Equations on Riemann surfaces

Recall that a Riemann surface is a complex 1-manifold $M$ with a complex structure $\Sigma$ (a class of analytically equivalent atlases on $X$). Here we consider equations that relate connections and Higgs fields with solutions on Riemann surfaces. Let $G=SU(2)$ (complex 2-matrices with determinant 1) or $SO(3)$ (real orthogonal 3-matrices with determinant 1), $\theta$ a Higgs field over $M$, and  $P$ a principal $G$-bundle over $M$.

Definition: The curvature of a principal $G$-bundle $P$ is the map
\[
\begin{array}{r c l}
F_\nabla\ :\ \mathcal A^0_M(P) & \to & \mathcal A^2_M(P), \\
\omega s & \mapsto & (d_\nabla \circ \nabla)(\omega s),
\end{array}
\]
where the extension $d_\nabla:\mathcal A^k_M(P)\to \mathcal A^{k+1}_M(P)$ is defined by the Leibniz rule, that is $d_\nabla (\omega\otimes s) = (d\omega)\otimes s +(-1)^k\omega \wedge \nabla s$, for $\omega$ a $k$-form and $s$ a smooth section of $P$.

Since we may write $\mathcal A^1 = \mathcal A^{1,0}\oplus\mathcal  A^{0,1}$ as the sum of its holomorphic and anti-holomorphic parts, respectively (see post "Smooth projective varieties as Kähler manifolds," 2016-06-16), we may consider the restriction of $d_\nabla$ to either of these summands.

Definition: For a vector space $V$, define the Hodge star $*$ by
\[
\begin{array}{r c l}
*\ :\ \bigwedge^k(V^*) & \to & \bigwedge^{n-k}(V^*), \\
e^{i_1}\wedge \cdots \wedge e^{i_k} & \mapsto & e^{j_1}\wedge\cdots \wedge e^{j_{n-k}},
\end{array}
\]
so that $e^{i_1}\wedge\cdots \wedge e^{i_k}\wedge e^{j_1}\wedge\cdots \wedge e^{j_{n-k}} = e^1\wedge \cdots \wedge e^n$. Extend by linearity from the chosen basis.

The dual of the generalized connection $d_\nabla$ is written $d_\nabla^* = (-1)^{m+mk+1}*d_\nabla *$, where $\dim(M)=m$ and the argument of $d_\nabla^*$ is in $\mathcal A^k_M$ (this holds for manifolds $M$ that are not necessarily Riemann surfaces as well).

Now we may understand some equations on Riemann surfaces. They all deal with the connection $\nabla$, its generalization $d_\nabla$, its curvature $F_\nabla$, and the Higgs field $\theta$. Below we indicate their names and where they are mentioned (and described in further detail).
\begin{align*}
\text{Hitchin equations} && \left.d_\nabla\right|_{A^{0,1}}\theta & = 0 && [2],\ \text{Introduction}\\
&& F_\nabla + [\theta,\theta^*] & = 0\\[10pt]
\text{Yang-Mills equations} && d^*_\nabla d_\nabla \theta + *[*F_\nabla,\theta] & = 0 && [1],\ \text{Section 4} \\
&& d_\nabla^*\theta & = 0 \\[10pt]
\text{self-dual Yang-Mills equation} && F_\nabla  - *F_\nabla & = 0 && [2],\ \text{Section 1}\\[10pt]
\text{Yang-Mills-Higgs equations} && d_\nabla *F_\nabla + [\theta,d_\nabla \theta] & = 0 && [4],\ \text{equation (1)} \\
&& d_\nabla * d_\nabla \theta & = 0
\end{align*}

