathfrak{g}}$ as $\Phi$, i.e.,
$\alpha\in\Lambda_{\mathfrak{g}}$ lies in $\Phi$ if $\alpha\cdot\alpha=-2$. We
write $\Phi=\Phi_{+}\amalg\Phi_{-}$, where $\alpha\in\Phi_{+}$ if
$\alpha=\sum{n_{i}C_{i}}$ with $n_{i}\leq 0$ and $\Phi_{-}=-\Phi_{+}$. Since
$q(X)=0$, every class in $H^{2}(X,\mathbb{Z})\simeq\operatorname{Pic}(X)$
corresponds to a unique line bundle over $X$ up to isomorphisms. Similar to
our earlier constructions, we could construct a Lie algebra bundle of ADE type
$\mathfrak{g}$ over $X$ as follows:

$\mathcal{E}^{\mathfrak{g}}=O_{X}^{\oplus
r}\mathop{\textstyle\bigoplus}\limits_{\alpha\in\Phi}O_{X}(\alpha).$

It is natural to ask whether this ADE bundle $\mathcal{E}^{\mathfrak{g}}$ over
$X$ can be descended to the original surface $X^{\prime}$ which admits the ADE
singularity $p$. Namely surfaces with an ADE singularity has a natural ADE
bundle over it. In [25], Friedman and Morgan showed that it is possible for
del Pezzo surfaces after small deformations of $\mathcal{E}^{\mathfrak{g}}$.
In [13], we gave a direct construction of these deformations, which also works
for general surfaces $X^{\prime}$ satisfying $q=p_{g}=0$.

In the simplest case where $p$ is an $A_{1}$ singularity, namely locally
$\mathbb{C}^{2}/\\{\pm 1\\}$, there is only one $(-2)$-curve $C_{1}$ in the
exceptional locus and we have $\mathcal{E}^{A_{1}}=O_{X}\oplus
O_{X}(-C_{1})\oplus O_{X}(C_{1})={\mathrm{E}nd}_{0}(O_{X}\oplus
O_{X}(C_{1}))$, the bundle of traceless endomorphisms of $O_{X}\oplus
O_{X}(C_{1})$. Restricting to $C_{1}\subset X$ we have

$(O_{X}\oplus O_{X}(C_{1}))|_{C_{1}}\simeq O_{\mathbb{P}^{1}}\oplus
O_{\mathbb{P}^{1}}(-2),$

which admits a nontrivial deformation as an extension

$0\rightarrow O_{\mathbb{P}^{1}}(-2)\rightarrow O_{\mathbb{P}^{1}}(-1)\oplus
O_{\mathbb{P}^{1}}(-1)\rightarrow O_{\mathbb{P}^{1}}\rightarrow 0.$

Using $p_{g}(X)=0$, we could lift this extension from $C_{1}$ to $X$. The
corresponding deformations of $\mathcal{E}^{A_{1}}$ would then be trivial
along $C_{1}$ and therefore can be descended to $X^{\prime}$ as a Lie algebra
bundle.

In the general case, we have the following result about these Lie algebra
bundles $\mathcal{E}^{\mathfrak{g}}$ through studying their minuscule
representation bundles in terms of $(-1)$-curves in $X$.

######  Theorem 8.1 ([13]).

Let $p$ be an ADE singularity of a surface $X^{\prime}$ with
$q(X^{\prime})=p_{g}(X^{\prime})=0$ and $X$ be its minimal resolution with
exceptional curve $C=\bigcup_{i=1}^{r}C_{i}$. Then

  1. $(i)$

given any
$(\varphi_{C_{i}})_{i=1}^{n}\in\Omega^{0,1}\big{(}X,\bigoplus_{i=1}^{n}O(C_{i})\big{)}$
with $\overline{\partial}\varphi_{C_{i}}=0$ for every $i$, it can be extended
to
$\varphi=(\varphi_{\alpha})_{\alpha\in\Phi^{-}}\in\Omega^{0,1}\big{(}X,\bigoplus_{\alpha\in\Phi^{-}}O(\alpha)\big{)}$
such that
$\overline{\partial}_{\varphi}:=\overline{\partial}+\operatorname{ad}(\varphi)$
is a holomorphic structure on $\mathcal{E}^{\mathfrak{g}}$. We denote this new
holomorphic bundle as $\mathcal{E}_{\varphi}^{\mathfrak{g}}$;

  2. $(ii)$

such a $\overline{\partial}_{\varphi}$ is compatible with the Lie algebra
structure;

  3. $(iii)$

$\mathcal{E}_{\varphi}^{\mathfrak{g}}$ is trivial on $C_{i}$ if and only if
$[\varphi_{C_{i}}|_{C_{i}}]\neq 0\in
H^{1}(C_{i},O_{C_{i}}(C_{i}))\cong\mathbb{C}$;

  4. $(iv)$

there exists $[\varphi_{C_{i}}]\in H^{1}(X,O(C_{i}))$ such that
$[\varphi_{C_{i}}|_{C_{i}}]\neq 0$;

  5. $(v)$

such a $\mathcal{E}_{\varphi}^{\mathfrak{g}}$ can descend to $X^{\prime}$ if
and only if $[\varphi_{C_{i}}|_{C_{i}}]\neq 0$ for every $i$.

