# Kevin Costello, Chern-Simons in Dimensions 4,5,6

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Why holomorphic CS is interesting: mirrors counts of curves through mirror symmetry, and shows up in construction of 4d integrable field theories.z
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In higher dimensions: non-topological field theories.
Setup: for $X$ a Calabi-Yau 3-fold, $A\in \Omega^{0, 1}(X, \lieg)$ a $\delbar$ connection, write a Lagrangian
\[
\mathrm{hCS}(A) = \int_X \Omega_X \wedge \CS(A)
.\]
Equations of motion imply flat connection, here $F^{0, 2}(A) = 0$, so a holomorphic instead of flat bundle.
Translation $\dd{}{\bar z_i}$ involves a BRST term.
For 4-manifolds, take $\Sigma_1 \times \Sigma_2$ a bundle of Riemann surfaces, where $\omega \in \Omega^1(\Sigma_2)$ has no zeros, and $A\in \Omega^1(\Sigma_1 \times \Sigma_2) \mod \Omega^{1, 0}( \Sigma_2)$.
The Lagrangian is $\int \omega \wedge \CS(A)$ for $\omega$ a 1-form, the equations of motion become $\omega F(A) = 0$.
Locally looks like $\int zF\wedge F$ if one writes $\omega = dz$.
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4d CS unifies integrable PDEs and field theories.
Basic example of integral PDEs: $G\in \Lie\Grp$ compact, $\sigma: \RR^2\to G$, asking $\sigma$ to be harmonic is an integrable PDE.
Importantly *lax*, so can build a connection that is flat iff the harmonic equation holds.
An important generalization: include a WZW term.
Interpret maps $\RR\times S^1\to G$ as $\RR\to \Loop G$, where the phase space is $\T\dual\Loop G$.
Conservation: $\poisbrack{H}{m(z)} = 0$.
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Given a Riemannian manifold with a closed 3-form, when is the harmonic map equation on $M$ integrable?
Traditional examples: Riemannian symmetric spaces and their deformations.
One example is $S^2$ with a specific metric, which also satisfies a Ricci flow condition.
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4d Chern-Simons: equations of motion become maps $\sigma: \RR^2\to \Bung\slice{\Sigma}$, a moduli of bundles over a Riemann surface trivialized at the poles of $\omega$ a 1-form.
This has a canonical metric and 3-form.
Theorem: the harmonic map equation for these maps is always integrable.
Proved painfully without reference to 3D CS.
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Ricci flow is closely tied to integrability.
For Riemannian manifolds with closed 3-form, modify the flow $\delta g_{uv} = \Ric_{uv} - ?$.
The "ancient" solutions correspond to something unique.
Consider the moduli of Ricci flows.
There is a natural flow on the moduli space $\mcm(\Sigma, \omega)$ given by computing periods of the form between its zeros?
Turns out to be proportional to a Ricci flow on this space.
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For 4d, mostly classical at this point.
Higher dimensions: get a similar Lagrangian
\[
\int \dz_1\dz_2 \Tr(\Ad A) + c \Tr(A* A * A) && A*B \da A\wedge B + c \dd{A}{z_1} \wedge \dd{B}{z_2} + \cdots
.\]
5d nonabelian CS is a supersymmetric sector of $M\dash$theory.
Holography: can match supersymmetric things for $N\, M2$ branes with computations in this 5d CS theory.
These are QM particles moving on a moduli space of rank $k$ instantons on $\RR^4$ with charge $N$.
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Most natural variant: holomorphic CS on a Calabi-Yau with $A\in \Omega^{0, 1}(X)$:
\[
\int \Omega_X \wedge \CS(A)
.\]
Problem: doesn't exist as a quantum theory due to gauge anomalies.
Apply Grothendieck-Hirzebruch-Riemann-Roch to show the canonical doesn't vanish:
\[
c_1 \Bung(X) = \int_X \Todd(\T X) \ch(\Ad_{\lieg}) 
.\]
Do some anomaly cancellation.
Famously not renormalizable.
Restricts gauge groups to $\SO_8$ or $G_2\times G_2$?
For $\sigma: \RR^4\to \SO_8$, Lagrangian involves Kähler potential.
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