Recall the definitions of $\theta$ and $\theta*$ from a previous post ("Higgs fields of principal bundles," 2016-08-24). Now we look at these equations in more detail. The first of the Hitchin equations says that $\theta$ has no anti-holomorphic component, or in other words, that $\theta$ is holomorphic. In the second equation, the Lie bracket $[\cdot,\cdot]$ of the two 1-forms is
\begin{align*}
[\theta,\theta^*] & = \left[\textstyle\frac12f(dz+i\ dy), \frac12\bar f(dz - i\ dy) \right] \\
& = \textstyle -\frac i4f\bar f\ dx\wedge dy + \frac i4 f\bar f\ dy \wedge dx -\frac i4 f\bar f\ dx\wedge dy +\frac i4 f\bar f\ dy\wedge dx \\
& = -i|f|^2\ dx\wedge dy.
\end{align*}
In the Yang-Mills and Yang-Mills-Higgs equations, we can simplify some parts by noting that, for a section $s$ of the complexification of $P\times_\ad \mathfrak g$,
\begin{align*}
d_\nabla (\theta\otimes s) & = \textstyle \frac12d_\nabla (fdx\otimes s) + \frac i2 d_\nabla (fdy \otimes s) \\
& = \textstyle \frac12 (df\wedge dx \otimes s - fdx \wedge \nabla s) +\frac i2 (df\wedge dy - fdy \wedge \nabla s) \\
& = \left(\frac i2\frac{\dy f}{\dy x} - \frac 12 \frac{\dy f}{\dy y}\right)dx\wedge dy \otimes s - \underbrace{\textstyle \frac12f(dx+idy)}_{\theta}\wedge \nabla s.
\end{align*}
The Hodge star of $\theta$ is $*\theta = \frac 12f(dy -idx)$, so
\begin{align*}
d_\nabla *(\theta\otimes s) & = \textstyle \frac12d_\nabla (fdy\otimes s) - \frac i2 d_\nabla (fdx \otimes s) \\
& = \textstyle \frac12 (df\wedge dy \otimes s - fdy \wedge \nabla s) -\frac i2 (df\wedge dx - fdx \wedge \nabla s) \\
& = \left(\frac 12\frac{\dy f}{\dy x} + \frac i2 \frac{\dy f}{\dy y}\right)dx\wedge dy \otimes s + \underbrace{\textstyle \frac12f(idx-dy)}_{i\theta}\wedge \nabla s.
\end{align*}
We could express $\nabla s = (s_1dx + s_2dy)\otimes s^1$, but that would not be too enlightening. Next, note the self-dual Yang-Mills equation only makes sense over a (real) 4-dimensional space, since the degrees of the forms have to match up. In that case, with a basis $dz_1=dx_1+idy_1, dz_2 = dx_2+idy_2$ of $\mathcal A^1$, we have
\begin{align*}
F_\nabla & = F_{12} dx_1\wedge dy_1 + F_{13} dx_1 \wedge dx_2 + F_{14} dx_1\wedge dy_2 + F_{23} dy_1\wedge dx_2 + F_{24} dy_1\wedge dy_2 + F_{34} dx_2\wedge dy_2, \\
*F_\nabla & = F_{12} dx_2\wedge dy_2 - F_{13} dy_1 \wedge dy_2 + F_{14} dy_1\wedge dx_2 + F_{23} dx_1\wedge dy_2 - F_{24} dx_1\wedge dx_2 + F_{34} dx_1\wedge dy_1.
\end{align*}
Then the self-dual equation simply claims that
\[
F_{12} = F_{34}
\hspace{1cm},\hspace{1cm}
F_{13} = -F_{24}
\hspace{1cm},\hspace{1cm}
F_{14} = F_{23}.
\]

Remark: This title of this post promises to talk about equations on Riemann surfaces, yet all the differential forms are valued in a principal $G$-bundle over $\R^2$ (or $\R^4$). However, since the given equations are conformally invariant (this is not obvious), and as a Riemann surface locally looks like $\R^2$, we may consider the solutions to the equations as living on a Riemann surface.

References:
[1] Atiyah and Bott (The Yang-Mills equations over Riemann surfaces)
[2] Hitchin (Self-duality equations on a Riemann surface)
[3] Huybrechts (Complex Geometry, Chapter 4.3)
[4] Taubes (On the Yang-Mills-Higgs equations)

Higgs fields of principal bundles

The goal here is to understand the setting of Higgs fields on Riemannian manifolds, in the manner of Hitchin. First we consider general topological spaces $X$ and groups $G$.