##  9 Relation to flag varieties of ADE type

In this section, we explain a relationship between our $G$-bundles over a
surface $X$ with a tautological $G$-bundle over the flag variety $G/B$ and its
cotangent bundle $G\times\mathfrak{n}/B$, as given in [16, Theorems 5, 6 and
7]. An ADE singularity of type $\mathfrak{g}$ is locally given by the
intersection of the transversal slice $S_{x}$ of a subregular nilpotent
element $x$ and the nilpotent variety $N(\mathfrak{g})$ of the complex Lie
algebra $\mathfrak{g}$. Recall that $N(\mathfrak{g})$ is the fiber over zero
of the adjoint quotient
$\mathfrak{g}\rightarrow\mathfrak{g/}G\simeq\mathfrak{t}/W$, where
$\mathfrak{t}$ is the Cartan subalgebra of $\mathfrak{g}$ and $W$ is the
corresponding Weyl group. Furthermore, the restriction of the adjoint quotient
$\mathfrak{g}\rightarrow\mathfrak{t}/W$ to the transversal slice $S_{x}$ is a
semiuniversal deformation of the corresponding $ADE$ singularity. This result
is conjectured by Grothendieck and proved by Brieskorn in 1970 [6]. After
that, Grothendieck defined a morphism
$G\times\mathfrak{b}/B\rightarrow\mathfrak{t}$ and gave a simultaneous
resolution of the adjoint quotient $\mathfrak{g}\rightarrow\mathfrak{t}/W$
using it. The restriction of the Grothendieck resolution to the above
transversal slice $S_{x}$ is also a simultaneous resolution [54]. In 1969,
Springer gave a resolution of singularities for the nilpotent variety
$N(\mathfrak{g})$ through $G\times\mathfrak{n}/B\rightarrow N(\mathfrak{g})$.
Note that $G\times\mathfrak{n}/B\cong T^{\ast}(G/B)$ is the cotangent bundle
of the flag variety $G/B$. The connection among these resolutions can be shown
in the following Brieskorn–Slodowy–Grothendieck diagram (here $\widetilde{S}$
is the minimal resolution of $S$ and $C=\bigcup_{i=1}^{r}C_{i}$ is the
exceptional locus with each $C_{i}$ irreducible component):

$\begin{array}[]{l}\begin{array}[]{*{20}c}{C=\bigcup_{i=1}^{r}C_{i}}&{\subset}&{\widetilde{S}}&{\longrightarrow}&{S=N(\mathfrak{g})\cap
S_{x}}\\\ \cap&{}\hfil&\cap&{}\hfil&\cap\\\
{G/B}&{\subset}&{G\times\mathfrak{n}/B}&{\longrightarrow}&{N(\mathfrak{g})}\\\
{}\hfil&{}\hfil&\cap&{\kern 1.0pt}&\cap\\\
{}\hfil&{}\hfil&{G\times\mathfrak{b}/B}&{\longrightarrow}&{\mathfrak{g}}\\\
{}\hfil&{}\hfil&\downarrow&{\kern 1.0pt}&\downarrow\\\
{}\hfil&{}\hfil&{\mathfrak{t}}&{\longrightarrow}&{\mathfrak{t}/W}.\\\
\end{array}\\\ \end{array}$

Under the above background, we consider the associated Lie algebra bundles
$G\times\mathfrak{g}/B$ over $G/B$ and
$G\times\mathfrak{n}\times\mathfrak{g}/B$ over $G\times\mathfrak{n}/B$
respectively. It is obvious that these bundles are trivial as the action of
$B$ on $\mathfrak{g}$ can extend to the whole $G$. We [16] describe natural
holomorphic filtration structures on these bundles explicitly.

As $B$ is a solvable Lie group, the associated representation bundle
$G\times\mathfrak{g}/B$ over $G/B$ is an iterated extension of holomorphic
line bundles. Also for the full flag variety $G/B$, we have
$\operatorname{Pic}(G/B)=\Lambda$, where $\Lambda$ is the weight lattice of
the Lie algebra $\mathfrak{g}$. Hence for every $\lambda\in\Lambda$, we can
associate a line bundle $L_{\lambda}$ over $G/B$. From the Borel–Weil–Bott
theorem, we can compute some particular cases of cohomology of line bundles
over $G/B$ easily. Note that our choice of $B$ gives rise to $\Phi_{+}$, the
set of positive roots, such that
$\mathfrak{b}=\mathfrak{t}\oplus\bigoplus_{\alpha\in\Phi_{+}}\mathfrak{g}_{\alpha}$.
We let $\\{\alpha_{1},\dots,\alpha_{r}\\}$ be the simple roots, then for any
root $\alpha\in\Phi$, we have

  1. $(i)$

$H^{i}(G/B,L_{\alpha})=0~{}~{}~{}$for any$~{}i\geq 2$;

  2. $(ii)$

$H^{1}(G/B,L_{\alpha})\cong\mathbb{C}$ if $\alpha=-\alpha_{i}$ for some simple
root $\alpha_{i}$ and $H^{1}(G/B,L_{\alpha})=0$ otherwise;

  3. $(iii)$

the restriction map $H^{1}(G/B,L_{-\alpha_{i}})\rightarrow
H^{1}(C_{i},L_{-\alpha_{i}}|_{C_{i}})\cong\mathbb{C}$ is an isomorphism for
every simple root $\alpha_{i}$. Hence $[\varphi_{-\alpha_{i}}|_{C_{i}}]\neq 0$
if and only if $[\varphi_{-\alpha_{i}}]\neq 0\in H^{1}(G/B,L_{-\alpha_{i}})$.

