# Simon Donaldson, The Chern-Simons Functional and Floer Homology 

:::{.remark}
Recent work: when a variety admits an Einstein metric, and $G_2$ manifolds.
For an account of the Chern-Simons paper, see Chern's *Complex manifolds without potential theory*, appendix on characteristic classes.
2+1: action functionals 
\[
\int e^{\mathrm{CS}} \mathrm{d} A
.\]
3+1: Floer homology $\HF(M^3)$.
:::

:::{.remark}
Consider $P\to Y^3 \in \Prin\bung$, define $\mca$ the affine space of functions, then $A_1-A_2\in \Omega^1(\ad P)$
Curvature $F(A) \in \Omega^2(\ad P)$.
Define a form $\Theta$
Chern-Simons functional: $\CS: \mca\to \RR$, descends to $\mca/\mcg\to S^1$ for $\mcg$ the gauge group.
Critical points are flat connections, Hessians at these points are quadratic forms corresponding to a bilinear form from earlier.
Alternative definition:
pick $X^4$ such that $\bd X = Y$ and set
\[
\CS(A) = \int_X \Tr(F(A)^2)
,\]
i.e. integrate the Chern-Weil form.
:::

:::{.remark}
Examples of functionals: for $\gamma\in \Loop M$,
\[
E(\gamma) \da \int \abs{\gamma'}^2
.\]
Sub-level sets will have compactness properties, Hessians are finite index at critical points.
The flow $-\grad E$ is a parabolic PDE, so a nonlinear heat equation.
But the Chern-Simons functional doesn't have these properties.
Very different!
:::

:::{.remark}
For symplectic manifolds:
\[
A(\gamma) = \int_{\DD} \omega && \text{where } \bd \DD = \gamma
.\]

For $M=\CC^n$, write as a Fourier series $A( \gamma) = \sum c_k \abs{\gamma'}^2$?
Fixed points of exact Hamiltonian diffeomorphisms $\phi$ correspond to critical points of a deformed functional $A_\phi$ on \( \Omega M \).
Arnold conjecture: $\size \Fix \phi \geq \sum \beta_i$ is bounded below by Betti numbers.
:::

:::{.remark}
For a torus, $\Loop T^{2n} \cong T^{2n} \times H^- \times H^+$ where the $H^\pm$ are vector spaces.
Arnold conjecture proved for torii by Conley and Zehnder, 1983.
Floer's insight: one can still "do Morse theory" with $A_\phi$ and $\CS$: while gradient flow isn't defined, flow lines between critical points do make sense.
:::

:::{.remark}
Introducing a Riemannian metric yields $\hodgestar: \Omega^2\to \Omega^1$.
Get connections $A_t$ over $Y^3$ as solutions to 
\[
\dd{A_t}{t} = \hodgestar F(A_t)
.\]
Solutions are Yang-Mills instantons on $Y\times \RR$ asymptotic to flat connections at $\pm \infty$.
These solve anti-self-dual equations
\[
F(A) = -\hodgestar F(A)
,\]
where here the star is a 4-dimensional version.
Connected to electromagnetism?
In symplectic case, gradient flow lines in $\Loop M$ are holomorphic curves in $M$.
:::

:::{.remark}
Usual story: chain complex with generators for each critical point, graded by index, $\mcm(p, q)$ the space of gradient flow lines $p\to q$, compute $\dim \mcm(p, q) = i(p) - i(q) -1$, and define a differential $\bd p = \sum \size \mcm(p, q)$.
Problem in infinite dimensions: index isn't well-defined, but the difference is e.g. when $G = \SU_2$.
Linearize the instanton equation.
Euler characteristic with respect to $\HF$ is twice the Casson invariant.
:::

:::{.remark}
If $Y \in \ZHS^3$, then the trivial flat connection is isolated.
Otherwise, Floer's construction works when you can avoid "reducible" flat connections, e.g. a nontrivial $\SO_3$ bundle over $Y^3$. 
For general $P\to Y$, need an equivariant version of Floer theory -- at present, this doesn't seem to exist.
:::

:::{.remark}
Toward a 3+1 TFT:
solutions to Yang-Mills on a 4-manifold give invariants by counting instantons in a 0-dimensional moduli space.
For $X$ a 4-manifold with $\bd X = Y$, these invariants $I(X)$ take values in $\HF(Y)$.
Sum over all flat connections $C_i$, and count number of connections asymptotic to $C_i$.
:::

:::{.remark}
Gluing formula: a type of surgery formula when $X = X_1 \glue{Y} X_2$ to compute $I(X) = I(X_1) I(X_2)$.
This uses the pairing $\HF_*(Y) \tensor \HF^*(Y)\to \ZZ$.
Proof: stretching the neck.
Analog in Morse theory moves intersections closer to critical points.
<!-- Xournal file: /home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/Chern/figures/2021-11-16_15-16.xoj -->

![](figures/2021-11-16_15-17-52.png)

:::

:::{.question}
Other Floer theories on $Y^3$: solutions to SW, combinatorial Heegard theory.
An outstanding problem: how are all of these Floer theories related?
:::

:::{.remark}
Floer's deepest work: work leading to his exact surgery sequence.
$K \injects Y\in \QHS^3$ a knot, take a tubular neighborhood diffeomorphic to a torus with a prefered meridian.
Do $+1$ surgery: cut out, glue meridian back along the diagonal.
Can also do $0$ surgery.
Get a LES in $\HF_*$ of the surgered pieces, using well-known cobordisms.
:::

:::{.remark}
Representation variety: moduli of flat connections!
Extending all of this to include surfaces: see bordered Floer theory, Lipshitz-Osvath-Thurston for Seiberg-Witten and Heegard cases.
Floer homotopy: see Manolescu, Bauer-Furuta.
:::

:::{.remark}
How to complexify this theory?
E.g. for $G = \SL_2(\CC)$ instead of $\SU_2$, or replace $Y$ with a Calabi-Yau threefold?
:::

:::{.remark}
For $Z$ a Calabi-Yau, have a nonvanishing holomorphic 3-form $\omega$, so define a $\CC\dash$valued function on $\mca$ the space of connections:
\[
F(A) = \int_Z \CS(A) \wedge \omega
.\]
Critical points are connections with $F^{0, 2} = 0$, i.e. $\delbar_A^2 = 0$.
See "holomorphic Casson invariants" by R. Thomas, counting holomorphic line bundles over $Z$.
More generally, counting coherent sheaves on $Z$ to generalize DT invariants.

:::

:::{.remark}
See Atiyah-Floer conjecture.
:::