Definition: Let $X$ be a topological space and $G$ a group. A principal bundle (or principal $G$-bundle) $P$ over $X$ is a fiber bundle $\pi:P\to X$ together with a continuous, free, and transitive right action $P\times G\to P$ that preserves the fibers. That is, if $p\in \pi^{-1}(x)$, then $pg\in \pi^{-1}(x)$ for all $g\in G$ and $x\in X$.

Now suppose we have a principal bundle $\pi:P\to X$, a representation $\rho$ of $G$, and another space $Y$ on which $G$ acts on the left. Define an equivalence relation $(p,y)\sim (p',y')$ on $P\times Y$ iff there is some $g\in G$ for which $p'=pg$ and $y'=\rho(g^{-1})y$. This is an equivalence relation. We will be interested in the adjoint representation (induced by conjugation).

Proposition: The projection map $\pi':P\times_\rho Y := (P\times Y)/\sim\ \to X$, where $\pi'([p,y]) = \pi(p)$, defines a vector bundle over $X$, called the associated bundle of $P$.

Recall a Lie group $G$ is a group that is also a topological space, in the sense that there is a continuous map $G\times G\to G$, given by $(g,h)\mapsto gh^{-1}$. The Lie algebra $\mathfrak g$ of the Lie group $G$ is the tangent space $T_eG$ of $G$ at the identity $e$. We will be interested in principal $G$-bundles $P\to \R^2$ and associated bundles $P\times_\ad \mathfrak g\to \R^2$, where $\ad$ is the adjoint representation of $G$.

Next, recall we had the space $\mathcal A^k_M$ of $k$-differential forms on $M$ (see post "Smooth projective varieties as Kähler manifiolds," 2016-06-16), defined in terms of wedge products of elements in the cotangent bundle $(TM)^* = T^*M$ of $M$. Now we generalize this to get differential forms over arbitrary vector bundles.

Definition: Let $E\to M$ be a vector bundle. Let
\begin{align*}
\mathcal A^k_M(E) & := \Gamma(E\otimes \textstyle\bigwedge^k T^*M) = \Gamma(E)\otimes_{\mathcal A^0_M}\mathcal A^k_M, \\
\mathcal A^{p,q}_M(E) & := \Gamma(E\otimes \textstyle\bigwedge^p (T^{1,0}M)^*\otimes \bigwedge^q (T^{0,1}M)^*) = \Gamma(E)\otimes_{\mathcal A^0_M}\mathcal A^{p,q}_M
\end{align*}
be the spaces of $k$- and $(p,q)$-differential forms, respectively, over $M$ with values in $E$.

Equality above follows by functoriality. Now we are close to understanding where exactly the Higgs field lives, in Hitchin's context.

Definition: Given a function $f:\C\to \C$, the conjugate of $f$ is $\bar f$, defined by $\bar f(z) = \overline{f(\bar z)}$.

Hitchin denotes this as $f^*$, but we will stick to $\bar f$. Finally, let $P$ be a $G$-principal bundle over $\R^2$ and $P\times_\ad \mathfrak g$ the associated bundle of $P$. Given $f\in \mathcal A^0_{\R^2}( (P\times_\ad \mathfrak g)\otimes \C)$, set
\begin{align*}
\theta & = \textstyle \frac12 f(dx+i\ dy) \in \mathcal A^{1,0}_{\R^2}((P\times_\ad\mathfrak g)\otimes \C) ,\\
\theta^* & = \textstyle \frac12 \bar f(dx-i\ dy) \in \mathcal A^{0,1}_{\R^2}((P\times_\ad\mathfrak g)\otimes \C),
\end{align*}
called a Higgs field over $\R^2$ and (presumably) a dual (or conjugate) Higgs field over $\R^2$. Note this agrees with the definition in a previous post ("Connections, curvature, and Higgs bundles," 2016-07-25).

References: Hitchin (Self-duality equations on a Riemann surface), Wikipedia (article on associated bundles, article on vector-valued differential forms)

What is a stack?

 Conference topic

This is from discussions at the 2016 West Coast Algebraic Topology Summer School (WCATSS) at The University of Oregon. Thanks to Piotr Pstragowski for explaining the material.

Definition: A groupoid is a category where all the morphisms are invertible. Alternatively, a groupoid is a set of objects $A$, a set of morphisms $\Gamma$, and a collection of maps as described by the diagram below.