Now we try to write the holomorphic structures on $G\times\mathfrak{g}/B$
explicitly. The filtration of the representation $\mathfrak{g}$ is given by
the Chevalley order of its weights, hence not unique and we will choose an
arbitrary one. Then the holomorphic structure $\overline{\partial}_{\varphi}$
on $G\times\mathfrak{g}/B$ can be written in an upper-triangular form with
respect to the holomorphic structure $\overline{\partial}$ on the graded
vector bundle associated to this filtration. Note that for a homogenous space
$G/P$, a vector bundle $V$ on $G/P$ is trivial if and only if the restriction
of $V$ to every Schubert line is trivial [47]. Back to our cases, the Schubert
lines in $G/B$ are given by $C_{i}=P_{\alpha_{i}}/B$, where $\alpha_{i}$’s run
through all the simple roots, and $P_{\alpha_{i}}$ is the parabolic subgroup
of $G$ corresponding to $\alpha_{i}$. The main result is as follows:

######  Theorem 9.1 ([16]).

For the Lie algebra bundle $(G\times\mathfrak{g}/B,\,[\,,\,])$ over $G/B$ with
holomorphic structure $\overline{\partial}_{\varphi}$ as above, we have:

  1. $(i)$

$\overline{\partial}_{\varphi}[\,,\,]=0$ if and only if
$\overline{\partial}_{\varphi}=\overline{\partial}+\sum_{\alpha\in\Phi^{-}}\operatorname{ad}(\varphi_{\alpha})$
with $\varphi_{\alpha}\in\Omega^{0,1}(G/B,L_{\alpha})$ for some
$\alpha\in\Phi^{-}$.

  2. $(ii)$

The bundle
$\bigl{(}G\times\mathfrak{g}/B,\overline{\partial}_{\varphi}=\overline{\partial}+\sum_{\alpha\in\Phi^{-}}\operatorname{ad}(\varphi_{\alpha})\bigr{)}$
is holomorphically trivial if and only if
$[\varphi_{-\alpha_{i}}|_{C_{i}}]\neq 0$ for every simple root $\alpha_{i}$.

  3. $(iii)$

The holomorphic structure of $(G\times\mathfrak{g}/B,\,[\,,\,])$ over $G/B$ is
$\overline{\partial}_{\varphi}=\overline{\partial}_{0}+\sum_{\alpha\in\Phi^{-}}\operatorname{ad}(\varphi_{\alpha})$
with $[\varphi_{-\alpha_{i}}]\neq 0$ for every simple root $\alpha_{i}$.

Consider the holomorphic structure of
$G\times\mathfrak{n}\times\mathfrak{g}/B$ over $G\times\mathfrak{n}/B\cong
T^{\ast}(G/B)$ when $\mathfrak{g}$ is of $ADE$ type. First, we know that
$G\times\mathfrak{n}\times\mathfrak{g}/B$ is an iterated extension of line
bundles over $G\times\mathfrak{n}/B$ as $B$ is solvable. And any line bundle
over $G\times\mathfrak{n}/B$ is the pull back of a line bundle over $G/B$
through the projection map $\pi\colon T^{\ast}(G/B)\cong
G\times\mathfrak{n}/B\rightarrow G/B$. Denote
$\mathfrak{L}_{\lambda}:=\pi^{\ast}L_{\lambda}$ to be the corresponding line
bundle over $G\times\mathfrak{n}/B$ for any weight $\lambda\in\Lambda$. Denote
$H^{i}(\lambda):=H^{i}(G\times\mathfrak{n}/B,\mathfrak{L}_{\lambda})$ for
convenience. Using cohomology of line bundles on the cotangent bundle of the
flag variety [7, 8, 28], we have:

  1. $(i)$

for any positive root $\alpha\in\Phi^{+}$, $H^{i}(\alpha)=0$ for all $i\geq
1$;

  2. $(ii)$

for any negative root $\alpha\in\Phi^{-}$, $H^{1}(\alpha)\neq 0$,
$H^{2}(\alpha)=0$;

  3. $(iii)$

the restriction map
$H^{1}(G\times\mathfrak{n}/B,\mathfrak{L}_{-\alpha_{i}})\rightarrow
H^{1}(G/B,L_{-\alpha_{i}})$ is surjective for every simple root $\alpha_{i}$.

######  Theorem 9.2 ([16]).

The holomorphic structure of
$(G\times\mathfrak{n}\times\mathfrak{g}/B,\,[\,,\,])$ over
$G\times\mathfrak{n}/B$ is
$\overline{\partial}_{\varphi}=\overline{\partial}+\sum_{\alpha\in\Phi^{-}}\operatorname{ad}(\varphi_{\alpha})$
with $[\varphi_{-\alpha_{i}}|_{G/B}]\neq 0\in H^{1}(G/B,L_{-\alpha_{i}})$ for
every simple root $\alpha_{i}$.

Since the minimal resolution $\widetilde{S}$ of the $ADE$ singular surface $S$
is contained in $G\times\mathfrak{n}/B$, we also consider the restriction of
the $\mathfrak{g}$-bundle $G\times\mathfrak{n}\times\mathfrak{g}/B$ from
$G\times\mathfrak{n}/B$ to $\widetilde{S}$. Note that $\widetilde{S}$ is the
minimal resolution of the $ADE$ singular surface $S=\mathbb{C}^{2}/\Gamma$. It
is obvious that this $\mathfrak{g}$-bundle over $\widetilde{S}$ is also an
iterated extension of line bundles. The Picard group of $\widetilde{S}$ is a
free abelian group generated by divisors dual to the irreducible curves
$C_{i}$ [22], i.e., $\operatorname{Pic}(\widetilde{S})=\mathbb{Z}\langle
D_{i}\rangle$ with each $D_{i}$ dual to $C_{i}$.

As before, we know that the irreducible curves $C_{i}=P_{\alpha_{i}}/B$ are
Schubert lines in $G/B$, where $\alpha_{i}$’s run through all the simple
roots. Now for any weight $\lambda$, we consider the restriction of the line
bundle $L_{\lambda}$ from $G/B$ to $C_{i}$ and it is easy to see that
$L_{\lambda}|_{C_{i}}\cong\mathcal{O}_{\mathbb{P}^{1}}(\langle\lambda,\alpha_{i}\rangle)$.
For the restriction of the line bundle
$\mathfrak{L}_{\lambda}=\pi^{\ast}L_{\lambda}$ from $G\times\mathfrak{n}/B$ to
$\widetilde{S}$, we know that for any root $\alpha=\sum n_{i}\alpha_{i}$,
$\mathfrak{L}_{\alpha}|_{\widetilde{S}}\cong\mathcal{O}_{\widetilde{S}}\bigl{(}\sum-
n_{i}C_{i}\bigr{)}$.

######  Theorem 9.3 ([16]).

The restriction of the $\mathfrak{g}$-bundle
$G\times\mathfrak{n}\times\mathfrak{g}/B$ from $G\times\mathfrak{n}/B$ to
$\widetilde{S}$ is

$\biggl{(}\mathcal{O}^{\oplus r}\oplus\bigoplus_{(\sum
n_{i}C_{i})^{2}=-2}\mathcal{O}\Bigl{(}\sum
n_{i}C_{i}\Bigl{)},\,\overline{\partial}_{\varphi}=\overline{\partial}+\sum_{\alpha\in\Phi^{-}}\operatorname{ad}(\varphi_{\alpha})\biggr{)}$

with $[\varphi_{-\alpha_{i}}]\neq 0\in
H^{1}\big{(}\widetilde{S},\mathcal{O}(C_{i})\big{)}\cong\mathbb{C}$ for every
simple root $\alpha_{i}$.

We note that the holomorphic structures described here have the same form as
the holomorphic structures constructed in Section 8.

##  10 Generalization to affine ADE bundles

In this section, we will generalize the results for $ADE$ bundles in Section 8
to affine $ADE$ bundles and obtain [15, Theorem 8]. For convenience, we will
call a curve $C=\cup C_{i}$ in a surface $X$ an $ADE$ $($resp. affine $ADE)$
curve of type $\mathfrak{g}$ $($resp. $\widehat{\mathfrak{g}})$ if each
$C_{i}$ is a smooth $(-2)$-curve in $X$ and the dual graph of $C$ is a Dynkin
diagram of the corresponding type. By Kodaira’s classification [34, 35, 36] of
fibers of relative minimal elliptic surfaces, every singular fiber is a affine
$ADE$ curve unless it is rational with a cusp, tacnode or triplepoint
$($corresponding to type $II$ or $III\big{(}\widehat{A}_{1}\big{)}$ or
$VI\big{(}\widehat{A}_{2}\big{)}$ in Kodaira’s notations$)$, which can also be
regarded as a degenerated affine $ADE$ curve of type $\widehat{A}_{0}$,
$\widehat{A}_{1}$ or $\widehat{A}_{2}$ respectively. We will not distinguish
affine $ADE$ curves from their degenerated forms since they have the same
intersection matrices. We also call the affine $ADE$ curves as Kodaira curves.

Suppose $C=\cup_{i=0}^{r}C_{i}$ is a affine $ADE$ curve of type
$\widehat{\mathfrak{g}}$ in $X$ with $C_{0}$ corresponding to the extended
root, then $\cup_{i=1}^{r}C_{i}$ will be the corresponding $ADE$ curve of type
$\mathfrak{g}$ and

$\Phi:=\biggl{\\{}\alpha=\Bigl{[}\sum_{i\neq 0}a_{i}C_{i}\Bigr{]}\in
H^{2}(X,\mathbb{Z})\,|\,\alpha^{2}=-2\biggr{\\}}$

is the root system of $\mathfrak{g}$. As before, we have a
$\mathfrak{g}$-bundle

$\mathcal{E}^{\mathfrak{g}}=O^{\oplus
r}\oplus\bigoplus_{\alpha\in\Phi}O(\alpha).$

Also, there exists unique $n_{i}$’s up to overall scalings such that $F:=\sum
n_{i}C_{i}$ satisfies $F\cdot F=0$. We know

$\Phi_{\widehat{\mathfrak{g}}}:=\\{\alpha+nF\,|\,\alpha\in\Phi,n\in\mathbb{Z}\\}\cup\\{nF\,|\,n\in\mathbb{Z},n\neq
0\\}$

is a affine root system and it decomposes into union of positive and negative
roots, i.e.,
$\Phi_{\widehat{\mathfrak{g}}}=\Phi_{\widehat{\mathfrak{g}}}^{+}\cup\Phi_{\widehat{\mathfrak{g}}}^{-}$,
where

$\displaystyle\Phi_{\widehat{\mathfrak{g}}}^{-}$ $\displaystyle=\Bigl{\\{}\sum
a_{i}C_{i}\in\Phi_{\widehat{\mathfrak{g}}}\,|\,a_{i}\geq 0\text{ for all
}i\Bigr{\\}}$
$\displaystyle=\\{\alpha+nF\,|\,\alpha\in\Phi^{+},\,n\in\mathbb{Z}_{\geq
1}\\}\cup\\{\alpha+nF\,|\,\alpha\in\Phi^{-},\,n\in\mathbb{Z}_{\geq
0}\\}\cup\\{nF\,|\,n\in\mathbb{Z}_{\geq 1}\\}$

and $\Phi_{\widehat{\mathfrak{g}}}^{+}=-\Phi_{\widehat{\mathfrak{g}}}^{-}$.
Then there is a canonically defined $\widehat{\mathfrak{g}}$-bundle
$\mathcal{E}^{\widehat{\mathfrak{g}}}$ over $X$,