To describe stacks, we compare them with sheaves. Both start out with a space $X$ and a topology on it, so that we may consider open sets $U$.

In addition to these conditions, there is a triple intersection condition for stacks that does not have an analogous one in sheaves. It is given by:

for every $U_i,U_j,U_k$ and $s_i,s_j,s_k\in \widehat{\mathcal F}(U_i), \widehat{\mathcal F}(U_j), \widehat{\mathcal F}(U_k)$, respectively, such that there exist isomorphisms $\varphi_{ij}:s_i|_{U_i\cap U_j}\to s_j|_{U_i\cap U_j}$, $\varphi_{jk}:s_j|_{U_j\cap U_k}\to s_k|_{U_j\cap U_k}$, and $\varphi_{ik}:s_i|_{U_i\cap U_k}\to s_k|_{U_i\cap U_k}$, the diagram below commutes:

Example: A Hopf algebroid may be viewed as a functor into groupoids, so that with the appropriate topology, it becomes a stack. Indeed, by definition a Hopf algebroid is a pair of $k$-algebras $(A,\Gamma)$ such that $(\Spec(A),\Spec(\Gamma))$ is a groupoid object in affine schemes, or in other words, is a functor from affine schemes into groupoids.

References: nLab (article on groupoids)

Morphisms of schemes

 Conference topic

This is from discussions at the 2016 West Coast Algebraic Topology Summer School (WCATSS) at The University of Oregon. Thanks to Zijian Yao for explaining the material.

Consider a morphism of schemes $\varphi:S'\to S$ and coherent sheaves $\mathcal F,\mathcal G$ over $S$. Consider also a map of sheaves $f:\mathcal F\to \mathcal G$ and a map $f'$ between the pullbacks of $\mathcal F$ and $\mathcal G$, as described by the diagram below.

There are two natural questions to ask.

1. When is $f' = \varphi^*f$?
2. If we start with $\mathcal G'$ over $S'$, when is $\mathcal G' = \varphi^*\mathcal G$?

To answer these questions, consider fiber products of schemes and projections from them, as given below.

Remark: If 1. is true, then $p_1^*(f') = p_2^*(f')$. If the previous statement is an equivalence, then $\varphi$ is a morphism of descent.

Remark: If 2. is true, then there exists $\alpha:p_1^*(\mathcal G') \to p_2^*(\mathcal G')$ such that $\pi_{32}^*(\alpha)\pi_{21}^*(\alpha) = \pi_{31}^*(\alpha)$ and $\pi^*(\Delta) = \alpha$. If the previous statement is an equivalence, then $\varphi$ is effective.

What is a scheme?

 Conference topic

This is from a problem session at the 2016 West Coast Algebraic Topology Summer School (WCATSS) at The University of Oregon. Thanks to Tyler Lawson for explaining the material.

Definition: Affine schemes are the category $\Ring^{op}$. An object $R\in \Ring$ becomes an object $\Spec(R)$ in affine schemes, and a ring map $R\to S$ becomes a map $\Spec(S)\to \Spec(R)$, where $\Spec$ denotes the set of prime ideals.

We try to think of $Spec(R)$ as a geometrical object.

Example: Let $k$ be a field and consider the ring
\[
R = k[x_1,\dots,x_n] / (f_1(x_1,\dots,x_n),\dots,f_r(x_1,\dots,x_n)).
\]
$\Spec(R)$ is supposed to be a substitute for the set of solutions to a system of equations
\begin{align*}
f_1(x_1,\dots,x_n) & = 0,\\
\vdots \hspace{.7cm}\\
f_r(x_1,\dots,x_n) & = 0.
\end{align*}

The scheme $\Spec(R)$ has a more precise definition. It consists of a set, a topology, and a sheaf. 

1. Set: The underlying set of the scheme $\Spec(R)$ is the set of prime ideals of $R$. For example:

• if $R = \C[x]$, then the prime ideals are $(x-\alpha)$ and $(0)$;
• if $R = \C[x,y]$, then the prime ideals are $(x-\alpha,y-\beta)$, irreducible polynomials $(f(x,y))$, and $(0)$.