$\displaystyle\mathcal{E}^{\widehat{\mathfrak{g}}}=\mathcal{E}^{\mathfrak{g}}\otimes\Bigl{(}\mathop{\textstyle\bigoplus}_{n\in\mathbb{Z}}O(nF)\Bigr{)}\oplus
O.$

We remark that if we remove the central extension part $O$ in
$\mathcal{E}^{\widehat{\mathfrak{g}}}$, then this is a loop algebra
$L\mathfrak{g}$ bundle over $X$ and the Lie algebra bundle structure is
independent of the choice of the affine root $C_{0}$ in $F$. Similar to
Section 8, we have the following results:

######  Theorem 10.1 ([15]).

Given any complex surface $X$ with $p_{g}=0$. If $X$ has a Kodaira curve
$C=\cup_{i=0}^{r}C_{i}$ of type $\widehat{\mathfrak{g}}$, then

  1. $(i)$

given any
$(\varphi_{C_{i}})_{i=0}^{r}\in\Omega^{0,1}\big{(}X,\bigoplus_{i=0}^{r}O(C_{i})\big{)}$
with $\overline{\partial}\varphi_{C_{i}}=0$ for every $i$, it can be extended
to
$\varphi=(\varphi_{\alpha})_{\alpha\in\Phi_{\widehat{\mathfrak{g}}}^{-}}\in\Omega^{0,1}\big{(}X,\bigoplus_{\alpha\in\Phi_{\widehat{\mathfrak{g}}}^{-}}O(\alpha)\big{)}$
such that
$\overline{\partial}_{\varphi}:=\overline{\partial}+\operatorname{ad}(\varphi)$
is a holomorphic structure on $\mathcal{E}^{\widehat{\mathfrak{g}}}$. We
denote the new bundle as $\mathcal{E}_{\varphi}^{\widehat{\mathfrak{g}}}$;

  2. $(ii)$

$\overline{\partial}_{\varphi}$ is compatible with the Lie algebra structure
on $\mathcal{E}^{\widehat{\mathfrak{g}}}$;

  3. $(iii)$

$\mathcal{E}_{\varphi}^{\widehat{\mathfrak{g}}}$ is trivial on $C_{i}$ if and
only if $[\varphi_{C_{i}}|_{C_{i}}]\neq 0\in
H^{1}(C_{i},O_{C_{i}}(C_{i}))\cong\mathbb{C}$;

  4. $(iv)$

there exists $[\varphi_{C_{i}}]\in H^{1}(X,O(C_{i}))$ such that
$[\varphi_{C_{i}}|_{C_{i}}]\neq 0$.

##  11 Deformability of Lie algebra bundles
and geometry of rational surfaces

In this section, we discuss the hidden geometry underlying the deformability
of the affine $E_{8}$ bundle over $X_{9}$, as stated in [14, Theorems 9 and
10]. For an ADE curve $C$ in a surface $X$ with $p_{g}=0$, the corresponding
Lie algebra bundle $\mathcal{E}^{\mathfrak{g}}$ over $X$ admits a deformation
which can be descended to the surface obtained by blowing down $C$ in $X$. On
the other hand, an affine ADE curve can never be blown down. Nevertheless, we
could explain the geometric meaning of such deformabilities as below.

Recall that we have an $E_{n}$ Lie algebra bundle $\mathcal{E}^{E_{n}}$ over
$X_{n}$ for $n\leq 8$. When $n=9$, $E_{9}$ is the affine Kac–Moody algebra of
$E_{8}$, i.e., $E_{9}=\hat{E}_{8}$. When $H^{2}(X,\mathbb{Z})$ has a
sublattice $\Lambda_{\mathfrak{g}}$ isomorphic to the root lattice of a simple
Lie algebra $\mathfrak{g}$, then our construction also gives a $\mathfrak{g}$
Lie algebra bundle $\mathcal{E}^{\mathfrak{g}}$ over $X$. Infinitesimal
deformations of $\mathcal{E}^{\mathfrak{g}}$ as a $\mathfrak{g}$-bundle are
parametrized by $H^{1}(X,\operatorname{ad}(\mathcal{E}^{\mathfrak{g}}))\simeq
H^{1}(X,\mathcal{E}^{\mathfrak{g}})\subset
H^{1}(X,\operatorname{End}(\mathcal{E}^{\mathfrak{g}}))$. We say $\mathcal{E}$
is $(i)$ fully deformable if there is a base $\Delta\subset\Phi$ of the root
system $\Lambda_{g}$ such that $H^{1}(X,O(\alpha))\neq 0$ for every
$\alpha\in\Delta$, $(ii)$ $\mathfrak{h}$-deformable with $\mathfrak{h}$ a Lie
subalgebra of $\mathfrak{g}$ if there exists a strict $\mathfrak{h}$-subbundle
of $\mathcal{E}$ which is fully deformable, $(iii)$ totally non-deformable if
$H^{1}(X,O(\alpha))=0$ for every $\alpha\in\Delta$, $(iv)$ deformable in
$\alpha$-direction for $\alpha\in\Phi$ if $H^{1}(X,O(\alpha))\neq 0$.

We proved the following results.

######  Theorem 11.1 ([14]).

On $X_{9}$, a blowup of $\mathbb{P}^{2}$ at $9$ points, these points are in
general position in $\mathbb{P}^{2}$ if and only if $\mathcal{E}^{E_{9}}$ is
totally non-deformable.

######  Theorem 11.2 ([14]).

If $-K$ is nef on $X_{9}$, then

  1. $(i)$

there exists an ADE curve $C\subset X_{9}$ of type $\mathfrak{g}$ if and only
if $\mathcal{E}^{E_{9}}$ is $\mathfrak{g}$-deformable;

  2. $(ii)$

there exists a affine ADE curve $C\subset X_{9}$ of type
$\widehat{\mathfrak{g}}$ if and only if $\mathcal{E}^{E_{9}}$ is
$\widehat{\mathfrak{g}}$-deformable;

  3. $(iii)$

$X_{9}$ admits an elliptic fibration structure with a multiple fiber of
multiplicity $m$ if and only if $\mathcal{E}^{E_{9}}$ is deformable in the
$(-mK)$-direction, but not in $(-m+1)K$-direction.

##  12 Cox rings of ADE surfaces

In this section, we explain another mysterious relationship between del Pezzo
surfaces $X_{n}$ and the Lie group $G$ of type $E_{n}$ discovered by Batyrev
and Popov [2] relating the Cox ring of $X_{n}$ and the flag variety
$G/P_{L}\subset\mathbb{P}(L)$. We also establish the the corresponding results
for any ADE surface in [45, Theorem 11]. Very loosely speaking, the Cox ring
of a variety $X$ with $q(X)=0$ and torsion free $H^{2}(X,\mathbb{Z})$ is the
sum of spaces of sections of all line bundles on $X$,

$\operatorname{Cox}(X)\sim\mathop{\textstyle\bigoplus}\limits_{[L]\in\operatorname{Pic}(X)}H^{0}(X,L),$

with the ring structure given by the tensor products of sections. As elements
in $\operatorname{Pic}(X)$ are only isomorphism classes of line bundles, in
order to define $\operatorname{Cox}(X)$ properly, one needs to fix a
collection of line bundles $L_{i}$’s with $i=1,\dots,b$ (with $b=b_{2}(X)$)
whose first Chern classes represent a basis of $H^{2}(X,\mathbb{Z})$, then the
correct definition of the Cox ring of $X$ with respect to this choice is

$\operatorname{Cox}(X)=\mathop{\textstyle\bigoplus}\limits_{n_{i}\in\mathbb{Z}}H^{0}\bigl{(}X,L_{1}^{\otimes
n_{1}}\otimes L_{2}^{\otimes n_{2}}\otimes\cdots\otimes L_{b}^{\otimes
n_{b}}\bigr{)}.$

Then the ring structure is simply given by the tensor products of holomorphic
sections.

When $X$ is a del Pezzo surface, the ample line bundle $K^{-1}$ defines a
grading on $\operatorname{Cox}(X)$. We have seen before that the geometry of a
degree $d$ del Pezzo surface is closely related to the Lie algebra $E_{n}$
with $n=9-d$. For example the number of lines in $X$ is equal to the dimension
of the fundamental representation $L$ for $n<8$. It turns out that the flag
variety $G/P_{L}$ corresponding to $L$, namely the unique closed $G$-orbit in
$\mathbb{P}(L)$, is closely related to the Cox ring of $X$ as follows: When
the degree $d$ of $X$ satisfies $d\leq 5$, Batyrev–Popov [2], Derenthal [18],
Serganova–Skorobogatov [52, 53] showed that there is a natural embedding of
the projective spectrum of the graded ring $\operatorname{Cox}(X)$ into
$G/P_{L}$:

$\operatorname{Proj}(\operatorname{Cox}(X))\hookrightarrow G/P_{L}.$

It was observed in [45] that this relationship can be easily generalized to
all ADE surfaces $X_{G}$. Recall that an ADE surfac $X_{G}$ of rank $n$ is a
blowup $X_{n+1}$ of $\mathbb{P}^{2}$ at $n+1$ distinct points, together with a
rational curve $C$ in $X_{n+1}$ whose class in $H^{2}(X_{n+1},\mathbb{Z})$ is
$(i)$ $h$ satisfying $h\cdot h=1$ for type $A$, $(ii)$ $f$ satisfying $f\cdot
f=0$ for type $D$ and $(iii)$ $l$ satisfying $l\cdot l=-1$ for type $E$. We
have $\langle K,C\rangle^{\bot}\subset H^{2}(X_{n+1},\mathbb{Z})$ is a root
lattice $\Lambda_{G}$ of corresponding ADE type. In particular, the blow down
of the $(-1)$-curve $l$ in an $E_{n}$-surface is a del Pezzo surface $X$ of
degree $d=9-n$ and
$\operatorname{Cox}(X)=\mathop{\textstyle\bigoplus}_{L\in\langle
l\rangle^{\bot}}H^{0}(X_{En},L)$ defined loosely as before. Notice that
$H^{2}(X,\mathbb{Z})\simeq\langle l\rangle^{\bot}\subset
H^{2}(X_{n+1},\mathbb{Z})$.

Similarly we define a generalization of the Cox ring for an ADE surface
$X_{G}=(X_{n+1},C)$ of type $G$ as follows:

$\operatorname{Cox}^{G}(X_{G})=\mathop{\textstyle\bigoplus}\limits_{L\in\langle
C\rangle^{\bot}}H^{0}(X_{n+1},L).$

We also define the flag variety of $G$ corresponding to the fundamental
representation $L$ as $G/P_{L}$, i.e., $G/P_{L}$ is the unique closed
$G$-orbit in $\mathbb{P}(L)$. Then we have the following statement:

######  Theorem 12.1 ([45]).

For any ADE surface $X_{G}=(X_{n+1},C)$ of type $G$, we have

$\operatorname{Proj}\bigl{(}\operatorname{Cox}^{G}(X_{G})\bigr{)}\hookrightarrow
G/P_{L},$

with $n\geq 4$ for type $E_{n}$ cases.