2. Topology: For every ideal $I\subset R$, the set $V(I) = \{P\subset R$ prime, $P\supset I\}$ is a closed set. Note that

\[
\bigcup_{n=1}^N V(I_n) = V\left(\bigcap_{n=1}^N I_n\right)
\hspace{1cm}\text{and}\hspace{1cm}
\bigcap_{\alpha\in I} V(I_\alpha) = V\left(\sum_{\alpha\in A} I_A\right).
\]
Geometrically, the closed sets are sets of points where one or more identities (like $f(x)=0$) can hold. For example, if $R=\C[x]$, then we have three different closed set types: $\Spec(C[x])$, $\emptyset$, or a finite union of $(x-\alpha_1,\dots, x-\alpha_n)$. Solutions to equations can be one of the following types below.

3. Sheaf: Let $X$ be a set with a topology. $\mathcal O_X$ is the sheaf for which:

• to each open set $U\subseteq X$ we get a ring $\mathcal O_X(U)$;
• to each containment $V\subseteq U\subseteq X$ of open sets, there exists a restriction map $\res_{UV}:\mathcal O_X(U)\to \mathcal O_X(V)$;
• the restriction maps are compatible, in the sense that $\res_{VW}\circ \res_{UV} = \res_{UW}$.

This is called the structure sheaf of $X$.

Say $R$ is our ring, $\Spec(R)$ our set of primes, and we have some open set $U\subseteq \Spec(R)$. We like to think of it in the following way:

• elements of $R$ are functions;
• elements of $\Spec(R)$ are points where we can evaluate a function $f\in P$ (or where the function vanishes);
• subsets $S\subset R$ are the sets $\{f\in R\ :\ f$ only vanishes at points outside $U\}$.

Note that $S$ is closed under multiplication. We localize $R$ at $S$ to get a set
\[
S^{-1}R = \left\{\left[\frac fs\right]\ :\ f\in R, s\in S\right\},
\]
for which $\mathcal O_X(U) = S^{-1}R$ (good enough for today's purposes). Now we have a triple $(\Spec(R),\tau,\mathcal O_X)$, for $\tau$ the Zariski topology, which we call a locally ringed space.

Definition: A scheme is a space $X$ with a topology and a sheaf of rings that is locally isomorphic to $\Spec(R)$.

Since the sheaf has the space $X$ and the topology (through the open sets) encoded in it, we may think of a scheme as a special type of sheaf. Also, isomorphism is meant in the category of locally ringed spaces.

Proposition: Morphisms of schemes $\Spec(R)\to \Spec(S)$ are the same as ring maps $S\to R$.

Example: In the Zariski topology, take $U\subseteq \Spec(k[x,y])$. Locally $U$ looks like it is covered by rings, though that may not be the case globally. Indeed:

Example: Consider projective space $\P^2$, where $[x:y:z] = [\lambda x: \lambda y:\lambda z]$. We may write
\[
\begin{array}{r c c c c c c}
\P^2 & = & U_0 & \cup & U_1 & \cup & U_2. \\
& & [1:y:z] & & [x:1:z] & & [x:y:1] \\
& & \Spec(k[y,z]) & & \Spec(k[x,z]) & & \Spec(k[x,y])
\end{array}
\]
How can we express $U_0\cap U_1$? This is left as an exercise.

Some facts about formal group laws

 Conference topic

Here we solve some problems from the 2016 West Coast Algebraic Topology Summer School (WCATSS) at The University of Oregon. Thanks to Piotr Pstragowski and Carolyn Yarnall for the solutions. First we recall some definitions.

Definition: Let $R$ be a commutative ring with unit. A formal group law $F$ over $R$ is an element $F\in R[[x,y]]$ satisfying

1. $F(x,y) = F(y,x)$ (symmetry), 
2. $F(x,0) =x$ and $F(0,y)=y$ (uniticity),
3. $F(F(x,y),z) = F(x,F(y,z))$ (associativity).

It follows from these three properties that $F(x,y)=x+y+($higher order terms$)$ for all $F$.

Proposition: For any formal group law $F(x,y)$ over $R$, $x$ has a formal inverse. That is, there exists an element $i(x)\in R[[x]]$ such that $F(x, i(x)) = 0$.