##  13 Conclusions

We have discussed several intriguing relationships between the geometry of
surfaces and Lie theory. A couple of these are related to physics, namely the
duality between F-theory and heterotic string theory and supergravity theory
in eleven dimensions. There is also a mysterious duality between the geometry
of del Pezzo surfaces and toroidal compactification of M-theory in physics, as
proposed by Iqbal, Neitzke and Vafa in [30]. We expect that more surprising
connections will be uncovered in the future.

### Acknowledgements

Authors thank Jia Jin Zhang for many discussions and collaborations over the
years. And we also want to thank the referees for providing constructive
comments and help in improving the contents of this paper. The work of N.
Leung described in this paper was substantially supported by grants from the
Research Grants Council of the Hong Kong Special Administrative Region, China
(Project No. CUHK14301619 and CUHK14306720) and a direct grant from CUHK
(Project No. CUHK4053400). The work of Y.X. Chen described in this paper was
supported by Natural Science Foundation of Shanghai (No.19ZR1411700).

## References

  * [1] Barth W., Peters C., Van de Ven A., Compact complex surfaces, Ergeb. Math. Grenzgeb. (3), Vol. 4, Springer-Verlag, Berlin, 1984.
  * [2] Batyrev V.V., Popov O.N., The Cox ring of a del Pezzo surface, in Arithmetic of Higher-Dimensional Algebraic Varieties (Palo Alto, CA, 2002), Progr. Math., Vol. 226, Birkhäuser Boston, Boston, MA, 2004, 85–103, arXiv:math.AG/0309111.
  * [3] Beauville A., Complex algebraic surfaces, 2nd ed., London Math. Soc. Stud. Texts, Vol. 34, Cambridge University Press, Cambridge, 1996.
  * [4] Beauville A., Counting rational curves on $K3$ surfaces, Duke Math. J. 97 (1999), 99–108, arXiv:alg-geom/9701019.
  * [5] Block F., Göttsche L., Refined curve counting with tropical geometry, Compos. Math. 152 (2016), 115–151, arXiv:1407.2901.
  * [6] Brieskorn E., Singular elements of semi-simple algebraic groups, in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, Gauthier-Villars, Paris, 1971, 279–284.
  * [7] Broer B., Line bundles on the cotangent bundle of the flag variety, Invent. Math. 113 (1993), 1–20.
  * [8] Broer B., Normality of some nilpotent varieties and cohomology of line bundles on the cotangent bundle of the flag variety, in Lie Theory and Geometry, Progr. Math., Vol. 123, Birkhäuser Boston, Boston, MA, 1994, 1–19.
  * [9] Bryan J., Leung N.C., Counting curves on irrational surfaces, in Surveys in Differential Geometry: Differential Geometry Inspired by String Theory, Surv. Differ. Geom., Vol. 5, Int. Press, Boston, MA, 1999, 313–339.
  * [10] Bryan J., Leung N.C., Generating functions for the number of curves on abelian surfaces, Duke Math. J. 99 (1999), 311–328, arXiv:math.AG/9802125.
  * [11] Bryan J., Leung N.C., The enumerative geometry of $K3$ surfaces and modular forms, J. Amer. Math. Soc. 13 (2000), 371–410, arXiv:alg-geom/9711031.
  * [12] Caporaso L., Harris J., Counting plane curves of any genus, Invent. Math. 131 (1998), 345–392, arXiv:alg-geom/9608025.
  * [13] Chen Y., Leung N.C., $ADE$ bundles over surfaces with $ADE$ singularities, Int. Math. Res. Not. 2014 (2014), 4049–4084, arXiv:1209.4979.
  * [14] Chen Y., Leung N.C., Deformability of Lie algebra bundles and geometry of rational surfaces, Int. Math. Res. Not. 2015 (2015), 11508–11519.
  * [15] Chen Y., Leung N.C., Affine $ADE$ bundles over surfaces with $p_{g}=0$, Math. Z. 284 (2016), 55–68, arXiv:1303.5578.
  * [16] Chen Y., Leung N.C., ADE bundles over ADE singular surfaces and flag varieties of ADE type, Int. Math. Res. Not. 2018 (2018), 3941–3958, arXiv:1811.02777.
  * [17] Clingher A., Morgan J.W., Mathematics underlying the F-theory/heterotic string duality in eight dimensions, Comm. Math. Phys. 254 (2005), 513–563, arXiv:math.AG/0308106.
  * [18] Derenthal U., Universal torsors of del Pezzo surfaces and homogeneous spaces, Adv. Math. 213 (2007), 849–864, arXiv:math.AG/0604195.
  * [19] Dolgachev I.V., Classical algebraic geometry: a modern view, Cambridge University Press, Cambridge, 2012.
  * [20] Donagi R., Principal bundles on elliptic fibrations, Asian J. Math. 1 (1997), 214–223, arXiv:alg-geom/9702002.
  * [21] Donagi R., Wijnholt M., ADE transform, Adv. Theor. Math. Phys. 24 (2020), 2043–2066, arXiv:1510.05025.
  * [22] Donten-Bury M., Cox rings of minimal resolutions of surface quotient singularities, Glasg. Math. J. 58 (2016), 325–355, arXiv:1301.2633.
  * [23] Fantechi B., Göttsche L., van Straten D., Euler number of the compactified Jacobian and multiplicity of rational curves, J. Algebraic Geom. 8 (1999), 115–133, arXiv:alg-geom/9708012.
  * [24] Friedman R., Morgan J., Witten E., Vector bundles and ${\mathrm{F}}$ theory, Comm. Math. Phys. 187 (1997), 679–743, arXiv:hep-th/9701162.
  * [25] Friedman R., Morgan J.W., Exceptional groups and del Pezzo surfaces, in Symposium in Honor of C.H. Clemens (Salt Lake City, UT, 2000), Contemp. Math., Vol. 312, Amer. Math. Soc., Providence, RI, 2002, 101–116, arXiv:math.AG/0009155.
  * [26] Göttsche L., A conjectural generating function for numbers of curves on surfaces, Comm. Math. Phys. 196 (1998), 523–533, arXiv:alg-geom/9711012.
  * [27] Göttsche L., Shende V., Refined curve counting on complex surfaces, Geom. Topol. 18 (2014), 2245–2307, arXiv:1208.1973.
  * [28] Hague C., Cohomology of flag varieties and the Brylinski–Kostant filtration, J. Algebra 321 (2009), 3790–3815, arXiv:0803.3424.
  * [29] Humphreys J.E., Introduction to Lie algebras and representation theory, Grad. Texts in Math., Vol. 9, Springer-Verlag, New York – Berlin, 1972\.
  * [30] Iqbal A., Neitzke A., Vafa C., A mysterious duality, Adv. Theor. Math. Phys. 5 (2001), 769–807, arXiv:hep-th/0111068.
  * [31] Julia B.L., Magics of ${\mathrm{M}}$-gravity, Fortschr. Phys. 49 (2001), 551–555, arXiv:hep-th/0105031.
  * [32] Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
  * [33] Klemm A., Maulik D., Pandharipande R., Scheidegger E., Noether–Lefschetz theory and the Yau–Zaslow conjecture, J. Amer. Math. Soc. 23 (2010), 1013–1040, arXiv:0807.2477.
  * [34] Kodaira K., On compact analytic surfaces. I, Ann. of Math. 71 (1960), 111–152.
  * [35] Kodaira K., On compact analytic surfaces. II, Ann. of Math. 77 (1963), 563–626.
  * [36] Kodaira K., On compact analytic surfaces. III, Ann. of Math. 78 (1963), 1–40.
  * [37] Kool M., Shende V., Thomas R.P., A short proof of the Göttsche conjecture, Geom. Topol. 15 (2011), 397–406, arXiv:1010.3211.
  * [38] Lee J., Leung N.C., Yau–Zaslow formula on $K3$ surfaces for non-primitive classes, Geom. Topol. 9 (2005), 1977–2012, arXiv:math.SG/0404537.
  * [39] Lee J., Leung N.C., Counting elliptic curves in $K3$ surfaces, J. Algebraic Geom. 15 (2006), 591–601, arXiv:math.SG/0405041.
  * [40] Leung N.C., ADE-bundles over rational surfaces, configuration of lines and rulings, arXiv:math.AG/0009192.
  * [41] Leung N.C., Xu M., Zhang J., Kac–Moody $\widetilde{E}_{k}$-bundles over elliptic curves and del Pezzo surfaces with singularities of type $A$, Math. Ann. 352 (2012), 805–828.
  * [42] Leung N.C., Zhang J., Moduli of bundles over rational surfaces and elliptic curves. I. Simply laced cases, J. Lond. Math. Soc. 80 (2009), 750–770, arXiv:0906.3900.
  * [43] Leung N.C., Zhang J., Moduli of bundles over rational surfaces and elliptic curves. II. Nonsimply laced cases, Int. Math. Res. Not. 2009 (2009), 4597–4625, arXiv:0908.1645.
  * [44] Leung N.C., Zhang J., Non-simply laced McKay correspondence and triality, Pure Appl. Math. Q. 8 (2012), 941–955.
  * [45] Leung N.C., Zhang J., Cox rings of rational surfaces and flag varieties of $ADE$-types, Comm. Anal. Geom. 23 (2015), 293–317, arXiv:1409.2325.
  * [46] Manin Yu.I., Cubic forms: algebra, geometry, arithmetic, 2nd ed., North-Holland Math. Libr., Vol. 4, North-Holland Publishing Co., Amsterdam, 1986.
  * [47] Pan X., Triviality and split of vector bundles on rationally connected varieties, Math. Res. Lett. 22 (2015), 529–547, arXiv:1310.5774.
  * [48] Pandharipande R., Thomas R.P., The Katz–Klemm–Vafa conjecture for $K3$ surfaces, Forum Math. Pi 4 (2016), e4, 111 pages, arXiv:1404.6698.
  * [49] Reid M., Undergraduate algebraic geometry, London Math. Soc. Stud. Texts, Vol. 12, Cambridge University Press, Cambridge, 1988.
  * [50] Reid M., La correspondance de McKay, Astérisque 276 (2002), 53–72, arXiv:math.AG/9911165.
  * [51] Segre B., The non-singular cubic surfaces, Oxford University Press, Oxford, 1942\.
  * [52] Serganova V.V., Skorobogatov A.N., Del Pezzo surfaces and representation theory, Algebra Number Theory 1 (2007), 393–419, arXiv:math.AG/0611737.
  * [53] Serganova V.V., Skorobogatov A.N., Adjoint representation of ${\mathrm{E}}_{8}$ and del Pezzo surfaces of degree 1, Ann. Inst. Fourier (Grenoble) 61 (2011), 2337–2360, arXiv:1005.1272.
  * [54] Slodowy P., Simple singularities and simple algebraic groups, Lecture Notes in Math., Vol. 815, Springer, Berlin, 1980.
  * [55] Tzeng Y.-J., A proof of the Göttsche–Yau–Zaslow formula, J. Differential Geom. 90 (2012), 439–472, arXiv:1009.5371.
  * [56] Yau S.-T., Zaslow E., BPS states, string duality, and nodal curves on $K3$, Nuclear Phys. B 471 (1996), 503–512, arXiv:hep-th/9512121.