Proof: Consider $F(x,y+z)$, with $|z| = n$. Note that
\begin{align*}
F(x,y+z) & = x+y +z +\sum_{i,j\geqslant 1} a_{ij} x^i(y+z)^j \\
& = x+y +z+\sum_{i,j\geqslant 1} a_{ij} x^i \sum_{k=0}^j \binom jk y^k z^{j-k} \\
& = x+y+z+\sum_{i,j\geqslant 1} a_{ij} x^i \left(y^j + \sum_{k=0}^{j-1} \binom jk y^k z^{j-k} \right)\\
& = x+y+z+\sum_{i,j\>1} a_{ij} x^i y^j  + \underbrace{\sum_{i,j\geqslant 1} a_{ij} x^i}_{\text{deg }\geqslant \ 1}\underbrace{\sum_{k=0}^{j-1} \binom jk y^k z^{j-k}}_{\text{deg }=\ k+n(j-k)\geqslant n}\\
& = F(x,y) + z + (\text{terms of deg }\geqslant\ n+1).
\end{align*}
First choose $z_1$ to be the negative of all the degree-1 terms of $F(x,0)$, so that $F(x,z_1)$ has terms of degree 2 and higher. Now choose $z_2$ to be the negative of all the degree-2 terms of $F(x,z_1)$, so $F(x,z_1+z_2)$ has terms of degree 3 and higher. Continue in this manner ad infinitum to get a formal inverse $\sum_i z_i$ (this will be a power series) of $x$. $\square$

Recall that we call $f_a(x,y) = x+y$ the additive formal group law and $F_m(x,y) = x+y+xy$ the multiplicative formal group law. Via the universal Lazard ring of formal group laws, these turn out to be the formal group laws of ordinary singular cohomology theory (additive) and complex $K$-theory $KU$ (multiplicative). Recall also nested notation: for $F$ a formal group law, we write
\begin{align*}
[1]_F(x) & = x, \\
[2]_F(x) & = F(x,x), \\
[3]_F(x) & = F(F(x,x),x), \\
[4]_F(x) & = F(F(F(x,x),x),x),
\end{align*}
and so on.

Definition: Let $F$ be a formal group law over $R$. A morphism of formal group laws is an element $\varphi\in R[[u]]$, giving a formal group law $\varphi F\in R[[x,y]]$ by $\varphi F(x,y):= F(\varphi(x),\varphi(y))$.

An isomorphism of formal group laws is a morphism where the formal power series $\varphi$ is an isomorphism.

Proposition: The additive formal group law and the multiplicative formal group law are not isomorphic over $F_p$.

Proof: We compare $[p]_{F_m}(x)$ and $[p]_{F_a}(x)$ and show they are not the same. If there were an isomorphism $\varphi$ between $F_a$ and $F_m$, we should have that
\[
F_m(x,x) = F_a(\varphi(x),\varphi(x)) = \varphi(F_a(x,x))
\ \ \implies\ \
[p]_{F_m}(x) = \varphi([p]_{F_a}(x)),
\]
since $\varphi$ is a homomorphism. However, we first see that
\[
[1]_{F_a}(x) = x
,\hspace{1cm}
[2]_{F_a}(x) = F_a(x,x) = 2x
,\hspace{1cm}
[3]_{F_a}(x) = F_a(F_a(x,x),x) = 3x,
\]
and so continuing this pattern we get that $[p]_{F_a}(x) = px = 0$ in $F_p$. Next, for the multiplicative formal group law we find that
\[
[1]_{F_m}(x) = x,
,\hspace{1cm}
[2]_{F_m}(x) = F_m(x,x) = 2x + x^2
,\hspace{1cm}
[3]_{F_m}(x) = F_m(2x+x^2,x) = 3x + 3x^2 + x^3.
\]
Here the pattern  is not immediate, but continuing these small examples we find that $[p]_{F_m}(x) = (x+1)^p-1 = 1+x^p-1 = x^p$ in $F_p$. An isomorphism sends only 0 to 0, but in this case $\varphi$ should send $x^p\neq 0$ to $0$, a contradiction. Hence no such isomorphism exists over $F_p$. $\square$